communication/review biomath 2 (2013), 1309257, 1–2 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum the algebraic structure of spaces of intervals contribution of svetoslav markov to interval analysis and its applications roumen anguelov department of mathematics and applied mathematics university of pretoria, pretoria, south africa e-mail: roumen.anguelov@up.ac.za in interval analysis addition of intervals is the usual minkowski addition of sets: a + b = {a + b : a ∈ a, b ∈ b}. the fact that the additive inverse generally does not exist has been a major obstacle in applications, e.g. constructing narrow enclosures of a solution, and possibly one of the most important mathematical challenges associated with the development of the theory of spaces of intervals. the work on this issue during the last 50-60 years lead to new operations for intervals, extended concepts of interval, setting the interval theory within the realm of algebraic structures more general than group and linear space. this theoretical development was paralleled by development of interval computer arithmetic. svetoslav markov was strongly involved in this major development in modern mathematics and he in fact introduced many of the main concepts and theories associated with it. these include: extended interval arithmetic [11], [5], [10], directed interval arithmetic [13], the theory of quasivector spaces [15], [16]. his work lead to practically important applications to the validated numerical computing as well as in the computations with intervals, convex bodies and stochastic numbers [12], [14], [18]. such advanced mathematical and computational tools are much useful under the conditions of extreme sensitivity that is often inheritably characteristic for biological processes as well as input biological parameters experimentally known to be in certain ranges [17]. one of the most important contributions to knowledge by svetoslav markov is in my view the embedding of the monoid structure of intervals and convex bodies into a group structure where the natural definition of multiplication by scalars is also extended in such a way that it is monotone with respect to inclusion, that is a ⊆ b =⇒ γa ⊆ γb, γ ∈ r. the obtained structure is called a quasivector space [16]. since the fundamental idea motivating this field is computations with sets and computing enclosures, the stated property cannot be really overemphasized. one should note that there have been several attempts to embed the quasi-linear space of convex bodies in a more computationally convenient algebraic structure. the most well known such attempt is radström’s embedding into a linear space [19]. however, this embedding fails to preserve precisely the monotonicity property mentioned above. hence, while the developed by radström theory is mathematically correct and elegant, it is quite irrelevant regarding the embedded set and in fact the field of interval analysis or more generally the field of setvalued computing. markov’s concept of quasivector space manages to capture and preserve the essential properties of computations with sets (like the stated monotonicity) while also providing a relatively simple structure for computing. indeed, the quasivector space is a direct sum of a vector(linear) space and symmetric quasivector space which makes the computations essentially as easy as computations in a linear space. a wide spectrum of applications is usually a testimony for the depth of an idea. the ideas of markov have certainly wide and far reaching implications. we focus on one particular direction of development, namely the citation: roumen anguelov, the algebraic structure of spaces of intervals: contribution of svetoslav markov to interval analysis, biomath 2 (2013), 1309257, http://dx.doi.org/10.11145/j.biomath.2013.09.257 page 1 of 2 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2013.09.257 r anguelov, the algebraic structure of spaces of intervals: contribution of svetoslav markov to interval analysis algebraic operations for interval-valued functions. i have had the privilege to have svetoslav markov as a teacher and as a collaborator. the algebraic structure of spaces of interval functions is the main topic of our more recent collaborative work. jointly with blagovest sendov we studied the operations for hausdorff continuous (hcontinuous) function. it turned out that the linear space structure of real functions can be extended to the space of h-continuous interval functions and it is actually the largest linear space of interval functions. hence the space of h-continuous functions has a very special place in interval analysis. further, we showed that the practically relevant set, in terms of providing tight enclosures of sets of real functions, is the set of dilworth continuous (d-continuous) interval valued function. using an earlier idea of svetoslav markov of abstract construction of interval space over a vector lattice, one can show that the set of d-continuous function is a quasi-linear space of intervals over the space of h-continuous functions. moreover, the space of h-continuous functions is precisely the linear space in the direct markov’s sum decomposition of the respective quasivector space. let us note that the space of h-continuous functions has applications in various areas of mathematics, e.g. real analysis [1], [2], approximation theory [20], validated computing [3], [6] as well as the general theory of pdes [9], [7], [4]. the issue of constructing enclosures is relevant to all mentioned applications. hence one can expect future developments in this research direction which involve the space of d-continuous functions and markov’s approach to computing with them. references [1] r anguelov, dedekind order completion of c(x) by hausdorff continuous functions, quaestiones mathematicae, vol. 27, 2004, pp. 153–170. [2] r anguelov, rational extensions of c(x) via hausdorff continuous functions, thai journal of mathematics, vol. 5 no. 2, 2007, pp. 261–272. [3] r anguelov, algebraic computations with hausdorff continuous functions, serdica journal of computing, vol. 1 no. 4, 2007, pp. 443–454. [4] r anguelov, d agbebaku, j h van der walt, hausdorff continuous solutions of conservation laws, in md todorov (editor),proceedings of the 4th international conference on application of mathematics in technical and natural sciences (amitans’12), (st constantine and helena, bulgaria), american institute of physics aip conference proceedings 1487, 2012, pp 151–158. [5] r anguelov r, s markov, extended segment analysis, freiburger intervall berichte 81/10, university of freiburg, 1981. [6] r anguelov, s markov, numerical computations with hausdorff continuous functions, in: t. boyanov et al. (eds.), numerical methods and applications 2006 (nma 2006), lecture notes in computer science 4310, springer, 2007, 279–286. [7] r anguelov, s markov, f minani, hausdorff continuous viscosity solutions of hamilton-jacobi equations, proceedings of the 7th international conference on large scale scientific computations, 3-8 june 2009, sozopol, bulgaria, lecture notes in computer science, 5910 (2010), pp. 241–248. [8] r anguelov, s markov and bl sendov, the set of hausdorff continuous functions the largest linear space of interval functions, reliable computing, vol. 12, 2006, pp. 337–363. [9] r anguelov, e e rosinger, solving large classes of nonlinear systems of pde’s, computers and mathematics with applications, vol. 53, 2007, pp. 491–507. [10] dimitrova, n., markov, s., popova, e., extended interval arithmetics: new results and applications. in: computer arithmetic and enclosure methods (eds. l. atanassova, j. herzberger), north-holland, amsterdam, 1992, 225–234. [11] s markov, a non-standard subtraction of intervals, serdica, 3, 1977, pp.359–370. [12] s markov, calculus for interval functions of a real variable. computing 22, 325–337 (1979). [13] s markov, on directed interval arithmetic and its applications, j. ucs 1 (7), 1995, 514–526. [14] s markov, an iterative method for algebraic solution to interval equations, applied numerical mathematics 30 (2–3), 1999, 225–239. [15] s markov, on the algebraic properties of convex bodies and some applications, j. convex analysis 7 (1), 2000, 129–166. [16] s markov, on quasilinear spaces of convex bodies and intervals, journal of computational and applied mathematics 162 (1), 93–112, 2004. [17] s markov, biomathematics and interval analysis: a prosperous marriage, in ch christov and md todorov (eds), proceedings of the 2d international conference on application of mathematics in technical and natural sciences (amitans’10), (21–26 june, sozopol, bulgaria), american institute of physics aip conf. proc. 1301, pp. 26–36. [18] s markov, r alt, stochastic arithmetic: addition and multiplication by scalars, applied numerical mathematics 50, 475– 488, 2004. [19] h radström, an embedding theorem for spaces of convex sets. proc. am.math. soc. 3, 165–169. (1952) [20] bl sendov, hausdorff approximations, kluwer academic, boston, 1990. biomath 2 (2013), 1309257, http://dx.doi.org/10.11145/j.biomath.2013.09.257 page 2 of 2 http://dx.doi.org/10.11145/j.biomath.2013.09.257 references original article biomath 1 (2012), 1209254, 1–5 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum an upstream finite volume scheme for a bone healing model yves coudière∗, mazen saad† and alexandre uzureau ∗ ∗ lmjl umr6629 cnrs-un université de nantes, nantes, france emails: yves.coudiere@univ-nantes.fr, alexandre.uzureau@univ-nantes.fr † lmjl umr6629 cnrs-un école centrale de nantes, nantes, france email: mazen.saad@ec-nantes.fr received: 12 july 2012, accepted: 25 september 2012, published: 17 october 2012 abstract—this paper is devoted to the introduction of a numerical scheme for a bone healing model and simulation of skull fractures. the mathematical model describes the evolution of mesenchymal stem cells, osteoblasts, bone matrix and osteogenic growth factor. we propose a numerical scheme based on an implicit finite volume method constructed on an orthogonal mesh. the efficiency and robustness of the scheme are shown in simulating a skull fracture in rats. keywords-bone healing; finite volume method; fracture fimulation i. introduction bone is a tissue with a remarkable ability to regenerate itself. but for large gap sizes bone fails to heal itself in a clinically reasonable period of time, this problem is called non-union or delayed union. approximately 5 − 10% of the 5.6 million fractures occurring annually in the united states develop into non-unions or delayed unions [1]. there exists different methods to help bone healing, such as autograft or synthesis materials. in europe, 1.5 million bone grafts are carried out to treat these fractures [2]. an exterior help often allows the complete bone healing but there is still clinical cases which don’t heal. for these complex cases, biology and medicine researchers would create ex vivo tissues that would then be reimplanted. the most common process therefore consists in the growth, in a bioreactor, of mesenchymal stem cells seeded on a synthesis material [3]. currently, this process yields good results only for two-dimensional bone growth in petri dishes. understanding the bone healing is fundamental to create tridimensional ex vivo bones and it is an important field of tissue engineering researches. there exists several mathematical models that simulate the bone healing [4], [5], [6]. in this paper, we propose one such mathematical model (based on the one developed in [7]), describe a numerical scheme to approximate its solution and show some numerical illustrations that simulate a healing process in skull bones. stability and convergence results for the numerical scheme are stated. ii. a model for population dynamics the bone healing is a complex phenomenon involving a cascade of cellular and tissue events [8], [9]. in this paper, we study a simplistic model but well-adapted to the bone growth in bioreactor describing the rates of change, with respect to time and space, of the concentrations in the mesenchymal stem cells s, the osteoblasts b, the extracellular bone matrix m and the osteogenic growth factor g. this dimensionless model is a simplified version of the model proposed by bailón-plaza and van citation: y. coudière, m. saad, a. uzureau, an upstream finite volume scheme for a bone healing model, biomath 1 (2012), 1209254, http://dx.doi.org/10.11145/j.biomath.2012.09.254 page 1 of 5 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.09.254 y. coudière et al., an upstream finite volume scheme for a bone healing model der meulen in [7] that reads ∂ts = f1(s, m, g) + div(λ(m)∇s︸ ︷︷ ︸ diffusion − v (m)χ(s)∇m︸ ︷︷ ︸ haptotaxis ), (1) ∂tb = f2(s, b, m, g), (2) ∂tm = f3(b, m), (3) ∂tg = f4(b, g) + div (λg∇g)︸ ︷︷ ︸ diffusion , (4) for t > 0 and x ∈ ω, where ω is an open bounded polyhedral and connected subset of rd (d = 2, 3). the reaction terms fi (i = 1, . . . 4) describe the exchange, production, decay, etc of the four populations of interest. the evolution of the stem cells is described by a diffusion term, a haptotaxis term (directional motility of the cells up a gradient of cellular adhesion sites, here the bone matrix) and a reaction term f1 describing the mitosis and the differentiation into osteoblasts of the stem cells f1(s, m, g) = α1 β21 + m 2 ms (1 − s)︸ ︷︷ ︸ mitosis − γ1 η1 + g gs︸ ︷︷ ︸ differentiation . the coefficient of diffusion λ and the velocity v of haptotaxis of the stem cells are non-linear functions. the function χ is given by χ(s) = s(1 − s). it allows to avoid an infinite accumulation of the stem cells due to the haptotaxis term. the osteoblasts are only regulated by a reaction term f2 because they are cling on the bone matrix. this term describes the mitosis, the decay of the osteoblasts and the osteoblastic differentiation f2(s, b, m, g) = α2 β22 + m 2 mb (1 − b)︸ ︷︷ ︸ mitosis + ρ γ1 η1 + g gs︸ ︷︷ ︸ differentiation − δ1b︸︷︷︸ removal . the term f3 describes the synthesis and the degradation of the bone matrix by the osteoblasts f3(b, m) = λ (1 − κm) b︸ ︷︷ ︸ synthesis and degradation . the rate of change of the growth factor is described by a diffusion term and a reaction term f4 describing the production by the osteoblasts and the decay of the growth factor f4(b, g) = γ2 (η2 + g) 2 gb︸ ︷︷ ︸ production − δ2g︸︷︷︸ decay . the parameters αi, βi, γi, ηi, δi (i = 1, 2), ρ, λ and κ are real positive numbers. these four equations are completed by the homogeneous neumann boundary conditions on s and g: (λ(m)∇s − v (m)χ(s)∇m) · n = 0, (λg∇g) · n = 0 for t > 0 and x ∈ ∂ω, where n is the outward unit normal of ∂ω; and by the data of initial conditions on s, b, m and g: s(0, x) = s0(x), b(0, x) = b0(x), m(0, x) = m0(x), g(0, x) = g0(x), for x ∈ ω. iii. the numerical scheme in order to simulate bone healing, we need a numerical scheme that approximate the solutions to the equations (1) to (4). the approximate solution must remain bounded in the region of physically bounded solutions s, b, m and g, defined by a = [0, 1]×[0, ρ γ1 δ1 ]× [0, 1 κ ] × [0, ργ1γ2 4δ1η2δ2 ] ⊂ r4 for ρ γ1 δ1 ≥ 1. hence the approximate solution converges towards a weak solution of the equations. the space discretization is based on an admissible mesh as defined in [10]. it is a finite family t of polygonal open convex subsets k of ω, called the control volumes such that ω̄ = ∪k∈t k̄, together with a finite family e of disjoint subsets of ω̄ consisting in non-empty open convex subsets σ of affine hyperplanes of rd, called the edges, and a family p = {xk , k ∈ t } of points in ω, called the centers verifying the following properties. • each σ ∈ e is contained in ∂k for some k ∈ t and for any k ∈ t , there exists a subset ek of e such that ∂k = ∪σ∈ek σ̄. for any edge σ ∈ e, either σ ⊂ ∂ω or σ = k̄ ∩ l̄ for some k 6= l in t . in the latter case, we denote σ = σkl, called the interfaces. we denote by e? ⊂ e the subset of all the interfaces and, for any k ∈ t , by n (k) = {l ∈ t , σkl ∈ ek ∩ e?} ⊂ t the neighbors of k. • for any k ∈ t , the point xk belongs to k. for any σkl ∈ e, the line (xk , xl) is orthogonal to σkl. biomath 1 (2012), 1209254, http://dx.doi.org/10.11145/j.biomath.2012.09.254 page 2 of 5 http://dx.doi.org/10.11145/j.biomath.2012.09.254 y. coudière et al., an upstream finite volume scheme for a bone healing model additionally, for any σkl ∈ e?, we denote by nkl and dkl, respectively, the unit vector normal to σkl outward of k and the distance |xk −xl|. the measure of k ∈ t is denoted by |k| and the (d − 1)-dimensional measure of σ ∈ e is denoted by |σ|. the time discretization is the sequence of discrete times tn = n∆t for n ∈ n where ∆t > 0 is a given time-step. the numerical scheme is obtained by using the finite volume method: equations (1) to (4) of the model are integrated on each control volume k and interval of time (tn, tn+1). by using the divergence theorem, we obtain the following scheme |k|(sn+1k − s n k ) − ∆t ∑ l∈n (k) λn+1kl sn+1l − s n k dkl |σkl| + ∆t ∑ l∈n (k) f (sn+1k , s n+1 l , v n+1 kl mn+1l − m n+1 k dkl )|σkl| = ∆t|k|f1(sn+1k , m n+1 k , g n+1 k ) |k|(bn+1k − b n k ) = |k|∆tf2(s n+1 k , b n+1 k , m n+1 k , g n+1 k ) |k|(mn+1k − m n k ) = |k|∆tf3(b n+1 k , m n+1 k ), |k|(gn+1k − g n k ) − ∆t ∑ l∈n (k) λg gn+1l − g n+1 k dkl |σkl| = |k|∆tf4(bn+1k , g n+1 k ), where the unknowns sn+1k , b n+1 k , m n+1 k and g n+1 k approximate 1|k| ∫ k s(tn+1, x)dx, 1|k| ∫ k b(tn+1, x)dx, 1 |k| ∫ k m(tn+1, x)dx and 1|k| ∫ k g(tn+1, x)dx. an implicit time stepping strategy is used for all the terms. the approximations of λ and v at an interface are calculated with an arithmetic mean between the two neighboring control volumes. the haptotaxis term is approximated by a flux f . although an upstream flux would be welladapted, it does not ensure a maximum principle on the discrete solution. consequently, the flux is defined such as follow: f (a, b, c) = c+ (χ↑(a) + χ↓(b)) − c− (χ↑(b) + χ↓(a)) where c+ = max(c, 0), c− = max(−c, 0), χ↑(a) =∫ a 0 χ′(s)+ds and χ↓(a) = − ∫ a 0 χ′(s)−ds. this flux verify the two classical properties of conservativity and consistency and an additional property of monotony: for any (a, c) ∈ r2, the mapping b ∈ r 7→ f (a, b, c) is non-increasing. this ensures the maximum principle. for this numerical scheme, we have proved the following results. theorem 3.1 (existence of an admissible solution): if the initial data is physically admissible, specifically if (s0k , b 0 k , m 0 k , g 0 k ) belongs to a for all control volumes k in t , then the discrete system of equations has a solution unt = (s n k , b n k , m n k , g n k ) for all n ∈ n, which is physically admissible: ∀n ≥ 0, ∀k ∈ t , (snk , b n k , m n k , g n k ) ∈ a. theorem 3.2 (convergence to a weak solution): if the initial data is physically admissible and the discrete equivalent of the h1 semi-norm [10] of b0 and m0 are bounded, then there exists a subsequence (uh) of discrete solutions that converges to a function u = (s, b, m, g) almost everywhere in [0, t ] × ω (for any t > 0). this function u is an admissible (u(t) ∈ a for all t > 0) weak solution of the model. iv. numerical simulation: a skull fracture in order to validate the interest of the model, we simulate the healing of a skull fracture in rats. it is wellsuited to our model because there is no cartilage in this kind of fractures although it is essential in many cases. the simulation corresponds to an experience presented in [11]. in this article, defects were created using a 2.3 mm outer diameter trephine in the parietal bones of 6 adult sprague-dawley rats and the rats were allowed to heal for 42 days. fig. 1: voronoi diagram used for the simulation of a skull fracture (5884 control volumes) to compute numerical solutions, we implement a simplified semi-implicit time-stepping technique with the newton’s method coupled with a biconjugate gradient method to solve the nonlinear system arising from the discretization. a difficulty in the implementation is to construct admissible meshes satisfying the orthogonality condition. structured rectangular meshes are admissible meshes but they can not be used for complex geometries, biomath 1 (2012), 1209254, http://dx.doi.org/10.11145/j.biomath.2012.09.254 page 3 of 5 http://dx.doi.org/10.11145/j.biomath.2012.09.254 y. coudière et al., an upstream finite volume scheme for a bone healing model like circular fractures for skull bones. we choose to use voronoi diagrams, that verify the property of orthogonality between the interfaces and their respective centers. but most mesh generators give voronoi diagrams with very small interfaces. to avoid this, we give a set of points well distributed in the domain, next we construct the voronoi diagram associated to these points (figure 1), which represents the centers of the control volumes. consequently, the number of interfaces is different for each control volumes. fig. 2: full geometry of the fracture. fig. 3: one quarter of the domain where the black area corresponds to the bone (m0(x, y) = 0.1 g.ml-1) and cellular cluster (s0(x, y) = 106 cells.ml-1 and g0(x, y) = 2 × 103 ng.ml-1). elsewhere, there is nothing initially. the geometry of the circular skull fracture is reported on figure 2. the symmetries about fracture line and bone axis implies that only one-quarter of the domain needs to be considered (figure 3). initially, the domain contains only the bone and two cell clusters along the broken bone made up of stem cells and growth factor (figure 3). the mesh used is a voronoi mesh made up of 5884 control volumes (figure 1) and the time-step is fixed at ∆t = 14 minutes and 24 seconds. after 3 days, we (a) bone matrix density 0 ≤ m ≤ 0.1 g.ml-1 (b) concentration of stem cells 0 ≤ s ≤ 2.29 × 103 cells.ml-1 (c) concentration of osteoblasts 0 ≤ b ≤ 9.9 × 105 cells.ml-1 (d) concentration of the growth factor 0 ≤ g ≤ 180 ng.ml-1 fig. 4: bone matrix density, concentrations of stem cells, osteoblasts and the growth factor at t = 3 days. biomath 1 (2012), 1209254, http://dx.doi.org/10.11145/j.biomath.2012.09.254 page 4 of 5 http://dx.doi.org/10.11145/j.biomath.2012.09.254 y. coudière et al., an upstream finite volume scheme for a bone healing model 0 0.173 0 0.05 0.1 x (cm) m (g/ml) 0 day 4 days 8 days 16 days 42 days fig. 5: bone matrix density along the line y = x as a function of distance to the origin (x, y) = (0, 0). observe the formation of osteoblasts (figure 4c) where the stem cells are initially concentrated, these osteoblasts synthesized a new bone matrix (figure 4a). the stem cells moved towards the center of the fracture (figure 4b). the osteoblasts trapped in the new bone become osteocytes. this model successfully simulates the evolution of the mineralization front (figure 5). at t = 42 days, since all stem cells disappear, we can consider that the healing is terminated. the defect heals approximatively 42% (close to the results of the article [11]), it is a nonunion fracture because the initial defect is too large. v. conclusion in this article, we proposes a model and a finite volume numerical method to simulate bone regeneration that is well-suited to the growth in bioreactor. the approximate solutions remain in a physically admissible region, and the convergence to a weak solution of the equation is guaranteed. the simulation of a skull fracture allows to validate this model for bone healing without cartilage. now, we plan to develop another model in order to simulate the growth in a bioreactor. it ought to include the flow environment and its interaction with the previous model. modeling this interaction is an important challenge in order to understand three-dimensional ex vivo tissue culture. references [1] c. laurencin, a. ambrosio, m. borden, and j. cooper jr, “tissue engineering: orthopedic applications”, annual review of biomedical engineering, vol. 1 no. 1, pp. 19–46, 1999. http://dx.doi.org/10.1146/annurev.bioeng.1.1.19 [2] c. t. l. d. et migonney v. b. bujoli, “ces matériaux qui réparent le corps”,cnrs le journal, no. 252-253, p. 26, 2011. 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[10] r. eymard, t. gallouët, and r. herbin, “finite volume methods”, handbook of numerical analysis, vol. 7, pp. 713–1018, 2000. http://dx.doi.org/10.1016/s1570-8659(00)07005-8 [11] g. cooper, m. mooney, a. gosain, p. campbell, j. losee, and j. huard, “testing the “critical-size”in calvarial bone defects: revisiting the concept of a critical-sized defect (csd)”, plastic and reconstructive surgery, vol. 125 no. 6, 1685, 2010. http://dx.doi.org/10.1097/prs.0b013e3181cb63a3 biomath 1 (2012), 1209254, http://dx.doi.org/10.11145/j.biomath.2012.09.254 page 5 of 5 http://dx.doi.org/10.1146/annurev.bioeng.1.1.19 http://dx.doi.org/10.1002/mabi.200400026 http://dx.doi.org/10.1016/j.jtbi.2007.11.008 http://dx.doi.org/10.1016/j.jbiomech.2004.10.019 http://dx.doi.org/10.1016/j.jtbi.2004.12.023 http://dx.doi.org/10.1006/jtbi.2001.2372 http://dx.doi.org/10.1097/00003086-199810001-00003 http://dx.doi.org/10.1016/s1570-8659(00)07005-8 http://dx.doi.org/10.1097/prs.0b013e3181cb63a3 http://dx.doi.org/10.11145/j.biomath.2012.09.254 introduction a model for population dynamics the numerical scheme numerical simulation: a skull fracture conclusion references original article biomath 1 (2012), 1209256, 1–5 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum structure of the global attractors in a model for ectoparasite-borne diseases attila dénes∗ and gergely röst∗ ∗ bolyai institute, university of szeged, aradi vértanúk tere 1, szeged, hungary emails: denesa@math.u-szeged.hu, rost@math.u-szeged.hu received: 15 july 2012, accepted: 25 september 2012, published: 19 october 2012 abstract—we delineate a mathematical model for the dynamics of the spread of ectoparasites and the diseases transmitted by them. we present how the dynamics of the system depends on the three reproduction numbers belonging to three of the four possible equilibria and give a complete characterization of the structure of the global attractor in each possible case depending on the reproduction numbers. keywords-ectoparasites; global dynamics; global attractors i. introduction ectoparasites (e.g. lice, fleas, mites) cause a serious problem in several parts in the world [2], [4]. ectoparasite infestations are often connected to the lack of hygiene and poor economical conditions, however, their presence is increasing in developed countries as well. the three louse species which transmit diseases are the head louse, the body louse and the pubic louse. these species are responsible for the spread of trench fever, epidemic typhus and relapsing fever. the flea species which most commonly affect humans are the cat, the rat and the human flea. fleas transmit plague, murine typhus, fleaborne spotted rickettsiosis. the transmission of these diseases is different from that of other vectorborne diseases, as it is carried out through the human contact network, which means that the spread of the vectors themselves is similar to that of a disease. in this paper we delineate a model for the dynamics of ectoparasite-borne diseases and we describe the structure of the global attractors in the different situations depending on the reproduction numbers. we assume the presence of one disease and one ectoparasite species which is a vector transmitting this particular disease. the human population is divided into three compartments: susceptibles (i.e. those who can be infested by both infectious and non-infectous vectors, denoted by s(t)), those who are infected by non-infectious parasites (denoted by t (t)) and those who are infested by infectious vectors (denoted by q(t)). we assume that someone infested by non-infectious vectors can transmit the parasites to susceptibles, while an individual infested by infectious vectors transmits both the parasites and the disease to susceptibles. an individual infested by infectious vectors transmits the infection to individuals infested by non-infectious vectors, i.e. a member of compartment t can move to compartment q upon adequate contact with someone from compartment q. we assume that a person is infected by the disease if and only he is infested by infectious parasites. we suppose that individuals infested by infected parasites transmit the disease at the same rate to susceptibles and to those who are infested by non-infected parasites. we denote this transmission rate by βq, and βt denotes the transmission rate for non-infectious vectors (to susceptibles). the rate of disinfestation is denoted by µ for the infected compartment and by θ for the non-infected compartment. we denote by b the natural birth and death rates, and we assume the disease is not fatal, thus the population size is constant. in the model equations we use mass action incidence. we have the following system of differential equations citation: a. dénes, g. röst, structure of the global attractors in a model for ectoparasite-borne diseases, biomath 1 (2012), 1209256, http://dx.doi.org/10.11145/j.biomath.2012.09.256 page 1 of 5 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.09.256 a. dénes et al., structure of the global attractors in a model for ectoparasite-borne diseases (with all the parameters assumed to be positive): s′(t) = −βt s(t)t (t) − βqs(t)q(t) + θt (t) + µq(t) + b − bs(t), t ′(t) = βt s(t)t (t) − βqq(t)t (t) − θt (t) − bt (t), q′(t) = βqs(t)q(t) + βqq(t)t (t) − µq(t) − bq(t). (1) it can easily be seen that any solution with nonnegative initial values remains non-negative for all forward time. we can suppose that n (t) = s(t) + t (t) + q(t) = 1 holds for the total population. the phase space of our system is x := {(s, t, q) ∈ r3+ : s + t + q = 1}. ii. equilibria, reproduction numbers by solving the algebraic equations 0 = −βt s∗t ∗ − βqs∗q∗ + θt ∗ + µq∗ + b − bs∗, 0 = βt s ∗t ∗ − βqq∗t ∗ − θt ∗ − bt ∗, 0 = βqs ∗q∗ + βqq ∗t ∗ − µq∗ − bq∗, we can determine the four equilibria of system (1): es = (1, 0, 0), et = ( b + θ βt , 1 − b + θ βt , 0 ) , eq = ( b + µ βq , 0, 1 − b + µ βq ) , eqt = ( θ − µ + βq βt , b + µ βq − θ − µ + βq βt , 1 − b + µ βq ) . reproduction numbers have a clear biological interpretation. we can obtain them by multiplying the number of new infections and the average length of the infectious period of an infectious agent newly introduced into a population currently being in one of the equilibria. by introducing an infested, non-infectious individual into a population in the equilibrium es , we obtain the reproduction number r1 = βt b + θ , by introducing an infested and infectious individual into the same equilibrium we obtain the reproduction number r2 = βq b + µ . calculating the expected number of secondary infections caused by the introduction of an infectious infested individual into a population in the equilibrium et gives the same reproduction number r2. if we introduce a non-infectious infested individual into a population in the equilibrium eq, we obtain the reproduction number r3 = βt (b + µ) βq(βq − µ + θ) . the following lemma is taken from [3]. lemma 2.1: the equilibrium es always exists. the equilibrium et exists if and only r1 > 1. the equilibrium eq exists if and only if r2 > 1. the equilibrium eqt exists if and only if r2 > 1 and r3 > 1. iii. structure of the global attractor here we recall the main result of [3]. theorem 3.1: let xq := {(s, t, 0) ∈ r3+ : s + t = 1} and xt := {(s, 0, q) ∈ r3+ : s + q = 1} denote the extinction sets for q and t , respectively. the four equilibria have the following global stability properties depending on the reproduction numbers: (i) equilibrium es is globally asymptotically stable if r1 ≤ 1 and r2 ≤ 1. (ii) equilibrium et is globally asymptotically stable on x \ xt if r1 > 1 and r2 ≤ 1. on xt , es is globally asymptotically stable. (iii) if r2 > 1, r3 ≤ 1 and r1 ≤ 1, then eq is globally asymptotically stable on x \ xq and es is globally asymptotically stable on xq. (iv) if r2 > 1, r3 ≤ 1 and r1 > 1, then eq is globally asymptotically stable on x \ xq and et is globally asymptotically stable on xq. (v) if r2 > 1, r3 > 1, then eqt is globally asymptotically stable on x \(xq ∪xt ), et is globally asymptotically stable on xq and eq is globally asymptotically stable on xt . an equilibrium e is said to be globally asymptotically stable on a set y if it is stable and for all y ∈ y the solution starting from y converges to e as t → ∞. following the notation of [1, 1.1.7], by m t we denote the set consisting of the states at time t of the solutions started from all of the points x ∈ m . definition 3.2: let a ∈ x be a compact invariant set. if a attracts each bounded subset of x, i.e. for any bounded subset m ⊂ x and any neighbourhood u of a there exists a t < ∞ such that m t ⊂ u for all t > t , then a is called the global attractor. definition 3.3: the ω-limit set of a point x ∈ x, denoted by ω(x) consists of those elements y of x for which there exists a real sequence {tn} such that tn ↗ ∞ and xtn → y as n → ∞. the α-limit set is defined similarly with tn ↘ −∞. in the following theorem we describe the structure of the global attractor for system (1) in the five cases listed in theorem 3.1. theorem 3.4: the global attractor a for system (1) has the following structure: (i) a = {es}. biomath 1 (2012), 1209256, http://dx.doi.org/10.11145/j.biomath.2012.09.256 page 2 of 5 http://dx.doi.org/10.11145/j.biomath.2012.09.256 a. dénes et al., structure of the global attractors in a model for ectoparasite-borne diseases fig. 1. representation of the flow and the attractors on the t q-plane. (ii) a = {es , et }∪γ1, where γ1 is a connecting orbit from es to et , which is actually the segment between es and et in the extinction space xq. (iii) a = {es , eq}∪γ2, where γ2 is a connecting orbit from es to eq, which is actually the segment between es and eq in the extinction space xt . (iv) a = {es , et , eq} ∪ γ1 ∪ γ2 ∪ γ3 ∪ a1, where γ3 is a connecting orbit from et to eq, and a1 is the domain surrounded by es , et , eq, γ1, γ2 and γ3 in the t qplane consisting of connecting orbits from es to eq. (v) a = {es , et , eq, eqt } ∪ γ1 ∪ γ2 ∪ γ4 ∪ γ5 ∪ a2, where γ4 is a connecting orbit from et to eqt , γ5 is a connecting orbit from eq to eqt , and a2 is the domain surrounded by es , et , eq, eqt , γ1, γ2, γ4 and γ5 in the t q-plane consisting of connecting orbits from es to eqt . proof: (i) as proved in theorem 3.1, es is globally asymptotically stable in case (i), which means that the global attractor is the singleton es in this case. for the proof of the remaining cases we reduce the system to two dimensions by substituting s with 1−t −q. we get the system t ′(t) = βt (1 − t (t) − q(t))t (t) − βqq(t)t (t) − θt (t) − bt (t), q′(t) = βq(1 − t (t) − q(t))q(t) + βqq(t)t (t) − µq(t) − bq(t) (2) and the four equilibria es = (0, 0), et = ( 1 − b + θ βt , 0 ) , eq = ( 0, 1 − b + µ βq ) , eqt = ( b + µ βq − θ − µ + βq βt , 1 − b + µ βq ) . by standard linearization, we calculate the eigenvalues and eigenvectors of the jacobian of the linearized system in the four equilibria. the details of the calculations are straightforward thus omitted, here we only discuss the results and implications. the eigenvalues of the jacobian biomath 1 (2012), 1209256, http://dx.doi.org/10.11145/j.biomath.2012.09.256 page 3 of 5 http://dx.doi.org/10.11145/j.biomath.2012.09.256 a. dénes et al., structure of the global attractors in a model for ectoparasite-borne diseases of the linearized equation around the equilibrium es are λ1 = −b−θ + βt = (b + θ)(r1 −1) with corresponding eigenvector (1, 0) and λ2 = −b−µ+βq = (b+µ)(r2− 1) with corresponding eigenvector (0, 1). linearizing at the equilibrium et , one finds the eigenvalues λ1 = b + θ − βt = (b + θ)(1 − r1) with the eigenvector (1, 0) and λ2 = −b − µ + βq = (b + µ)(r2 − 1) with the eigenvector( (βq + βt )(b + θ − βt ) βt (βq + βt − 2b − θ − µ) , 1 ) . linearization around the steady state eq gives the following eigenvalues of the jacobian: λ1 = b + µ − βq = (b + µ)(1 − r2) with the eigenvector (0, 1) and λ2 = −θ+µ−βq +(b+µ)βt /βq = (r3−1)βt /(r2r3) with the eigenvector (1, 0). finally, if we linearize the system around the equilibrium eqt , we obtain the eigenvalues λ1 = b + µ − βq = (b + µ)(1 − r2) with corresponding eigenvector( (βq + βt )(βq(βq + θ) − bβt − µ(βq + βt )) βt (b(βq + βt ) − βq(2βq + θ) + (2βq + βt )µ) , 1 ) and λ2 = θ − µ + βq − (b + µ)βt /βq = (1 − r3)βt /(r2r3) with corresponding eigenvector (1, 0). (ii) if r1 > 1 and r2 < 1 then es has the stable eigenvector (0, 1) and the unstable eigenvector (1, 0). this means that es has a one-dimensional stable manifold which coincides with the invariant extinction space xt and a one-dimensional unstable manifold which coincides with the segment (es , et ) of the extinction space xq, while both of the eigenvectors at et are stable. γ1 is the connecting orbit from es to et lying in xq. if r2 = 1 then the second eigenvalue at es is equal to zero. in this case, the equation for q′(t) takes the form q′(t) = −βqq2(t) < 0 on xt , which means that all solutions started from xt tend to es . thus es has the same one-dimensional stable and unstable sets as in the case r2 6= 1. from theorem 3.1 we know that all solutions started from x \ xt tend to et , thus et has a two-dimensional stable set. (iii) if r1 < 1, r2 > 1 and r3 < 1 then es has the stable eigenvector (1, 0), and the unstable eigenvector (0, 1), while (0, 1) and (1, 0) are both stable eigenvectors for eq. if r1 = 1 then the equation for t ′(t) takes the form t ′(t) = −βt t 2(t) < 0, on the invariant extinction space xq. this means that all solutions on the center manifold belonging to the zero eigenvalue (which coincides with xq) tend to es . if r3 = 1 then the jacobian of the linearized system at eq has a zero eigenvalue with eigenvector (1, 0). the line q = 1 − b + µ βq is invariant: if we substitute 1 − (b + µ)/βq into the equation for q′(t) we get q′(t) = 0. this means that for r3 = 1 the center manifold belonging to the zero eigenvalue coincides with this line. for r3 = 1, the equation for t ′(t) has the form t ′(t) = −βt t 2(t) < 0 on this line, which means that all solutions started from this line tend to the equilibrium eq. γ2 is the connecting orbit from es to eq lying in xt . this shows the statement of (iii). (iv) in this case, the first eigenvector belonging to es loses its stability, while the same vector becomes a stable eigenvector for et . thus, es has two unstable eigenvectors and et has the stable eigenvector (1, 0) and an unstable eigenvector. from theorem 3.1 we know that any solution started from the one-dimensional unstable manifold of et tends to eq, from which the existence of a heteroclinic orbit γ3 from et to eq follows. the situation for eq is the same as in case (iii). we have to show that a1 consists of heteroclinic orbits from es to eq. let us take an arbitrary point p ∈ a1. from theorem 3.1 we know that ω(p) = {eq}. the negative limit set α(p) exists and is non-empty as the backward orbit is bounded by γ1 ∪ γ2 ∪ γ3. from the poincaré–bendixson theorem we know that α(p) can only be an equilibrium point (as there are no periodic orbits). we can exclude eq as it has a two-dimensional stable manifold. the unstable manifold of et coincides with xq, which is invariant and a1 ∩ xq = ∅, thus α(p) = {es}. (v) in this case, again, es has two unstable eigenvectors and thus a two-dimensional unstable manifold. similarly to the previous case, et has a stable and an unstable eigenvector, thus having a one-dimensional stable manifold and a one-dimensional unstable manifold. the eigenvector (1, 0) for eq is unstable, which means that eq has a one-dimensional stable manifold and a one-dimensional unstable manifold. eqt has two stable eigenvectors and thus a two-dimensional stable manifold. from theorem 3.1 we know that all solutions started from x \ (xt ∪ xq) tend to eqt , thus there exists a connecting orbit γ4 from et to eqt and a connecting orbit γ5 from eq to eqt . similarly to case (iv) we can show that the domain a2 consists of connecting orbits from es to eqt . if neither is available on your system, please use the font closest in appearance to times. avoid using bit-mapped fonts if possible. true-type 1 or open type fonts are preferred. please embed symbol fonts, as well, for math, etc. iv. conclusion we described the global attractor in all possible cases. depending on the three reproduction numbers, the global attractor might have the following structure: biomath 1 (2012), 1209256, http://dx.doi.org/10.11145/j.biomath.2012.09.256 page 4 of 5 http://dx.doi.org/10.11145/j.biomath.2012.09.256 a. dénes et al., structure of the global attractors in a model for ectoparasite-borne diseases • a singleton • a one-dimensional set consisting of two equilibria and a connecting orbit • a two-dimensional set consisting of three or four equlibria and connecting orbits between them. the biological interpretation of our results is the following. the reproduction numbers ri (i = 1, 2, 3) completely determine whether the infectious or the non-infectious parasites can invade a human population. this is mathematically expressed in the structure of the global attractors that we described. if r1 ≤ 1 and r2 ≤ 1, then the population is safe from any parasites. the implication for the control of the infection and infestation is that to eradicate the disease only, we have to decrease r2 to be less than 1, which is possible by reducing βq or increasing µ. to eliminate all the parasites, besides decreasing r2 we also have to decrease r1 (possible by reducing βt or increasing θ). decreasing only r1 is not enough for the elimination of the parasites. the reproduction number r3 is a threshold parameter which shows whether all the parasites become infectious or both infectious and noninfectious parasites can be present in the population. the transmission rates βq and βt can be effectively reduced by vigorous monitoring and isolation of infested individuals, while µ and θ can be increased by disinfestation treatment of individuals. acknowledgment research supported by european research council starting grant nr. 259559, otka k75517 and bolyai scholarship of hungarian academy of sciences. references [1] n. p. bhatia, g. p. szegö, dynamical systems: stability theory and applications, springer-verlag, 1967. [2] p. brouqui, d. raoult, “arthropod-borne diseases in homeless", ann. n.y. acad. sci., vol. 1078, pp. 223–235, 2006. http://dx.doi.org/10.1196/annals.1374.041 [3] a. dénes, g. röst, “global dynamics for the spread of ectoparasite-borne diseases", submitted. [4] l. houhamdi, p. parola, d. raoult, “les poux et les maladies transmises à l’homme", med. trop., vol. 65, pp. 13–23, 2005. [5] h. l. smith, h. r. thieme, dynamical systems and population persistence, graduate studies in mathematics, vol. 118, ams, providence, 2011. biomath 1 (2012), 1209256, http://dx.doi.org/10.11145/j.biomath.2012.09.256 page 5 of 5 http://dx.doi.org/10.1196/annals.1374.041 http://dx.doi.org/10.11145/j.biomath.2012.09.256 introduction equilibria, reproduction numbers structure of the global attractor conclusion references www.biomathforum.org/biomath/index.php/biomath covid-19 research communication modelling the impacts of lockdown and isolation on the eradication of covid-19 joel n. ndam department of mathematics, university of jos, p.m.b 2084, jos, nigeria ndamj@unijos.edu.ng received: 28 july 2020, accepted: 10 september 2020, published: 26 october 2020 abstract—a model describing the dynamics of covid-19 is formulated and examined. the model is meant to address the impacts of lockdown and social isolation as strategies for the eradication of the pandemic. local stability analysis indicate that the equilibria are locally-asymptotically stable for r0 < 1 and r0 > 1 for the disease-free equilibrium and the endemic equilibrium respectively. numerical simulations of the model equations show that lockdown is a more effective strategy in the eradication of the disease than social isolation. however, strict enforcement of both strategies is the most effective means that could end the disease within a shorter period of time. keywords-covid-19; isolation; lockdown; disease-free equilibrium; endemic equilibrium i. introduction the coronavirus disease, otherwise known as covid-19, has been ravaging the world since the beginning of the year 2020. the disease which started in wuhan, china in late december 2019, rapidly spread to almost all parts of the world by march 2020. consequently, it was declared a global health emergency by the world health organisation (who) on march 11, 2020 ([1], [2]). some of the immediate non-pharmaceutical measures taken in order to contain the spread of the pandemic includes quarantine, isolation, lockdown, border closure, wearing of face masks and other sanitary measures, including regular washing of hands and use of hand sanitizers ([4], [7]). medical experts have acknowledged the effectiveness of these measures in curbing the spread of the disease ([6], [3]). however, the question remains as to how we can quantify the effectiveness of these non-pharmaceutical measures. some mathematical models have been used to address some of these strategies, including the work of frost et al [5] who examined the effects of lockdown on the control of covid-19 in some african countries and predicted when the pandemic would end under different lockdown regimes. anguelov et al [9] also examined the effects of lockdown on the control of the disease, especially the different levels of lockdown intensity in south africa. lockdowns and social isolation have been widely used as copyright: c© 2020 ndam. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: joel n. ndam, modelling the impacts of lockdown and isolation on the eradication of covid-19, biomath 9 (2020), 2009107, http://dx.doi.org/10.11145/j.biomath.2020.09.107 page 1 of 8 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2020.09.107 joel n. ndam, modelling the impacts of lockdown and isolation on the eradication of covid-19 s i q r (α + δ)i (α + δs)qαs αr η λ βsi σi (1−γ)ωq θr ωγq fig. 1. flow diagram for the transmission of the pandemic. means of curbing the rapid spread of the pandemic [10], as well as curtailing the importation of the disease [11]. in the current work, we intend to examine the impacts of lockdown and social isolation in the eradication of the pandemic. we shall also try to establish which strategy is more effective in containing the disease in comparison with others. this, we believe, can guide the implementation of policies for optimal results. the remaining part of this paper are organised as follows: section 2 is dedicated to the mathematical formulation of the model, stability analysis of equilibria will be the subject of section 3, while numerical simulations and discussion will be considered in section 4 and the conclusion will be done in section 5. ii. formulation of the model we construct a mathematical model which focuses on the investigation of the impacts of lockdown and isolation as strategies for containing the transmission of covid-19. the population is divided into the following compartments: the susceptibles s(t), the infected i(t), the isolated q(t) and the recovered r(t), with the total population given by n(t) = s(t) + i(t) + q(t) + r(t). the flow diagram of the transmission dynamics based on these compartments is shown on figure 1. from the flow diagram, some isolated persons may escape, and thus move back into the infected compartment at the rate ωγq and there is importation of infections at the rate η. following the flow chart in figure 1, the governing equations for the dynamics of the disease become ds dt =λ + θr−βsi −αs di dt =η + (βs −α−δ −σ) i + ωγq dq dt =σi − (α + δs + ω)q dr dt =(1 −γ)ωq− (α + θ)r (1) with the initial conditions s(0) > 0,i(0) > 0,q(0) ≥ 0,r(0) ≥ 0. the parameter λ, is the recruitment rate of susceptible individuals (taken as the average birth rate per 1,000 in nigeria), η is the per capita rate of importation of infected persons into the population, α is the natural death rate of individuals in all the compartments, β is the effective rate of infection, σ is the isolation effort, while 0 < γ < 1 is the proportion of isolated individuals who escape from isolation at the rate ω, θ is the rate at which the recovered lose immunity and become susceptible again, δ and δs are the disease induced death rates of the infected and the isolated respectively. all the parameters are positive and each of them is taken as a rate per day. biomath 9 (2020), 2009107, http://dx.doi.org/10.11145/j.biomath.2020.09.107 page 2 of 8 http://dx.doi.org/10.11145/j.biomath.2020.09.107 joel n. ndam, modelling the impacts of lockdown and isolation on the eradication of covid-19 the feasible region for the system (1) is given by ψ = { (s,i,q,r) �r4+ : s + i + q + r ≤ λ α } , which is positively invariant. the basic reproduction ratio r0 of the model is obtained using the next generation matrix procedure. hence, we express the governing equations of the infected compartments as dxi dt = fi(x) −vi(x), (2) where fi are the new infections in compartment i and vi are the rates of transfer of infections in and out of the compartments. from (1), we obtain f1 = βsi,f2 = 0 and v1 = (α + δ + σ)i −ωγq−η, v2 = (α + δs + ω)q−σi. hence, at the dfe, we have f = ( βλ α 0 0 0 ) ,v = ( α + δ + σ −ωγ −σ α + δs + ω ) and |v | = (α + δ + σ)(α + δs + ω) −σωγ. the basic reproduction number, r0 is the spectral radius of the matrix (fv −1), given by r0 = βλ(α + δs + ω) α[(α + δ + σ)(α + δs + ω) −σωγ] (3) provided (α + δ + σ)(α + δs + ω) > σωγ. iii. existence and local stability of equilibria here, we examine the local stability of the disease-free equilibrium (dfe) as well as determine the existence and stability of the endemic equilibrium. a. disease-free equilibrium the jacobian of the system (1) evaluated at the dfe yields the matrix je0=   −α −βλ α 0 θ 0 βλ α −(α+δ+σ) ωγ 0 0 σ −(α+δs+ω) 0 0 0 (1 −γ)ω −(α+θ)   with eigenvalues λ1 = −α, λ2 = −(α + θ), λ3 = 1 2 [ (a−b−c)+ √ (a−b+c)2 +4σωγ ] , λ4 = 1 2 [ (a−b−c)− √ (a−b+c)2 +4σωγ ] , where a = βλ α ,b = α + δ + σ and c = α + δs + ω. hence the dfe, e0, is locally-asymptotically stable if b > a + σωγ c ⇒ bc > ac + σωγ. ∴ (α+δ+σ)(α+δs +ω)−σωγ > βλ α (α+δs +ω) ⇒ βλ(α + δs + ω) α[(α + δ + σ)(α + δs + ω) −σωγ] = r0 < 1, and we have the following result: theorem 1. the disease-free equilibrium (dfe) of the model (1) is locally-asymptotically stable for r0 < 1, that is, b > a + σωγ c and unstable otherwise. b. endemic equilibrium the endemic equilibrium (s∗,i∗,q∗,r∗) is obtained in terms of i∗ as s∗ = λ α + βi∗ + σθ(1 −γ)ωi∗ c(α + θ)(α + βi∗) , q∗ = σi∗ c , r∗ = σ(1 −γ)ωi∗ c(α + θ) . substituting the values of s∗, q∗ and r∗ in the equation for i in (1), we obtain (ar0 −β)i∗2 + (br0 −α)i∗ + cr0 = 0, (4) biomath 9 (2020), 2009107, http://dx.doi.org/10.11145/j.biomath.2020.09.107 page 3 of 8 http://dx.doi.org/10.11145/j.biomath.2020.09.107 joel n. ndam, modelling the impacts of lockdown and isolation on the eradication of covid-19 parameter description estimated value λ recruitment rate of susceptible individuals 0.0375 estimated [14] θ rate at which the recovered lose immunity 0.2 assumed β effective rate of infection 0.14 estimated[15] α natural death rate of susceptible individuals 0.015 assumed σ isolation efforts 0.4 [3] ω rate of recovery of isolated infectives 0.12 assumed γ proportion of isolated individuals that recovered 0.27 estimated [8] δ disease induced death rate of the infectives 0.021 estimated [15] δs disease induced death rate of the isolated infectives 0.02 assumed η rate of importation of infected individuals 0.1 assumed table i description of baseline parameters for model (1). where a= σθ(1 −γ)ω λ(α + θ)(α + δs + ω) , b= 1+ η λ , c = αη βλ . based on the properties of roots of equation (4), as discussed extensively by ouifki and banasiak [13], we have the following results: theorem 2. the model equation (1) has (i) no endemic equilibrium when r0 > max { α b , β a } . (ii) a unique endemic equilibrium when α b < r0 < β a or r0 < min { α b , β a } . (iii) two endemic equilibria when β a < r0 < α b . we now establish the stability of the endemic equilibrium using the eigenvalues of the jacobian matrix evaluated at (s∗,i∗,q∗,r∗), given by je∗=   −(α+βi∗) −βs∗ 0 θ βi∗ βs∗−b ωγ 0 0 σ −c 0 0 0 (1−γ)ω −(α+θ)   (5) letting h = α + βi∗ and φ = α + θ, we obtain the characteristic polynomial of the matrix (5) as λ4 + a1λ 3 + a2λ 2 + a3λ + a4 = 0, (6) where a1 = b + c + h + φ −βs∗, a2 = β 2s∗i∗ + (b + h)c + (h + φ)b +(c + h)φ −β(c + h + φ)s∗ −σωγ, a3 = (bh + β 2s∗i∗)(c + φ) + (b + h)cφ −β(ch + hφ + cφ)s∗ − (h + φ)σωγ a4 = ( bh + β2s∗i∗ ) cφ + σθωγβi∗ −(βcs∗ + σωγ)hφ −βσωθi∗. from (6), the necessary and sufficient conditions for the stability of an endemic equilibrium can be determined by the nature of roots of the quartic equation, based on the routh-hurwitz criteria as summarised in the theorem below: theorem 3. an endemic equilibrium is locallyasymptotically stable if (i) b + c + h + φ > βs∗ (ii) β2s∗i∗+(b+h)c+(h+φ)b+(c+h)φ > β(c + h + φ)s∗ + σωγ (iii) ( bh + β2s∗i∗ ) cφ +σθωγβi∗ > (βcs∗ + σωγ)hφ + βσωθi∗ and (iv) a1a2a3 > a23 + a 2 1a4. using the approach of wangari, et al [12], we take the effective rate of infection, β, as the bifurcation parameter, and obtain βc = ar0 1 + λ 4αηr0 (br0 −α)2 (7) replacing β with βc in (3) and solving for r0 yields the expression for the critical reproduction biomath 9 (2020), 2009107, http://dx.doi.org/10.11145/j.biomath.2020.09.107 page 4 of 8 http://dx.doi.org/10.11145/j.biomath.2020.09.107 joel n. ndam, modelling the impacts of lockdown and isolation on the eradication of covid-19 fig. 2. time-course solution of model equations with relaxed lockdown. fig. 3. time-course solution with total lockdown and moderate isolation, η = 0.0. number as rc = α b2λ [bλ + 2η(ak − 1) +2 √ η(ak − 1)(aηk + bλ −η)], (8) where k = λ(α+δs+ω) α[(α+δ+σ)(α+δs+ω)−σωγ] , and a and b are as previously defined. hence the critical reproduction number is obtained as rc = 0.47, based on the parameters defined in table i. consequently, the disease can be controlled, when r0 < rc < 0.47. iv. simulations and discussion an siqrs model has been constructed and analysed theoretically. in this section, we carry out the numerical simulation of the model equations to confirm the theoretical results. the simulation is carried out based on the parameter values in table i. the results are depicted on figures 2 to biomath 9 (2020), 2009107, http://dx.doi.org/10.11145/j.biomath.2020.09.107 page 5 of 8 http://dx.doi.org/10.11145/j.biomath.2020.09.107 joel n. ndam, modelling the impacts of lockdown and isolation on the eradication of covid-19 fig. 4. time-course solution with relaxed lockdown and strict social isolation, σ = 0.70,η = 0.1. fig. 5. time-course solution with relaxed social isolation and total lockdown , σ = 0.1,η = 0.0. 6 below. figure 2 depicts the case of enforcement of social isolation with relaxed lockdown, while figure 3 shows the effects of a total lockdown and social isolation, hence the disease can be eradicated. the scenario depicted in figure 4 is that of strict enforcement of social isolation and relaxed lockdown. under this situation, incidence of the disease reduces in the population with lower risk than the case depicted in figure 2.figure 5 on the other hand, shows the effect of relaxing social isolation while observing total lockdown. under this scenario, cases of the disease decline considerably, but persists in the population for a long period of time. the results shown in figures 2 to 6 indicate that the disease can only be eradicated under a total lockdown regime. however, a strict enforcement of both strategies could lead to the eradication of the disease within a shorter period of time as shown in figure 6. biomath 9 (2020), 2009107, http://dx.doi.org/10.11145/j.biomath.2020.09.107 page 6 of 8 http://dx.doi.org/10.11145/j.biomath.2020.09.107 joel n. ndam, modelling the impacts of lockdown and isolation on the eradication of covid-19 fig. 6. time-course solution with strict social isolation and total lockdown , σ = 0.70,η = 0.0. v. conclusion an siqrs model for the transmission of covid-19 is investigated with a view to determining the impacts of lockdown and social isolation on the eradication of the disease. conditions for the existence and local stability of the equilibria were determined.the results indicate that the model has two endemic equilibrium points, thus indicating the possibility of occurrence of a backward bifurcation. numerical results show that lockdown is more effective than social isolation in the containment of the pandemic. however, when lockdown is enforced for a long time, it could affect the economy of the nation adversely, but it can still be used effectively by implementing in segments. that is, enforce the lockdown only in disease endemic areas. nevertheless, enforcing both strategies together yield the best result as can be seen in figure 6. declaration of competing interest the author declares that they have no known competing interest or personal relationship that could have influenced the work reported in this paper. acknowledgments the author acknowledges the assistance of patricia azike in the construction of the model flow diagram. references [1] who (2020). coronavirus disease 2019 (covid-19) situation report-59, 19 march, 2020. 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[15] nigeria centre for disease control (ncdc)(2020). covid-19 situation report 173, august 19, 2020. biomath 9 (2020), 2009107, http://dx.doi.org/10.11145/j.biomath.2020.09.107 page 8 of 8 http://dx.doi.org/10.11145/j.biomath.2020.09.107 introduction formulation of the model existence and local stability of equilibria disease-free equilibrium endemic equilibrium simulations and discussion conclusion references www.biomathforum.org/biomath/index.php/biomath original article mechanotransduction caused by a point force in the extracellular space bradley j. roth department of physics, oakland university rochester, mi, usa roth@oakland.edu received: 16 january 2018, accepted: 19 october 2018, published: 27 october 2018 abstract—the mechanical bidomain model is a mathematical description of biological tissue that focuses on mechanotransduction. the model’s fundamental hypothesis is that differences between the intracellular and extracellular displacements activate integrins, causing a cascade of biological effects. this paper presents analytical solutions of the bidomain equations for an extracellular point force. the intraand extracellular spaces are incompressible, isotropic, and coupled. the expressions for the intraand extracellular displacements each contain three terms: a monodomain term that is identical in the two spaces, and two bidomain terms, one of which decays exponentially. near the origin the intracellular displacement remains finite and the extracellular displacement diverges. far from the origin the monodomain displacement decays in inverse proportion to the distance, the strain decays as the distance squared, and the difference between the intraand extracellular displacements decays as the distance cubed. these predictions could be tested by applying a force to a magnetic nanoparticle embedded in the extracellular matrix and recording the mechanotransduction response. keywords-analytical solution; extracellular matrix; integrin; intracellular cytoskeleton; mathematical model; mechanotransduction; mechanical bidomain model; point source. i. introduction mechanotransduction is the process by which biological tissues grow and remodel in response to mechanical signals. one cause of mechanotransduction might be a cascade of biological responses triggered by activation of integrin molecules in the cell membrane [2], [3], [16]. a force acting on the extracellular matrix is transmitted to the cytoskeleton via these integrins, thereby coupling the intraand extracellular spaces. much research on mechanotransduction is qualitative, but to predict quantitatively how tissue responds to applied forces we need a mathematical model [12]. many studies in mechanobiology analyze individual cells and molecules, but to describe tissues and organs we require a macroscopic model that averages over the cellular and molecular scales. yet, this macroscopic model must predict the activation of integrin molecules. one mathematical model that describes mechanotransduction is the mechanical bidomain model copyright: c© 2018 roth. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: bradley j. roth, mechanotransduction caused by a point force in the extracellular space, biomath 7 (2018), 1810197, http://dx.doi.org/10.11145/j.biomath.2018.10.197 page 1 of 7 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2018.10.197 bradley j. roth, mechanotransduction caused by a point force in the extracellular space fig. 1. a schematic illustration of the mechanical bidomain model. the green springs represent the intracellular cytoskeleton, the blue the extracellular matrix, and the red the integrins. the figure illustrates a two-dimensional version of the model, but this article analyzes a three-dimensional version. [11], [15]. it predicts displacements of the intraand extracellular spaces individually. the difference between the intraand extracellular displacements results in a force on the integrins that couple the two spaces. a schematic illustration of the model is shown in figure 1. one of the most important properties of a mathematical model is how it responds to a point source. often complicated responses can be expressed as a convolution of the point source response, so knowing how tissue responds to a point force provides insight into its general behavior. in this paper, i derive analytical expressions describing how the mechanical bidomain model responds to a point source in the extracellular space. experimentally, this could be approximated by, for instance, applying a magnetic force on a superparamagnetic nanoparticle [7], [8]. magnetic tweezers [5] have been used to exert forces on single cells or individual molecules. the technique, however, could be applied to intact tissue where a nanoparticle is embedded in the extracellular matrix. when a force is exerted by the nanoparticle it pulls on the matrix, which stretches the integrins embedded in the membranes of nearby cells, triggering mechanotransduction [9]. ii. methods i assume the intraand extracellular spaces are incompressible and isotropic, and their strains are small and linear. incompressibility implies that the intracellular displacement u and the extracellular displacement w are both divergenceless. i use spherical coordinates (r, θ, φ) with the force applied at the origin and acting along the z axis (θ = 0). by symmetry there are no displacements or derivatives in the φ direction. in that case u and the intracellular strain �i are related by [10] �irr = ∂ur ∂r , (1) �iθθ = 1 r ∂uθ ∂θ + ur r , (2) �iφφ = uθ r cotθ + ur r , (3) �irθ = 1 2 ( 1 r ∂ur ∂θ + ∂uθ ∂r − uθ r ) , (4) with analogous relationships in the extracellular space. the intracellular stress τi and the intracellular strain are related by τirr = −p + 2ν�irr, (5) biomath 7 (2018), 1810197, http://dx.doi.org/10.11145/j.biomath.2018.10.197 page 2 of 7 http://dx.doi.org/10.11145/j.biomath.2018.10.197 bradley j. roth, mechanotransduction caused by a point force in the extracellular space τiθθ = −p + 2ν�iθθ, (6) τiφφ = −p + 2ν�iφφ, (7) τirθ = 2ν�irθ, (8) where p is the intracellular pressure and ν is the intracellular shear modulus. similar stress-strain relationships exist for the extracellular pressure q and extracellular shear modulus µ. the equations of mechanical equilibrium are [10], [15] − ∂p ∂r + 2ν [ ∂�irr ∂r + 1 r ∂�irθ ∂θ + 1 r ( 2�irr − �iθθ − �iφφ + cotθ �irθ )] = k (ur − wr) , (9) − 1 r ∂p ∂θ + 2ν [ ∂�irθ ∂r + 1 r ∂�iθθ ∂θ + 1 r ( (�iθθ − �iφφ) cotθ + 3�irθ )] = k (uθ − wθ) , (10) − ∂q ∂r + 2µ [ ∂�err ∂r + 1 r ∂�erθ ∂θ + 1 r ( 2�err − �eθθ − �eφφ + cotθ �erθ )] + fδ (r) cosθ = −k (ur − wr) , (11) − 1 r ∂q ∂θ + 2µ [ ∂�erθ ∂r + 1 r ∂�eθθ ∂θ + 1 r ( (�eθθ − �eφφ) cotθ + 3�erθ )] − fδ (r) sinθ = −k (uθ − wθ) , (12) where k is the integrin spring constant coupling the two spaces, f is the force applied to the extracellular space, and δ(r) is the delta function. i assume that the displacements and pressures go to zero at large r. to picture the problem physically, imagine that in figure 1 a point in the extracellular matrix (one of the blue dots) is pulled to the right by an attached nanoparticle. this force would displace the extracellular matrix (blue springs), which would stretch the integrins coupling the two spaces (red springs). the integrins would then pull on the cytoskeleton, causing the intracellular space to be displaced. iii. results equations 9-12 were solved using the method of undetermined coefficients. the solution is ur = f 8π (ν + µ) cosθ{ 2 r − 4σ2 r3 + 4 [ σ2 r3 + σ r2 ] e− r σ } , (13) uθ = f 8π (ν + µ) sinθ{ − 1 r − 2σ2 r3 + 2 [ σ2 r3 + σ r2 + 1 r ] e− r σ } , (14) wr = f 8π (ν + µ) cosθ{ 2 r + ν µ 4σ2 r3 − 4 ν µ [ σ2 r3 + σ r2 ] e− r σ } , (15) wθ = f 8π (ν + µ) sinθ{ − 1 r + ν µ 2σ2 r3 − 2 ν µ [ σ2 r3 + σ r2 + 1 r ] e− r σ } , (16) p = 0, (17) q = f 4π cosθ r2 . (18) biomath 7 (2018), 1810197, http://dx.doi.org/10.11145/j.biomath.2018.10.197 page 3 of 7 http://dx.doi.org/10.11145/j.biomath.2018.10.197 bradley j. roth, mechanotransduction caused by a point force in the extracellular space each expression for the displacement contains a monodomain term (first term in the brace) that is the same in the intraand extracellular spaces, and two bidomain terms that are different in the two spaces (one is -ν/µ times the other). the first bidomain term is proportional to σ2, where σ = √ νµ k(ν+µ) is a length constant characteristic of the mechanical bidomain model [15]. the exponential in the second bidomain term decays with length constant σ. the displacements (eqs. 13-16) have interesting properties as r goes to zero. if you expand the exponential as a taylor series, you will find that the terms in the expression for the intracellular displacement that are singular at the origin cancel and it remains finite there. the extracellular displacement, however, diverges at the origin as 1/r as expected for a delta function source in the extracellular space. at large distances (r � σ) bidomain terms decay more rapidly than monodomain terms. the fundamental hypothesis of the mechanical bidomain model is that mechanotransduction depends on the difference u w [15]. the monodomain terms are the same in the two spaces and do not contribute to u w; only the bidomain terms generate the displacement difference that drives mechanotransduction, ur−wr = f 8πµ cosθ { − 4σ2 r3 +4 [ σ2 r3 + σ r2 ] e− r σ } , uθ−wθ = f 8πµ sinθ { − 2σ2 r3 +2 [ σ2 r3 + σ r2 + 1 r ] e− r σ } . for r � σ the exponentials are negligible and the difference in displacements falls as 1/r3. figure 2 shows the extracellular displacement, w, the intracellular displacement, u, and their difference, u w, in the plane corresponding to a constant angle φ. near the source, u w resembles -w. far from the source, u w is small compared to u and w individually. fig. 2. the extarcellular displacement, w, the intracellular displacement, u, and their difference, u-w. the calculation assumes ν = µ. the black dot indicates the position of the point source, corresponding to an applied force f acting to the right. biomath 7 (2018), 1810197, http://dx.doi.org/10.11145/j.biomath.2018.10.197 page 4 of 7 http://dx.doi.org/10.11145/j.biomath.2018.10.197 bradley j. roth, mechanotransduction caused by a point force in the extracellular space fig. 3. ur, wr, ur wr, and �irr as functions of r/σ, for θ = 0; ur is indicated by short dashes, wr by long dashes, ur wr by a solid line, and �irr by dash-dot. all quantities are normalized so that the intracellular displacement and strain are equal to one at the origin. figure 3 plots the intraand extracellular displacements and their difference along the direction of the applied force. it also shows the intracellular strain, �irr. at large distances, the displacements fall as 1/r, the strain as 1/r2, and the difference in the displacements as 1/r3. this result is a testable prediction. if mechanotransduction depends on the strain it decays relatively slowly, as 1/r2. if, however, mechanotransduction depends on u w it decays relatively rapidly, as 1/r3. iv. discussion most biomechanical models treat tissue as a single phase: a monodomain. these mathematical models are often valuable tools for predicting tissue displacements, stresses, and strains [4]. if, however, mechanotransduction is triggered by activation of integrins, and integrins are activated by differences between the displacements of the intraand extracellular spaces, then a bidomain model is essential for predicting where mechanotransduction occurs. the activation of integrins could in principle be determined by measuring the intraand extracellular displacements individually, and then taking their difference. in practice, however, this difference is very small compared to the displacements themselves, and a better strategy would be to measure a mechanotransduction effect caused by integrin activation, such as tissue growth, remodeling, or genetic changes associated with these processes. the monodomain solution for a point source is ur = wr = f8π(ν+µ) 2 cosθ r and uθ = wθ = − f 8π(ν+µ) sinθ r . this solution is the same as the expression for the velocity caused by a point force in an incompressible fluid at low reynolds number [10], sometimes referred to as a stokeslet. when σ is small the stokeslet approximates the displacements in the intraand extracellular spaces, but it provides no information about where mechanotransduction occurs because it contributes nothing to u w. the monodomain term can be represented in fig. 3 as a line that matches the u and w curves at large radii, and is extrapolated back linearly at smaller radii. a key parameter in the model is the length constant σ, which depends on the bidomain constant k coupling the intraand extracellular spaces. in monolayers of stem cells, σ is about 150 microns [1], which is larger than a cell and much larger than a nanoparticle, implying that a macroscopic model should be valid. the mechanical bidomain model has many similarities to the electrical bidomain model [6] used to describe pacing and defibrillation of the heart. my analysis of the mechanical bidomain model’s response to a point force is analogous to the calculation of the transmembrane potential produced by a point current using the electrical bidomain model [13]. in the electrical model, unequal anisotropy ratios for the intraand extracellular conductivities plays a crucial role in determining the transmembrane potential distribution. similar effects might arise in the mechanical model if it were made anisotropic. what experiment can test the predictions of this biomath 7 (2018), 1810197, http://dx.doi.org/10.11145/j.biomath.2018.10.197 page 5 of 7 http://dx.doi.org/10.11145/j.biomath.2018.10.197 bradley j. roth, mechanotransduction caused by a point force in the extracellular space model? one suggestion is to grow a large cluster of epithelial cells, with a magnetic particle at its center. alternatively, tissue engineering techniques could be used to grow cells in an extracellular substrate containing a magnetic particle. then, a force could be applied to the particle, and the mechanotransduction response could be imaged by monitoring a second messenger activated by the integrins, or the turning on of a gene associated with cell growth. the bidomain model has several limitations. it assumes a linear relationship between displacement and strain, which is only appropriate for small strains [10]. in my solution, the extracellular displacement and strain diverge at the origin, so the small strain assumption is violated there. however, the delta function is an approximation that breaks down on a distance scale similar to the radius of the magnetic nanoparticle used to exert the force. as long as the strains are small at this scale, the linear approximation should be valid. i assume the stress-strain relationships are linear, whereas in tissue these relationships can be nonlinear [4]. if the strains are small enough, however, a linear approximation should suffice. i assume that the tissue is isotropic, but tissues such as muscle are anisotropic and the model needs to be extended to account for anisotropy. i assume both the intraand extracellular spaces are incompressible. because both spaces contain mostly water, the incompressible assumption should be accurate [14]. my model is for steady-state. if the applied force varies with time, the solution might be invalid over short times because of the propagation of sound waves, or over long times because of viscoelasticity or tissue growth and remodeling. finally, and fundamentally, i assume that mechanotransduction depends on the difference in the displacements, u w. if it depends on other factors, such as the intracellular stress or strain, or some microscopic behavior that is not included in this macroscopic model, the results might not describe mechanotransduction correctly. the model could be extended to avoid some of my limiting assumptions, but in that case an analytical solution might not exist. analytical solutions can provide insight into the model behavior and are valuable even when the model is only an approximation. moreover, analytical solutions are useful for testing limiting cases of complex models and for evaluating the accuracy of numerical methods. v. conclusion the mechanical bidomain model makes testable predictions about where mechanotransduction occurs. in particular, the model predicts that the distribution of mechanotransduction in response to a point source in the extracellular space falls off with distance more rapidly if mechanotransduction is driven by the difference in the intraand extracellular displacements, and less rapidly if mechanotransduction is driven by intraor extracellular strain. this prediction could be tested by measuring how the tissue responds to a force applied using a magnetic nanoparticle embedded in the extracellular space. references [1] auddya d, roth bj (2017) a mathematical description of a growing cell colony based on the mechanical bidomain model. j phys d 50:105401. [2] chiquet m (1999) regulation of extracellular matrix gene expression by mechanical stress. matrix biology 18:417-426. [3] dabiri be, lee h, parker kk (2012) a potential role for integrin signaling in mechanoelectrical feedback. prog biophys mol biol 110:196-203. [4] fung yc (1981) biomechanics: mechanical properties of living tissues. springer, new york. [5] gosse c, croquette v (2002) magnetic tweezers: micromanipulation and force measurement at the molecular level. biophys j 82:3314-3329. [6] henriquez cs (1993) simulating the electrical behavior of cardiac tissue using the bidomain model. crit rev biomed eng 21:1-77. [7] hughes s, mcbain s, dobson j, el haj aj (2007) selective activation of mechanosensitive ion channels using magnetic particles. j r soc interface 5:855-863. [8] ingber de (2009) from cellular mechanotransduction to biologically inspired engineering. ann biomed eng 38:1148-1161. [9] kresh jy, chopra a (2011) intercellular and extracellular mechanotransduction in cardiac myocytes. pflugers arch eur j physiol 462:75-87. biomath 7 (2018), 1810197, http://dx.doi.org/10.11145/j.biomath.2018.10.197 page 6 of 7 http://dx.doi.org/10.11145/j.biomath.2018.10.197 bradley j. roth, mechanotransduction caused by a point force in the extracellular space [10] love aeh (1944) a treatise on the mathematical theory of elasticity. dover, new york. [11] roth bj (2013) the mechanical bidomain model: a review. isrn tissue engineering 2013:863689. [12] schwarz us (2017) mechanobiology by the numbers: a close relationship between biology and physics. nat rev mol cell biol 18:711-712. [13] sepulveda ng, roth bj, wikswo jp (1989) current injection into a two-dimensional anisotropic bidomain. biophys j 55:987-999. [14] sharma k, roth bj (2014) how compressibility influences the mechanical bidomain model. biomath 3:141171. [15] sharma k, al-asuoad n, shillor m, roth bj (2015) intracellular, extracellular, and membrane forces in remodeling and mechanotransduction: the mechanical bidomain model. journal of coupled systems and multiscale dynamics 3:200-207. [16] sun y, chen cs, fu j (2012) forcing stem cells to behave: a biophysical perspective of the cellular microenvironment. annu rev biophys 41:519-542. biomath 7 (2018), 1810197, http://dx.doi.org/10.11145/j.biomath.2018.10.197 page 7 of 7 http://dx.doi.org/10.11145/j.biomath.2018.10.197 introduction methods results discussion conclusion references original article biomath 2 (2013), 1312302, 1–5 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum multiscale analysis of ionic transport in periodic charged media claudia timofte faculty of physics, university of bucharest bucharest, romania claudiatimofte@yahoo.com received: 23 september 2013, accepted: 30 december 2013, published: 23 january 2014 abstract—a macroscopic model for describing the ion transport in periodic charged porous media is rigorously derived. our results can serve as a tool for biophysicists to analyze the ion transport through protein channels. also, such a model is useful for describing the flow of electrons and holes in a semiconductor device. keywords-homogenization; ion transport; the periodic unfolding method. i. introduction and setting of the problem the goal of this paper is to obtain the effective behavior of a system of coupled partial differential equations appearing in the modeling of transport phenomena in periodic charged porous media or in the modeling of the flow of holes and electrons in semiconductors. such a system is known in the literature as the poissonnernst-planck system, for the case of ion flow through membrane channels, or as the van roosbroeck model, for the case of the transport of holes and electrons in semiconductors. for the physical aspects behind these models and for a review of the recent literature, we refer to [8], [13], [14], [15], [16] and [19]. let us briefly describe the geometry of the problem. we assume, as it is customary in the literature, that the porous medium has a periodic microstructure. more precisely, for n ≥ 2, we consider a smooth bounded connected open set ω in rn, with | ∂ω |= 0. in what follows, we shall only consider the natural cases n = 2 or n = 3. we assume that the unit cell y = (0, 1)n consists of two smooth parts, the fluid part yf and, respectively, the solid part ys, which are supposed to be open, nonempty and disjoint sets such that y = yf ∪ys and yf ∩ys = γ. we suppose that the solid part has a lipschitz boundary and does not intersect the boundary of the basic cell y . therefore, the fluid zone is connected. let ε < 1 be a real parameter taking values in a sequence of positive numbers converging to zero. for each k ∈ zn, let y ks = k + ys and kε = {k ∈ zn | εy ks ⊆ ω}. we denote by ωsε = ⋃ k∈kε εy ks the solid part, by ωε = ω \ ωsε the fluid part and we set θ = ∣∣y \ys∣∣ . it is easy to see that if the solid part is not allowed to reach the fixed exterior boundary of the domain ω, the intersection between the outer boundary ∂ω and the citation: claudia timofte, multiscale analysis of ionic transport in periodic charged media, biomath 2 (2013), 1312302, http://dx.doi.org/10.11145/j.biomath.2013.12.302 page 1 of 5 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2013.12.302 c timofte, multiscale analysis of ionic transport in periodic charged media interior boundary γε of the porous medium is empty, i.e.: γε ∩∂ω = ∅. in such a periodic porous medium, we shall consider, at the microscale, the poisson-nernst-planck system, with suitable boundary and initial conditions. the nernstplanck equations, describing the diffusion in the fluid phase, are coupled with the poisson equation characterizing the electrical field which can influence the ionic transfer. more precisely, if we denote by [0,t], with t > 0, the time interval of interest, we shall be interested in analyzing the asymptotic behavior, as ε → 0, of the solution of the following system:  −∆ φε = c+ε − c−ε + d in (0,t) × ωε, −∇φε ·ν = εσεg(φε) on (0,t) × γε, ∇φε ·ν = 0 on (0,t) ×∂ω, ∂tc ± ε −∇· (∇c±ε ± c±ε ∇φε) = f±(c+ε ,c−ε ) in (0,t) × ωε, (∇c±ε ± c±ε ∇φε) ·ν = 0 on (0,t) × γε, (∇c±ε ± c±ε ∇φε) ·ν = 0 on (0,t) ×∂ω, c±ε (0,x) = c ± 0 (x) in ωε. (1) here, ν is the unit outward normal to ωε, φε represents the electrostatic potential, c±ε are the concentrations of the ions (or the density of electrons and holes in the particular case of van roosbroeck model), d ∈ l∞(ω) is the given doping profile, f± is a reaction term and g is a nonlinear function which takes into account the effect of the electrical double layer phenomenon arising at the interface γε. we assume that σε = σ (x ε ) , with σ(y) being a y -periodic, bounded, smooth real function such that σ(y) ≥ δ > 0, and g is a continuously differentiable function, monotonously increasing and such that g(0) = 0. also, we shall suppose that there exist c ≥ 0 and r, with 0 ≤ r ≤ n/(n− 2) for n = 3 and 0 ≤ r < ∞ for n = 2, such that | g′(s) |≤ c(1+ | s |r−1), ∀s ∈ r. let us notice that this hypothesis concerning the smoothness of the nonlinearity g can be relaxed by using a regularization technique, such as yosida approximation (see [18]). also, the results of this paper can be obtained, under our assumptions, without imposing any growth condition (see [17]). in practical applications, based on the gouy-chapman theory, one can use the grahame equation (see [2], [8] and [9]) in which g(s) = k1 sinh(k2s), k1,k2 > 0. for the case of lower potentials, sinh(x) can be expanded in a power series of the form sinh(x) = x + x3 3! + ... and one can use the approximations sinh x ≈ x or sinh x ≈ x + x3/3!. concerning the reaction terms, we deal, as in [13], with the linear case in which f±(c+ε ,c − ε ) = ∓(c + ε − c − ε ), but we can also address by our techniques the more general case in which f±(c+ε ,c − ε ) = ∓(a εc+ε − b εc−ε ), with aε(x) = a (x ε ) , bε(x) = b (x ε ) , where a(y) and b(y) are y -periodic, bounded, smooth real functions such that a(y) ≥ a0 > 0, b(y) ≥ b0 > 0. for the case of other nonlinear reaction rates f± and more general functions g, see [6], [10], [11] and [18]. we assume that the initial data are non-negative and bounded independently of ε and∫ ωε (c+0 − c − 0 + d)dx = ε ∫ γε σεg(φε)ds. (2) we also assume that the potential φε has zero mean value in ωε. let us mention that, for simplifying the notation, we have suppressed in system (1) some constant physical relevant parameters. we consider here only two oppositely charged species, i.e. positively and negatively charged particles, with concentrations c±ε , but all the results can be generalized for the case of n species. for the case in which we consider different scalings in (1), see [13] and [18]. also, let us remark that we can treat the case in which the electrostatic potential is defined all over the domain ω, with suitable transmission conditions at the interface γε (see, for instance, [8] or [16]). biomath 2 (2013), 1312302, http://dx.doi.org/10.11145/j.biomath.2013.12.302 page 2 of 5 http://dx.doi.org/10.11145/j.biomath.2013.12.302 c timofte, multiscale analysis of ionic transport in periodic charged media from the nernst-planck equation, it is easy to see that the total mass m = ∫ ωε (c+ε + c − ε )dx is conserved and suitable physical equilibrium conditions hold true, at the microscale and, also, at the macroscale (see, for details, [8] and [18]). using similar arguments as in [8] or [13], we can prove the well posedness of problem (1) in suitable function spaces and we can obtain proper energy estimates. the high complexity of the geometry and of the governing equations implies that an asymptotic procedure becomes necessary for describing the solution of such a problem. using the periodic unfolding method recently introduced by d. cioranescu, a. damlamian, g. griso, p. donato and r. zaki (see [3], [5] and [4]), we can prove that the asymptotic behavior of the solution of our problem is governed by a new coupled system of equations (see (3)-(5)). in particular, the evolution of the macroscopic electrostatic potential is governed by a new law, similar to grahame’s law (see [8] and [9]). an advantage of this approach is that we can avoid the use of extension operators and, therefore, we can deal in a rigorous manner with media which are less regular than those usually considered in the literature (composite materials and biological tissues are highly heterogeneous media with not very smooth interfaces, in general). similar problems have been considered, using different techniques, in [8], [13] or [15]. as already mentioned, our approach is based on a new method, i.e. the periodic unfolding method, which allows us to consider very general heterogeneous media. another novelty brought by our paper consists in dealing with a general nonlinear boundary condition for the electrostatic potential and with more general reaction terms. the rest of the paper is organized as follows: in section 2, we formulate our main convergence result, while section 3 is devoted to the proof of this result. the paper ends with some conclusions and a few references. ii. the main result using the periodic unfolding method, we are allowed to pass to the limit in the weak formulation of problem (1) and to obtain the effective behavior of the solution of our microscopic model. theorem 1. the solution (φε, c+ε ,c − ε ) of system (1) converges, as ε → 0, to the unique solution (φ, c+,c−) of the following macroscopic problem in (0,t) × ω:  −div (d0 ∇φ) + 1 | yf | σ0g = c + − c− + d, ∂ c± ∂ t − div(d0 ∇c± ±d0 c±∇φ) = f±0 , (3) with the boundary conditions on (0,t) ×∂ω:{ d0 ∇φ ·ν = 0, (d0 ∇c± ±d0 c±∇φ) ·ν = 0 (4) and the initial conditions c±(0,x) = c±0 (x), ∀x ∈ ω. (5) here, σ0 = ∫ γ σ(y)ds, f±0 (c +,c−) = ∓(c+ − c−) and d0 = (d0ij) is the homogenized matrix, defined as follows: d0ij = 1 |yf| ∫ yf ( δij + ∂χj ∂yi (y) ) dy , in terms of the functions χj, j = 1, ...,n, solutions of the cell problems  χj ∈ h1per(yf ) , ∫ yf χj = 0, −∆ χj = 0 in yf, (∇χj + ej) ·ν = 0 on γ, (6) where ei, 1 ≤ i ≤ n, are the vectors of the canonical basis in rn. iii. proof of the main result we shall only sketch the proof of our main convergence result. for details, we refer to [18]. let us consider now the equivalent variational formulation of problem (1): find (φε, c+ε , c − ε ), with  φε ∈ l∞(0,t; h1(ωε)), c±ε ∈ l∞(0,t; l2(ωε)) ∩l2(0,t; h1(ωε)), ∂tc ± ε ∈ l2(0,t; (h1(ωε)) ′ ) (7) such that, for any t > 0 and for any ϕ1, ϕ2 ∈ h1(ωε), (φε, c + ε , c − ε ) satisfy:∫ ωε ∇φε ·∇ϕ1 dx− ∫ γε ∇φε ·νϕ1 dσ = biomath 2 (2013), 1312302, http://dx.doi.org/10.11145/j.biomath.2013.12.302 page 3 of 5 http://dx.doi.org/10.11145/j.biomath.2013.12.302 c timofte, multiscale analysis of ionic transport in periodic charged media ∫ ωε (c+ε − c − ε + d) ϕ1 dx, (8) 〈∂tc±ε , ϕ2〉(h1)′ ,h1 + ∫ ωε (∇c±ε ± c ± ε ∇φε) ·∇ϕ2dx = ∫ ωε f±(c+ε ,c − ε )ϕ2 dx (9) and c±ε (0,x) = c ± 0 (x) in ωε. (10) there exists a unique weak solution (φε, c+ε , c − ε ) of problem (8)-(10) (see [8], [13] or [18]). moreover, exactly like in [13], we can prove that the concentration fields are non-negative, i.e. are bounded from below uniformly in ε. also, the concentration fields are bounded from above uniformly in ε. under the above hypotheses, by standard techniques, we can show that there exists a constant c ∈ r+, independent of ε, such that the following a priori estimates hold true: ‖φε‖l2((0,t )×ωε) + ‖∇φε‖l2((0,t )×ωε) ≤ c max 0≤t≤t ‖c−ε ‖l2(ωε) + max 0≤t≤t ‖c+ε ‖l2(ωε)+ ‖∇c−ε ‖l2((0,t )×ωε) + ‖∇c + ε ‖l2((0,t )×ωε)+ ‖∂tc−ε ‖l2(0,t ;(h1(ωε))′ ) + ‖∂tc + ε ‖l2(0,t ;(h1(ωε))′ ) ≤ c. as already mentioned, we are interested in obtaining the limit behavior, as ε → 0, of the solution (φε, c+ε , c−ε ) of problem (8)-(10). our approach is based on the periodic unfolding method introduced by d. cioranescu, a. damlamian, g. griso, p. donato and r. zaki (see [3] and [5]). this approach has the advantage that we do not need to use extension operators like in [8] or [13]. using the properties of the unfolding operator tε introduced in [3] and [5] and the above a priori estimates, we can easily prove that there exist φ ∈ l2(0,t; h1(ω)), φ̂ ∈ l2((0,t) × ω; h1per(yf )), c± ∈ l2(0,t; h1(ω)), ĉ± ∈ l2((0,t) × ω; h1per(yf )), such that, up to a subsequence, tε(φε) ⇀ φ weakly in l2((0,t) × ω; h1(yf )), tε(∇φε) ⇀ ∇φ +∇yφ̂ weakly in l2((0,t)×ω×yf ), tε(c±ε ) → c ± strongly in l2((0,t) × ω; h1(yf )), tε(∇c±ε ) ⇀ ∇c ±+∇yĉ± weakly in l2((0,t)×ω×yf ). for proving theorem 1, let us take, first, in the poisson equation (8), the test function ϕ1(t,x) = ψ0(t,x) + εψ1(t,x, x ε ), with ψ0 ∈d((0,t); c∞(ω)) and ψ1 ∈d((0,t) × ω; h1per(yf )). unfolding each term by using the operator tε and passing to the limit with ε → 0, we obtain (see, for details, [18]): t∫ 0 ∫ ω×yf (∇φ(t,x)+ ∇yφ̂(t,x,y)) (∇ψ0(t,x) + ∇yψ1(t,x,y)) dxdy dt+ σ0 t∫ 0 ∫ ω gψ0(t,x) dxdt = t∫ 0 ∫ ω×yf (c+(t,x) − c−(t,x)+ d(x))ψ0(t,x)dxdy dt. (11) then, by density, it follows that (11) holds true for any ψ0 ∈ l2(0,t; h1(ω)) and ψ1 ∈ l2((0,t) × ω; h1per(yf )). taking ψ0(t,x) = 0, we obtain  −∆y φ̂(t,x,y) = 0 in (0,t) × ω ×yf, ∇y φ̂ ·ν = −∇x φ(t,x) ·ν on (0,t) × ω × γ, φ̂(t,x,y) periodic in y. by linearity, we get φ̂(t, x, y) = n∑ j=1 χj(y) ∂φ ∂xj (t, x), (12) where χj, j = 1, n, are the solutions of the cell problems (6). taking ψ1(t,x,y) = 0, integrating with respect to x and using (12), we easily get the macroscopic problem for the electrostatic potential φ. now, taking in the nernst-planck equation (9) the test function ϕ2(t,x) = ψ0(t,x) + εψ1(t,x, x ε ), biomath 2 (2013), 1312302, http://dx.doi.org/10.11145/j.biomath.2013.12.302 page 4 of 5 http://dx.doi.org/10.11145/j.biomath.2013.12.302 c timofte, multiscale analysis of ionic transport in periodic charged media with ψ0 ∈d((0,t); c∞(ω)) and ψ1 ∈d((0,t) × ω; h1per(yf )), unfolding each term by using the operator tε and passing to the limit with ε → 0, we get (see [18]) − t∫ 0 ∫ ω×yf (c±(t,x)) ∂tψ0(t,x) dxdy dt+ t∫ 0 ∫ ω×yf (∇c±(t,x) + ∇yĉ±(t,x,y))(∇xψ0(t,x)+ ∇yψ1(t,x,y)) dxdy dt = t∫ 0 ∫ ω×yf f±0 (c +,c−)ψ0(t,x) dxdy dt. (13) by standard density arguments, we see that (13) holds true for any ψ0 ∈ l2(0,t; h1(ω)) and ψ1 ∈ l2((0,t)× ω; h1per(yf )). taking, first, ψ0(t,x) = 0, and, then, ψ1(t,x,y) = 0, we obtain exactly the macroscopic problem for the concentrations c±. since φ and c± are uniquely determined (see [13] and [18]), the whole sequences of microscopic solutions converge to a solution of the unfolded limit problem and this completes the proof of theorem 1. iv. conclusion using the periodic unfolding method, the macroscopic behavior of the solution of a system of equations describing the ion transport in periodic charged media is analyzed. our model is relevant for studying the ion transport through protein channels or the flow of electrons and holes in a semiconductor device. references [1] g. allaire, a. mikelić and a. piatnitski, ”homogenization of the linearized ionic transport equations in rigid periodic porous media”, journal of mathematical physics, vol. 51 (12), pp. 123103.1-123103.18, 2010. 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[19] w. r. van roosbroeck, ”theory of flow of electrons and holes in germanium and other semiconductors”, bell system technical j., vol. 29, pp. 560-607, 1950. http://dx.doi.org/10.1002/j.1538-7305.1950.tb03653.x biomath 2 (2013), 1312302, http://dx.doi.org/10.11145/j.biomath.2013.12.302 page 5 of 5 http://dx.doi.org/10.1137/080713148 http://dx.doi.org/10.1137/100817942 http://dx.doi.org/10.1007/s10958-011-0443-2 http://dx.doi.org/10.1007/s10659-013-9427-4 http://dx.doi.org/10.1021/cr60130a002 http://dx.doi.org/10.1016/j.jmaa.2012.01.052 http://dx.doi.org/10.4310/cms.2011.v9.n3.a3 http://dx.doi.org/10.1007/s11565-007-0018-9 http://dx.doi.org/10.1002/j.1538-7305.1950.tb03653.x http://dx.doi.org/10.11145/j.biomath.2013.12.302 introduction and setting of the problem the main result proof of the main result conclusion references www.biomathforum.org/biomath/index.php/biomath original article new properties of the attenuated v-line transform for breast cancer detection with compton cameras hanqiu tan1, rim gouia-zarrad2 1department of mathematics and applied mathematics, virginia commonwealth university po box 842014, richmond, virginia, usa tanh4@mymail.vcu.edu 2department of mathematics and statistics, american university of sharjah sharjah 26666, uae rgouia@aus.edu received: 21 july 2017, accepted: 14 november 2017, published: 11 december 2017 abstract—in the recent past, compton camera became an attractive alternative to the anger camera in scintimammography, known as nuclear medicine breast imaging or molecular breast imaging. this novel imaging modality leads to the use of the v-line transform, which integrates a function along coupled rays with a common vertex. in previous works the attenuation phenomena was mostly neglected. however, in scintimammography ignoring the effect of the attenuation of photon can significantly degrade the quality of the reconstruction image. in this paper, we introduce the attenuated v-line transform and establish a new integral relation between the attenuated v-line transform and the exponential radon transform. the results are not only interesting as original mathematical discoveries, but also can be useful in challenging applications e.g., in breast imaging for tumor detection close to the chest wall. keywords-v-line transform; attenuated v-line transform; imaging of breast cancer; breast cancer detection; compton camera. i. introduction according to the american cancer society, in the united states, breast cancer is one of the most commonly diagnosed cancer among women and it has the second highest mortality rate [1]. statistic shows that one in eight women will have breast cancer in her lifetime [1]. despite the high chance of breast cancer related death, early diagnosis and timely treatment can decrease the fatality rate [2]. breast cancer screening has made significant advancement in recent years. currently, x-ray mammography and ultrasonography are the two main techniques applied for breast cancer detection. according to lee et al., the widespread use of mammographic screening has led to nearly 30% decrease in breast cancer mortality since 1990 copyright: c© 2017 tan et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: hanqiu tan, rim gouia-zarrad, new properties of the attenuated v-line transform for breast cancer detection with compton cameras, biomath 6 (2017), 1711147, http://dx.doi.org/10.11145/j.biomath.2017.11.147 page 1 of 7 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2017.11.147 hanqiu tan, rim gouia-zarrad, new properties of the attenuated v-line transform for ... fig. 1. single-head compton camera. [3]. ultrasound is an important clinical adjunct technique to mammography by providing details of the lesions. however, these two techniques are not free from limitations. smith and andropoulou claim in [4] that despite its effectiveness for detection of breast cancer among women 50 and 69 years old, mammography suffers radiation risk and diminished sensitivity in dense breasts of younger women. several new techniques have been developed to improve the preciseness of breast cancer screening. one effective invention is the scintimammography. scintimammography, known as nuclear medicine breast imaging has the benefit of enabling breast cancer detection among young women by using radioactive materials and electronically collimated camera called compton camera. the compton camera utilizes the compton scattering effect to locate the radioactive source located inside the breast. a compton camera consists of two parallelpositioned detectors (see fig. 1). incident gamma rays are scattered in the scatter detector and subsequently detected by the absorb detector. in both interactions, the energies e1 and e2 and positions u1 and u2 are recorded. the angle φ can be found as follows (e.g. see [5], [6]) cosφ = 1− mc2e1 (e1 + e2)e2 , where m is the mass of the electron and c is the speed of light. the information data u1, u2 and φ can be used to locate the gamma source somewhere on the cone surface in 3d (see fig. 2) and the two semilines with common vertex in 2d (see fig. 3). another limitation of the conventional tomography imaging system is its low efficiency when close to the detector edge, which leads to failure in detecting tumors close to the chest wall (e.g. see [7]). new compton camera, such as c-smm system, overcomes this problem by placing the detector directly on the chest. it allows the higher efficiency and sensitivity towards the area close to the chest wall [7]. on the other hand, compton cameras have the flexibility to use a wide range of radio-pharmaceutical energies. in many detecting process, two compton cameras are used with one below and one above the breast. this system is called the dual-head compton camera system. in [8], hruska et al mentioned that the dual-head compton camera system simultaneously acquires the superior and inferior views, thus it provides the views of the breast in both the craniocaudal position and mediolateral oblique position. theoretically speaking, dual-head compton camera is more sensitive in detecting breast tumors than the singlehead compton camera, as it gives two separate images [9]. however, the preliminary results from [8] explained that due to the symmetry position, the results from the two cameras are very similar. several works, e.g. [7], [8], [9], [10], concentrated on the single-head compton camera to detect breast cancer (see fig. 1). in all these works the attenuation phenomena was neglected. however, in scintimammography ignoring the effect of the attenuation of photon can significantly degrade the quality of the reconstruction image. this problem has been recently studied by [11]. the authors studied the attenuated v-line transform in 2 dimensions using circular harmonic expansions and derived an analytic inversion approach in the case of vertices on a circle. in this paper we introduce the attenuated v-line transform and present some of its new properties on a class of v-lines with vertical central axis and vertex on the x-axis. we start by detailing the problem in section 2 before providing the definitions and recalling the important results about the v-line radon transform in section 3. at last, section 4 introduces the concept of the attenuated biomath 6 (2017), 1711147, http://dx.doi.org/10.11145/j.biomath.2017.11.147 page 2 of 7 http://dx.doi.org/10.11145/j.biomath.2017.11.147 hanqiu tan, rim gouia-zarrad, new properties of the attenuated v-line transform for ... fig. 2. the compton camera. v-line transform in r2 and present an integral relations between the attenuated v-line transform and the exponential radon transform. ii. formulation of the problem in 2d scintimammography, f represents the distribution of radiotracer concentration inside the breast. before the γ rays emitted from the radioactive source arrive at the detector, they are attenuated by an attenuation coefficient µ which is a real function on r2. in the case of uniformly attenuating medium µ can be approximated as a constant in the domain of the function f. therefore the data may be modeled by a set of exponential weighted v-line integrals of f over two semilines l+ and l− with common vertex (u,0) , vertical central axis and a half opening angle φ, called attenuated v-line projections v [f]µ(u,φ) (see fig. 3). after making all the measurements for all possible φ and all vertexes (u,0), one obtains a two-dimensional family of v [f]µ(u,φ) data. the problem of image reconstruction in 2d scintimammography requires the inversion of v [f]µ, i.e. finding f from the measured data v [f]µ. our goal is to present some new properties of the attenuated v-line transform v [f]µ on a class of v-lines with vertical central axis and vertex on the x-axis. iii. notations and preliminaries let f, be compactly supported function in the half space r × (0,∞) ∈ s(r2) and let θ = (cosφ,sinφ)ᵀ ∈ s1 := {v ∈ r2, |v| = 1}. v (u,φ) simply consists of two half-lines l+ and l− with common vertex (u,0). we denote by φ the half opening angle of the v-lines and µ ∈ r the attenuation constant. definition 3.1: the attenuated v-line transform of f at point (u,φ) ∈ r× (0, π 2 ) is defined by v [f]µ(u,φ) = ∫ v (u,φ) f(v)eµl(u,v)dl, where l(u, v) = |(u,0) − (x,y)| is the distance from the vertex (u,0) to the point with coordinate (x,y), dl is the length element on the v-line v (u,φ). fig. 3. geometrical setup of the v-line v (u,φ). the v-line transform v [f](u,φ) ∈ r × (0, π 2 ) is thus given by v [f]µ=0 (e.g. see [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]). definition 3.2: the exponential radon transform tµf on the unit cylinder z = s1 × r is defined by tµf(θ,s) = ∫ r eµtf(sθ + tθ⊥)dt where θ⊥ = (−sinφ,cosφ) ∈ s1. the classical radon transform rf(θ,s) ∈ z = s1 ×r is thus given by tµ=0. the fourier transform generated by a function f(x,y) with respect to the first argument is denoted by f̂(λ,y) = ∫ r f(x,y)e−iλxdx where (λ,y) ∈ r×r. biomath 6 (2017), 1711147, http://dx.doi.org/10.11145/j.biomath.2017.11.147 page 3 of 7 http://dx.doi.org/10.11145/j.biomath.2017.11.147 hanqiu tan, rim gouia-zarrad, new properties of the attenuated v-line transform for ... iv. the v-line transform and its inversions the inversion problem of the v-line transform has been studied by many authors. in the following we only recall some of the important results useful for breast cancer detection with compton cameras. theorem 4.1: (projection-slice theorem for the v-line radon transform) consider a function f ∈ c∞(r2) compactly supported in the half space r× (0,∞). then we have cosφv̂ [f](λ,φ) = 2f(c)[f̂](λ,λtanφ), where f(c) is the fourier-cosine transform with respect to the second argument, f̂ is the fourier transform of f with respect to the first argument and v̂ [f](λ,φ) is the fourier transform of v [f] with respect to the first argument. theorem 4.2: consider a function f ∈ c∞(r2) compactly supported in the half space r×(0,∞) ∈s(r2). we can write f̂(λ,y) = |λ| π ∫ ∞ 0 costy √ t2 + 1 v̂ [f](λ,t) dt, where f̂ and v̂ [f](λ,φ) are the fourier transforms of f and v [f] respectively in the first argument and t = tanφ. proof: see [22], [23]. an alternative inversion formula is based on the results of the classical radon transform. theorem 4.3: consider a function f ∈ c∞(r2) compactly supported in the half space r×(0,∞) ∈s(r2). then we have v [f](u,φ) = rfs(θ,ucosφ) where fs is an even extension of f obtained by symmetry with respect to the x-axis and θ = (cosφ,sinφ)ᵀ ∈ s1. proof: we can write the v-line transform v [f](u,φ) = ∫ ∞ 0 f(u + tsinφ,tcosφ)dt + ∫ ∞ 0 f(u− tsinφ,tcosφ)dt. we use the substitution y = tcosφ to obtain v [f](u,φ) = ∫ ∞ 0 f(u + y tanφ,y) dy cosφ + ∫ ∞ 0 f(u−y tanφ,y) dy cosφ . we change y′ = −y in the first integral v [f](u,φ) = ∫ 0 −∞ f(u−y tanφ,−y) dy cosφ + ∫ ∞ 0 f(u−y tanφ,y) dy cosφ . using an approach similar to [24], we consider an even extension obtained by symmetry with respect to the x-axis denoted by fs: fs(x,y) = { f(x,y) 0 ≤ y f(x,−y) 0 > y or we can write it as fs(x,y) = f(x, |y|). now we can combine the two integrals v [f](u,φ) = 1 cosφ ∫ ∞ −∞ fs(u−y tanφ,y)dy. we recognize the integral as the integral along the line perpendicular to (cosφ,sinφ)ᵀ with signed distance ucosφ from the origin. in fact, we can explain the above conclusion using the definition of the classical radon transform, rfs(θ,s) = ∫ r fs(sθ + tθ ⊥)dt, where (θ,s) ∈ s1 ×r, we can write rfs ((cosφ,sinφ) ᵀ,ucosφ) =∫ r fs(u+cos 2 φ−tsinφ,ucosφsinφ+tcosφ)dt. with the change of variables y = ucosφsinφ + tcosφ, biomath 6 (2017), 1711147, http://dx.doi.org/10.11145/j.biomath.2017.11.147 page 4 of 7 http://dx.doi.org/10.11145/j.biomath.2017.11.147 hanqiu tan, rim gouia-zarrad, new properties of the attenuated v-line transform for ... we can write rfs ((cosφ,sinφ) ᵀ,ucosφ) = ∫ r fs(u−y tanφ,y) dy cosφ = 1 cosφ ∫ r fs(u−y tanφ,y)dy. corollary 4.4: an exact solution of the inversion problem for the v-line transform is given by the formula fs(v)= 1 2πi ∫ π 0 h(∂tv [f]) ( < v.θ > cosφ ,φ ) dφ, (1) where h is the hilbert transform defined by hg(t) = 1 2π ∫ r sgn(r)ĝ(r)eirtdr, ĝ(r) is the fourier transform of g(t) and sgn(r) is the sign function. proof: the filtered backprojection formula is used to invert the classical radon transform (see [25]). we can conclude that the knowledge of the vline transform can be transferred into the knowledge of the classical radon transform of fs the original function and its mirror with respect to xaxis. using the uniqueness inversion of the classical radon transform, fs can be uniquely recovered and consequently f (see [24] for numerical simulations). v. the attenuated v-line transform theorem 5.1: consider a function f ∈ c∞(r2) compactly supported in the half space r×(0,∞). then we have cosφv̂ [f]µ(λ,φ)= ∫ ∞ −∞ f̂(λ, |y|)e µ|y| cos φe−iλy tanφdy where f̂ is the fourier transform of f with respect to the first argument and v̂ [f]µ(λ,φ) the fourier transform of v [f]µ with respect to the first argument. proof: v (u,φ) simply consists of two halflines l+ and l− with common vertex (u,0), so we can write the attenuated v-line transform v [f]µ(u,φ) = ∫ l+ f(x,y)eµl((u,0),(x,y))dl + ∫ l− f(x,y)eµl((u,0),(x,y))dl, v [f]µ(u,φ) = ∫ ∞ 0 f(u + tsinφ,tcosφ)eµtdt + ∫ ∞ 0 f(u−tsinφ,tcosφ)eµtdt. we use the substitution y = tcosφ to obtain v [f]µ(u,φ) = ∫ ∞ 0 f(u + y tanφ,y)e µy cos φ dy cosφ + ∫ ∞ 0 f(u−y tanφ,y)e µy cos φ dy cosφ . we change y′ = −y in the first integral cosφv [f]µ(u,φ)= ∫ ∞ −∞ f(u−y tanφ, |y|)e µ|y| cos φ dy. this result is further simplified using the fourier transform with respect to the first argument to obtain cosφv̂ [f]µ(λ,φ)= ∫ ∞ −∞ f̂(λ, |y|)e µ|y| cos φe−iλy tanφdy. theorem 5.2: consider a function f ∈ c∞(r2) compactly supported in the half space r×(0,∞). then we have∫ π 0 v [f]µ(u,φ)dφ = eµ|u| |u| ∗f(u), where u = (u,0). proof:∫ π 0 v [f]µ(u,φ)dφ = ∫ π 0 ∫ ∞ 0 f(u+tsinφ,tcosφ)eµtdtdφ + ∫ π 0 ∫ ∞ 0 f(u−tsinφ,tcosφ)eµtdtdφ. biomath 6 (2017), 1711147, http://dx.doi.org/10.11145/j.biomath.2017.11.147 page 5 of 7 http://dx.doi.org/10.11145/j.biomath.2017.11.147 hanqiu tan, rim gouia-zarrad, new properties of the attenuated v-line transform for ... changing φ to −φ in the second integral and using 2π-periodicity of sine and cosine functions, we obtain∫ π 0 v [f]µ(u,φ)dφ = ∫ 2π 0 ∫ ∞ 0 f(u+tcosφ,tsinφ)eµtdtdφ. let y = (tcosφ,tsinφ), we get∫ π 0 v [f]µ(u,φ)dφ = ∫ r2 f(u + y)eµ|y| dy |y| . we can write it in the convolution form∫ π 0 v [f]µ(u,φ)dφ = eµ|u| |u| ∗f(u). this formula is the starting point for reconstruction method of ρ-filtered layergram type, see ([25], chapter v.6). the authors plan to address this problem in future work. using the dual operator t]µ as defined t]µg(u) = ∫ s1 eµu·x ⊥ g(x, u · x)dx, we derive a new integral relation between the attenuated v-line transform and the exponential radon transform. corollary 5.3: consider a function f ∈ c∞(r2) compactly supported in the half space r× (0,∞). then we have∫ π 0 (v [f]µ + v [f]−µ) (u,φ)dφ = t ] −µtµf(u). proof:∫ π 0 (v [f]µ + v [f]−µ) (u,φ)dφ = ( 2 cosh(µ|u|) |u| ) ∗f(u). ∫ π 0 (v [f]µ + v [f]−µ) (u,φ)dφ = t ] −µtµf(u). the last equality is due to ([25], chapter ii.6). acknowledgment a part of this paper was written during the first author’s one semester long visit to the american university of sharjah (aus). the work was supported by the american university of sharjah (aus) research grant frg2. references [1] c. desantis, j. ma, l. bryan, and a. jemal, “breast cancer statistics, 2013,” ca: a cancer journal for clinicians, vol. 64, no. 1, pp. 52–62, 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[25] f. natterer, the mathematics of computerized tomography. siam, 2001. biomath 6 (2017), 1711147, http://dx.doi.org/10.11145/j.biomath.2017.11.147 page 7 of 7 http://dx.doi.org/10.11145/j.biomath.2017.11.147 introduction formulation of the problem notations and preliminaries the v-line transform and its inversions the attenuated v-line transform references www.biomathforum.org/biomath/index.php/biomath original article some properties of the blumberg’s hyper-log-logistic curve roumen anguelov∗, nikolay kyurkchiev†, svetoslav markov‡ ∗ department of mathematics and applied mathematics, university of pretoria pretoria, south africa roumen.anguelov@up.ac.za † faculty of mathematics and informatics, university of plovdiv paisii hilendarski plovdiv, bulgaria nkyurk@uni-plovdiv.bg ‡ institute of mathematics and informatics, bulgarian academy of sciences sofia, bulgaria smarkov@bio.bas.bg received: 16 june 2018, accepted: 31 july 2018, published: 14 august 2018 abstract—the paper considers the sigmoid function defined through the hyper–log–logistic model introduced by blumberg. we study the hausdorff distance of this sigmoid to the heaviside function, which characterises the shape of switching from 0 to 1. estimates of the hausdorff distance in terms of the intrinsic growth rate are derived. we construct a family of recurrence generated sigmoidal functions based on the hyper–log–logistic function. numerical illustrations are provided. keywords-hyper–log–logistic model, heaviside function, hausdorff distance, upper and lower bounds mathematics subject classifications (2010) 41a46; 68n30 i. introduction the logistic function belongs to the important class of smooth sigmoidal functions arising from population and cell growth models. the logistic function was introduced by pierre françois verhulst [1]–[3], who applied it to human population dynamics. verhulst proposed his logistic equation to describe the mechanism of the self-limiting growth of a biological population. a number of models have been proposed to provide growth curve from 0 to 1 (or to some carrying capacity) of different shape, e.g. gompertz [4], pearl [5], von bertalanffy [6], richards [7], nelder [8], blumberg [9], turner and al. [10], schnute [11], tsoularis [12], tsoularis and wallace [13]. analysis of continuous growth models in terms copyright: c©2018 anguelov et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: roumen anguelov, nikolay kyurkchiev, svetoslav markov, some properties of the blumberg’s hyper-log-logistic curve, biomath 7 (2018), 1807317, http://dx.doi.org/10.11145/j.biomath.2018.07.317 page 1 of 8 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2018.07.317 r. anguelov, n. kyurkchiev, s. markov, some properties of the blumberg’s hyper-log-logistic curve of generalized logarithm and exponential functions can be found in [14]. a very good kinetic interpretation of log–logistic dose–time response curves is given in [15] (see also [16]). in artificial neural networks, [17], the sigmoid functions are used as activation or transfer function between two states, usually 0 and 1. in all of their application, the shape of the sigmoid functions is essential factor determining the properties of the underlying biological, chemical or artificial system. an important characteristic related to the shape of a sigmoid is how far it deviates from the heaviside function, also referred to as step-function, binary switch, or binary activation depending on the context. as shown in [18]-[19], an appropriate measure of this deviation is the hausdorff distance of the sigmoid to the interval heaviside function. some approximation and modelling aspects are discussed in [20]–[23]. in this paper we discuss the hausdorff distance of the hyper–log–logistic sigmoid curve to the interval heaviside function. ii. the blumberg hyper–log–logistic model in 1968 blumberg [9] introduced a modified verhulst logistic equation, the so called hyper–log– logistic equation: dn(t) dt = knα(1 −n)γ, (1) where k is the rate constant and α and γ are shape parameters. the equation (1) is consistent with the verhulst logistic model when α = γ = 1. we will consider the following modification of the hyper–log–logistic equation (1) (see for instance [12]: dn(t) dt = kn 1−1 β (1 −n)1+ 1 β (2) where β is a shape parameter. for β → ∞ the equation (2) reduces to the verhulst equation. the equation (2), in essence, provides parametric interpolation between the logistic equation (β →∞) and second order kinetics (β = 1). an explicit form of the solution is derived as follows. let the function n(t) be defined by the following nonlinear equation:( n 1 −n )1 β = 1 + kt β . (3) after differentiation of both sides of eq. (3), we have 1 β ( n 1 −n )1 β −1 n ′(1 −n) + nn ′ (1 −n)2 = k β . from here it follows that n ′ = kn 1−1 β (1 −n)1+ 1 β and, therefore, the function n(t) satisfies the hyper–log–logistic differential equation (2). the equation (3) can be rewritten as: n(t) = 1 − 1 1 + ( 1 + kt β )β . (4) further, we see that n(0) = 1 2 . (5) since equation (2) satisfies the conditions for local existence and uniqueness while n > 0, the function n(t) given in (4) is a unique solution of equation (2) satisfying the condition (5). the function is defined on [ β k , +∞ ) . the definition can be extended in a unique way on the rest of the t-axis as zero. iii. preliminaries as stated in the introduction, our main interest is the hausdorff distance from the hyper–log– logistic function in (4) to the interval heaviside function. we recall here the relevant definitions. definition 1. the interval heaviside function is defined as [24]: h(t) =   0, if t < 0, [0, 1], if t = 0, 1, if t > 0. (6) biomath 7 (2018), 1807317, http://dx.doi.org/10.11145/j.biomath.2018.07.317 page 2 of 8 http://dx.doi.org/10.11145/j.biomath.2018.07.317 r. anguelov, n. kyurkchiev, s. markov, some properties of the blumberg’s hyper-log-logistic curve definition 2. [28] the one–sided hausdorff distance −→ρ (f,g) between two interval functions f,g on ω ⊆ r, is the one–sided hausdorff distance between their completed graphs f(f) and f(g) considered as closed subsets of ω × r. more precisely, −→ρ (f,g) = sup b∈f(g) inf a∈f(f) ||a−b||, where || · || is a norm in r2. we recall that completed graph of an interval function f is the closure of the graph of f as a subset of ω×r. if the graph of an interval function f equals f(f), then the f is called s-continuous. the hausdorff distance ρ(f,g) = max{−→ρ (f,g),−→ρ (g,f)} defines a metric in the set of all s-continuous interval functions. the topological and algebric structure of the space of s-continuous functions and its subspaces is studied in [24]–[27]. in this paper we apply only the concept of the one–sided hausdorff distance. iv. main results our main interest is characterizing the shape of n as a switching curve from 0 to 1. to this end, we use as a characteristic the one–sided hausdorf distance from n to h as in [19]. the following theorem gives upper and lower bounds for −→ρ (n,h). theorem 3. the one–sided hausdorff distance −→ρ (n,h) from the function n given in (4) to the heaviside function h given in (6) satisfies the following inequalities for k > 0: dl := 1 2 + k < −→ρ (n,h) < 1 1 + √ 1 + k =: dr. (7) proof: first we consider the interval [0, +∞). taking into account the sigmoid shape of the function n(t) in (4), the one–sided hausdorff distance from n to the heaviside function h on the interval [0, +∞) is a root of the equation n(t) = 1 − t, or, equivalently, f(t) := ( 1 + kt β )β + 1 − 1 t = 0. (8) clearly, f is an increasing function of t ∈ [0, +∞). hence, if (8) has a root, then it is unique. we use the well-known inequalities 1 + α < ( 1 + α x )x < eα, (9) where α ∈ r, x > 1 and α + x > 0. using the first inequality in (9) we have f(t) > 1 + kt + 1 − 1 t = kt2 + 2t− 1 t the positive root of the quadratic in the numerator is −1 + √ 1 + k k = 1 1 + √ k + 1 = dr. then f(dr) > 0. (10) using the second inequality in (9) we have f(t) < ekt + 1 − 1 t . hence, f(dl) = f ( 1 k + 2 ) < e k k+2 + 1 −k − 2 < (k + 1) ( e 1− 2 k+2 k + 1 − 1 ) . for the derivative of ϕ(k) = e 1− 2 k+2 k + 1 − 1 we have ϕ′(k) =e 1− 2 k+2 2 (k + 2)2 1 k + 1 − 1 (k + 1)2 e 1− 2 k+2 =− k2 + 2 (k + 1)2(k + 2)2 e 1− 2 k+2 < 0. therefore ϕ is a decreasing function of k. using that k > 0 we have f(dl) < (k + 1)ϕ(k) < (k + 1)ϕ(0) = 0. (11) biomath 7 (2018), 1807317, http://dx.doi.org/10.11145/j.biomath.2018.07.317 page 3 of 8 http://dx.doi.org/10.11145/j.biomath.2018.07.317 r. anguelov, n. kyurkchiev, s. markov, some properties of the blumberg’s hyper-log-logistic curve fig. 1. the model (5) for β = 21, k = 20; h–distance = 0.109948, dl = 0.0454545, dr = 0.179129. since f is an increasing function, the inequalities (10) and (11) imply that (8) has a unique root in the interval (dl,dr). secondly, we consider the interval (−∞, 0]. similarly to the interval [0, +∞), using the shape of the sigmoid, the hausdorff distance from n to h is a root of the equation n(−θ) = θ, or, equivalently, g(θ) := ( 1 − kθ β )β + 1 − 1 1 −θ = 0. (12) clearly, g is a decreasing function of θ ∈ [0, min{β k , 1}]. hence, if (12) has a root, then it is unique. using the first inequality in (9) we have g(θ) > 2 −kθ − 1 1 −θ . then g(dl) = g ( 1 k + 2 ) > 2 − k k + 2 − k + 2 k + 1 = k (k + 1)(k + 2) > 0. (13) using the second inequality in (9) we have g(θ) < e−kθ + 1 − 1 1 −θ . then g(dr) = g ( 1 1 + √ 1 + k ) < e − k 1+ √ 1+k + 1 − 1 + √ 1 + k √ 1 + k = 1 √ 1 + k (√ 1 + ke1− √ 1+k − 1 ) (14) it is easy to see that the function φ(k) = √ 1 + ke1− √ 1+k − 1 is decreasing. indeed, φ′(k) = 1 2 √ 1+k e1− √ 1+k− √ 1 + k 1 2 √ 1+k e1− √ 1+k = 1 2 √ 1+k e1− √ 1+k(1− √ 1+k) < 0. hence, g(dr) in (14) is also e decreasing function of k. using that k > 0 we have g(dr) < 1 √ 1 (√ 1e1− √ 1 − 1 ) = 0. (15) since g is a decreasing function of θ, the inequalities (13) and (15) imply that (12) has a unique root in the interval (dl,dr). this completes the proof. the model (4) for β = 21, k = 20 is visualized on fig. 1. from the equations (8) and (12) as biomath 7 (2018), 1807317, http://dx.doi.org/10.11145/j.biomath.2018.07.317 page 4 of 8 http://dx.doi.org/10.11145/j.biomath.2018.07.317 r. anguelov, n. kyurkchiev, s. markov, some properties of the blumberg’s hyper-log-logistic curve fig. 2. the model (5) for β = 61, k = 41; h–distance = 0.0660383, dl = 0.0232558, dr = 0.133677. fig. 3. comparison between function v (t) (dashed) and n(t) (red) at fixed k = 20 and β = 21. well as the inequalities (7) we have: −→ρ (n,h) = 0.109948, dl = 0.0454545, dr = 0.179129. the model (4) for β = 61, k = 41 is visualized on fig. 2. the estimates (7) of the one–sided hausdorff distance of the blumberg sigmoidal function to the heaviside function, match those obtained for the vehulst sigmoidal function. this should not surprise us. we already mentioned that the equation (2) is consistent with the verhulst logistic model when β → +∞. as it is known, the verhulst logistic function is of the form v (t) = 1 1 + e−kt . a comparison between function v (t) and n(t) at fixed k = 20 and β = 21 is shown in fig. 3. the hausdorff distance from the verhulst function to the interval heaviside function by is studied in detail in [19], [24]. specifically, in the article [19], one may find more accurate estimates. the hyper–log–logistic function can be used to recurrently generate a family of sigmoidal function: ni+1(t) = 1− 1 1+ ( 1+ k β ( t−1 2 +ni(t) ))β , (16) i = 0, 1, 2, . . . , with ni+1(α) = 1 2 , i = 0, 1, 2, . . . , (17) where n0(t) = n(t) – the function given in biomath 7 (2018), 1807317, http://dx.doi.org/10.11145/j.biomath.2018.07.317 page 5 of 8 http://dx.doi.org/10.11145/j.biomath.2018.07.317 r. anguelov, n. kyurkchiev, s. markov, some properties of the blumberg’s hyper-log-logistic curve fig. 4. the recurrence generated sigmoidal hyper–log–logistic functions: n0(t) (red); d0 = 0.21821, n1(t) (green); d1 = 0.134208, n2(t) (dashed); d2 = 0.095564 and n3(t) (thick); d3 = 0.0749788. (4). we refer to this family shortly as recurrence generated sigmoidal hyper–log–logistic functions. the recurrence generated sigmoidal hyper–log– logistic functions: n0(t),n1(t),n2(t) and n3(t) for k = 4 and β = 21 are visualized on fig. 4. this type of family of functions can find application in the field of debugging and test theory [39]–[40]. further, the results can be of interest to specialists working in the field of constructive approximation by superposition of sigmoidal functions [29]–[38]. v. conclusions in the areas of population dynamics, chemical kinetics or neural networks it is important to study the shape of the involved sigmoidal curve, since it relates to the fundamental properties of the respective system. in order to study the shape usually the curve is divided into lag phase, growth phase and saturation phase, [41]. these are defined in different ways in the literature, but in essence in the lag phase and in the saturation phase there is little or no growth, while most of the growth occurs in the growth phase. hence the latter one is also called exponential phase. in [19] the hausdorff distance to the interval heaviside function is considered as a rigorously defined characteristic of the shape. one may consider that the points, where the value of the one–sided hausdorff distance is attained, are precisely the points dividing the curve into the three mentioned segments. then, the timelength of the growth phase is exactly twice the value of this distance. in this paper we study the properties of the hyper-log-logistic curve produced by the blumberg model through the one–sided hausdorff distance of this curve to the interval heaviside function. lower and upper estimates of this distance are derived in terms of the intrinsic growth parameter and some possible applications are discussed. acknowledgments ra has been supported by the nrf/dst sarchi chair on mathematical models and methods in bioengineering and bioscience. nk has been supported by the project fp17-fmi008 of department for scientific research, paisii hilendarski university of plovdiv. references [1] p.-f. verhulst, notice sur la loi que la population poursuit dans son accroissement, correspondance mathematique et physique 10 (1838) 113–121. biomath 7 (2018), 1807317, http://dx.doi.org/10.11145/j.biomath.2018.07.317 page 6 of 8 http://dx.doi.org/10.11145/j.biomath.2018.07.317 r. anguelov, n. kyurkchiev, s. markov, some properties of the blumberg’s hyper-log-logistic curve [2] p.-f. verhulst, recherches mathematiques sur la loi d’accroissement de la population (mathematical researches into the law of population growth increase), nouveaux memoires de l’academie royale des sciences et belles-lettres de bruxelles 18 (1845) 1–42. 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covid-19 roumen anguelov, jacek banasiak, chelsea bright, jean lubuma, rachid ouifki department of mathematics and applied mathematics university of pretoria received: 30 april 2020, accepted: 10 may 2020, published: 11 may 2020 abstract—the paper draws attention to the asymptomatic and mildly symptomatic cases of covid-19, which, according to some reports, may constitute a large fraction of the infected individuals. these cases are often unreported and are not captured in the total number of confirmed cases communicated daily. on the one hand, this group may play a significant role in the spread of the infection, as asymptomatic cases are seldom detected and quarantined. on the other hand, it may play a significant role in disease extinction by contributing to the development of sufficient herd immunity. i. introduction in the current covid-19 pandemic we have a rather overwhelming situation of an enormous amount of data produced worldwide everyday and no reliable way of making sense of it or making any motivated predictions. to a large extent, this situation is created by the fact that data are passively collected, mostly in terms of number of cases, severity, recovered and deaths, as recorded in clinics and hospitals. as this information is publicly available, it is distributed worldwide on various official and private social media platforms, possibly making little contribution to the understanding of the disease and its epidemiological characteristics, and preventing any reliable predictions of what lies ahead. here we would like to draw attention to the role of asymptomatic or mildly symptomatic infections on the epidemic dynamics. for covid19 the severe symptomatic percentage of infections requiring hospitalization increases with age from 0.1% for those under 5 years old to 27.3% for the over 80 age group. ceteris paribus, the number of infectious needing hospitalization depends on the age structure of the population. south africa’s population is relatively young, e.g. 82.9% of the population is under 50. using the detailed age structure given in [8, page 10] and the percentages of severe symptomatic cases given in [3, table 1], the average percentage of covid-19 infectious people needing hospitalization is 4.02%. this could mean, when the infection follows locally/internally determined dynamics, i.e. it is no longer fueled by imported cases, a relatively small copyright: c© 2020 anguelov at al.. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: roumen anguelov, jacek banasiak, chelsea bright, jean lubuma, rachid ouifki, the big unknown: the asymptomatic spread of covid-19, biomath 9 (2020), 2005103, http://dx.doi.org/10.11145/j.biomath.2020.05.103 page 1 of 9 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2020.05.103 r. anguelov, j. banasiak, c. bright, j. lubuma, r. ouifki, the big unknown: the asymptomatic spread ... fraction of the infectious will be hospital cases and counted as such. possibly, a larger number will be seen at clinics and outpatient rooms, but it is quite likely that a very large fraction of infections in the population will remain undetected. in [3], it is noted that the data from china and repatriating flights suggest that 40% 50% of infections were not detected as ”cases”. this means a large fraction of the population will experience the infection asymptomatically or with mild symptoms and will not seek medical attention. hence, the counting of confirmed cases, if not interpreted appropriately, may present quite a distorted epidemiological assessment. ii. on the relative size of asymptomatic covid-19 spread it was suggested in [9] that, via testing of random samples of the population, one can determine the prevalence of the infection and possibly of immunity of the general population not treated by the health system. such study was carried out in the municipality of gangelt in germany, where a random sample of the population was tested for the virus and for antibodies. in [11], an intermediate result is given from the study. approximately 500 people from a total sample of 1000 were tested. an existing immunity of approximately 14 percent was recorded and 2 individuals were found infected at the time of testing. a total of approximately 15 percent were recorded to have been infected. a mortality rate of 0.37 percent was calculated from the total infections in gangelt. this contrasts the mortality rate of 1.98 percent in germany calculated by johns hopkins university, 5 times higher than the mortality rate in gangelt. the mortality rate based on the total population in gangelt is currently 0.06 percent. the lower mortality rate in gangelt is indicative of the fact that the study in gangelt considers all infected people in the sample, including those with asymptomatic and mild symptoms. hence, the study suggests a lower lethality of the virus than previously thought. that is, testing only/mostly those with medium to severe symptoms may give a distorted view on the spread of the disease and the mortality rate. the authors of [11] also state that it is possible to achieve herd immunity as the virus does not lie dormant in the body after recovery, immunity is estimated to last 6-18 months, and the epidemic dies out when 60-70 percent are immune. it is also suggested that lower initial viral load may result in less severe symptoms and, at the same time, development of immunity. a study in iceland, reported in [4], [6], suggests that almost all infections are either asymptomatic or mildly symptomatic, with about 50 percent of positive cases asymptomatic. this is significant, since, as of 11 april 2020, 10 percent of iceland’s population has been tested, the highest percentage in comparison to any other country. current data show that, as of 10 april 2020, the fatality rate in iceland is 0.41 percent [5], close to that of gangelt. in addition, since mid-march, the frequency of the virus among those without co-morbidities or symptoms is either decreasing or stable, suggesting increased prevalence of immunity. publications highlighting the significant role of asymptomatic cases in the covid-19 epidemiology include a recent study, published in the british medical journal [2], stating that 130 of 166 new infections (78 percent) in china, identified on 1 april 2020, were asymptomatic. this is further supported by a study in [1], where it is stated that blanket testing in a isolated village in italy, with a population of approximately 3000, recorded a drop, within 10 days, of 90 percent of people with symptoms due to isolating those who were symptomatic and asymptomatic. iii. mathematical model and associated observable variables the main focus of health authorities, government and media is the current burden of the disease represented as the number of so called ”confirmed cases”, these typically being symptomatic patients seeking medical assistance. this is rightfully so, as the present need of medical assistance as well as reducing immediate future demand are issues requiring urgent attention and action. however, as the review in the previous section suggests, biomath 9 (2020), 2005103, http://dx.doi.org/10.11145/j.biomath.2020.05.103 page 2 of 9 http://dx.doi.org/10.11145/j.biomath.2020.05.103 r. anguelov, j. banasiak, c. bright, j. lubuma, r. ouifki, the big unknown: the asymptomatic spread ... fig. 1. compartmental flow chart for covid-19 the asymptomatic cases and unreported cases involving mild symptoms constitute a significant, if not dominant part, of infections. considering the relatively younger population of south africa, the asymptomatic cases are likely to be a larger fraction of all infections than in europe or usa. hence, these are likely to have a strong and decisive impact on the long term epidemiological dynamics of covid-19 and its eventual results in terms of the loss of life and economic burden. in general, the impact of this, yet invisible, covid-19 epidemiological component can be expected to result in various changes in the confirmed cases, which might be difficult to explain via the confirmed cases count only. for example, quarantine measures based only on symptomatic cases are not likely to produce the expected results. further, asymptomatic cases may contribute strongly to building herd immunity, leading to unexpected (but desired) decline in cases. the proposed model is of the well known seir type, the main difference being that the infectious are structured according to the severity of symptoms. this model is also used in [10], but with additional compartments related to intervention. here our focus is on the epidemiological dynamics of the infection. the flow chart is given in figure 1. the susceptibles (s), due to infections with the force determined by the standard incidence with coefficient βc, move to the compartment of the exposed (e). as an exposed individual becomes infectious (waiting time 1 σ ), he/she moves either to compartment a (asymptomatic or mild symptoms), or to the compartment i of those with medium to severe symptoms who are likely to seek medical attention. some of the latter (transfer rate δi) will require hospitalization (compartment h). from compartments i and h the two other exits are to the recovered compartment r (rates γi and γh, resp.)) or death (rates αi, αh, resp.). the flow chart is implemented as the system of equations (1)–(7). ds dt = −βc(i + a)s −λ(t)s, (1) de dt = βc(i + a)s + λ(t)s −σe, (2) da dt = (1−ρ)σe −γaa, (3) di dt = ρσe − (δi + αi + γi)i, (4) biomath 9 (2020), 2005103, http://dx.doi.org/10.11145/j.biomath.2020.05.103 page 3 of 9 http://dx.doi.org/10.11145/j.biomath.2020.05.103 r. anguelov, j. banasiak, c. bright, j. lubuma, r. ouifki, the big unknown: the asymptomatic spread ... dh dt = δii − (αh + γh)h, (5) dra dt = γaa, drih dt = γii + γhh, (6) dd dt = αii + αhh. (7) the compartments represent fractions/percentage of the population so that the force of infection given in the right hand side of equation (1) represents standard incidence, where β represents the probability of infection at contact and c is the number of contacts per person. the product βc sometimes is referred to as the number of sufficient contacts per person, where sufficient contact means contact in which transmission of the infection occurs. the recent interventions of government, like lockdowns and sanitary measures, aim to reduce precisely this parameter. hence, in principle, it may be used for testing such interventions. the parameter λ is a function of time used to account for infections brought in by people who travelled abroad. it is mostly relevant at the beginning of the infection, before the borders are strictly controlled. we take λ to be a smooth and monotone decreasing function of the time t, such that λ(t) = { 2.18×10−6 for t ∈ [0,8] 0 for t ≥ 9 the parameter ρ, which reflects that ratio of split of the output of e between compartments a and i, is the main interest of this paper. the three equations (6)–(7) can be decoupled from the system and, indeed, it is useful to do so in the theoretical analysis or for computing the solution. however, the variables ra, rih and d are convenient for representing the total number of cases in each category. more specifically, the total number of symptomatic cases, past and present, is given by i + h + rih + d. similarly the total number of all cases, past and present, is a + i + h + r + d, where r = ra + rih. typical graphs of the outcome from the system (1)–(7) are presented in figure 2. the values of most parameters are given in table 1. the parameter of most critical importance is the coefficient βc. many of the interventions, like table i parameter values γa γi γh δi αi αh σ ρ 1 14 0.9 14 0.1 0.09 14 0.005 14 0.2 7 1 0.4 improved hygiene, social distancing and quarantining, can be modelled via their impact on the value of βc. for the simulation in figure 2 we use βc = 0.17. then, the basic reproduction ratio is r0 = βc ( ρ δi + αi + γi + 1−ρ γa ) = 2.3848, and satisfies 2 < r0 < 3 as estimated by many health agencies so far [7], [12]. as common in mathematical modelling, not all variables of the model are observable. in fact, typically only a function (observation operator) of the model variables is observable. in this model, the observable variables are daily recruitment (new cases) into compartment i as well as the current values of i, h, rih and d. the data on the daily recruitment rate into i tend to oscillate significantly due to many random factors, e.g. due to various reasons a case can be counted a day early or a day late. hence, it is more appropriate to consider the cumulative distribution of the recruitment rate ρσe into the i compartment. it is easy to see that∫ t 0 ρσe(θ)dθ = i(t) + h(t) + rih(t) + d(t), that is, this cumulative distribution is exactly the total number of symptomatic cases, past and present. we consider this number to be approximately represented by the total confirmed cases as reported daily. hence, we consider this sum as the observable variable and will adjust the model to fit the data available for it in the next section. this variable is represented by the blue line on figure 2. similarly, the total number of infections, symptomatic and asymptomatic, that is a(t)+i(t)+h(t)+r(t)+d(t), is the cumulative distribution of the total infective recruitment σe. it is represented on figure 2 by the magenta line. clearly the total number of infections is well above biomath 9 (2020), 2005103, http://dx.doi.org/10.11145/j.biomath.2020.05.103 page 4 of 9 http://dx.doi.org/10.11145/j.biomath.2020.05.103 r. anguelov, j. banasiak, c. bright, j. lubuma, r. ouifki, the big unknown: the asymptomatic spread ... fig. 2. typical progression of the infection, without interventions. the observed cases. the distance between the two is determined by the parameter ρ. in order to illustrate the significance of the distance between the two curves, we note that when the graph of active infective cases a + i + h (the red line) picks up, the total confirmed cases are at point b, that is at about 22% of the population. if these were all the cases, then one should deduce that we are at point c on the curve of total cases. in actual fact, the total cases at that time would be at point d, at about 55% of the population. the immune population r (the black curve) is at that time about 36% of the population and is the factor stopping the further increase of the active cases. indeed, it is easy to calculate that at that time the susceptibles are at about 42%, so that effective reproduction ratio at that time is r0 ×s = 2.3848×0.42 ≈ 1. figure 2 is not intended to provide accurate prediction. many of the parameters are not precisely known, most notably the parameter ρ, and it does not reflect any interventions currently taking place. it is intended as a qualitative illustration of the role of the asymptomatic compartment in any predictions and on directing some research effort in estimating its size. one way is to try to establish the already existing immunity in the population by testing for antibodies. the research work in [11] has precisely this goal. similar testing has been initiated in other countries as well. due to the many specific factors, this data is likely to be country specific. hence, tests need to be carried out in south africa as well. iv. modelling the covid-19 spread in south africa the first case of covid-19 in south africa was confirmed on 5 march 2020. until 20 march or so, the confirmed cases were dominated by individuals who travelled abroad. in view of the high prevalence of asymptomatic cases, it is highly likely that many more asymptomatic infective travellers entered south africa. since our model is based on deterministic differential equations, it can provide accurate representation of the infection spread only when the numbers are large. we initiate the model on 19 march, when there were 150 confirmed cases. we assume that at that time there were also 300 asymptomatic cases outside the health system records. further, it is also assumed that there was influx of infective south africans returning from abroad (as represented by λ(t)) until the lockdown came into effect. the influx from abroad is essential to explain the fast growth rate of confirmed cases until the lockdown. if this growth was a biomath 9 (2020), 2005103, http://dx.doi.org/10.11145/j.biomath.2020.05.103 page 5 of 9 http://dx.doi.org/10.11145/j.biomath.2020.05.103 r. anguelov, j. banasiak, c. bright, j. lubuma, r. ouifki, the big unknown: the asymptomatic spread ... fig. 3. covid in south africa: lockdown fig. 4. covid in south africa: morum mutatio result only of local infection, this would imply a basic reproduction ratio far outside the indicated range. for the initial stage, with the given value of λ until the lockdown, we obtain a good fit with βc = 0.2 resulting in r0 = 2.8. a country-wide lockdown was implemented as from midnight of 26 march. this resulted in a significant drop in the increase of daily new confirmed cases. in fact, for the first two and half weeks or so, these remained almost constant. the data and the simulation are presented in figure 3. until about 13 april the model is fitted to the data with βc = 0.07 resulting in r0 ≈ 1, visible from the fact that the graph of confirmed cases (the blue line) is approximately linear. however, there is a visible exponential increase of confirmed cases as from 13 april, deviating significantly from the near straight line for the first two and a half biomath 9 (2020), 2005103, http://dx.doi.org/10.11145/j.biomath.2020.05.103 page 6 of 9 http://dx.doi.org/10.11145/j.biomath.2020.05.103 r. anguelov, j. banasiak, c. bright, j. lubuma, r. ouifki, the big unknown: the asymptomatic spread ... fig. 5. covid in south africa: alert level 4 weeks. there could be reasons of different nature. it was widely reported in the uk media that, as the time under lockdown increases, the amount of pedestrian and motor traffic is also increasing. this lower level of compliance with the regulations is referred to as a lockdown fatigue. in south africa, the regulations of the lockdown were amended a few times, which might also be a contributing factor. another contributing factor could be black market activities related to the prohibition of sales of tobacco and alcohol. addiction to either of the two could be a strong driving factor of illegal social interactions. while the reasons are not clear, the fact of the switch from approximately linear growth to exponential growth in the observable variable i+h+rih+d about two and half weeks or so into the lockdown can be clearly seen in the available data. as we do not know the precise reason, we will refer to the point of change of the lockdown efficiency as morum mutatio (behaviour change). on figure 4 we present simulations, where the model is fitted to the period after 13 april with βc = 0.137 and associated basic reproduction number r0 = 1.92. we note that in this setting, due to asymptomatic cases, by the end of april 2020 the total number of cases could be over 14000 with 7500 or so who have already acquired immunity a factor which will begin to play a significant role in the further dynamics. the lockdown, also called alert level 5, is changed as from 1 may to alert level 4, which allows for some economic activity. it is expected that the basic reproduction ratio will not significantly increase as this would require a new lockdown. figure 5 indicates that if the infection progresses with the same basic reproduction ratio, in 3 weeks the total cases are expected to cover nearly 0.1% of the population, producing immunity which surpasses the number of recorded/confirmed cases. quantitative predictions beyond the mentioned period are not likely to be accurate due to the many unknowns. on the one hand, the parameters, most notably ρ, are yet to be precisely determined. on the other hand, one cannot predict what further actions health authorities or governments may take, or how human behaviour and interactions may change. nevertheless, in order to illustrate the significance of ρ, we run long term simulations for ρ = 0.4 (figure 6), as in the simulations so far, and with ρ = 0.2 (figure 7). in addition to the other graphs, in these figures we present the biomath 9 (2020), 2005103, http://dx.doi.org/10.11145/j.biomath.2020.05.103 page 7 of 9 http://dx.doi.org/10.11145/j.biomath.2020.05.103 r. anguelov, j. banasiak, c. bright, j. lubuma, r. ouifki, the big unknown: the asymptomatic spread ... fig. 6. long term dynamics with ρ = 0.4 fig. 7. long term dynamics with ρ = 0.2 graph of the active symptomatic infections (dashed red line). one can observe that while the graphs of the other presented variables are more or less the same, the graphs of the active symptomatic infections (i +h) and the total symptomatic cases (i + h + rih + d) are quite different. the graph of active symptomatic cases peaks at 5.41% for ρ = 0.4 (figure 6), while for ρ = 0.2 this peak is at 2.71% (figure 7). further, the saturation level of the total symptomatic cases (i +h +rih +d the blue line) is at about 30% for ρ = 0.4 and at about 16% for ρ = 0.2. the size of the symptomatic infections is a critical variable which needs to be below a certain level so that the health system can cope. we recall that a fraction of the symptomatic infective individuals would need hospitalization and a fraction of them will need critical care and possibly ventilabiomath 9 (2020), 2005103, http://dx.doi.org/10.11145/j.biomath.2020.05.103 page 8 of 9 http://dx.doi.org/10.11145/j.biomath.2020.05.103 r. anguelov, j. banasiak, c. bright, j. lubuma, r. ouifki, the big unknown: the asymptomatic spread ... tors. hence, knowing the ratio of symptomatic to asymptomatic infections is of crucial importance for determining the time and the size of the peak of active infective cases in any relevant setting and, therefore, inform an appropriate action. v. conclusion in this paper we suggest that further covid19 epidemiological research does not just need more data, but it also needs data which goes beyond the case count captured by the health system. the importance of testing for the virus cannot be doubted as it is an important tool of reducing the spread through quarantine measures, thus reducing βc or equivalently r0 and flattening the curve of active infective cases. however, the long term dynamics is strongly impacted by the level of immunity acquired by the population. it is built through both symptomatic and asymptomatic infections. since the latter ones, while likely to be the majority of the cases, are mostly unrecorded, they represent a significant unknown factor for the long term epidemiological dynamics. the only relevant testing that we are aware of, is the testing for antibodies, under the assumption that the presence of antibodies can provide immunity for 6-18 months, as suggested by some authors. we fitted the model to data for south africa on the total number of confirmed cases (the blue curve). part of our future research work is to focus on better estimation of the unknown parameter ρ. to this end, we will be following closely all research effort on testing for antibodies, as this would provide data which the curve of the recovered (the black curve) can be fitted to. references [1] michael day, covid-19: identifying and isolating asymptomatic people helped eliminate virus in italian village. bmj 2020;368:m1165. https://doi.org/10.1136/bmj. m1165 [2] michael day, covid-19: four fifths of cases are asymptomatic, china figures indicate. bmj 2020;369:m1375. https://doi.org/10.1136/bmj.m1375 [3] neil m ferguson, et al. (imperial college covid-19 responce team), impact of non-pharmaceutical interventions (npis) to reduce covid-19 mortality and healthcare demand. https://doi.org/10.25561/77482 [4] daniel f. gudbjartsson et al., early spread of sarscov-2 in the icelandic population, medrxiv. https://www. medrxiv.org/content/10.1101/2020.03.26.20044446v2 [5] iceland coronavirus tracker map. https://visalist.io/ emergency/coronavirus/iceland-country/iceland [6] john p. a. ioannidis, cathrine axfors, despina g. contopoulos-ioannidis, population-level covid-19 mortality risk for non-elderly individuals overall and for nonelderly individuals without underlying diseases in pandemic epicenters. medrxiv preprint. https://doi.org/ 10.1101/2020.04.05.2005436 [7] ying liu, albert a. gayle, annelies wilder-smith, joacim rocklöv, the reproductive number of covid19 is higher compared to sars coronavirus, journal of travel medicine 27(2) (2020). https://doi.org/10.1093/ jtm/taaa021 [8] government department: statistics of south africa, midyear population estimates, statistical release p0302, 2019. www.satassa.gov.za [9] josé lourenço, robert paton, mahan ghafari, moritz kraemer, craig thompson, peter simmonds, paul klenerman, sunetra gupta, fundamental principles of epidemic spread highlight the immediate need for large-scale serological surveys to assess the stage of the sars-cov2 epidemic, preprint. https://doi.org/10.1101/2020.03.24. 20042291 [10] biao tang, et al., estimation of the transmission risk of the 2019-ncov and its implication for public health interventions, journal of clinical medicine 9 (2020), 462. https://doi.org/10.3390/jcm9020462 [11] h. streeck et al. vorläufiges ergebnis und schlussfolgerungen der covid-19 case-clusterstudy (gemeinde gangelt) [12] world health organization, coronavirus disease 2019 (covid-19) situation report – 46. (2020) https://www.who.int/docs/default-source/coronaviruse/ situation-reports/20200306-sitrep-46-covid-19.pdf? sfvrsn=96b04adf 2 biomath 9 (2020), 2005103, http://dx.doi.org/10.11145/j.biomath.2020.05.103 page 9 of 9 https://doi.org/10.1136/bmj.m1165 https://doi.org/10.1136/bmj.m1165 https://doi.org/10.1136/bmj.m1375 https://doi.org/10.25561/77482 https://www.medrxiv.org/content/10.1101/2020.03.26.20044446v2 https://www.medrxiv.org/content/10.1101/2020.03.26.20044446v2 https://visalist.io/emergency/coronavirus/iceland-country/iceland https://visalist.io/emergency/coronavirus/iceland-country/iceland https://doi.org/10.1101/2020.04.05.2005436 https://doi.org/10.1101/2020.04.05.2005436 https://doi.org/10.1093/jtm/taaa021 https://doi.org/10.1093/jtm/taaa021 www.satassa.gov.za https://doi.org/10.1101/2020.03.24.20042291 https://doi.org/10.1101/2020.03.24.20042291 https://doi.org/10.3390/jcm9020462 https://www.who.int/docs/default-source/coronaviruse/situation-reports/20200306-sitrep-46-covid-19.pdf?sfvrsn=96b04adf_2 https://www.who.int/docs/default-source/coronaviruse/situation-reports/20200306-sitrep-46-covid-19.pdf?sfvrsn=96b04adf_2 https://www.who.int/docs/default-source/coronaviruse/situation-reports/20200306-sitrep-46-covid-19.pdf?sfvrsn=96b04adf_2 http://dx.doi.org/10.11145/j.biomath.2020.05.103 introduction on the relative size of asymptomatic covid-19 spread mathematical model and associated observable variables modelling the covid-19 spread in south africa conclusion references original article biomath 1 (2012), 1209253, 1–7 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum on waves and distributions in population dynamics ∗ ∗ dedicated to the 80th anniversary of acad. prof. blagovest sendov — a man with memorable contribution to bulgarian biological mathematics nikolay k. vitanov† and zlatinka i. dimitrova ‡ † institute of mechanics-bas acad. g. bonchev st., bl. 4, 1113 sofia, bulgaria email: vitanov@imbm.bas.bg ‡“g. nadjakov” institute of solid state physics-bas blvd. tzarigradsko chaussee 72, 1784 sofia, bulgaria email: zdim@issp.bas.bg received: 12 july 2012, accepted: 25 september 2012, published: 16 october 2012 abstract—we discuss dynamics of spatially distributed interacting populations. the following two cases are discussed: (i.) migration waves for the case or negligible random fluctuations of the populations densities and (ii.) probability distributions of the spatially averaged populations densities for the case of significant random fluctuations of these densities. for each of the cases we obtain in general a system of differential equations that is treated analytically (when possible) or numerically. the obtained results are discussed from the point of view of population dynamics. keywords-interacting populations; nonlinear waves; method of simplest equation; density fluctuations; probability density functions for populations densities i. introduction in many cases the dynamics of interacting populations is studied on the basis of models consisting of equations that contain only time dependence of the population densities [1]-[4]. these models are very useful for understanding the complex dynamics of the interacting populations but they neglect two important aspects of this dynamics: (i.) the possible influence of spatial characteristics of the environment; and (ii.) the possible fluctuations of the population densities caused by different factors. below we shall investigate two kinds of population dynamics models that account for each of these effects. first of all we shall discuss the dynamics of spatially distributed populations and this will be a continuation of our previous work [5], [6]. then we shall show that by appropriate averaging the spatial model can be reduced to model in which the population densities depend only on the time. this model is valid not only for the cases of small values of the densities of the populations and because of this we shall discuss the influence of random fluctuations of the population densities for arbitrary values of the densities. the result of the action of these fluctuations is that instead of equations for the trajectories of the populations in the phase space of the population densities we shall write and solve equations for the probability density functions of the densities of the interacting populations. ii. model equations a. spatially distributed populations let us consider an two-dimensional area s where n competing populations are present. the density of each population is ρi(x,y,t) = ∆ni∆s , where ∆ni is the number of the individuals of the i-th population that are present in the small area ∆s at the moment t. now let a movement of population members through the borders of citation: n. vitanov, z. dimitrova, on waves and distributions in population dynamics, biomath 1 (2012), 1209253, http://dx.doi.org/10.11145/j.biomath.2012.09.253 page 1 of 7 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.09.253 n. vitanov et al., on waves and distributions in population dynamics the area ∆s be possible and let ~ji(x,y,t) be the current of this movement. then (~ji · ~n)δl is the net number of members from i-th population, crossing a small border line δl with normal vector n. let the density changes (other then these caused by the border crossings) be summarized by the function ci(ρ1,ρ2, . . . ,ρn,x,y,t). then the change of the density of members of the ith population in the studied area is described by the equation ∂ρi ∂t + div~ji = ci. (1) below we shall discuss the case where ~ji has the form of linear multicomponent diffusion. in this case ~ji = − n∑ k=1 dik(ρi,ρk,x,y,t)∇ρk, (2) where dik is the diffusion coefficient. eq. (2) accounts for the possibility that the spatial motion of the population members happens not only as consequence of gradient of density of the own population but also as consequence of gradients of the densities of the other populations. we shall not specify the kind of the function ci as we shall consider the general case of relative small populations densities that allow us to write ci as taylor series expansion around the zero values of all population densities as follows ci(ρ1,ρ2, . . . ,ρn ) = ∞∑ n1=0 ∞∑ n2=0 . . . ∞∑ nn =0 α(i)n1,n2,...,nn ρ n1 1 ρ n2 2 . . .ρ nn n , (3) where the constant coefficients α (i) n1,n2,...,nn are as follows α(i)n1,n2,...,nn = 1 n1!n2! . . .nn ! × ∂cn1+n2+...+nni ∂ρn11 ∂ρ n2 2 . . .∂ρ nn n |ρ1=ρ2=...=ρn =0 . (4) we shall discuss the one-dimensional case and in addition we shall assume that the diffusion coefficients dik are constants. then the substitution of eqs. (2) and (3) in (1) leads to the following system of nonlinear pdes for the studied n interacting populations: ∂ρi ∂t − n∑ k=1 dik ∂2ρk ∂x2 = ∞∑ n1=0 ∞∑ n2=0 . . . ∞∑ nn =0 α(i)n1,n2,...,nn ρ n1 1 ρ n2 2 . . .ρ nn n . (5) for the case of 1 population the system (5) is reduced to the equation ∂ρ ∂t − d ∂2ρ ∂x2 = ∞∑ n1=0 αn1ρ n1. (6) b. spatially averaged equations below we shall apply spatial averaging to the system of equations (5) similar to the averaging used in the optimum theory of turbulence [7] [8]. in the general two-dimensional case let a quantity q(x,y,t) be defined in an large two-dimensional plane area s with acreage | s |. then by definition the spatial average of q is q(t) = 1 | s | ∫ ∫ s dx dy q(x,y,t). (7) then q(x,y,t) can be separated in spatial averaged part q and the rest quantity q(x,y,t): q(x,y,t) = q(t) + q(x,y,t). (8) let | s | be large enough so that the plane average of any product of the rest quantities vanish: qi = qiqj = qiqjqk = . . . = 0. in addition we shall assume that∫ ∫ s dx dy∇ 2q has finite and small value such that ∇2q = 1|s| ∫ ∫ s dx dy∇ 2q → 0. the application of the averaging to eq. (1) in presence of the assumptions given by eqs. (2), (3) and (4) (note that in this case we have two spatial dimensions) leads to the system of odes as follows (i = 1, 2, . . . ,n): dρi dt = ∞∑ n1=0 ∞∑ n2=0 . . . ∞∑ nn =0 α(i)n1,n2,...,nn ρ n1 1 ρ n2 2 . . .ρ nn n . (9) we note that eq. (9) follows directly from eq. (5) in the spatially homogeneous case. for the case of one population the equation becomes dρ dt = ∞∑ n1=0 αn1ρ n1. (10) we note that the equations of the kind of eqs.(9) and (10) are often used as model equations in population dynamics not only for small values of population densities but also for large values of these densities, i.e., for large ρi. iii. traveling waves let us discuss the simplest case of one population described by eq. (6). first we introduce the travelingwave coordinate ξ = x−vt where v is the velocity of the wave. in addition we shall assume that the polynomial biomath 1 (2012), 1209253, http://dx.doi.org/10.11145/j.biomath.2012.09.253 page 2 of 7 http://dx.doi.org/10.11145/j.biomath.2012.09.253 n. vitanov et al., on waves and distributions in population dynamics non-linearity in eq.(6) is up to order l. we rescale the coefficients in eq.(6) as follows: d† = −d/v; α†n1 = αn1/v. (11) then eq.(6) becomes: dρ dξ + d† d2ρ dξ2 + l∑ n1=0 α†n1ρ n1 = 0. (12) below we shall obtain exact solution of eq. (12) by application of the modified method of simplest equation for obtaining exact solutions of nonlinear pdes. a. application of the modified method of simplest equation to eq. (12) the method of simplest equation [9]-[11] is based on the fact that after application of appropriate ansatz some npdes can be reduced to odes of the kind p ( r(ξ), dr dξ , d2r dξ2 , . . . ) = 0, (13) and for some equations of the kind (13) particular solutions can be obtained which are finite series r(ξ) = p∑ i=1 ai[φ(ξ)] i, (14) constructed by solution φ(ξ) of more simple equation referred to as simplest equation. the simplest equation can be the equation of bernoulli, equation of riccati, etc. the substitution of eq. (14) in eq. (13) leads to the polynomial equation p = σ0 + σ1φ + σ2φ2 + . . . + σrφr = 0, (15) where the coefficients σi, i = 0, 1, . . . ,r depend on the parameters of the equation and on the parameters of the solutions. equating all these coefficients to 0, i.e., by setting σi = 0, i = 1, 2, . . . ,r, (16) one obtains a system of nonlinear algebraic equations. each solution of this system leads to a solution of kind (14) of eq. (13). in order to ensure non-trivial solution by the above method we have to ensure that σr contains at least two terms. to do this we have to balance the highest powers of φ that are obtained from the different terms of the solved equation of kind (13). as a result of this we obtain an additional equation between some of the parameters of the equation and the solution. this equation is called balance equation [12]-[15]. let us now apply the methodology to eq.(12). let us search a solution as finite series ρ(ξ) = p∑ i=0 ai[φ(ξ)] i, (17) where φ(ξ) is a solution of the riccati equation dφ dξ = aφ2 + bφ + c, (18) i.e., φ(ξ) = − b 2a − θ 2a tanh [ θ(ξ + ξ0) 2 ] . (19) the substitution of eq.(17) in eq.(12) and the balance of the largest powers of φ that arise from the different terms of eq.(12) lead to the balance equation p(l − 1) = 2. (20) thus we have the possibilities: p = l = 2 or p = 1; l = 3. below we discuss the first possibility, namely p = l = 2. in such a way we shall obtain exact traveling-wave solution of the equation ∂ρ ∂t − d ∂2ρ ∂x2 = α0 + α1ρ + α2ρ 2, (21) which will be of the kind ρ(ξ) = a0 − a1 { b 2a + θ 2a tanh [ θ(ξ + ξ0) 2 ]} + a2 { b 2a + θ 2a tanh [ θ(ξ + ξ0) 2 ]}2 . (22) the parameters of the solutions are determined by the following system of kind (16) 6d†a2a 2 + α†2a 2 2 = 0, ad†(a1a + 5a2b) + a2a + α † 2a1a2 = 0, d†[3a1ab + 4a2(2ac + b 2)] + a1a+ 2a2b + α † 1a2 + α † 2(2a0a2 + a 2 1) = 0, d†[a1(2ac + b 2) + 6a2bc] + α † 1a1+ 2α†2a0a1 + a1b + 2a2c = 0, α † 0 + α † 1a0 + α † 2a 2 0 + a1c + d †(a1bc + 2a2c 2) = 0. (23) now we have 2 possibilities: α†0 6= 0 and α † 0 = 0 (which is more close to the classical population dynamics models that usually do not possess terms independent on the population density). biomath 1 (2012), 1209253, http://dx.doi.org/10.11145/j.biomath.2012.09.253 page 3 of 7 http://dx.doi.org/10.11145/j.biomath.2012.09.253 n. vitanov et al., on waves and distributions in population dynamics 1) case α†0 6= 0: for this case the solution of the system (23) is as follows: α † 0 = 625α†1 2 d† 2 − 36 2500α†2d †2 , a0 = − 75d† 2 b2 + 30d†b − 3 + 25α†1d † 50α†2d † , a1 = − 3[(25d† 2 b2 − 1)(5d†b + 1)] 250α†2cd † , a2 = − 3(25d† 2 b2 − 1) 5000α†2c 2d† 3 , a = 25d† 2 b2 − 1 100cd†2 . (24) 2) case α†0 = 0: for this case the solution of the system (23) is as follows: d† = 6 25α†1 , a0 = − 36b2 + 60bα†1 + 25α † 1 2 100α†1α † 2 , a1 = − (36b2 − 25α†1 2 )(6b + 5α†1) 600cα†1α † 2 , a2 = − (36b2 − 25α†1 2 )2 14400cα†1α † 2 , a = 36b2 − 25α†1 2 144c . (25) the obtained solutions describe kink waves that can be considered as traveling waves of change of the value of the population density of the studied population. appropriate values of the boundary conditions at ξ = ±∞ can ensure that ρ(ξ) is non-negative everywhere. we note that the parameters of the solved eq. (21) (after the rescallings (11)) are d† and α†0,1,2. the first relationship from (24) connects these 4 parameters. then eq.(22) is exact solution of eq. (21) only when the mentioned above 4 parameters are bounded by the corresponding relationship. this means that only 3 of these 4 parameters are free. iv. statistical distributions and exit time eqs. (9) and (10) are typical equations for description of dynamics of dynamical systems. the general case of such equations is dxi dt = xi(x1,x2, . . . ,xn ); i = 1, 2, . . . ,n. (26) for such kind of systems there exists a theory that allows us to characterize some system properties in the case when the system is under the action of random perturbations. pontryagin, andronov and vitt [16] developed such theory for random impulses that occur after every interval of time τ and each impulse causes the phase point of the dynamical system described by eqs. (26) to jump through a distance a along a random direction. let us first consider the case of single population and one spatial dimension. for the case when a tends to 0 together with τ in such a way that the ratio a/τ tends to finite limit b it is possible to obtain an equation for the probability density function f(x,t) as follows: ∂f ∂t + ∂ ∂x [ x(x)f ] = b 2 ∂2f ∂x2 . (27) for the general case of n populations the equation for the probability density function becomes ∂f ∂t + n∑ i=1 ∂ ∂xi [ xi(x1,x2, . . . ,xn )f ] = 1 2 n∑ i=1 n∑ j=1 bij ∂2f ∂xi∂xj , (28) where bij are again coefficients that characterize the random impulses. another kind of problem that can be solved by this approach is to calculate the mathematical expectation of the exit time. let us again first discuss the case of one population and one spatial dimension. we have a phase point that is inside the interval [�1,�2] (�1 < �2) and the system is under the influence of the same random perturbations as described above. the exit time is the time for which the phase point that was inside the above interval at t = 0 will leave this interval through �1 or through �2. if we denote as f(x) the mathematical expectation for the exit time then f(x) is a solution of the equation [16] b 2 d2f dx2 + x(x) df dx + 1 = 0, (29) with boundary conditions f(�1) = f(�2) = 0. for the case of many populations the zero boundary conditions are on the entire border of the multidimensional phase space area that has to be exited and the equation for the probability density function of the exit time is 1 2 n∑ i=1 n∑ j=1 bij ∂2f ∂xi∂xj + n∑ i=1 x(x1,x2, . . . ,xn ) ∂f ∂xi +1 = 0. (30) biomath 1 (2012), 1209253, http://dx.doi.org/10.11145/j.biomath.2012.09.253 page 4 of 7 http://dx.doi.org/10.11145/j.biomath.2012.09.253 n. vitanov et al., on waves and distributions in population dynamics -8 -6 -4 -2 0 2 4 6 8 ρ 0 0.05 0.1 0.15 0.2 0.25 f (a) -5 0 5 ρ 0 0.05 0.1 0.15 0.2 f (b) -5 0 5 ρ 0 0.5 1 f (c) fig. 1. several profiles of f (ρ) from eq.(33) (in the figures ρ is denoted as ρ). figure (a): b = 2; α0 = α2 = α3 = 0; α1 = −0.3. in this case the model equation has a single fixed point ρ = 0 and at this fixed point the maximum of the probability density function is centered. figure (b): α0 = α2 = 0; α1 = 0.3; α3 = −0.05; b = 3. here the fixed points are 3. the two maxima of the p.d.f. distribution are centered around the two stable fixed points and the minimum is centered on the unstable fixed point ρ = 0. figure (c): α0 = 0.05; α1 = 0.3; α2 = 0.01; α3 = −0.05; b = 3. here the fixed points are 3 again but one of the two stable fixed points is more preferred which can be seen from the larger peak of p.d.f. function at that point. let us now apply this theory to eqs. (9) and (10). for the case of single population let us be interested in the case when after a long time the probability density function f becomes stationary and depends only on the spatial coordinate x. in this case x(x) = ∑l n1=0 αn1ρ ni . 0 2 4 6 8 10 ρ 0 5 10 15 20 f q fig. 2. exit time expectations for q = 0 (which means extinction of the population) calculated on the basis of eq.(36). ρ on the horizontal axis is equal to ρ from eq.(36). for all curves b = 2. solid line: α0 = 0.01; α1 = −0.1; α2 = α3 = 0. dashed line: α0 = 0.01; α1 = −0.2; α2 = α3 = 0. dot-dashed line: α0 = 0.1; α1 = −0.2; α2 = 0; α3 = −0.1; dotted line: α1 = 0.01; α1 = 0.2; α3 = 0.1; α3 = −0.1. as we see the negative values of α1,3 make extinction be expected sooner whereas the positive values of the other two parameters can delay the extinction. the integration of eq.(27) leads to f(ρ) l∑ n1=0 αn1ρ n1 = b 2 df dρ + c1, (31) where c1 is a constant of integration. this constant is equal to 0 if following condition is fulfilled 1 f(0) df dρ |ρ=0= 2α0 b (32) with c1 = 0 we can continue the integration of eq.(7) and the result is f(ρ) = c b exp  2 b l∑ n1=0 αn1 ρn1+1 n1 + 1   , (33) where the constant of integration c is determined by the normalization condition∫ ∞ −∞ dρ f(ρ) = 1. (34) eq. (32) in combination with eq. (33) mean that∑l n1=0 αn1 = 0. in addition f(ρ) must tend to 0 when ρ → ±∞. the dominant term at large values of ρ is αlρ l. then αl must be negative (to ensure f → 0 at large positive values of ρ and l must be odd (to ensure f → 0 at large negative values of ρ). several examples of f(ρ) are shown in fig.1. let us now calculate the exit time expectation on the basis of eq.(29). we calculate the distribution for exit from the initial position ρ to a position q < ρ. one integration of eq.(29) leads to the equation df dρ = exp(−ψ(ρ)) ( c1 + ∫ ∞ ρ dξ 2 b exp(ψ(ξ)) ) . (35) biomath 1 (2012), 1209253, http://dx.doi.org/10.11145/j.biomath.2012.09.253 page 5 of 7 http://dx.doi.org/10.11145/j.biomath.2012.09.253 n. vitanov et al., on waves and distributions in population dynamics the solution is searched in presence of two requirements: 1) f(ρ = q) = 0 , 2) f(ρ,q) increases in the slowest possible manner as ρ → ∞ (i.e. df dρ from eq. (35) is as small as possible). the second requirement leads to c1 = 0, and the integration of eq.(35) leads to the result fq(ρ) = ∫ ρ q dξ exp ( − 2 b l∑ n1=0 αn1 ξn1+1 n1 + 1 ) × [ 2 b ∫ ∞ ξ dη exp ( 2 b l∑ n1=0 αn1 ηn1+1 n1 + 1 )] . (36) fig. 2 shows the dependence of the exit time expectation on the population density and coefficients of the model equation for the case l = 3. the theory can be easily applied for the case of system of many interacting populations but even in the simplest one-dimensional case the integral from eq.(36) must be calculated numerically. v. conclusion in this paper we have discussed two aspects of population dynamics. first we have presented a model of the space-time dynamics of the system of interacting population in 2 spatial dimensions. for the most simple case of one spatial dimension and for one population we have obtained exact traveling-wave solution of the model nonlinear pde by means of the recently developed modified method of simplest equation for obtaining exact and approximate solutions of nonlinear pdes. the obtained exact solution describes the spreading of changes of the population density in the space. the generalization of this theory to the case of many populations is straightforward and describes the spreading of coupled waves of changes of densities of the studied populations. this research will be reported elsewhere. the second discussed aspect of the population dynamics was connected to the influence of the random fluctuations on the population densities. the presence of fluctuations leads to description in terms of probability density functions for the population densities. the discussed general theory is illustrated again for the simplest possible case of one population in two aspects: calculation of probability density functions and calculation of the expected extinction time. the minima and maxima of the obtained probability density functions are exactly at the fixed points of the corresponding non-perturbed model system of differential equations. the expected extinction time strongly depends on the coefficients of the model equations. because of the lack of space we do not discuss the case of more than one population here. this case will be reported elsewhere. acknowledgment this research was partially supported by the fund of scientific researches of republic of bulgaria under contract do 02-338/22.12.2008 in the scope of which the averaging applied in section ii b has been developed and used. references [1] j. d. murray, lectures on nonlinear differential equation models in biology, oxford, england, oxford university press, 1977. [2] z. i. dimitrova, n. k. vitanov, “influence of adaptation on the nonlinear dynamics of a system of competing populations”, phys. lett. a, vol. 272, pp. 368–380, 2000. http://dx.doi.org/10.1016/s0375-9601(00)00455-2 [3] z. i. dimitrova, n. k. vitanov, “adaptation and its impact on the dynamics of a system of three competing populations”, physica a, vol. 300, pp. 91–115, 2001. http://dx.doi.org/10.1016/s0378-4371(01)00330-2 [4] n. k. vitanov, z. i. dimitrova, h. kantz, “on the trap of extinction and its elimination”, phys. lett. a, vol. 349, pp. 350–355, 2006. http://dx.doi.org/10.1016/j.physleta.2005.09.043 [5] n. k. vitanov, i. p. jordanov, z. i. dimitrova. “on nonlinear dynamics of interacting populations: coupled kink waves in a system of two populations”, commun. nonlinear sci. numer. simulat., vol 14, pp. 2379–2388, 2009. http://dx.doi.org/10.1016/j.cnsns.2008.07.015 [6] n. k. vitanov, i. p. jordanov, z. i. dimitrova, “on nonlinear population waves”, applied mathematics and computation, vol. 215, pp. 2950–2964, 2009. http://dx.doi.org/10.1016/j.amc.2009.09.041 [7] n. k. vitanov, “convective heat transport in a fluid layer of infinite prandtl number: upper bounds for the case of rigid lower boundary and stress-free upper boundary”, eur. phys. j. b, vol. 15, pp. 349–355, 2000. http://dx.doi.org/10.1007/s100510051136 [8] n. k. vitanov, “upper bounds on the convective heat transport in a rotating fluid layer of infinite prandtl number: case of large taylor numbers”, eur. phys. j. b, vol. 23, pp. 249– 266, 2001. http://dx.doi.org/10.1007/s100510170075 [9] n. a. kudryashov, “simplest equation method to look for exact solutions of nonlinear differential equations”, chaos solitons & fractals, vol. 24, pp. 1217-1231, 2005. http://dx.doi.org/10.1016/j.chaos.2004.09.109 [10] n. a. kudryashov, “exact solitary waves of the fisher equation”, phys. lett. a, vol. 342, pp. 99-106, 2005. http://dx.doi.org/10.1016/j.physleta.2005.05.025 [11] n. a. kudryashov, n. b. loguinova, “extended simplest equation method for nonlinear differential equations”, commun. nonlinear sci. numer. simulat., vol. 14, pp. 3507–3529, 2009. http://dx.doi.org/10.1016/j.cnsns.2009.01.023 [12] n. k. vitanov, z. i. dimitrova, h. kantz, “modified method of simplest equation and its application to nonlinear pdes”, applied mathematics and computation vol. 216, pp. 2587– 2595, 2010. http://dx.doi.org/10.1016/j.amc.2010.03.102 biomath 1 (2012), 1209253, http://dx.doi.org/10.11145/j.biomath.2012.09.253 page 6 of 7 http://dx.doi.org/10.1016/s0375-9601(00)00455-2 http://dx.doi.org/10.1016/s0378-4371(01)00330-2 http://dx.doi.org/10.1016/j.physleta.2005.09.043 http://dx.doi.org/10.1016/j.cnsns.2008.07.015 http://dx.doi.org/10.1016/j.amc.2009.09.041 http://dx.doi.org/10.1007/s100510051136 http://dx.doi.org/10.1007/s100510170075 http://dx.doi.org/10.1016/j.chaos.2004.09.109 http://dx.doi.org/10.1016/j.physleta.2005.05.025 http://dx.doi.org/10.1016/j.cnsns.2009.01.023 http://dx.doi.org/10.1016/j.amc.2010.03.102 http://dx.doi.org/10.11145/j.biomath.2012.09.253 n. vitanov et al., on waves and distributions in population dynamics [13] n. k. vitanov, “modified method of simplest equation: powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear pdes”, commun. nonlinear sci. numer. simulat., vol. 16, pp. 1176–1185, 2011. http://dx.doi.org/10.1016/j.cnsns.2010.06.011 [14] n. k. vitanov, “application of simplest equations of bernoulli and riccati kind for obtaining exact traveling wave solutions for a class of pdes with polynomial nonlinearity”, commun. nonlinear sci. numer. simulat., vol. 15, pp. 2050-2060, 2010. http://dx.doi.org/10.1016/j.cnsns.2009.08.011 [15] n. k. vitanov, z. i. dimitrova, k. n. vitanov, “on the class of nonlinear pdes that can be treated by the modified method of simplest equation. application to generalized degasperis processi equation and b-equation”, commun. nonlinear sci. numer. simulat., vol. 16, pp. 3033–3044, 2011. http://dx.doi.org/10.1016/j.cnsns.2010.11.013 [16] l. s. pontryagin, a. a. andronov, a. a. vitt, “on statistical considerations of dynamical systems”, jetp vol. 3, pp. 165– 180, 1933. biomath 1 (2012), 1209253, http://dx.doi.org/10.11145/j.biomath.2012.09.253 page 7 of 7 http://dx.doi.org/10.1016/j.cnsns.2010.06.011 http://dx.doi.org/10.1016/j.cnsns.2009.08.011 http://dx.doi.org/10.1016/j.cnsns.2010.11.013 http://dx.doi.org/10.11145/j.biomath.2012.09.253 introduction model equations spatially distributed populations spatially averaged equations traveling waves application of the modified method of simplest equation to eq. (12) case 0†=0 case 0†= 0 statistical distributions and exit time conclusion references original article biomath 2 (2013), 1309087, 1–9 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum calculating hyphal surface area in models of fungal networks laurens bakker∗, andrew poelstra∗,† ∗ center for interdisciplinary research in the mathematical and computational sciences (irmacs) † department of mathematics simon fraser university, burnaby, canada. emails: {laurens bakker, andrew poelstra}@sfu.ca received: 5 may 2013, accepted: 8 september 2013, published: 18 september 2013 abstract—filamentous fungi grow efficient nutrient transportation networks which are highly resilient to attacks by grazers. understanding them may benefit the design of human-built networks, where such properties are sought after. we recently developed a mathematical model that improved previous 2-dimensional studies by representing the space in a 3-dimensional face-centered cubic lattice. while the model focused on structural aspects (hyphal orientation, branching, and fusion), these are closely tied to functional aspects, that is, the handling of nutrients. in this paper, we refine our previous model by modelling the hyphal network as a set of cylindrical tubes connecting spherical junction points, and calculating the exact local hyphal surface area. in further development of the model, this will allow the refinement and incorporation of existing nutrient consumption models—in particular, how nutrients are used for turgor maintenance at a particular network location. keywords-fungal networks; mathematical biology; mathematical modelling i. introduction fungi, like all kingdoms of life, are economically and ecologically critical. fungi have two modes of growth, and a number of mathematical models have been developed to better understand and control their growth. in yeasts, a minority of fungi, cells separate after they divide. in filamentous fungi cells do not physically separate and are kept together in filaments called hyphae. these hyphae extend at their tips and branch, forming a network, the mycelium, that penetrates the environment and absorbs resources. in this paper, we focus on the mathematical modelling of filamentous fungi. a large body of research has been devoted to modelling filamentous fungi, as covered by prosser in the late 70s [10], summarized by kotov and reshetnikov in 1990 [9], or reviewed more recently by davidson in 2007 [3]. models can be intuitively divided based on the scale they focus on, since fungi can cover hectares due to indeterminate growth1 while their basic machinery operates on the micrometer scale [3]. in this paper, we focus on the micrometer scale, which is of particular interest when attempting to understand how the function and structure of fungi give rise to specific patterns of growth. models at this scale can be further divided into three types [1]: macroscopic models, focusing on quantities at the colony-level (e.g., overall growth rate, biomass density), microscopic models, explicitly representing every hypha, and intermediate models which represent quantities over several hyphae and are classically reaction-diffusion models. in order to precisely understand the complex coupling of structure and function in fungi, the most accurate mathematical abstraction is offered by microscopic models. 1in some cells, including most animal cells, division terminates after a fixed number of times. in contrast, the division of the cells in a fungal mycelium is indeterminate, as they can divide an infinite number of times if resources permit. citation: laurens bakker, andrew poelstra, calculating hyphal surface area in models of fungal networks, biomath 2 (2013), 1309087, http://dx.doi.org/10.11145/j.biomath.2013.09.087 page 1 of 9 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2013.09.087 l bakker, a poelstra, calculating hyphal surface area in models of fungal networks a. contribution of the paper boswell and colleagues developed a model at the micrometer scale in which hyphae are explicitly represented [2]. this model was two-dimensional, and thus created only planar fungal networks (i.e. there cannot be a hypha on ‘top’ of another). we recently introduced a three-dimensional model to better reproduce the structure of fungi [4]. the move to a three-dimensional model makes it particularly challenging to compute the surface area of the mycelium, and the cost of turgor maintenance. fungi work to generate osmotic forces, consuming resources to generate turgor pressure, and drive water uptake and the bulk flow of cytoplasm towards the growing hyphal tips [7], [8]. hyphae regulate local osmotic pressure (and thereby water uptake) by transporting ions from the environment into the hypha or vice versa. the cost of ion transportation will be a function of the osmotic pressure gradient and the surface area exposed to this gradient. thus the local turgor maintenance cost depends on the available hyphal surface area. this cost needs to be taken into account to determine which hyphae have a surplus of nutrients available for growth or branching. the key contribution of this paper lies in calculating the exact local hyphal surface area in the model, which improves on both [2] and [4] and can be further used to increase the accuracy of fungal network models. b. organization of the paper in section ii, we explain how the physical space is discretized in our model. as our focus is on the cost of turgor maintenance, we refer the interested reader to [2], [4] for the rules detailing the orientation, branching, and fusion of hyphae in the model. having described a discretization of space, in section iii we fix the remaining free parameters (e.g., lattice width) and choose our coordinate systems. we see that the lattice width is restricted by the cross-section radius of the hyphae. our hyphae are constructed by connecting cylinders and spheres. in section iv we compute the surface area of these hyphae by determining how much of the surface of the spheres is not covered by cylinders, and how much of the surface of the cylinders is not covered by other cylinders. ii. discretized space dividing the space into chunks involves two questions: dimensionality and degrees of freedom. firstly, (a) (b) fig. 1. the plane is discretized into elements of equal size. the elements can be square (left), allowing the hypha to change direction by 90◦, or hexagonal (right), allowing a change of 60◦. the dimensionality asks whether the fungus should be modelled as spreading in a slice of soil (2 dimensions), or spreading as it naturally does in a volume (3 dimensions). limiting a model to two dimensions means that hyphae cannot be overlapping, thereby forcing the structure to be artificially planar. therefore, we investigate the 3dimensional case, which allows for overlapping hyphae. secondly, the degrees of freedom must be sufficient so that an extending hypha can change direction with a realistic angle. as illustrated in figure 1(a), if the cells were square then the hyphae can extend to the cell in front (angle of 0◦), the cell on the right or on the left (angle of 90◦). thus, the hyphae would either keep its direction or change it by 90◦. an angle of 60◦ was suggested as a more accurate description [2], leading to the hexagonal cells depicted in figure 1(b). in fact, 60◦ is the smallest allowable angle in a uniform discretization of 2d (or 3d) space. obtaining an angle of 60◦ between neighbours is straightforward in two dimensions: the hexagonal grid shown in figure 1(b) is the only space-filling configuration of equal-sized cells that achieves this. in three dimensions, there are several arrangements that produce 60◦ angles between neighbours. such arrangements fall under two schemes: hexagonal close packing (hcp) and face centred cubic (fcc) arrangement [5], [6]. of these, fcc is the only viable candidate (figure 2(a)), since the asymmetry in hcp can prevent a hypha from growing straight ahead, which is its most common direction (figure 2(b)). fcc is an extension from the 2d hexagonal grid, in which horizontal hexagonal layers are stacked on top of each other but displaced slightly to allow for a tight packing. this extension can be intuitively understood by induction. biomath 2 (2013), 1309087, http://dx.doi.org/10.11145/j.biomath.2013.09.087 page 2 of 9 http://dx.doi.org/10.11145/j.biomath.2013.09.087 l bakker, a poelstra, calculating hyphal surface area in models of fungal networks (a) (b) fig. 2. the local neighbourhood in a face centered cubic (fcc) arrangement is symmetric along each edge between the central cell and a neighbour (a). this is not the case for the local neighbourhood in a hexagonal close packing (hcp) arrangement, which is symmetric only along some edges between the central cell and a neighbour (b). as a consequence, a fungal tip that grew onto the central cell from the right below (red line) would not be able to grow exactly straight ahead. fig. 3. construction of a face centred cubic arrangement by induction. iii. discretization in 3d assume there is a discrete model simulating the growth of a mycelium in three dimensions. the model must take into account the amount of resource needed for hyphal wall maintenance for each discrete point in the lattice to determine where excess resource permits growth. in the fcc lattice chosen in section ii, each vertex of the lattice is assigned the part of the hyphal surface that lies within its voronoi cell. therefore, knowing the cost of hyphal wall maintenance requires computing the surface area represented by each vertex of the fcc lattice. formally, given a subgraph g of an fcc lattice with a sphere of radius r at each vertex and a cylinder of radius r and length ∆x at each edge, we want to compute the external surface area of the union of all these shapes. a. notation from here on, we will use spherical (θ ∈ (−π,π],φ ∈ [−π/2,π/2]) coordinates, where θ represents the angle between a vector and the positive x axis and φ represents the angle between a vector and the positive side of the x,y plane; and cylindrical coordinates (θ ∈ (−π,π],z ∈ r), where θ is the same as before and z represents the height above the x,y plane. b. adequately discretizing the space in order to ‘charge’ maintenance costs to individual vertices, the hyphal surface area needs to be computed for the voronoi cells associated with these vertices. this puts restrictions on the size of the spheres and cylinders that are put at vertices and edges, respectively. if their radius r is too large in relation to the voronoi cell size ∆x, a fraction of a hypha’s surface area may be assigned to a cell that it does not logically belong to (see figure 4). therefore, ∆x should be chosen such that a hypha between two vertices, say vi and vj, falls entirely within the boundary between the associated pair of voronoi cells and does not ‘invade’ other cells. theorem 1. hypha connecting two vertices vi and vj falls entirely within the associated voronoi cells when r ≤ (√ 3/4/3 ) ∆x. proof: the intersection of a hypha (represented by a cylinder) with the boundary face between vi and vj is a circle of diameter 2r. the centre of this circle is at the intersection of edge eij and the boundary face. without loss of generality, pick any diameter of the circle and put two vertices, vk and v`, as close as possible to either end of the diameter (see figure 5), but at distance ≥ ∆x from biomath 2 (2013), 1309087, http://dx.doi.org/10.11145/j.biomath.2013.09.087 page 3 of 9 http://dx.doi.org/10.11145/j.biomath.2013.09.087 l bakker, a poelstra, calculating hyphal surface area in models of fungal networks fig. 4. an example case when ∆x is too small in relation to r: when ∆x = 2r, the highlighted surface area is assigned to the voronoi cell indicated by the dotted line, but it logically belongs to the voronoi cells of the two vertices that lie within the hypha. vi vj vkv` fig. 5. the cylinder between vi and vj needs to fit through the boundary between their voronoi cells, which runs between the centroids of 4vivjvk and 4vivjv`. vi and vj. this maximally constrains the boundary size, which is the distance between the centroids of 4vjvivk and 4vivjv`. c. from two to three dimensions in 2 dimensions the surface area (i.e., surface length since it is 1-dimensional) within a particular hexagonal voronoi cell vi with a set hi of edges occupied by hyphae is equal to ∆x + πr when |hi| = 1, or {2 · |hi| · ∆x 2︸ ︷︷ ︸ surface length + max(max j (∠hi[j],hi[j + 1] −π), 0) ·r︸ ︷︷ ︸ arc of the circle − |hi|∑ k=1 2r cot ( ∠hi[k],hi[k − 1] 2 ) ︸ ︷︷ ︸ overlap between hyphae } (1) where hi is a clockwise ordered circular list, i.e. hi[0] = hi[|hi|]. this is illustrated in figure 6. πr/3 2r/ √ 3 fig. 6. total hyphal surface length consists of the cylinder (associated with edges) surface length (2|hi|∆x/2 = 2∆x) and the portion of the sphere (associated with a vertex; the dashed circle) not covered by cylinders (πr/3), less the overlap between cylinders (2r/ √ 3). in three dimensions, the first term of (1) needs to be adjusted only to account for the extra dimension: 2 · |hi|·πr ∆x2 . the second and third term generalise to three dimensions, although a one-pass scan (as in (1)) does not suffice anymore. the method we will use for both the portion of the sphere to be added and the portion of the overlapping cylinders to be subtracted are similar. for a generic sphere (cylinder), construct the refinement of all overlaps with (other) cylinders. then for every one of the 212 possible neighbourhoods, mark the appropriate areas in the refinement as ‘exposed’. store the resulting areas in a lookup table with the neighbourhood (represented as a binary string) as lookup key. the second and third terms of (1) are to be replaced by lookups in these tables. the construction of the refinements used to create these lookup tables is detailed in the next section. iv. area calculation in section iii, we described a discrete model simulating the growth of a mycelium in three dimensions. in order to compute the surface area contained in a voronoi cell in the lattice, we required ‘maps’ of which parts of the surface of a central sphere is covered by which neighbours’ cylinder, and of which part of the surface of a neighbour’s cylinder is covered by other neighbours’ cylinders. in this section, we describe these ‘maps’ and compute the required areas. a. central sphere for every hypha that is connected to a vertex vi, a hemisphere of that vertex’s sphere is covered, and therefore removed from the surface area. what remains biomath 2 (2013), 1309087, http://dx.doi.org/10.11145/j.biomath.2013.09.087 page 4 of 9 http://dx.doi.org/10.11145/j.biomath.2013.09.087 l bakker, a poelstra, calculating hyphal surface area in models of fungal networks c 1 4 3 5 2 6 fig. 7. vertex labelling congruent with directions ϑk; opposite directions (−ϑk) omitted. fig. 8. area: πr2 fig. 9. area: πr2/3 is a hemisphere, lune, spherical triangle or spherical quadrangle (e.g.,figure 7–figure 10). each neighbour vj divides vi’s sphere into two hemispheres along a circular boundary that is perpendicular to eij. the refinement of all divisions associated with a neighbour of vi is constructed as follows. fig. 10. area: πr2/12 the six directions ϑk along which neighbours are placed are numbered in figure 7. we denote the direction opposite of ϑk (mirrored in c) as −ϑk. the intersection of the boundaries corresponding to neighbours vj and vj′ are given by ±(ϑj × ϑj′ ).1. these intersections lie along one of seven directions ϕk: ϕ1 := ϑ3 ×ϑ1 = ϑ5 ×ϑ1 = ϑ5 ×ϑ3 ϕ2 := ϑ1 ×ϑ2 = ϑ1 ×ϑ6 = ϑ2 ×ϑ6 ϕ3 := ϑ4 ×ϑ1 ϕ4 := ϑ5 ×ϑ4 = ϑ6 ×ϑ5 = ϑ6 ×ϑ4 ϕ5 := ϑ3 ×ϑ2 = ϑ4 ×ϑ2 = ϑ4 ×ϑ3 ϕ6 := ϑ3 ×ϑ6 ϕ7 := ϑ2 ×ϑ5 the actual refinement is constructed by connecting the intersection points ϕk along the associated circular boundaries, to produce a polygonal partition of the sphere’s surface area. theorem 2. the partition of the surface area of vi’s sphere as induced by its neighbours’ cylinders consists of 24 triangles of equal size. proof: we first observe that the partition will consist of polygonal regions, where each polygon vertex is an intersection point ϕk. at ±ϕ3, ±ϕ6 and ±ϕ7, we have the intersection of two circles, producing four π/2 angles each, 24 such angles total. (these angles can be calculated explicitly or derived from symmetry of the lattice.) each pair of 1alternatively, the intersections could also be derived from the convex hull of the neighbour positions (each ϕk corresponds to the centre of one of its faces), or from the voronoi diagramme of the neighbour directions on the sphere (each ϕk corresponds to one of the voronoi diagramme’s vertices). the refinement is the delaunay triangulation of the ϕk. biomath 2 (2013), 1309087, http://dx.doi.org/10.11145/j.biomath.2013.09.087 page 5 of 9 http://dx.doi.org/10.11145/j.biomath.2013.09.087 l bakker, a poelstra, calculating hyphal surface area in models of fungal networks these intersection points is separated by a circle, so that each polygon in the partition has at most one of these intersections as a vertex (and therefore at most one interior angle of extent π/2). therefore the partition has at least 24 polygons. at each of ±ϕ1, ±ϕ2, ±ϕ4 and ±ϕ5, three circles intersect, producing six angles of extent π/3, 48 such angles total. in total, the intersection points are responsible for 72 angles, which are the interior angles of the polygons in the partition. this gives an upper bound of 24 polygons, which is achieved when every one is a triangle. this, combined with our earlier observation that we have at least 24 polygons, tells us that the partition consists of exactly 24 triangles. finally, since the area of a spherical triangle is entirely determined by its angles, which are π/2,π/3,π/3 for each triangle, every triangle has the same area. to determine the topology, we notice that each cylinder directly touches six intersection points. for any of the remaining eight points, say, φ, either (a) every triangle with φ as a vertex is covered, or (b) none of them are. which case applies for a given vertex is geometrically obvious. using this, we determine the topology of the refinement, and create figure 12. this figure also encodes the relationships between a neighbour and the areas of the sphere it covers: each neighbour lies at the center of a circle that its cylinder would cover, e.g. ϑ2 lies at the centre of the circle through ϕ2, ϕ7, −ϕ5, −ϑ5, −ϕ2, −ϕ7, ϕ5 and ϑ5. a cylinder in direction ϑ2 would cover all areas inside this circle, and a cylinder in the opposite direction (−ϑ2) would cover all areas outside this circle. b. hyphal cylinder the cylinder that is put at every hypha eij not only covers part of the surface of the spheres belonging to the cells it connects, but it also covers (and is covered by) parts of the surface of other cylinders connected to these cells. the cylinder associated with another hypha eik,j 6= k connected to one end of the cylinder intersects it along an elliptical boundary ζk that is defined by the plane through eij ×eik, eik ×eij and eij + eik. as was done for the central sphere, the refinement of all divisions associated with an edge eik needs to be constructed. cutting the cylinder associated with ϑ1 along the plane θ = 0, we see that these intersections between the cylinder associated with eij and those associated with eik,j 6= k correspond to sine waves (figure 11). specifically, if a hypha ϑi has coordinates (θ,φ), the intersection of its cylinder with that of ϑ1 will have the form ζi = tan (average{φ,π/2}) cos (x−θ) x ∈ [θ −π/2,θ + π/2] (2) which follows immediately from converting to ϑ1’s cylindrical coordinates. the refinement of these boundaries is a subdivision of the cylinder into 30 triangles and 6 quadrangles, of 11 different sizes. these sizes can be calculated as sums of definite integrals over the sine waves associated with the edges. the symmetrical nature of the lattice supports that areas which appear equal actually are equal. the only feature of this diagram not geometrically clear is that the 3-way intersections marked ◦ are true; i.e., they are not just three 2-way intersections very close together. it suffices to verify one 3-way intersection. this is because the diagram will look identical no matter which hypha we use as a base, up to relabelling—by replacing ϑ1 by ϑ′3, ϑ ′ 4 or ϑ6, respectively, the other three intersections marked with ◦ can be positioned at the •. we see that the leftmost ◦ is a true 3-way intersection, since when x = 0, the three curves have the value ϑ3 : tan (π/6) sin (π/2) = √ 1/3 ϑ4 : tan (π/4) sin ( cos−1 (√ 2/3 )) = √ 1/3 ϑ6 : tan (π/3) sin ( cos−1 (√ 8/9 )) = √ 1/3 all intersection points lie along one of 26 directions. below are the coordinate specifications for the intersection points in the interval θ ∈ [ cos−1 (√ 2/3 ) , cos−1 ( − √ 1/3 )] (in cylindrical coordinates), the top left quarter of figure 13. the remaining directions follow by symmetry (vertical planes of symmetry along ϑ4 and ϕ3). biomath 2 (2013), 1309087, http://dx.doi.org/10.11145/j.biomath.2013.09.087 page 6 of 9 http://dx.doi.org/10.11145/j.biomath.2013.09.087 l bakker, a poelstra, calculating hyphal surface area in models of fungal networks 1 2 3 4 5 6 7 8 9 10 11 ζ−1 ζ4 ζ3 ζ5 ζ2 ζ6 ζ−4 ζ−5 ζ−3 ζ−6 ζ−2 • ◦◦ ◦ 3 5 ϕ2 6 ϕ4−4ϕ −3 6 ϕ4−2ϕ −4 −3ϕ −5 6 ϕ2−3ϕ −2 −3ϕ −5 −6ϕ fig. 11. the cylinder associated with ϑ1, when cut in two along the vertical plane θ = 0, illustrates that the area calculation consists entirely of intersections of areas below a sine wave. the 11 areas numbered in red are the only ones for which the area needs to be calculated. the others follow by symmetry. like in figure 12, intersection points ijϕ are labelled. to distinguish the labels from those used in figure 12, the intersection is labelled using the indices of two of the arcs that cross (ζi,ζj). i j ϕ: ( θ , z ) −5 −6ϕ ∼ −ϕ5: ( cos−1 ( − √ 1/3 ) , √ 2 ) −2 −3ϕ: ( cos−1 ( − √ 1/3 ) , √ 2/9 ) −5 2 ϕ = −54 ϕ = 2 4 ϕ ∼ −ϕ6: ( cos−1 (√ 2/3 ) , 1 ) −3 6 ϕ: ( cos−1 (√ 2/3 ) , 1/3 ) −2 6 ϕ = 2−6ϕ = 2 −2ϕ = 6 −6ϕ = −ϕ2: ( cos−1 (1/3) , 0 ) −4 4 ϕ = −ϕ3: ( cos−1 ( − √ 1/3 ) , 0 ) −6 =4 ϕ = −6−3ϕ = 4 −3ϕ ∼ −ϑ5: ( π/2 , √ 1/3 ) 2 −3ϕ: ( cos−1 (√ 1/3 ) , √ 2/9 ) 4 −2ϕ: ( cos−1 ( − √ 2/27 ) , 1/3 ) the actual refinement is constructed by connecting the intersection points along the associated circular boundaries. by a similar process to the one used to create figure 12, we can determine the topology of the refinement, and create figure 13. this figure also encodes the relationships between a neighbour eik and the areas of eij’s cylinder it covers: each neighbour lies at the center of a circle that it covers, e.g. ϑ2 lies at the centre of the circle through ϑ′5, ϕ1, ϑ5, ϕ ′ 1, ϕ16, ϕ20 and ϕ22. a cylinder in direction ϑ2 would cover all areas inside this circle. unlike with the sphere, a cylinder in the opposite direction (ϑ′2) would not cover all areas outside this circle; rather, ϑ′2 has its own circle (through ϑ3, ϕ1, ϕ13, ϕ18, ϑ′3, ϕ17, ϕ11 and ϕ ′ 1). biomath 2 (2013), 1309087, http://dx.doi.org/10.11145/j.biomath.2013.09.087 page 7 of 9 http://dx.doi.org/10.11145/j.biomath.2013.09.087 l bakker, a poelstra, calculating hyphal surface area in models of fungal networks ϕ1 ϕ1 ϕ1 ϕ1 ϕ1 ϕ1 −ϕ1 −ϕ4 ϕ4 −ϕ5 ϕ5 ϕ2 −ϕ2 ϕ7 −ϕ7 ϕ6 −ϕ6 −ϕ3 ϕ3 −ϑ2−ϑ6 −ϑ4 −ϑ1 −ϑ3−ϑ5 ϑ1 ϑ3 ϑ5ϑ2 ϑ6 ϑ4 fig. 12. topology of the refinement of hemisphere intersections (ϕk) induced by the 11 neighbours (ϑk in blue) of the central cell. −ϑk or −ϕk denote ϑk or ϕk reflected in the origin, respectively. it has been projected from ϕ1 onto the plane θ = π/2, and rotated counterclockwise by π/2 to emphasise the symmetry. v. conclusion we have computed the surface area for paths in a fcc lattice, which model hyphae growing in a mycelium. this improves on earlier models in that it is 3d and can therefore model non-planar mycelia, the typical case in nature. by using a hexagonal lattice rather than a cubic one, we allow our hyphae to grow in straight lines or to change direction by as little as 60◦, giving a realistic range of motion while still being computationally feasible. acknowledgment we would like to thank the modeling of complex social systems program (mocssy) and philippe giabbanelli and dr. veselin jungic at the irmacs center. philippe in particular was instrumental in bringing the two authors together and provided support and mentoring throughout. we are indebted to richard dorrell, from the department of biochemistry (university of cambridge), for numerous discussions on the biology of fungi. references [1] m. bezzi and a. ciliberto. modeling growth of filamentous microorganisms. comments on theoretical biology, 8:563–585, 2003. [2] g. p. boswell, h. jacobs, k. ritz, g. m. gadd, and f. a. davidson. the development of fungal networks in complex environments. bulletin of mathematical biology, 69:605–634, 2007. http://dx.doi.org/10.1007/s11538-005-9056-6 [3] f. a. davidson. mathematical modelling of mycelia: a question of scale. fungal biology reviews, 21:30–41, 2007. http://dx.doi.org/10.1016/j.fbr.2007.02.005 [4] p. j. giabbanelli and l. bakker. a mathematical model of fungal networks accounting for hyphal orientation, branching and fusion. proceedings of frontiers in biophysics, page 2, 2012. [5] t. c. hales. sphere packing i. discrete and computational geometry, 17:1–51, 1997. http://dx.doi.org/10.1007/bf02770863 [6] t. c. hales. sphere packing ii. discrete and computational geometry, 18:135–149, 1997. http://dx.doi.org/10.1007/pl00009312 [7] l. l. m. heaton, e. lópez, p. k. maini, m. d. fricker, and n. s. jones. growth-induced mass flows in fungal networks. proc. roy. soc. b., 277(1698):3265–3274, 2010. [8] l. l. m. heaton, e. lópez, p. k. maini, m. d. fricker, and n. s. jones. advection, diffusion, and delivery over a network. phys. rev. e, 86:021905, aug 2012. http://dx.doi.org/10.1103/physreve.86.021905 [9] k. kotov and s. v. reshnetnikov. a stochastic model for early mycelial growth. mycol. res., 94(5):577–586, 1990. http://dx.doi.org/10.1016/s0953-7562(09)80655-2 [10] j. prosser. mathematical modelling of mycelial growth, pages 359–384. cambridge university press, 1979. biomath 2 (2013), 1309087, http://dx.doi.org/10.11145/j.biomath.2013.09.087 page 8 of 9 http://dx.doi.org/10.1007/s11538-005-9056-6 http://dx.doi.org/10.1016/j.fbr.2007.02.005 http://dx.doi.org/10.1007/bf02770863 http://dx.doi.org/10.1007/pl00009312 http://dx.doi.org/10.1103/physreve.86.021905 http://dx.doi.org/10.1016/s0953-7562(09)80655-2 http://dx.doi.org/10.11145/j.biomath.2013.09.087 l bakker, a poelstra, calculating hyphal surface area in models of fungal networks ϑ4 −ϑ4 ϑ3 ϑ2 −ϑ5 −ϑ6 ϕ2 ϕ3 ϕ1 −ϕ2 −ϕ3 −ϕ1 6 −4ϕ 5 4 ϕ −3 6 ϕ4−2ϕ −4 −3ϕ −2 5 ϕ 5 6 ϕ −5 6 ϕ 2 −3ϕ −2 −3ϕ −2 3 ϕ 5 −6ϕ −5 2 ϕ −6 3 ϕ 2 3 ϕ −5 −6ϕ −ϑ3 −ϑ2 ϑ5 ϑ6−ϑ1 1 3 4 5 6 7 9 10 2 8 11 fig. 13. topology of the refinement of cylinder boundary intersections (ϕk) induced by the 11 neighbours (ϑk in blue) of the central cell. −ϑk or ϕk denote ϑk or ϕk reflected in the origin, respectively. it has been projected onto the plane from above ϑ1. biomath 2 (2013), 1309087, http://dx.doi.org/10.11145/j.biomath.2013.09.087 page 9 of 9 http://dx.doi.org/10.11145/j.biomath.2013.09.087 introduction contribution of the paper organization of the paper discretized space discretization in 3d notation adequately discretizing the space from two to three dimensions area calculation central sphere hyphal cylinder conclusion references original article biomath 1 (2012), 1209255, 1–7 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum multiregional sir model with infection during transportation diána h. knipl∗ and gergely röst† ∗analysis and stochastics research group – hungarian academy of sciences, bolyai institute, university of szeged, szeged, hungary email: knipl@math.u-szeged.hu † bolyai institute, university of szeged, szeged, hungary email: rost@math.u-szeged.hu received: 12 july 2012, accepted: 25 september 2012, published: 18 october 2012 abstract—we present a general epidemic model to describe the spread of an infectious disease in several regions connected by transportation. we take into account that infected individuals not only carry the disease to a new place while traveling from one region to another, but transmit the disease during travel as well. we obtain that a model structured by travel time is equivalent to a large system of differential equations with multiple delays. by showing the local lipschitz property of the dynamically defined delayed feedback function, we obtain existence and uniqueness of solutions of the system. keywords-epidemic spread; transportation model; dynamically defined delay; lipschitz continuity i. introduction we consider an arbitrary n number of regions which are connected by transportation, and present an sir based model which describes the spread of infection in the regions and also during travel between them. we show that our model is equivalent to a system of functional differential equations x′(t) = f(xt), (1) where t ∈ r, t ≥ 0 and x : r � r3n. we use the notation xt ∈ c, xt(θ) = x(t + θ) for θ ∈ [−σ, 0], where for σ > 0, we define our phase space c = c([−σ, 0], r3n) as the banach space of continuous functions from [−σ, 0] to r3n, equipped with the usual supremum norm || · ||. in the sequel we use the notation |v| for the euclidean table i variables and key model parameters (j, k ∈ {1, . . . n}) sj , ij , susceptible, infected, recovered, all rj , nj individuals in region j sk,j , ik,j , susceptible, infected, recovered, all rk,j , nk,j individuals during travel from region k to j λj incidence in region j λk,j incidence during travel from region k to j αj recovery rate of infected individuals in region j αk,j recovery rate during travel from region k to j µj,k travel rate from region j to region k τk,j duration of travel from region k to j norm of any vector v ∈ rm for m ∈ z+. in order to obtain the general existence and uniqueness result for the system, we prove that f : c � r3n satisfies the local lipschitz condition on each bounded subset of c, that is, for every m > 0 there exists a constant k = k(m) such that the inequality |f(φ) − f(ψ)| ≤ k||φ − ψ|| holds for every φ,ψ ∈ c with ||φ||, ||ψ|| ≤ m. the paper is organised as follows. in section 2 we introduce our model, then we obtain the compact form of the system in section 3. section 4 concerns with the proof of the local lipschitz condition for various types of incidence (new cases per unit of time). citation: d. knipl, g. röst, multiregional sir model with infection during transportation, biomath 1 (2012), 1209255, http://dx.doi.org/10.11145/j.biomath.2012.09.255 page 1 of 7 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.09.255 d. knipl et al., multiregional sir model with infection during transportation ii. model description we formulate a dynamical model describing the spread of an infectious disease in n regions and also during travel from one region to another. divide the entire populations of the n regions into the disjoint classes sj, ij, rj, j ∈ {1, . . .n}, where sj(t) ij(t), rj(t), j ∈ {1, . . .n} denote the number of susceptible, infected and recovered individuals at time t in region j. for the total population in region j at time t, we use the notation nj(t) = sj(t) + ij(t) + rj(t). the incidence in region j is denoted by λj ( sj(t),ij(t),rj(t) ) , model parameter αj represents the recovery rate of infected individuals in region j. we denote the travel rate from region j to region k by µj,k and we set µj,j = 0 for j,k ∈ {1, . . .n}. let sk,j, ik,j, rk,j denote susceptible, infected and recovered travelers, where lower index-pair {k,j}, j,k ∈ {1, . . .n} indicates that individuals are traveling from region k to region j. let τk,j > 0 denote the time required to complete the travel from region k to region j, which is assumed to be fixed. to describe the disease dynamics during travel, for each t∗ we define sk,j(u; t∗), ik,j(u; t∗), rk,j(u; t∗), j,k ∈ {1, . . .n} as the densities of individuals with respect to u who started travel at time t∗ and belong to classes sk,j, ik,j, rk,j, where u ∈ [0,τk,j] denotes the time elapsed since the beginning of the travel. then sk,j(τk,j; t − τk,j), ik,j(τk,j; t − τk,j), rk,j(τk,j; t − τk,j) express the inflow of individuals arriving from region k to compartments sj, ij, rj at time t, respectively. let nk,j(u; t∗) = sk,j(u; t∗) + ik,j(u; t∗) + rk,j(u; t∗) denote the total density of individuals during travel from region k to j, where j,k ∈ {1, . . .n}. the total density is constant during travel, i.e. nk,j(u; t∗) = nk,j(0, t∗) for all u ∈ [0,τk,j]. during the course of travel from region k to j, λk,j ( sk,j(u; t∗), ik,j(u; t∗),rk,j(u; t∗) ) represents the incidence, and let αk,j denote the recovery rate. all variables and model parameters are listed in table i. based on the assumptions formulated above, we obtain the following system of differential equations for the disease transmission in region j, j ∈ {1, . . .n}:  ṡj(t) = −λj(·) − ( n∑ k=1 µj,k ) sj(t) + n∑ k=1 sk,j(τk,j; t − τk,j), i̇j(t) = λj(·) − ( n∑ k=1 µj,k ) ij(t) − αjij(t) + n∑ k=1 ik,j(τk,j; t − τk,j), ṙj(t) = αjij(t) − ( n∑ k=1 µj,k ) rj(t) + n∑ k=1 rk,j(τk,j; t − τk,j). (lj) for each j,k ∈ {1, . . .n} and for each t∗, the following system (tk,j) describes the evolution of the densities during the travel from region k to region j which started at time t∗:   d du sk,j(u; t∗) = −λk,j(·), d du ik,j(u; t∗) = λk,j(·) − αk,jik,j(u; t∗), d du rk,j(u; t∗) = αk,jik,j(u; t∗). (tk,j) for sake of simplicity, in systems (lj) and (tk,j) we use the notations λj(·) and λk,j(·) for the incidences, where these functions are meant to be evaluated at the appropriate points. for u = 0, the densities are determined by the rates individuals start their travels from region k to region j at time t∗. hence, the initial values for system (tk,j) at u = 0 are given by  sk,j(0; t∗) = µk,jsk(t∗), ik,j(0; t∗) = µk,jik(t∗), rk,j(0; t∗) = µk,jrk(t∗). (iv tk,j) notice that µj,j = 0 for j ∈ {1, . . .n} implies that for each t∗, it holds that sj,j(u; t∗) = ij,j(u; t∗) = rj,j(u; t∗) ≡ 0, as there is no travel from region j to itself. since travel from region k to region j takes τk,j units of time to complete, we need to assure that there exists a unique solution of system (tk,j) on [0,τk,j] (see proposition iv.1). now we turn our attention to the terms sk,j(τk,j; t−τk,j), ik,j(τk,j; t−τk,j), rk,j(τk,j; t−τk,j), j,k ∈ {1, . . .n} in biomath 1 (2012), 1209255, http://dx.doi.org/10.11145/j.biomath.2012.09.255 page 2 of 7 http://dx.doi.org/10.11145/j.biomath.2012.09.255 d. knipl et al., multiregional sir model with infection during transportation system (lj), which give the inflow of individuals arriving to classes sj, ij, rj at time t, upon completing a travel from region k. at time t, these terms are determined by the solution of system (tk,j) at u = τk,j with initial values (iv tk,j) for t∗ = t − τk,j, since individuals who left region k with rate µk,j at time t − τk,j will enter region j at time t. next we specify initial values for system (lj) at t = 0. since for k ∈ {1, . . .n}, travel from region k to region j takes τk,j units of time to complete, arrivals to region j at time t are determined by the state of the classes of region k at t−τk,j, via the solution of system (tk,j) and initial values (iv tk,j). thus, we set up initial values as follows:   sj(θ) = ϕs,j(θ), ij(θ) = ϕi,j(θ), rj(θ) = ϕr,j(θ), (iv lj) where θ ∈ [−τ, 0] for τ := maxj,k∈{1,...n} τk,j, moreover ϕs,j, ϕi,j and ϕr,j are continuous functions for j ∈ {1, . . .n}. iii. the compact form of the system for each j,k ∈ {1, . . .n} and t∗ ≥ 0, we define y(u) = yt∗k,j(u) = (sk,j(u; t∗), ik,j(u; t∗), rk,j(u; t∗)) t and g = gk,j = (gs,gi,gr)t , where y : [0,τk,j] � r3, g : r3 � r3 and gs(y) = −λk,j(y1,y2,y3), gi(y) = λk,j(y1,y2,y3) − αk,jy2, gr(y) = αk,jy2. then for each j,k and t∗, system{ y′(u) = g(y(u)), y(0) = y0 (2) is a compact form of system (tk,j) with initial values (iv tk,j) for y0 = (µk,jsk(t∗), µk,jik(t∗), µk,jrk(t∗))t . let y(u, 0; y0) denote the solution of system (2) at time u with initial value y0. the feasible phase space is the nonnegative cone c+ = c([−τ, 0], r3n+ ) of the banach space of continuous functions from [−τ, 0] to r3n with the sup norm. for every j,k ∈ {1, . . .n}, let hk,j : r3n � r3 be defined by hk,j = (hs,k,j, hi,k,j, hr,k,j)t , where hs,k,j(v) = µk,jv3k−2, hi,k,j(v) = µk,jv3k−1, hr,k,j(v) = µk,jv3k. for φ ∈ c+, we use the notation ŷφ(−τk,j )(u) = y(u, 0; hk,j(φ(−τk,j))). furthermore we define wk : c+ � r3n as wk(φ) = ( ŷφ(−τk,1)(τk,1), . . . ŷφ(−τk,n)(τk,n) )t . let x(t) = (s1(t), i1(t), r1(t), . . .sn(t),in(t),rn(t))t for t ≥ 0, and f = (fs,1,fi,1,fr,1, . . .fs,n,fi,n,fr,n)t , where for j ∈ {1, . . .n}, fs,j(x) = −λj(x3j−2,x3j−1,x3j) − ( n∑ k=1 µj,k ) x3j−2, fi,j(x) = λj(x3j−2,x3j−1,x3j) − αjx3j−1 − ( n∑ k=1 µj,k ) x3j−1, fr,j(x) = αjx3j−1 − ( n∑ k=1 µj,k ) x3j. clearly the union of systems (lj) with initial conditions (iv lj), j ∈ {1, . . .n} can be written in a closed form as   x′(t) = f(x(t)) + n∑ k=1 wk(xt) =: f(xt), x0 = ϕ, (3) where f : c+ � r3n and for ϕ ∈ c+, ϕ := (ϕs,1,ϕi,1,ϕr,1, . . . ,ϕs,n,ϕi,n,ϕr,n)t . iv. the local lipschitz property this section is devoted to the proof of the general existence and uniqueness result of system (3). first, we obtain the following simple result. proposition iv.1. assume that λk,j possesses the local lipschitz property on each bounded subset of r3. moreover, assume that λk,j(q1,q2,q3) ≥ 0 and λk,j(0,q2,q3) = 0 hold for q1,q2,q3 ≥ 0. then there exists a unique solution of system (2) which continuously depends on the initial data, and for u ∈ [0,τ] and y0 ≥ 0 the following inequality holds componentwise: 0 ≤ y(u, 0; y0) ≤ √ 3 |y0|. proof: the local lipschitz condition guarantees the existence of a unique solution which continuously depends on the initial data (see picard-lindelöf theorem in chapter ii, theorem 1.1 and chapter v, theorem 2.1 in [1]). we have also seen that nk,j(u; t∗) is constant for all u in the maximal interval of existence, moreover from the nonnegativity condition of λk,j it follows that biomath 1 (2012), 1209255, http://dx.doi.org/10.11145/j.biomath.2012.09.255 page 3 of 7 http://dx.doi.org/10.11145/j.biomath.2012.09.255 d. knipl et al., multiregional sir model with infection during transportation nonnegative initial data give rise to nonnegative solution. hence we obtain 0 ≤ nk,j(0; t∗) = nk,j(u; t∗), 0 ≤ sk,j(0; t∗) + ik,j(0; t∗) + rk,j(0; t∗) = sk,j(u; t∗) + ik,j(u; t∗) + rk,j(u; t∗) = µk,j(sk(t∗) + ik(t∗) + rk(t∗)), 0 ≤ sk,j(u; t∗), ik,j(u; t∗),rk,j(u; t∗) ≤ µk,j(sk(t∗) + ik(t∗) + rk(t∗)), (4) where we used (iv tk,j). using the definition of y, (4) implies that the inequality 0 ≤ ( y(u, 0; y0) ) 1 , ( y(u, 0; y0) ) 2 , ( y(u, 0; y0) ) 3 ≤ (y0)1 + (y0)2 + (y0)3 ≤ √ 3 √ ((y0)1) 2 + ((y0)2) 2 + ((y0)3) 2 holds for u ∈ [0,∞), where y0 = ( (y0)1, (y0)2, (y0)3 )t is the initial value and we used the arithmetic-quadratic mean inequality. we conclude that the solution exists on [0,τ] and is bounded. now we prove that if we assume that λj and λk,j possess the local lipschitz property, then f is also locally lipschitz continuous. lemma iv.2. let us suppose that for all j,k ∈ {1, . . .n}, λj and λk,j possess the local lipschitz property, λk,j(q1,q2,q3) ≥ 0 and λk,j(0,q2,q3) = 0 hold for q1,q2,q3 ≥ 0. then f satisfies the local lipschitz condition on each bounded subset of c+. proof: we claim that for every m > 0 there exists a constant k = k(m) such that the inequality |f(φ) −f(ψ)| ≤ k||φ−ψ|| holds for every φ,ψ ∈ c+ with ||φ||, ||ψ|| ≤ m. fix indices j,k ∈ {1, . . .n}. for ||ψ|| ≤ m it holds componentwise that 0 ≤ ψ(−τk,j) ≤ m, so due to the continuity of hk,j, there exists a constant l h k,j(m) such that 0 ≤ hk,j(ψ(−τk,j)) ≤ lhk,j is satisfied componentwise. for y0 = hk,j(ψ(−τk,j)) proposition iv.1 implies that there exists a jk,j = jk,j(lhk,j) = jk,j(m) such that the inequality |ŷψ(−τk,j )(u)| ≤ jk,j holds for u ∈ [0,τ] (for instance we can choose jk,j = √ 3lhk,j). the local lipschitz property of hk,j follows from its definition. we assumed that λk,j is lipschitz continuous, this implies the lipschitz continuity of g. let khk,j = k h k,j(m) be the lipschitz constant of hk,j on the set {v ∈ r3n : |v| ≤ m}, we denote the lipschitz constant of g = gk,j on the set {v ∈ r3 : |v| ≤ jk,j} by k g k,j = k g k,j(j) = k g k,j(m). for any ||φ||, ||ψ|| ≤ m, it holds that |φ(−τk,j)|, |ψ(−τk,j)| ≤ m. since solutions of (2) can be expressed as y(u, 0; y0) = y0 +∫u 0 g(y(r, 0; y0)) dr, we have∣∣ŷφ(−τk,j )(u) − ŷψ(−τk,j )(u)∣∣ = ∣∣∣∣hk,j(φ(−τk,j)) + ∫ u 0 g(ŷφ(−τk,j )(r)) dr − ( hk,j(ψ(−τk,j)) + ∫ u 0 g(ŷψ(−τk,j )(r)) dr )∣∣∣∣ ≤ |hk,j(φ(−τk,j)) − hk,j(ψ(−τk,j))| + ∫ u 0 ∣∣g(ŷφ(−τk,j )(r)) − g(ŷψ(−τk,j )(r))∣∣ dr ≤ khk,j||φ − ψ|| + ∫ u 0 k g k,j ∣∣ŷφ(−τk,j )(r) − ŷψ(−τk,j )(r)∣∣ dr (5) for u ∈ [0,τ]. define γ(u) = ∣∣ŷφ(−τk,j )(u) − ŷψ(−τk,j )(u)∣∣ for u ∈ [0,τ]. then (5) gives γ(u) ≤ khk,j||φ − ψ|| + k g k,j ∫ u 0 γ(r) dr, and from gronwall’s inequality we have γ(u) ≤ khk,j||φ − ψ||e k g k,ju. (6) applying the definition of wk, we arrive to the inequality |(wk(φ))j − (wk(ψ))j| = ∣∣ŷφ(−τk,j )(τk,j) − ŷψ(−τk,j )(τk,j)∣∣ ≤ khk,je k g k,jτk,j||φ − ψ||, where we used (6) at u = τk,j. it is straightforward that wk has the lipschitz condition for any k ∈ {1, . . .n}, kwk = kwk (m) =√∑n j=1 ( khk,je k g k,jτk,j )2 is a suitable choice for the lipschitz constant. finally, the assumption that λj is lipschitz continuous for any j ∈ {1, . . .n} implies the lipschitz continuity of f, so let kf = kf (m) be the lipschitz constant of f on the set {v ∈ r3n : |v| ≤ m}. then for any ||φ||, ||ψ|| ≤ m, |φ(0)|, |ψ(0)|, |φ(−τ)|, |ψ(−τ)| ≤ m hold and thus |f(φ) − f(ψ)| ≤ |f(φ(0)) − f(ψ(0))| + n∑ k=1 |wk(φ) − wk(ψ)| ≤ kf||φ − ψ|| + n∑ k=1 kwk||φ − ψ||. biomath 1 (2012), 1209255, http://dx.doi.org/10.11145/j.biomath.2012.09.255 page 4 of 7 http://dx.doi.org/10.11145/j.biomath.2012.09.255 d. knipl et al., multiregional sir model with infection during transportation |λ1(p) − λ1(q)| = β1 ∣∣∣∣ q1q2q1 + q2 + q3 − p1p2p1 + p2 + p3 ∣∣∣∣ = β1 ∣∣∣∣ q1q2q1 + q2 + q3 − q1p2q1 + q2 + q3 + q1p2q1 + q2 + q3 − q1p2 q1 + p2 + q3 + q1p2 q1 + p2 + q3 − q1p2 q1 + p2 + p3 + q1p2 q1 + p2 + p3 − q1p2 p1 + p2 + p3 + q1p2 p1 + p2 + p3 − p1p2 p1 + p2 + p3 ∣∣∣∣ ≤ β1 (∣∣∣∣ q1q2q1 + q2 + q3 − q1p2q1 + q2 + q3 ∣∣∣∣ + ∣∣∣∣ q1p2q1 + q2 + q3 − q1p2q1 + p2 + q3 ∣∣∣∣ + ∣∣∣∣ q1p2q1 + p2 + q3 − q1p2q1 + p2 + p3 ∣∣∣∣ + ∣∣∣∣ q1p2q1 + p2 + p3 − q1p2p1 + p2 + p3 ∣∣∣∣ + ∣∣∣∣ q1p2p1 + p2 + p3 − p1p2p1 + p2 + p3 ∣∣∣∣ ) = β1 ( |q2 − p2| ∣∣∣∣ q1q1 + q2 + q3 ∣∣∣∣ + |p2 − q2| ∣∣∣∣ q1p2(q1 + q2 + q3)(q1 + p2 + q3) ∣∣∣∣ +|p3 − q3| ∣∣∣∣ q1p2(q1 + p2 + q3)(q1 + p2 + p3) ∣∣∣∣ +|p1 − q1| ∣∣∣∣ q1p2(q1 + p2 + p3)(p1 + p2 + p3) ∣∣∣∣ + |q1 − p1| ∣∣∣∣ p2p1 + p2 + p3 ∣∣∣∣ ) ≤ β1 (2|q2 − p2| + |p3 − q3| + 2|p1 − q1|) ≤ 5β1|q − p| (7) hence kf + ∑n k=1 √∑n j=1 ( khk,je k g k,jτk,j )2 is a suitable choice for k, the lipschitz constant of f for the set {ψ ∈ c+ : ||ψ|| ≤ m}. the assumptions of lemma iv.2 on the incidences λj(sj(t),ij(t),rj(t)) and λk,j(sk,j(u; t∗), ik,j(u; t∗),rk,j(u; t∗)) can be fulfilled by several choices on the type of disease transmission. for instance, let βj > 0 be the transmission rate in region j and let βtk,j > 0 denote the transmission rates during travel. for j,k ∈ {1, . . .n} and for q = (q1,q2,q3) ∈ r3, define λj(q) = −βj q1 q1 + q2 + q3 q2, λk,j(q) = −βtk,j q1 q1 + q2 + q3 q2. this implies that λj and λk,j have the form λj(sj,ij,rj) = −βj sj nj ij, λk,j(sk,j, ik,j,rk,j) = −βtk,j sk,j nk,j ik,j, (8) which is called standard incidence. now we prove the following existence-uniqueness theorem. theorem iv.3. with the incidences λj and λk,j defined in (8), there exists a unique solution of system (3). proof: recall theorem 3.7 from [2]: suppose that f satisfies the local lipschitz property on each bounded subset of c+ = c+([−τ, 0], r3n+ ), moreover let m > 0. there exists a > 0, depending only on m such that if φ ∈ c+ satisfies ||φ|| ≤ m, then there exists a unique solution x(t) = x̂(t, 0; φ) of (3), defined on [−τ,a]. in addition, if k is the lipschitz constant for f corresponding to m, then max −τ≤η≤a |x(η, 0; φ) − x(η, 0; ψ)| ≤ ||φ − ψ||eka holds for ||φ||, ||ψ|| ≤ m. we showed in lemma iv.2 that the local lipschitz continuity of f follows from the local lipschitz property of the incidences and the nonnegativity condition of λk,j. the latter condition clearly holds, hence it is sufficient to prove that the incidences defined in (8) possess the local lipschitz property. as one may observe, the definition of the λj-s and λk,jbiomath 1 (2012), 1209255, http://dx.doi.org/10.11145/j.biomath.2012.09.255 page 5 of 7 http://dx.doi.org/10.11145/j.biomath.2012.09.255 d. knipl et al., multiregional sir model with infection during transportation s (j,k ∈ {1, . . .n}) only differ in a constant multiplier, hence it is sufficent to prove the local lipschitz property only for one of them, i.e. for λ1. moreover, we prove this property only on the nonnegative cone r3+, which is invariant under systems (3) and (2). for p, q ∈ r3+, by (7) we obtain the lipschitz constant k = 5β1, where we used that for any a,b,c > 0, it holds that a a+b+c < 1. remark 1. it follows from the proof of theorem iv.3 that the incidences λj and λk,j defined in (8) also satisfy the global lipschitz property, meaning that the lipschitz constant k arises independently of m. in this case, the solution of system (3) exists on [0,∞). another natural choice for the incidences can be the following: for q = (q1,q2,q3) ∈ r3 and for j,k ∈ {1, . . .n}, let λj(q) = −βjq1q2, λk,j(q) = −βtk,jq1q2, which leads to the mass action-type disease transmission, therefore λj and λk,j have the form λj(sj,ij,rj) = −βjsjij, λk,j(sk,j, ik,j,rk,j) = −βtk,jsk,jik,j. (9) theorem iv.4. with incidences λj and λk,j defined in (9), there exists a unique solution of system (3). proof: similarly as by theorem iv.3, it is enough to show that λj and λk,j satisfy the local lipschitz property, we detail the proof only for λ1 and consider the nonnegative subspace r3+. for any m > 0, for any p, q ∈ r3+ such that |p|, |q| ≤ m, we obtain |λ1(p) − λ1(q)| = | − β1p1p2 + β1q1q2| ≤ β1|p1p2 − q1q2| ≤ β1|p1p2 − p1q2 + p1q2 − q1q2| ≤ β1(|p1p2 − p1q2| + |p1q2 − q1q2|) ≤ β1(p1|p2 − q2| + q2|p1 − q1|) ≤ 2mβ1|q − p|, so we can choose k(m) = 2mβ1. remark 2. although the global lipschitz property does not hold for λj and λk,j defined in (9), it is possible to show that the solution of (3) is bounded and hence exists on [0,∞). v. conclusion the topic of epidemic spread of infectious diseases via transportation networks has recently been examined in several studies (see [3], [4], [5], [6], [7]), although these works mostly consider only two connected regions. we introduced a dynamic model which describes the spread of an infectious disease in and between n regions which are connected by transportation. we used the commonly applied sir model as a basic epidemic building block in the regions and also during the travel. the model formulation led to a system structured by travel time, which turned out to be equivalent to a system of differential equations with multiple dynamically defined delays. we showed that under local lipschitz conditions on the infection terms within the regions and during travel, the usual existence and uniqueness results hold. recent epidemics like the 2002-2003 sars outbreak and the 2009 pandemic influenza a(h1n1) highlighted the importance of the global air travel network in the study of epidemic spread. during long distance travel such as intercontinental flights, a single infected individual may infect several other passengers during the flight, and since the progress of these diseases is fast, even a short delay (a fraction of a day) arising due to transportation may play a significant role in the disease dynamics. in this paper we illustrated by proving an existence and uniqueness result that such epidemiological situations can be studied in the framework of delay differential equations. acknowledgment dhk was partially supported by the hungarian research fund grant otka k75517 and the támop4.2.2/b-10/1-2010-0012 program of the hungarian national development agency. rg was supported by european research council stg nr. 259559, hungarian research fund grant otka k75517 and the bolyai research scholarship of hungarian academy of sciences. references [1] p. hartman, “ordinary differential equations”, classics in applied mathematics, vol. 38, siam, 2002. [2] h. l. smith, an introduction to delay differential equations with applications to the life sciences, springer, 2010. [3] d. h. knipl, “fundamental properties of differential equation with dynamically defined delayed feedback”, preprint, 2012. [4] d. h. knipl, g. röst, j. wu,“epidemic spread of infectious diseases on long distance travel networks”, preprint, 2012. [5] j. liu, j. wu, y. zhou, “modeling disease spread via transportrelated infection by a delay differential equation”, rocky mountain j. math., vol. 38 no 5, pp. 1225–1541, 2008. biomath 1 (2012), 1209255, http://dx.doi.org/10.11145/j.biomath.2012.09.255 page 6 of 7 http://dx.doi.org/10.11145/j.biomath.2012.09.255 d. knipl et al., multiregional sir model with infection during transportation [6] y. nakata, “on the global stability of a delayed epidemic model with transport-related infection”, nonlinear analysis series b: real world applications, vol. 12 no 6, pp. 3028–3034 2011. http://dx.doi.org/10.1016/j.nonrwa.2011.05.004 [7] y. nakata, g. röst, “global analysis for spread of an infectious disease via global transportation”, submitted, 2012. biomath 1 (2012), 1209255, http://dx.doi.org/10.11145/j.biomath.2012.09.255 page 7 of 7 http://dx.doi.org/10.1016/j.nonrwa.2011.05.004 http://dx.doi.org/10.11145/j.biomath.2012.09.255 introduction model description the compact form of the system the local lipschitz property conclusion references www.biomathforum.org/biomath/index.php/biomath original article inverse problem of the holling-tanner model and its solution adejimi adesola adeniji, igor fedotov, michael y. shatalov department of mathematics and statistics, tshwane university of technology adejimi.adeniji@gmail.com, fedoptovi@tut.ac.za, shatalovm@tut.ac.za received: 8 august 2018, accepted: 5 december 2018, published: 19 december 2018 abstract—in this paper we undertake to consider the inverse problem of parameter identification of nonlinear system of ordinary differential equations for a specific case of complete information about solution of the holling-tanner model for finite number of points for the finite time interval. in this model the equations are nonlinearly dependent on the unknown parameters. by means of the proposed transformation the obtained equations become linearly dependent on new parameters functionally dependent on the original ones. this simplification is achieved by the fact that the new set of parameters becomes dependent and the corresponding constraint between the parameters is nonlinear. if the conventional approach based on introduction of the lagrange multiplier is used this circumstance will result in a nonlinear system of equations. a novel algorithm of the problem solution is proposed in which only one nonlinear equation instead of the system of six nonlinear equations has to be solved. differentiation and integration methods of the problem solution are implemented and it is shown that the integration method produces more accurate results and uses less number of points on the given time interval. keywords-parameter estimation, goal function, absolute error curves, inverse method, hollingtanner model, least square method, differentiation method, integration method i. introduction the numerical evaluation of known coefficient of a dynamical system i.e. the problem of dynamical system identification, is one of the most important problem of the mathematical biology [1], ecology [2], [3], [4], etc. usually, to identify a dynamics of a system, it is necessary to have certain statistical information for time values about the unknown functions of this system. in the present paper we consider the inverse problem of parameter identification of the holling-tanner predator-prey model [5], [6]. this model is widely used in mathematical biology, for example, in the study of transmissible disease [7]. several investigations have been done by various researchers on the mite-spider-mite, lynx-hare and sparrow-hawksparrow competition [8], [9], [10]. in [11], the authors proposed a method consisting in the direct integration of a given dynamical system with the subsequent application of quadrature rules and the least square method [12], [13] provided that there copyright: c© 2018 adeniji et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: adejimi adesola adeniji, igor fedotov, michael y. shatalov, inverse problem of the holling-tanner model and its solution, biomath 7 (2018), 1812057, http://dx.doi.org/10.11145/j.biomath.2018.12.057 page 1 of 9 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2018.12.057 a.a. adeniji, i. fedotov, m.y. shatalov, inverse problem of the holling-tanner model and its solution is complete statistical information about the unknown function. in this paper, we assume that the complete information about the competing species is available and the two methods of solution, differentiation and integration methods, are proposed. the problem of the holling-tanner model identification has its specifics, because it nonlinearly depends on the unknown parameters. it is possible to transform this model to a new form where the equations of the system linearly depends on the set of new parameters. these new parameters are not independent and we need to consider the constraint between the parameters, which are nonlinear. the holling-tanner model has only one constraint and hence, can be simply treated by a novel method developed by the authors. the theoretical considerations are accompanied by numerical examples where the developed algorithm is tested for both differentiation and integration methods of solution. it is shown that the integration methods is more accurate than the differentiation one and needs less amount of experimental information. ii. main results in our paper we consider the holling-tanner model [9], [14] described by the following system of equations:   ẋ = b1x− b2x2 − b3 x·yb4+x, ẏ = b5y − b6 y 2 x , t = 0, x(0) = x0, y(0) = y0 (1) where x = x(t), y = y(t), ẋ = dx(t) dt , ẏ = dy(t) dt , t is time and b1, · · ·b6 are positive constant parameters [15]. initial conditions for this system are formulated so that at t = 0 : x(t = 0) = x0 > 0 and y(t = 0) = y0 > 0. the main results relating to solution of this initial value problem were obtained in [10], [16], [17], [18] as • solution of the initial value problem (1) {x(t),y(t)} with positive initial conditions is positive, i.e. x(t) > 0 and y(t) > 0 for t ≥ 0. • initial value problem (1) has the positive steady-state solution [15] (x̃, ỹ) which corresponds to either stable focus or stable node critical point depending on b1, · · ·b6 so that: x̃= b1b6−b3b5−b2b4b6 + √ ∆ 2b2b6 > 0, (2) ỹ = b5(b1b6−b3b5−b2b4b6 + √ ∆) 2b2b 2 6 > 0. where ∆ = (b1b6−b3b5− b2b4b6)2 +4b1b2b4b26. • initial value problem (1) has unstable steadystate solution (˜̃x, ˜̃y) = ( b1 b2 , 0), which corresponds to the saddle critical point. iii. on solvability of identification problem assume that solution of initial problem (1), x(t) and y(t) is given on the finite time interval t ∈ [0,t] with initial t = 0 and terminal t = t time instants in n + 1 equispaced time instants ti = t n i ∈ [0,t]: xi = x(ti), yi = y(ti) (i = 0, · · · ,n) (3) lets us formulate the identification problem for parameters b1, · · ·b6 from the known solution (3) this problem can be solved if the conditions of the following theorem are satisfied: theorem 1. parameters b1,b2,b3,b4 of model (1) can be identified by the least squares method if (n + 1) × 1-vector columns [xi] , [ x2i ] , [ x3i ] , [ẋi] , [xi,yi] are linearly independent. parameters b5 and b6 of the above mentioned model can be identified by the mentioned method if (n + 1) × 1-vector columns [yi] and [ y2i x1 ] are linearly independent. proof: by multiplying the first equation of system (1) by (b4 + x) and grouping the resulting terms we obtain c1(−x3(t)) + c2(−x(t)y(t)) + c3(−ẋ(t)) +c4(x(t)) + c5(x 2(t)) + (−x(t)ẋ(t)) = 0, (4) biomath 7 (2018), 1812057, http://dx.doi.org/10.11145/j.biomath.2018.12.057 page 2 of 9 http://dx.doi.org/10.11145/j.biomath.2018.12.057 a.a. adeniji, i. fedotov, m.y. shatalov, inverse problem of the holling-tanner model and its solution where c1 = b2, c2 = b3, c3 = b4, c4 = b1b4, c5 = b1−b2b4 are new unknown parameters. it is easy to check that the parameters c1,c3,c4,c5 satisfy the following constrains: c1c23 + c3c5 − c4. considering x(t) and y(t) in time instants t = ti we obtain the following overdetermined system of n + 1 linear algebraic equations: c1 ~f1 +c2 ~f2 +c3 ~f3 +c4 ~f4 +c5 ~f5−~f6 = 0, (5) where ~f1 = [f1i] = [ −x3i ] , ~f2 = [f2i] = [−xiyi] , ~f3 = [f3i] = [−ẋi] , ~f4 = [f4i] = [xi] , ~f5 = [f5i] = [ x2i ] , and ~f6 = [f6i] = [xiẋi] are (n + 1) × 1-vector columns. hence, the unknown parameters c1, c2, c3, c4 and c5 can be found by, for example, the least squares method [19] by means of the constrained minimization of function g1: g1 = g1(c1,c2,c3,c4,c5,λ) = = 1 2 (c1 ~f1+c2 ~f2+c3 ~f3+c4 ~f4+c5 ~f5−~f6)t (c1 ~f1+c2 ~f2+c3 ~f3+c4 ~f4+c5 ~f5−~f6) + λ(c1c 2 3 + c3c5 −c4) −→ min (6) this problem can be solved providing that vectors ~f1, · · · , ~f5 are linearly independent in (6). the last term contains the lagrange multiplier λ and the constraint between coefficients c1, · · · ,c5. moreover, the second equation of system (1) can be rewritten in time instants t = ti as the following overdetermined system of n + 1 linear algebraic equations: c6 ~f7 + c7 ~f8 − ~f9 = 0, (7) where ~f7 = [f7i] = [yi] , ~f8 = [f8i] = [ −y2i xi ] , ~f9 = [f9i] = [ẏi] ,c6 = b5,c7 = b6. that is why coefficients c6,c7 can be found by application of the least square method by means of minimization of function g2 g2 =g2(c6,c7) = 1 2 ( c6 ~f7 + c7 ~f8 + ~f9 )t (c6 ~f7 + c7 ~f8 + ~f9) −→ min (8) this problem can be solved providing that vectors ~f7 and ~f8 are linearly independent of (8). remark 2. in vectors ~f3, ~f6 the component ẋi, and in vector ~f9 the components ẏi are calculated by means of numerical differentiation of xi,yi with respect to time t and that is why the proposed method is called the differential method of identification. corollary 3. parameters b1,b2,b3,b4 of the model (1) can be identified by the least square method [19] if (n + 1) × 1-vector columns[∫ ti 0 x(τ)dτ ] , [∫ ti 0 x2(τ)dτ ] , [∫ ti 0 x3(τ)dτ ] , [xi −x0] , [∫ ti 0 x(τ)y(τ)dτ ] are linearly dependent. parameters b5 and b6 of the abovementioned model can be identified by the abovementioned method if (n + 1)×1-vector columns [∫ ti 0 y(τ)dτ ] and [∫ ti 0 y2(τ) x(τ) dτ ] are linearly dependent. proof: integrating expression (4) with respect to time t ∈ [0,t] we obtain c1 ( − ∫ t 0 x3(τ)dτ ) + c2 ( − ∫ t 0 x(τ)y(τ)dτ ) + c3 (x0 −x(t)) + c4 (∫ t 0 x(τ)dτ ) + c5 (∫ t 0 x2(τ)dτ ) − ( 1 2 (x2(t) −x20) ) = 0. (9) integrating second equation of system 5 with respect to time t ∈ [0,t] we have c6 (∫ t 0 y(τ)dτ ) + c7 ( − ∫ t 0 y2(τ) x(τ) dτ ) −c3 (y(t) −y0) = 0. (10) biomath 7 (2018), 1812057, http://dx.doi.org/10.11145/j.biomath.2018.12.057 page 3 of 9 http://dx.doi.org/10.11145/j.biomath.2018.12.057 a.a. adeniji, i. fedotov, m.y. shatalov, inverse problem of the holling-tanner model and its solution performing all integrations in (9) and (10) from 0 to tj ∈ [0,t] we obtain the following overdetermined systems of n + 1 linear algebraic equations c1 ~g1 + c2 ~g2 + c3 ~g3 + c4 ~g4 + c5 ~g5 − ~g6 = 0, c6 ~g7 + c7 ~g8 − ~g9 = 0, (11) where ~g1 = [ − ∫ ti 0 x3(τ)dτ ] , ~g2 = [ − ∫ ti 0 x(τ)y(τ)dτ ] , ~g3 = [x0 −xi] , ~g4 = [∫ ti 0 x(τ)dτ ] , ~g5 = [∫ ti 0 x2(τ)dτ ] , ~g6 = [ 1 2 (x2i −x 2 0) ] , ~g7 = [∫ ti 0 y(τ)dτ ] , ~g8 = [ − ∫ ti 0 y2(τ) x(τ) dτ ] , ~g9 = [yi −y0] are the (n + 1)×1-vector columns. now applying the method used in theorem 1 we prove the corollary. remark 4. in vector ~g1, ~g2, ~g4, ~g5, ~g7, ~g8 the integrals are calculated by means of numerical integration of xi,yi and their combinations with respect to time t and that is why the proposed method is called the integration method of identification. remark 5. note that expressions (5), (7) and (11) are linear with respect to unknown constants c1, · · · ,c7. direct use of the constraint minimization using the lagrange multiplier with constraint: c1c 2 3 + c3c5 −c4 = 0 (12) produces nonlinear system of equations for determination of six unknowns c1,c2,c3,c4,c5,λ. thus the search is performed in six-dimensional space of parameters and hence this method substantially complexifies the solution procedure. determination of parameters and c6 and c7 needs solution of linear system of two algebraic equations. in the next section we describe an original problem solution algorithm reducing the search space dimension to one and using only linear matrix manipulations in the process of solution, which substantially simplifies and accelerates the problem solution. iv. solution of the parameter identification problem there are four original independent parameters (b1,b2,b3,b4) in the first equation of (1). first four cparameters (c1,c2,c3,c4) depend on bparameters so that there is one-to-one correspondence between them. the parameter c5 depends on the first four c-parameter as follows: c5 = c4 c3 −c1c23. (13) hence, it is possible to consider (c1,c2,c3,c4) as independent parameters and introduce new name for the dependent parameter c5 = −λ. the novel algorithm will be considered in detail for the differentiation method of solution, i.e. with ~f1,··· ,9 vector columns(see expression (5) and (7). the integration method of solution uses the same algorithm in which ~f1,··· ,9 vector columns are changed to ~g1,··· ,9 -ones (see (11)). parameter λ will be selected from the given interval λ ∈ [λmin,λmax] and substituted in goal function g3 which is composed as follows g3 = g3(c1,c2,c3,c4,λ) = 1 2 ( c1 ~f1+c2 ~f2+c3 ~f3+c4 ~f4−(λ~f5+ ~f6) )t ( c1 ~f1+c2 ~f2+c3 ~f3+c4 ~f4−(λ~f5+ ~f6) ) (14) and subjected to minimization. in expression (14), parameter λ is considered as constant at every minimization and minimization itself is performed with respect to parameters c1,c2,c3,c4. solution of this problem is given by the following formula c(λ) = [c1(λ), c2(λ), c3(λ), c4(λ)] t = ( (lt1 l1) −1lt1 ) r(λ), (15) where l1 = [ ~f1 ~f2 ~f3 ~f4 ]t , r(λ) = λ~f5 + ~f6. (16) in expression (15) it is possible to calculate 1 × (n +1)vector row ( (lt1 l1) −1lt1 ) only once and biomath 7 (2018), 1812057, http://dx.doi.org/10.11145/j.biomath.2018.12.057 page 4 of 9 http://dx.doi.org/10.11145/j.biomath.2018.12.057 a.a. adeniji, i. fedotov, m.y. shatalov, inverse problem of the holling-tanner model and its solution after that perform its multiplication by (n +1)×1vector row r(λ), which is very fast operation. components of vector c(λ) and c5 = −λ are substituted in the constraint (12) to obtain the following nonlinear scalar equation c1(λ)c 2 3 (λ) −λc3(λ) −c4(λ) = 0, (17) which is solved with respect to λ. all roots of equation (17) are found (sometimes to find all the roots it is necessary to expand the interval λ ∈ [λmin,λmax] to the left or to the right or to both sides). after finding a particular root λ the corresponding b-parameters are calculated as follows: b1(λ) = c4(λ) c3(λ) , b2(λ) = c1(λ), b3(λ) = c2(λ), b4(λ) = c3(λ). (18) (see (4)). the estimations of b-parameters are obtained from the proper selection of root λ = λ̄: b̄1 = b1(λ̄), b̄2 = b2(λ̄), b̄3 = b3(λ̄), b̄4 = b4(λ̄) (19) (one of the criteria of the correct choice of λ̄ must be positiveness of all estimated b̄ parameters, see numerical examples). parameters b5 and b6 are estimated by means of minimization of the goal function of equation 8. solution of this problem is given by the formulas:[ b̄5 b̄6 ] = (lt2 l2) −1lt2 ~f9 (20) where l2 = [ ~f7 ~f8 ] is (n + 1)×2matrix, (see (7)). expression (15)-(20) give solution to the identification problem by means of the differentiation method. to find solution of the problem by the integral method it is necessary to consider vectors ~g1,··· ,9 (see expression 11) instead of ~f1,··· ,9. in the next section you will find more information about application of the differentiation and integration methods. v. numerical examples let us solve the initial problem of equation 1 with the following parameters: b1 = 0.2 b2 = 0.01 b3 = 0.05 b4 = 1 b5 = 0.062 b6 = 0.0223 (21) and initial conditions: [x0 y0] t = [10 5] t . the stable critical point has coordinates (x̃, ỹ) ≈ (7.77064, 21.4066) (see equation 2) and it is the stable focus (eigenvalues of the linearized system in the vicinity of the critical point are ν1,2 ≈ −0.0138 ± 0.0735i, where i2=−1). the unstable saddle has coordinates (˜̃x, ˜̃y)=(20,0). numerical solution x = x(t) on the time interval t ∈ [0,t = 150] in n + 1 = 25 points is shown in figure 1 and solution y = y(t) is shown in figure 2. performing solution by means of the differential fig. 1. graph of solution x = x(t) fig. 2. graph of solution y = y(t) method in accordance with the described algorithm we obtain nonlinear equation (12) from which the parameters are calculated: λ1 ≈ −0.3282, λ2 ≈ −0.1091 and λ3 ≈ 0.2087. as we see, only λ3 parameter can be selected from three biomath 7 (2018), 1812057, http://dx.doi.org/10.11145/j.biomath.2018.12.057 page 5 of 9 http://dx.doi.org/10.11145/j.biomath.2018.12.057 a.a. adeniji, i. fedotov, m.y. shatalov, inverse problem of the holling-tanner model and its solution table i values of b-parameters, corresponding to different roots of equation (12) for n + 1 = 25 (differentiation method) original values λ1 ≈ −0.3282 λ2 ≈ −0.1091 λ3 ≈ 0.2087 b1 = 0.2000 −0.0580 −0.0621 0.2129 b2 = 0.0100 −0.0150 −0.0045 0.0108 b3 = 0.0500 −0.0295 −0.0026 0.0491 b4 = 1.0000 −18.0380 −10.5185 0.3906 roots, because λ1 and λ2 generate the negative values of b-parameters. the relative error of the b-parameters corresponding to λ3-parameter are as follows: error%(b1) ≈ 6.437% error%(b2) ≈ 7.725% error%(b3) ≈ 1.832% error%(b4) ≈ 60.943% (22) estimation of parameters b5 and b6 gives coincidence with the original values of the parameters in four decimals with the following relative errors: error%(b5) ≈ 0.029% error%(b6) ≈ 0.028% (23) comparison of original graphs with graphs obtained by numberical solution of initial problem (1) with the same initial conditions but with estimated parameters is shown in figure 3 and figure 4. as we see the estimated parameters gives quite good estimation of the process dynamics. the estimated values of the steady states are as follows (˜̃x, ˜̃y) ≈ (7.7143, 21.4282) with relative errors: error%(x̃) ≈ 0.102% error%(ỹ) ≈ 0.101% (24) estimation of the parameters with n + 1 = 49 points gives λ1 ≈ −0.2585, λ2 ≈ −0.0878, λ3 ≈ 0.1914 and the following values of parameters (see table 2) fig. 3. graph of original solution x = x(t) (dots) and solution with estimated parameters (solid line) fig. 4. graph of original solution y = y(t) (dots) and solution with estimated parameters (solid line) as we see, only λ3 parameter can be selected from the three roots, because λ1 and λ2 generate the negative values of b-parameters. one can see the substantial improvement of the parameters estimations. the relative errors of the b-parameters biomath 7 (2018), 1812057, http://dx.doi.org/10.11145/j.biomath.2018.12.057 page 6 of 9 http://dx.doi.org/10.11145/j.biomath.2018.12.057 a.a. adeniji, i. fedotov, m.y. shatalov, inverse problem of the holling-tanner model and its solution table ii values of b-parameters, corresponding to different roots of equation (12) for n + 1 = 49 (differentiation method) original values λ1 ≈ −0.2585 λ2 ≈ −0.0878 λ3 ≈ 0.1914 b1 = 0.2000 −0.0405 −0.0485 0.2010 b2 = 0.0100 −0.0119 −0.0036 0.0101 b3 = 0.0500 −0.0271 −0.0022 0.0500 b4 = 1.0000 −18.3099 −10.9975 0.9553 corresponding to λ3parameter are as follows: error%(b1) ≈ 0.496% error%(b2) ≈ 0.583% error%(b3) ≈ 0.087% error%(b4) ≈ 4.469% (25) estimation of parameters b5 and b6 gives coincidence with the original ones in four decimals with the following relative errors: error%(b5) ≈ 0.002% error%(b6) ≈ 0.002% (26) comparison of original graphs with graphs obtained by numerical solution of initial problem (1) with the same initial conditions but with estimated parameters is shown in figure 5 and figure 6. fig. 5. graph of original solution x = x(t) (dots) and solution with estimated parameters (solid line) as we see the estimated parameters give very good estimation of the process dynamics. the fig. 6. graph of original solution y = y(t) (dots) and solution with estimated parameters (solid line) estimated values of the steady states are as follows (˜̃x, ˜̃y) ≈ (7.7077, 21.4102) with relative errors: error%(x̃) ≈ 0.017% error%(ỹ) ≈ 0.017% (27) absolute errors in calculation of x = x(t) and y = y(t) in the differentiation method for n + 1 = 25 and n + 1 = 49 points are shown in figure 7 and figure 8. performing solution by means of the integration method in accordance with the described algorithm we obtain three roots of nonlinear equation (12): λ1 ≈ −0.2391, λ2 ≈ −0.0725, λ3 ≈ 0.1899. as we see, only λ3 parameter can be selected from the three roots, because λ1 and λ2 generate the negative values of b-parameters. the relative errors of the b-parameters corresponding to λ3biomath 7 (2018), 1812057, http://dx.doi.org/10.11145/j.biomath.2018.12.057 page 7 of 9 http://dx.doi.org/10.11145/j.biomath.2018.12.057 a.a. adeniji, i. fedotov, m.y. shatalov, inverse problem of the holling-tanner model and its solution table iii values of b-parameters, corresponding different roots of equation (12) for n + 1 = 25 (integration method) original values λ1 ≈ −0.2391 λ2 ≈ −0.0725 λ3 ≈ 0.1899 b1 = 0.2000 −0.0302 −0.0386 0.1999 b2 = 0.0100 −0.0115 −0.0032 0.0100 b3 = 0.0500 −0.0282 −0.0022 0.0500 b4 = 1.0000 −18.1074 −10.6869 0.9997 fig. 7. absolute errors of calculation for n +1 = 25 points (differentiation method) fig. 8. absolute errors of calculation for n +1 = 49 points (differentiation method) parameter are as follows: error%(b1) ≈ 0.052% error%(b2) ≈ 0.044% error%(b3) ≈ 0.059% error%(b4) ≈ 0.033% (28) fig. 9. absolute errors of calculation for n +1 = 25 points (integration method) . estimation of parameters b5 and b6 gives coincidence with the original values of b-parameter in four decimals with the following relative errors: error%(b5) ≈ 0.008% error%(b6) ≈ 0.007% (29) comparison of original graphs with graphs obtained by numerical solution of initial problem (1) with the same initial conditions but with estimated parameters are visually indistinguishable from figure 5 and figure 6. absolute errors in calculation of x = x(t) and y = y(t) in the integration method for n + 1 = 25 points are shown in figure 9. the parameters are estimated with very high accuracy at n + 1 = 25 points. the estimated values of the steady states are as follows (˜̃x, ˜̃y) ≈ (7.7062, 21.4061) with relative errors: error%(x̃) ≈ 0.002% error%(ỹ) ≈ 0.002% (30) biomath 7 (2018), 1812057, http://dx.doi.org/10.11145/j.biomath.2018.12.057 page 8 of 9 http://dx.doi.org/10.11145/j.biomath.2018.12.057 a.a. adeniji, i. fedotov, m.y. shatalov, inverse problem of the holling-tanner model and its solution vi. conclusion two methods of solution of the inverse problem on parameter identification of the holling-tanner model with complete information are discussed. these are the differentiation and integration methods of solution. the conditions are indicated at which all parameters of the model can be identified. the main disadvantage of the conventional method of constraint minimization by means of the lagrange multipliers is that the method generates a system of six nonlinear equations with unknown initial guess values. proposed is the novel method of the problem solution in which the six dimensional space of search is reduced to one dimensional space and the procedure of the initial guess value is performed by fast vector multiplication. numerical examples of the proposed algorithm implementation are demonstrated for the differentiation and integration methods. it is shown that the integration method generates more accurate results than the differentiation one. the integration method also needs less number of points on the fixed time interval to produce accurate results than the differentiation method. vii. acknowledgment the authors acknowledge the department of mathematics and statistics of the tshwane university of technology towards the research. references [1] m yu shatalov, as demidov, and ia fedotov. estimating the parameters of chemical kinetics equations from the partial information about their solution. theoretical foundations of chemical engineering, 50(2):148–157, 2016. [2] dmitrii logofet. matrices and graphs stability problems in mathematical ecology: 0. crc press, 2018. [3] john pastor. mathematical ecology of populations and ecosystems. john wiley & sons, 2011. [4] richard mcgehee and robert a armstrong. some mathematical problems concerning the ecological principle of competitive exclusion. journal of differential equations, 23(1):30–52, 1977. [5] gilles clermont and sven zenker. the inverse problem in mathematical biology. mathematical biosciences, 260:11–15, 2015. [6] andreas kirsch. an introduction to the mathematical theory of inverse problems, volume 120. springer science & business media, 2011. [7] mainul haque and ezio venturino. the role of transmissible diseases in the holling–tanner predator–prey model. theoretical population biology, 70(3):273–288, 2006. [8] james t tanner. the stability and the intrinsic growth rates of prey and predator populations. ecology, 56(4):855–867, 1975. [9] david j wollkind, john b collings, and jesse a logan. metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit trees. bulletin of mathematical biology, 50(4):379–409, 1988. [10] peter a braza. the bifurcation structure of the holling–tanner model for predator-prey interactions using two-timing. siam journal on applied mathematics, 63(3):889–904, 2003. [11] m shatalov and i fedotov. on identification of dynamic system parameters from experimental data. 2007. [12] michael shatalov, igor fedotov, and stephan v joubert. novel method of interpolation and extrapolation of functions by a linear initial value problem. in buffelspoort time 2008 peer-reviewed conference proceedings. buffelspoort time 2008 peer-reviewed conference proceedings, 2008. [13] joubert s.v shatalov m, greeff j.c and fedotov i. parametric identification of the model with one predator and two prey species. buffelspoort time2008 peerreviewed conference proceedings,, 2008. [14] rashi gupta. dynamics of a holling–tanner model. american journal of engineering research (ajer), page 2, 2017. [15] rui peng and mingxin wang. positive steady states of the holling–tanner prey–predator model with diffusion. proceedings of the royal society of edinburgh section a: mathematics, 135(1):149–164, 2005. [16] guirong liu, sanhu wang, and jurang yan. positive periodic solutions for neutral delay ratio-dependent predator-prey model with holling-tanner functional response. international journal of mathematics and mathematical sciences, 2011, 2011. [17] sze-bi hsu and tzy-wei huang. global stability for a class of predator-prey systems. siam journal on applied mathematics, 55(3):763–783, 1995. [18] shanshan chen and junping shi. global stability in a diffusive holling–tanner predator–prey model. applied mathematics letters, 25(3):614–618, 2012. [19] charles l lawson and richard j hanson. solving least squares problems, volume 15. siam, 1995. biomath 7 (2018), 1812057, http://dx.doi.org/10.11145/j.biomath.2018.12.057 page 9 of 9 http://dx.doi.org/10.11145/j.biomath.2018.12.057 introduction main results on solvability of identification problem solution of the parameter identification problem numerical examples conclusion acknowledgment references www.biomathforum.org/biomath/index.php/biomath original article a new class of activation functions. some related problems and applications nikolay kyurkchiev1,2 1faculty of mathematics and informatics university of plovdiv paisii hilendarski 24 tzar asen str., 4000 plovdiv, bulgaria nkyurk@uni-plovdiv.bg 2institute of mathematics and informatics bulgarian academy of sciences acad. g. bonchev str., bl. 8, 1113 sofia, bulgaria received: 21 january 2020, accepted: 3 may 2020, published: 17 may 2020 abstract—the cumulative distribution function (cdf) of the discrete two–parameter bathtub hazard distribution has important role in the fields of population dynamics, reliability analysis and life testing experiments. also of interest to the specialists is the task of approximating the heaviside function by new (cdf) in hausdorff sense. we define new activation function and family of new recurrence generated functions and study the ”saturation” by these families. in this paper we analyze some intrinsic properties of the new topp–leone–g–family with baseline ”deterministic–type” (cdf) – (ntlg– dt). some numerical examples with real data from biostatistics, population dynamics and signal theory, illustrating our results are given. it is shown that the study of the two characteristics ”confidential curves” and ”super saturation” is a must when choosing the right model. some related problems are discussed, as an example to the approximation theory. keywords-two–parameter bathtub hazard distribution; ”saturation” by: new activation function and family of new recurrence generated functions; topp–leone–g–family with baseline ”deterministic–type” (cdf) – (ntlg–dt); heaviside function; hausdorff distance; upper and lower bounds i. introduction and preliminaries definition 1. define the following deterministic (cdf) based on two–parameter bathtub hazard distribution [2]: mβ(t) = 1 −qe tβ−1, (1) where 0 < q < 1; β > 0, t > 0. definition 2. the shifted heaviside step function copyright: c© 2020 kyurkchiev. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: nikolay kyurkchiev, a new class of activation functions. some related problems and applications, biomath 9 (2020), 2005033, http://dx.doi.org/10.11145/j.biomath.2020.05.033 page 1 of 10 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2020.05.033 nikolay kyurkchiev, a new class of activation functions. some related problems and applications is defined by ht0 (t) =   0, if t < t0, [0, 1], if t = t0, 1, if t > t0 (2) definition 3. [3] the hausdorff distance (the h– distance) ρ(f,g) between two interval functions f,g on ω ⊆ r, is the distance between their completed graphs f(f) and f(g) considered as closed subsets of ω ×r. more precisely, ρ(f,g) = max{ sup a∈f(f) inf b∈f(g) ||a−b||, sup b∈f(g) inf a∈f(f) ||a−b||}, wherein ||.|| is any norm in r2, e. g. the maximum norm ||(t,x)|| = max{|t|, |x|}; hence the distance between the points a = (ta,xa), b = (tb,xb) in r2 is ||a−b|| = max(|ta − tb|, |xa −xb|). definition 4. we define the following activation function: a(t; β) = qe −tβ −qe tβ qe −tβ + qe tβ . (3) definition 5. define the following family of new recurrence generated functions ai+1(t; β) = ai(t + ai(t; β); β), i = 0, 1, 2, . . . ; a0(t; β) = a(t; β). (4) based on the function a(t; β). in [1] bantan, jamal, chesneau and elgarhy introduced a new power topp–leone–g–family (ntl–g) of distribution with (cdf) f(t) = e αβ ( 1− 1 g(t) ) ( 2−eβ ( 1− 1 g(t) ))α (5) where α,β ∈ r+ and g(t) is a (cdf) of a baseline continuous distribution. the following result shows some inequalities involving f(t) (see, proposition 1 [1]): e αβ ( 1− 1 g(t) )( 2 −g(t)β )α ≤f(t)≤2αeαβ ( 1− 1 g(t) ) . (6) in this paper we study some properties of the new topp–leone–g–family with baseline ”deterministic–type” (cdf) – (ntlg–dt); g(t) = 1 −qe t−1, where 0 < q < 1. definition 6. we define the following corresponding (cdf): q(t) = e αβ ( 1− 1 1−qet−1 ) ( 2 −eβ ( 1− 1 1−qet−1 ))α (7) where α,β ∈ r+ and 0 < q < 1. ii. main results when studying the intrinsic properties of the family mβ(t), it is also appropriate to study the ”saturation” to the horizontal asymptote. in this section we give upper and lower estimates for the one–sided hausdorff approximation of the heaviside step–function ht0 (t) by means of family (1), where t0 is the level of the ”median”. a. the case β = 1. let t0 is the unique positive root of the nonlinear equation m1(t0) − 12 = 0. the one–sided hausdorff distance d between ht0 (t) and the function (1) satisfies the relation m1(t0 + d) = 1 −qe (t0+d)−1 = 1 −d. (8) the following theorem gives upper and lower bounds for d theorem 1. let β = 1, q < 2 e 2( e1.052.1 −1) ≈ 0.971975. (9) then, for the one–sided hausdorff distance d between ht0 (t) and the (cdf) – (1) the following inequalities hold: dl = 1 2.1(1+ 1 2 ln 2 q ) <d< ln(2.1(1+ 1 2 ln 2 q )) 2.1(1+ 1 2 ln 2 q ) =dr. (10) biomath 9 (2020), 2005033, http://dx.doi.org/10.11145/j.biomath.2020.05.033 page 2 of 10 http://dx.doi.org/10.11145/j.biomath.2020.05.033 nikolay kyurkchiev, a new class of activation functions. some related problems and applications proof. in order to express d in terms of q, let us examine the function f(d) = m1(t0 + d) − 1 + d. from f ′(d) > 0 we conclude that function f(d) is strictly monotonically increasing. consider then the function g(d) = − 1 2 + (1 + 1 2 ln 2 q )d, which approximates function f with d → 0 as o(d2) (see, fig. 1). in addition g′(d) > 0. we look for two reals dl and dr such that g(dl) < 0 and g(dr) > 0 (leading to g(dl) < d < g(dr)). from (9) we have g ( dl = 1 2.1(1 + 1 2 ln 2 q ) ) < 0, g ( dr = ln(2.1(1 + 1 2 ln 2 q )) 2.1(1 + 1 2 ln 2 q ) ) > 0 proving the estimates (10). for example, for β = 1, q = 0.1 we have dl = 0.190639 < d = 0.230226 < 0.31596 = dr and for β = 1, q = 0.9 we have dl = 0.340317 < d = 0.355551 < 0.36682 = dr. b. the case β 6= 1. for given β 6= 1 the one–sided hausdorff distance d satisfies the relation mβ(t0 + d) = 1 −qe (t0+d) β−1 = 1 −d. (11) the reader may formulate the corresponding approximation problem following the ideas given in theorem 1, and will be omitted. we illustrate the ”saturation” with the (cdf) – (1) for various β and fixed q = 0.1 (see, fig. 2) fig. 1. the functions f(d) and g(d) for a) β = 1, q = 0.1; b) β = 1, q = 0.9. fig. 2. a) β = 1, q = 0.1; t0 = 0.263156; hausdorff distance d = 0.230226; b) β = 2, q = 0.1; t0 = 0.512988; hausdorff distance d = 0.208046; c) β = 3, q = 0.1; t0 = 0.640823; hausdorff distance d = 0.181048; d) β = 6, q = 0.1; t0 = 0.800514; hausdorff distance d = 0.127635. iii. some applications. it is well known that in many cases the existing modifications to the classical logistic and gompertz models do not give very reliable results in approximating ”specific data”. we examine the following ”specific datasets”: biomath 9 (2020), 2005033, http://dx.doi.org/10.11145/j.biomath.2020.05.033 page 3 of 10 http://dx.doi.org/10.11145/j.biomath.2020.05.033 nikolay kyurkchiev, a new class of activation functions. some related problems and applications fig. 3. the fitted model (1). example 1. we analyze the following data [4] data communication := {{0.584, 0.027}, {0.649, 0.147},{0.909, 0.187},{1.039, 0.303}, {1.558, 0.453},{2.208, 0.527},{2.792, 0.580}, {3.052, 0.627},{3.312, 0.657},{4.091, 0.707}, {4.740, 0.753},{5, 0.780},{5.390, 0.827}, {7.078, 0.853},{7.597, 0.877},{8.961, 0.903}, {9.091, 0.927},{10.195, 0.950},{22.078, 0.980}, {24.610, 1}}; the cdf mβ(t) for β = 0.484411 and q = 0.82547 is visualized on fig. 3. example 2. analysis of ”data nicotine” [5] data nicotine := {{0.11, 0.021},{0.21, 0.053},{0.31, 0.063}, {0.41, 0.105},{0.51, 0.2},{0.61, 0.274}, {0.71, 0.358},{0.81, 0.495},{0.91, 0.632}, {1.01, 0.726},{1.11, 0.832},{1.21, 0.905}, {1.31, 0.942},{1.41, 0.958},{1.51, 0.974}, {1.61, 0.979},{1.71, 0.989},{1.81, 1}, {1.9, 1},{2, 1}}; after that using the model mβ(t) for β = 1.98567 and q = 0.485475 we obtain the fitted model (see, fig. 4). example 3. analysis of data ”biomass produced by paesilomyces lilacinus 6029” [6]. after that using the model m∗β(t) = ωmβ(t) for ω = 10.521, β = 0.805824 and q = 0.97915 we obtain the fitted model (see, fig. 5). fig. 4. the fitted model (1). fig. 5. the fitted model. the new activation function. we define the following activation function: a(t; β) = qe −tβ −qe tβ qe −tβ + qe tβ . (12) in antenna-feeder technique most often occurred signals are of types shown on fig. 6 – fig. 7. for β even, the corresponding approximation using model (7) is shown in fig. 6. for β odd, the corresponding approximation using new activation function a(t; β) is shown in fig. 7. a family of recurrence generated functions based on the a(t; β). let us consider the following family of recurrence generated functions ai+1(t; β) = ai(t + ai(t; β); β), i = 0, 1, 2, . . . ; a0(t; β) = a(t; β), (13) based on the function a(t; β). let for instance β = 1. biomath 9 (2020), 2005033, http://dx.doi.org/10.11145/j.biomath.2020.05.033 page 4 of 10 http://dx.doi.org/10.11145/j.biomath.2020.05.033 nikolay kyurkchiev, a new class of activation functions. some related problems and applications fig. 6. the function a(t;β); β = 4, q = 0.01, t0 = 0.587335; hausdorff distance d = 0.138899; β = 6, q = 0.01, t0 = 0.701333; hausdorff distance d = 0.111603; β = 8, q = 0.01, t0 = 0.766378; hausdorff distance d = 0.0992629; β = 10, q = 0.01, t0 = 0.808266; hausdorff distance d = 0.0867535; β = 16, q = 0.01, t0 = 0.87543; hausdorff distance d = 0.0632673. fig. 7. the function a(t;β); β = 3, q = 0.01, t0 = 0.491867; hausdorff distance d = 0.152538; β = 7, q = 0.01, t0 = 0.737794; hausdorff distance d = 0.107003; β = 13, q = 0.01, t0 = 0.848962; hausdorff distance d = 0.073086. fig. 8. the recurrence generated family: a0(t) (blue), a1(t) (red) and a2(t) (dashed). the recurrence generated family: a0(t),a1(t) and a2(t) is visualized on fig. 8. some properties of the new topp–leone–g– family with baseline ”deterministic–type” (cdf) – (ntlg–dt) q(t)(7). we study the hausdorff approximation of the heaviside step function ht0 (t) where t0 is the ”median” by families of the new topp–leone–g– family with baseline ”deterministic–type” (cdf) – (ntlg–dt). the obtained two-sides estimations (see proposition 1. [1] ) in particular case with usage of the baseline ”deterministic–type” (cdf) for α = 0.9; β = 0.3; q = 0.1 e αβ ( 1− 1 1−qet−1 ) ( 2 − ( 1 −qe t−1 )β)α ≤ q(t) (14) ≤ 2αeαβ ( 1− 1 1−qet−1 ) are given in fig. 9 a. let t0 is the value for which q(t0) = 12 . the hausdorff distance d between the function ht0 (t) and q(t) satisfies the relation q(t0 + d) = 1 −d. (15) biomath 9 (2020), 2005033, http://dx.doi.org/10.11145/j.biomath.2020.05.033 page 5 of 10 http://dx.doi.org/10.11145/j.biomath.2020.05.033 nikolay kyurkchiev, a new class of activation functions. some related problems and applications fig. 9. a) the two-sides estimations (14) for α = 0.9; β = 0.3; q = 0.1; b) the model q(t) for α = 0.9; β = 0.3; q = 0.1, t0 = 0.0852097; h–distance d = 0.116811 for fixed α = 0.9; β = 0.3; q = 0.1 we find t0 = 0.0852097 and from the nonlinear equation (15) we have d = 0.116811 (see, fig. 9 b). from fig. 9 it can be seen that these estimations can be used as ”confidence bounds”, which are extremely useful for the specialists in the choice of model for cumulative data approximating in areas of biostatistics, population dynamics, growth theory, debugging and test theory, computer viruses propagation, financial and insurance mathematics. for other results, see [8]–[53], [59]. iv. concluding remarks. the results obtained in this article can be successfully continued. 1. for example, we study the hausdorff approximation of the heaviside step function ht0 (t) where t0 is the ”median” by families of the new topp–leone–g–family q1(t) with baseline ”deterministic–inverse–type” (cdf) – (ntlg–dit) g(t) = qe 1 t −1, where 0 < q < 1, q1(t) = e αβ ( 1− 1 qe 1 t −1 )  2 −eβ ( 1− 1 qe 1 t −1 )  α (16) fig. 10. a) the two-sided bounds (17) for α = 0.6; β = 0.1; q = 0.4; b) the model q1(t) for α = 0.6; β = 0.1; q = 0.4, t0 = 0.698075; h–distance d = 0.153113 the obtained two-sided bounds (see proposition 1. [1] ) in particular case with usage of the baseline ”deterministic–inverse–type” (cdf) for α = 0.6, β = 0.1, q = 0.4, e αβ ( 1− 1 qe 1 t −1 ) ( 2 −qβ ( e 1 t −1 ))α (17) ≤ q1(t) ≤ 2αe αβ ( 1− 1 qe 1 t −1 ) are given in fig. 10 a. example 4. storm worm one of the most biggest cyber threats of 2008. we analyze the following data [7] data storm ids := {{1, 0.843}, {4, 0.926},{5, 0.954},{6, 0.967}, {7, 0.976},{8, 0.981},{9, 0.985}, {10, 0.991},{22, 0.995},{38, 0.997}, {51, 0.998},{64, 0.9985},{74, 0.999}, {83, 1},{100, 1},{367, 1}} biomath 9 (2020), 2005033, http://dx.doi.org/10.11145/j.biomath.2020.05.033 page 6 of 10 http://dx.doi.org/10.11145/j.biomath.2020.05.033 nikolay kyurkchiev, a new class of activation functions. some related problems and applications fig. 11. the fitted model q1(t). the cdf q1(t) for α = 0.14146, β = 161.078891, q = 0.9 is visualized on fig. 11. exploring both features ”confidential curves” and ”super saturation” is a must when choosing the right model. 2. following the ideas given in [54]–[56] we consider the following new differential model:  dy(t) dt = ky(t)s(t) = ky(t)qe t−1 y(t0) = y0 (18) where k > 0 and 0 < q < 1. the general solution of the differential equation (18) is of the following form: y(t) = y0e k q ei(et lnq)−k q ei(lnq) (19) where ei(.) is the traditional exponential integral. the new ”growth” function y(t) and the ”input function” s(t) = qe t−1 are visualized on fig. 12– fig. 13. example 4. we will analyze a sample of experimental data obtained by the biologist t. carlson in 1913 about the development of saccharomyces culture in nutrient medium (see, for example [58], [57]). after that using the model m∗(t) = ωe k q ei(et lnq)−k q ei(lnq) for k = 0.293574, fig. 12. the ”growth” function y(t)–(red) and s(t)–(green) for k = 12.6; q = 0.14; y0 = 0.01. fig. 13. the ”growth” function y(t)–(red) and s(t)–(green) for k = 1.1; q = 0.906; y0 = 0.1. fig. 14. the fitted model m∗(t). biomath 9 (2020), 2005033, http://dx.doi.org/10.11145/j.biomath.2020.05.033 page 7 of 10 http://dx.doi.org/10.11145/j.biomath.2020.05.033 nikolay kyurkchiev, a new class of activation functions. some related problems and applications fig. 15. the fitted model m∗(t). q = 0.999983 and ω = 30.114 we obtain the fitted model (see, fig. 14). example 5. analysis of data ”biomass produced by paesilomyces sinclairi ascomycota”. after that using the model m∗(t) for ω = 0.305247, k = 3.01914 and q = 0.83 we obtain the fitted model (see, fig. 15). the general solution y(t) has been applied widely in life testing experiments and debugging theory. acknowledgment the author would like to thank the anonymous referees for their valuable comments. this paper is supported by the national scientific program ”information and communication technologies for a single digital market in science, education and security (ictinses)”, financed by the ministry of education and science. references [1] r. bantan, f. jamal, ch. chesneau, m. elgarhy, a new power topp-leone generated family of distributions with applications, entropy, 21, 12, 1177, 2019. 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[59] n. kyurkchiev, selected topics in mathematical modeling: some new trends (dedicated to academician blagovest sendov (1932-2020)), lap lambert academic publishing, 2020; isbn: 978-620-2-51403-3. biomath 9 (2020), 2005033, http://dx.doi.org/10.11145/j.biomath.2020.05.033 page 10 of 10 http://dx.doi.org/10.11145/j.biomath.2020.05.033 introduction and preliminaries main results the case =1. the case =1. some applications. concluding remarks. references www.biomathforum.org/biomath/index.php/biomath original article a class of individual–based models mirosław lachowicz institute of applied mathematics and mechanics university of warsaw warsaw, poland m.lachowicz@mimuw.edu.pl received: 9 february 2018, accepted: 12 april 2018, published: 15 april 2018 abstract—we discuss a class of mathematical models of biological systems at microscopic level — i.e. at the level of interacting individuals of a population. the class leads to partially integral stochastic semigroups — [25]. we state general conditions guaranteeing the asymptotic stability. in particular under some rather restrictive assumptions we observe that any, even non–factorized, initial probability density tends in the evolution to a factorized equilibrium probability density — [16]. we discuss possible applications of the general theory such as redistribution of individuals — [10], thermal denaturation of dna [7], and tendon healing process — [11]. keywords-individual–based models; markov jump processes; integro–differential equations, stochastic semigroups, stability. i. microscopic scale in the present paper we review the general class of individual–based models in biology developed in ref. [15] — see also [3], [14], [16] and references therein. we show that the class corresponds to the partially integral stochastic semigroups and under some more restrictive assumptions leads to the stability result. we discuss possible applications of the general theory such as redistribution of individuals — [10], thermal denaturation of dna [7], and tendon healing process — [11]. we consider the general equation that defines the evolution of a number n of individuals of biological populations — cf. refs. [3], [14], [15] and references therein. each individual n (n ∈ {1, . . . ,n}) is characterized by its inner (microscopic ) state un ∈ u , where u is a borel set in rd, d ∈ {1, 2, 3, . . .}. the variable un related to the individual n may have various meanings: it may be any vector parameter that characterize an individual biological state of any of the individuals. in particular it may also contain an information of a subpopulation to which the individuals belongs (a discrete component) — see [3], [14], [15]. in the general setting (u,b,µ) is a σ–finite measure space. in some applications u is a product of a discrete set and a lebesgue–measurable subset (e.g. a closed bounded interval) in the discrete– continuous picture or a discrete set in the discrete– discrete case and the measure µ is a product of the copyright: c© 2018 lachowicz et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: mirosław lachowicz, a class of individual–based models, biomath 7 (2018), 1804127, http://dx.doi.org/10.11145/j.biomath.2018.04.127 page 1 of 8 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2018.04.12 7 mirosław lachowicz, a class of individual–based models counting measure and the counting measure (in the discrete–discrete picture) or the lebesgue measure (in the discrete–continuous picture). the n1–individual changes its state at random times. we consider the possibility of the following stochastic changes • without interactions, • due to interaction with the n2–individual, n2 ∈{1, . . . ,n}, n2 6= n1 , • due to interaction with the n2 and n3 individuals, • ... • due to interaction with n2, n3, ..., nm individuals, where m is an integer 2 ≤ m ≤ n. consider interactions of the given individual with m− 1 individuals, m = 1, . . . ,m. assumption 1. the rate of interaction between the individual with state un1 and the individuals with states un2 , ..., unm is given by the measurable function a[m] = a[m](un1, . . . ,unm), such that 0 ≤ a[m](un1,un2, . . . ,unm) ≤ a [m] + , (1) for all un1,un2, . . . ,unm ∈ u , where a [m] + < ∞ is a constant. assumption 2. the transition into state v of an n1–individual with state un1 , due to the interaction with individuals of n2,...,nm with states un2 ,...,unm, respectively, is described by the measurable function a[m] = a[m] (v; un1, . . . ,unm) ≥ 0 , where∫ u a[m] ( v ; un1,un2, . . . ,unm ) dµ(v) = 1 , (2) for all un1,un2, . . . ,unm ∈ u . the stochastic model (at the microscopic level) is determined by the functions a[m] and a[m]. l (n) 1 is the space equipped with the norm ‖f‖ l (n) 1 =∫ un ∣∣∣f(u1, . . . ,un)∣∣∣dµ(u1) . . . dµ(un ) . if n = 1 we simply write l1. given n, m, and a[m], a[m], for m = 1, . . . ,m, we consider the stochastic system that is defined by the markov jump process of n individuals through the following generator λ acting on densities λ = λ+ − λ− = m∑ m=1 ( λ[m] + − λ[m]− ) , λ[m] +f ( t,u1,u2, ...,un ) = cn,m ∑ 1≤n1,...,nm≤n ni 6=nj ∀i6=j ∫ u a[m] ( un1 ; v,un2, . . . ,unm ) × a[m] ( v,un2, . . . ,unm ) × f ( t,u1, . . . ,un1−1,v,un1+1, . . . ,un ) dµ(v) , λ[m]−f ( t,u1,u2, ...,un ) = cn,m ∑ 1≤n1,...,nm≤n ni 6=nj ∀i6=j a[m] ( un1,un2, . . . ,unm ) × f ( t,u1, . . . ,un ) , on un , where cn,m = 1 (m−1)! ( n m−1) are normalizing constants. assume that the system is initially distributed according to f ∈ l(n)1 and time evolution is described by the following (linear) equation — the modified liouville equation , ∂ ∂t f = λf , ; f ∣∣∣ t=0 = f . (3) with the initial data f ∣∣∣ t=0 = f . (4) we refer here to the liouville equation in the sense of particle dynamics: eq. (3) plays a similar role as the liouville equation in kinetic theory — see ref. [6] (cf. also [18]). the generator λ is the difference between the gain term and loss terms λ = λ+ − λ−, where • the gain term λ+ is a sum of terms describing the changes from state v of n1–individual into un1 due to the interaction with n2, ..., nm individuals with states un2 , ..., unm, respectively for 2 ≤ m ≤ m and the term (m = 1) describing the direct changes of state v of n1– individual into un1 without interactions; • the loss term λ− is a sum of terms describing the changes from state un1 of n1–individual into another state due to the interaction with biomath 7 (2018), 1804127, http://dx.doi.org/10.11145/j.biomath.2018.04.127 page 2 of 8 http://dx.doi.org/10.11145/j.biomath.2018.04.12 7 mirosław lachowicz, a class of individual–based models n2, ..., nm individuals with states u2, ..., um, respectively for 2 ≤ m ≤ m or without interactions for m = 1. it is easy to see that under assumptions 1—2 the operator λ is a bounded in l(n)1 . thus the cauchy problem has the unique solution f(t) = exp ( tλ ) f (5) in l(n)1 for all t ≥ 0. the solution is nonnegative for nonnegative initial data and the l(n)1 –norm is preserved for any t > 0. therefore { exp ( tλ )} t≥0 defines a continuous (linear) semigroup of markov operators that is a stochastic semigroup in the sense of ref. [21]. actually we may note that here we have even a group. in ref. [15] (see also [14], [3] and the references therein) the limit n → ∞ was studied. under suitable assumption eq. (3) results in a nonlinear kinetic equation referred to the corresponding mesoscopic level. moreover in various cases macroscopic limits can be obtain. in the next section (section ii) we show that{ exp ( tλ )} t≥0 is a partially integral stochastic semigroup (see refs. [26], [25]) and, under some additional assumptions, leads to a stability result. in section iii we review some possible applications. ii. asymptotic behaviour in order to formulate the time asymptotic result we can refer to the notion of the partially integral stochastic semigroups — see refs. [26], [25]) — and the lower function theorem by lasota and yorke — see [22] (theorem 2; cf. also corollary iv.16 in ref. [28]). using similar strategy as in refs. [16] we prove a more general result that may be related to a general class of microscopic systems in the form given by eq. (5) under reasonably general assumptions 1 and 2. lemma ii.1. let assumptions 1 and 2 be satisfied. assume moreover that a[m] is non–zero, for some m ∈{1, . . . ,m}. then { exp ( t λ )} t≥0 is a partially integral stochastic semigroup. proof: let γ be the operator given by γf = λf + a+f , for f ∈ l(n)1 , where a + = max m=1,...,m a[m]. then γf ≥ λ+f ≥ max { 0, λf } (6) and exp ( t λ ) f = exp ( −a+ t ) exp ( t γ ) f , (7) for any probability density f on un . by eqs. (6) and (7), for any probability density f on un , we have exp ( t λ ) f ( u1, . . . ,un ) ≥ t n n! exp ( −a+ t )( λ+ )n f ( u1, . . . ,un ) ≥ cn,m(t) ∫ un k[m] ( u1, . . . ,un,v1, . . . ,vn ) ×f ( v1, . . . ,vn ) dµ(v1) . . . dµ(vn ) , (8) where cn,m(t) is a constant that depends on m,n and t > 0 and k[m] is a complicated function that depends on a[m] and a[m], k[m] ( u1, . . . ,un,v1, . . . ,vn ) = = a[m] ( u1; v1,u2, . . .um ) a[m] ( v1,u2, . . .um ) ×a[m] ( u2; v2,u3, . . .um+1 ) a[m] ( v2,u3, . . .um+1 ) × . . . a[m] ( un ; vn,v1, . . .vm−1 ) ×a[m] ( vn,v1, . . .vm−1 ) . thus (see [26]) { exp ( t λ )} t≥0 is a partially integral stochastic semigroup. we note that the above result does not need any additional assumption. the class of partially integral stochastic semigroups is particularly important (see [25], [26], [28]) in the analysis of asymptotic behaviour of stochastic semigroups. to state the asymptotic stability we need however a stronger assumption theorem ii.2. let assumptions 1 and 2 be satisfied. additionally, for some m ∈{1, . . . ,m} we assume that a[m] is nonzero, and there exists a measurable nonnegative function h on u such that∫ u h(u) dµ(u) > 0 , (9) biomath 7 (2018), 1804127, http://dx.doi.org/10.11145/j.biomath.2018.04.127 page 3 of 8 http://dx.doi.org/10.11145/j.biomath.2018.04.12 7 mirosław lachowicz, a class of individual–based models and a[m] ( v,u1, . . . ,um ) a[m] ( u1, . . . ,um ) ≥ h(v) , (10) for any v,u1,u2, . . . ,um ∈ u. then the stochastic semigroup { exp ( tλ )} t≥0 is asymptotically stable. proof: as in the proof of lemma ii.1, for any probability density f on un , we have exp ( t λ ) f ( u1, . . . ,un ) ≥ t n n! exp ( −a+t )( λ+ )n f ( u1, . . . ,un ) ≥ cn,m(t) n∏ i=1 h(ui) × ∫ un f ( v1, . . . ,vn ) dµ(v1) . . . , dµ(vn ) = cn,m(t) n∏ i=1 h(ui) , (11) where cn,m(t) is a constant that depends on m,n and t > 0. let t0 > 0 be fixed, e.g. t0 = 1. we have exp ( t λ ) f = exp ( t0 λ ) exp ( (t− t0) λ ) f . by eq. (11) we obtain exp ( t λ ) f ( u1, . . . ,un ) ≥ cn,m(t0) n∏ i=1 h(ui) ∥∥∥ exp ((t− t0) λ)f∥∥∥ l (n) 1 , (12) for t > t0. keeping in mind that exp ( (t−t0) λ ) f is a probability density, for each t > t0 we conclude exp ( t λ ) f ( u1, . . . ,un ) ≥ `(u1, . . . ,un ) , (13) where `(u1, . . . ,un ) = cn,m(t0) n∏ i=1 h(ui) depends on n but does not depend on t and f . by assumptions 1 and 2 it follows that ` ∈ l(n)1 . moreover by eq. (9)∫ un `(u1, . . . ,un ) dµ(u1) . . . dµ(un ) > 0 , (14) and ` is a lower function in the sense of lasota and yorke [22]. in fact the following condition holds lim t→∞ ∥∥∥( exp(t λ) f − `)−∥∥∥ l (n) 1 = 0 , for every probability density f , where x− = 0 if x ≥ 0 and x− = −x if x < 0. thus by the lower function theorem of lasota and yorke the semigroup { exp(t λ) } t≥0 is asymptotically stable. as a by–product of theorem ii.2 we obtain the uniqueness of an equilibrium (stationary) solution corresponding to eq. (3). the identification of possible equilibrium solutions is an essential step in studying macroscopic limits corresponding to the microscopic models — see [3]. in a particular case referred to a microscopic system in ref. [16], under some (rather strong) assumption, it is shown that any, even non– factorized, initial probability density tends in the evolution to a factorized equilibrium probability density. such a situation one can refer to as asymptotic annihilation of initial correlations in the system. on the other hand it was also shown that if the mentioned assumptions are not satisfied — a number of equilibrium states could be large and no annihilation is observed. the possible relationships between micro-, mesoand macroscales were discussed in ref. [3] (see also references therein) — chapter 8 and in particular subsection 8.3.3. iii. applications there are many possible applications of the general theory presented in section i. the stochastic systems that corresponds at the macroscopic level to standard logistic growth were considered in ref. [16]. the parameter un, n ∈ {1, 2, . . . ,n}, describing the microscopic (individual) state of n–individual, may be related to its activity (cf. refs. [4], [3]). the parameter may also describe dominance [13] or social state (c.f. refs. [1], [5], [9]). the references mentioned above refer to the mesoscopic (kinetic) description whereas ref. [16] to microscopic (individual–based) one. usually the importance of the microscopic approach may be particularly visible in a case when the number of interacting entities of the system is not huge which is typical for biological systems. in such cases the kinetic (mesoscopic) description not always may be properly justified. biomath 7 (2018), 1804127, http://dx.doi.org/10.11145/j.biomath.2018.04.127 page 4 of 8 http://dx.doi.org/10.11145/j.biomath.2018.04.12 7 mirosław lachowicz, a class of individual–based models a. redistribution of individuals an application of the general model at the microscopic level of the redistribution of individuals in a closed domain featuring, as an example, an elevator — see [10]. the modeling bases on experiments performed in order to elucidate the interactions between pedestrians — see [12] and the references in [10]. in these experiments the inflow of persons into a spatially restricted area, e.g. an elevator, was studied, featuring the process inverse to evacuation. in ref. [10] an n–th individual, n = 1, . . . ,n, is characterized by its position un. the model and analysis presented in [10] have a preliminary nature and still have to be developed. b. thermal denaturation of dna in ref. [7] some aspects of deoxyribonucleic acid dna thermal denaturation process (cf. [23], [24], [2]) were considered. ref. [7] is a continuing the idea of [8] where a preliminary mesoscopic (kinetic) model was discussed. in ref. [7] a new, more adequate, model that takes into account at the individual (microscopic) level the time evolution of the probability distribution of the state of all hydrogen bonds. two types of bonds (with two and three hydrogen bonds) and the direct dependence on the temperature are included in the model. the base pairs a–t, c–g are numbered by the discrete variable n ∈ {1, 2, . . . ,n}, and the continuous variable u ∈ [0,∞[ representing the stretching of the distance between the two connected base is used. the variable u is called stretching parameter. every base pair (’bond’) n is then characterized by the variables un. the discrete variable belongs to one of two subsets of bonds: j2 — two hydrogen bonds connecting a and t and j3 — three hydrogen bonds connecting c and g j2 ∪j3 = j, j2 ∩j3 = ∅ . according to the biological knowledge it is assumed that the three hydrogen bonds are more resistant to heating than the two hydrogen bonds. the probability densities f = f(t,u1, . . . ,un ) that describe the distribution of the variable u1, . . . ,un at all bonds is considered. c. tendon healing process in ref. [11] a kinetic model of collagen remodeling occurring in latter stage of tendon healing process was proposed and studied. the model is an integro–differential equation describing the alignment of collagen fibers in a finite time. an important feature of tendon structure is the collagen fibers orientation. in the healthy tendon they are aligned. the result of the tendon injury is a disturbance of the parallel structure. the healing process consists in the reconstruction of parallel structure. scars that may be formed during the healing process cause no proper alignment of collagen fibers. one of the most important indicators of the success of the treatment of tendon injury is the degree of alignment of collagen fibers. the model in [11] refers to the function g(t,x,v) that describes a statistical state of collagen, i.e. the probability density g = g(t,x,v) to find a collagen fiber at the instant of time t > 0 at point x ∈ d and with orientation v ∈ v, where d, v are domains in rd. thus the model has a mesoscopic nature. we consider the following equation ∂ ∂t g(t,x,v) = ∫ v ∫ d ( tg(y,v; x,w)g(t,x,w) −g(t,x,v)tg(y,w; x,v) ) dydw, (15) where tg(y,v; x,w) describes the transition probability from the orientation w ∈ v at x ∈ d to the orientation v ∈ v at x caused by an adaptation to the orientation at y ∈ d. the model bases on a proper choice of the function tf that in general may depend on both collagen distribution g and tenocytes density c. in ref. [11] a simplified case of constant (uniform) tenocytes density was considered. a realistic definition is tg(y,v; x,w) = β(y,v; x,w)g γ(t,y,v) , where γ > 0 describes the strength of influence of collagen fibers from neighborhood on collagen fiber in considered point. the bigger is the γ the stronger the influence is. that choice leads to the biomath 7 (2018), 1804127, http://dx.doi.org/10.11145/j.biomath.2018.04.127 page 5 of 8 http://dx.doi.org/10.11145/j.biomath.2018.04.12 7 mirosław lachowicz, a class of individual–based models following general class of equations ∂ ∂ t g(t,x,v) =∫ d ∫ v ( β(y,v; x,w) gγ(t,y,v) g(t,x,w) −g(t,x,v)β(y,w; x,v)gγ(t,y,w) ) dydw. (16) the function β(y,v,x,w) is related to the interaction between the collagen fiber with orientation w at point x with collagen fiber with orientation v located at point y and describes the transition from orientation w to orientation v. in ref. [11] we show that the solutions may exist globally in time or may blow–up in a finite time depending on initial data. the latter behavior is related to the healing of injury without the scar formation in a finite time: a full alignment of collagen fibers occurs. the approach of [11] may be related to the mesoscopic scale. one may however propose a stochastic individually based (microscopic) model following the idea stated in section i. the number n may be then related to the number of collagen fibers that are taken into account in the modeling process. the system is described in terms of a markov jump process and the related linear evolution equations as in section i. the equation describes the evolution of probability densities, with microscopic representation of the system of n interacting agents. we consider the interactions of a given agent with γ agents. the system is initially distributed according to the probability density f ∈ l(n)1 . the time evolution is described by eq. (3) where λ is the generator that takes the form λf ( t,x1,v1, . . . ,xn,un ) = cn,γ ∑ 1≤n1,...,nγ+1≤n ni 6=nj ∀i6=j ( ∫ d×v a ( xn1,vn1 ; y,w,xn2,vn2, . . . ,vnγ+1,vnγ+1 ) ×a ( y,w,xn2,vn2, . . . ,xnγ+1,vnγ+1 ) ×f ( t,x1,v1, . . . ,xn1−1,vn1−1,y,w, xn1+1,vn1+1, . . . ,xn,vn ) dy dw −a ( xn1,vn1, . . . ,xnγ+1,vnγ+1 ) f ( t,x1,v1, . . . ,xn,vn )) . in the limit n → ∞, the (linear) modified liouville equation (3) yields, [15], [3], a nonlinear integro–differential equation that can be related to the mesoscopic description ∂ ∂t f(t,u) = g[f](t,u) −f(t,u)l[f](t,u) , u = (x,v) ∈ d×v , (17) where g[f] is the gain term , given by g[f](t,x,v) = ∑ {} ∫( d×v )γ+1 a ( x,v; y,w,{x2,v2, . . . ,xγ+1,uγ+1} ) ×a ( y,w,{x2,u2, . . . ,xγ+1,uγ+1} ) ×f(t,y,w)f(t,x2,v2) . . .f(t,xγ+1,vγ+1) dy dw dx2, dv2, . . . dxγ+1, dvγ+1 , and fl[f] is the loss term , defined as l[f](t,x,v) = ∑ {} ∫( d×v )γ a ( x,v,{x2,v2, . . . ,xγ+1,vγ+1}) ×f(t,x2,v2) . . .f(t,xγ+1,vγ+1) ×dx2 dv2 . . . dxγ+1 dvγ+1 , and ∑ {} means the sum over all permutation of variables within {}. it is easy to see that the global (in time) existence and uniqueness of solutions f = f(t) to eq. (17) in l(1)1 follows. one may now state the theorem (cf. [15], [3]) that defines the links between the solutions to eq. (3) and to eq. (17) or, in other words, that defines the transition from the microscopic level to the mesoscopic level. the mathematical properties of eq. (17) are different than those of eq. (16). the possible rich behavior of solutions of eq. (16), see [11], [17], leading to blow–ups in a finite of time are not possible in the case of solutions of eq. (17). on the other hand in some limit (approximating ”delta –function”) the solutions of eq. (16) may be approximated by the solutions of eq. (17). this defines the relationship between a stochastic, individually–based (microscopic) description and eq. (16). biomath 7 (2018), 1804127, http://dx.doi.org/10.11145/j.biomath.2018.04.127 page 6 of 8 http://dx.doi.org/10.11145/j.biomath.2018.04.12 7 mirosław lachowicz, a class of individual–based models a formal series with respect to γ = 1, 2, . . . , results in the following nonlinear kinetic (mesoscopic) equation alternative to eq. (16) ∂ ∂ t g(t,x,v) =∫ d ∫ v ( β(y,v; x,w) g(t,y,v) 1−g(t,y,v) g(t,x,w) −g(t,x,v) β(y,w; x,v) g(t,y,w) 1−g(t,y,w) ) dy dw. (18) at present, the mathematical theory of eq. (18) is missing. iv. conclusions in section i we review the general class of microscopic models that are able to describe interactions between individuals of a biological population. the class refers to the stochastic semigroups. in section ii we show the methods that leads to the asymptotic stability under some rather restrictive assumptions. on the other hand the asymptotic behaviour in the general case is still an open problem. the important technical tool could be the fact that the semigroup is partially integral (without any particular additional assumption). this may be treated as a preliminary step towards the description of macroscopic (”hydrodynamic ”) limits that seems to be essential part of the program of giving full description on various scales starting from microscopic, then mesoscopic and finally — macroscopic. in a very simple case considered in [16] — the macroscopic equation was obtain from the mesoscopic equation by the averaging with respect to microscopic variable. in the general case it is far from being solved. therefore we may believe that the methods of the present paper can indicate the possible further research. in section iii we review some important applications. they show that the general framework is suitable to describe various systems in which the interactions between individuals are essential. we point out some new equations that result in various limits that can be interesting for further mathematical studies. acknowledgment the author was supported by the national science centre poland grant 2017/25/b/st1/00051. references [1] g. ajmone marsan, n. bellomo and m. egidi, towards a mathematical theory of complex socio– economical systems by functional subsystems representation, kinetic rel. models, 1:2 (2008), 249–278; doi:10.3934/krm.2008.1.249. 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[28] r. rudnicki, models and methods in mathematical biology, in polish, impan, warszawa 2014. biomath 7 (2018), 1804127, http://dx.doi.org/10.11145/j.biomath.2018.04.127 page 8 of 8 http://dx.doi.org/10.11145/j.biomath.2018.04.12 7 microscopic scale asymptotic behaviour applications redistribution of individuals thermal denaturation of dna tendon healing process conclusions references a model for the early covid-19 outbreak in china with case detection and behavioural change biomath https://biomath.math.bas.bg/biomath/index.php/biomath b f biomath forum original article a model for the early covid-19 outbreak in china with case detection and behavioural change julien arino1, khalid el hail2,∗, mohamed khaladi3, aziz ouhinou2 1department of mathematics university of manitoba, winnipeg, manitoba, canada julien.arino@umanitoba.ca 0000-0001-6409-5027 2department of mathematics, faculty of sciences and techniques, university of sultan moulay slimane, beni-mellal, morocco elhail.kha@gmail.com 0000-0001-5166-9386 a.ouhinou@usms.ma 0000-0002-0206-4935 3laboratory of mathematics and population dynamics – ummisco, faculty of sciences semlalia, cadi ayyad university, marrakech, morocco khaladi@uca.ac.ma 0000-0002-7703-5637 received: november 5, 2022, accepted: december 20, 2022, published: january 27, 2023 abstract: we investigate a model of the early stage of the covid-19 epidemic comprising undetected infected individuals as well as behavioural change towards the use of self-protection measures. the model is fitted to china data reported between 22 january and 29 june 2020. using fitting results, we then consider model responses to varying screening intensities. keywords: covid-19, behavioural change, screening intensity, protective measures i. introduction at the time of writing, the first known covid-19 human case is one with onset on 8 december 2019, in wuhan, china [1, 2], although there is evidence that the disease have been spreading earlier; see [3] for a timeline of early spread. on 20 january 2020, studies confirmed human-to-human transmission through respiratory droplets [4]. there is now an unprecedentedly large body of work on the worldwide covid-19 outbreak; however, many epidemiological features such as per capita transmissibility, screening and diseaserelated death rates are still ambiguous and, to a large extent, seem quite dependent on the location under consideration, with outbreak intensities varying greatly from country to country. parameters may vary from region to region depending, for instance, on control measures taken by policymakers, availability of personal protective equipment, hospitalisation, demographic pyramid, life activities and cultural aspects. many works consider the early spread of covid-19 in china, which at the start of the pandemic had the most data since it had the most cases; the list is far too extensive to detail here and we list just a few. in [5], the authors estimated the basic reproduction number to be up to r0 = 3.58 at the beginning of the outbreak in china. using the official counts of confirmed cases, r0 was suggested in [6] to be on average 4.6, and, by assuming presymptomatic and mildly symptomatic infectious individuals to be twenty or forty times the reported number of infected cases, the mean of r0 was copyright: © 2022 julien arino, khalid el hail, mohamed khaladi, aziz ouhinou. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. *corresponding author citation: julien arino, khalid el hail, mohamed khaladi, aziz ouhinou, a model for the early covid-19 outbreak in china with case detection and behavioural change, biomath 11 (2022), 2212207, https://doi.org/10.55630/j.biomath.2022.12.207 1/8 https://biomath.math.bas.bg/biomath/index.php/biomath mailto:julien.arino@umanitoba.ca https://orcid.org/0000-0001-6409-5027 mailto:elhail.kha@gmail.com https://orcid.org/0000-0001-5166-9386 mailto:a.ouhinou@usms.ma https://orcid.org/0000-0002-0206-4935 mailto:khaladi@uca.ac.ma https://orcid.org/0000-0002-7703-5637 https://creativecommons.org/licenses/by/4.0/ https://doi.org/10.55630/j.biomath.2022.12.207 arino et al, a model for the early covid-19 outbreak in china with case detection and behavioural change estimated to be 3.2 or 2.6, respectively; daily infection mortality and recovery rates were also estimated. in addition, in the early stage of the covid-19 epidemic in china, [1] suggested r0 to be approximately 2.2 and the incubation period to have a mean of 5.2 days. because infection severity differs greatly in infected individuals, some individuals are infectious while presymptomatic. together with asymptomatic infections, this means that some individuals may have “evaded” screening, despite contributing to the spread of the disease [6–9]. besides isolating detected active cases and their known immediate contacts, healthcare authorities worldwide did their best to educate the population about covid-19 severity, its mode of transmission and convince people to use all available preventive measures. accordingly, in this work we take into consideration the population response to education campaigns. we use a system of differential equations to model the covid-19 epidemic with, including burden-dependent behavioural change. we calibrate some of the parameters so that the model fits the covid-19 data in china from 22 january until 29 june 2020. this article is organised as follows. in section ii, the mathematical model of covid-19 transmission is derived. section iii presents numerical results, which are then discussed further in section iv. ii. mathematical modelling the mathematical model considered in this paper comprises seven epidemiological compartments, s, e, is, ie, jd, jt and r, as well as an auxiliary variable, a, used to account for awareness of the disease. the flow diagram is shown in figure 1; let us elaborate on this structure. s are susceptible individuals in the classical sense, while e denotes educated susceptible individuals using self-protective measures against the infection. infected individuals are divided into four compartments. is and ie are, respectively, non-educated and educated undetected infectious individuals; individuals in both of these compartments who get screened move, upon detection, into the isolation compartment jd, where they wait for recovery or a potential hospitalisation [10]. if their infection goes undetected, upon recovery or death, they progress directly to the removed compartment. the fourth infected compartment, jt , contains infected cases who are under treatment in hospital. both jd and jt are isolated and as a consequence, they are not infectious to others. the compartment r is for table i: state variables. variable definition s susceptible individuals e educated susceptible individuals is undetected infectious non-educated individuals ie undetected infectious educated individuals jd isolated infected individuals jt hospitalised infected individuals r removed individuals a disease awareness (auxiliary variable) removals due to recovery or death. table i summarises the definition of all state variables. we use the auxiliary variable a to represent awareness of the disease. awareness is based on available information: known (detected) cases, hospitalisations and deaths from the disease. in the case of covid-19, many people used personal protection equipment and practiced social distancing when they became aware of the presence of the disease. this was further reinforced by stringent social distancing and confinement measures imposed or recommended by authorities. however, even if they are aware of the presence of the disease and with strong or even coercive governmental policies, not all individuals follow public health recommendations or orders. we therefore assume that susceptible individuals become educated (and therefore follow public health recommendations) at the rate e(a) = e0a 2 a∗2 + a2 , giving a hill functional form [11] and described in [12, 13]. here a∗, is the awareness level producing half of the maximum education response e0 to campaign efforts; see [11, 14] for more details. we model individuals flow between different compartments using the following system of differential equations s′ = −λs −e(a)s (1a) e′ = e(a)s − (1−ε)λe (1b) i′s = λs − (α + e(a) + δ)is (1c) i′e = (1−ε)λe + e(a)is − (α + δ)ie (1d) j′d = (1−θ)α(is + ie)− (γ1 + w)jd (1e) j′t = θα(is + ie)−γ2jt + wjd (1f) r′ = δ(ie + is) + γ1jd + γ2jt (1g) a = jd + (1 + pγ2)jt , (1h) with initial conditions (s0,e0,is0,ie0,jd0,jt 0,r0) ∈ r7+. biomath 11 (2022), 2212207, https://doi.org/10.55630/j.biomath.2022.12.207 2/8 https://doi.org/10.55630/j.biomath.2022.12.207 arino et al, a model for the early covid-19 outbreak in china with case detection and behavioural change s e is ie jd jt ra e (a )s e (a )i s λs (1−ε)λe δie (1−θ)αis θαi s (1 −θ )α ie θαie δis γ 1 j d w j d γ 2 j t fig. 1: flow diagram of the model. dark red compartments are infectious, blue represents awareness. plain arcs are flows of individuals between compartments, dashed lines indicate the influence of compartments on a, dotted lines show the flows on which a acts. the force of infection takes the form λ = β is + (1−ε)ie n , where β is the per capita transmission rate per unit time, ε ∈ [0,1] is the efficacy of self-protective measures, α is the detection rate and θ is the treatment rate. table ii summarises the parameters used. table ii: definition of parameters. param. definition β transmission rate α detection rate δ natural recovery rate θ proportion of individuals needing hospitalisation after detection γ1 recovery rate for individuals in self-isolation γ2 removal rate for individuals under treatment w hospitalisation rate for self-isolating individuals p proportion of deaths among removed individuals a∗ burden level producing half maximum of education response e0 maximum of education response ε efficacy of self-protective means let us briefly comment on some characteristics of the model. in a standard way as in [11], one can show that system (1) is well-posed and has a unique positive solution whenever the initial condition is positive. by construction, the total population n = s + e + is + ie + jd + jt + r is constant. the disease-free equilibrium (dfe) is x∗ = (n,0,0,0,0,0,0) and at x∗ awareness is a = 0 and thus e = 0. to apply the next generation matrix method [15], we focus on the infected compartments. although still infected, individuals in jd and jt no longer contribute to the infection and can be considered as having been removed. as a consequence, the infected compartments considered for the computation are is and ie. we get f = ( λs (1−ε)λe ) and v = ( (α + e + δ)is −eis + (α + δ)ie ) . therefore, the next generation matrix near x∗ is fv −1, where f = ( β (1−ε)β 0 0 ) and v = ( α + δ 0 0 α + δ ) . hence, the basic reproduction number for (1) is given by r0 = ρ ( fv −1 ) = β α + δ , where ρ(·) is the spectral radius. using the method in [16], it is also possible to derive a final size relation. biomath 11 (2022), 2212207, https://doi.org/10.55630/j.biomath.2022.12.207 3/8 https://doi.org/10.55630/j.biomath.2022.12.207 arino et al, a model for the early covid-19 outbreak in china with case detection and behavioural change iii. parameter estimation and numerical simulations a. parameter estimation we now estimate the parameters that are used in numerical simulations. first, let us establish the initial conditions used. we take the initial time to be 8 december 2019, when the first known patient developed symptoms of covid-19 [2]. the initial susceptible population is 1.438 billion, the estimated total population of china at the time [17]. at this very early stage of the pandemic, people did not yet know about the disease and had thus not changed their behaviour to use adequate self-protection measures. thus, initially, the educated compartments e0 and ie0 are empty. also, there were no reported recoveries or deaths of the new disease [2]. table iii summarises the initial conditions considered for (1) in simulations. table iii: initial conditions on 8 december 2019 (b stands for billion). comparts0 e0 is0 ie0 jd0 jt 0 r0 ment value 1.438b 0 1 0 1 0 0 sequencing of the sars-cov-2 genome was accomplished early on, in january 2020, and as a consequence, pcr tests followed in the same month. however, because of limitations in test processing capacities, many jurisdictions, including china, have, at least at times, imposed criteria that individuals had to satisfy in order to be tested. during the period considered, in china and wuhan in particular, two different sets of criteria were used [18–21]. most of the time, a restrictive set of criteria was in effect, requiring individuals to show many symptoms in order to be considered as suspected cases and therefore be eligible for testing, leading to what we also refer to as normal screening. during this period, we use the detection rate αr. then, between 12 and 19 february 2020, the criteria for screening were temporarily changed from the restrictive set to a milder set requiring less symptoms, implying that far more tests were carried out. during that short time period of intense screening, we use the detection rate αm ≥ αr. almost all parameters in the model are fitted. however, for the proportion of deaths among removed individuals, we use the estimation in [22], namely, 0.04. to estimate parameters, we use data on cases in china between 22 january 2020 and 29 june 2020 as reported in [22]. we use the python optimize module to fit our model to cumulative and active cases (see figure 2) and calibrate the parameters. note that for active cases, this means we fit jd(t) +jt (t). table iv presents the parameter values found by that process. table iv: parameter values found by fitting. parameter value remark/source β 0.347 fitted (αr,αm) (0.1,0.274) fitted as a step function δ 0.058 fitted θ 0.023 fitted γ2 0.092 fitted γ1 9.16 × 10−6 fitted p 0.04 [22] ε 0.662 fitted w 0.099 fitted e0 0.389 fitted a∗ 36432.82 fitted the transmission rate obtained is 0.347 day−1, which is close to the mean value estimated in [23]. the time between infection and detection is calibrated to be α−1r = 10 days, while the value α −1 m = 3.649 days is found during the intense screening period between 12 and 19 february 2020. the present study suggests a hospitalisation rate of w = 0.099 for 98% of detected individuals, while the remaining detected individuals go directly to hospitals, which is consistent with the results in [1], where the authors conclude that most patients were hospitalised after at least 5 days and that this delay can go up to 14 days. on the other hand, [10] reports an average recovery time after symptoms onset of 24.7 days and the mean time to death to be 17.8 days, while our fitting suggests a mean period between detection and hospital discharge by recovery or death of γ1 (γ1 + w)2 + w γ1 + w ( 1 γ1 + w + 1 γ2 ) = 20.832 days for 98% of detected individuals and 1/γ2 = 10.86 days for 2% of detected individuals, finally obtaining a mean time from detection to removal of 20.632 days. using parameter values in table iv, the basic reproduction number is estimated to be r0 = 2.118, close to the value 2.2 obtained in [1]. b. numerical simulations figure 2 shows the evolution of the cumulative number and number of active cases reported by the chinese government from 22 january to june 29 2020, as well as the result of fitting model (1) to this data, displaying good agreement with the real data. overall, our simulation results are in accordance with both real data and published findings. we obtained that biomath 11 (2022), 2212207, https://doi.org/10.55630/j.biomath.2022.12.207 4/8 https://doi.org/10.55630/j.biomath.2022.12.207 arino et al, a model for the early covid-19 outbreak in china with case detection and behavioural change jan -18 feb -02 feb -17 mar -03 mar -18 apr -02 apr -17 may -02 may -17 jun -01 jun -16 jul01 date 0 20000 40000 60000 80000 cu m ul at iv e ca se c ou nt real data fitted model (a) cumulative cases jan -18 feb -02 feb -17 mar -03 mar -18 apr -02 apr -17 may -02 may -17 jun -01 jun -16 jul01 date 0 10000 20000 30000 40000 50000 60000 nu m be r o f a ct iv e ca se s real data fitted model (b) active cases fig. 2: actual data and fitted solution from 22 january to 29 june 2020. dots are the real data and dotted lines are obtained from simulations. (a) cumulative number of reported cases; (b) active reported cases. the per capita per day transmissibility rate is about β = 0.346 days−1, giving a basic reproduction number of r0 = 2.11. the obtained values are fairly consistent with the approximations in [1]. figure 3 shows the percentage of undetected infected individuals (including asymptomatic, mild and symptomatic individuals) among the total number of covid19 cases. we observe, in the beginning, an increase of the percentage of undeclared infected individuals, reaching 66% during the outbreak and remaining above 50% until 3 february 2020. this significant percentage ensured that infection continued despite the isolation of detected cases, explaining in part the persistence of transmission of covid-19 during the early stages of the epidemic. to get more insight into the impact of the screening protocol change, we investigate the effect of the timing de c-0 4 de c-1 1 de c-1 8 de c-2 5 jan -01 jan -08 jan -15 jan -22 jan -29 fe b-0 5 fe b-1 2 fe b-1 9 fe b-2 6 ma r-0 4 ma r-1 1 ma r-1 8 ma r-2 5 ap r-0 1 ap r-0 8 ap r-1 5 ap r-2 2 ap r-2 9 date 0% 20% 40% 60% 80% 100% pe rc en ta ge undetected cases detected cases fig. 3: percentage repartition of undetected and detected infected individuals among the total number of infected individuals. of the modification of that protocol on the intensity of the covid-19 outbreak as shown in figure 4. we use the parameter values in table iv, but use αr until the day of the modification and αm afterwards. we observe that the sooner we adopt a less stringent set of symptoms needed to trigger testing, the lower the percentage of undetected infected individuals, leading to a reduction of outbreak intensity. figure 4a shows that the timing of screening criteria change strongly affects the burden of the disease. a change on 22 january 2020, for instance, leads to a burden equal to about a third of the burden that is observed when no change in screening criteria occur. this fact is explained by figure 4b, where, with policy change on 22 january, the percentage of hidden infected individuals declines immediately and exponentially, while it does not with criteria modification on 12 february. this implicitly confirms that the presymptomatic period contains hidden infectious individuals who contributed to the persistent transmission in the early stages of the covid-19 epidemic. we furthermore deduce that increasing the detection rate α early substantially helps to control the covid-19 epidemic. on the contrary, we observe that a late screening intensity increase after 12 february does not have remarkable effects in dampening the disease intensity. this might be due to behavioural changes of individuals or effectiveness of preventive measures. figures 5, 6 and 7 address the sensitivity of the dynamics of the covid-19 outbreak to the rate α of detection and the efficiency ε of self-protective measures. we assume no change in the screening strategy; other parameter values are taken from table iv. in figure 5, biomath 11 (2022), 2212207, https://doi.org/10.55630/j.biomath.2022.12.207 5/8 https://doi.org/10.55630/j.biomath.2022.12.207 arino et al, a model for the early covid-19 outbreak in china with case detection and behavioural change jan -18 jan -28 feb -07 feb -17 feb -27 mar -08 mar -18 mar -28 apr -07 apr -17 apr -27 may -07 may -17 may -27 jun -06 jun -16 jun -26 jul06 date 0 10000 20000 30000 40000 50000 nu m be r o f a ct iv e ca se s criteria modification on jan-24 criteria modification on feb-02 criteria modification on feb-12 criteria modification on feb-22 no criteria modification (a) active cases jan -18 jan -28 feb -07 feb -17 feb -27 mar -08 mar -18 mar -28 apr -07 apr -17 apr -27 may -07 may -17 may -27 jun -06 jun -16 jun -26jul06 date 0% 10% 20% 30% 40% 50% 60% pe rc en ta ge o f u nd et ec te d ca se s criteria modification on jan-24 criteria modification on feb-02 criteria modification on feb-12 criteria modification on feb-22 no criteria modification (b) percentage undetected cases fig. 4: effect of the timing of the relaxation of the criteria for screening, i.e., of the change from αr to αm. all dates in 2020. (a) number of detected active cases. (b) percentage of undetected cases among infected individuals. we observe that the outbreak peak is very sensitive to parameters α and ε. figure 6a and 7a show how serious the epidemic would be with a low detection rate (see near the ε axis). figure 6b is a zoom of figure 6a, around the region in the (ε,α)-space close to the fitted parameters for china. this confirms that the parameters found lie in a region where solutions are quite sensitive to parameter variations, confirming the significant sensitivity to parameter α observed in figure 5. iv. discussion in this work, we present a simple model for the spread of covid-19 taking into account undetected cases, the isolation of detected cases and education favouring the use of protective measures. we fitted this model to chinese data corresponding to the period from jan -18 fe b-0 7 fe b-2 7 ma r-1 8 ap r-0 7 ap r-2 7 ma y-1 7 jun -06 jun -26 date 0 10000 20000 30000 40000 50000 nu m be r o f a ct iv e ca se s = 0.56 = 0.63 = 0.7 = 0.76 (a) sensitivity to ε jan -18 fe b-0 7 fe b-2 7 ma r-1 8 ap r-0 7 ap r-2 7 ma y-1 7 jun -06 jun -26 date 0 20000 40000 60000 80000 100000 120000 140000 nu m be r o f a ct iv e ca se s = 0.02 = 0.08 = 0.14 = 0.21 (b) sensitivity to α fig. 5: number of active cases when (a) the efficiency ε of protective measures and (b) detection rates α vary, with all other parameters as in table iv. all dates in 2020. the start of the epidemic to the end of june 2020. the model does a good job of fitting that data, as can be seen in figure 2. in order to obtain this fit, though, we introduced two different values of the intensity α of screening: αm for a period of intense screening corresponding to a loose definition of symptoms required for screening and αr for a period with more restrictive set of symptoms leading to lower testing rates. the calibrated value αm = 0.438 is consistent with the results in [18–21] reporting the enhancement of the detection process on 12 february. taking the calibrated values, we then explore in more detail the effect of changing the intensity of screening. we saw in figure 3 that stringent criteria for screening giving a detection parameter αr = 0.17 led to an extended time period during which over 54% of the infected individuals evaded detection. these biomath 11 (2022), 2212207, https://doi.org/10.55630/j.biomath.2022.12.207 6/8 https://doi.org/10.55630/j.biomath.2022.12.207 arino et al, a model for the early covid-19 outbreak in china with case detection and behavioural change 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 nu m be r o f c as es a t t he p ea k 1e8 (a) sensitivity to ε and α 0.62 0.64 0.66 0.68 0.70 0.06 0.08 0.10 0.12 0.14 ( , ) for china o 20000 30000 40000 50000 60000 70000 80000 90000 nu m be r o f c as es a t t he p ea k (b) zoom around parameters found for china fig. 6: sensitivity of the number of active infected cases at the peak to the detection rate α and efficacy ε of protective measures. undetected infectious individuals may not know about their infection and keep interacting with the population causing new cases even among loved ones [8]. figure 4 strengthens the findings of figure 3 and emphasises the effect of detection strategy change. thus, the requirement that individuals show a large number of symptoms in order to be tested might have contributed to a longer persistence of the outbreak in china. interestingly, modification of screening intensity after 12 february 2020 does not have much effect (figure 4). this may be because, as the epidemic was well established at the time, public awareness of the crisis had increased concomitantly with an expansion of the set of public protection measures, leading to an increase in 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.2 0.4 0.6 0.8 1.0 100 101 102 103 104 105 106 107 108 109 nu m be r o f c as es a t t he p ea k (a) sensitivity to ε and α (logarithmic scale) 0.62 0.64 0.66 0.68 0.70 0.06 0.08 0.10 0.12 0.14 the case of china o 103 104 105 106 nu m be r o f c as es a t t he p ea k (b) zoom around parameters found for china fig. 7: sensitivity of the number of active infected cases at the peak to the detection rate α and efficacy ε of protective measures (logarithmic scale). uptake of a wider variety of measures. besides detecting infected individuals before illness onset and isolating them, thereby reducing the chance of transmission of the disease to susceptible individuals, reporting the real number of infected individuals alerts the population about the actual danger presented by the disease. this means that more individuals, including undetected infected individuals, change their behaviour and consider all possible actions to protect themselves or others from the infection. figure 5 considers the sensitivity of the covid-19 outbreak dynamics to the efficacy of self-protective measures and detection rates, when these parameters are near the parameter values found for china. it shows that the outbreak is biomath 11 (2022), 2212207, https://doi.org/10.55630/j.biomath.2022.12.207 7/8 https://doi.org/10.55630/j.biomath.2022.12.207 arino et al, a model for the early covid-19 outbreak in china with case detection and behavioural change sensitive to both parameters, with a particularly marked sensitivity to α, the rate of detection. the contour plot in figure 6 confirms this: movement along the (selfprotective measures) ε axis induces less variation than movement along the (detection) α axis. altogether, this highlights that good detection, for instance by deploying more tests in highly affected areas and using strategies favouring the tracing of infected individuals, has a significant effect on early spread. according to figure 6, this provides more capacity to control spread than behavioural changes and efficacy of protective measures whose use is made obligatory when detection rates are low. it would be interesting to study an optimal control problem considering the 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original article the impact of infected t lymphocyte burst rate and viral shedding rate on optimal treatment scheduling in a human immunodeficiency virus infection anuraag bukkuri moffitt cancer center anuraag.bukkuri@moffitt.org received: 27 april 2020, accepted: 17 august 2020, published: 12 september 2020 abstract—we consider a mathematical model of human immunodeficiency virus (hiv) infection dynamics of t lymphocyte (t cell), infected t cell, and viral populations under reverse transcriptase inhibitor (rti) and protease inhibitor (pi) treatment. existence, uniqueness, and characterization of optimal treatment profiles which minimize total amount of drug used, viral, and infected t cell populations, while maximizing levels of t cells are determined analytically. numerical optimal control experiments are also performed to illustrate how burst rate of infected t cells and shedding rate of virions impact optimal treatment profiles. finally, a sensitivity analysis is performed to detect how model input parameters contribute to output variance. keywords-optimal control, hiv, reverse transcriptase inhibitors, protease inhibitors msc: 92c50, 93c15 i. introduction hiv, also known as the human immunodeficiency virus, is a virus which impairs the immune system by entering vital cells (e.g. dendritic cells, microglial cells, cd4+ t cells, and macrophages) through cd4 and coreceptors on the cell membrane and destroying them [cunningham et al. 2010; cenker et al. 2017]. once the cd4+ levels drop to a critical level, cell-mediated immunity is lost and viral load increases, which leads to the development of acquired immunodeficiency syndrome (aids), the final stage of hiv infection. when this occurs, due to low immunity, the patient’s body becomes a breeding ground for opportunistic infection and aids defining cancers such as kaposi sarcoma, non-hodgkin lymphoma and cervical cancers. without treatment, average survival time after hiv infection is 9-11 years [unaids 2007]. pioneering research done by scientists in the copyright: c© 2020 bukkuri. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: anuraag bukkuri, the impact of infected t lymphocyte burst rate and viral shedding rate on optimal treatment scheduling in a human immunodeficiency virus infection, biomath 9 (2020), 2008173, http://dx.doi.org/10.11145/j.biomath.2020.08.173 page 1 of 12 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2020.08.173 a. bukkuri, the impact of infected t lymphocyte burst rate and viral shedding rate on optimal ... fig. 1. cellular structures and processes involved in cell-to-cell transmission of hiv. panels a-g show different pathways for hiv (green dots) cell-cell transmission between donor (green) and target (pink) cells. 1990s determined that direct cell-to-cell contact is an important, and perhaps the predominant, contributor to the propagation of hiv within a host. specifically, hiv spreads by entering immune cells and subverting intercellular communication to enable their own viral replication through structures such as tunneling nanotubes and filopodia. immunological synapses and phagocytosis of infected cells are also hijacked by hiv and used as gateways to infect target cells. furthermore, hiv is able to elicit fusion between infected donor and target cells, forming horrific infected cell clusters with a very high capacity for viral reproduction and survival. there are many mechanisms by which cell-cell transfer of hiv occurs, some of which are summarized in figure 1 from [bracq et al. 2018]. for more details on how these mechanisms work, refer to [bracq et al. 2018]. the main treatments for hiv are a class of drugs called antiretrovirals (art). often, these drugs are given in combination to suppress viral replication and reduce plasma hiv viral load in a treatment called highly active antiretroviral therapy (haart) [autran et al. 1997; komanduri et al. 1998; lederman et al. 1998]. two common such arts often used in haart are reverse transcriptase inhibitors and protease inhibitors [arts et al. 2012]. reverse transcriptase inhibitors work to terminate transcription, thus inhibiting viral replication. they do this in one of two ways: nucleoside/nucleotide reverse transcriptase inhibitors incorporate into the nascent dna strand during reverse transcription, whereas non-nucleoside reverse transcriptase inhibitors bind to non-catalytic enzyme sites and is usually mediated through steric hindrance that impedes structural changes in hiv reverse transcriptase [gulnik et al. 1995; gotte et al. 2000]. protease inhibitors are usually substrate-based inhibitors which are designed specifically against the viral protease based on its crystal structure. they act on this viral protease, inhibiting maturation of new viral particles. by doing this, they attack the already formed hiv before the next cycle of infection begins [michaud et al. 2012]. figure 2, reproduced from [michaud et al. 2012], depicts the hiv life cycle with various biomath 9 (2020), 2008173, http://dx.doi.org/10.11145/j.biomath.2020.08.173 page 2 of 12 http://dx.doi.org/10.11145/j.biomath.2020.08.173 a. bukkuri, the impact of infected t lymphocyte burst rate and viral shedding rate on optimal ... fig. 2. depiction of hiv life cycle with entry, protease, rt, and integrase inhibitor drug targets antiretroviral drug targets to more clearly elucidate how they work. currently, physicians primarily use three markers to assess the stage of the disease and create treatment strategies: cd4 t cell count, viral load, and drug resistance [penta 2004]. through optimal control and sensitivity analyses, we propose the inclusion of two more criteria into this assessment: viral shedding rate and infected t cell burst rate. medically, it is possible to measure the rate of viral shedding using polymerase chain reaction (pcr) from collected samples as done in [tronstein et al. 2011]. burst rate of infected t cells can be measured using standard burst rate analysis techniques, such as through viral inhibitors, washouts of infected cells [dimitrov et al. 1993; eckstein et al. 2001], quantitative image analyses with in situ hybridization, quantitative competitive rt-pcr of bulk tissue or single cells at limiting dilution [chun et al. 1997; haase et al. 1996; hockett et al. 1999; chen et al. 2007; hyman et al. 2009]. in this paper, we first summarize a model of hiv infection with rti and pi treatment developed by [mobisa et al. 2018]. then, in section 3, a detailed optimal control analysis will be performed, including existence, uniqueness, and analytical characterization results. in section 4, parameter estimates are provided and several numerical simulations were performed, specifically analyzing the impact of changes to the burst rate of infected cells and shedding rate of virions on optimal treatment profiles. in section 5, a sensitivity analysis is performed to quantify the amount of impact each parameter value has on healthy t cell, infected t cell, and viral population dynamics. ii. model description the model for hiv infection dynamics among t cells (t), infected t cells (u), and free virus particles (v) under rti (u) and pi (v) treatment is taken from a model created by [mobisa et al. 2018] and is reproduced below: dt dt = rt ( 1 − t tmax ) − (1 −u)βv t −(1 −v)αtu −µt du dt = (1 −u)βv t + (1 −v)αtu −ku dv dt = wku − cv biomath 9 (2020), 2008173, http://dx.doi.org/10.11145/j.biomath.2020.08.173 page 3 of 12 http://dx.doi.org/10.11145/j.biomath.2020.08.173 a. bukkuri, the impact of infected t lymphocyte burst rate and viral shedding rate on optimal ... in this model, the t cells are produced at rate r and die naturally at rate µ; the growth of the t cells is bounded with the enforcement of a carrying capacity. the healthy t cells become infected by the free virus and actively infected t cells via the mass infection terms βv t and αtu, respectively. the infected t cells die naturally at a rate of k. furthermore, these cells produce free viruses, as captured by the first term in the viral equation, which are then removed from circulation at a rate c per virus. note that the viral decline depends on the amount and efficacy of the rti and pi treatments. iii. optimal control analysis to determine best treatments strategies for hiv infection, we formulate our problem as an optimal control one. we aim to maximize the number of healthy t cells, while minimizing the number of infected t cells, free viruses, and drugs u and v used over a fixed therapy horizon [0,t]. our control class are piecewise continuous functions defined for all t such that 0 ≤ u(t) ≤ 1 where u(t) = 1 represents maximal rti treatment and u(t) = 0 represents no rti treatment. similarly, v(t) = 0 represents no pi treatment and v(t) = 1 represents maximal pi treatment. we can then describe the class of admissible controls as u(t) = u(t),v(t) piecewise continuous s.t. 0 ≤ u(t),v(t) ≤ 1,∀t ∈ [0,t]. now, we define our objective functional and optimal control problem. for a fixed therapy interval [0,t], maximize the objective functional j(u,v) = ∫ t 0 at−bu−fv− 1 2 φ1u 2− 1 2 φ2v 2dt (1) over all lebesgue-measurable functions u : [0,t] → [0,umax], v : [0,t] → [0,vmax], subject to the above ode dynamics and initial conditions of t(0) = 1000, u(0) = 1, v(0) = 0.1. note that the objective functional is quadratic, instead of linear, in u and v to ensure that the resulting optimal controls can take on a continuum of values from 0 to 1, rather than being binarily confined to either 0 or 1. a. existence of optimal control using the theory developed by [fleming et al. 1975], we can determine the existence of an optimal control for our state system. specifically, boundedness of solutions of the system for finite time is required for existence and uniqueness of an optimal control. using the technique of supersolutions and keeping in mind that t ≤ tmax, upper bounds on the solutions of the state system are found. consider the following system: dt̄ dt = rt̄ dū dt = βv̄ tmax + αūtmax dv̄ dt = wkū note that the supersolutions t̄ , ū, and v̄ of this system are bounded on a finite time interval. we can also re-write this system in matrix form as follows: t̄ū v̄   ′ =  r 0 00 αtmax βtmax 0 wk 0    t̄ū v̄   , (2) where ′ = d dt . since this is a linear system in finite time with bounded coefficients, the supersolutions t̄, ū, and v̄ are uniformly bounded. we can now prove existence of an optimal control as done in [bukkuri 2019]. theorem 1: there exists optimal controls u and v that maximizes the objective functional j(u,v) if the following conditions are met: 1) the class of all initial conditions with controls u and v such that u and v are lebesgue integrable functions on [0,t] with values in the admissible control set along with each state equation being satisfied is not empty 2) the admissible control set is closed and convex 3) the right hand side of the state system is continuous, is bounded above by a sum of the bounded control and the state, and can be written as a linear function of u and v biomath 9 (2020), 2008173, http://dx.doi.org/10.11145/j.biomath.2020.08.173 page 4 of 12 http://dx.doi.org/10.11145/j.biomath.2020.08.173 a. bukkuri, the impact of infected t lymphocyte burst rate and viral shedding rate on optimal ... with coefficients depending on time and the state variables 4) the integrand of the functional is concave on the admissible control set and is bounded above by c3−c2|u|θ−c1|v|ψ, where c2,c1 > 0, and θ,ψ > 1. proof: first, from an existence result in [lukes et al. 1982], since our state system has bounded coefficients and any solutions are bounded on the finite time interval, we obtain the existence of the solution of the state system. second, the admissible control set is closed and convex, by definition. for the third condition, the right hand side of the state system is continuous since each term with a denominator is nonzero. moreover, the system is bilinear in the controls and can be rewritten as ~f(t, ~x,u,v) = ~γ(t, ~x), (3) where ~x = (t,u,v ) and ~γ is a vector-valued function of ~x. since the solutions are bounded, we have |~f(t, ~x,u,v)| ≤ ∣∣∣∣∣∣  r 0 00 αtmax βtmax 0 wk 0    tu v   ∣∣∣∣∣∣ ≤ c1|~x|, (4) where c1 depends on the coefficients of the system. also, note that the integrand of j(u,v) is concave on the admissible control set. the existence of optimal control follows from the fact that at − bu − fv − 1 2 φ1u 2 − 1 2 φ2v 2 ≤ c3 − c2|u|θ − c1|v|ψ, where c2,c1 > 0, and θ,ψ > 1, since t(t) ≤ tmax. b. characterization of optimal control now, we will characterize the optimal control pair (u,v) using a version of the pontryagin maximum principle (pmp), as done in [bukkuri 2019; bukkuri 2020]. first, let’s define the lagrangian associated with j(u,v): l = at − bu −fv − 1 2 φ1u 2 − 1 2 φ2v 2 + λ1 ( rt ( 1 − t tmax ) − (1 −u)βv t − (1 −v)αtu −µt ) + λ2((1 −u)βv t + (1 −v)αtu −ku) + λ3(wku − cv ) + ν1(t)u + ν2(t)(1 −u) + δ1(t)v + δ2(t)(1 −v) here, the ν and δ terms have been added in as penalties for non-optimal dosing patterns, i.e. ν1(t)u = ν2(t)(1−u) = δ1(t)v = δ2(t)(1−v) = 0 for the optimal controls (u∗,v∗). theorem 2: given optimal controls u∗ and v∗ and solutions of the corresponding state system, there exist adjoint variables λi for i = 1, 2, 3 such that dλ1 dt =− ∂l ∂t = − [ a+λ1 ( uα(v−1)−r ( t tmax −1 ) − tr tmax −µ +v β(u−1) ) +λ2(−uα(v−1))−v β(u−1) ] , dλ2 dt =− ∂l ∂u = −[−b+λ1(tα(v−1))+λ2(−k−tα(v−1)) +λ3(kw)], dλ3 dt =− ∂l ∂v = −[−f +λ1(tβ(u−1))+λ2(−tβ(u−1))−λ3(c)], where λi(t) = 0 for i = 1, 2, 3 by the pmp transversality condition. moreover, u∗ is given by: u∗ = min ( max ( 0, λ1tv β−λ2tv β b1 ) , 1 ) , (5) while v∗ is similarly given by: v∗ = min ( max ( 0, λ1tuα−λ2tuα b2 ) , 1 ) . (6) biomath 9 (2020), 2008173, http://dx.doi.org/10.11145/j.biomath.2020.08.173 page 5 of 12 http://dx.doi.org/10.11145/j.biomath.2020.08.173 a. bukkuri, the impact of infected t lymphocyte burst rate and viral shedding rate on optimal ... table i parameters used in numerical simulations parameter meaning estimated value tmax carrying capacity of healthy t cells 1500 cells/mm3 r production rate of uninfected t cells 0.03 cells/day µ natural death rate of uninfected t cells 0.02 cells/day k death rate of infected t cells due to viral lysis 0.24 cells/day c shedding rate of virions 2.4/day β viral infection rate by free virions 2.4 ∗ 10−5/mm3 α cellular infection rate 2.4 ∗ 10−5/mm3 w burst rate of infected t cells greater than 0 u input function for rti treatment 0-1 v input function for pi treatment 0-1 proof: to maximize the lagrangian (with respect to the optimal control pair variables), we differentiate l with respect to u and v. from this, we get: ∂l ∂u = −b1u + λ1tv β −λ2tv β + ν1(t) −ν2(t) (7) thus, the representation of u∗ is λ1tv β−λ2tv β b1 . ∂l ∂v = −b2v + λ1tuα−λ2tuα + δ1(t) −δ2(t) (8) thus the representation of v∗ is λ1tuα−λ2tuα b2 . since 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1, the explicit control profiles given in equations 5 and 6 are determined. moreover, since both state and adjoint solutions are l∞-bounded, the right side of the adjoint and state equations are lipschitz for those solutions. this ensures that the solutions to the optimality system are unique (and thus the optimal controls are unique), assuming the final time is not very large. a rigorous proof of such an argument can be found in [burden et al. 2004] and [fister et al. 1998]. iv. numerical simulations we now aim to numerically implement our optimal control solutions to visualize optimal drug treatment strategies. the optimality system can be thought of as a two-point boundary value problem, which was solved using a fourth-order iterative runge-kutta scheme, as done in [jung et al. 2002]. in this scheme, a forward sweep of the state equations with initial guesses for u and v was performed. then, a backward calculation using the adjoint equation and an update of the controls was made as done by [duda 1997] and [deininger et al. 2003]. this was iteratively performed until convergence was obtained. the parameter values in table i represent standard hiv infection values, were obtained from [mobisa et al. 2018], and were used in the following optimal control simulations. with these parameter values and the aforementioned initial conditions, several optimal control profiles were ran. the following objective functional weighting terms were used (derived from scaling the healthy and infected t cells and virus populations appropriately): a = 0.00125, b = 0.01, f = 1, φ1 = φ2 = 0.2. the following initial conditions were also tested, but did not lead to any qualitatively different results, so their results are omitted here: t(0) = 1000, u(0) = 0, v(0) = 0.001; t(0) = 800, u(0) = 10, v(0) = 0.001; t(0) = 900, u(0) = 5, v(0) = 0.01. first, we run a control simulation. in this simulation, we let w = 1000 and c = 2.4. as explained in [mobisa et al. 2018], w varies greatly and we shall later explore how optimal treatment protocols change when the burst rate is much lower or higher. the value of 2.4 for c is based on viral infectivity assays which found that hiv-1 strains biomath 9 (2020), 2008173, http://dx.doi.org/10.11145/j.biomath.2020.08.173 page 6 of 12 http://dx.doi.org/10.11145/j.biomath.2020.08.173 a. bukkuri, the impact of infected t lymphocyte burst rate and viral shedding rate on optimal ... fig. 3. hiv control simulation fig. 4. impact of burst rate of infected t cells (w) iiib and rfii which lost half of their infectivity in 4-6 h at 37◦c [layne et al. 1989]. however, it’s clear that different strains of hiv in different individuals and at different temperatures may lead to different viral shedding rates. we shall soon see the impact of lower viral shedding rates on optimal protocols. figure 3 is a depiction of optimal rti and pi treatment for our control case. note that, in the simulations below, 100 time steps is equivalent to one day; thus, the following simulations were run for over 600 days. from this, it’s clear to see that the optimal treatment calls for full intensity of both rti and pi treatment for about 290 days. now, let’s take a look at what happens when we change the burst rate. in figure 4, the upper image represents w = 1 and the lower image is for w = 100, 000. in the case of the low burst rate, we notice that the optimal profiles does not change much–the only change is that it takes slightly longer to reach the maximal dosage for both rti and pi treatment. however, it is clear that the high burst rate has a significantly different profile. specifically, we see the emergence of bang-bang controls for both rti and pi treatment; moreover, we notice that the rti treatment is given for slightly longer (≈ 20 days) in the beginning than the pi treatment. both treatments then have a drug holiday until ≈ day 210, after which both drugs are given at full potency for ≈ 80 days, before all treatment is stopped. let’s now consider the impact of different shedding rates, for the following values for c: 0.1, 0.2, 0.3, and 5. in figure 5, the upper three images represent c values of 0.1, 0.2, and 0.3, respectively, while the bottom image represents a c value of 5. in these simulations, we again see a departure from our control optimal treatment recommendations. first, consider the high shedding rate value. in this case, optimal treatment recommends full dosage treatment of rti and pi for ≈ 130 days instead of ≈ 280 days, before ceasing all treatment. now, consider the lower shedding rates. when c is 0.1, we notice that the rti treatment is given at full dosage for ≈ 190 days, before switching to rapid bang-bang controls (between full and no treatment) until ≈ 300 days. a drug vacation is then given for ≈ 90 days, before giving the rti biomath 9 (2020), 2008173, http://dx.doi.org/10.11145/j.biomath.2020.08.173 page 7 of 12 http://dx.doi.org/10.11145/j.biomath.2020.08.173 a. bukkuri, the impact of infected t lymphocyte burst rate and viral shedding rate on optimal ... fig. 5. impact of shedding rate of virions (c) treatment at full potential for ≈ 20 more days, then ceasing treatment. for the pi treatment, we notice that the treatment is prescribed at full potency for the first ≈ 80 days, before it gradually decays to no treatment over the next ≈ 20 days. the treatment again picks up around day 210, and switches to rapid bang-bang controls like the rti until about day 300. then we note the presence of bang-bang controls (mostly remaining at no treatment, though) during the rti drug holiday, before continuing full treatment at ≈ day 380 for around 30 days, then ceasing all treatment. when the c value is 0.2, the optimal protocols change drastically again. the rti treatment is prescribed at full potency for the first ≈ 145 days, before declining to 0 treatment over the next 5 days or so. a drug holiday is then taken for around the next 250 days, after which the treatment is gradually increased (over ≈ 90 days) until the end of the considered therapy horizon. the pi treatment is given at full potential for the first ≈ 90 days; it then rapidly declines and stays at no treatment for the rest of the 600 day interval. finally, let’s consider what happens when the shedding rate is 0.3. in this case, the optimal profile includes an rti treatment which stays at full potency for the entire interval. the pi treatment is given at its maximum for ≈ 190 days, before rapidly declining to no treatment, and then gradually increasing back to full potency (over an ≈ 80 day period) after about 10 days, and staying at this maximum level for the rest of the therapy horizon. thus, from the above numerical experiments, we can clearly see that the standard optimal treatment protocol, which calls for full treatment of both rti and pi for ≈ 290 days, can drastically change depending on each patient’s specific infected t cell burst rate and viral shedding rate. v. sensitivity analysis to further assess the impact of parameter values on hiv dynamics we perform a sensitivity analysis on our ode system using the sobol-martinez method. other than u and v, which were given ranges of 0 to 1, all other parameters were given biomath 9 (2020), 2008173, http://dx.doi.org/10.11145/j.biomath.2020.08.173 page 8 of 12 http://dx.doi.org/10.11145/j.biomath.2020.08.173 a. bukkuri, the impact of infected t lymphocyte burst rate and viral shedding rate on optimal ... fig. 6. sensitivity analysis of hiv with rti and pi treatments biomath 9 (2020), 2008173, http://dx.doi.org/10.11145/j.biomath.2020.08.173 page 9 of 12 http://dx.doi.org/10.11145/j.biomath.2020.08.173 a. bukkuri, the impact of infected t lymphocyte burst rate and viral shedding rate on optimal ... a range of 10% lower to 10% higher than the values indicated in the table in section 5. the sobol-martinez method uses variance decomposition techniques to measure the contributions of input parameters to output variance. the algorithm outlined by [zhang et al. 2015] was implemented here. the left panel of each image is the first order sobol index: the contribution to the output variance by the single input alone. the right panel is the total sobol index: the contribution to the output variance caused by a model input, including the first-order effects as well as all higher-order interactions. note that, in the figure 6, uther is equivalent to u and vther is equivalent to v. also, all parameters with sensitivity orders less than 0.01 were omitted from the figures. first, consider the healthy t cells. in this case, we see three main parameters which impact the t cell population: the burst rate of the infected t cells, the rti treatment, and the viral shedding rate. the other parameters which had minor effects on t cell population dynamics are the natural death rate of healthy t cells and the viral infection rate by free virions. thus, the most effective treatments, solely in terms of the cd4+ cell population, are the rti treatment and those that target the burst rate of infected t cells and the shedding rate of virions. considering the infected t cells, we note that almost all the parameters are effective in changing its population dynamics–thus the sensitivity analysis cannot be very enlightening for treatment planning. however, when we consider the viral graph, we see that the the shedding rate is most significant, followed by the burst rate of infected cells. thus, for the overall desired dynamics, we deem that further medical research into the development of drugs which specifically target the shedding rate of virions and the burst rate of infected cells is advised, as these are the most effective ways of minimizing the virus and infected t cell population, while maximizing the healthy t cell count. these results are in accordance with our numerical optimal control experiments, which show great changes in optimal treatment protocols when shedding and burst rate parameters are changed. vi. conclusion in this paper, we performed optimal control and sensitivity analyses of an hiv model with reverse transcriptase inhibitor and protease inhibitor antiretroviral treatments. proofs of existence, uniqueness, and analytical characterization of the optimal control profile were given. numerical optimal control simulations were performed and it was found that burst rate of infected t cells and viral shedding rate have great impacts on optimal control profiles, often suggesting intricate bang-bang controls for the treatment. we hope that the results of this analysis will help physicians more effectively assess and treat hiv patients. specifically, we hope physicians will measure and include viral shedding rates and infected t cell burst rates into treatment consideration. moreover, we 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pharmacology models. cpt pharmacometrics syst. pharmacol. 4, 69-79. biomath 9 (2020), 2008173, http://dx.doi.org/10.11145/j.biomath.2020.08.173 page 12 of 12 http://dx.doi.org/10.11145/j.biomath.2020.08.173 introduction model description optimal control analysis existence of optimal control characterization of optimal control numerical simulations sensitivity analysis conclusion references original article biomath 3 (2014), 1404131, 1–11 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum on nonlinear dynamics of the stat5a signaling protein elena nikolova, ivan jordanov, nikolay vitanov institute of mechanics bulgarian academy of sciences, sofia, bulgaria emails: elena@imbm.bas.bg, i jordanov@email.bg, vitanov@imbm.bas.bg received: 2 september 2013, accepted: 13 april 2014, published: 29 may 2014 abstract—in this paper we model dynamics of cross talk between mek/erk and jak/stat signaling pathways by means of nonlinear ordinary differential equations. the considered system of four ordinary differential equations is reduced to one ordinary differential equation, representing dynamics of the phosphorylated stat5a signaling protein. we show that the diffusion together with the corresponding biochemical reactions is likely to play a critical role in governing the dynamical behavior of the considered signaling protein. by the modified method of simplest equation to the described reaction-diffusion equation we obtain an analytical solution which explains drop and jump propagation of the stat5a protein concentration. keywords-stat5a signaling protein; pdes; modified method of simplest equation; analytical solution; drop and jump propagation; i. introduction the idea to modeling proteins interactions in the intracellular environment by spatial-temporal systems is based on the following circumstance: while the dynamics of the interacting proteins and their molecular pathways and networks can be described by reaction models, the heterogeneous distributions of the protein concentrations are not taken into consideration. it is proved, however, that similar inhomogeneous protein distributions in the form of cellular jump or drop propagation play an important role in the control of the main processes in the cell. in this way the cellular complexity appears to be space-temporal, expressed mathematically by reaction-diffusion models. in fact, the traditional approximation scheme of a well-stirred reactor is most used and somewhat plausible simplification. however, the concentration gradients of cell enzymes that modulate signal transduction refute this simplification [1–4]. the role of diffusion in reaction-diffusion systems of the cell becomes significant when reactions are relatively faster (but not so very) than diffusion rates. sometimes the term ‘molecular crowding’ is used to denote more specific type of spatial distribution [5, 6]. the biochemical essence of this phenomenon is based on the fact that the state of phosphorylation of target molecules with spatially separated membrane-localized protein kinases and cytosolic phosphatases depends essentially on diffusion. then the very high protein density in the intracellular space, or so called molecular crowding, mentioned above can enhance the spatial effect. consequently, molecular crowdcitation: elena nikolova, ivan jordanov, nikolay vitanov, on nonlinear dynamics of the stat5a signaling protein, biomath 3 (2014), 1404131, http://dx.doi.org/10.11145/j.biomath.2014.04.131 page 1 of 11 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2014.04.131 e nikolova et al., on nonlinear dynamics of the stat5a signaling protein ing can also alter protein activities and break down classical reaction kinetics [5]. therefore here we introduce reaction-diffusion modeling and its computational tools to concrete example of erk (extracellular signal-regulated kinase) and stat (signal transducers and activators of transcription) protein interaction. in [7] the cross talk between the mek/erk and jak/stat signaling pathways has been modeled in a form of a system of nonlinear ordinary differential equations for the protein concentrations (jak is an abbreviation of janus kinase, and mek is an abbreviation of mapk/erk kinase). next, in the same paper the spatial modeling of the mentioned interaction has been performed, by introducing an appropriate diffusion-reaction scheme. by the accomplished stability analysis in [7] the authors concluded that in terms of the turing bifurcation in erk and stat dynamical model the mentioned above crowding effect can be interpreted as a process of stabilization of the dissipative structures inherent to the considered intracellular process. ii. themethod of the simplest equation there are many approaches for obtaining analytic solutions of nonlinear partial differential equations [8–10]. in this paper we will use the modified method of the simplest equation. the method is a modified version of the method of the simplest equation, created by n. a. kudryashov [11, 12] that is based on the fact that after application of an appropriate ansatz a large class of nonlinear pdes can be reduced to nonlinear odes of the kind (p means polynomial) p(f(ξ), df dξ , d2 f dξ2 , ...) = 0 (1) and for some equations of the kind (1) particular solutions can be obtained which are finite series f(ξ) = n∑ i=0 ai[φ(ξ)] i (2) constructed by the solution φ(ξ) of a simpler equation referred to as the simplest equation. the simplest equation can be the equation of bernoulli, equation of riccati, etc. the substitution of (2) in (1) leads to the polynomial equation p = σ0 + σ1φ + σ2φ 2 + ... + σrφ r = 0 (3) where the coefficients σi, i = 0, 1, ..., r depend on the parameters of the equation and on the parameters of the solutions. equating all these coefficients to zero, i.e., by setting σi = 0, i = 0, 1, ..., r (4) one obtains a system of nonlinear algebraic equations. each solution of this system leads to a solution of kind (2) of (1). in order to obtain a non-trivial solution by the above method we have to ensure that σr contains at least two terms. to do this within the scope of the modified method of the simplest equation we have to balance the highest powers of φ that are obtained from the different terms of the solved equation of kind (1). as a result of this we obtain an additional equation between some of the parameters of the equation and the solution. this equation is called a balance equation [13,14]. we note that the method of the simplest equation and its modified version are closely connected to the problem for obtaining meromorphic solutions of nonlinear partial differential equations [15, 16]. by the methodology described in [15, 16] one can obtain other interesting classes of solutions of nonlinear pdes such as rational solutions for example. in addition we stress that by means of the traveling wave ansatz one reduces the nonlinear pde to a nonlinear ode and after this if an appropriate simplest ode exists then a particular solution can be obtained that usually depends on as many parameters of the problem as possible. in many cases such particular solutions are among the few possible exact analytic solutions of the studied nonlinear pde. iii. the interaction between erk and stat signaling pathways a. a reaction model and its reduction the specificity of biological responses is often achieved in a combinatorial fashion through the biomath 3 (2014), 1404131, http://dx.doi.org/10.11145/j.biomath.2014.04.131 page 2 of 11 http://dx.doi.org/10.11145/j.biomath.2014.04.131 e nikolova et al., on nonlinear dynamics of the stat5a signaling protein concerted interaction of signaling pathways [17]. here we will discuss the eventual interaction between the units of mek/erk and jak/stat signaling pathways. in cell signaling the pathways are understood as networks of recurrent biochemical reactions, by which the information transmission (in a signal form) is accomplished. the disturbance of the intracellular signal transmission from the membrane receptors to the nucleus genes is assumed as a general reason for a cancer diseases progress. in [17] the authors discussed the basic features of interaction domains, and suggested that rather simple binary interactions can be used in sophisticated ways to generate complex cellular responses. moreover, in [18], the protein stat was found to play important roles in numerous cellular processes including immune responses, cell growth and differentiation, cell survival and apoptosis, and oncogenesis. the stat target genes include socs/cis, a class of inhibitory proteins that interfere with stat signaling through several mechanisms. (socs is an abbreviation of suppressor of cytokine signaling and cis means cytokine inducible sh2 domain containing). the protein socs/cis can inhibit the stat phosphorylation and block the access of stat to receptors of jaks or both [19]. on the other hand, socs-3 can bind to sequester the ras-gap proten (the ras proteins are members of the mek/erkpathway)[20]. however, this is not only the way of interaction between stat and erk pathways. in [21–22], the authors suggested that the stat5 functional capacity of binding dna could be influenced by the mitogen-activated protein kinase (mapk)-pathway. later on, in [23] the interactions between stat5a and erk1 (or erk2) signaling proteins was considered. a simple biochemical diagram of this interaction is presented in fig. 1. according to fig. 1 in [7] the following system of ordinary differential equations for the kinetics of stat5a/s phosphorylation and erk activation was constructed: de1 dt = −k1e1 s1 + k2e2, d s1 dt = −k1e1 s1 + k3 s2 + i de2 dt = k1e1 s1 − k2e2, d s2 dt = k1e1 s1 − k3 s2 − i (5) fig. 1. biochemical diagram of the interaction between stat5a and erk proteins. here e1, e2, s1 and s2 are state variables, representing concentrations of erk-inactive, erk-active, stat5a-non-phosphorylated and stat5a-phosphorylated proteins respectively. the following initial conditions are put on the system (5): e1(0) = 10 −3, s1(0) = 10 −2, e2(0) = 0, s2(0) = 0 they correspond to the initial concentrations of the above–mentioned proteins in mm units [27]. moreover, k1 is proportional to the frequency of collisions of erk and stat5a protein molecules and presents a rate constant of reactions of associations; k2 and k3 are constants of exponential growths and disintegration; i is an inhibitor source respectively. the source i inhibits the phosphorylation of non-phosphorylated stat5a. more concrete interpretation of the inhibitor i can be given by considering the role of thee socs proteins in linking the jak/stat pathway. biological responses elicited by the jak/stat pathway are modulated by inhibition of jak (and respective attenuation of stat) by a member of the socs proteins [24, 25]. in [7] the inhibitor i is presented mathematically in the following manner: biomath 3 (2014), 1404131, http://dx.doi.org/10.11145/j.biomath.2014.04.131 page 3 of 11 http://dx.doi.org/10.11145/j.biomath.2014.04.131 e nikolova et al., on nonlinear dynamics of the stat5a signaling protein i = k0σ (6) where σ is a constant concentration of socs proteins and k0 is a reaction rate constant of inhibition respectively. it is clear that the considered interaction between erk and stat pathways can occur only if σ is sufficiently small, i.e. phosphorylation of the protein stat5a exists. therefore we assume the concentration of socs proteins to be sufficiently small. in addition in [7] the authors reduced (5) to the following two-dimensional system: de2 dt = k1 e0s 0 − (k1s 0 + k2)e2 − k1 e0 s2 + k1e2 s2 d s2 dt = −k0σ + k1 e0s 0 − k1s 0e2 (7) −(k1 e0 + k3)s2 + k1e2 s2 taking into account that only two equations of the four ones are independent. this means that between the concentrations e1. e2, s1 and s2 the following relations exist: e1 + e2 = e0, s1 + s2 = s 0 (8) where e0 and s 0 are initial values of the sums of corresponding concentrations of inactive and active erks and non-phosphorylated and phosphorylated stats. we assume for (7) that the inequality e2 << s2 holds, in view of the fact that the amount of erk molecules is essentially smaller than the amount of stat5a ones [21–23]. in this way the inequality e2/s2 = � << 1 will be valid, and (7) can be written as: � de2 dt = k1 e0s 0 − (k1s 0 + k2)e2 − k1 e0 s2 + k1e2 s2 d s2 dt = −k0σ + k1 e0s 0 − k1s 0e2 (9) −(k1 e0 + k3)s2 + k1e2 s2 next, in accordance with the qssa (quasisteady-state approximation) theorem [26] we consider the first equation of (9) to be linear with respect to e2 and we treat it as an attached system [26], i.e. we take into consideration that e2 is a sufficiently small ”constant“. further according to the requirements of the qssa theorem we prove that the attached system has a stable steady state (then well-known lyapunov definition of stability is satisfied). after replacing the steady state value e02 = k1 e0(s 0 − s2) k1(s 0 − s2) + k2 > 0 (10) in the second equation of (9) (the degenerate system), the quasi-stationary approximation of (7) (or (5)) is obtained in the form d s2 dt = k21 e0 s 2 2 − 2k 2 1 e0s 0 s2 + k 2 1 e0s 2 0 k1 s2 − k1s 0 − k2 − (11) −(k1 e0 + k3)s2 + k1 e0s 0 − k0σ next, for our convenience we rewrite (11) in the following manner: d s dt = as2 − bs + c ds − e − f s + g (12) where we denote the variable s2 only by s. according to (11) the new coefficients a = k21 e0, b = 2k 2 1 e0s 0, c = k 2 1 e0s 2 0, d = k1, e = k1s 0 + k2, f = k1 e0 + k3, (13) g = k1 e0s 0 − k0σ are positive because we assumed that the initial concentration of the stat5a protein is large, the erk concentration is smaller than the stat concentration, but bigger than the concentration of the socs proteins, and k0, k1, k2 and k3 are positive constants. biomath 3 (2014), 1404131, http://dx.doi.org/10.11145/j.biomath.2014.04.131 page 4 of 11 http://dx.doi.org/10.11145/j.biomath.2014.04.131 e nikolova et al., on nonlinear dynamics of the stat5a signaling protein b. stability analysis of the steady state solution of (12) the fixed points of (12) can be found by solving the nonlinear equation as2 − bs + c ds − e − f s + g = 0 (14) they correspond to the following stationary concentrations s0: s01,2 = −(ef + gd − b) 2(a − df) (1 ∓ (15) ∓ √ 1 − 4(a − f d)(c − eg) (ef + gd − b)2 ) at ds − e , 0, which are positive, because a − df < 0, (16) and ef + gd − b > 0, c − eg < 0, (17) if the socs concentration σ is sufficiently smaller than the total erk (e0) and stat (s 0) concentrations as we assumed in the previous paragraph. according to the nonlinear character of the first term of (14) we can present it in the following form: as2−bs+c ds−e = (ds−e)(c1 s+c2 ) ds−e = c1 s + c2 (18) the new coefficients c1 and c2 can be found by the equality as2 − bs + c = dc1 s 2 + (dc2 − ec1)s − ec2 (19) or c1 = a d , c2 = − c e (20) moreover, the following relationship among the coefficients of the basic form exists too. b = cd e + ae d (21) we replace the coefficients (20) in (18) and put it in (14). in this way we obtain the following steady state of (12): s03 = d(c − eg) e(a − df) (22) which is positive in view of (16) and (17). however, it will be valid only if (21) is satisfied. taking into account (13), the last condition will hold if the coefficients k1 or k2 are sufficiently small (almost approaching zero), i.e. associations between erk and stat5a protein molecules or inactivation of active erk molecules are very slow processes. let us now analyze the stability of this equilibrium. for the purpose we postulate the substitution s = s0 + η, where η is a small variation (perturbation) from the steady state value s0 the corresponding variational equation has the form: dη dt = wη (23) where w = (2as0 − b)(ds0 − e) − d(as20 − bs0 + c) (ds0 − e)2 − f (24) it is easy to show that for s0 = s01,2 the coefficient w will have negative value. this is a sufficient condition for verification of the asymptotically stable character of the corresponding equilibrium states. on contrary, for s0 = s03, w will have positive value, i.e. the corresponding equilibrium state will be unstable. therefore we proved that the smallest (s01) and biggest (s 0 2) steady states are stable, and the intermediate steady state s03 is unstable, respectively. biomath 3 (2014), 1404131, http://dx.doi.org/10.11145/j.biomath.2014.04.131 page 5 of 11 http://dx.doi.org/10.11145/j.biomath.2014.04.131 e nikolova et al., on nonlinear dynamics of the stat5a signaling protein finally, the results in this section can be summarized as follows: 1) the dynamical behavior of the interaction between erk and stat5a proteins (in particular, between mek/erk and jak/stat signaling pathways) near its quasi-stationary state determines only by the behavior of the stat5a signaling protein; 2) stabilization of the considered process in time will observe at smallest (about 10−3mm ) and biggest (over 10−2mm) stat5a concentrations. destabilization of the same process will occur at intermediate steady state concentrations of the stat5a protein, which will lead to initiation of some pathological process (for example, cancerogenesis). iv. space-temporalmodeling of the stat5a signaling protein a. a reaction-diffusion model of the stat5a signaling protein. stability analysis of the inhomogeneous distribution of the stat5a concentration. we introduce systems with distributed variables (reaction-diffusion models) when the connections between neighbor points of the space are taken into account by the diffusion law of molecular motion from the higher to lower concentrations. here we take into account the reaction-diffusion effect, described in the introduction of this article to the reaction model of the stat5a signaling protein. as a result we obtain the following one-dimensional model with distributed parameters ∂s ∂t = as2 − bs + c ds − e − f s + g + ds ∂2 s ∂r2 (25) where r is the space coordinate from the cell membrane to the nucleus, and ds is the diffusion coefficient of the stat5a concentration. in order to analyze qualitatively and solve quantitatively (25), it is necessary to fix initial and boundary conditions for the function s(r, t) in the form: s(r, 0) = 0, s|r=0r=l = 0 where l is the distance between the membrane and nucleus. next, we consider the equation (25) and search for solutions of the kind of traveling waves: s = s(ξ) = s(r−vt) , where v is the velocity of the wave. in this way, by introducing the travelingwave coordinate ξ = r − vt, the equation (25) transforms to the following ordinary differential equation of second order: ds s ′′ + vs′ + as2 − bs + c ds − e − f s + g = 0 (26) and next in the following system of ordinary differential equations of first order: s′ = y (27) y′ = 1 ds (−vx − as2 − bs + c ds − e + f s − g) where ′ denotes d/dξ. by analogy with the previous paragraph the fixed points of (27) can be found by solving the equation (14). next we will analyze stability of the fixed points of (27) (in particular of (26) or (25)) in the phase plane (s, s’). for the purpose we introduce the substitutions s = s0 + $ and y = y0 + ζ, where $ and ζ are small variations (perturbations) from the inhomogeneous equilibrium (s0, s′0). we substitute the last expressions in (27), and obtain the following variational equations: d$ dξ = ζ dζ dξ = − w ds $− v ds ζ where w is a coefficient, presented by (24). the corresponding characteristic equation has the form: µ2 + ςµ + τ = 0 (28) where biomath 3 (2014), 1404131, http://dx.doi.org/10.11145/j.biomath.2014.04.131 page 6 of 11 http://dx.doi.org/10.11145/j.biomath.2014.04.131 e nikolova et al., on nonlinear dynamics of the stat5a signaling protein ς = v ds , τ = w ds (29) are routh-hurwitz coefficients. according to the routhhurwitz stability criterion they must have positive values to ensure the stable character of the corresponding steady state. here the diffusion coefficient ds must be positive and the wave velocity v must be different from 0. thereby the stability of inhomogeneous steady state of (27) (or (26)) is determined by signs in front of the coefficients v and w. we can observe a stable steady state solution of (27) if v > 0. if v < 0 the corresponding steady state will be always unstable. thus, at v > 0 the smallest (s01) and biggest (s 0 2) steady states are unstable in view of the negative values of w, and the intermediate steady state s03 is stable because w is positive. finally, we can conclude that the average steady state concentrations of the stat5a protein are indicative for the dissipative structures existence, but too low and too high ones are not. in this sense, the well-known crowding effect will hold, and only increase or decrease of the steady state distributions could assure the dissipative structure stability in the cell. b. traveling wave solutions of (26). application of the modified method of simplest equation the equation (26) (in particular (25)) will have solutions for such values of s, at which its nonlinearity can be reduced to a polynomial nonlinearity. the reduction of a non-linearity to a weak non-linearity is a commonly used approach in the nonlinear dynamics. in this way, here we develop the last three terms of (26) in a taylor series centered in s = 0. we retain only the terms up to cubic power and obtain the following approximation of (26): ds ∂2 s ∂ξ2 + v ∂s ∂ξ + αs3 + βs2 + χs + δ = 0 (30) where the new coefficients have the form: α = ade2 − bd2 e3 + cd3 e4 , β = a e − bd e2 + cd2 e3 χ = f − be + cd e2 , δ = g − c e (31) in this case we have not interested in the concrete values of the above–given coefficients, because their signs (positive or negative) do not affect our further analytical investigations. in addition, the cubic polynomial approximation means that we accept a weak non–linearity (but not linearization) of (26), i.e. d (the rate constant k1) is sufficiently smaller than e (the rate constant k2) to assure the approximation validity. indeed, the last inequality follows from the biochemical consideration that the processes of erk inactivation and stat dephosphorylation are faster than that of erk and stat interaction. the last is of molecular recognition type [21–23]. next we will consider the wave propagation of the stat5a density. for the purpose we will apply the methodology from section i i to (31). in order to solve (31) we constrict a solution as finite series s(ξ) = n∑ i=0 aiφ i, ( dφ dξ )2 = r∑ =0 cφ  (32) where φ is the solution of a simpler equation (referred to as simplest equation according to section i i), and ai and c j are parameters that we will determine below. we substitute (33) in (31) and obtain an equation that contains powers of φ. next, we balance the highest power arising from the second derivative in (31) with the highest power arising in the term containing s3 in the same equation. the resulting balance equation is r = 2n + 2, n = 2, 3, ... in the simplest case, if n = 2, then r = 6. here we will use an equation of kind of bernoulli as simplest equation. for the purpose we assume that c0 = c1 = c3 = c5 = 0; c2 = p2; c4 = 2pq; c6 = q2 , 0, and search for a solution in the form: s(ξ) = a0 + a1φ + a2φ 2, dφ dξ = pφ + qφ3 (33) biomath 3 (2014), 1404131, http://dx.doi.org/10.11145/j.biomath.2014.04.131 page 7 of 11 http://dx.doi.org/10.11145/j.biomath.2014.04.131 e nikolova et al., on nonlinear dynamics of the stat5a signaling protein we note that at another choice of the parameters c j we can obtain another kind of dφ dξ , but we aim to use some simpler equation which has an exact solution according to the modified method of simplest equation. the substitution of (34) in (31) leads to the following system of relationships for the parameters of the solution (the system is of kind (4)): 8dsa2q2 −αa32 = 0 3αa1a22 + 3dsa1q 2 = 0 −α((2a0a2 + a21)a2 + 2a 2 1a2 + a 2 2 ∗ a0)+ +2va2q + 12dsa2 pq −βa22 = 0 4dsa1 pq + va1q −α(4a0a1a2 + (2a0a2 + a21)a1)− −2βa1a2 = 0 4dsa2 p2 + 2va2 p −χa2 −α(a20a2 + 2a0a 2 1+ +(2a0a2 + a21)a0 −β(2a0a2 + a 2 1) = 0 (34) −3αa20a1 − 2βa0a1 + va1 p −χa1 + dsa1 p 2 = 0 −βa20 −χa0 −αa 3 0 −δ = 0 the system (35) implies that a1 = 0 and q is a free parameter. the solution of (35) is: a0 = 1 6 3√ λ α − 2 3 3χα−β2 α 3√ λ − β 3α, a2 = 2 √ 2αds q α p = − √ 2(−ds 3√ λ2 + 12dsχα− (35) −4dsβ2 + v √ 2αds 3√ λ)/(12ds √ αds 3√ λ) where λ = 36χβα− 108α2 − 8β3 + +12 √ 3(4χ3α−χ2β2 − 18χβαδ + 27δ2α2 + 4δβ3)α the expression for the solitary wave depends on the solution of the differential equation in (34) and it is given by: s(ξ) = a0 + a2 √ p exp [2p(ξ + c)] 1 − q exp [2p(ξ + c)] (36) fig. 2. graph of the solution s(ξ) at α = 1; β = 1; χ = 100; δ = 10; k = 1; ds = 11. for the case p > 0, q < 0 and s(ξ) = a0 + a2 √ p exp [2p(ξ + c)] 1 + q exp [2p(ξ + c)] (37) for the case p < 0, q > 0, where a0, a2, p are given by (36). the values (37) or (38) describe ’kink’ waves, which can be interpreted as fronts of some density propagation of the stat5a signaling protein in the intracellular space, as we will show in the next paragraph of the paper. two examples of these kinks are shown in fig. 2 and fig. 3. we note, however, that the figures give rather an illustrative idea of the possible spatial–temporal behavior of the considered object due to the lack of experimental data for the model parameters. c. discussions in section i i i we demonstrated that the lowest and highest steady state values of the stat5a concentration are realized practically in the homogeneous case corresponding to the absence of appropriate response in the form of erk or socs protein production initiated by the cell signaling. biomath 3 (2014), 1404131, http://dx.doi.org/10.11145/j.biomath.2014.04.131 page 8 of 11 http://dx.doi.org/10.11145/j.biomath.2014.04.131 e nikolova et al., on nonlinear dynamics of the stat5a signaling protein fig. 3. graph of the solution s(ξ) at α = 1; β = −1; χ = 100; δ = 10; k = −1; ds = 11. then stabilization of the stat5a signaling protein occurs. when, however, similar response (erk or socs proteins increase) is available, the stat5a concentration dramatically increases or decreases near the nucleus membrane and some inhomogeneous distribution of the stat5a molecules takes place in the cytosol. this means that the diffusion involves in the process and it can be presented by (25) (or (26)). in this case the stat5a concentration will propagate in the intracellular space as a ”kink“ wave, as we demonstrated in the previous paragraph. we choose the wave propagates along the positive direction of axis , i.e. from the cell membrane to the nucleus one according to the direction of the signal transduction. equation (25) (in particular (26) or (27) in view of the introduced traveling-wave ansatz) has the same steady state values for s as in the homogeneous case which we considered in section i i i. however in inhomogeneous case, presented by (25) or (26), the lowest and the highest steady states are unstable, and the intermediate one is stable. therefore the steady state values in inhomogeneous case change the type of their character in comparison with the homogeneous one. in this way at additional cell signals, initiating increase of the active erk molecules (but in absence of additional socs protein production), the stat5a concentration will increase from the lowest destabilized value s01 to stabilized one s03, and density (concentration) jump will be observed. a similar example is shown in fig.2. on the other hand at additional new signal (ligand), which can inhibit the stat phosphorylation (for example, the socs protein production increases, but there is not erk protein production), the stat5a concentration can decrease from its highest destabilized value s02 to stabilized one s03, and density (concentration) drop will be realized in the intracellular space (fig. 3). thereby the inhomogeneous effects observed in the stat5a protein dynamics are predetermined by the levels of the erk and socs proteins. v. conclusion in this paper, we have re-considered the reaction model of the cross talk between erk and stat signaling pathways. by applying a well-known qssa theorem to the described model we have demonstrated that near the quasi-stationary state of the considered process its dynamics determines by the behavior of the phosphorylated stat5a protein. moreover, if the stat5a concentration is homogeneous distributed in the cell cytoplasm its possible minimum or maximum levels will support stabilization of the cell signaling. but, if some inhomogeneous distribution of the stat5a protein appears in the intracellular space as a result of external signals 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[27] v. kotev, s. nikolov, stability analysis of time delay model of crosstalk between erk and stat5a interaction, bioautomation, vol. 7, pp. 90–98, 2007. biomath 3 (2014), 1404131, http://dx.doi.org/10.11145/j.biomath.2014.04.131 page 11 of 11 http://dx.doi.org/10.1161/atvbaha.110.207464 http://dx.doi.org/10.11145/j.biomath.2014.04.131 introduction the method of the simplest equation the interaction between erk and stat signaling pathways a reaction model and its reduction stability analysis of the steady state solution of (12) space-temporal modeling of the stat5a signaling protein a reaction-diffusion model of the stat5a signaling protein. stability analysis of the inhomogeneous distribution of the stat5a concentration. traveling wave solutions of (26). application of the modified method of simplest equation discussions conclusion references www.biomathforum.org/biomath/index.php/biomath original article optimal control analysis of combined chemotherapy-immunotherapy treatment regimens in a pkpd cancer evolution model anuraag bukkuri university of minnesota bukku001@umn.edu received: 21 september 2019, accepted: 13 february 2020, published: 2 march 2020 abstract—the author constructs a mathematical model capturing tumor-immune dynamics, incorporating the evolution of drug resistance, pkpd (pharmacokinetics and pharmacodynamics) of administered drugs, and immunotherapy possibilities. numerical simulations are performed to analyze the model under a variety of treatment possibilities. a sensitivity analysis is performed to determine the parameters contributing the most to the variance in effector cell, resistant, and sensitive tumor cell populations. then, a detailed optimal control analysis is performed, along with a numerical simulation of optimal treatment profiles for a hypothetical patient. keywords-optimal control, oncology, chemotherapy, immunotherapy, pharmacokinetics, pharmacodynamics msc: 92c45, 92c40, 92c50, 34h05 i. introduction mathematical modeling has the potential to significantly impact the treatment of cancer through optimization of drug administration schedules. currently, clinical treatments are administered in an ad hoc way, modifying the treatment along the way, after observing how the patient is responding to it. since it is not possible to clinically test all possibilities for drug dosage, combinations, sequence, timing, and duration for a patient, mathematical modeling is able to help determine the optimal treatment profile for each patient, leading to truly personalized cancer treatment [rockne et al. 2019]. in recent years, immunotherapy has become a viable and promising treatment option for cancer patients. this biological treatment focuses on using the immune system to fight cancerous cells. though the immune system can naturally prevent/slow cancer growth, cancer cells have evolved many ways to bypass the body’s immune system such as having genetic mutations which make them harder to detect by the immune system, having surface proteins which deactivate immune cells, and changing the healthy cells surrounding the tumor so that they interfere with the immune response to cancer. immunotherapies elicit or amcopyright: c© 2020 bukkuri. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: anuraag bukkuri, optimal control analysis of combined chemotherapy-immunotherapy treatment regimens in a pkpd cancer evolution model, biomath 9 (2020), 2002137, http://dx.doi.org/10.11145/j.biomath.2020.02.137 page 1 of 12 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2020.02.137 anuraag bukkuri, optimal control analysis of combined chemotherapy-immunotherapy treatment ... plify the immune response in these cases to fight against the cancer more effectively in many ways. for example, immune checkpoint inhibitors block immune checkpoints, leading to a stronger immune response, adoptive cell therapy boosts the natural ability of t cells to fight against cancerous ones, and monoclonal antibodies help the immune cells recognize cancerous cells more accurately [kruger et al. 2019]. chemotherapy is another common cancer treatment plan. like immunotherapy, there are many drugs with different mechanisms chemotherapy subsumes including doxorubicin, novantrone, and epirubicin. each of these drugs targets and attempts to kill cells which are growing and dividing rapidly, like those characteristic of cancer. however, a marked side effect of this is that the drug also kills many healthy cells in the process, especially those fast-growing cells in the bone marrow, hair, and skin [schirrmacher 2019]. current medical literature is focused on attempting to combine these two treatments. though much work has been done in this area, the question still remains: what is the best way to combine these treatments to ensure tumor remission while also attempting to minimize side effects? this paper attempts to address this question in a few ways. in section 2, a mathematical model capturing the key components of our system is presented. in section 3, parameter value estimates will be provided and simulations with varying amounts of continuous infusion chemotherapy will be presented. in section 4, a sensitivity analysis will be performed, allowing us to understand which parameter values impact the variance in the effector cell and tumor cell populations the most. in section 5, the author analyzes the existence and characterization of the optimal control. in section 6, numerical simulations depicting the optimal control profile under different treatment side effect combinations for a hypothetical patient will be provided. ii. model creation the model presented below can be thought of to consist of two parts: the first three equations describe tumor-immune interactions. separate equations are given for sensitive and resistant (to the chemotherapy drug) tumor cells, and a term is provided which converts the sensitive cells to resistant cells. for greatest generality, immunotherapy is included by adding to the effector cell population. the last two equations form the second part of the model, capturing the pharmacokinetics and pharmacodynamics of chemotherapy drug administration. this component is linked to the first set of equations via the effect of the chemotherapy drug on the sensitive tumor cell population. below is our model: ds dt =as ( 1− tt cmax ) −qes−s ( c γ 2 c γ 2 +wic50 ) −ls −dts (1) dr dt =ar(1− tt cmax )−qer+ls−dtr (2) de dt =s+pe tt g + tt −mett−µec1−dee +s2v(t) (3) dc1 dt =−(k1+k2)c1+ s1u(t) v1 (4) dc2 dt =k12 v1 v2 c1−k2c2 (5) the terms u(t) and v(t) are termed controls for our treatment: u(t) refers to the chemotherapy treatment, and v(t) refers to the immunotherapy treatment. in this model, s and r represent the drugsensitive and drug-resistant tumor cells, tt represents the total number of tumor cells (s + r), e represents the effector cells, and c1 and c2 represent the drug concentrations in the plasma and at the tumor site, respectively. below, we will explain the functional forms of each of the equations above: ds/dt: the first term represents the normal dynamics of these cells, taking into account the carrying capacity (captured by cmax). the second term captures the killing rate of tumor cells by effector cells. the third term is the killing of sensitive tumor cells by the drug; ic50 is the biomath 9 (2020), 2002137, http://dx.doi.org/10.11145/j.biomath.2020.02.137 page 2 of 12 http://dx.doi.org/10.11145/j.biomath.2020.02.137 anuraag bukkuri, optimal control analysis of combined chemotherapy-immunotherapy treatment ... median inhibitory concentration, and w scales between the measurements in vitro to in vivo (we can assume this is 1); γ is provided to help scale the hyperbolic curve for pkpd data fitting purposes. the fourth term is used to incorporate drug resistance–l represents the mutation rate from sensitive to resistant tumor cells. finally, the last term captures the natural death rate of the cancer cells. dr/dt: the first, second, and last terms serve the same purpose as in the ds/dt equation. the ls captures the transferal of sensitive to resistant cells via mutation. note that we assume identical base growth dynamics for the sensitive and resistant cells, as well as identical interactions with effector cells. de/dt: the first term is the rate of influx of effector cells into the region of tumor localization and the second term captures the tumor activation of effector cells while ensuring a maximum rate at which effector cells are produced and de is their death rate. the third term is the death of effector cells due to interaction with the tumor cells. the fourth term is added to incorporate drug toxicity. finally, the last term, s2v(t), captures the administration of the immunotherapy treatment, in which v(t) is the input function ranging from 0 to 1 and s2 is a scaling term. since each immunotherapy treatment mechanistically works differently, the generalization made here is that the immunotherapy effectively increases the number of effector cells at the tumor site. dc1/dt: here, k1 is the elimination from the plasma, and k2 is the elimination from the effect compartment. u is the input function (ranging from 0 to 1) for administration protocol of the chemotherapy treatment (with s1 serving as a scaling factor) and v1 is the volume of distribution of the compartment. dc2/dt: k12 is the link process between the plasma component and the tumor, v1 and k2 are as above, v2 is the effect component for c2(t). together, these last two equations incorporate the pharmacokinetic components of drug administration. iii. impacts of continuous infusion of chemotherapy the parameter values used in model simulations are given in table i. the most standard clinical treatment for cancer is chemotherapy administration. as such, in this section, numerical simulations are performed for various chemotherapy treatment situations. initial conditions of 300 effector cells and 10,000 sensitive cancer cells and an s1 value of 100 were used for all simulations below. three simulations were performed for varying intensities of chemotherapy treatment. the results can be seen in figure 1. fig. 1. chemotherapy numerical simulations biomath 9 (2020), 2002137, http://dx.doi.org/10.11145/j.biomath.2020.02.137 page 3 of 12 http://dx.doi.org/10.11145/j.biomath.2020.02.137 anuraag bukkuri, optimal control analysis of combined chemotherapy-immunotherapy treatment ... parameter meaning estimated value source a natural growth rate of cancer cells 0.616/day depillis cmax carrying capacity of cancer cells 9.804 ∗ 108cells depillis q killing rate of cancer cells by effector cells 1.101 ∗ 10−7/cell ∗ day lai estimation w scaling between in vitro and in vivio 1 assumption ic50 median drug inhibitory concentration 10 depends on drug γ hyperbolic curve modification 1 n/a l conversion rate of sensitive to resistant cancer cells 10−9 birkhead dt natural death rate of cancer cells 0.17/day liao; lai de natural death rate of effector cells 0.0412/day kuznetsov s effector cells at tumor site 1.3 ∗ 105cells/day kuznetsov p tumor activation of effector cells 0.1245/day kuznetsov g maximum rate of effector cell production 2.019 ∗ 107cells kuznetsov m death of effector cells due to cancer cells 3.422 ∗ 10−10/cell ∗ day kuznetsov µ side effect of chemotherapy on effector cells 2 ∗ 10−2 liters/g*day estimation k1 elimination from plasma 1.6/day iliadis k2 link process between plasma and tumor 0.8/day iliadis u input function for chemotherapy protocol depends on control n/a v input function for immunotherapy protocol depends on control n/a v1 volume of the plasma compartment 25 iliadis v2 volume of the tumor compartment 15 iliadis k12 link process between plasma and tumor compartments 0.4/day iliadis table i parameters used in numerical simulations in this figure, the top panel corresponds to a u value of 0.2, or 20% of the maximal chemotherapy administration possible. the middle panel represents 50% and the bottom panel is 100%. there are a few things to notice in these figures. first, consider the key similarity: in all three treatment scenarios, the effector cell population reached the same equilibrium of 4∗106 cells at similar times. note that, in these simulations, a low side effect of the chemotherapy drug on the immune system was assumed. the critical differences lie in the populations of sensitive and resistant cells. we see a common trend in that the higher the dose of chemotherapy, the faster the sensitive cancer cell population goes extinct, but the larger the final equilibrium of resistant cancer cells is. since, in this model, we are dealing with a rapidly evolving cancer cell population, regardless of the intensity of continuous infusion chemotherapy, resistance is ensured to occur, without complete remission of the cancer. thus, to more rigorously determine which aspects of the cancer to attack, a sensitivity analysis was performed. iv. sensitivity analysis a sensitivity analysis was performed on the above ode model using the sobol-martinez method. parameters which cannot be modified with treatments were given a range 10% higher and 10% lower than the original value used in the original numerical simulations. the sobolmartinez method is based on variance decomposition techniques to provide a rigorous measure of the contributions of the input to output variance. the algorithm outlined by zhang et al. was implemented for the sensitivity analysis given in figures 2, 3 and 4. the images on the left are the first order sobol indices: the contribution to the output variance by the single model input alone. the second image is the total sobol index, one which measures the contribution to the output variance caused by a model input, including the first-order effects as well as all higher-order interactions. as one can see, in the case of the effector cells, the biomath 9 (2020), 2002137, http://dx.doi.org/10.11145/j.biomath.2020.02.137 page 4 of 12 http://dx.doi.org/10.11145/j.biomath.2020.02.137 anuraag bukkuri, optimal control analysis of combined chemotherapy-immunotherapy treatment ... fig. 2. sensitivity analysis: effector cells fig. 3. sensitivity analysis: resistant cancer cells most sensitive parameter value for the first 50 or so time steps is s, a quantity that is increased by immunotherapy treatments such as dendritic cell therapy. however, after that time, until 400 time steps, the µ parameter (side effects of the drug) is just as sensitive, if not more sensitive. since the side effects of immunotherapy do not directly impact the effector cells usually, this can be interpreted as the side effects of chemotherapy impacting the effector cells even more than the potential benefits of the immunotherapy. medically, from the effector cell side, it would seem that the best way to progress would be to give a high dose of dendritic cell therapy, for example, at the beginning, then work to limit the side effects of other chemotherapy drugs that are given. in the case of the resistant cells, one can see biomath 9 (2020), 2002137, http://dx.doi.org/10.11145/j.biomath.2020.02.137 page 5 of 12 http://dx.doi.org/10.11145/j.biomath.2020.02.137 anuraag bukkuri, optimal control analysis of combined chemotherapy-immunotherapy treatment ... fig. 4. sensitivity analysis: sensitive cancer cells that all three parameter values shown are quite sensitive. considering the total sobol index, it seems that, for most of the time, a (growth rate of the cancer) and q (killing rate of tumor by effector cells) switch off between being the most and second most sensitive parameter values, while the mutation rate (l) becomes more important, relative to the other parameters, as time progresses. this implies that, for the resistant cells, it might be most effective to alternate some treatment that directly reduces the cancer’s ability to grow and immunotherapy for the first 200 time steps, and then just give immunotherapy for the last 200 time steps. however, it’s hard to directly reduce natural tumor growth rate using a drug. there is, though, a burgeoning area of medical research focused on the elimination of the tudor-sn protein from cancer cells using crispr-cas9 gene editing techniques. it is known that tudor-sn is the main influencer of cancer cell growth and thus, silencing this protein would suppress the rapid cellular proliferation characteristic of cancers. a recent study showed that when tudor-sn was removed from human cells, the levels of several micrornas increase, putting brakes on genes that encourage cell growth. this significantly slowed down cell progression from the preparatory to cell division phase [elbarbary 2017]. as suggested by the sensitivity analysis of the resistant cancer cell output values, using such a treatment based on the suppression of tudor-sn in conjunction with immunotherapy would be the most effective treatment to reduce the resistant tumor cell population. for the sensitive cells, it’s clear that, at the beginning 50 time steps, a is the most sensitive parameter. this then quickly switches to q being the most important parameter, with the ic50 value of the drug becoming more sensitive as time progresses. medically, this implies that it’s best to give immunotherapy throughout the treatment, while introducing increasing amounts of chemotherapy over time. thus, overall, it seems that the parameters a, q, s, and l are the parameters that impact the dynamics the most. medically, it then makes sense to target these parameter values. though it is often not possible to target all of these parameters in a treatment plan, our analysis suggests the plausibility of a combined chemotherapyimmunotherapy treatment plan. so, to determine what the best combination and timing of chemotherapy-immunotherapy administered protocols are, we perform an optimal control analysis. biomath 9 (2020), 2002137, http://dx.doi.org/10.11145/j.biomath.2020.02.137 page 6 of 12 http://dx.doi.org/10.11145/j.biomath.2020.02.137 anuraag bukkuri, optimal control analysis of combined chemotherapy-immunotherapy treatment ... v. optimal control here, we formulate the problem of determining the most effective treatment regimen as an optimal control one. specifically, we desire to minimize the amount of drug given and the tumor size over a fixed therapy interval [0,t] while maximizing the number of effector cells, thus maintaining an effective tumor-immune balance as done in [bukkuri 2019]. drug toxicity is accounted for by the u(t) term in the integral, denoting the total drug dosage over the given treatment period as a penalty term. we choose as our control class piecewise continuous functions defined for all t such that 0 ≤ u(t) ≤ 1 where u(t)= 1 represents maximal chemotherapy and u(t)= 0 represents no chemotherapy. thus, we depict the class of admissible controls as u1(t) = u(t) piecewise continuous s.t. 0 ≤ u(t) ≤ 1,∀t ∈ [0,t] u2(t) = u(t),v(t) piecewise continuous s.t. 0 ≤ u(t),v(t) ≤ 1,∀t ∈ [0,t] now, we define our objective functional and optimal control problem. we specifically consider two cases: one treatment (j1) with just chemotherapy and another treatment (j2) with a combined chemotherapy-immunotherapy (e.g. dendritic cell therapy) regimen. for a fixed therapy horizon [0,t], maximize the objective functional j1(u)= ∫ t 0 αe(t)−β1s(t))−β2r(t)−bu(t))dt j2(u,v)= ∫ t 0 αe(t)−β1s(t))−β2r(t) −b1u(t)−b2v(t))dt over all lebesgue-measurable functions u : [0,t] → [0,umax] subject to the above ode dynamics and initial conditions. theorem 1: consider the objective functional j1(u) subject to the state system in section 2. assume that: • there exists an admissible pair ( ~k,u(t)) • n( ~k,u1,t) is convex in u1 for each ( ~k,t) • u1 is closed and bounded • ∃ a number θ s.t. ||~k|| ≤ θ∀t ∈ [t0, t1] and all admissible pairs ( ~k,u(t)) then there exists an optimal control pair ( ~k∗,v∗) that maximizes j1(u). moreover, for sufficiently small t, the optimal control system has a unique solution. proof. an admissible pair ( ~k,v(t)) is needed for the existence of optimal control. since the system in equations (1) – (5) (our original set of equations) has bounded coefficients and any solutions are bounded on the finite time interval, using a result from lukes [lukes 1982], we obtain the existence of the solution of the system described by equations (1) – (5). now, we need n( ~k,u1,t) to be convex in u1 for each ( ~k,t). we define: w1 =(αe −β1s −β2r−bu1 + γ1, as ( 1− tt cmax ) −nes−s ( c γ 2 c γ 2+wic50 ) −ls−dts, ar ( 1− tt cmax ) −ner + ls −dtr, s + pe tt g + tt −mett −µec1 −dee, − (k1 + k2)c1 + s1u1 v1 , k12 v1 v2 c1 −k2c2) for some γ1 ≤ 0 and u1 ∈ u1, w2 =(αe −β1s −β2r−bu2 + γ2, as ( 1− tt cmax ) −nes−s ( c γ 2 c γ 2+wic50 ) −ls−dts, ar ( 1− tt cmax ) −ner + ls −dtr, s + pe tt g + tt −mett −µec1 −dee, − (k1 + k2)c1 + s1u2 v1 , k12 v1 v2 c1 −k2c2) for some γ2 ≤ 0 and u2 ∈ u1. we must now prove biomath 9 (2020), 2002137, http://dx.doi.org/10.11145/j.biomath.2020.02.137 page 7 of 12 http://dx.doi.org/10.11145/j.biomath.2020.02.137 anuraag bukkuri, optimal control analysis of combined chemotherapy-immunotherapy treatment ... that for every λ ∈ [0,1] we have w3 = λw1 + (1−λ)w2 ∈ n( ~k,u1, t). to do this, let z1 =λ(αe −β1s −β2r−bu1 + γ1) + (1−λ)(αe −β1s −β2r−bu2 + γ2) = αe −β1s −β2r−b((1−λ)u2 + λu1) + λγ1 + (1−λ)γ2 and define γ3 = z1 − (αe −β1s −β2r) + bu3 for u3 = (1−λ)u2 + λu1. then, γ3 = λγ1 + (1−λ)γ2 ≤ 0, since γ1,γ2 ≤ 0 and λ ∈ [0,1]. then, we see that z2 =λ ( as ( 1− tt cmax ) −nes−s ( c γ 2 c γ 2 + wic50 ) − ls −dts ) + (1−λ) ( as ( 1− tt cmax ) −nes −s ( c γ 2 c γ 2 + wic50 ) − ls −dts ) =as ( 1− tt cmax ) −nes−s ( c γ 2 c γ 2+wic50 ) −ls−dts z3 =λ(ar(1− tt cmax )−ner + ls −dtr) + (1−λ) ( ar ( 1− tt cmax ) −ner+ls−dtr ) = ar ( 1− tt cmax ) −ner + ls −dtr z4 =λ ( s+pe tt g+tt −mett−µec1−dee ) +(1−λ) ( s+pe tt g+tt −mett−µec1−dee ) = s + pe tt g + tt −mett −µec1 −dee z5 =λ ( −(k1 + k2)c1 + s1u1 v1 ) + (1−λ) ( −(k1 + k2)c1 + s1u2 v1 ) = −(k1 + k2)c1 + s1u3 v1 z6 =λ ( k12 v1 v2 c1 −k2c2 ) +(1−λ) ( k12 v1 v2 c1 −k2c2 )) = k12 v1 v2 c1 −k2c2 combining this information, we find a v3 ∈ [0,1] and γ3 ≤ 0 such that λw1 + (1−λ)w2 =( αe −β1s −β2r−bu3 + γ3, as ( 1− tt cmax ) −nes−s ( c γ 2 c γ 2+wic50 ) −ls−dts, ar ( 1− tt cmax ) −ner + ls −dtr, s + pe tt g + tt −mett −µec1 −dee, −(k1 + k2)c1 + s1u3 v1 , k12 v1 v2 c1 −k2c2 ) therefore, λw1 + (1 − λ)w2 ∈ n( ~k,u1, t). thus, n( ~k,u1, t) is convex in u1. the next requirement for the existence of optimal control is that u is closed and bounded. this is true, by definition. finally, there exists a number θ s.t. ||~k|| ≤ θ∀t ∈ [t0, t1] and all admissible pairs ( ~k,u(t)). the boundedness argument is analogous to those previously performed [fister et al. 1998, burden et al. 2004, fister et al. 2005]. a similar argument can be applied for the existence of an optimal control for j2(u,v). for the analysis, we shall continue with the j2(u,v) objective so that the proofs are more biomath 9 (2020), 2002137, http://dx.doi.org/10.11145/j.biomath.2020.02.137 page 8 of 12 http://dx.doi.org/10.11145/j.biomath.2020.02.137 anuraag bukkuri, optimal control analysis of combined chemotherapy-immunotherapy treatment ... insightful. note that j1 is identical to the j2 objective, but with a zero v term. since we’ve proven the existence of optimal controls to maximize the j1 and j2 functionals, first order necessary conditions for optimality can be determined by a version of the pontryagin maximum principle. for the characterization of the optimal control, we first define the hamiltonian associated with j2(u,v) and the system of odes as follows: h =αe −β1s −β2r−b1u +λ1 ( as ( 1− tt cmax ) −nes−s ( c γ 2 c γ 2 +wic50 ) − ls−dts ) +λ2 ( ar ( 1− tt cmax ) −ner+ls−dtr ) +λ3 ( s+pe tt g+tt −mett−µec1−dee +s2v(t) ) + λ4 ( −(k1 + k2)c1 + s1u(t) v1 ) + λ5 ( k12 v1 v2 c1 −k2c2 ) the existence of an optimal control for the state system given in section 2 associated with the objective j1(u) can be determined from the filippovcesari theorem. for the theorem, the following notation shall be used: ~k =   s r e c1 c2   and n( ~k,u1, t)=( αe(t)−β1s(t)−β2r(t)−b1u(t)−b2v(t)+φ, as ( 1− tt cmax ) −nes−s ( c γ 2 c γ 2 +wic50 ) − ls −dts, ar(1− tt cmax )−ner + ls −dtr, s + pe tt g + tt −mett −µec1 −dee, − (k1 + k2)c1 + s1u(t) v1 , k12 v1 v2 c1 −k2c2 ) where φ ≤ 0 and u ∈ u1. regarding the uniqueness of the controls, similar proofs are given in [burden et al. 2004, fister et al. 1998]. the additional constraint that t must be small is due to the fact that the state system is moving forward in time while the adjoint system is moving backwards. theorem 2: given optimal controls u∗ and v∗ and solutions of the corresponding state system, there exist adjoint variables λi for i = 1,2,3,4,5 satisfying the following: dλ1 dt = − ∂h ∂s dλ2 dt = − ∂h ∂r dλ3 dt = − ∂h ∂e dλ4 dt = − ∂h ∂c1 dλ5 dt = − ∂h ∂c2 where λi(t) = 0 for i=1,2,3,4,5 by the pmp transversality condition. furthermore, from the optimality condition, u∗ is given by:{ 0, if s1λ4 v1 −b1 < 0 1, if s1λ4 v1 −b1 > 0 while v∗ is similarly given by:{ 0, if s2λ3 −b2 < 0 1, if s2λ3 −b2 > 0 proof. from the hamiltonian, the derivatives of the adjoints were calculated, and the following is seen. u∗ is given by:  0, if s1λ4 v1 −b1 < 0 1, if s1λ4 v1 −b1 > 0 singular, if s1λ4 v1 −b1 = 0 biomath 9 (2020), 2002137, http://dx.doi.org/10.11145/j.biomath.2020.02.137 page 9 of 12 http://dx.doi.org/10.11145/j.biomath.2020.02.137 anuraag bukkuri, optimal control analysis of combined chemotherapy-immunotherapy treatment ... while v∗ is similarly given by:  0, if s2λ3 −b2 < 0 1, if s2λ3 −b2 > 0 singular, if s2λ3 −b2 = 0 to determine the representation of the control, we first prove that both controls are bang-bang via proof by elimination. if both controls are singular on [0,t] and take time derivatives, then λ3(t) = 0 and λ4(t) = 0 on [0,t] due to the continuity of λ3(t) and λ4(t). this is a contradiction of the fact that λ3 = b2 s2 and λ4 = b1v1 s1 since we’re assuming the presence of a tumor and some side effects of the drug. thus, it’s not possible for both controls to be singular. now, let’s analyze the possibility of one of the controls being singular. for simplicity, we will work with the v∗ control, but the same argument applies to the u∗ control. if the v∗ control is singular, λ3 = b2 s2 on [0,t]. taking a time derivative of s2λ3 −b2 = 0, we get s2λ′3(t) = 0, implying that λ′3 = 0 and that λ3(t) is constant. however, since λ3(t) is continuous on [0,t] and λ3(t) = 0 by the transversality condition, then λ3(t) = 0 for any subset on [0,t]; however, again, λ3 = b2 s2 , a nonzero quantity for any meaningful therapy. thus, we conclude that both controls must be bang-bang. note that this agrees with current medical knowledge that chemotherapy is best given in bang-bang controls and hence why chemotherapy is given in cycles: a dose of one or more drugs followed by several days or weeks without treatment. this ensures that the normal cells have enough time to recover from the drug side effects. a similar argument can be given for immunotherapy, though the side effects are less direct than those for chemotherapy intuitively, this makes sense since it’s stating that no treatment will be used when the v1 is high (i.e. essentially that the tumor/normal compartment ratio is high) and the maximum effect of the drug is low compared to the patient’s treatment tolerance level. on the other hand, if the patient can tolerate great side effects, the tumor size is large, and there exists a high drug efficacy, maximum treatment should be given. vi. simulated patients we can think of our optimality system as a twopoint boundary value problem, which we solve using a fourth-order iterative runge-kutta scheme, as done in [jung et al. 2002]. in this scheme, we perform a forward sweep of the state equations with initial guesses for u and v, before performing a backward calculation using the adjoint equation and an update of the controls. this method is done iteratively until convergence is obtained. below are simulations of a few hypothetical patients. we specifically analyze the cases of low and high side effects for chemotherapy and immunotherapy. the weighting terms in the objective functional were fixed at α = 0.03, b1 = 0.5, and b2 = 0.6. s1 was fixed at 100 and s2 was fixed at 100,000. the table below shows the values of b1 and b2 used to represent high and low side effects for chemotherapy and immunotherapy: side effect chemotherapy (b1) immunotherapy (b2) low 1000 10,000 high 1,000,000 100,000,000 table ii b1 and b2 values used in figure 5 depictions of all four possible combinations of side effects for chemotherapy and immunotherapy over 700 days are presented. note that, in accordance with clinical oncology practices, the optimal controls in all side effect scenarios include bang-bang controls for both chemotherapy and immunotherapy. the key differences in optimal control protocols occur in the first 400 days of treatment. in the case of low side effects for both chemotherapy and immunotherapy, it is advised that chemotherapy be delivered for the first 150 days and immunotherapy be delivered for the last 380 days. this is in stark contrast to the other three scenarios, in which chemotherapy and immunotherapy treatments do biomath 9 (2020), 2002137, http://dx.doi.org/10.11145/j.biomath.2020.02.137 page 10 of 12 http://dx.doi.org/10.11145/j.biomath.2020.02.137 anuraag bukkuri, optimal control analysis of combined chemotherapy-immunotherapy treatment ... fig. 5. numerical simulations of optimal control profiles for a hypothetical patient not overlap for the first 400 days. for example, consider low side effects for chemotherapy and high side effects for immunotherapy. in this case, chemotherapy should be prescribed for the first 150 days and immunotherapy should be given for the next 250 days. when chemotherapy has high side effects for the patient, it is effectively removed from the optimal treatment plan (for the first 400 days). instead immunotherapy is given for the last 380 days (if it has a low side effect) or the last 250 days (if it has a high side effect). past day 400, optimal treatment protocols in all four cases are to give 70 days of chemotherapy followed by 70 days of a drug vacation and to also give 70 days of immunotherapy followed by 70 days of a drug vacation–note that these two treatments overlap and are often given together. vii. conclusion in this paper, a model was created capturing dynamics among tumor cells sensitive and resistant to chemotherapy, effector cells, and chemotherapy pkpd. evolution of resistance to chemotherapy was taken into count. parameter values were estimated from medical literature and numerical simulations were performed displaying dynamics under control and treatment cases. then, sobol-martinez sensitivity analyses were performed which found the growth rate of cancer cells, killing of cancer cells by the immune system, effector cells in the tumor compartment, and the resistance emergence in cancer cells to be the parameters which impact the dynamics of effector cells, sensitive tumor cells, and resistant tumor cells the most. next, a characterization of the optimal treatment profile was given analytically, along with proofs for the existence and uniqueness of such a protocol. numerical optimal control was then performed to illustrate the optimal treatment schedule for a hypothetical cancer patient. this showed a profile of bang-bang chemotherapy and immunotherapy controls, often with overlapping treatment regimens. the author hopes that the model and analysis will help inspire further medical research to be conducted in creating drugs which reduce the natural growth rate of cancer cells, as well biomath 9 (2020), 2002137, http://dx.doi.org/10.11145/j.biomath.2020.02.137 page 11 of 12 http://dx.doi.org/10.11145/j.biomath.2020.02.137 anuraag bukkuri, optimal control analysis of combined chemotherapy-immunotherapy treatment ... as the clinical use of combined chemotherapyimmunotherapy treatments in accordance with the optimal protocols presented. references [1] birkhead bg, rankin em, gallivan s, dones l, rubens rd (1987). a mathematical model of the development of drug resistance to cancer chemotherapy. eur. j. cancer clin. oncology, 23(9), 1421-1427. [2] bukkuri a (2019). optimal control analysis of combined anti-angiogenic and tumor immunotherapy. open journal of mathematical sciences, 3, 349-357. [3] burden t, ernstberger j, fister r (2004). optimal control applied to immunotherapy. discrete and continuous dynamical systems, 4(1), 135–146. [4] de pillis lg, gu w, radunskaya ae (2006). mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations. journal of theoretical biology, 238(4):841–862. https://doi.org/10.1016/j.jtbi.2005.06.037 [5] elbarbary ra, miyoshi k, myers jr, du p, ashton jm, tian b, maquat le (2017). tudor-sn-mediated endonucleolytic decay of human cell micrornas promotes g1/s phase transition.; science (new york, n.y.); vol 356(6340). [6] fister r, donnelly, j (2005). immunotherapy: an optimal control theory approach, math. biosci. and engrg., 2(3), 499-510. [7] fister r, lenhart s, mcnally j (1998). optimizing chemotherapy in an hiv , elec. j.de., 32, 1–12. [8] iliadis a, barbolosi d (2000). optimizing drug regimens in cancer chemotherapy by an efficacy–toxicity mathematical model [9] jung e, lenhart s, feng z (2002). optimal control of treatments in a two-strain tuberculosis model, discrete contin.dyn. syst. ser., 2, 473-482. [10] kruger s, ilmer m, kobold s. et al. (2019). advances in cancer immunotherapy 2019 – latest trends. j exp clin cancer res 38, 268 [11] kuznetsov va, makalkin ia, taylor ma, perelson as (1994). nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. bull. math. biol., 56(2): 295—321. [12] lai x, friedman a (2017). combination therapy of cancer with cancer vaccine and immune checkpoint inhibitors: a mathematical model. plos one 12(5): e0178479. https://doi.org/10.1371/journal.pone.0178479 [13] liao kl, bai xf, friedman a (2014). mathematical modeling of interleukin-27 induction of anti-tumor t cells response. plos one 9(3):e91844 [14] lukes d (1982). differential equations: classical to controlled, math. sci. engrg. 162. [15] rockne rc, hawkins-daarud a, swanson kr, sluka jp, glazier ja, macklin p, hormuth d, jarrett am, da fonseca lima eab, oden j. et al. (2019). the 2019 mathematical oncology roadmap, physical biology 16(4), 041005. [16] schirrmacher v. (2019). from chemotherapy to biological therapy: a review of novel concepts to reduce the side effects of systemic cancer treatment. int. j. oncol. 54(2), 407-419. [17] zhang xy, trame mn, lesko lj, schmidt, s (2015). sobol sensitivity analysis: a tool to guide the development and evaluation of systems pharmacology models. cpt pharmacometrics syst. pharmacol. 4, 6979. biomath 9 (2020), 2002137, http://dx.doi.org/10.11145/j.biomath.2020.02.137 page 12 of 12 http://dx.doi.org/10.11145/j.biomath.2020.02.137 introduction model creation impacts of continuous infusion of chemotherapy sensitivity analysis optimal control simulated patients conclusion references www.biomathforum.org/biomath/index.php/biomath original article bi-stable dynamics of a host-pathogen model roumen anguelov∗†, rebecca bekker∗, yves dumont∗‡ ∗department of mathematics and applied mathematics, university of pretoria, south africa roumen.anguelov@up.ac.za, rebeccaabekker@gmail.com † associate member of the institute of mathematics and informatics bulgarian academy of sciences, sofia , bulgaria ‡ cirad, umr amap, pretoria, south africa amap, university of montpellier, cirad, cnrs, inra, ird, montpellier, france yves.dumont@cirad.fr received: 8 august 2018, accepted: 2 january 2019, published: 23 january 2019 abstract—crop host-pathogen interaction have been a main issue for decades, in particular for food security. in this paper, we focus on the modeling and long term behavior of soil-borne pathogens. we first develop a new compartmental temporal model, which exhibits bi-stable asymptotical dynamics. to investigate the long term behavior, we use lasalle invariance principle to derive sufficient conditions for global asymptotic stability of the pathogen free equilibrium and monotone dynamical systems theory to provide sufficient conditions for permanence of the system. finally, we develop a partially degenerate reaction diffusion system, providing a numerical exploration based on the results obtained for the temporal system. we show that a traveling wave solution may exist where the speed of the wave follows a power law. keywords-host-pathogen; bi-stability; monotone dynamical system; lasalle invariance principle; partially degenerate reaction-diffusion; traveling wave i. introduction the global food supply is currently experiencing pressure from climate change and ever increasing demand. another major concern is the increasing impact of pathogens. an estimated 16% of the global crop yield is lost to various pathogens annually [3], [10], [14]. consequently there has been an increase in research of botanical pathogens and the resulting diseases, with foliar pathogens being the focus of the majority of published work. one important difference between foliar and soil-borne pathogens is the environment wherein each occurs. foliar pathogens have to contend with external factors such as wind, radiation and varying temperatures. however, the soil environment dampens the effects of such factors, although the inherent opacity of soil poses a number of challenges of its own. added to these are challenges relating to capturing the direct and indirect influence of the environment on processes such as survival, copyright: c©2019 anguelov et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: roumen anguelov, rebecca bekker, yves dumont, bi-stable dynamics of a host-pathogen model, biomath 8 (2019), 1901029, http://dx.doi.org/10.11145/j.biomath.2019.01.029 page 1 of 17 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2019.01.029 r. anguelov, r. bekker, y. dumont, bi-stable dynamics of a host-pathogen model dispersal and germination of pathogens, tissue growth, spatial distribution and susceptibility of hosts [12]. to some extent this paper is motivated by an early work of gilligan [6], [7], [8], where he proposed a seir type compartmental model for the propagation of a soil-borne plant disease. this model includes a diffusion term, suggesting movement of infectious individuals. although correct in certain contexts, this is unlikely in plant populations. in the model proposed here we consider compartments for the pathogen, where infection/infestation occurs when pathogen attaches to susceptible roots. in the spatio-temporal model we consider diffusion of the unattached pathogen, which we believe is a more biologically realistic assumption. the paper is organized as follows. in the next section we construct the temporal host-pathogen model highlighting the assumptions on which it is based. section 3 deals with the equilibria of the system. sufficient conditions for extinction and persistence of the pathogen are presented in section 4 and 5 respectively. the spatio-temporal model is numerically considered in section 6. ii. the host-pathogen model we consider a population of susceptible host plants with a constant recruitment rate λ, and a pathogen present in the soil. as usual, the compartments of susceptible and infective/infested hosts are denoted by s and i, respectively, and n = s + i is the total host population. the natural decay rate of the host is d per time unit, and infected hosts have an addition decay rate of α per time unit. we assume that the pathogen is dependent on its host for nutrients or energy, and as such has an expected off–host death rate δ per time units. after coming into contact with a susceptible host it attaches at rate ρ, and grows at intrinsic growth rate of λ, restricted by the carrying capacity γi of the infected/infested roots. the attached pathogen (compartment a) detaches from their hosts at a rate of σ per time unit. the unattached or free pathogen (compartment f ) is responsible for new infections/infestations. it is assumed that if the population of free pathogen is large, the transmission rate from s to i depends solely on a constant β and the level of susceptible hosts present. this type of incidence is called saturation incidence and we use the specific form βf m+f . using a saturating infestation rate is motivated by biological observations that increasing the free pathogen beyond a certain level no longer increases infestation proportionally. from a mathematical point of view, if only mass action principle is applied e.g. βfs, then, since f can potentially be very large, s would decrease very rapidly, which is unrealistic. for simplicity, the attachment rate (transfer from f to a is just mass action principle, namely ρfs. however, the growth in the a compartment is limited through the carrying capacity γi = γ(n − s). since s cannot decrease unrealistically quickly then a cannot increase unrealistically quickly. the flow fig. 1. flow chart of the host-pathogen model chart is given in figure 1. the model is a system of odes presented below: da dt = λa(γi −a) −σa + ρfs df dt = −δf + σa−ρfs ds dt = λ −ds − βf m + f s di dt = βf m + f s − (α + d)i (1) using the notation x = (a,f,s,i)t the model biomath 8 (2019), 1901029, http://dx.doi.org/10.11145/j.biomath.2019.01.029 page 2 of 17 http://dx.doi.org/10.11145/j.biomath.2019.01.029 r. anguelov, r. bekker, y. dumont, bi-stable dynamics of a host-pathogen model (1) is written as ẋ = f(x) (2) with f(x) =   λa(γi −a) −σa + ρfs −δf + σa−ρfs λ −ds − βf m+f s βf m+f s − (α + d)i   . the local existence and uniqueness of solutions of (1) in r4+ = {x ∈ r4 : x ≥ 0} follows from the fact that f is continuously differentiable on r4+. the vector field defined by f points inwards on the boundary of r4+. hence, r 4 + is positively invariant. in order to obtain global existence of solutions it remains to show that all solutions initiated in r4+ are bounded. adding the equations for ds dt and di dt we have the inequalities λ−(d+α)(s+i)≤ d(s+i) dt ≤ λ−d(s+i), (3) which do not depend on the other coordinates of x. hence, for any solution we have that the interval[ λ α + d , λ d ] is a global attractor for s(t) + i(t). more precisely, we have λ α + d ≤ lim inf t→+∞ (s(t) + i(t)) ≤ lim sup t→+∞ (s(t) + i(t)) ≤ λ d . (4) since s and i are also nonnegative, they are bounded. using (4) it is easy to obtain that the rest of the coordinates of any solution x(t) are also bounded. in fact, since the bounds in (4) do not depend on the initial condition, one obtains that (1) defines a dynamical system on r4+, which is dissipative. iii. equilibria in the absence of pathogen the population of the host is s and it is governed by the third equation in the model (1). it has an asymptotically stable equilibrium at λ d with basin of attraction s ∈ [0, +∞). the resulting equilibrium of the model (1) we call pathogen free equilibrium (pfe), that is, pfe = ( 0, 0, λ d , 0 ) . (5) the basic reproduction number/ratio, r0, is a threshold quantity, which is often used to characterize the properties of epidemiological models. it is popularly defined as the number of new infections caused by a single infectious individual in a wholly susceptible population. its precise definition is that it is the spectral radius of the next generation matrix calculated at an asymptotically stable equilibrium of the population in the absence of disease, [17]. such equilibrium is usually referred to as disease free equilibrium. due to the nature of the model in this paper, and as mentioned above, we use the term pathogen free equilibrium (pfe). the model (1) has a unique pathogen free equilibrium given by (5). following the method in [17] for the computation of r0, the compartment vector x in the model (1) is decomposed into three-dimensional vector of pathogen related compartments y = (a,f,i)t and one pathogen free compartment s and we have ẏ = f(y,s) −v(y,s), where f(y,s) =  λa(γi −a) + ρsf0 βfs m+f   , v(y,s) =   σaδf −σa + ρfs (α + d)i   . the next generation matrix is ∂f ∂y ( 0, λ d )( ∂v ∂y ( 0, λ d ))−1 = 1 m(δ + ρλ d )  ρm λd ρm λd 00 0 0 β λ d β λ d 0   . thus r0 = ρλ dδ + ρλ . biomath 8 (2019), 1901029, http://dx.doi.org/10.11145/j.biomath.2019.01.029 page 3 of 17 http://dx.doi.org/10.11145/j.biomath.2019.01.029 r. anguelov, r. bekker, y. dumont, bi-stable dynamics of a host-pathogen model it is clear that r0 < 1 regardless of the values of the parameters (as long as they are positive). therefore, it follows from [17, theorem 2] that the pfe is always asymptotically stable. the asymptotic behavior of the solutions of (1) depends on the existence of pathogen-endemic equilibria or other invariant sets and their properties. it is interesting to remark that for the model under consideration r0 is not a threshold quantity at all. in order to obtain all equilibria, we set right hand side in model (1) equal to zero, λa(γi −a) −σa + ρfs =0, −δf + σa−ρfs =0, λ −ds − βf m + f s =0, βf m + f s − (α + d)i =0. (6) from the third equation in (6) we obtain an expression for s in terms of f : s = λ(m + f) (d + β)f + dm . (7) then, using (7), from the second equation in (6) we obtain an expression for a: a = σ−1(δ + ρs)f = σ−1 ( δ + ρλ(m + f) (d + β)f + dm ) f. (8) similarly, using (7), the fourth equation in (6) gives an expression for i in terms of f : i = βfs (α + d)(m + f) = λβf (α + d)((d + β)f + dm) . (9) substituting these expressions into the first equation and excluding the case f = 0, we obtain a cubic equation about f in the form −a1f 3 + a2f 2 + a3f −a4 = 0, (10) where a1 = λ(α + d)[δ(d + β) + ρλ] 2, a2 = −2λ(α + d)[δ(d + β) + ρλ](δd + ρλ)m −δ(α + d)(d + β)2σ2 +λγλσβ[δ(d + β) + ρλ], a3 = λγσλβ(δd+ρλ)m−2δ(α+d)(d+β)dmσ2 −λ(α + d)(δd + ρλ)2m2, a4 = δ(α + d)σ 2d2m2. clearly, a1 > 0 and a4 > 0, while the signs of a2 and a3 may vary depending on the values of the parameters. however, it is easy to see that for any signs of a2 and a3 there are always either two sign changes in the sequence of the coefficients of (10) or no sign changes at all. hence the equation has either two positive roots or no positive roots. when these roots exist we denote them by f1 and f2, with f1 ≤ f2. the respective equilibria of the model (10) are denoted by ee1 = (a1,f1,s1,i1) t and ee2 = (a2,f2,s2,i2)t . from the expressions (7), (8) and (9) we see that ee1 > 0 and ee2 > 0. further, we can also see from the expressions (7)–(9) that s is a decreasing function of f , while i and a are increasing functions of f . hence we have a1 ≤ a2, i1 ≤ i2 and s1 ≥ s2. (11) the two positive roots of (10) appear simultaneously as a double root f1 = f2, which then splits into two simple roots. hence, in the bifurcation state when f1 = f2, the equilibrium ee1 = ee2 appears and then splits into the two distinct equilibria ee1 and ee2. since the constant term of (10) is strictly positive, this bifurcation is bounded away from the pfe. the pfe does not undergo any bifurcation and, as mentioned above, it is always asymptotically stable. we perform numerical simulations using nonstandard finite difference schemes [2] to solve systems (1) and (33). in all numerical simulations of model (1) we observe two qualitatively different cases and the transition (bifurcation) from one to the other. the biomath 8 (2019), 1901029, http://dx.doi.org/10.11145/j.biomath.2019.01.029 page 4 of 17 http://dx.doi.org/10.11145/j.biomath.2019.01.029 r. anguelov, r. bekker, y. dumont, bi-stable dynamics of a host-pathogen model one case is when pfe is the only equilibrium of the system. in this case we observe that all solution converge to pfe, that is pfe is globally asymptotically stable on r4+. an illustrative example is given figure 2 with value of the parameters given in table i. table i parameter values used in figure 2 parameter value parameter value λ 1.1000 λ 0.5000 γ 0.5000 β 1.0000 σ 0.4000 ρ 0.2000 δ 0.1000 m 10.000 α 0.5100 d 0.5000 the second case is when the model has two positive equilibria. in the simulations presented in figure 3, ee2 is stable and attracting, while ee1 is unstable (saddle point). table ii contains the parameter values used for the simulations in figure 3. the solutions that are initiated below ee1 in the (a,f,i)-space converge to the pfe. solutions φ1 and φ2 are initiated at ee1 with altered value of s, s0 = 2 and s0 = 20 respectively. these values are below and above the pfe value of s, respectively. we observe that φ1 converges to the pfe, while φ2 increases and eventually converges to ee2. the unstable equilibrium is typically very close to the pfe, so that the basin of attraction of the pfe is relatively small. nevertheless, it contains the whole nonnegative s-axis. table ii parameter values used in figure 3. parameter value parameter value λ 1.0000 λ 0.5000 γ 0.9000 β 5.0000 σ 0.4000 ρ 0.1000 δ 0.1000 m 100.000 α 0.0450 d 0.1000 due to the complexity of the model, we could not obtain theoretically a general result for the observed properties of the positive equilibria, or alternatively the global asymptotic stability of the pfe. however, we derive in the next two sections sufficient conditions for the two practically important properties: extinction and persistence of the pathogen. iv. sufficient conditions for global asymptotic stability of pfe we prove sufficient conditions for global asymptotic stability of pfe using lasalle’s invariance principle [11, theorem 2]. theorem 1. the pfe of model (1) is globally asymptotically stable on r4+ if either condition a) or condition b) below hold: a) λβγ2λ2 16d2(α + d)δm ≤ 1 (12) b) β < α + d and λβ2γ2λ2 4d2δm(β + α + d)2 ≤ 1. (13) proof: taking into account the inequalities (4), it is sufficient to consider the system (1) on the domain ω = { x ∈ r4+ : s + i ≤ λ d } . we consider on ω the function v (x) = a + f + ξ 2 i2, where ξ is a positive constant with value yet to be determined. we have v̇ (x) = ȧ + ḟ + ξii̇ = λa(γi −a) − ξ(α + d)i2 + ξ βf m + f si −δf. since the first term is a quadratic of a it obtains it largest value when a = 1 2 γi. using also that f ≥ 0 and si ≤ ( s + i 2 )2 ≤ λ2 4d2 , we have v̇ (x) ≤ ( λγ2 4 − ξ(α + d) ) i2 + ( ξβλ2 4d2m −δ ) f (14) biomath 8 (2019), 1901029, http://dx.doi.org/10.11145/j.biomath.2019.01.029 page 5 of 17 http://dx.doi.org/10.11145/j.biomath.2019.01.029 r. anguelov, r. bekker, y. dumont, bi-stable dynamics of a host-pathogen model fig. 2. illustration of the case of global asymptotic stability of the pfe. the coefficient of i2 is nonpositive if and only if ξ ≥ λγ2 4(α + d) (15) similarly, the coefficient of f is nonpositive if and only if ξ ≤ 4d2mδ βλ2 (16) a constant ξ satisfying (15) and (16) exists if and only if λγ2 4(α + d) ≤ 4d2mδ βλ2 , or, equivalently, λβγ2λ2 16d2(α + d)δm ≤ 1, (17) that is if and only if condition a) holds. hence, if a) holds, we can select ξ such that v̇ (x) ≤ 0 for x ∈ ω. let e = {x ∈ ω : v̇ (x) = 0}. according to lasalle invariance principle, all solutions converge to the largest invariant set m of (1) which is contained in e. if the inequality in (17) is strict, then ξ can be selected in such a way that the coefficients in (14) are negative. hence, v̇ (x) = 0 only if i = f = 0 and then a = 0 as well. hence, e = { (0, 0,s, 0) : 0 ≤ s ≤ λ d } . (18) the largest invariant set in e given in (18) is m = {pfe}. if the inequality in (17) holds as equality, then the right hand side of (14) is zero irrespective of the values of i and f . however, if i or f is not zero, then the respective inequalities leading to (14) should be satisfied as equalities. this process leads enlargement of e in (18) by an isolated point ( γλ 4d , 0, λ 2d , λ 2d ) and we again have m = {pfe}. therefore, by the lasalle invariance principle all solutions converge to pfe. b) the condition β < α+d provides for slightly sharper upper bound of si and hence slightly weaker condition on the parameters. biomath 8 (2019), 1901029, http://dx.doi.org/10.11145/j.biomath.2019.01.029 page 6 of 17 http://dx.doi.org/10.11145/j.biomath.2019.01.029 r. anguelov, r. bekker, y. dumont, bi-stable dynamics of a host-pathogen model fig. 3. if solutions of model (1) are initiated ‘close’ to the pfe convergence to this equilibrium occurs. solutions that are initiated from the ‘smaller’ endemic equilibrium converge to either the pfe or the ‘larger’ equilibrium, dependent on the initial density of the susceptible compartment. from the last equation of the model (1) we have di dt ≤ βs − (α + d)i ≤ β(s + i) − (β + α + d)i ≤ βλ d − (β + α + d)i. therefore lim sup t→+∞ i(t) ≤ βλ d(β + α + d) . hence, in the investigation of the asymptotic behavior of the system we can consider only the subset of ω where i ≤ βλ d(β + α + d) . the inequality β < α + d implies that βλ d(β + α + d) ≤ λ 2d . hence, si ≤ ( λ d − i ) i ≤ ( λ d − βλ d(β + α + d) ) βλ d(β + α + d) = β(α + d)λ2 (β + α + d)2d2 . then, following the same method as for a), in place of (17) we obtain the second inequality in b). the parameter values given in table i satisfy both condition (a) and condition (b) of theorem 1. hence, the global asymptotic stability of pfe observed earlier in figure 2 can be deduced from either (a) or (b). another illustration is given on figure 4. the model parameters, given in table iii, satisfy condition (a), but not condition (b) in theorem 1. this is sufficient to deduce the global asymptotic stability of pfe seen in figure 4. we need to remark the conditions (a) and (b) are each only sufficient, but not necessary for the biomath 8 (2019), 1901029, http://dx.doi.org/10.11145/j.biomath.2019.01.029 page 7 of 17 http://dx.doi.org/10.11145/j.biomath.2019.01.029 r. anguelov, r. bekker, y. dumont, bi-stable dynamics of a host-pathogen model table iii parameter values used in figure 4 parameter value parameter value λ 0.9000 λ 0.5000 γ 0.5000 β 1.0000 σ 0.2000 ρ 0.3000 δ 0.2900 m 5.000 α 0.0600 d 0.1500 global stability of the pfe. the parameter values given in table iv satisfy neither of the conditions in theorem 1. yet, global asymptotic stability of pfe can be observed in the simulations presented in figure 5. table iv parameter values used in figure 5 parameter value parameter value λ 1.1000 λ 0.6900 γ 0.6000 β 1.5000 σ 0.4000 ρ 0.5000 δ 0.2000 m 5.000 α 0.0100 d 0.3000 v. sufficient conditions for persistence we derive sufficient conditions for persistence using the theory of monotone systems. the system (1) is not monotone. we construct an auxiliary system about the vector y = (a,f,i)t which is monotone. from the third equation of (1) we have λ − (d + β)s ≤ ds dt ≤ λ −ds (19) then, it follows that for every solution of (1) it holds λ β + d ≤ lim inf t→+∞ s(t) ≤ lim sup t→+∞ s(t) ≤ λ d . hence, for the asymptotic properties of the solutions of (1) it is sufficient to consider the subset of ω where λ β + d ≤ s ≤ λ d . (20) in this subdomain we consider the following system for y = (a,f,i)t : dy dt = h(y) :=   λa(γi −a) −σa−δf + σa− ρλ d f βf m+f λ β+d − (α + d)i   . (21) let us recall that function h is said to satisfy the kamke condition if hi is increasing in yj for i 6= j. the kamke condition implies that the respective system of odes is monotone with respect to the initial condition or shortly monotone, [16, section 3.1]. the jacobian of h is jh =  λγi−2λa−σ 0 λγaσ −δ − ρλd 0 0 βmλ (β+d)(m+f)2 −(α+d)   . since the nondiagonal entries of jh are nonnegative, the system (21) is monotone. moreover, since the system is irreducible in the interior of r3+, it is strongly monotone, [16, theorem 4.1.1]. let us recall that a system of the form (21) is called strongly monotone if for any two solution y(1) and y(2) y(1)(0) < y(2)(0) =⇒ y(1)i (t) < y (2) i (t), t > 0, i = 1, 2, 3. our interest in the system (21) is motivated by the fact that its solutions provide lower bounds for the coordinates a, f , and i of the solutions of (1). this will be shown later by using differential inequalities given in [18] for systems of odes with quasi-monotone right hand side. hence, we carry out first the asymptotic analysis of (21). to find the equilibria of (21), we set the right hand side to zero: λa(γi −a) −σa = 0 (22) −δf + σa− ρλ d f = 0 (23) βf m + f λ β + d − (α + d)i = 0 (24) the origin, 0 is an equilibrium. to find the nonzero equilibria, we multiply the first equation by σ λ and add it to the second one to eliminate a. we biomath 8 (2019), 1901029, http://dx.doi.org/10.11145/j.biomath.2019.01.029 page 8 of 17 http://dx.doi.org/10.11145/j.biomath.2019.01.029 r. anguelov, r. bekker, y. dumont, bi-stable dynamics of a host-pathogen model fig. 4. illustration of the global stability of the pfe of the host-pathogen model when only condition (a) in theorem 1 holds. multiply the third equation by γσ α+d and add it to the second one to eliminate i. hence, we obtain ϕ(y) := σ λ h1(y) + h2 + γσ α + d h3(y) = (25) − ( δ+ ρλ d ) f− σ2 λ + γσβλf (α+d)(β+d)(m +f) = 0 (26) equation (26) is equivalent to the quadratic equation ( δ + ρλ d ) f 2+( m ( δ + ρλ d ) + σ2 λ − γσβλ (β + d)(α + d) ) f + mσ2 λ = 0. (27) the equation (27) has two positive real roots if and only if the coefficient of f is negative and the discriminant is positive, that is m ( δ+ ρλ d ) + σ2 λ − γσβλ (β+d)(α+d) < 0, (28) ∆ = ( m ( δ+ ρλ d ) + σ2 λ − γσβλ (β+d)(α+d) )2 − 4mσ2 λ ( δ + ρλ d ) > 0 (29) assuming that conditions (28)–(29) hold, we denote by f̃1 and f̃2, f̃1 < f̃2, the roots of (27) and by ẽ1 = (ã1, f̃1, ĩ1)t and ẽ2 = (ã2, f̃2, ĩ2)t the corresponding equilibria of (21). where ai = 1 σ ( δ + ρλ d ) fi, i = 1, 2, (30) ii = 1 γ (ai + σ) , i = 1, 2. (31) considering the expressions (30) and (31, we have 0 << ẽ1 << ẽ2. theorem 2. let conditions (28)–(29) hold. then for every solution y(t) of (21) we have y(0) > ẽ1 =⇒ lim inf t→+∞ y(t) ≥ ẽ2. biomath 8 (2019), 1901029, http://dx.doi.org/10.11145/j.biomath.2019.01.029 page 9 of 17 http://dx.doi.org/10.11145/j.biomath.2019.01.029 r. anguelov, r. bekker, y. dumont, bi-stable dynamics of a host-pathogen model fig. 5. illustration that pfe of the host-pathogen model may be globally asymptotically stable when neither of the condition in theorem 1 holds. proof: it is easy to see that the equilibrium 0 is asymptotically stable, indeed the eigenvalues of jh(0), ξ1 = −σ, ξ2 = − ( δ + ρλ d ) and ξ3 = −(α + d) are all negative. we consider the order interval [0, ẽ1]. it follows from [16, theorem 2.2.2], that the solutions initiated in this interval, excluding the end points, either all converge to 0 or all converge to ẽ1. since 0 is asymptotically stable, this implies that all solutions converge to 0. the jacobian of h at ẽ1 after some simplifications is: jh(ẽ1) =  −λa ∗ 0 λγa∗ σ −δ − ρλ d 0 0 βmλ (β+d)(m+f̃1)2 −(α + d)   , since the nondiagonal entries of jh(ẽ1) are nonnegative and the matrix is irreducible, it follows from the theory of nonnegative matrices [4, theorem 2.1.4] that jh(ẽ1) has eigenvector v with positive coordinates and associated eigenvalue ξ, which is a real number. since ẽ1 is repelling in [0, ẽ1], we have that ξ ≥ 0. we will show that ξ > 0. assume the opposite, namely ξ = 0. then it is easy to compute that w = ( σ γã1 , 1, σγ α + d )t is a left eigenvector. then, using the expression for ϕ in (25) and that h(ẽ1) = 0 we have ∇ϕ(ẽ1) = wtjh(ẽ1) = 0. the first and the third coordinates of ∇ϕ(y) are always zero since ϕ does not depend on a and i. the fact that ∂ϕ(ẽ1) ∂f = 0 implies that the equation (26), or, equivalently (27, has a double root. this contradicts (29). therefore ξ > 0. next, we consider the order interval [ẽ1, ẽ2]. again following [16, theorem 2.2.2], that the solutions initiated in this interval, excluding the end points, either all converge to ẽ1 or all converge to ẽ2. considering that ẽ1 is repelling in the direction of the positive vector v, we conclude that all solutions converge to ẽ2. let y(t) be any solution of (21) such that y(0) > ẽ1. consider the solution of (21) with initial condition z(0) = min{y(0), ẽ2}. since z(0) ∈ [ẽ1, ẽ2] and z(0) > ẽ1, we have lim t→+∞ z(t) = ẽ2. biomath 8 (2019), 1901029, http://dx.doi.org/10.11145/j.biomath.2019.01.029 page 10 of 17 http://dx.doi.org/10.11145/j.biomath.2019.01.029 r. anguelov, r. bekker, y. dumont, bi-stable dynamics of a host-pathogen model then using that y(0) ≥ z(0) and the monotonicity of the system (21) we have y(t) ≥ z(t), t ≥ 0. therefore lim inf t→+∞ y(t) ≥ lim t→+∞ z(t) = ẽ2. using theorem 2 for the auxiliary system (21) we prove the following theorem for the original model (1). theorem 3. let conditions (28) and (29) hold. then, for any solution of (1) we have that if a(0) > ã1, f(0) > f̃1, i(0) > ĩ1, (32) then lim inf t→+∞ a(t) ≥ ã2, lim inf t→+∞ f(t) ≥ f̃2, lim inf t→+∞ i(t) ≥ ĩ2. proof: as discussed earlier, it is sufficient to consider the subdomain, where (20) holds. let x(t) be a solution of (2) such that a(0), f(0), i(0) satisfy (32) and s(0) ∈ [ λ β+d , λ d ] . from (19) it follows that s(t) ∈ [ λ β+d , λ d ] for t ≥ 0. then the coordinates a(t), f(t), and i(t) satisfy d dt  a(t)f(t) i(t)   =  f1(x)f2(x) f4(x)   ≥ h  a(t)f(t) i(t)   using that h is quasi-monotone and applying [18, chapter 2, section 12.x], we obtain that the vector function (a(t),f(t),i(t))t is bounded below by the solution of (21) with the same initial condition at t = 0. then, theorem 2 implies that ẽ2 is a lower bound for the limit inferior of (a(t),f(t),i(t))t as t → +∞, which proves the theorem. theorem 3 shows that if the initial invasion is sufficiently large the pathogen establishes itself at a level above ẽ2. we should remark that it is not necessary to have initially all compartments a, f and i above ẽ1. it is sufficient that at some future time they all exceed ẽ1. for example, and as it can be also expected from biological point of view, the initial infection/infestation is brought in the compartment f . if this initial value of f is sufficiently large that a and i increase above ã1 and ĩ1, while f is still above f̃1, the pathogen will persist eventually at least at a level of ẽ2. theorem 3 is illustrated numerically by figure 6. the initial conditions of the solutions in this figure were chosen specifically so that (a0,f0,i0) ≥ ẽ1. clearly any solution initiated at or above the level of ẽ1 ≈ (4.091, 1.7241, 0.2164) persists at a non-zero level above ẽ2 for all time; and in fact converges to an equilibrium of model (1), at least for the parameter values in table v. we verify that the data in table v satisfies conditions (28)-(29) of theorem 3. indeed, the left hands sides of (28) and (29) are respectively negative and positive, so that inequalities in (28) and (29) hold. theorem 3 gives only sufficient conditions for persistence of pathogen. in figure 7, we show that the pathogen persist, in fact, the solutions converge to ee2, even though the initial conditions do not satisfy the requirements of theorem 3. table v parameter values used in figures 6 and 7 parameter value parameter value λ 0.9000 λ 1.0000 γ 23.52536 β 5.0000 σ 1.0000 ρ 0.9000 δ 0.9000 m 10.000 α 0.0010 d 0.5500 our numerical investigations of model (1) have revealed that in addition to the pfe which is always locally attracting, there exists an equilibrium close to the pfe which is repelling, and an attracting equilibrium which is removed from the pfe. although we have not proven this mathematically, the important properties of the model, namely extinction and persistence, have been proven under certain conditions. vi. the spatio-temporal host-pathogen model the model in section ii looks only at the temporal progression of an infection, and although this biomath 8 (2019), 1901029, http://dx.doi.org/10.11145/j.biomath.2019.01.029 page 11 of 17 http://dx.doi.org/10.11145/j.biomath.2019.01.029 r. anguelov, r. bekker, y. dumont, bi-stable dynamics of a host-pathogen model fig. 6. illustration of persistence of the infection as proven in theorem 3. observe that solutions of model (1) originating at the level of ẽ1 remain non-zero for all time. approach is acceptable under certain assumptions (such as a pathogen entering an entire field in a uniform manner), the model can be modified slightly to accommodate the spatial movement of pathogens through the field. diffusion has been used to model spatial spread in theoretical ecology since the latter half of the twentieth century [9], [15], with its use for modelling fungal growth being justified by the “observation that tip growth occurs to fill space and to capture nutrients” [5]. davidson (1998) also noted that fungal growth “is, in the main, directed from areas of high hyphal density to areas of low hyphal density”, and included diffusion in his model with the warning ‘that this flux should not be viewed as the movement of existing biomass, but rather the propensity of new biomass to grow away from high density areas’. we reiterate this warning, and include diffusion to model the spatial growth of off-host pathogen in search of new hosts, with µ denoting the diffusion constant. our host-pathogen spatio-temporal model is defined as follows:  ∂a ∂t = λa(γi −a) −σa + ρfs ∂f ∂t = −δf + σa−ρfs + µ∆f ∂s ∂t = λ −ds − βf m + f s ∂i ∂t = βf m + f s − (α + d)i with a(x, 0) ≥ 0, f(x, 0) ≥ 0, s(x, 0) ≥ 0, i(x, 0) ≥ 0, ∂f ∂x (−l,t) = 0 = ∂f ∂x (l,t). (33) if the initial condition is spatially uniform, and taking the boundary conditions into account, then the solution is also spatially uniform, reducing it to a solution of the corresponding temporal system. the properties and long-term behaviour of the temporal model have been theoretically proven in section ii, and this chapter devotes itself to numerical investigation of the behaviour of biomath 8 (2019), 1901029, http://dx.doi.org/10.11145/j.biomath.2019.01.029 page 12 of 17 http://dx.doi.org/10.11145/j.biomath.2019.01.029 r. anguelov, r. bekker, y. dumont, bi-stable dynamics of a host-pathogen model fig. 7. illustration of persistence of the infection when initial conditions do not satisfy conditions (32) in theorem 3. solutions of system (33). our interest is mainly in the practically relevant case where the pathogen is introduced in one location, and studying the dynamics of its propagation. in order to solve model (33) numerically, we use non-standard discretization coupled with a second order centralspace discretization [2]. a. numerical investigations 1) under the conditions for asymptotic stability obtained by application of lasalle’s invariance principle: the parameter values given in table vi satisfy the conditions (12) which ensures the pfe of the temporal model is globally asymptotically stable. the details are given in section iv. we investigate whether the solutions of the spatiotemporal model behave in a similar fashion, using the diffusion constant µ = 0.01. indeed, although convergence occurs over a long time period, the addition of diffusion does not result in observable change in the asymptotic properties of the steady state. in figure 8, even assuming that the initial population has free pathogen over a quarter of the field, this does not result in the infection spreading. table vi parameter values used in figure 8 parameter value parameter value λ 1.0000 λ 0.4000 γ 0.2000 β 1.0000 σ 0.0100 ρ 0.4000 δ 0.1000 m 100.000 α 0.0205 d 0.1000 table vii parameter values used in figure 9 parameter value parameter value λ 1.0000 λ 0.7000 γ 0.4000 β 1.0000 σ 0.1000 ρ 0.2000 δ 0.2000 m 100.00 α 0.0100 d 0.2000 in figure 9, we also provide an example where the conditions (12) are not satisfied and yet there is convergence to pfe (see also table vii). 2) parameter values for persistence of the infection: a monotone system, constructed to approximate the temporal host pathogen model from below was proven to admit two interior equilibria biomath 8 (2019), 1901029, http://dx.doi.org/10.11145/j.biomath.2019.01.029 page 13 of 17 http://dx.doi.org/10.11145/j.biomath.2019.01.029 r. anguelov, r. bekker, y. dumont, bi-stable dynamics of a host-pathogen model fig. 8. the time evolution of pathogen and disease through the field at different times, using the parameter values that satisfy the conditions for stability of the pfe that were obtained by the application of lasalle’s invariance principle. fig. 9. the parameter values in table vii do not satisfy condition for asymptotic stability of the pfe, however convergence to pfe is observed. biomath 8 (2019), 1901029, http://dx.doi.org/10.11145/j.biomath.2019.01.029 page 14 of 17 http://dx.doi.org/10.11145/j.biomath.2019.01.029 r. anguelov, r. bekker, y. dumont, bi-stable dynamics of a host-pathogen model fig. 10. when solutions of model (33) are initiated with pathogen and infectious hosts at the level of ee2, on the left boundary, a field of completely susceptible hosts will experience a travelling infection front. this front connects ee2 and the pfe. in section v, denoted ẽ1 and ẽ1, with ẽ1 < ẽ1. additional conditions were derived, under which the pathogen persists. indeed, it was found that solutions initiated at or above ẽ1 and satisfying conditions (28) and (29), page 9, remain non-zero for all t ≥ 0. for the parameter values in table viii on page 15, these conditions are satisfied. indeed, we have m γσ ( δ + ρλ d ) + σ γλ − βλ (β + d)(α + d) ≈−0.4204 < 0, ∆ ≈ 0.0053 > 0. the equilibrium, ẽ1 of the lower approximating system is: ẽ1 ≈ (4.091, 1.7241, 0.2164). the persistence of pathogen is illustrated in figure 10. in fact, the solutions converge to ee2, although the stability properties of ee1 and ee2 have not been proven. how does the inclusion of diffusion affect this phenomenon? solutions initiated at the level of ee2 at the boundary exhibit a travelling infection front, the movement of which is driven by the increase in attached pathogen and infested hosts by the diffusion of the free pathogen (figure 10). this behaviour suggests a possible control strategy: if the speed of the front can be sufficiently decreased, a percentage of the field would be saved from disease. table viii parameter values used in figure 10. parameter value parameter value λ 0.9000 λ 1.0000 γ 23.52536 β 5.0000 σ 1.0000 ρ 0.9000 δ 0.9000 m 10.000 α 0.0010 d 0.5500 to this end, we investigate the relationship between µ and the wave speed c. the parameter values in table viii were again used, and a solution with (a0,f0,i0) taking the value of ee2 on the biomath 8 (2019), 1901029, http://dx.doi.org/10.11145/j.biomath.2019.01.029 page 15 of 17 http://dx.doi.org/10.11145/j.biomath.2019.01.029 r. anguelov, r. bekker, y. dumont, bi-stable dynamics of a host-pathogen model fig. 11. the speed of the infection front for different values of µ, for solutions of model (33) initiated with levels of inoculum and disease at the level of ee2 on the left boundary. the parameter values in table viii were used. clearly the equation c(µ) = 0.01088µ0.4189 fits the data well. left boundary was considered. the diffusion constant µ was taken to be in the interval [10−7, 10−1], which results in c ∈ (0, 4.5 × 10−3]. an equation of the form c(µ) = aµb was fitted to the data in figure 11, and the fitting process reveals a ∈ (0.010770, 0.011) and b ∈ (0.416, 0.4218) with 95% confidence. in fact, a = 0.01088 and b = 0.4189. literature indicates that the value of b should be higher, with gilligan [8] and metz, mollison and van den bosch [13] finding the wave speed to be proportional to the square root of the diffusion constant; that is c ∝ √ µ. although b < 0.5 the equation fits the data well, and since sse = 8.59 × 10−6 its use in making predictions would be justified. the coefficient of determination r2 = 0.9933 indicating that 99.33% of the variance of the data is explained by the equation. vii. conclusion in this work, we have derived a host-pathogen model where the pfe is always las and may coexist with endemic equilibria. we provided sufficient conditions for pfe being globally asymptotically stable and for persistence of the pathogen, using two different approaches, lasalle invariance principle approach and monotone system approach. we show that these results can be extended to the spatio-temporal system, where we add diffusion in the free pathogen compartment. we also show numerically that a bi-stable travelling wave solution can exist between pfe and a stable endemic equilibrium, here ee2. we show that the speed c of this traveling wave is of the form aµb, where µ is the diffusion parameter. further theoretical investigations are needed, in order to be able to derive appropriate control strategies to avoid invasion of the pathogen in the whole crop. references [1] r. anguelov, y. dumont, j. lubuma (2012), mathematical modelling of sterile insect technology for control of anopheles mosquito, computers & mathematics with biomath 8 (2019), 1901029, http://dx.doi.org/10.11145/j.biomath.2019.01.029 page 16 of 17 http://dx.doi.org/10.11145/j.biomath.2019.01.029 r. anguelov, r. bekker, y. dumont, bi-stable dynamics of a host-pathogen model applications 64(3), 374-389. https://doi.org/10.1016/j.camwa.2012.02.068 [2] r. anguelov, y. dumont, and j. m.-s. lubuma (2012). on nonstandard finite difference schemes in biosciences, aip conf. proc. 1487(1), 212-223. https://doi.org/10.1063/1.4758961 [3] d.p. bebber, t. holmes, s.j. gurr (2014), the global spread of crop pests and pathogens, glob. ecol. biogeogr 23(12), 1398-1407. https://doi.org/10.1111/geb.12214 [4] a. berman, r.j. plemmons, nonnegative matrices in the mathematical sciences, siam, 1994. [5] n.j. cunniffe (2007), dispersal of soil-borne plant pathogens and efficacy of biological control, phd thesis, university of cambridge. [6] c.a. gilligan (1985), construction of temporal models: iii. disease progress of soil-borne pathogens. in: c.a. gilligan (ed.), mathematical modelling of crop disease, london, academic press. [7] c.a. gilligan (1990), mathematical modeling and analysis of soilborne pathogens. in: j. kranz (ed.) epidemics of plant diseases: mathematical analysis and modeling, heidelberg, springer-verlag, 96-142. [8] c.a. gilligan (1995), modelling soil-borne plant pathogens: reaction-diffusion models, canadian journal of plant pathology 17, 96-108. [9] e.e. holmes, m.a. lewis, j.e. banks, r.r. veit (1994), partial differential equations in ecology: spatial interacpartial differential equations in ecology: spatial interactions and population dynamics, ecology 75, 17-29. [10] state of the world’s plants report 2016, royal botanic gardens, kew, https://stateoftheworldsplants.org/2016/ [11] j. lasalle (1960), some extensions of liapunov’s second method, ire transactions on circuit theory 7(4), 520-527. [12] j.d. macdonald, the soil environment. in: c.l. campbell, d.m. benson (eds), epidemiology and management of root disease, springer-verlag, 1994, 82-116. [13] j.a.j. metz, d. mollison, f. van den bosch (2000), the dynamics of invasion waves. in: u. dieckmann, r. law, j.a.j metz (eds), the geometry of ecological interactions: simplifying spatial complexity, cambridge university press, 482-512, https://doi.org/10.1017/cbo9780511525537.027 [14] e.c. oerke (2006). crop losses to pests, j.agric. sci. 144(1), 31-43. [15] a. okubo, s.a. levin, diffusion and ecological problems: modern perspectives, springer, 2001. [16] h.l. smith, monotone dynamical system, ams, 1995. [17] p. van den driessche, j. watmough (2002), reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, mathematical biosciences 180, 29-48, https://doi.org/10.1016/s0025-5564(02)00108-6 [18] w. walter, differential and integral inequalities, springer, 1970. biomath 8 (2019), 1901029, http://dx.doi.org/10.11145/j.biomath.2019.01.029 page 17 of 17 https://doi.org/10.1016/j.camwa.2012.02.068 https://doi.org/10.1063/1.4758961 https://doi.org/10.1111/geb.12214 https://stateoftheworldsplants.org/2016/ https://doi.org/10.1017/cbo9780511525537.027 https://doi.org/10.1016/s0025-5564(02)00108-6 http://dx.doi.org/10.11145/j.biomath.2019.01.029 introduction the host-pathogen model equilibria sufficient conditions for global asymptotic stability of pfe sufficient conditions for persistence the spatio-temporal host-pathogen model numerical investigations under the conditions for asymptotic stability obtained by application of lasalle's invariance principle under the conditions for persistence of the pathogen conclusion references communication/review biomath 1 (2012), 1210017, 1–5 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum blagovest sendov – pioneer of mathematical modeling in bulgaria svetoslav markov institute of mathematics and informatics-bas acad. g. bonchev st., block 8, 1113 sofia, bulgaria email: smarkov@bio.bas.bg i. my teacher prof. sendov in february 2012 prof. blagovest sendov turned 80. i first met him in 1963 as a student in mathematics at the department of physics and mathematics of sofia university. i was fascinated by his lectures in numerical methods. his first lecture was devoted to mathematical modeling. on several realistic case studies prof. sendov revealed what was to me the philosophy of science. this lecture was crucial for my orientation in mathematical research. i was deeply impressed and i started to collect material for a book based on the ideas of prof. sendov, which i published later on [14]. during the years that followed, mathematical modeling became my main topic of interest. thanks to prof. sendov, i was equipped with many useful ideas, tools and insights that i used in the study of real-life problems. prof. sendov’s “philosophy” included a deep understanding of the mechanisms of the underlying real processes, the mathematical description of these processes using contemporary mathematical theories and the solution of the formulated mathematical problems using advanced numerical and computational tools. an important area of mathematical applications is the one of biology. in the sequei i shall try to briefly survey prof. sendov’s achievements in the field of mathematical modeling in biology and also mention his contributions to the field of “interval analysis”, which field is tightly related to sendov’s main topic of interest “approximation theory” [25]. prof. sendov possesses the enormous ability to formulate difficult tasks and problems, playing a key role in mathematics and its applications, and requiring years of efforts to be resolved. at his weekly seminar on “mathematical modeling” he used to pose such difficult problems and to make us young collaborators enthusiastic about working on them. he never pressed anybody of us to work on something particular, but he waited that everyone himself/herself chooses a topic of interest. ii. the contributions of prof. blagovest sendov to biomathematics in the period 1965–1971 prof. sendov collaborated actively with prof. dr. roumen tsanev, an excellent molecular biologist, and also a very competent mathematician. in the summer of 1965 prof. sendov and dr. tsanev began jointly to study a hypothetical mechanisms for cellular proliferation, differentiation and carcinogenesis, suggested by dr. tsanev. the two scientists wanted to establish whether a mechanism for cellular activity based on interrelated genes is logically feasible. they decided to use the newly installed computer in the institute of mathematics at the bulgarian academy of sciences to study a model of cellular activity based on a network of genes interrelated on the basis of equations describing the synthesis of mrna, controlled by dnaprotein interactions and programming the ribosomes for the synthesis of proteins. during the next several years they had many discussions on the formulation of a suitable mathematical model. prof. sendov tried many formulations and performed multiple computer experiments. citation: s. markov, blagovest sendov – pioneer of mathematical modeling in bulgaria, biomath 1 (2012), 1210017, http://dx.doi.org/10.11145/j.biomath.2012.10.017 page 1 of 5 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.10.017 s. markov, blagovest sendov – pioneer of mathematical modeling in bulgaria the results of this active collaboration with lengthy discussions and computer experiments led to several modifications of the models, which were reported in a series of joint papers [1]–[10]. these papers are devoted to modeling of different biological objects such as epidermis and liver, or different processes such as cellular activity, cellular differentiation and carcinogenesis. the main result of the investigation was that the process has to be controlled by an independent carrier of genetic information, which is not necessarily semantically connected to the genetic information. this independent carrier of information was postulated as an epigenetic code. the mathematical model of living cells in a multi-cellular organism, based on the existence of an epigenetic code, was able to explain uniformly the processes of embryonic development, cytodifferentiation, vegetative reproduction, somatic embryogenesis, carcinogenesis and even the emergence of new forms of natural selection. all this is explained in [10] with more than 300 references to experimental results showing good agreement with the results produced by the mathematical model. during the next several years a number of scientific papers are published, amongst them four papers in the journal of theoretical biology, a survey paper in “uspehi mathematicheskih nauk” and a monograph in russian. the studies of sendov and tsanev can be subdivided into four steps. the first step was to construct a mathematical model of living cells, based on the concept of jacob and monod for the existence of conjugate operons, working as a flip-flop. an operon has two states: repressed as inactive and derepressed as active. the mathematical realization of such flip-flop is by formulating a well-known set of nonlinear differential equations. the interaction between the different cells in an organism and the role of the nuclear membrane are important for eukaryotic cells. to this end a special variable for the diffusion through the membrane has been introduced. this variable depends on the functional state of the cell. it is known that when the eukaryotic cell enters in the mitotic cycle, the diffusion through its dissolved membrane stops. so they added to the system of differential equations, describing the activity of a cell, some additional differential equations with discontinuous right-hand side. the goal of this first model was to find out whether it is possible to choose the constants in the mathematical model of a system of synchronized cells, which divide and interact between themselves by the substances going through the membranes, in such a way that they reach a homeostasis. the result of the computer experiment is that the model is adequate under a suitable choice of the model parameters. the results obtained were in good agreement with the observed reaction of real tissues, such as the epidermal tissue, which was studied extensively by dr tsanev. on the second stage a model of non-synchronized cells imitating the liver was constructed [11]. this model demonstrated a good agreement with experimental data, especially the reaction after a “partial hepatectomy”. all this was achieved on the basis of the idea for repression and derepression. the third stage was to model the mechanism of cytodifferentiation. here the two scientists introduce the mechanisms of blocking and deblocking of the operons. in this situation, every operon has four different states: repressed-blocked, repressed-deblocked, derepressed-blocked and derepressed-deblocked. only in the state derepressed-deblocked, the operon is active. to test these mechanisms, they constructed a mathematical model of a set of cells, which interacted between themselves. every cell in the model has eight operons, one mitotic, responsible for the division of the cell, and seven functional operons. every operon, when active, produces substances which may repress or deblock another operon. this interconnection between the operons is prescribed in the model by a matrix and represents the meaning of the epigenetic code. in other words, the existence of an epigenetic code means that the operons in a cell form a genetic network. different types of cells in a organism have the same genetic information and differ by the set of blocked operons. to define the interaction between the cells, a special geometrical arrangement of the cells has been proposed and the place of a new cell produced after division is prescribed. to make it simple, a population of cells called “cylindros” has been created. to avoid the three dimensional geometry, an infinite cylinder is considered and a plane intersection of it is studied. thus the cylindros was a ring of two-dimensional cells on the plane with an interior space in the middle. the diffusion of all substances produced in the cells was possible only in the middle space and between the neighbor cells. in this way, the interaction between different cells was fulfilled. after division of a cell, the two new cells stayed on the same place on the ring as neighbors. choosing a particular matrix for the genetic net in the cylindros, all important behaviors as embryonic development, cytodifferenciation, vegetative reproduction, somatic embryogenesis and carcinogenesis have been biomath 1 (2012), 1210017, http://dx.doi.org/10.11145/j.biomath.2012.10.017 page 2 of 5 http://dx.doi.org/10.11145/j.biomath.2012.10.017 s. markov, blagovest sendov – pioneer of mathematical modeling in bulgaria demonstrated. mathematically the cylindros was a system of ordinary differential equations of first order with discontinuous right hand sides and the number of the equations in this system depended on the time. on the fourth stage of the study prof. sendov analyzes the system of ordinary differential equations for stability with respect to the number of the equations [11]–[12]. he shows that if the system is not stable, which means that the number of the equations goes to infinity, this case can be physiologically interpreted as cancerogenesis. the collaborative work of sendov and tsanev is a typical example of tight interaction between the two sciences biology and mathematics. on one side biology benefits from mathematics, on the other side mathematics also benefits, as novel interesting problems and suitable tools for their solution appear. such a tool is the analysis of hausdorff-continuous functions and interval analysis. the contributions of prof. sendov to hausdorff approximations are well known. here i would like to mention some of his contributions to interval analysis. iii. novel tools for mathematical modeling in many instances sendov–tsanev models make use of partially continuos functions of one variable, that is functions which are continuous in certain subintervals of their domain and have “jumps” in between. a typical such function is the heaviside step function (function “jump”) s(x) = {0, x < 0; 1, x ≥ 0}. another famous function is the dirac’s function: δ(x) = {0, x 6= 0; 1, x = 0}. the interest of prof. sendov to such functions is well known. it can be assumed that it is this interests that led him to numerous results about these functions, and many scientific papers and books, in particular the integrated “theory of hausdorff approximations” developed by him [25]. sendov’s approach to such functions is to consider their complete graphs by filling the jumps by intervals. thus the complete graphs of the functions s and δ are interval functions defined resp. as follows: s(x) = {0, x < 0; [0, 1], x = 0; 1, x > 0}, δ(x) = {0, x 6= 0; [0, 1], x = 0}. then sendov measures the distance between the complete graphs and the approximating function using the hausdorff distance between the graphs as plane sets. a famous result obtained by prof sendov in 1964 is that the heaviside step function and the dirac’s function (and any bounded function in fact) can be approximated by algebraic polynomials of order n in hausdorff metric with accuracy: c.(ln n/n), c = const. an important role in sendov’s theory of hausdorff approximations is played by the so-called hausdorff continuous functions, which are a special case of interval functions. in this way the theory of hausdorff aproximations is tightly related to interval analysis. incidentally, the first difficult problem that was given to me by prof. sendov in my master thesis about finding a numerical algorithm for the hausdorff approximation of the heaviside step function using algebraic polynomials. at that time we performed numerical computations related to this problem using computer [13]. many papers have been devoted to this problem, untill successfully solved [16]. the numerical approximation of functions like s and δ still caue difficulties on contemporary computer platforms [15]. the investigations in the field of interval analysis were initiated by prof. sendov during 1976-1980. interval functions have often been discussed in relation to hausdorff approximations at sendovs seminar on approximation theory held regularly at the institute of mathematics since 1964. numerical computations related to the best polynomial hausdorff approximations of certain interval functions require special attention to round-off errors. in 1975 sendov, who was my phd supervisor, gave me reprints of papers by t. sunaga, h. ratschek and g. schroeder on interval arithmetic and differentiation of interval functions. i was very impressed by these papers, especially by the famous paper by t. sunaga, which i studied thoroughly and later on i published (jointly with k. okumura) a review of this extraordinary paper [17]. in the years to follow prof. sendov actively worked in the field of interval analysis and published several papers on the so-called s-limit and s-derivative of interval functions — terms that are tightly related to the theory of hausdorff approximations, [18]–[20]. in this papers sendov established a theory for analysis of interval functions. his studies have been continued by some of his many collaborators and during the years a lot of scientific papers have been published. here i would like to mention some of the developments about the relations between hausdorff continuous functions and interval analysis obtained in the past decade thanks to a new property established in 2004 by roumen anguelov. the above mentioned novel property is that the set of hausdorff-continuous functions is complete after dedekind with respect to the familiar order relation. let us note that the familiar functional spaces such as the space of continuous functions, the sobolev spaces etc, with very few exceptions, are incomplete w.r.t. the order relation. thus, a possibility appeared to solve a number of open problems in real analysis and the general theory of pde or to improve previous results using biomath 1 (2012), 1210017, http://dx.doi.org/10.11145/j.biomath.2012.10.017 page 3 of 5 http://dx.doi.org/10.11145/j.biomath.2012.10.017 s. markov, blagovest sendov – pioneer of mathematical modeling in bulgaria hausdorff continuous functions. an important result is the introduction of algebraic operations with hausdorffcontinuous functions and their application for numerical computations. let us mention that hausdorff continuous functions form a special class of interval functions. it is wellknown that interval functions do not form a linear space. thus it is an important fact that the algebraic operations for addition and multiplication by scalars in the set of continuous functions can be extended over the set of hausdorff-continuous functions in such a way that they form a linear space. in several papers (jointly with prof. sendov) it has been shown that the space of hausdorff continuous functions is the largest linear space of interval functions [21]–[23]. the obtained results have been applied to numerical computations [24]. iv. the conference biomath-2012 during the years prof. sendov supported the biomathematical research in the bulgarian academy of sciences. he co-chaired the first international conference biomath 1995 [26] and took active part in many biomathematical workshops and seminars held at the academy. the present conference biomath-2012 [27] proves that the pioneering work of prof. sendov in the field of mathematical modelling is alive. our wish is the biomath conferences to become a forum for biologists and mathematicians, chemists and physicists, computer scientists and others, working together as partners in research connected with living organisms and the applications of the results to medicine, biology, ecology, agriculture and elsewhere. it is natural in such collaborations that the leading ideas come from the biologists. however, success depends on the abilities of both sides, especially when complicated mathematical models are involved. in this article we tried to outline part of prof. sendov’s activities related to mathematical modelling. the interested reader may find more about prof sendov’s activities in the article [27] where also some of his contributions to bulgarian education are presented in some detail. let us wish prof. sendov good health and still more future success in his scientific and social activities. references [1] tsanev, r. and bl. sendov: a model of the regulatory mechanism of cellular proliferation, c. r. acad. bulgare sci., 19, (1966), no. 9, 835–838. [2] tsanev, r. and bl. sendov: a model of the regulatory mechanism of cellular multiplication, j. theoret. biol., new york, 12, (1966), 327–341. [3] sendov, bl. and r. tsanev: modeling of the regulatory mechanism of the cellular proliferation in the liver, central. biochem. lab. bas, 3, (1968), 21–35 (in bulgarian). [4] tsanev, r. and bl. sendov: computer studies on the mechanism controlling cellular proliferation, in: effects on radiation on cellular proliferation and differentiation. vienna, int. atomic energy agency. 1968, 453–461. [5] sendov, bl. and r. tsanev: computer simulation of the regenerative processes in the liver, j. theoret. biol., new york, 18, (1968), 90–104. [6] sendov, bl. and r. tsanev: computer simulation of the regulatory mechanisms of cellular proliferation, inform. processing, amsterdam, 68 (1969), 1506–1507. [7] tsanev, r. and bl. sendov: a model of cancer studies by a computer, j. theoret. biol. 23, (1969), 124–134. http://dx.doi.org/10.1016/0022-5193(69)90071-x [8] tsanev, r. and bl. sendov: a possible mechanism for cellular differentiation, c. r. acad. bulgare sci., 22, (1969), no. 12, 1433–1436. [9] sendov, bl., r. tsanev and e. mateeva: a mathematical model of the regulation of cellular proliferation of the epidermis, izv. math. inst., bas, 11, (1970), 221–246. [10] tsanev, r. and bl. sendov: possible molecular mechanism for cell differentiation in multicellular organisms, j. theoret. biol., new york, 30, (1971), 337–193. [11] sendov, bl.: mathematical models of the cellular proliferation and differentiation, uspehi math. nauk, moscow, 31 no. 3, (1976), 255–256 (in russian). [12] sendov, bl.: mathematical models of the processes for cellular proliferation and differentiation, publ. moscow university, 1976 (in russian). 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[25] sendov, bl., hausdorff approximations, kluwer, 1990. http://dx.doi.org/10.1007/978-94-009-0673-0 [26] http://www.biomath.bg/1995/biomath95.php [27] http://www.biomath.bg/2012 [28] kenderov, p, a. andreev, s. dimova, s. markov, academician blagovest sendov at 80, in: j. revalski (ed.) mathematics and education in mathematics, sofia, umb, 7–22 (in bulgarian). biomath 1 (2012), 1210017, http://dx.doi.org/10.11145/j.biomath.2012.10.017 page 5 of 5 http://dx.doi.org/10.1007/978-94-009-0673-0 http://www.biomath.bg/1995/biomath95.php http://www.biomath.bg/2012 http://dx.doi.org/10.11145/j.biomath.2012.10.017 my teacher prof. sendov the contributions of prof. blagovest sendov to biomathematics novel tools for mathematical modeling the conference biomath-2012 references www.biomathforum.org/biomath/index.php/biomath original article reaction networks reveal new links between gompertz and verhulst growth functions svetoslav markov institute of mathematics and informatics, bulgarian academy of sciences smarkov@bio.bas.bg received: 8 january 2019, accepted: 16 april 2019, published: 20 april 2019 abstract—new reaction network realizations of the gompertz and logistic growth models are proposed. the proposed reaction networks involve an additional species interpreted as environmental resource. some natural generalizations and modifications of the gompertz and the logistic models, induced by the proposed networks, are formulated and discussed. in particular, it is shown that the induced dynamical systems can be reduced to one dimensional differential equations for the growth (resp. decay) species. the reaction network formulation of the proposed models suggest hints for the intrinsic mechanism of the modeled growth process and can be used for analyzing evolutionary measured data when testing various appropriate models, especially when studying growth processes in life sciences. keywords-dynamical growth model; logistic function; gompertz function; sigmoidal function; dynamical system; reaction network, first integral; conservation equation. i. introduction sigmoidal functions are an useful tool for modeling measurenent data for the study of evolutionary growth processes in life sciences [9], [10]. when studying the time evolution of biological growth processes we are often given a set of measured data of the form (ti,yi), i = 1, ...,n, where yi is an experimentally obtained value at time moment ti. we then have to choose a model function y = f(t) that fits the measured data. more specifically, function f is chosen from a family of functions depending on some parameters and the fitting process consists in finding a suitable parameter set so that a good approximation (fit) is achieved. the definition of the family of modeling functions is a major challenge. to achieve a good fit, we need to choose a family that indicates (incorporates, reflects) the “mechanism” (law) of the physical process generating the experimentally measured data set. in practice, the modeling function f is chosen either from a family of explicitly defined functions, or f is defined as a solution to a class of dynamical systems. in many situations the intrinsic mechanism of the growth process is little or not known. in such situations, a good idea is to look for a dynamical model that consists of a system of reaction equations induced copyright: c© 2019 markov. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: svetoslav markov, reaction networks reveal new links between gompertz and verhulst growth functions, biomath 8 (2019), 1904167, http://dx.doi.org/10.11145/j.biomath.2019.04.167 page 1 of 14 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2019.04.167 s. markov, reaction networks reveal new links between gompertz and verhulst growth functions by some chemical reaction network via reaction kinetic principles, such as mass action kinetics [22], [25], [5]. in this way we readily have a physico-chemical interpretation of the dynamical model and its constituents such as reacting species, rate parameters, order of interactions, sigmoidal shape of the graph of the growth variable, lag time, etc. what then remains is to give a meaning of the chemical terms relative to the studied realworld biological process. reaction networks are well known for a number of basic dynamical growth models, such as the saturation-decay models, epidemiological models, population predatorprey type models, demographic models, etc. undoubtedly, a reaction network formulation of a dynamical model, if possible, contributes to a better understanding of the mechanism of the specific physical process and to possible improvements of the existing mathematical model. in the present work we propose a reaction network realization of the gompertz model [8]. the proposed reaction network suggests an interpretation of the reaction mechanism of the gompertz model in terms of population dynamics theory and reveals its relation to the verhulst logistic growth model. the proposed reaction network conforms with some of the already suggested interpretations based on the differential formulation of the model and recent studies in cancer research [7], [34]. in particular, our reaction mechanism involves two species, one for the population size/volume plus an additional species presenting the nutrient resources of the environment. in addition, some natural generalizations and modifications of the gompertz model induced by the reaction network are proposed, discussed and compared to classical logistic and gompertz models. the next section provides a brief introduction to reaction network theory, in order to fix notations and for the exposition to be more self-contained. section iii introduces the gompertz models, while its reaction network realization and generalizations are discussed in section iv. section v considers reaction networks and respective models with logistic decay of the resource species. it is followed by some notes on applications and concluding remarks in the last two sections of the paper. ii. growth models and reaction networks we focus our attention on growth functions (models) formulated as solutions to differential equations or systems of differential equations. in the latter case we speak of dynamical system that may have several variables apart from the one considered as growth function. without loss of generality we shall consider growth functions f(t) defined in the interval t = [0,∞) with nonnegative values f(t) ≥ 0. in many situations the dynamical system suggests some “insight” for the “inner mechanism” that controls the behavior of the solutions and for the physical meaning of the parameters involved in the system. the “mechanism” of the process is especially well presented when the dynamical model has a realization in the form of a reaction network [5], [22]. a reaction network is a set of (elementary) reactions. a reaction is defined by a set of species that are either reactants (reagents) or products or both. for example, let a reaction have three species s, p , x and let species s, x be reactants, whereas p and x be products, then the reaction is written symbolically in the form: s + x k−→ p + x, (1) k > 0 denoting the rate of the reaction. applying mass action (ma) kinetics, reaction (1) is uniquely “translated” into a system of three differential equations, one for the mass (concentration) of each species involved. the differential equations in the dynamical system are then called “reaction equations” (although this term is often used in a much broader sense in the literature). in our case, the reaction network (1) is translated as a dynamical system of three reaction equations for the corresponding masses (concentrations) s = s(t), p = p(t), x = x(t) of the species s, p , x: s′ = −ksx, p′ = ksx, x′ = 0, biomath 8 (2019), 1904167, http://dx.doi.org/10.11145/j.biomath.2019.04.167 page 2 of 14 http://dx.doi.org/10.11145/j.biomath.2019.04.167 s. markov, reaction networks reveal new links between gompertz and verhulst growth functions where s′ = ds/dt, p′ = dp/dt, x′ = dx/dt denote derivatives with respect to the time variable t. the solutions of this system are the familiar exponential decay s, the saturation growth p and the constant catalyst x. on the same example we can demonstrate the converse process — passing from a dynamical system to a reaction network. this process is not unique, if at all possible. indeed, from the third equation x′ = 0 we have x = const = c, so that the dynamical system reduces to two equations of the form s′ = −acs, p′ = acs. a possible realization of this system in terms of reaction network is then s → p , with a rate parameter equal to ac, which is distinct from the reaction network (1) as the catalyst species x is now missing. let us note that the presence of species x can be important, especially if x participates in some other reaction(s). a. the logistic model let us consider the verhulst logistic model [33]. the logistic growth function is usually defined as solution to the differential equation x′ = kx(1 − x k ), (2) wherein k, k are positive parameters, resp. called (intrinsic) reaction rate and carrying capacity. for our purposes it is more convenient to consider (2) in the following form: x′ = kx(c−x), (3) k and c being positive parameters. the solutions of equations (2) and (3) coincide up to an affine transformation of the time variable of the form t∗ = at, a = const. the two solutions are same (a = 1) whenever k = c = 1. we can say that the two forms (2) and (3) are equivalent in the above sense. equation (3) can be “recast” into two differential equations by introducing a new variable s = c−x. then we have x′ = kxs, s′ = −x′ = −ksx, so, we can write (3) as the dynamical system: s′ = −ksx, x′ = ksx. (4) system (4) is equivalent to equation (3) in the sense that the solutions for x in both dynamical systems coincide. the form of system (4) suggests that the new variable s can be interpreted as mass of some new “intermediate” species s. looking for a suitable reaction network involving two species s, x with masses/concentrations resp. s, x, satisfying the dynamical system (4), we may consider the following reaction network [14]: s + x k−→ 2x, (5) where k > 0 is the reaction rate and 2x is an abbreviation of x +x. indeed, applying the mass action principle to reaction network (5) we obtain the dynamical reaction system (4). conversely, due to s′+x′ = 0, and consequently s + x = const = c > 0, (6) we have s = c−x, (7) which substituted in the second equation of (4) gives the differential equation (3). assuming in (4) initial conditions s(0) = a, x(0) = b, we have s + x = c = a + b. due to s > 0 and 0 < x < c, we have x′ = kx(c−x) > 0, showing that the solution x is monotone increasing and tends asymptotically to c = a+b with t →∞, thus justifying the interpretation of the number c as environmental carrying capacity. species s can be interpreted as the resource (food) consumed (used, uptaken) by species x in order to reproduce itself. note that species x appears on both sides of reaction network (5), playing thus simultaneously the roles of a reactant and a product, so x is a catalyst. as the catalysts x reproduces itself, reaction network (5) is called autocatalytic. for the second derivative of x we obtain x′′ = kx′(c−x) + kx(−x′) = kcx′ − 2kxx′ = kx′(c− 2x). this shows that, for x(0) = b < c/2 = (a + b)/2, biomath 8 (2019), 1904167, http://dx.doi.org/10.11145/j.biomath.2019.04.167 page 3 of 14 http://dx.doi.org/10.11145/j.biomath.2019.04.167 s. markov, reaction networks reveal new links between gompertz and verhulst growth functions that is b < a, an inflection occurs when x = (a + b)/2. thus, function x has a sigmoidal shape when increasing from x(0) < c/2 up to approaching asymptotically the value c. we wish to note again that every reaction network induces a differential reaction system in a unique way, but there is no uniqueness in the opposite direction, that is there can be many distinct reaction networks that induce the same differential systems. the logistic model offers an illustrative example. indeed, consider the reaction network: x k1−→←− k2 x + x. (8) this so-called “reversible reaction” consists of two elementary reactions that can be written equivalently as x k1−→ x +x, x +x k2−→ x. applying ma kinetics, we obtain the differential equation x′ = k1x−k2x2, which is of the same type as equation (3). in such situations it is up to the modeler to choose the reaction network that offers a more adequate interpretation of the particular modeling situation. note that reaction network (5) involves two species, whereas (8) uses only one, so it is important to decide how many species are involved in the physical process. a dynamical process may involve intermediate species whose mass is zero at the beginning and at the end of the process. such species remain hidden as they cannot be easily measured. reaction network (8) is reversible, something considered normal for chemical reactions where the so-called “principle of microscopic reversibility” takes place. on the other hand, reaction network (5) is irreversible, which is a normal situation when modeling complex biological systems such as organs, organisms and populations. a (chemical) reaction network formulation of a dynamical model, if possible, contributes to a deeper understanding of the mechanism of the particular real-word biological process and to possible improvements of the dynamical model whenever necessary. in section v we propose one more reaction network that induces the logistic differential equation (3). b. generalized verhulst growth models the theory of reaction networks offers a powerful tool not only to understand the mechanism of classical models, but also to construct various modifications of existing models in order to adapt the model better to particular real-world phenomena. we next illustrate this idea on a modification and generalization of the logistic growth model. the verhulst model is generalized by the reaction network formulated in the following proposition. proposition 1. the autocatalytic reaction network: x + n∑ i=1 si k−→ x + mx, (9) where n,m ≥ 1 are integers, induces the following dynamical growth model x′ = kmx n∏ i=1 ( ci − x m ) , (10) where ci > 0 are constants. proof: applying the ma law to reaction network (9) we obtain: s′i = −kx n∏ j=1 sj, i = 1, ...,n, (11) x′ = mkx n∏ i=1 si. (12) from (11)–(12) we have for every i = 1, ...,n s′i + x ′/m = 0, implying si + x/m = ci, where ci > 0 are constants. substituting si = ci−x/m, i = 1, ...,n in (12) we obtain (10). special cases. for n = 1, m = 1 we obtain the special case of reaction network (5) inducing verhulst logistic equation (3). biomath 8 (2019), 1904167, http://dx.doi.org/10.11145/j.biomath.2019.04.167 page 4 of 14 http://dx.doi.org/10.11145/j.biomath.2019.04.167 s. markov, reaction networks reveal new links between gompertz and verhulst growth functions for n = m = 2 we obtain the reaction network x + s1 + s2 k−→ x + 2x. (13) for n = m and si = s, i = 1, ...,n, we obtain reaction network ns + x k−→ x + nx. (14) the special cases (13) and (14) are proposed in [1]. discussion. a possible “biochemical” interpretation of model (9) is as follows: the model takes into account the interaction between several species, such as various types of foods and other environmental resources (water, air, light, etc). in chemistry it is unlikely that more than three species interact simultaneously [22]. however, in models related to biology and social sciences, this restriction can be relaxed. iii. the gompertz growth model: general notes the gompertz growth model is used in numerous applications. for a review and classification of various formulations of the gompertz model see e.g. [32]. next, we briefly recall some familiar facts related to the gompertz model. the gompertz growth function is often defined as a solution x = x(t) to the differential equation x′ = kx(c− ln x), (15) where k > 0 and c are parameters. similarly to the case of the logistic differential equation (3), equation (15) can be recast into a system of two differential equations, e.g. s′ = −ks, x′ = ksx, (16) as discussed in works related to a special class of dynamical systems, called s-systems [4], [29], [30]. it is remarkable that, similarly to the logistic case, a new variable s = s(t) appears in system (16), apart from the growth function x = x(t) in the single equation (15). instead of system (16), some authors use a slightly different model: s′ = −s, x′ = ksx, (17) see e.g. [7]; other authors make use of the form s′ = −ks, x′ = sx, (18) see, e.g, [34]. in order to discuss the three slightly different model forms (16), (17) and (18), let us introduce some notation that is independent on the notation used in the available literature. each one of the three systems (16), (17) and (18), consists of two differential equations. one of these equations is uncoupled, involving only one unknown function, which is exponentially decreasing. this function will be called decay function or briefly d-function, in our case, this is function s = s(t). the uncoupled equation for the d-function will be called decay equation or d-equation. the other equation, named growth equation or g-equation, involves, apart from the dfunction, one more function, named growth function or g-function, which is increasing in time. in this work the g-function is denoted by x. let us now discuss the differences between the three systems (16), (17) and (18). system (16) involves two equal rate parameters in both equations, system (17) has a fixed rate equal to 1 in the d-equation and a (free) rate parameter k > 0 in the g-equation. system (18) has a (free) rate parameter in the d-equation and a fixed rate 1 in the g-equation. using gompertz-type models of the forms (17) and (18) one accepts the possibility of distinct rate parameters. in the present work we also adopt this assumption and shall explicitly denote the two rate parameters by different symbols, namely k1,k2: s′ = −k1s, x′ = k2sx. (19) dynamical system (19) induces the relation s′/k1 + x ′/(k2x) = 0, which can be written as s′/ρ+x′/x = 0, ρ = k1/k2. integrating (obtaining a first integral) leads to the “conservation” relation: s ρ + ln x = const = c. (20) relation (20) gives an expression for the variable s in terms of the variable x, namely s = ρ(c− ln x). (21) biomath 8 (2019), 1904167, http://dx.doi.org/10.11145/j.biomath.2019.04.167 page 5 of 14 http://dx.doi.org/10.11145/j.biomath.2019.04.167 s. markov, reaction networks reveal new links between gompertz and verhulst growth functions using relation (21) we can establish a relation between the parameters of the single gompertz equation (15), and those of the 2d-system (19). let functions s, x be solutions to system (19). hence, (21) holds. substituting s = ρ(c − ln x) in the equation for x in (19) we obtain x′ = k2xs = k2xρ(c− ln x) = k1x(c− ln x). we thus obtain gompertz equation (15) with parameters k1 and c. if initial conditions s(0) = a, x(0) = b are given, then c = a/ρ + ln b, so that the gompertz equation resulting from the system (19) with rate constants k1,k2 is x′ = k1x(a/ρ + ln b− ln x) or, equivalently, x′ = k1x(a/ρ + ln b/x). conversely, let g-function x be solution to (15) and x(0) = b. define function s = ρ(c − ln x), where ρ > 0 and c are parameters to be determined, so that: x′ = k ρ xs, and, s′ = ρ(c− ln x)′ = −ρ x′ x = −ρ kxs x = −ks. hence, s,x satisfy (19) with k1 = k and k2 = k/ρ. note that for fixed parameters c,x(0) the initial condition s(0) for the decay function is determined from s(0) = ρ(c− ln b). the above discussion on the relation between the gompertz equation (15) and system (19) can be summarized in the following proposition: proposition 2 [1]. let functions s,x be solutions to system (19), with initial conditions s(0) = s0, x(0) = x0, then x is a solution to a differential equation of the form (15), with parameters k = k1 and c = k2s(0)/k1 + ln(x(0)) and initial condition x(0) = x0. conversely, if function x is a solution to (15), x(0) = x0, then for any ρ > 0 functions s,x, where s = ρ(c − ln x), satisfy a dynamical system of the form (19) with k1 = k and k2 = k/ρ. remarks. 1) relation (21) gives an expression for the variable s in terms of variable x. relation (21) reminds of the analogous relation (7) for the logistic resource variable, resp. the interpretation of the parameter c as an environmental carrying capacity. 2) from the d-equation s′ = −k1s we have s(t) = s(0)e−k1t. then from the second equation in (19) we have x′ x = k2s = k2s(0)e −k1t, an expression known as gompertz law of mortality. some authors call the expression of the form x′/x per capita population growth rate. 3) the second part of proposition 2 shows that representing the gompertz equation (15) as a system of two differential equations (19) introduces one free parameter. it can be taken as ρ or as a = s(0), either one being a function of the other through a = ρ(c−ln b). 4) the appearance of a new variable (the dfunction s) in the gompertz model (19) suggests that this variable is mass/concentration of some particular species s depending on the modelling situation. the biological meaning of species s in systems of the form (19) is much discussed in the literature, see e.g. [7], [34]. 5) system (19) belongs to the class of “ssystems” [2], [3], [4], [29], [30], it can be considered as “recast” form of equation (15). in the literature on s-systems one can find theorems that generalize proposition 2. our next aim is to consider the dand gfunctions s = s(t), x = x(t) as masses (concentrations) of two reacting species s, x and to formulate reaction network(s) involving the two species s, x, that generate system (19) under the ma principle. as shown above, gompertz model can be formulated equivalently in several forms. we shall next focus on the gompertz model in the form (19) as expressing reactions between two species. biomath 8 (2019), 1904167, http://dx.doi.org/10.11145/j.biomath.2019.04.167 page 6 of 14 http://dx.doi.org/10.11145/j.biomath.2019.04.167 s. markov, reaction networks reveal new links between gompertz and verhulst growth functions so let us look at the variables s,x as masses (concentrations) of two species s, resp. x. the first equation (s′ = −k1s) of system (19) indicates no interaction between species s and x, but merely an exponential decay of s with a rate parameter k1. the second equation (x′ = k2sx) suggests that s and x interact as s+x. due to the independent decay of s, the latter observation may lead to the conclusion that no realization of the gompertz model as a reaction network is possible, as stated in [1]. however, there exists such a realization as formulated in the next section. iv. reaction network realization of the gompertz model and generalizations a. main result the gompertz model can be formulated by means of a reaction network as follows. proposition 3. the reaction network involving species s,x,p : s k1−→ p s + x k2−→ 2x + s (22) induces gompertz reaction equations (19), resp. gompertz differential equation (15) for the masses/concentrations s,x of species s,x. proof: applying the mass action law to reaction network (22) yields the dynamical system: s′ = −k1s, p′ = k1s, x′ = k2sx. (23) system (23) incorporates system (19) plus an additional reaction equation for the by-product p (p′ = k1s); the first two reaction equations for s and p are uncoupled representing a saturationdecay mechanism. thus the reaction for p can be either ignored or used depending on the particular modeling situation. hence, system (23) is equivalent to system (19) as far as species s and x are concerned. as we know, system (19) induces gompertz differential equation (15) in the sense of proposition 2. this proves the proposition. remark. in the special case k1 = 1, we obtain dynamical system (17), if k2 = 1 we obtain system (18), and if k1 = k2 we obtain dynamical system (19). discussion. the second reaction in network (22) is based on the logistic reaction network (5) with the modification that species s catalyzes the reproduction of species x. in certain applications when the mass of s is much greater than the mass of x, the reaction network says that the exhaustion of s is due not only to species x, but also to other factors leading to transformation of s into a third species p . in other applications it may be the case that species x uses s as a resource (food) for its reproduction but takes care to sustain the mass of s. depending on the modeling situation reaction s → p can be replaced by some other suitable d-reaction, e.g. s → ø meaning an outflow of resource s from the modeled system. other possibilities of the decay mode of s, such as a logistic decay (s +z → 2z) will be explored later. b. generalized gompertz-type models based on reaction networks the second reaction in (22) unifies two processes: a) reproduction of species x at the expense of an uptake of the resource s, and b) recover of species s by a reproduction process catalysed by species x. these two processes can be formulated separately as follows: s k1−→ p s + x k2−→ 2x s + x k2−→ 2s + x (24) indeed, reaction network (24) induces the following dynamical system: s′ = −k1s−k2sx + k2sx = −k1s, p′ = k1s, x′ = k2sx, which is identical with (23). discussion. reaction network (24) involves the logistic reaction network (5): s + x k2−→ 2x. in biomath 8 (2019), 1904167, http://dx.doi.org/10.11145/j.biomath.2019.04.167 page 7 of 14 http://dx.doi.org/10.11145/j.biomath.2019.04.167 s. markov, reaction networks reveal new links between gompertz and verhulst growth functions the logistic model species x consumes species s and reproduces itself. the consumption of s is compensated by the third reaction: s + x k2−→ 2s + x , indicating that species x catalyzes the reproduction of species s. note that instead of reaction s → p , a reaction s → ø can be used, depending on the real-word modeling situation. from chemical point of view the rate parameters in the reactions with s + x in the left-hand side should be equal, however in certain modeling situations the rate parameters may be considered distinct, which leads to a more flexible generalized model of gompertz type. more specifically the following model generalizes model (22): s k1−→ p s + x k2−→ 2x s + x k3−→ 2s + x (25) indeed, reaction network (25) coincides with (24) if k3 = k2. the above generalization (25) of reaction network (22) can be naturally extended by replacing the logistic reaction in (25) by the generalized logistic reaction (9) as follows: proposition 4. the reaction network involving species s,s1,s2, ...,sn,p,x: s k1−→ p x + n∑ i=1 si k2−→ x + mx s + x k3−→ 2s + x (26) where n,m are positive integers, generalizes gompertz model (22). proof: follows from proposition 1 and assuming k3 = k2. depending on the modeling situation the generalized logistic reaction network (9): x + n∑ i=1 si k2−→ x + mx used in (26) can be replaced by some special case such as (13), (14). 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 t 0.0 0.2 0.4 0.6 0.8 1.0 x( t) c = 1.0, c1 = 0.5, c2 = 0.5, x(0) = 0.1 dx/dt = kgx( lnx), kg = 3.50 dx/dt = kmx(c1 x2 )(c2 x 2 ), km = 19.90 dx/dt = klx(c x), kl = 8.95 _cfig. 1. graphs of the solutions to models (3), (13) and (15). the rate constants and initial conditions are equal in order to compare the different sigmoidal shapes of the growth functions. fig. 1 represents the graphs of the solutions to models (3), (13) and (15). the rate constants and initial conditions are chosen to be equal in order to compare the shapes of the growth functions. c. a mixed verhulst-gompertz model with decaytype resource uptake our approach to formulate verhulst and gompertz models in terms of reaction networks reveals an important link between these two classical growth models. in order to pass from the verhulst logistic model (5) to gompertz model (22) we need to perform two steps in the modification of the reaction networks: a) add a reaction s k1−→ p , and b) add a species s in the right-hand side of (5) in order to obtain the second reaction (s + x k2−→ 2x + s) of the gompertz reaction network (22), providing thus a sustainability of the resource species s. this observation suggests to take a closer look at the first step a) in the above modification process, namely adding a decay reaction to the logistic one, leading to the reaction network: s k1−→ p, s + x k2−→ 2x. (27) biomath 8 (2019), 1904167, http://dx.doi.org/10.11145/j.biomath.2019.04.167 page 8 of 14 http://dx.doi.org/10.11145/j.biomath.2019.04.167 s. markov, reaction networks reveal new links between gompertz and verhulst growth functions proposition 5. reaction network (27) involving species s,p,x induces the following “mixed verhulst-gompertz” dynamical system for the masses s,p,x: s′ = −k1s−k2xs, p′ = k1s, x′ = k2sx, (28) where k1,k2 are positive parameters. dynamical system (28) generates the following mixed verhulst-gompertz differential equation for the gfunction x: x′ = k2x(c−x− ln xρ), ρ = k1/k2, (29) where c = s(0) + x(0) + ln x(0)ρ. proof: reaction network (27) induces (28) according to mass action kinetics. from (28) we obtain s′ + x′ = −k1s = −k1 x′ k2x . hence, s′ + x′ + ρ x′ x = 0, where ρ = k1/k2. integrating yields the “conservation” relation s + x + ln xρ = const = c, (30) wherein c = s(0) + x(0) + ln x(0)ρ. relation (30) allows us to express s in terms of x: s = c−x− ln xρ. substituting this expression in equation x′ = k2xs we obtain (29). this proves the proposition. equation (29) suggests that the reaction mechanism (27) presents an intermediate step between the logistic and the gompertz growth models. more specifically, the main steps in the construction of the gompertz reaction mechanism, starting from the logistic one, are as follows. a construction of the gompertz model in three steps starting from the logistic model. the reaction network approach offers a simple presentation of the consecutive steps leading from the logistic to the gompertz model. here is a list of three “elementary” steps in the construction of the gompertz model, starting from the logistic one. step 1: s + x k−→ 2x. logistic model: x′ = kx(c−x). step 2: s k1−→ p, s + x k2−→ 2x. mixed v-g model: x′ = k2x(c−x− ln xρ). step 3: s k1−→ p, s + x k2−→ 2x + s. gompertz model: x′ = k1x(c− ln x). the above steps suggest a variety of combinations of different mechanisms that can replace the “elementary” reactions, for example the exponential decay reaction s k1−→ p can be replaced by logistic decay. fig. 2 represents the graphs of the solutions to models (3), (27) and (15). the rate constants and initial conditions are chosen to be equal in order to compare the shapes of the growth functions. it is observed that the solution of the mixed verhulst model is “between” the solutions of the logistic and the gompertz models. fig. 2. graphs of the solutions to models (3), (27) and (15). the rate constants and initial conditions are equal in order to compare the different sigmoidal shapes of the growth functions. d. another reaction network realizations of the verhulst logistic model in section 3 we considered the reaction network (5) generating the logistic model (3). below we propose another reacrion network that generates the logistic model. biomath 8 (2019), 1904167, http://dx.doi.org/10.11145/j.biomath.2019.04.167 page 9 of 14 http://dx.doi.org/10.11145/j.biomath.2019.04.167 s. markov, reaction networks reveal new links between gompertz and verhulst growth functions proposition 6. the reaction network involving species s,x,p : s + x k1−→ p + x s + x k2−→ 2x + s (31) induces the following dynamical system for the masses/concentrations s,x of species, resp. s,x: s′ = −k1sx, x′ = k2sx, (32) where k1,k2 are positive rate parameters. dynamical system (32) generates the verhulst differential equation (2) for the growth function x. proof: applying the mass action law to reaction network (31) yields the dynamical system: s′ = −k1sx, p′ = k1sx, x′ = k2sx. the above system incorporates system (32) plus an additional reaction equation for a by-product p (p′ = k1sx). the latter equation is uncoupled and can be ignored obtaining thus system (32). due to s′/k1 + x′/k2 = 0, and consequently s/k1 + x/k2 = const, we can write s + ρx = const = c > 0, ρ = k1/k2, or s = c−ρx. (33) the constant c can be determined from the initial condititions for s,x, namely c = s(0) + ρx(0). substituting the expression (33) in x′ = k2xs gives the logistic differential equation: x′ = k2x(c−ρx). formally the above equation obtains the form (2) for c = 1,k = 1/ρ. this proves the proposition. remark. from chemical point of view it is uncommon that the two reactions in (31) have distinct rate parameters, providing that the reactants involved (s,x) are the same. however from biological perspective the two reactions in network (31) can be considered to be of different nature, hence they may have distinct rates. in addition, reaction network (31) has some methodological value, demonstrating the difference between the logistic and the gompertz models. namely, the difference consists in the decay equation, showing the independence of the decay reaction s k1−→ p , resp. of the resource species s, on x in the case of the gompertz model towards the dependant (catalyzed) decay s + x k1−→ p + x of species s on species x in the case of the logistic model. v. reaction network models with logistic decay function for the resource species as shown above the logistic reaction network in the form (31) has two reactions: i) a growth reaction s + x−→2x + s inducing a logistic growth of the g-species x, and ii) a decay reaction of the form s + x −→ p + x representing a logistic decay (consumption, uptake, outflow) of the dspecies (resource, food) s. for comparison, the mixed logistic reaction network (27) involves a dreaction describing an exponential decay: s−→p , resp. s′ = −ks, inducing the exponential solution s(t) = s0 exp(−kt). a logistic decay process results from a logistic d-reaction of the form s + z−→2z. such a process represents a consumption of the resource s by a competitive autocatalytic species z inducing thereby a sigmoidal “logistictype” decay function s(t). we next apply the logistic decay mode of the d-function to the mixed verhulst model (27). a. reaction network model using logistic reactions for the gand d-species consider the reaction network s + z k1−→ 2z, s + x k2−→ 2x. (34) proposition 7. reaction network (34) involving species s,z,x induces the following mixed verhulst-gompertz differential equation for the gfunction x: x′ = k2x(c−x− c1xρ), (35) biomath 8 (2019), 1904167, http://dx.doi.org/10.11145/j.biomath.2019.04.167 page 10 of 14 http://dx.doi.org/10.11145/j.biomath.2019.04.167 s. markov, reaction networks reveal new links between gompertz and verhulst growth functions where ρ = k1/k2, c,c1 = const, or, equivalently, x′ = k2x exp(−k1t)/(1 + exp(−k1t), (36) in (35) and (36, k1,k2 are the positive rate parameters in (34). proof: reaction network (34) induces the following dynamical system: s = −k1sz −k2sx, z = k1sz, (37) x = k2sx. adding the three equations of dynamical system (37) we obtain: s′ + z′ + x′ = 0. hence, s+z +x = const = c = s(0) +z(0) +x(0). (38) from the equations for z′ and x′ in (37), we obtain the relation z′ k1z = x′ k2x (= s), hence, ln z k1 − ln x k2 = const = c∗, yielding ln(zk2/xk1 ) = c, or zk2/xk1 = const = c∗∗. we thus obtain zk2 = c∗∗xk1 , or z = c∗∗xρ, ρ = k1/k2. substituting z in (38), we obtain s = c−x−z = c−x− c∗∗xρ. substituting this expression for s in the third equation of (37) we obtain differential equation (35). to prove the non-autonomous equation (36) it is enough to note that the sigmoidal logistic-type uptake of the resource s by species z does not depend on x. remark. model (34) is symmetrical with respect to species x and z in the sense that both species can exchange their places in the reactions. due to this symmetry the equations for the gfunction z are similar to those of g-function x. b. reaction network model using gompertz-type growth reaction and verhulst-type decay reaction as shown above the gompertz reaction network model has two important constituents: i) a growth reaction s + x−→2x + s inducing a logistic growth of the g-function, and ii) an additional reaction of the form s −→ p representing the decay (consumption, uptake, outflow) of the resource (food) species s. in the gompertz reaction network the form of the d-reaction describes an exponential decay: s(t) = s0 exp(−kt). we next suggest that the d-reaction mat represent a consumption of the resource s by a competitive species following a sigmoidal “logistic-mode”, that is the d-species s is consumed by an autocatalyst z according to the d-reaction s + z−→2z. we shall next apply this consumption mode to the mixed verhulst-gompertz model (27). consider the reaction network s + z k1−→ 2z, s + x k2−→ 2x + s. (39) proposition 8. reaction network (39) involving species s,z,x induces the following logistic differential equation for the g-function x: x′ = k2x(c−µxρ), (40) where k1,k2 are the positive rate parameters in (39), ρ = k1/k2 and c,µ = const. proof: reaction network (39) induces the following dynamical system: s = −k1sz, z = k1sz, (41) x = k2sx. adding the first two equations of dynamical system (41) we obtain: s′ + z′ = 0. hence, s + z = const = c. from the equations for z′ and x′ in (41) we obtain the relation z′ k1z = x′ k2x (= s), biomath 8 (2019), 1904167, http://dx.doi.org/10.11145/j.biomath.2019.04.167 page 11 of 14 http://dx.doi.org/10.11145/j.biomath.2019.04.167 s. markov, reaction networks reveal new links between gompertz and verhulst growth functions hence, ln z k1 = ln x k2 + ln µ, µ = const, yielding z(1/k1) = µx(1/k2)), or z = µxρ. substituting z in s = c−z, we obtain s = c−z = c−µxρ, which substituted in third equation of (41), gives the differential equation (40). remark. model (39) is another more general formulation of the logistic model. indeed, for k1 = k2 equation (40) is similar to the logistic differential equation (2). another interesting conclusion is that all forms of the logistic reaction equation can be written in the form: s′ = −k1s(c−s), x′ = k2sx. (42) the above “recast” form of the logistic equation shows that the d-equation is uncoupled from the gequation and its solution can be given in the form: s(t) = c 1 + ek1(t−γ) , where γ is determined by the initial condition. then from the second equation in (42) we obtain x′ x = k2c 1 + ek1(t−γ) (43) formula (43) can be interpreted as “logistic mortality law” or “logistic per capita population growth rate”. vi. notes on applications gompertz-type models have been widely used to simulate the kinetics of various natural phenomena such as the growth of various species (microorganisms, animals, plants) and bio-products formation, e.g. methane [31], hydrogen [6], [24], etc. numerous research articles are devoted to application of gompertz-type models in tumor growth [7], [26], [20], [21], [34]. gompertz models find application in software reliability models (gsrm, gmsrm) [28], [27], [23]. gompertz-type modeling and data fitting problems stimulate various mathematical and computational studies, such as applications of new cumulative distribution functions, transformations (type i–type iii) to construct families of sigmoidal functions and new activation functions using ’correcting amendments’ [17], [18], [19]. for other related works the reader may consult [11]–[14]. vii. concluding remarks modeling growth processes in life sciences is an important scientific area [9], [10]. dynamical growth processes are often described by sigmoidal growth functions, such as saturation, logistic, gompertz, etc., many of them being solutions to dynamical systems. reaction network theory ia an important tool for generating dynamical growth models providing thereby useful interpretation of the intrinsic mechanism of the biological process. in this work we focus our attention on logistic and gompertzian-type growth models from the perspective of the reaction networks theory. there are well-known reaction network realizations for a number of dynamical growth models, such as saturation, logistic, epidemic, etc., however, to our knowledge no such realization is known for the gompertz model. in this paper a reaction network inducing the gompertz model is proposed. the proposed reaction network involves an additional reaction for the uptake of the resource species. we also propose several reaction networks inducing dynamical models that generalize the gompertzian one. discussed are important links between the gompertz and the logistic model. our method can be considered as an extension of the work of previous authors who “recast” the gompertz differential equation into a dynamical system of two differential equations involving thereby an additional variable (species) that can be interpreted as “resource” or “food” consumed by the growth variable (species). discussed is also how the induced dynamical systems can be reduced to onedimensional differential equations for the growth (resp. decay) species, by finding a first integral leading to a conservation equation.. the proposed reaction networks are simple and may seem trivial, biomath 8 (2019), 1904167, http://dx.doi.org/10.11145/j.biomath.2019.04.167 page 12 of 14 http://dx.doi.org/10.11145/j.biomath.2019.04.167 s. markov, reaction networks reveal new links between gompertz and verhulst growth functions but are of some importance to those who construct new models to study biological growth processes whose underlying mechanism is unknown. the proposed reaction network realization of gompertz growth model can be interpreted from the perspective of demographic and socio-economic sciences. it is remarkable that the gompertz reaction network comprises a reaction equation describing biological activity that is characteristic for highly organized biological organs, organisms or populations. this explains why using the gompertz model in demographic studies and cancer research is so successful. the reaction network approach clearly explains the close links between the gompertz model and the verhulst logistic model. acknowledgments. the author is grateful to dr. n. kyurkchiev and dr. m. borisov for stimulating discussions on the subject and numerous encouragements. the author is grateful to the anonymous reviewer for his careful reading of the manuscript, critical remarks and suggested improvements. references [1] anguelov, r., borisov m., iliev a., kyurkchiev n., markov s., on the chemical meaning of some growth models possessing gompertziantype property. math meth appl sci. 2017;112 https://doi.org/10.1002/mma.4539 [2] bajzer z., marusic m, vuk-pavlovic s., conceptual frameworks for mathematical modelling of tumor growth dynamics. math comput model. 1996;23:31-46. 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(2017) the use of gompertz models in growth analyses, and new gompertz-model approach: an addition to the unified-richards family. plos one 12(6): e0178691. https://doi.org/10.1371/journal.pone.0178691 [33] verhulst p.f., notice sur la loi que la population poursuit dans son accroissement. corresp math phys. 1838;10:113-121. [34] west, j., z. hasnain, p. macklin, p. k. newton, an evolutionary model of tumor cell kinetics and the emergence of molecular heterogeneity driving gompertian growth, siam rev soc ind appl math. 2016 ; 58(4): 716736. https://doi.org/10.1137/15m1044825. biomath 8 (2019), 1904167, http://dx.doi.org/10.11145/j.biomath.2019.04.167 page 14 of 14 http://dx.doi.org/10.11145/j.biomath.2019.04.167 introduction growth models and reaction networks the logistic model generalized verhulst growth models the gompertz growth model: general notes reaction network realization of the gompertz model and generalizations main result generalized gompertz-type models based on reaction networks a mixed verhulst-gompertz model with decay-type resource uptake another reaction network realizations of the verhulst logistic model reaction network models with logistic decay function for the resource species reaction network model using logistic reactions for the gand d-species reaction network model using gompertz-type growth reaction and verhulst-type decay reaction notes on applications concluding remarks references investigating the role of mobility between rural areas and forests on the spread of zika biomath https://biomath.math.bas.bg/biomath/index.php/biomath b f biomath forum original article investigating the role of mobility between rural areas and forests on the spread of zika kifah al-maqrashi1, fatma al-musalhi2,∗, ibrahim m. elmojtaba1, nasser al-salti3 1department of mathematics, sultan qaboos university, muscat, oman kifahhsm@gmail.com elmojtaba@squ.edu.om 0000-0001-6018-4671 2center for preparatory studies, sultan qaboos university, muscat, oman fatma@squ.edu.om 0000-0003-1108-3879 3department of applied mathematics and science, national university of science and technology, muscat, oman alsalti@nu.edu.om 0000-0001-9726-4624 received: july 8, 2021, accepted: december 14, 2022, published: december 22, 2022 abstract: a mathematical model of zika virus transmission, incorporating human movement between rural areas and nearby forests, is presented to investigate the role of human movement in the spread of zika virus infections in human and mosquito populations. proportions of both susceptible and infected humans living in rural areas are assumed to move to nearby forest areas. direct, indirect, and vertical transmission routes are incorporated for all populations. a mathematical analysis of the proposed model is presented. the analysis starts with normalizing the proposed model. the positivity and boundedness of solutions to the normalized model are then addressed. the basic reproduction number is calculated using the nextgeneration matrix method and its relation to the three routes of disease transmission has been presented. the sensitivity analysis of the basic reproduction number to all model parameters is investigated. the analysis also includes the existence and stability of disease-free and endemic equilibrium points. bifurcation analysis is also carried out. finally, numerical solutions to the normalized model are obtained to confirm the theoretical results and demonstrate human movement’s role in disease transmission in human and mosquito populations. keywords: zika, vertical transmission, basic reproduction number, stability analysis, sensitivity analysis, bifurcation analysis i. introduction zika is an arboviral disease in the genus flavivirus closely related to yellow fever, west nile (wn), and dengue (den) viruses. it was first identified in 1947 in zika forest in uganda during sylvatic yellow fever surveillance in a sentinel rhesus monkey [1]. in 1954, it was reported in humans for the first time in nigeria [2]. the zika epidemic was stated as a public health emergency of international concern (pheic) by the world health organization (who) on february 1st, 2016 [3]. it has attracted global attention since its worldwide spread among tropical and subtropical regions. in yap island, micronesia in 2007, the first zika outbreak occurred among humans [4]. during 20132014 the largest epidemic of zika ever reported was in french polynesia [4]. since 2014, the zika virus (zikv) has continued spreading to other pacific islands [2]. it reached southern and central america after 2015 and brazil and the caribbean were highly affected by zikv [4]. local transmission of zikv was realized in 34 countries by march 2016 [5]. zikv is transmitted primarily to the human popucopyright: © 2022 kifah al-maqrashi, fatma al-musalhi, ibrahim m. elmojtaba, nasser al-salti. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. *corresponding author citation: kifah al-maqrashi, fatma al-musalhi, ibrahim m. elmojtaba, nasser al-salti, investigating the role of mobility between rural areas and forests on the spread of zika, biomath 11 (2022), 2212149, https://doi.org/10.55630/j.biomath.2022.12.149 1/14 https://biomath.math.bas.bg/biomath/index.php/biomath mailto:kifahhsm@gmail.com mailto:elmojtaba@squ.edu.om https://orcid.org/0000-0001-6018-4671 mailto:fatma@squ.edu.om https://orcid.org/0000-0003-1108-3879 mailto:alsalti@nu.edu.om https://orcid.org/0000-0001-9726-4624 https://creativecommons.org/licenses/by/4.0/ https://doi.org/10.55630/j.biomath.2022.12.149 al-maqrashi et al, investigating the role of mobility between rural areas and forests on the spread of zika lation by bites of infected female aedes mosquitoes. analysts have found 19 species of aedes mosquitoes competent of carrying zika infection, but the foremost common is the tropical privateer, aedes aegypti. the vector (mosquito) can pass into the human population through biting after taking a blood meal from an infected human. in addition, sexual interaction, perinatal transmission, and blood transfusion are other routes of spreading zikv between humans even months after infection. a pregnant woman can pass zika to her baby, which can cause genuine birth defects. infection with zika increases the chances of the infant developing injury with microcephaly as reported in [6] and guillian syndrome as reported in [2] from infected mothers [4]. in february 2016, france registered the first sexually transmitted case of zikv [3]. zika disease is characterized by mild symptoms including fever, headache, maculopapular rash, joint and muscle pain, conjunctivitis, etc. the clinical symptoms duration is within two to seven days after the bites [3]. most reports show that zika is a self–limiting febrile disease that could be misidentified as dengue or chikungunya fever [7]. the prevention of mosquito bites and control of vectors by using insecticide, eradication of adult and larval breeding areas is the only possible treatment available till now [8]. understanding virus transmission and disease epidemiology through mathematical modelling are of great importance for disease management. several mathematical models have been developed to study the dynamics and propose control strategies for the transmission of zikv disease. in [3], the authors proposed a zika mathematical model by assuming the standard incidence type interaction of human-to-human transmission of the illness. also, they extended their work to include optimal control programs (insecticide-treated bed nets, mosquito-repulsive lotions, and electronic devices) to reduce the biting rate of vectors, and to decline the spread of the disease among the human population. in [8], authors proposed a zika mathematical model including the applications of prevention, treatments, and insecticide as the best way to minimize the spread of zikv disease. in [9], researchers suggested a multifold zika mathematical model. they considered the transmission of the zikv in the adult population and infants either directly by vector bites or through vertical transmission from mothers. the model shows that asymptomatic individuals magnify the disease weight in the community. it also indicated that postponing conception, coupled with aggressive vector control and personal protection use, decrease the cases of microcephaly and transmission of zikv. globally, the survival of around 1.6 billion rustic people depends on products obtained from local forests, in whole or in part. those individuals live adjacent to the forest and have had simple survival conditions and livelihoods for many generations. they depend on those natural and wild resources to meet their needs [10]. in this paper, a mathematical model of zikv is constructed to demonstrate the specific and realistic conditions, where the nearby movement of humans may contribute to the spread of virus infections. this happens when an infected human with mild symptoms, moves from rural areas to nearby forest areas looking for work or food. additionally, the movement of a susceptible human can affect the spread of infections via contagious mosquitoes in the forest. hence, in this paper, we have split the vector compartment, based on mosquito location, into rural areas and nearby forest areas. human movement between rural areas and their interaction with vector populations are illustrated in figure 1. in addition, sexual and vertical transmissions in the human population are considered. also, vertical transmission from a contaminated female mosquito to its offspring, is suggested as a component that guarantees the upkeep of zikv. the paper is organized as follows. the model formulation is described in section 2. the model analysis includes the positivity, boundedness of the solution, basic reproduction number, and sensitivity analysis are discussed in section 3. furthermore, stability analysis and bifurcation analysis are presented. a numerical analysis of the model using assumed baseline parameters is given in section 4 to illustrate the effects of highly sensitive parameters on the human population. finally, the conclusion is given in section 5. ii. model description in this section, we introduce a model for zikv transmission between humans and vectors in rural areas and nearby forests. we begin the description of the model with the human compartments. we split the human population into susceptible sh, symptomatic ih and recovered rh. susceptible humans sh can get infected with zika via three main routes [11]: via a mosquito bite (vector transmission), via sexual transmission or blood transfusion (direct transmission), or by being passed from a mother to a newborn child (vertical transmission). zika causes nearly no mortality among humans and has been a public health crisis for a relatively short pebiomath 11 (2022), 2212149, https://doi.org/10.55630/j.biomath.2022.12.149 2/14 https://doi.org/10.55630/j.biomath.2022.12.149 al-maqrashi et al, investigating the role of mobility between rural areas and forests on the spread of zika fig. 1: illustrated figure for human movement and their interactions with vector populations of zikv. riod of time, so we assume the total human population remains constant: sh + ih + rh = nh. we split the vector population into the rural population (sv, iv) and the nearby forest population (su, iu). the overall vector populations at time t are sv + iv = nv and su + iu = nu . rural and forest mosquitoes are assumed to be only infected by infectious humans. the infection period of mosquitoes ends when the mosquitoes die. as mosquitoes travel distances of no more than a few kilometres, forest mosquitoes will have a direct interaction only with the human population moving from rural areas to the forest. hence, we assume that a proportion κ1 of the susceptible individuals may get infected by infectious mosquitoes that live in forests and nearby rural areas iu due to their movement to forest areas, and a proportion κ2 of the infected individuals are assumed to move from rural areas to the nearby forests such that κ1 > κ2 and hence they may infect mosquitoes that live in forests. the proportion (1 −κ1) of susceptible humans who stay in the rural areas can get infected by infectious mosquitoes that live in rural areas iv, and a proportion (1 − κ2) of infected individuals who stay in the rural areas may infect mosquitoes that live in rural areas. moreover, a proportion (1 − κ1) of susceptible individuals can also get the infection by interaction with (1 −κ2) of infectious humans (symptomatic), through sexual transmission or other direct routes. we assume that a fraction ε1 of newborns are affected and enter the symptomatic class. evidence suggests that the fraction is about 2/3 [12]. we also assume that zikv is transmitted vertically in the vector population [13] and this is the main pathway it survives in the colder months. we incorporate vertical transmission ε2,ε3 of the zikv in both vector populations, respectively. the set of non-linear differential equations that represents the proposed mathematical model is given by: s′h = µhnh −µhε1ih − (1 −κ1)β1θ1iv sh nh −κ1β2θ1iu sh nh − (1 −κ1)(1 −κ2)λih sh nh −µhsh i′h = µhε1ih + (1 −κ1)β1θ1iv sh nh + κ1β2θ1iu sh nh + (1 −κ1)(1 −κ2)λih sh nh − (γ + µh)ih r′h = γih −µhrh s′v = µv nv −µv ε2iv − (1 −κ2)β1θ2sv ih nh −µv sv i′v = µv ε2iv + (1 −κ2)β1θ2sv ih nh −µv iv s′u = µunu −µuε3iu −β2θ2κ2su ih nh −µusu i′u = µuε3iu + β2θ2κ2su ih nh −µuiu (1) with non negative initial conditions sh(0), ih(0), rh(0), sv(0), iv(0), su(0), iu(0). in addition, the parameters and their values of the system are defined in table i. let sh = sh nh , ih = ih nh , rh = rh nh , sv = sv nv , iv = iv nv , su = su nu , iu = iu nu , such that sh + ih + rh = 1, sv + iv = 1, su + iu = 1. thus, the considered model (1) have been normalized and rewritten as follows: s′h = µh −µhε1ih −κ1β1θ1α1iv sh −κ1β2θ1α2iush −κ1κ2λihsh −µhsh i′h = µhε1ih + κ1β1θ1α1iv sh + κ1β2θ1α2iush + κ1κ2λihsh − (γ + µh)ih r′h = γih −µhrh s′v = µv −µv ε2iv −κ2β1θ2sv ih −µv sv i′v = µv ε2iv + κ2β1θ2sv ih −µv iv s′u = µu −µuε3iu −β2θ2κ2suih −µusu i′u = µuε3iu + β2θ2κ2suih −µuiu (2) biomath 11 (2022), 2212149, https://doi.org/10.55630/j.biomath.2022.12.149 3/14 https://doi.org/10.55630/j.biomath.2022.12.149 al-maqrashi et al, investigating the role of mobility between rural areas and forests on the spread of zika fig. 2: progression diagram of the proposed zikv model. table i: parameters used in the model (1). parameter symbol value per day natural death/birth rate of humans µh 1/(68.5*365) [14] natural death/birth rate of mosquitoes in rural areas µv [0.025-0.125] [15, 16] natural death/birth rate of mosquito in forest areas µu [0.025-0.125] [15, 16] biting rate of rural mosquitoes on humans β1 [0.3-1.5] [17] biting rate of forest mosquitoes on humans β2 [0.3-1.5] [17] transmission probability from an infectious mosquito to a susceptible human θ1 [0.1–0.75] [17] transmission probability from an infectious human to a susceptible mosquito θ2 [0.3–0.75] [17] direct (sexual) transmission rate between humans λ [0.01-0.47] [18] recovery rate of humans γ [0.07-0.33] [16] probability of vertical transmission in humans ε1 0.67 [12] probability of vertical transmission in rural mosquitoes ε2 0.06 [19] probability of vertical transmission in forest mosquitoes ε3 0.06 [19] fraction of susceptible humans moving from rural to forest areas κ1 [0-0.5] fraction of infected humans moving from rural to forest areas κ2 [0-0.5] where κ1 = (1 −κ1), κ2 = (1 −κ2), α1 = nv nh , α2 = nu nh , and with non-negative initial condition x(0) := ( sh(0),ih(0),rh(0), sv (0),iv (0),su (0),iu (0) )t . iii. model analysis in this section, the positivity of solutions, the positive invariant set, and the basic reproduction number are discussed. also, sensitivity analysis and results related to stability analysis and bifurcation analysis are presented. a. positivity of solutions and positively invariant set it is clear that model (1) together with the given non-negative initial condition has a unique solution. next, we show that all solutions remain non-negative for all t ∈ [0,∞) for arbitrary choice of initial conbiomath 11 (2022), 2212149, https://doi.org/10.55630/j.biomath.2022.12.149 4/14 https://doi.org/10.55630/j.biomath.2022.12.149 al-maqrashi et al, investigating the role of mobility between rural areas and forests on the spread of zika ditions to have an epidemiological convincing result. the following theorem demonstrates the positivity and boundedness of state variables: theorem 1. the solutions sh(t), ih(t), rh(t), sv (t), iv (t), su (t) and iu (t) of system (2) with non-negative initial conditions sh(0), ih(0), rh(0), sv (0), iv (0), su (0), iu (0) remain non-negative for all time t > 0 in a positively invariant closed set ω := { (sh,ih,rh,sv ,iv ,su,iu ) t ∈ r7+ : 0 6 sh(t),ih(t),rh(t), sv (t),iv (t),su (t),iu (t) 6 1 } . proof: assume that the initial conditions of the system (2) are non-negative. let t1 > 0 be the first time at which there exists at least one component which is equal to zero and other components are non-negative on [0, t1). in the following, we will show that none of the components can be zero at t1. let’s first assume that sh(t1) = 0 and other components are non-negative on [0, t1). now, s′h can be written as s′h = µh(1−ε1)+µhε1rh−m1sh−µh(1−ε1)sh, where m1 = κ1β1θ1α1iv + κ1β2θ1α2iu + κ1κ2λih > 0. then, at t1, we have sh(t) dt ∣∣∣∣ t=t1 = µh(1 −ε1) + µhε1rh(t1) > 0, which means that sh(t) is strictly monotonically increasing at t1, that is sh(t) < sh(t1) for all t ∈ (t1 − ε,t1), where ε > 0. since sh(t1) = 0, then, sh(t) < 0 on (t1−ε,t1). this leads to a contradiction. therefore, sh(t) cannot be zero at t1. now, we assume that ih(t1) = 0 and other components are non-negative. then ih(t) dt ∣∣∣∣ t=t1 = sh(t1) ( κ1β1θ1α1iv (t1) + κ1β2θ1α2iu (t1) ) > 0, which means that ih(t) is strictly monotonically increasing at t1. hence, we also get a contradiction. next, assume that rh(t1) = 0 and other components are non-negative. then rh(t) dt ∣∣∣∣ t=t1 = γih(t1) > 0, which again leads to a contradiction. similarly, one can prove that the remaining components of vector populations sv (t), iv (t), su (t), iu (t) cannot be zero at t1. hence, from the above, we conclude that such a point t1 at which at least one component is zero does not exist. hence, all components remain non-negative for all time t > 0. for the positively invariant closed set ω, we first note that the set ω is said to be positively invariant if the initial conditions are in ω implies that( sh(t),ih(t),rh(t),sv (t),iv (t), su (t),iu (t) )t ∈ ω. let φ(t) = (φ1(t), φ2(t), φ3(t)) t , where φ1(t) = sh(t) + ih(t) + rh(t), φ2(t) = sv (t) + iv (t), φ3(t) = su (t) + iu (t). then φ′(t) =   µh −µhφ1(t)µv −µv φ2(t) µu −µu φ3(t)   . now, solving for φ1, φ2 and φ3, we get φ1(t) = 1 − (1 − φ1(0)) e−µht, φ2(t) = 1 − (1 − φ2(0)) e−µv t, φ3(t) = 1 − (1 − φ3(0)) e−µut, where φ1(0) = sh(0) + ih(0) + rh(0), φ2(0) = sv (0) + iv (0), φ3(0) = su (0) + iu (0). it is straightforward to conclude that φ1(t) 6 1 if φ1(0) 6 1, φ2(t) 6 1 if φ2(0) 6 1, φ3(t) 6 1 if φ3(0) 6 1. thus, we have 0 6 sh(t),ih(t),rh(t),sv (t),iv (t), su (t),iu (t) 6 1 and hence the set ω is positively invariant set. moreover, the set ω is a globally attractive set since if φi(0) > 1 then lim t→∞ φi(t) = 1 for i = 1, 2, 3. biomath 11 (2022), 2212149, https://doi.org/10.55630/j.biomath.2022.12.149 5/14 https://doi.org/10.55630/j.biomath.2022.12.149 al-maqrashi et al, investigating the role of mobility between rural areas and forests on the spread of zika b. the basic reproduction number the model (2) has a disease-free equilibrium (dfe): z0 := (s0h, 0, 0,s 0 v , 0,s 0 u, 0) ∈ ω, where s0h = s 0 v = s 0 u = 1. the number of new infections produced by a typical infected individual in a population at dfe is called the basic reproduction number r0 which can be obtained by applying the next generation method [20]. the next-generation matrix is: pq−1 =  λκ1κ2 γ + µh(1 −ε1) κ1α1β1θ1 µv (1 −ε2) κ1α2β2θ1 µu (1 −ε3) κ2β1θ2 γ + µh(1 −ε1) 0 0 κ2β2θ2 γ + µh(1 −ε1) 0 0   , where p is the jacobian of the transmission matrix which describes the production of new infections, whereas q is the jacobian of the transition matrix which describes changes in state and they are given by p =  λκ1κ2 κ1α1β1θ1 κ1α2β2θ1κ2β1θ2 0 0 κ2β2θ2 0 0   and q =  γ + µh(1 −ε1) 0 00 µv (1 −ε2) 0 0 0 µu (1 −ε3)   . the basic reproduction number r0 is the dominant eigenvalue of pq−1, which can be expressed as: r0 = 1 2 ( rhh + √ r2hh + 4(rhv + rhu ) ) , where rhh = κ1κ2λ γ + µh(1 −ε1) , rhv = α1κ1κ2β 2 1θ1θ2 µv (1 −ε2) ( γ + µh(1 −ε1) ), rhu = κ1κ2α2β 2 2θ1θ2 µu (1 −ε3) ( γ + µh(1 −ε1) ). note that rhh represents the contribution to the reproduction number due to human-to-human transmission, rhv represents the contribution to the reproduction number due to interaction between human and vector in a rural area, and rhu represents the contribution to the reproduction number due to interaction between human and vector in the forest area. the square root arises from the two “generations” required for an infected vector or host to “reproduce” itself [20]. moreover, the threshold of the disease occurs at r0 = 1 ⇐⇒ rhh +rhv +rhu = 1. also, it can be easily proven that r0 < 1 implies rhh + rhv + rhu < 1, which means the disease to die out, all the transmission routes represented by rhh, rhv and rhu need to be reduced. clearly, this will also imply that r0 > 1 whenever rhh, rhv or rhu is greater than one. c. sensitivity analysis of the basic reproduction number a fundamental and valuable numeric value for the study of infectious disease dynamics is the basic reproduction number r0, since it predicts whether an outbreak will be expected to continue (when r0 > 1) or die out (when r0 < 1). sensitivity analysis of the basic reproduction number allows us to determine which model parameters have the most impact on r0. a highly sensitive parameter leads to a high quantitative variation in r0. moreover, sensitivity analysis highlights the parameters that must be attacked by intervention and treatment strategies. here, we adopt the elasticity index (normalized forward sensitivity index) [21], er0p , which computes the relative change of r0 with respect to any parameter p as follows: er0p = p r0 lim ∆p→0 ∆r0 ∆p = p r0 ∂r0 ∂p . (3) evidently, the corresponding model parameters will affect the basic reproduction number either positively or negatively. the positive sign of the sensitivity indices of the parameter denotes the increase of the basic reproduction number r0 as that parameter changes, whereas the negative sign of the sensitivity indices of the parameter denotes the decrease of the basic reproduction number r0 as that parameter changes. moreover, the magnitude denotes the relative importance of the spotlight parameter. the biting rate of mosquitoes is a significant parameter of the epidemiology of the parasite or pathogen since it is directly related to the basic reproduction rate and affects the dynamics of disease transmission in both areas. it varies depending on the local abundance of vectors, vector host preferences, and host attractiveness [22]. mosquitoes adjust their preferences depending on the availability of a specific host species to enhance their reproductive success [22]. it is also affected by environmental factors such as temperature, humidity, and larval food sources. the terms (1−κ1)(1−κ2)β21 and κ1κ2β22 in r0 suggest that biomath 11 (2022), 2212149, https://doi.org/10.55630/j.biomath.2022.12.149 6/14 https://doi.org/10.55630/j.biomath.2022.12.149 al-maqrashi et al, investigating the role of mobility between rural areas and forests on the spread of zika the signs of the elasticity indices of the proportion of susceptible humans, moving from rural to forest areas κ1 and the proportion of infected humans moving from rural to forest areas κ2, depend on the biting rates of mosquitoes β1 and β2. to investigate this dependence, we use relation (3) to obtain the following relations which describe the elasticity indices of κ1 and κ2, respectively, in terms of both β1 and β2, taking all other parameters to be fixed, namely, κ1 = 0.3, κ2 = 0.2, α1 = 2, α2 = 3, ε1 = 0.67, ε2 = 0.06, ε3 = 0.06, µh = 0.00004, µv = 0.07, µu = 0.07, λ = 0.235, γ = 0.16, θ1 = 0.33, θ2 = 0.3: e r0 κ1 = 0.3 ( −0.5874 + 1 4 −1.9325 − 60.1774β22 + 22.5665β21√ 0.6763 + 42.1241β22 + 6.7699β 2 1 ) 0.4112 + 1 2 √ 0.6763 + 42.1241β22 + 6.7699β 2 1 e r0 κ2 = 0.2 ( −0.5140 + 1 4 −1.6909 − 52.6552β22 − 33.8497β21√ 0.6763 + 42.1241β22 + 6.7699β 2 1 ) 0.4112 + 1 2 √ 0.6763 + 42.1241β22 + 6.7699β 2 1 clearly, the above two equations show that the signs of the sensitivity indices of κ1 and κ2 depend on the values of β1 and β2. this dependence is illustrated in figure 3 based on the range of values of β1 and β2 given in table i. figure 3 shows three different regions for the signs of the indices depending on the values of β1 and β2, namely, a region of negative indices, a region of positive indices, and a region of negative index for κ1 and a positive index for κ2. these three regions are all feasible for low values of the rural mosquito’s biting rate β1 up to a certain limit, taken here to be around β1 = 0.75. in this case, the region of positive indices is only feasible for higher values of the forest mosquito’s biting rate β2, taken here to be higher than 0.8. this means that the movement of susceptible and infected humans from rural to forest areas will have the effect of increasing the basic reproduction number if the disease transmission in the forest areas is relatively higher than in the rural areas. however, for higher values of β1, the region of positive indices is not feasible even for high values of β2. the only feasible positive index is for κ2, whereas the index of κ1 remains negative for all values of β2 since the disease transmission in the rural area is high. using β1 = 0.35 and β2 = 0.9, which correspond to the region of positive indices, we calculate the elasticity fig. 3: signs of the sensitivity indices of the movement rates κ1 and κ2 in terms of the biting rates β1 and β2. table ii: sensitivity indices and their interpretation. para. value sens. ind. interpretation β1 0.35 0.36629 β1 by 10% r0 by 37% β2 0.9 0.38926 β2 by 10% r0 by 39% λ 0.235 0.24444 λ by 10% r0 by 24% κ1 0.3 0.01138 κ1 by 10% r0 by 1% κ2 0.2 0.08773 κ2 by 10% r0 by 9% θ1 0.33 0.37778 θ1 by 10% r0 by 38% θ2 0.3 0.37778 θ2 by 10% r0 by 38% γ 0.16 -0.62217 γ by 10% r0 by 62% α1 2 0.18315 α1 by 10% r0 by 18% α2 3 0.19463 α2 by 10% r0 by 19% µh 0.00004 -0.00005 µh by 10% r0 by 0.005% µv 0.07 -0.18315 µv by 10% r0 by 18% µu 0.07 -0.19463 µu by 10% r0 by 19% ε1 0.67 0.000104 ε1 by 10% r0 by 0.01% ε2 0.06 0.01169 ε2 by 10% r0 by 0.11% ε3 0.06 0.01242 ε3 by 10% r0 by 12% indices for all parameters. the obtained values and their interpretations are listed in table ii. clearly, the most efficacious parameter is the biting rate of forest mosquitoes on humans β2, i.e., it has a strong positive impact on the value of r0. also, the biting rate of forest mosquitoes on humans β1 has a positive impact on r0. the transmission probabilities per bite – per human θ1 and per mosquito θ2 – have a positive influence on the value of r0. similarly, one can note that the proportions of movement for susceptible humans κ1 and infected κ2 have a small positive effect on r0, with κ2 having a higher index than κ1. on the biomath 11 (2022), 2212149, https://doi.org/10.55630/j.biomath.2022.12.149 7/14 https://doi.org/10.55630/j.biomath.2022.12.149 al-maqrashi et al, investigating the role of mobility between rural areas and forests on the spread of zika other hand, the recovery rate of humans γ has the most negative sensitivity index, as it will decrease r0 by 62% when it increases by 10%. clearly, there is a very small positive effect of the vertical transmission of humans and both vectors ε1,ε2,ε3 with ε3 having the highest index among them. using the parameter values listed in table ii, the basic reproduction number is estimated to be r0 = 2.093. this value is close to the one obtained in [23], in which the authors have estimated from notification data that the basic reproduction number for zikv in rio de janeiro is r0 = 2.33. d. local stability of the dfe here we discuss the local stability of the dfe by finding the eigenvalues of the linearized system. the following theorem is devoted to the local stability of the dfe, i.e., the disease would be eliminated under certain conditions. theorem 2. if r0 ≤ 1, the dfe of the model (2) is locally asymptotically stable. if r0 > 1, it is unstable. proof: the linearized matrix of the system (2) at the disease-free equilibrium z0 is: jz0 =  −µh −µhε1 −λκ1κ2 0 0 0 −γ −µh(1 −ε1) + λκ1κ2 0 0 0 γ −µh 0 0 −β1κ2θ2 0 −µv 0 β1κ2θ2 0 0 0 −β2κ2θ2 0 0 0 β2κ2θ2 0 0 −κ1α1β1θ1 0 −κ1α2β2θ1 κ1α1β1θ1 0 κ1α2β2θ1 0 0 0 −µv ε2 0 0 −µv (1 −ε2) 0 0 0 −µu −µuε3 0 0 −µu (1 −ε3)   . it is clear that the system has three negative eigenvalues which are `1 = −µv , `2 = −µu and `3 = −µh with multiplicity two. the remaining eigenvalues can be found from the characteristic equation k(`) = 0, where k(`) is given by: k(`) = `3 + k1` 2 + k2` + k3, with k1 = ξ(1 −rhh) + µv (1 −ε2) + µu (1 −ε3), k2 = ξµv (1 −ε2)(1 −rhh −rhv ) + ξµu (1 −ε3)(1 −rhh −rhu ) + µv µu (1 −ε2)(1 −ε3), k3 = ξµv µu (1 −ε2)(1 −ε3)(1 −rhh −rhv −rhu ), where ξ = ( γ + µh(1 −ε1) ) . it is clear that k3 > 0 if rhh + rhv + rhu < 1 which also implies that k1 > 0 and k2 > 0. hence, in order to use routh’s stability criterion [24] to show that the roots of the above characteristic equation have negative real parts, it remains to show that k1k2 − k3 is positive, that is: k1k2 −k3 = 2ξµv µu (1 −ε2)(1 −ε3)(1 −rhh) + ξ2µv (1 −ε2)(1 −rhh)(1 −rhh −rhv ) + ξ2µu (1 −ε3)(1 −rhh)(1 −rhu −rhv ) + ξµ2v (1 −ε2) 2(1 −rhh −rhv ) + ξµ2u (1 −ε3) 2(1 −rhh −rhu ) + µ2v µu (1 −ε2) 2(1 −ε3) + µv µ 2 u (1 −ε2)(1 −ε3) 2. clearly, k1k2 −k3 > 0 if and only if rhh +rhv < 1 and rhh + rhu < 1. therefore, by routh’s stability criterion, the roots of the characteristic equation k(`) = 0 have negative real parts, and hence we conclude that the dfe is locally asymptotically stable whenever r0 ≤ 1. otherwise, it is unstable. e. global stability of the dfe when the solution of the dynamical system (2) approaches a unique equilibrium point regardless of initial conditions then the equilibrium point is globally asymptotically stable. the global stability of the dfe will ensure that the disease is eliminated under all initial conditions. in this regard, we state and prove the following theorem: theorem 3. if r0 ≤ 1, the disease-free equilibrium z0 is globally asymptotically stable on the compact set ω. proof: applying castillo-chavez theorem [25], consider the following two compartments: x(t) =   sh(t) rh(t) sv (t) su (t)   , y (t) =  ih(t)iv (t) iu (t)   , biomath 11 (2022), 2212149, https://doi.org/10.55630/j.biomath.2022.12.149 8/14 https://doi.org/10.55630/j.biomath.2022.12.149 al-maqrashi et al, investigating the role of mobility between rural areas and forests on the spread of zika which describe the uninfected and infected individuals of the system (2), respectively. so that system (2) can be written as: dx dt = f(x,y ), dy dt = g(x,y ), g(x, 0) = 0, where f(x,y ) and g(x,y ) are the corresponding right hand side of system (2). to guarantee the global asymptotic stability of the dfe, according to the castillo-chavez theorem, the following two conditions must be satisfied: (h1) for dx dt = f(x, 0), x0 = (1, 0, 1, 1)t is globally asymptotically stable. (h2) ĝ > 0, where ĝ(x,y ) = ay −g(x,y ) and a = dy g(x 0, 0) is an metzler matrix ∀(x,y ) ∈ ω. to check the first condition, we find: f(x, 0) =   −µhsh + µh −µhrh −µv sv + µv −µusu + µu   . solving the system of odes in (h1), we obtain the following behavior of each component: sh(t) = 1 + sh(0)e −µht =⇒ lim t→∞ sh(t) = 1, rh(t) = rh(0)e −µht =⇒ lim t→∞ rh(t) = 0, sv (t) = 1 + sv (0)e −µv t =⇒ lim t→∞ sv (t) = 1, su (t) = 1 + su (0)e −µut =⇒ lim t→∞ su (t) = 1. hence, the first condition is satisfied. now, to check the second condition, we first find: a =  −ξ + λκ1κ2 κ1α1β1θ1 κ1α2β2θ1κ2β1θ2 −µv (1 −ε2) 0 κ2β2θ2 0 −µu (1 −ε3)   , where ξ = ( γ + µh(1 −ε1) ) . then, ĝ(x,y ) = ay −g(x,y ) = (κ1α1β1θ1iv + κ1α2β2θ1iu + κ1κ2λih)(1 −sh)β1κ2θ2ih(1 −sv ) β2κ2θ2ih(1 −su )   since 0 6 sh 6 1, 0 6 sv 6 1 and 0 6 su 6 1 then ĝ > 0 for all (x,y ) ∈ ω. thus, z0 is globally asymptotically stable provided that r0 ≤ 1. f. existence of endemic equilibrium the endemic equilibrium is the state where the infection cannot be totally eradicated and the disease progression persists in a population at all times but in relatively low frequency. here, we discuss the existence of endemic equilibrium. theorem 4. for model (2) there exists an endemic equilibrium z∗ ∈ ω whenever r0 > 1. proof: let z∗ := (s∗h,i ∗ h,r ∗ h,s ∗ v ,i ∗ v ,s ∗ u,i ∗ u ) be the endemic equilibrium of the model (2) such that: s∗h = µh − (γ + µh)i∗h µh , r∗h = γi∗h µh , s∗v = µv (1 −ε2) µv (1 −ε2) + κ2β1θ2i∗h , i∗v = κ2β1θ2i ∗ h µv (1 −ε2) + κ2β1θ2i∗h , s∗u = µu (1 −ε3) µu (1 −ε3) + κ2β2θ2i∗h , i∗u = κ2β2θ2i ∗ h µu (1 −ε3) + κ2β2θ2i∗h , and i∗h satisfies the following equation: q1i ∗4 h + q2i ∗3 h + q3i ∗2 h + q4i ∗ h = 0, where q1 = β1β2λθ 2 2κ2κ2(γ + µh), q2 = ξβ1β2κ2κ2θ 2 2µh(1 −rhh) + ξβ2κ2θ2µv (1 −ε2)(γ + µh)(rhh + rhv ) + ξκ2β1θ2µu (1 −ε3)(γ + µh)(rhh + rhu ), q3 = ξβ2κ2θ2µhµv (1 −ε2)(1 −rhh −rhv ) + ξβ1θ2κ2µhµu (1 −ε3)(1 −rhh −rhu ) + ξµv µu (1 −ε2)(1 −ε3)(γ + µh) (rhh + rhv + rhu ), q4 = ξµhµv µu (1 −ε2)(1 −ε3) (1 −rhh −rhv −rhu ), where ξ = ( γ + µh(1 −ε1) ) . solving the above equation we get i∗h = 0, which corresponds to the dfe (z0) and the remaining roots satisfy the cubic equation: q1i ∗3 h + q2i ∗2 h + q3i ∗ h + q4 = 0. clearly, if rhh + rhv + rhu > 1, then the above equation has a positive root, since q1 > 0 and q4 < 0. biomath 11 (2022), 2212149, https://doi.org/10.55630/j.biomath.2022.12.149 9/14 https://doi.org/10.55630/j.biomath.2022.12.149 al-maqrashi et al, investigating the role of mobility between rural areas and forests on the spread of zika fig. 4: bifurcation figure when λ is taken as a bifurcation parameter of system (2) with a bifurcation value λ∗ = 0.2132 at r0 = 1 and by fixing parameter α1 = 2, α2 = 3, κ1 = 0.3, κ2 = 0.2, ε1 = 0.67, ε2 = 0.06, ε3 = 0.06, µh = 0.00004, µv = 1/14, µu = 1/14, β1 = 0.15, β2 = 0.1, γ = 0.16, θ1 = 0.33, θ2 = 0.3. now, note that q3 can be written in terms of q2 as follows: q3 = ξ ( µhθ2 ( β2κ2µv (1 −ε2) + β1κ2µu (1 −ε3) ) + µv µu (1 −ε2)(1 −ε3)(γ + µh) (rhh + rhv + rhu ) ) − µh γ + µh ( q2 − ξβ1β2κ2κ2θ22µh(1 −rhh) ) . to ensure the uniqueness of the positive roots, we apply descartes’s sign rule [26]. there exists a unique positive root when q2 > 0 regardless of the sign of q3 and this happens if rhh < 1 and rhh + rhu + rhv > 1. however, when q2 < 0 and rhh + rhu + rhv > 1 there exist at least one positive root. note that the existence of three positive roots is only possible when q2 < 0 and q3 > 0. g. bifurcation analysis when the stability of a system is changed as a parameter changes causing the emergence or disappearance of new stable points, then the system is said to undergo bifurcation. in this section, we prove that system (2) has transcritical bifurcation. the proof is based on the sotomayor theorem described in [27]. let f be defined as the right-hand side of the system (2) and z = (sh,ih,rh,sv ,iv ,su,iu ) t . at r0 = 1, we can check that the constant term of the characteristic equation of jz0 is zero which implies that jz0 has a simple zero eigenvalue. here, we choose λ as a bifurcation parameter such that the bifurcation value corresponding to r0 = 1 is given by: λ∗ = (γ + µh(1 −ε1))(1 −rhv −rhu ) κ1κ2 . solving j(z0,λ∗)v = 0, where v = (v1,v2,v3,v4,v5,v6,v7) t is a nonzero right eigenvector of j(z0,λ∗) corresponding to the zero eigenvalue, we obtain: v =   − γ + µh γ µh γ 1 − β1κ2θ2µh µv γ(1 −ε2) β1κ2θ2µh µv γ(1 −ε2) − β2κ2θ2µh µuγ(1 −ε3) β2κ2θ2µh µuγ(1 −ε3)   v3, v3 6= 0. next we find the corresponding nonzero left eigenvector w = (w1,w2,w3,w4,w5,w6,w7)t , which satisfies jt (z0,λ∗) w = 0. we get: w =   0 1 0 0 α1β1κ1θ1 µv (1 −ε2) 0 α2β2κ1θ1 µu (1 −ε3)   w2, w2 6= 0. model (2) can be written as dz/dt = f(z), where f(z) is the right hand side of the model. now, we check the conditions of the sotomayor theorem and begin with finding fλ(λ∗,z0): fλ(λ ∗,z0) = (0, 0, 0, 0, 0, 0, 0)t . so, the first condition is satisfied: wtfλ(λ ∗,z0) = 0. biomath 11 (2022), 2212149, https://doi.org/10.55630/j.biomath.2022.12.149 10/14 https://doi.org/10.55630/j.biomath.2022.12.149 al-maqrashi et al, investigating the role of mobility between rural areas and forests on the spread of zika next, we find the jacobian of fλ(λ∗,z) as follows: dfλ(λ ∗,z) =  −κ1κ2ih −κ1κ2sh 0 0 0 0 0 κ1κ2ih κ1κ2sh 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   . checking the second condition, we have: wtdfλ(λ ∗,z0)v = κ1κ2w2v2 6= 0. finally, we check the third condition by finding d2f(λ∗,z0), where d2 denotes the matrix of the partial derivatives of each component of df(z) and we get: d2f(λ∗,z0)(v,v) =  −2λκ1κ2v1v2 − 2α1κ1β1θ1v1v5 − 2κ1α2β2θ1v1v7 2λκ1κ2v1v2 + 2α1κ1β1θ1v1v5 + 2κ1α2β2θ1v1v7 0 −2κ2β1θ2v4v2 2κ2β1θ2v4v2 −2κ2β2θ2v6v2 2κ2β2θ2v6v2   thus, wt ( d2f(λ∗,z0)(v,v) ) = 2v1(α2β2θ1κ1v7 + α1β1θ1κ1v5 + λκ1κ2) + 2β1θ2κ2v2v4w5 + 2β2θ2κ2v2v6w7. by substituting the values of v’s and w’s, we get: wt ( d2f(λ∗,z0)(v,v) ) = − ( 2µh(γ + µh)(γ + µh(1 −ε1)) γ2 + 2β32κ1κ 2 2θ 2 2θ1α2 γ2µ2u (1 −ε3)2 ) w2v 2 3, which is nonzero since w2 and v3 are nonzero. hence, the system (2) experiences a transcritical bifurcation at z0 as the parameter λ passes through the bifurcation value λ = λ∗. the bifurcation diagram is created by using the matcont package [28] and is illustrated in figure 4. this leads us to establish the following theorem: theorem 5. model (2) undergoes transcritical bifurcation at the dfe (z0) when the parameter λ passes through the bifurcation value λ = λ∗. remark. we can establish the local stability of endemic equilibrium using the above calculations. we note that based on theorem 4 in [20], a and b are given by: a = 1 2 wt ( d2fλ(λ ∗,z0)(v,v) ) = 1 2 n∑ i,j,k=1 vivjwk ∂2fi ∂xj∂xk (λ∗,z0), b = wtdfλ(λ ∗,z0)v = n∑ i,j=1 viwj ∂2fi ∂xj∂λ (λ∗,z0). according to the calculations in this section, it is clear that b 6= 0 and a < 0 if w2 is positive. thus, there exists δ > 0 such that the endemic equilibrium z∗ is locally asymptotically stable near z0 for 0 < λ < δ. moreover, according to castillo-chavez and song [29] the direction of the bifurcation of the system (2) at r0 = 1 is forward (supercritical bifurcation). iv. numerical analysis in this section, the forgoing theoretical results are confirmed by presenting the numerical results of the zika sir-si model (2). the asymptotic behavior of the model is characterized by solving the system numerically using the numerical simulations of matlab with baseline parameters listed in table i with appropriate initial conditions. we assume that the human population size is 200000, the rural mosquito population size is 400000, and the forest mosquito population size is 600000. these values are obtained by taking into consideration desired conditions or from literature. phase diagram for the case r0 < 1 is illustrated in figure 5. here, the biting rates of rural and forest mosquitoes on humans are taken to be β1 = β2 = 0.3. figure 5 shows that all populations reach the diseasefree equilibrium with the disease disappearing from vector populations faster than the human population. for the case r0 > 1, we consider q2 > 0 and q3 > 0 and the biting rate of rural mosquitoes and of forest mosquitoes on humans to be β1 = 0.35, β2 = 0.8, respectively. the unique endemic equilibrium for this case is given by: z∗ := (0.05867, 0.00054, 0.940789, 0.99808, 0.00192, 0.99890, 0.00109) and the disease dynamics are illustrated in figure 6. it shows that the solution exhibits oscillations before reaching its steady state. now, we present numerical simulations for the effects of variations of κ1 and κ2. in figure 7, we change the biomath 11 (2022), 2212149, https://doi.org/10.55630/j.biomath.2022.12.149 11/14 https://doi.org/10.55630/j.biomath.2022.12.149 al-maqrashi et al, investigating the role of mobility between rural areas and forests on the spread of zika fig. 5: phase diagram when r0 < 1 with parameter values taken to be α1 = 2, α2 = 3, κ1 = 0.3, κ2 = 0.2, ε1 = 0.67, ε2 = 0.06, ε3 = 0.06, µh = 0.00004, µv = 0.07, µu = 0.07, β1 = 0.3, β2 = 0.3, λ = 0.01, γ = 0.16, θ1 = 0.2, θ2 = 0.3. fraction of susceptible humans moving to forest area κ1 and fix the other parameters as listed in table ii. we note that increasing the values of κ1 leads to a slight increase in the maximum of both infected humans and infected vectors in forest areas. the infections reach their maximum and their endemic steady states slightly earlier as the movement of susceptible humans increases. figure 8 illustrates the effect of varying the proportion of infected humans moving to forest area κ2 and fixing all other parameters. it shows that increasing the proportion of infected humans moving to forest areas has the effect of increasing the number of infected vectors in forest areas and the number of infected humans. the time it takes to reach the maximum fig. 6: phase diagram when r0 > 1 with parameter values taken to be α1 = 2, α2 = 3, κ1 = 0.3, κ2 = 0.2, ε1 = 0.67, ε2 = 0.06, ε3 = 0.06, µh = 0.00004, µv = 0.025, µu = 0.025, β1 = 0.35, β2 = 0.8, λ = 0.235, γ = 0.07, θ1 = 0.33, θ2 = 0.3. number of infections remains the same for the vector population and it becomes slightly earlier for the human population as κ2 increases. note that when κ2 = 0 the number of infected mosquitoes in the forest area reaches zero, which means that the disease will disappear from the forest since the model assumes that infected humans are the only source of infection for the vector population in the forest. however, there have been some reports of zikv being found in non-human primates, raising the possibility that they could act as reservoirs [30, 31]. these sources of infection will be considered in future works. v. conclusion a mathematical model of zikv disease including human movement and three transmission routes, namely, biomath 11 (2022), 2212149, https://doi.org/10.55630/j.biomath.2022.12.149 12/14 https://doi.org/10.55630/j.biomath.2022.12.149 al-maqrashi et al, investigating the role of mobility between rural areas and forests on the spread of zika 0 50 100 150 t 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 i h (t ) 1 =0.25 1 =0.3 1 =0.35 1 =0.4 0 50 100 150 t 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 i u (t ) 1 =0.25 1 =0.3 1 =0.35 1 =0.4 fig. 7: number of infected populations for different values of κ1, where other parameters are fixed κ2 = 0.2, α1 = 2, α2 = 3, ε1 = 0.67, ε2 = 0.06, ε3 = 0.06, µh = 0.00004, µv = 0.07, µu = 0.07, β1 = 0.35, β2 = 0.9, λ = 0.235, γ = 0.16, θ1 = 0.33, θ2 = 0.3. human-to-human transmission, vector transmission, and vertical transmission has been proposed. the model has been analyzed and studied to investigate the role of human movement from rural areas to forest areas on the spread of zikv. the positivity of the solution and the boundedness of the invariant region were discussed. the basic reproduction number r0 was computed and expressed in terms of reproduction numbers related to the interactions between humans rhh, between human and vector in rural area rhv and between human and vector in forest area rhu . it was found that the threshold of the disease which occurs at r0 = 1 is equivalent to rhh + rhv + rhu = 1 and hence all transmission routes need to be controlled to reduce the spread of the disease. 0 50 100 150 200 250 300 t 0 0.05 0.1 0.15 i h (t ) 2 =0 2 =0.05 2 =0.1 2 =0.15 0 50 100 150 200 250 300 t 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 i u (t ) 2 =0 2 =0.05 2 =0.1 2 =0.15 fig. 8: number of infected populations for different values of κ2, where other parameters are fixed κ1 = 0.3, α1 = 2, α2 = 3, ε1 = 0.67, ε2 = 0.06, ε3 = 0.06, µh = 0.00004, µv = 0.07, µu = 0.07, β1 = 0.35, β2 = 0.9, λ = 0.235, γ = 0.16, θ1 = 0.33, θ2 = 0.3. sensitivity analysis of r0 was carried out and it showed that r0 is sensitive to almost all model parameters either positively or negatively, except the parameters κ1 and κ2, representing the proportions of susceptible and infected humans moving to forest areas, respectively, where their signs of sensitivity indices were found to depend on the biting rates when fixing all other parameters. this dependence has been calculated and illustrated graphically. it has been found that the indices are both positive when the forest mosquito’s biting rate is high and the rural mosquito’s biting rate is small up to a certain limit. however, this positive effect on r0 was found to be very small, with κ2 having a higher effect than κ1. the most positive influential parameters are the biting rate biomath 11 (2022), 2212149, https://doi.org/10.55630/j.biomath.2022.12.149 13/14 https://doi.org/10.55630/j.biomath.2022.12.149 al-maqrashi et al, investigating the role of mobility between rural areas and forests on the spread of zika of rural and forest mosquitoes on humans, while the recovery rate of humans has the most negative impact. then, the local and global stability of the diseasefree equilibrium was derived whenever r0 is less than unity. furthermore, the system was shown to possess a unique endemic equilibrium under certain conditions, and it is locally asymptotically stable when r0 is greater than unity since the direction of the bifurcation was found to be forward. the bifurcation analysis was presented both analytically and graphically. finally, numerical simulations were presented to demonstrate the obtained theoretical results. they confirmed that the human movement from rural areas to forests has a small effect on increasing the infected human and vector populations, with the movement of infected humans having a higher effect than the movement of susceptible humans. references [1] g. s. campos, a. c. bandeira, s. i. sardi, “zika virus outbreak, bahia, brazil”, emerging infectious diseases, 21(10):18851886, 2015. 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https://doi.org/10.3390/v13102088 https://doi.org/10.55630/j.biomath.2022.12.149 introduction model description model analysis positivity of solutions and positively invariant set the basic reproduction number sensitivity analysis of the basic reproduction number local stability of the dfe global stability of the dfe existence of endemic equilibrium bifurcation analysis numerical analysis conclusion references www.biomathforum.org/biomath/index.php/biomath original article modeling, analysis and simulations of mers outbreak in saudi arabia nofe al-asuoad, meir shillor department of mathematics and statistics oakland university rochester, mi, usa nalasuoa@oakland.edu, shillor@oakland.edu received: 5 october 2017, accepted: 27 february 2018, published: 9 march 2018 abstract—this work describes a continuous differential equations model for the dynamics of middle eastern respiratory syndrome (mers) and provides its computer simulations. it is a continuation of our previous paper al-asuoad et al. (biomath 5, 2016) and it extends the simulations results provided there, which were restricted to the city of riyadh, to the whole of saudi arabia. in addition, it updates the results for the city of riyadh itself. using an optimization procedure, the system coefficients were obtained from published data, and the model allows for the prediction of possible scenarios for the transmission and spread of the disease in the near future. this, in turn, allows for the application of possible disease control measures. the model is found to be very sensitive to the daily effective contact parameter, and the presented simulations indicate that the system is very close to the bifurcation of the stability of the disease free equilibrium (dfe) and appearance of the endemic equilibrium (ee), which indicates that the disease will not decay substantially in the near future. finally, we establish the stability of the dfe using only the stability number rc, which simplifies and improves one of the main theoretical results in the previous paper. keywords-mers model; stability of dfe and ee; simulations; sensitivity analysis; i. introduction this work uses the mathematical model constructed in [2] to study the dynamics of the middle east respiratory syndrome (mers) in saudi arabia. it also expands the study that was performed there of the disease in the city of riyadh, since new data became available since the publishing of the paper. the aim of this work is to provide the health care community and related authorities with a predictive tool that allows to assess various mers scenarios and the effectiveness of various intervention practices. mers is a new respiratory disease caused by the newly discovered middle east respiratory syndrome coronavirus (mers-cov). the first case of the disease was reported in saudi arabia in june 2012, when a 60-year-old man died of progressive respiratory and renal failure 11 days copyright: c© 2017 al-asuoad et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: nofe al-asuoad, meir shillor, modeling, analysis and simulations of mers outbreak in saudi arabia, biomath 7 (2017), 1802277, http://dx.doi.org/10.11145/j.biomath.2018.02.277 page 1 of 18 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2018.02.277 nofe al-asuoad, meir shillor, modeling, analysis and simulations of mers outbreak in saudi arabia after hospital admission. he had a 7-day history of fever, cough, expectoration and shortness of breath [26], [30]. in september 2012 a case of a 49-year old man from qatar with pneumonia and kidney failure, who was treated in an intensive care unit at a london hospital, was reported. he had a history of travel to saudi arabia. further laboratory tests revealed a positive mers-cov infection [6]. retrospectively, the infection was found in stored respiratory and serum samples on two deceased patients from jordan, where in april 2012 an outbreak of acute respiratory illness occurred in a public hospital [17]. in 2003, a previously unknown coronavirus, the severe acute respiratory syndrome coronavirus (sars-cov), caused a global outbreak of pneumonia that resulted in approximately 800 deaths [22]. mers-cov that has effects similar to those of sars-cov, is classified as a coronavirus, which is a family of single-stranded rna viruses. this family includes viruses that cause mild illness such as common cold as well as severe illness such as sars in humans. mers-cov is a beta coronavirus which has not been identified in humans before 2012 and is different from any coronaviruses (including sars-cov) that have been found in humans or animals [10], [31]. within a year from its discovery, a total of 130 mers-cov cases were identified, 58 of which died, which means that the case fatality rate is 45%, much higher than sars-cov, which has a case fatality of 15% [24], [27]. up to date (august 31, 2017) 2067 confirmed cases of mers-cov have been reported worldwide, out of these 1679 were reported from saudi arabia where the case fatality rate has been 40.6% [28]. the infection has been a global threat due to continuous outbreaks in the arabian peninsula and international spread to 26 countries including qatar, jordan, united arab emirates, united kingdom, the philippines, united states and other countries, by infected travelers [14]. in 2015, a large outbreak happened in south korea, which was the first outbreak outside the arabian peninsula [19], [23], and 186 people were infected, 38 of whom died. after intensive search, camels were found to have a high rate of anti-mers-cov antibodies, which indicates that they were infected with the virus. then, definite evidence of camel-to-human transmission of the virus has been reported recently [5], [29]. moreover, there is clear evidence that the infection is transmitted from person to person upon close contact, including from patients to healthcare workers [4], [12]. the incubation period from exposure to the development of clinical disease is from five to 14 days. mers-cov is typically characterized by cough, fever, sore throat, chills, myalgia and shortness of breath [11], [13]. one-third of the patients had also gastrointestinal symptoms such as vomiting and diarrhea. the common complications of the mers-cov infection include pneumonia, acute respiratory distress syndrome and respiratory failure. although it is known that asymptomatic infection occurs, the percentage of patients who have it is unknown, yet, [3], [7]. no specific treatment is available for the merscov infection. currently, the management of the disease is done by supportive therapy that minimizes the symptoms. some patients require mechanical ventilation or extra-corporal membrane oxygenation. since no vaccine exists for mers [9], [18], once a case is identified, the individual and those connected to them are being isolated to minimize the spread of the disease. part of the content of this article can be found in the recent doctoral dissertation [1] where additional information can also be found. to help assessing the threat of the spread of mers, we constructed a mathematical model as a tool to predict possible future scenarios and the effectiveness of various intervention procedures. the model is in the form of a coupled system of five nonlinear ordinary differential equations (odes) for the susceptible, asymptomatic, clinically symptomatic, isolated and recovered populations. it is of a rather standard mseir type (see, e.g., [8], [15], [16], [21] and the many references therein). the novelty in this work lies in the theoretical proof biomath 7 (2017), 1802277, http://dx.doi.org/10.11145/j.biomath.2018.02.277 page 2 of 18 http://dx.doi.org/10.11145/j.biomath.2018.02.277 nofe al-asuoad, meir shillor, modeling, analysis and simulations of mers outbreak in saudi arabia of the global stability of the endemic equilibrium (when it exists), and simulations for the whole of saudi arabia. first, we mathematically analyze the model and provide a new proof of the global stability of disease free equilibrium (dfe) and of the endemic equilibrium (ee), when it exists (and then the dfe becomes unstable). moreover, the new proof of the global stability of the ee is simpler and more elegant than the one in proposition 4 in [2], since it uses only the effective reproduction number or the stability control parameter rc. second, the model is used for simulations of the overall outbreak in saudi arabia, and it also extends the study of mers done in [2] of the city of riyadh, as more data has been collected since its publication. indeed, currently there exists data that spans 1550 days, [25]. for the whole of saudi arabia, the model parameters were fitted to the data using the first 865 days, then runs for 1690 days, until nov. 4, 2020, were performed, allowing for the prediction of the disease spread in the next three years. in the previous paper, we fitted our model to the daily reported cumulative cases of mers data for riyadh for the period from nov. 4, 2013 to march 17, 2016 (865 days). here, we fitted the model to the data from nov. 4, 2013 to july 11, 2017 (1346 days). the model was found to be very sensitive to the scaled contact parameter that is directly related to the number of individuals a person is in contact with each day. nevertheless, the simulations provide a very good fit with what has been observed and are similar in their predictions of the near future, say the next two years. the rest of the paper is structured as follows. the model is described in section ii, where its compartmental structure and flow chart are also provided. the stability analysis of the dfe and ee is done in section iii, where the local and global stability of the dfe are studied. our new results on the global stability of the ee are summarized in proposition 4 and proved using the lyapunov method and lasalle’s principle. the description of the simulation results can be found in section iv. first the optimized parameters for the baseline for saudi arabia are presented and the simulation results depicted. then, the extended study of riyadh is described. the sensitivity of the model to the contact parameter β is done in section v, which is one of the main characteristics of the model. in section vi we depict graphically the errors, i.e., the difference between the data and model predictions. the paper concludes in section vii where the results are summarized and some unresolved issues indicated. ii. the model we use the basic model of mers dynamics developed in [2], where full details of the model and its underlying assumptions can be found so, the description of the model here follows very closely the one in [2]. the model represents the disease dynamics of five populations of individuals: susceptible s(t), asymptomatic e(t), symptomatic i(t), isolated j(t), and recovered r(t), where the time t is measured in days. also, n(t) denotes the total population at time t and is given by n(t) = s(t) + e(t) + i(t) + j(t) + r(t). (1) the model flow diagram is depicted in fig. 1. p s µs ? e µe ? i ke µi �� γi @@r ? r σ1i σ2j d1i d2j µr hhj j µj ? 6 @@r fig. 1: compartmental structure and flow chart for the model the mathematical model for the mers disease (the basic model in [2]), which consists of five rate equations for the dynamics of s,e,i,j and r, is as follows. biomath 7 (2017), 1802277, http://dx.doi.org/10.11145/j.biomath.2018.02.277 page 3 of 18 http://dx.doi.org/10.11145/j.biomath.2018.02.277 nofe al-asuoad, meir shillor, modeling, analysis and simulations of mers outbreak in saudi arabia model 1: find five functions (s,e,i,j,r), defined on [0,t] and values in r+ ∪ {0}, such that for 0 < t ≤ t , the following hold: ds dt = p − s(βi + �eβe + �jβj) n −µs, (2) de dt = s(βi + �eβe + �jβj) n − (k + µ)e, (3) di dt = ke − (γ + σ1 + d1 + µ)i, (4) dj dt = γi − (σ2 + d2 + µ)j, (5) dr dt = σ1i + σ2j −µr, (6) together with the initial conditions s(0) = s0, e(0) = e0, i(0) = i0, (7) j(0) = j0, r(0) = r0. here, we denote by s0 > 0 the initial population before the disease outbreak, and e0,i0,j0 and r0 are nonnegative populations that satisfy (1) at t = 0. it is appropriate, within the context of saudi arabia to assume that that initially s0 = n(0) > 0, and the others vanish, meaning that at first there are only susceptibles in the population. however, for the sake of generality, we assume that the initial populations are nonnegative. the rate of change of the susceptible population s(t) is given in equation (2), where p represents the recruitment rate and is assumed to be a constant. we denote by β (1/day) the effective contact rate. the dimensionless parameters �e and �j are the transmission coefficients from asymptomatic and symptomatic individuals, respectively. thus, the second term on the righthand side of equation (2) describes the rate at which the susceptibles become infected with the virus as a result of contact with asymptomatic, infected, and isolated individuals. the population’s natural death rate is µ (1/day). equation (3) describes the rate of change of the asymptomatic population e(t). these individuals carry the virus but have not yet developed clinical symptoms of mers, which means that they can infect susceptibles unintentionally. following [2], k (1/day) is the rate of development of clinical symptoms in asymptomatic population. the last term on the right-hand side of (3) describes both the mortality rate due to development of clinical symptoms at rate k and the natural mortality rate. we turn now to equations (4) and (5). it is assumed that infectives are isolated at rate γ (1/day). the parameters σ1,σ2 (1/day) denote the recovery rate of symptomatic and isolated populations, while d1,d2 (1/day) are the disease-induced death for symptomatic and isolated populations, respectively. the first term on the right-hand side of the equation (4) represents the asymptomatic individuals who developed clinical symptoms and become infected, while the first term on the right-hand side of the equation (5) represents the isolatedinfected individuals. finally, the rate of change of the recovered population r(t) is given in equation (6) where the first and second terms on the righthand side represent the recover-infected and the recover-isolated individuals, respectively. we note that since there is no data, yet, about possible reinfection of the recovered, we assume that they are permanently immune. we recall that µ denotes the natural death rate. thus, if a person has a a life expectancy of 80 years, then the natural death rate µ is 0.000034 per day. it was assumed that in the absence of disease, the total saudi population was n = p µ = 32 million for p = 1088 people and µ = 0.000034 per day and the total population of riyadh was n = p µ = 5 million for p = 170 people and µ = 0.000034 per day. a full description of the variables, parameters, and the parameters’ values considered in the model can be found in table (i). finally, the cumulative cases of mers up to time t, were obtained from the expression ct(t) = ∫ t 0 (ke(τ)) dτ, (8) with the initial value ct(0) = 0, while the cumulative recovered from the disease up to time t, were obtained from cr(t) = ∫ t 0 (σ1i(τ) + σ2j(τ))) dτ, (9) biomath 7 (2017), 1802277, http://dx.doi.org/10.11145/j.biomath.2018.02.277 page 4 of 18 http://dx.doi.org/10.11145/j.biomath.2018.02.277 nofe al-asuoad, meir shillor, modeling, analysis and simulations of mers outbreak in saudi arabia with the initial value cr(0) = 0. similarly, the cumulative deaths induced by the disease up to time t, were obtained from d(t) = ∫ t 0 (d1i(τ) + d2j(τ))) dτ, (10) with the initial value d(0) = 0. we note, for the sake of completeness, that the following results, in addition to those mentioned and improved in section iii, were established in [2]: the existence and uniqueness, positivity and boundedness of the solutions. indeed, it was found that all the trajectories of the system lie in the set ω = {(s,e,i,j,r) ∈ r5+ : 0 ≤ s + e + i + j + r = n ≤ p µ +n0}, which is invariant and compact. iii. stability analysis we first analyze the stability of the disease-free equilibrium (dfe) and of the endemic equilibrium (ee), both locally and then globally. however, we note that the local stability of the ee has already been done in [2]. these results improve considerably the results there, and also simplify them as they show that the effective reproduction or control number rc controls the stability, and this closes a gap described there. thus, there is no need for the basic stability number r0 that was introduced there. we have, rc = λ1 = �eβ d1 + βk d1d2 + �jβkγ d1d2d3 . (11) here, λ1 is the largest eigenvalue of the jacobian matrix j(p0) given shortly, and d1 = k + µ, d2 = γ + d1 + σ1 + µ, (12) d3 = σ2 + d2 + µ. a. local stability of the dfe we begin with the local stability. it is straightforward to see that the dfe is given by p0 = (s0, 0, 0, 0, 0), and s0 = p/µ. in our previous paper ( [2]), the local stability of the p0 was proved in term of r0. here, we used the routh-hurwitz criterion (see e.g., [21]) to prove the following stability result using rc. our local result is the following. proposition 2: the disease-free equilibrium of the model is locally asymptotically stable when rc < 1 and is unstable when rc > 1. proof: the jacobian matrix of the system at the disease-free equilibrium p0 = (pµ , 0, 0, 0, 0), when γ > 0, is given by j(p0) =   −µ −�eβ −β −�jβ 0 0 −d1 + �eβ β �jβ 0 0 k −d2 0 0 0 0 γ −d3 0 0 0 σ1 σ2 −µ   . the characteristic equation is (λ + µ)2(λ3 + aλ2 + bλ + c) = 0, where a = d1 + d2 + d3 − �eβ = 1 d3d2 ( d3(d3d2 +d 2 2 +kβ) +�jβkγ+d1d2d3(1−rc)) , b = d1d2 + d1d3 + d2d3 −((d2 + d3)�eβ + kβ) = d3d2 + d3 d2 kβ + �jβkγ ( 1 d2 + 1 d3 ) +(d1(d2 + d3))(1 −rc), c = d1d2d3 (1 −rc) . now, since (λ + µ)2 = 0, there are two equal and negative eigenvalues, λ1,2 = −µ. the remaining three eigenvalues are determined from the cubic equation λ3 + aλ2 + bλ + c = 0. it follows from the routh-hurwitz criterion ( [21]) that the solutions of this equation have negative real parts when a > 0,b > 0,c > 0, and ab > biomath 7 (2017), 1802277, http://dx.doi.org/10.11145/j.biomath.2018.02.277 page 5 of 18 http://dx.doi.org/10.11145/j.biomath.2018.02.277 nofe al-asuoad, meir shillor, modeling, analysis and simulations of mers outbreak in saudi arabia c. clearly, a,b,c > 0 when rc < 1. it remains to show that ab > c. we have, ab = 3d1d2d3 (1 −rc) +(1−rc)(d21(1−rc)(d2 +d3)+d1(d 2 2 +d 2 3)) +�jβkγ ( 3+ ( 1 d2 + 1 d3 ) + �jβkγ d3d2 +2d1(1−rc) ) +d3d2(d2 + d3) +kβ ( d3 ( kβ d22 + d3 d2 +2 ) +d1(1−rc) ( 2d3 d2 +1 )) + k2β2�jγ d2 ( 1 d3 + 2 d2 ) . it is seen that ab > c since 3d1d2d3 (1 −rc) > d1d2d3 (1 −rc). thus, ab > c, when rc < 1, and it follow from the routh-hurwitz criterion that all the eigenvalues have negative real parts in this case. we conclude that p0 is locally asymptotically stable. b. global stability of the disease-free equilibrium the global stability of the disease-free equilibrium is shown in the following proposition, based on the construction of an appropriate lyapunov function and the use of lasalle’s invariance principle. proposition 3: the disease-free equilibrium of the model is globally asymptotically stable in r5+ when rc ≤ 1. proof: to show the global stability of the disease-free equilibrium p0, we construct the following lyapunov function l(e,i,j) = ω1e + ω2i + ω3j, in which we only considered the variables representing the infected components of the model, where, ω1 = �ed2d3 + kd3 + �jkγ, ω2 = d1(d3 + �jγ), ω3 = �jd1d2. next, we let λ = βi + �eβe + �jβj n , (13) where n = s + e + i + j + r. calculating the derivative of l along the solution (e(t),i(t),j(t)) of the system (3)–(5), we obtain dl dt = ω1 de dt + ω2 di dt + ω3 dj dt = ω1(sλ−d1e)+ω2(ke−d2i)+ω3(γi−d3j) = ω1sλ−ω1d1e+ω2ke−ω2d2i+ω3γi−ω3d3j = ω1λs −d1d2d3(i + �ee + �jj) = ω1λs −d1d2d3 n(βi + β�ee + β�jj) nβ = ω1λs − λnd1d2d3 β = λnd1d2d3 β ( ω1λsβ λnd1d2d3 − 1 ) = λnd1d2d3 β ( sβ(�ed2d3 +kd3 +�jkγ) nd1d2d3 − 1 ) ≤ λnd1d2d3 β ( β(�ed2d3 +kd3 +�jkγ) d1d2d3 −1 ) = λnd1d2d3 β ( �eβ d1 + βk d1d2 + �jβkγ d1d2d3 −1 ) = λnd1d2d3 β (rc − 1) ≤ 0. therefore, since all the parameters are nonnegative, dl dt ≤ 0 when rc ≤ 1. we note that dl/dt = 0 if and only if e = i = j = 0 i.e., it vanishes only at the disease-free equilibrium. therefore, if we let γ = {(s,e,i,j,r) ∈ r5+ : dl dt ≤ 0}, then the largest compact and invariant set in γ is the singleton {p0}. by lasalle’s invariance principle ( [20]), every solution of the equations (2)-(6), with initial conditions in ωn , approaches p0 as t → ∞, whenever rc ≤ 1. this completes the proof. biomath 7 (2017), 1802277, http://dx.doi.org/10.11145/j.biomath.2018.02.277 page 6 of 18 http://dx.doi.org/10.11145/j.biomath.2018.02.277 nofe al-asuoad, meir shillor, modeling, analysis and simulations of mers outbreak in saudi arabia c. global stability of the endemic equilibrium the global asymptotic stability of the endemic equilibrium when it exists, is also proved by constructing an appropriate lyapunov function and using lasalle’s invariance principle. we note that it follows from the local stability analysis (see [2][proposition 5]) that the ee exists and is unique only when rc > 1. therefore, we deal with the case rc > 1. theorem 4: the the endemic equilibrium p∗ = (s∗,e∗,i∗,j∗,r∗) exists and is globally asymptotically stable when rc > 1, and does not exist when rc < 1. proof: to show the global stability of the endemic equilibrium p∗, we consider change of variables and construct the following lyapunov function l, l(w1,w2,w3,w4,w5) = w∗21 + w ∗2 2 + w ∗2 3 + w ∗2 4 + w ∗2 5 , where, w∗1 = s − p λ + µ , w∗2 = e − λs d1 , w∗3 = i − ke d2 , w∗4 = j − γi d3 , w∗5 = r− σ1i + σ2j µ . we note that l(0, 0, 0, 0, 0) = 0 and l(w1,w2,w3,w4,w5) is positive. calculating the derivative of l about the system (2)–(6), we obtain dl dt =2 ( s − p λ + µ ) ds dt + 2 ( e − λs d1 ) de dt + 2 ( i − ke d2 ) di dt + 2 ( j − γi d3 ) di dt + 2 ( r− σ1i + σ2j µ ) dr dt . then, using the equations, we obtain( s− p λ+µ ) ds dt =ps−(λ+µ)s2− p2 λ+µ +(λ+µ)s p λ+µ ,( e− λs d1 ) de dt = λse−d1e2− (λs)2 d1 +d1e λs d1 ,( i− ke d2 ) di dt =kei−d2i2− (ke)2 d2 +d2i ke d2 , ( j− γi d3 ) dj dt =γij−(d3)j2− (γi)2 d3 +d3j γi d3 ,( r− σ1i+σ2j µ ) dr dt = (σ1i+σ2j)r −µr2− (σ1i+σ2j) 2 µ +µr (σ1i+σ2j) µ . it follows that dl dt =− 2(λ+µ) ( s− p λ+µ )2 −2d1 ( e− λs d1 )2 − 2d2 ( i − ke d2 )2 − 2d3 ( j − γi d3 )2 − 2µ ( r− σ1i + σ2j µ )2 . hence, each term of dl/dt < 0, and thus the largest invariant set at which dl/dt = 0 is the equilibrium point p∗ and it follows from lasalle’s invariant principle [20] that p∗ is globally asymptotically stable. iv. numerical simulations we turn to describe the numerical simulations of the model. we used the same numerical algorithm that was developed and implemented in maple in [2]. then, we run extensive numerical simulations, using the values of the parameters given in table i for the baseline simulations. a number of other sets of parameters were also used, as explained below. the simulations were run for both the city of riyadh and the whole of saudi arabia. those for riyadh were an extension of the simulation in [2], biomath 7 (2017), 1802277, http://dx.doi.org/10.11145/j.biomath.2018.02.277 page 7 of 18 http://dx.doi.org/10.11145/j.biomath.2018.02.277 nofe al-asuoad, meir shillor, modeling, analysis and simulations of mers outbreak in saudi arabia since new cases were found since the last day of simulation reported there, and the additional data has been incorporated into the simulations below. the simulations of saudi arabia were new and motivated by the fact that the model predictions in [2] were very close to what subsequently has been observed in the field. the data currently available for the outbreaks of mers in riyadh and in saudi arabia was from november 4, 2013 till february 1, 2018, a total of 1550 days (∼ 51 months). the parameters p and µ, which were not associated with the disease, were readily available for both places. the other model parameters were obtained by fitting the numerical solutions to the data of the first 865 days of the disease breakout using the optimization routine lsqcurvefit in matlab. this generated the baseline case parameters that are provided in table i. then, we run the simulations for a longer time and observed the model predictions in the following 540 days for which data were available but not used in the parameters fitting. this provided an insight into how well the model predicted the disease dynamics. we would like to point out here, as was noted in [2], that the additional data was found to fit very well into the model, without any need to change the previously fitted parameters, and we describe these results in detail below. we first present the baseline simulations for the whole country, and this is completely a novel addition to the literature. then, we study the disease spread in the city of riyadh, where additional information was provided. finally, we perform a reduced sensitivity analysis for the model with respect to the scaled effective contact number β, which shows that the simulation results are extremely sensitive to its value. we discuss it in section v. a note on the optimization for the model parameters. the optimization program found a number of local minima that provided, for the cases of saudi arabia and riyadh, results in which rc had a value close to one, both below and above one. we chose the baseline simulations in both cases to be those with rc < 1, since these lead to a slightly better fit. however, below we depict simulation results for saudi arabia with either rc = 0.99704 for the baseline case, in which the dfe is stable and attracting, or rc = 1.004, in which the ee is asymptotically stable and the dfe unstable. similarly, for the city of riyadh, we used the baseline case with rc = 0.9928, which is related to a stable and attracting dfe, or rc = 1.0045 that has an unstable dfe and stable and attracting ee. these are directly related to the sensitivity of the model to β and as we explain below, it was found that a change of 0.3% leads to the change in the stability, hence in the disease dynamics. we note in the cases when rc < 1, when there in ee, the decay to the disease free equilibrium to the dfe is slow, over a period of more than a hundred years, and since our interest in this work is only the next few years, we do not depict the long time results. a. baseline simulations – saudi arabia in this subsection, we describe the baseline simulations for the whole of saudi arabia with rc = 0.99704. then, for the sake of completeness, below we describe another set of simulations with rc > 1. both agree well with the data in the short term (1550 days ≈51 months), but differ in long term behavior, as was to be expected, since the baseline is associated with rc < 1 in which there is no ee, while the second set with rc > 1 is associated with stable and attracting ee. however, both values are very close to 1. the daily reported new cases of mers were obtained from the saudi arabian ministry of health website [25]. more specifically, we considered the period of 1550 days from nov. 4, 2013 to february 1, 2018. a nonlinear least square fit, using lsqcurvefit a matlab function contained in the optimization toolboxwas performed to obtain the model parameter values. as was noted above, we fitted the basic model parameters of (2)–(6) to the data from nov. 4, 2013 until mar. 17, 2016 (a period 865 days) using reasonable initial guesses for the parameter values and obtained better estimates of the same parameters from the biomath 7 (2017), 1802277, http://dx.doi.org/10.11145/j.biomath.2018.02.277 page 8 of 18 http://dx.doi.org/10.11145/j.biomath.2018.02.277 nofe al-asuoad, meir shillor, modeling, analysis and simulations of mers outbreak in saudi arabia table i: model baseline parameters for saudi arabia and riyadh parameter description saudi arabia riyadh s susceptible population s(0) = 31,999,990 s(0) = 4,999,990 e asymptomatic population e(0) = 0 e(0) = 0 i symptomatic population i(0) = 10 i(0) = 10 j isolated population j(0) = 0 j(0) = 0 r recoverd population r(0) = 0 r(0) = 0 p recruitment rate of susceptible individuals 1088 170 β effective contact rate 0.1818 0.1222 �e reduction factor in transmission rate by exposed per day 0.3688 0.2956 �j reduction factor in transmission rate by exposed per day 0.1 0.0901 k rate of development of clinical 0.6937 0.1529 symptoms in asymptomatic population µ natural death rate 0.000034 0.000034 d1 disease-induced death for symptomatic population 0.0191 0.0110 d2 disease-induced death for isolated population 0.1260 0.0516 σ1 recovery rate in symptomatic population 0.0336 0.02913 σ2 recovery rate in isolated population 0.2472 0.1098 γ isolation rate 0.1577 0.1335 optimization fit, which are given in table i. the results of model fitting, which is the baseline, are depicted in fig. 2. 0 200 400 600 800 0 200 400 600 800 1000 1200 time in days c u m u la ti v e c a s e s o f m e r s fig. 2: mers model parameters fit to daily reported cumulative new cases data red points obtained from saudi arabian ministry of health website during the first 865 days of the disease outbreak. the solid blue line represents the baseline model prediction. the estimated parameters are provided in table i. we next describe the simulations of the mers model, equations (2)–(6) with the initial conditions s(0) = 31, 999, 990, e(0) = 0,i(0) = 10,j(0) = 0,r(0) = 0. this choice of these initial conditions was made based on the data or the lack of it on nov. 4, 2013 when it became available and when the simulations start. the results of the numerical simulation, depicted in fig. 3, show a very good agreement between the model predictionssmooth colored curvesand the observed data -red dots (red curves on this scale). we emphasize again that the curve fitting was done on the first 865 days and the next 683 days are the model predictions, and they agree with the field date very well, indeed. then, in the figure we depict the model prediction for another three years, or 1006 days, until nov. 4, 2020, which means cumulative results for 2555 days of simulations. in the simulations, fig. 3, the cumulative number of: infected cases is depicted in the top (t), the recovered in the middle (m), and the deaths on the bottom (b). the model predicts, that if the epidemic continues its current trajectory, by nov. 4, 2020 (another 33 months), there will be about 2200 new cases (m), the cumulative recovered will be about 1449 (m), and the cumulative deaths will be around 760, (b). we note that there is an under-reporting issue with the cumulative number of recovered (m) for biomath 7 (2017), 1802277, http://dx.doi.org/10.11145/j.biomath.2018.02.277 page 9 of 18 http://dx.doi.org/10.11145/j.biomath.2018.02.277 nofe al-asuoad, meir shillor, modeling, analysis and simulations of mers outbreak in saudi arabia table ii: model parameter for saudi arabia, the case rc = 1.004. parameter parameters value units p 1088 individual/day β 0.1284 1/day �e 0.0490 �j 0.1077 k 0.2915 1/day µ 0.000034 1/day d1 0.00490 1/day d2 0.1072 1/day σ1 0.0490 1/day σ2 0.1077 1/day γ 0.0542 1/day the first 183 days, since the data was not available, so the number was set as zero and this explains why the whole red graph is below the blue curve. however, by raising the red dots curve to agree with the blue curve on day 183 led to a very good fit on the cumulative recovered, too. although in this case the dfe is stable and attracting, and the disease will die out in about 20 years, we did not show the long term behavior since at this stage it seems not to be very relevant. it is seen that if mers continues in the current trajectory, in the next three years one can expect another 548 cases or so in the whole country. although the number is not large relatively to the size of the population of the whole country, the possible epidemic-like spread of the disease must be taken into account by the authorities. we return to this point in the conclusions section. next, as was noted above, running the optimization program with different initial conditions yielded a number of sets of values, related to local minima of the optimization function. so for the sake of completeness, we present simulation results with somewhat different parameters, provided in table ii in which case rc = 1.004. thus, the ee is stable and attracting, and the disease cannot be eradicated. we note that in the case when the dfe is asymptotically stable we had β = 0.1818 and here fig. 3: saudi baseline simulations of cumulative cases of mers (t) green curve; cumulative number of recovered (m) blue curve; and cumulative number of death (b) brown curve. the red dots are the field data. the run was for 2555 days (∼ 84 months). we have β = 0.1284, which is very interesting and this point is discussed further in section 5, when we study the sensitivity with respect to β. biomath 7 (2017), 1802277, http://dx.doi.org/10.11145/j.biomath.2018.02.277 page 10 of 18 http://dx.doi.org/10.11145/j.biomath.2018.02.277 nofe al-asuoad, meir shillor, modeling, analysis and simulations of mers outbreak in saudi arabia since in this case rc = 1.004, the endemic equilibrium exists and is asymptotically stable. indeed, we found that the ee values p∗ = (s∗,e∗,i∗,j∗,r∗) were p∗ = (31, 772, 111; 27; 51; 13; 113, 994). note that the population of the country was taken as |p∗| = 31, 886, 195. what these numbers mean is that when the disease is close to the ee, one would have on each day about 27 asymptomatics, 51 people with the disease symptoms, 13 isolated, and 113, 994 had just recovered. clearly these numbers, if mers would takes such a turn, pose significant challenges to the authorities and the whole society. the eigenvalues corresponding to the jacobian matrix j(p∗) were found to be − 0.000034, −0.399388, −0.225192 − 0.000017 ± 0.0001127i, indicating that the ee is locally stable and attracting, as was claimed above. we solved the system with the same initial conditions as above. the simulations results are depicted in fig. 4, where the run was for 2555 days (∼ 84 months). we note that the endemic equilibrium is approach in about 80 years. the cumulative infected cases of mers are depicted in the upper left (t), the cumulative number recovered in the upper right (m), and the cumulative number of deaths on the bottom (b). if the epidemic switches to such a trajectory, by the year 2020 (another 33 months), the model predicts a bit over 4000 new cases, the cumulative death will be around 2000, and there will be about 2000 recovered. it seems, comparing the two simulations, that the fit for rc = 0.99704 is better than the one with rc = 1.004, however, visually it is also related to the vertical scales in the figures. the difference in the fit is actually quite small, although the difference in the predictions is larges. the simulations in saudi arabia show that eventually the disease will either disappear, or stabilize. at this stage it is impossible do decide which is fig. 4: saudi simulations with rc = 1.004. cumulative cases of mers (t), cumulative number of recovered (m), and cumulative number of death (b). the red dots are the field data. the run was for 2555 days (∼ 84 months). the case based on the model predictions. however, in either case, in the next few years mers will be spreading, possibly between 620 and 1240 deaths biomath 7 (2017), 1802277, http://dx.doi.org/10.11145/j.biomath.2018.02.277 page 11 of 18 http://dx.doi.org/10.11145/j.biomath.2018.02.277 nofe al-asuoad, meir shillor, modeling, analysis and simulations of mers outbreak in saudi arabia per year. this we would like the authorities to be aware of it. b. baseline simulations riyadh we turn to the simulations of riyadh, and as was noted above, these include more data that became available since our paper [2]. moreover, the parameters are somewhat different than those in the article. indeed, as was the case with saudi arabia, we present simulations with two sets of parameters obtained by the optimization subroutine. the first, which seems to be a better fit with the field data has rc = 0.9928, while the other one, whose fit is slightly worse has rc = 1.0045. therefore, the first simulations are in the case when mers will eventually disappear, while in the second case the endemic equilibrium exists and the disease will persist. in the first case, the values of the parameters obtained from the fit for rc = 0.9928 are given in table i. we solved the mers model (2)–(6) with the initial conditions s(0) = 4, 999, 990, e(0) = 0,i(0) = 10,j(0) = 0,r(0) = 0. the results of numerical simulation, depicted in fig. 5, seem to agree well with the observed data. again, we note that the fit was found using the field data reported in [25] for the first 865 days, and the very good agreement in the following 683 days just supports the model predictions. the cumulative infected cases of mers are depicted on the upper left (t) of fig. 5, the cumulative number of recovered in the upper right (m), and the cumulative number of deaths on the bottom (b). if the epidemic continues at its current trajectory, by 2020 (another 33 months), the model predicts about 840 new cases, the cumulative death will be around 260, and there will be about 580 recovered. as was done above, we now present another simulation in the case when rc = 1.0045. the values of the parameters obtained from the optimization fit are given in table iii. we solved the mers model with the same initial conditions and simulation results are depicted fig. 5: riyadh baseline simulations of cumulative cases of mers (t) green curve; cumulative number of recovered (m) blue curve; and cumulative number of death (b) brown curve. the red dots are the field data. the run was for 2555 days (∼ 84 months). in fig. 6. the model predicts that if the epidemic would proceed on this trajectory, by the year 2020 (another 33 months), there would be about 1440 biomath 7 (2017), 1802277, http://dx.doi.org/10.11145/j.biomath.2018.02.277 page 12 of 18 http://dx.doi.org/10.11145/j.biomath.2018.02.277 nofe al-asuoad, meir shillor, modeling, analysis and simulations of mers outbreak in saudi arabia table iii: model parameter for riyadh, the case rc = 1.0045. parameter parameters value units p 170 individual/day β 0.04 1/day �e 0.7105 �j 0.7107 k 0.3165 1/day µ 0.000034 1/day d1 0.0077 1/day d2 0.0426 1/day σ1 0.0089 1/day σ2 0.0449 1/day γ 0.0316 1/day new cases, the cumulative death will be around 740, and there will be about 680 recovered. when comparing with the predictions of the case with rc < 1 above, it is found that there would be 480 more deaths in the next three years. we conclude that wether the dfe is stable and attracting and eventually the disease will disappear or the ee is stable and attracting and the disease will be active for along time, in the near future, say the next three years, one can expect at least a hundred to two hundred and fifty deaths per year in the city of riyadh. however, these scenarios depend crucially on the effective contact number β. so we turn to discuss this dependence next. v. sensitivity to β this section deals with the considerable sensitivity of the model to the scaled contact number β. first, we describe the mathematical aspects, then we remark on the possible disease control implications of this model sensitivity. we note that we already performed a similar study in [2] for the city of riyadh, and it was found that the system was very sensitive with respect to β. here, we perform it for the whole country of saudi arabia and very similar results are obtained. for the sake of completeness, we do it for the city of riyadh too. this sensitivity may have considerable policy implications. we selected three typical examples that predict very different scenarios, with very close values of fig. 6: riyadh simulations with rc = 1.0045. cumulative cases of mers (t), cumulative number of recovered (m), and cumulative number of death (b). the red dots are the field data. the run was for 2555 days (∼ 84 months). the contact rate β, and we run the simulations for over seven year, actually 2600 days (since the beginning of mers in saudi arabia). we used the biomath 7 (2017), 1802277, http://dx.doi.org/10.11145/j.biomath.2018.02.277 page 13 of 18 http://dx.doi.org/10.11145/j.biomath.2018.02.277 nofe al-asuoad, meir shillor, modeling, analysis and simulations of mers outbreak in saudi arabia fig. 7: saudi arabia. simulation results of the first case, for the cumulative reported cases of mers (t) and the cumulative number of deaths (b). parameter values used are β = 0.1818 – solid curves; β = 0.1824 –dashed; and β = 0.1830 – dash-dot curves. baseline value β = 0.1818 (table i), and the two additional values β = 0.1824 and β = 0.1830. the choice was such that the stability numbers were rc < 1, rc = 1 and rc > 1, respectively, but the first and last with values very close to 1. the simulations are depicted in fig. 7, where the case with β = 0.1824 is depicted in solid curves, β = 0.1824 in dashed curves, and β = 0.1830 in dash-dot curves. the predicted cumulative cases of mers at the end of the seven years (t) were found to be about 2200, 4200 and 9580, respectively. the cumulative deaths were found to be about 760, 1450 and 3280, respectively. a noticeable difference among the three cases was found, indeed, the numbers more than doubled from the first to the second and from the second to the third case, while the difference between consecutive values of β was just 0.3%. this clearly indicates that the model is very sensitive to the value of β. as was pointed out in [2], it is very unlikely that this just a mathematical model. this belief is also supported by the description in the literature on the virulent spread of mers in confined places. we conducted a similar study of seven years for the city of riyadh, with results depicted in fig. 8. we used three scenarios with the values β = 0.1222 (solid lines), β = 0.1231 (dashed lines), and β = 0.1240 (dash-doted lines). the choice was based on the same considerations as for the whole country. the results were about 850, 1740 and 4442 cumulative cases of mers, respectively, shown in fig. 8 (t); and about 265, 540, and 1365 cumulative deaths, respectively, shown in fig. 8 (b). it is seen that changes of 0.7% lead to the doubling of the cumulative cases of mers and of the cumulative numbers of deaths. again, this reinforces the issue of the extreme sensitivity of the model to the scaled contact number. vi. the model errors in this short section we provide a graphic representations of the errors, which are the differences between the data points and the model solution results. these are depicted in figs. 9 to 11. in fig. 9 we show the difference between the cumulative reported cases of mers in saudi arabia, for the 865 days used to find the system coefficients using the optimization routine in matlab. it represents the errors in the results depicted in fig. 2 above. next, fig. 10 presents the errors for saudi arabia in the cumulative numbers of reported cases and the deaths. these are the details presented in fig. 3 above. finally, the errors between the data and model simulations for the city of riyadh in the cumulative numbers of reported cases and the deaths are depicted in fig. 11. these are taken biomath 7 (2017), 1802277, http://dx.doi.org/10.11145/j.biomath.2018.02.277 page 14 of 18 http://dx.doi.org/10.11145/j.biomath.2018.02.277 nofe al-asuoad, meir shillor, modeling, analysis and simulations of mers outbreak in saudi arabia fig. 8: riyadh: simulation results for the cumulative reported cases of mers (left) and the cumulative number of deaths (right). parameter values used are β = 0.1222 – dashed curves, β = 0.1231 – solid curves, and β = 0.124 – dashed dot curves. from the results in fig. 5. it seems that the errors do not have any noticeable pattern, with mean about zero, which reinforces our confidence in the model predictions. vii. conclusions this work deals with the possible trajectories of the mers disease in saudi arabia and in the city of riyadh. it is a continuation of the study in [2] where the basic model was constructed and simulations of the outbreak of mers in riyadh were conducted. our aim in this work was two-fold. first, we established the local stability of the dfe and the 0 100 200 300 400 500 600 700 800 900 time in days -80 -60 -40 -20 0 20 40 60 80 c u m u la ti v e r e p o rt e d c a se s o f m e r s the difference between the data and the solution fig. 9: the difference between the data and the model solution in fig.2 0 200 400 600 800 1000 1200 1400 1600 time in days -150 -100 -50 0 50 100 150 200 c u m u la ti v e r e p o rt e d c a se s o f m e r s the difference between the data and the solution 0 200 400 600 800 1000 1200 1400 1600 time in days -60 -40 -20 0 20 40 60 80 c u m u la ti v e n u m b e r o f d e a th the difference between the data and the solution fig. 10: saudi arabia: the difference between the data and the solution. biomath 7 (2017), 1802277, http://dx.doi.org/10.11145/j.biomath.2018.02.277 page 15 of 18 http://dx.doi.org/10.11145/j.biomath.2018.02.277 nofe al-asuoad, meir shillor, modeling, analysis and simulations of mers outbreak in saudi arabia 0 200 400 600 800 1000 1200 1400 1600 time in days -80 -60 -40 -20 0 20 40 60 80 c u m u la ti v e r e p o rt e d c a se s o f m e r s the difference between the data and the solution 0 200 400 600 800 1000 1200 1400 1600 time in days -60 -50 -40 -30 -20 -10 0 10 20 30 c u m u la ti v e n u m b e r o f d e a th the difference between the data and the solution fig. 11: riyadh: the difference between the data and the solution. global stability of the both the dfe and the ee by using only the effective reproduction number or stability control number rc, and this improves the theoretical results in [2], section 3.3. the analysis of the stability of the model’s equilibrium points can be found in section 3. the second and more important aim was to simulate and predict the disease outbreak in saudi arabia in the very near future, actually, the next two years. we fit the model parameters to a part of the available data, from the first 865 days since the disease was identified, to see how does the model compare with the data for the next 683 days for which data is available, and then used the model to predict the disease outcomes for the next three years, assuming that its trajectory remains the same. it was seen that for both saudi arabia and the city of riyadh, the model predictions for the last 683 days were excellent. nevertheless, considering the future predictions of the model some caution is in order. indeed, the baseline simulations for saudi arabia, which agreed very well with the data for the 1550 days since the disease was identified, were with the control number rc = 0.99704. however, another parameter fit, which was almost as good, was with rc = 1.004. in the first case there was no endemic equilibrium (ee), while in the second case the ee was found to be asymptotically stable, and these explain why the predictions for the next three years somewhat diverge. in the first case the model predictions for the next three years, until nov. 4, 2020, there will be about 2200 new cases, the cumulative recovered will be about 1449, and the cumulative deaths will be around 760, fig. 3. in the second case the model predictions were over 4000 new cases, the cumulative death will be around 2000, and there will be about 2000 recovered. the difference in the predictions is noticeable, although in a country of 32 million population these do not seem to be too divergent. however, at this stage of the research it is not clear which scenario will play itself in the long run, the one with rc = 0.99704 in which the disease dies (although in 20 years or so) or the one with rc = 1.004, in which the disease is endemic and lingers for a long time. nevertheless, both predictions seem very reasonable and only time will tell which would be closer to field observations. we stress again that these observations depend crucially on the assumption that the disease continues its current trend. similar observations were found for riyadh, provided in section 5. one of the main mathematical features of the model, already pointed out in [2], is its considerable sensitivity to the value of the scaled contact number β. the number measures the probability that one contact between a susceptible individual and a sick one results in infection, and therefore it includes the rate at which people meet each other. indeed, as was seen in section 5, figs. 7 biomath 7 (2017), 1802277, http://dx.doi.org/10.11145/j.biomath.2018.02.277 page 16 of 18 http://dx.doi.org/10.11145/j.biomath.2018.02.277 nofe al-asuoad, meir shillor, modeling, analysis and simulations of mers outbreak in saudi arabia and 8, for saudi arabia and for riyadh, changes of 0.3% and 0.7%, respectively, led to quite different predictions, increases of more than 100%. this sensitivity may have considerable policy implications. in settings where many people congregate and contact is high, the value of may be β higher with possibly severe outbreaks of mers. acknowledgement. the authors would like to thank the reviewers for their comments that improved the presentation of the work, and made it easier to read. references [1] n al-asuoad, “mathematical 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http://dx.doi.org/10.1016/s0140-6736(15)60454-8 http://dx.doi.org/10.1016/s0140-6736(15)60454-8 http://dx.doi.org/10.11145/j.biomath.2018.02.277 introduction the model stability analysis local stability of the dfe global stability of the disease-free equilibrium global stability of the endemic equilibrium numerical simulations baseline simulations – saudi arabia baseline simulations riyadh sensitivity to the model errors conclusions references www.biomathforum.org/biomath/index.php/biomath review article one and two-phase cell cycle models katarzyna pichór∗, ryszard rudnicki† ∗institute of mathematics, university of silesia katowice, poland katarzyna.pichor@us.edu.pl †institute of mathematics, polish academy of sciences katowice, poland rudnicki@us.edu.pl received: 19 february 2019, accepted: 26 may 2019, published: 1 june 2019 abstract—in this review paper we present deterministic and stochastic one and two-phase models of the cell cycle. the deterministic models are given by partial differential equations of the first order with time delay and space variable retardation. the stochastic models are given by stochastic iterations or by piecewise deterministic markov processes. we study asymptotic stability and sweeping of stochastic semigroups which describe the evolution of densities of these processes. we also present some results concerning chaotic behaviour of models and relations between different types of models. keywords-cell cycle; transport equation; stochastic operator and semigroup; asymptotic stability; markov process i. introduction the cell cycle is a series of events that take place in a cell leading to its replication. it is regulated by a complex network of protein interactions. for example, a relatively simple mathematical model of mammalian cell-cycle control consists of eighteen differential equations [32]. usually the cell cycle is divided into four phases [2], [18], [31]. the first one is the growth phase g1 with synthesis of various enzymes. the duration of the phase g1 is highly variable even for cells from one species. the dna synthesis takes place in the second phase s. in the third phase g2 a significant protein synthesis occurs, which is required during the process of mitosis. the last phase m consists of nuclear division and cytoplasmic division. we consider also g0 phase (quiescence). a cell can enter the g0 phase from g1 and may remain quiescent for a long period of time, possibly indefinitely, or after some period of time it can go back to the g1 phase. the schematic model of the cell cycle is given in fig. 1. there are several mathematical models of the cell cycle. one can consider four-phase models [6], but the most popular are one or two-phase models. in one-phase models, we put together phases g1, s, g2, and m and neglect the phase g0. the second category are two-phase models. biologists used to divide the whole cycle into interphase, which consists of g1, s and g2, and the mitotic phase m. from a mathematical point of view it is copyright: c©2019 pichór et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: katarzyna pichór, ryszard rudnicki, one and two-phase cell cycle models, biomath 8 (2019), 1905261, http://dx.doi.org/10.11145/j.biomath.2019.05.261 page 1 of 15 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2019.05.261 katarzyna pichór, ryszard rudnicki, one and two-phase cell cycle models g0g1a s g2 b m fig. 1. schematic model of the cell cycle. better to divide the cell cycle into the resting (or growth) phase a = g1 with a random duration ta and the proliferating phase b which consists of the phases s, g2 and m, and has an almost constant duration tb. in these models the phase g0 is also neglected. there are also two-phase models with cells in the proliferating state (phases g1, s, g2 and m) and in the quiescent state [4], [41]. the mathematical models of the cell cycle are based on the concept of maturity, also called the physiological age. the maturity can be the size of a cell, its volume or dna content. the core of the theory was formulated in the late sixties [24], [43]. a lot of new models appeared in the eighties and we can divide them into two groups. the first group contains discrete-time models (generational models) which describe the relation between the initial maturity of mother and daughter cells. in this group we have one-phase models [20] and two-phase models [57], [58]. the second group is formed by continuoustime models characterizing the time evolution of distribution of cell maturity. we consider two types of continuous-time one-phase models: with the division of a cell when the cell maturity has a given level [24], [42], [43], or with division at a random maturity (including size structured models) [5], [7], [11], [15], [29], [34], [52], [59]. the second type consists of two-phase models [1], [8], [27], [39], [56]. the above mentioned models describe the distribution of maturity in the whole population. we also investigate models given by piecewise deterministic markov processes [54], which describe the evolution of maturity of consecutive descendants of a single cell [22], [29], [38]. such models seem to be the most suitable for the description of the cell cycle because they do not include environmental components. this paper provides an introduction to one and two-cycle models, with particular emphasis on models given by piecewise deterministic markov processes. we also formulate some results concerning their long-time behaviour and compare asymptotic properties of discrete and continuous time models. the evolution of distribution of maturity in the discrete time models and in models given by piecewise deterministic markov processes is described by stochastic (markov) operators and semigroups [21], [47]. the results concerning asymptotic stability and sweeping of stochastic semigroups are based on papers [35], [36], [37]. we also present results concerning chaotic properties of some maturity structured models [49], [50]. ii. asymptotic properties of stochastic operators and semigroups now we introduce the notion of a stochastic operator and a stochastic semigroup. then we present some results concerning asymptotic stability, sweeping and foguel alternative for stochastic semigroups. these results will be applied to models of cell cycle presented in the next sections. let a triple (x, σ,µ) be a σ-finite measure space. denote by d the subset of the space l1 = l1(x, σ,µ) which consists of all densities d = {f ∈ l1 : f ≥ 0, ‖f‖ = 1}. a linear operator p : l1 → l1 is called stochastic if p(d) ⊆ d. a family {p(t)}t≥0 of linear operators on l1 is called a stochastic semigroup if it is a strongly continuous semigroup and all operators p(t) are stochastic. now, we introduce some notions which characterize the asymptotic behaviour of iterates of stochastic operators pn, n = 0, 1, 2, . . . , and stochastic semigroups {p(t)}t≥0. the iterates of stochastic operators form a discrete-time semigroup and we can also biomath 8 (2019), 1905261, http://dx.doi.org/10.11145/j.biomath.2019.05.261 page 2 of 15 http://dx.doi.org/10.11145/j.biomath.2019.05.261 katarzyna pichór, ryszard rudnicki, one and two-phase cell cycle models use the notation p(t) = pt for their powers so that we formulate most of the definitions and results for both types of semigroups without distinguishing them. a stochastic semigroup {p(t)}t≥0 is asymptotically stable if there exists a density f∗ such that lim t→∞ ‖p(t)f −f∗‖ = 0 for f ∈ d. (1) from (1) it follows immediately that f∗ is invariant with respect to {p(t)}t≥0, i.e. p(t)f∗ = f∗ for each t ≥ 0. a stochastic semigroup {p(t)}t≥0 is called sweeping with respect to a set b ∈ σ if for every f ∈ d lim t→∞ ∫ b p(t)f(x) µ(dx) = 0. in order to formulate a result concerning asymptotic stability of a stochastic semigroup we need the following definition. a stochastic semigroup {p(t)}t≥0 is called partially integral if there exists a measurable function q(t, ·, ·) : x ×x → [0,∞) called kernel such that∫ x ∫ x q(t,x,y) µ(dx) µ(dy) > 0 for some t > 0 and p(t)f(y) ≥ ∫ x q(t,x,y)f(x) µ(dx) for f ∈ d. (2) theorem 2.1 ([35]): let {p(t)}t≥0 be a continuous-time partially integral stochastic semigroup. assume that the semigroup {p(t)}t≥0 has a unique invariant density f∗. if f∗ > 0 a.e., then the semigroup {p(t)}t≥0 is asymptotically stable. the next result concerns the foguel alternative [21], that is, when a stochastic semigroup {p(t)}t≥0 is asymptotically stable or sweeping from all compact sets. we assume additionally that (x,ρ) is a separable metric space, σ = b(x) is the σ-algebra of borel subsets of x and that the semigroup {p(t)}t≥0 is partially integral with the kernel q which satisfies the following condition: (k) for every x0 ∈ x there exist ε > 0, t > 0 and a measurable function η ≥ 0 such that ∫ η(y) µ(dy) > 0 and q(t,x,y) ≥ η(y)1b(x0,ε)(x) for x,y ∈ x, (3) where 1b(x0,ε) is the characteristic function of b(x0,ε) = {x ∈ x : ρ(x,x0) < ε}. we define condition (k) for a stochastic operator p in the same way, remembering the notation p(t) = pt. condition (k) is satisfied if, for example, for every point x ∈ x there exist t > 0 and y ∈ x such that the kernel q(t, ·, ·) is continuous in a neighbourhood of (x,y) and q(t,x,y) > 0. we need an auxiliary definition. we say that a stochastic semigroup {p(t)}t≥0 overlaps supports if for every f,g ∈ d there exists t > 0 such that µ(supp p(t)f ∩ supp p(t)g) > 0. the support of any measurable function f is defined up to a set of measure zero by the formula supp f = {x ∈ x : f(x) 6= 0}. theorem 2.2: assume that {p(t)}t≥0 satisfies (k) and overlaps supports. then {p(t)}t≥0 is sweeping or {p(t)}t≥0 has an invariant density f∗ with a support a and there exists a positive linear functional α defined on l1(x, σ,µ) such that (i) for every f ∈ l1(x, σ,µ) we have lim t→∞ ‖1ap(t)f −α(f)f∗‖ = 0, (4) (ii) if y = x\a, then for every f ∈ l1(x, σ,µ) and for every compact set f we have lim t→∞ ∫ f∩y p(t)f(x) µ(dx) = 0. (5) in particular, if {p(t)}t≥0 has an invariant density f∗ with the support a and x \ a is a subset of a compact set, then {p(t)}t≥0 is asymptotically stable. the proof of theorem 2.2 is based on theorems on asymptotic decomposition of stochastic operators [36, theorem 1] and stochastic semigroups [36, theorem 2] and it is given in [38]. another consequence of [36, theorem 2] it the following. biomath 8 (2019), 1905261, http://dx.doi.org/10.11145/j.biomath.2019.05.261 page 3 of 15 http://dx.doi.org/10.11145/j.biomath.2019.05.261 katarzyna pichór, ryszard rudnicki, one and two-phase cell cycle models corollary 1: assume that a continuous-time stochastic semigroup {p(t)}t≥0 satisfies condition (k) and has no invariant densities. then {p(t)}t≥0 is sweeping from compact sets. iii. rubinow-type models in all models in this section we consider a sequence of consecutive descendants of a single cell in a single line. one of the oldest models of cell cycle, introduced by rubinow [43], is based on the concept of maturity and maturation velocity. maturity is a real variable m from the interval [0, 1] which describes the position of a cell in the cell cycle. a new born cell has maturity 0 and a cell splits at maturity 1. in rubinow’s model m grows according to the equation m′ = v, where the maturation velocity v can depend on m and also on other factors such as time, the size of the population, temperature, light, environmental nutrients, ph, etc. if we neglect environmental factors, resource limitations, and stochastic variation, then we can assume that v depends only on m and all cells have identical cell cycles, in particular, they have the same cell cycle length l. however, experimental observations concerning cell populations, cultured under identical conditions for each member, revealed high variability of l in the population. it means that the population is heterogeneous with respect to cell maturation velocities and therefore mathematical models of the cell cycle should take into account maturation velocities. a model of this type was proposed by lebowitz and rubinow [24]. in their model the cell cycle is determined by its maturation velocity v, which is fixed at the birth of the cell and constant during the cell cycle. the relation between the maturation velocities of mother’s v and daughter’s cells v̄ is given by a probability transition function p(v,dv̄). denote by tn the time, when a cell from the nth-generation splits. then the maturity and the maturation velocity of a cell from the nth-generation are described by a stochastic process ξ(t) = (m(t),v(t)), t ∈ [tn−1, tn). the process ξ(t), t ≥ 0, has jumps at points t0, t1, t2, . . . and between jumps the pair (m(t),v(t)) satisfies the following system of differential equations{ m′(t) = v(t), v′(t) = 0. (6) at jump points we have m(tn) = 0 and p(v(tn) ∈ b |v(t−n ) = v) = p(v,b) for each n ∈ n and each borel subset b of (0,∞). since v(t) is constant in the interval (tn−1, tn), we have tn − tn−1 = 1/v(tn−1). it is not difficult to check that ξ(t), t ≥ 0, is a homogeneous markov process. observe that there is an increasing sequence of random times (tn), called jump times, such that the sample paths (trajectories) of ξ(t) are defined in a deterministic way in each interval (tn, tn+1). a process which has such properties is called piecewise deterministic. the lebowitz and rubinow model can be identified with a one-dimensional stochastic billiard on the interval [0, 1]. namely, consider a particle moving in the interval [0, 1] with a constant velocity. we assume that when the particle hits the boundary points 0 and 1, it changes its velocity according to the probability measures p0(−v,b) = p(v,b) and p1(v,−b) = p(v,b), respectively, where v > 0 and b is a borel subset of (0,∞). observe that the pdmp defined in the lebowitz–rubinow model is given by ξ(t) = { (m(t),v(t)), if v(t) > 0, (1 −m(t),−v(t)), if v(t) < 0, where m(t) and v(t) represent position and velocity of the moving particle at time t. asymptotic properties of the general onedimensional stochastic billiard were studied in [30]. based on that paper we briefly present properties of the lebowitz–rubinow model: for v, v̄ ∈ (0, 1], p(v,dv̄) = q(v, v̄) dv̄, where∫ 1 0 q(v, v̄) dv̄ = 1. the process ξ(t) = (m(t),v(t)) induces a stochastic semigroup {p(t)}t≥0 on the space l1(x, σ,µ), where x = (0, 1]2, σ = b(x), and dµ = dm × dv. the infinitesimal generator a of the semigroup biomath 8 (2019), 1905261, http://dx.doi.org/10.11145/j.biomath.2019.05.261 page 4 of 15 http://dx.doi.org/10.11145/j.biomath.2019.05.261 katarzyna pichór, ryszard rudnicki, one and two-phase cell cycle models {p(t)}t≥0 is given by the formula af(m,v) = −v ∂f ∂m (m,v). the functions f from the domain of the operator a satisfy the boundary condition: v̄f(0, v̄) = ∫ 1 0 vq(v, v̄)f(1,v) dv. observe that if this semigroup has an invariant density f∗, then af∗ = 0. thus f∗ does not depend on m. set h(v) = vf∗(v). then h is a fixed point of the stochastic operator k on l1[0, 1] given by kh(v̄) = ∫ 1 0 q(v, v̄)h(v) dv. (7) since the boundary condition contains a kernel operator, one can check that the semigroup {p(t)}t≥0 is partially integral. we assume that k is irreducible which is equivalent to say that∫ (0,1]\b (∫ b q(v, v̄) dv ) dv̄ > 0 for each measurable set b ⊆ (0, 1] of the lebesgue measure 0 < |b| < 1, see [55, p. 334]. from irreducibility it follows that if an invariant density f∗ exists then it is unique and f∗(v) > 0 for va.e. the question of the existence of an invariant density is nontrivial. if for example we assume that there exist c > 0 and γ > 0 such that q(v, v̄) ≤ c|v̄|γ for v, v̄ ∈ (0, 1], (8) then an invariant density exists [30]. it means that irreducibility of k and condition (8) imply asymptotic stability of the semigroup. now we consider the case when the semigroup {p(t)}t≥0 has no invariant density. assume that the kernel q is continuous and bounded. then the semigroup satisfies condition (k) and from corollary 1 it follows that the semigroup is sweeping from compact sets. it means that lim t→∞ ∫ ε 0 ∫ 1 0 p(t)f(m,v) dmdv = 1 (9) for every density f and every ε > 0. the sweeping property in this case means that the length of the cell cycle tends to infinity in the sense of distribution, which is not so a rare phenomenon in tissue cells. for example if q ≡ 1, then the semigroup has no invariant density, and consequently is sweeping from compact sets. it is interesting that in this example we have p(t)f(m,v) ∼ c |v| (log t)−1 as t →∞ for v ≥ ε and m ∈ [0, 1], where c is some constant. the lebowitz–rubinow model is a special case of the rotenberg model [42]. in the rotenberg model the maturation velocity can also change during the cell cycle. a new born cell inherits the initial maturation velocity from its mother according to a transition probability p(v,dv̄), as in the lebowitz–rubinow model. during the cell cycle it can change its maturation velocity with intensity ϕ(m,v), i.e., a cell with parameters (m,v) can change the maturation velocity in a small time interval of length ∆t with probability ϕ(m,v)∆t + o(∆t). we suppose that if (m,v) is the state of the cell at the moment of changing of the maturation velocity, then a new maturation velocity is drawn from a distribution p(m,v,dv̄). the process ξ(t) = (m(t),v(t)) describing consecutive descendants of a single cell is a pdmp which has jumps when cells split and random jumps during their cell cycles. between jumps the pair (m(t),v(t)) satisfies system (6). if a jump is at the moment of division, then it is given by the same formula as in the lebowitz–rubinow model. if a jump is during the cell cycle, then m(tn) = m(t − n ) and p(v(tn)∈b |m(t−n ) =m, v(t − n ) =v) =p(m,v,b) for each borel subset b of (0,∞). sample graphs of maturity in the rubinow, lebowitz-rubinow and rotenberg models are presented in fig. 2. iv. bell-anderson-type models now we consider one-phase models which are based on two main assumptions: the maturity m grows with velocity g(m) and a cell can splits at a rate ϕ(m), i.e. a cell with maturity m divides during a small time interval of length ∆t with biomath 8 (2019), 1905261, http://dx.doi.org/10.11145/j.biomath.2019.05.261 page 5 of 15 http://dx.doi.org/10.11145/j.biomath.2019.05.261 katarzyna pichór, ryszard rudnicki, one and two-phase cell cycle models 1 m t a) 1 m t b) 1 m t c) fig. 2. sample graphs of maturity in the models: a) rubinow, b) lebowitz-rubinow, c) rotenberg. probability ∆p = ϕ(m)∆t + o(∆t). the maturity of the daughter cell m is a function of the maturity of the mother cell m, i.e. m = h(m). we assume that g : [0,∞) → (0,∞) is a c1 function which increases sublinearly, ϕ: [0,∞) → [0,∞) is a continuous function, and h is a positive c1 function such that h′(m) > 0. for example if m is the volume of a cell, then h(m) = m/2. first we consider a discrete-time model which describes the relation between the initial maturities of the mother and daughter cells. since ∆p = ϕ(m) ∆t+o(∆t), ∆m = g(m) ∆t+o(∆t), we have ∆p = ϕ(m) g(m) ∆m + o(∆m), while g(m) = exp { − ∫ m m0 ϕ(s) g(s) ds } is the survival function, where m0 is the initial cell maturity. let q(m) = m∫ 0 ϕ(r) g(r) dr. then g(m) = eq(m0)−q(m). we assume that limm→∞q(m) = ∞, which guaranties that each cell splits with probability one. let ξ be the maturity of the cell at the moment of division and let η be a positive random variable with density e−x. since p(ξ>m) =eq(m0)−q(m) = p(η>q(m)−q(m0)), we have p(q(ξ) > q(m)) = p(q(m0) + η > q(m)) and therefore the random variables q−1(q(m0) + η) and ξ have the same distribution. it is easy to check that the random variable ξ has the density λ′(m)q′(λ(m))eq(m0)−q(λ(m)) for m ≥ h(m0), where λ(m) = h−1(m). if we assume that the distribution of the initial maturity of mother cells has a density f, from the above formula we infer that the initial maturity of the daughter cells has the density pf(m) = ∫ λ(m) 0 λ′(m)q′(λ(m))eq(y)−q(λ(m))f(y)dy. (10) then p is a stochastic operator on the space l1[0,∞). in the continuous version of the above model we consider a sequence of consecutive descendants of a single cell. the maturity of cells can be described by the following homogeneous piecewise deterministic markov process ξ(t). let tn be the time when a cell from the nth-generation divides. if tn−1 ≤ t < tn, then the maturity satisfies the equation ξ′(t) = g(ξ(t)). the process ξ(t) has a jump at the moment of the division of the cell: ξ(tn) = h(ξ(t − n )). if ξ(tn−1) = m0, then the cumulative distribution function f of tn − tn−1 is given by f(t) = 1 −g(π(t,m0)), where π(t,m0) is the solution at time t of the equation m′ = g(m) with the initial condition m(0) = m0. maturity structured models have been investigated in many papers. usually, we are interested in the behaviour of the density of the maturity biomath 8 (2019), 1905261, http://dx.doi.org/10.11145/j.biomath.2019.05.261 page 6 of 15 http://dx.doi.org/10.11145/j.biomath.2019.05.261 katarzyna pichór, ryszard rudnicki, one and two-phase cell cycle models u(t,m). we should underline that u(t, ·) is not density in our probabilistic sense because the integral of u with respect to m may be different than one. since in these models we consider the whole population, beside the rate of division ϕ there is also the rate of death µ. these models coincide with the model given by the process ξ(t), when ϕ = µ and the rate of division in the definition of ξ(t) equals 1 2 ϕ. we briefly present some examples of these models. the basic model was proposed by bell and anderson [7]. they assume that the size of the cell is a number m ∈ (mmin, 1), 0 < mmin < 1. in order not to cross 1, it is assumed that ∫ 1 mmin ϕ(m) = ∞. this model was studied and generalized, for example, in [11], [15], [59]. versions of this model with unequal division were investigated in [3], [16], [19], [52]. the papers [5], [33] are devoted to a general model that includes also age structure. there were also considered versions with unbounded growth of cells [29], [53]. v. two-phase model we start with a short biological description of two phase-cell cycle models. the cell cycle is divided into the resting and proliferating phase. the duration of the resting phase is random variable ta which depends on the maturity of a cell. the duration tb of the proliferating phase is almost constant. therefore, we assume that tb = τ, where τ is a positive constant. a cell can move from the resting phase to the proliferating phase with rate ϕ(m). we assume that cells age with unitary velocity and mature with a velocity g1(m) in the resting phase and with a velocity g2(m) in the proliferating phase. the age variable a in the proliferating phase is assumed to range from a = 0 at the moment of entering the proliferating phase to a = τ at the point of cytokinesis. the maturity of the daughter cell m is a function of the maturity of the mother cell m, i.e. m = h(m) (see fig. 3). we consider a version of the model studied in [38]. now we collect the assumptions concerning the model: 1 a m aτ m a m m h (1) (2) (3) (4) fig. 3. evolution of maturity of a mother cell: (1) – resting phase; (2) – proliferating phase and a daughter cell: (3) – resting phase; (4) – proliferating phase. (m1) ϕ is a continuous function such that ϕ(m) = 0 for m ≤ mp and ϕ(m) > 0 for m > mp , where mp > 0 is the minimum cell size of which it can enter the proliferating phase, (m2) h: [mp ,∞) → [0,∞) is a c1 function such that h′(m) > 0, (m3) g1 : [0,∞) → (0,∞) and g2 : [mp ,∞) → (0,∞) are c1 functions which increase sublinearly, (m4) lim m→∞ m∫ 0 ϕ(r) g1(r) dr = ∞. denote by πi(t,m0) the solution of the equation m′(t) = gi(m(t)), i = 1, 2, (11) with the initial condition m(0) = m0 ≥ 0. now, we introduce two auxiliary functions. let λ(m) =π2(−τ,h−1(m)) and q(m) = m∫ 0 ϕ(r) g1(r) dr. according to (m4) lim m→∞ q(m) = ∞, which guaranties that each cell enters the proliferating phase with probability one. under this notation the initial maturity of the daughter cells has density pf, if f is the analogous density of the mother cells, where p is the stochastic operator given by (10). now we present some results concerning the operator p . theorem 5.1: the operator p satisfies the foguel alternative, i.e. p is asymptotically stable or sweeping from compact sets. theorem 5.2: let α(m) = q(λ(m)) − q(m). the following conditions hold: biomath 8 (2019), 1905261, http://dx.doi.org/10.11145/j.biomath.2019.05.261 page 7 of 15 http://dx.doi.org/10.11145/j.biomath.2019.05.261 katarzyna pichór, ryszard rudnicki, one and two-phase cell cycle models (a) if lim inf m→∞ α(m) > 1, then p is asymptotically stable, (b) if α(m) ≤ 1 for sufficiently large m, then p is sweeping from each bounded interval, (c) if inf α(m) > −∞, then the operator p is completely mixing, i.e. lim n→∞ ‖pnf −png‖ = 0 for f,g ∈ d. theorem 5.1 was proved in [38]. the results from theorem 5.2 were proved, respectively, (a) in [14], (b) in [25], and (c) in [45]. the sweeping property in this case means that the maturity of cells statistically tends to infinity, which also means that the length of the cell cycle tends to infinity. now we consider a continuous version of the model. the cell cycle can be described as a piecewise deterministic markov process. we consider a sequence of consecutive descendants of a single cell. let sn be the time, when a cell from the nthgeneration enters a resting phase and tn = sn + τ be the time of its division. if tn−1 ≤ t < tn then the state ξ(t) = (a(t),m(t), i(t)) of the n-th cell is described by the age a(t), maturity m(t) and the index i(t), where i = 1 if the cell is in the resting phase and i = 2 if it is in the proliferating phase. random moments t0,s1, t1,s2, t2, . . . are called jump times. between jump times the parameters change according to the following system of equations:   a′(t) = 1, m′(t) = gi(t)(m(t)), i′(t) = 0. (12) the process ξ(t) changes at the jump points according to the following rules: a(sn) = 0, m(sn) = m(s − n ), i(sn) = 2, a(tn) = 0, m(tn) = h(m(t − n )), i(tn) = 1. if m(tn−1) = m0, then the cumulative distribution function f of sn − tn−1 is given by f(t) = 1 −eq(m0)−q(π1(t,m0)). (13) then ξ(t) is a homogeneous markov process. if the distribution of ξ(0) is given by a density 2 a m m = π1(a,0) aτ m mp m = π2(a,mp ) 1-phase 2-phase fig. 4. the set x function f(0,a,m,i), i.e. a measurable function of (a,m,i) such that p(ξ(0) ∈ a× i) = ∫∫ a f(0,a,m,i) dadm for any borel set a and i = 1, 2, then ξ(t) has a density f(t,a,m,i). having a homogeneous markov process ξ(t) with the property that if the random variable ξ(0) has a density f0, then ξ(t) has a density ft, we can define a stochastic semigroup {p(t)}t≥0 corresponding to ξ(t) by p(t)f0 = ft. the proper choice of the space x of the values of the process ξ(t) plays an important role in investigations of the process and the semigroup {p(t)}t≥0. we define x = x1 ∪x2, where x1 = {(a,m, 1) : m ≥ π1(a, 0), a ≥ 0}, x2 = {(a,m, 2) : m ≥ π2(a,mp), a ∈ [0,τ]}, σ = b(x) and µ is the product of the twodimensional lebesgue measure and the counting measure on the set {1, 2} (see fig. 4). we need two additional assumptions: ψ(m) = h(π2(τ,m)) < m for m ≥ mp (14) and h′(π2(τ,m̄))g2(π2(τ,m̄))g1(m̄) 6= g1(h(π2(τ,m̄)))g2(m̄) (15) for some m̄ > mp . condition (14) guarantees that with a positive probability each cell will have a descendant with biomath 8 (2019), 1905261, http://dx.doi.org/10.11145/j.biomath.2019.05.261 page 8 of 15 http://dx.doi.org/10.11145/j.biomath.2019.05.261 katarzyna pichór, ryszard rudnicki, one and two-phase cell cycle models a sufficiently small maturity in some generation and thanks to that property each two states from the interior of the set x communicate. condition (15) seems to be technical but if h′(π2(τ,m))g2(π2(τ,m))g1(m) = g1(h(π2(τ,m)))g2(m) for all m ≥ mp , then all descendants of a single cell in the same generation have the same maturity at a given time t. it means that the cells have synchronous growth and we cannot expect the model to be asymptotically stable. in particular, if g1 ≡ g2 and h(m) = m/2, then (15) reduces to 2g2(m) 6= g2(2m) for some m > π2(τ,mp ). a similar condition appears in many papers concerning size-structured models [5], [11], [15], [52], [54]. the following results are proved in [38]. theorem 5.3: the semigroup {p(t)}t≥0 satisfies the foguel alternative, i.e. {p(t)}t≥0 is asymptotically stable or sweeping from compact sets. the proof of this result is based on theorem 2.1 and corollary 1. theorem 5.4: if the operator p given by (10) has an invariant density and ϕ(m) ≥ ε > 0 for sufficiently large m, then the semigroup {p(t)}t≥0 is asymptotically stable. if p has no invariant density and ϕ is a bounded function, then the semigroup {p(t)}t≥0 is sweeping from compact sets. according to theorem 5.2 and theorem 5.4 we have the following alternative. corollary 2: if lim inf m→∞ (q(λ(m)) −q(m)) > 1 and there is ε > 0 such that ϕ(m) ≥ ε for sufficiently large m, then the semigroup {p(t)}t≥0 is asymptotically stable. if q(λ(m)) −q(m) ≤ 1 for sufficiently large m and ϕ is bounded, then the semigroup {p(t)}t≥0 is sweeping. remark 1: one can give an example of the operator p which is asymptotically stable but the semigroup {p(t)}t≥0 is sweeping. such a case can happen when limm→∞ϕ(m) = 0. the explanation of this phenomenon is that in this example the rate of entering the proliferating phase is very small for large m. then the mean length of the resting phase can be large and more and more mature cells dominate the population as t → ∞. in [48], [60] one can find the comparison of the discrete time model presented here with a two-phase model of maturity structured population considered in the paper [27] and briefly presented in the next section. vi. two-phase population models now we recall a two-phase maturity structured model of a cellular population from the paper [27]. the model is based on the same biological assumption as that of section v, but we include also the mortality rates µr(m) and µp(m) in both phases. denote by r(t,m,a) and p(t,m,a) the maturity-age distribution of resting and proliferating cells, respectively. we also assume that the rate of entering the proliferating phase ϕ depends on m and the total number of cells in the resting phase r̄(t) = ∫ r(t,m) dm, where r(t,m) =∫ r(t,m,a) da. the time evolution of p and r is described by the following system of equations: ∂r ∂t + ∂r ∂a + ∂(g1(m)r) ∂m = −(µr(m) + ϕ(r̄,m))r, ∂p ∂t + ∂p ∂a + ∂(g2(m)p) ∂m = −µp(m)p with the boundary conditions r(t, 0,m) = 2(h−1(m))′p(t,τ,h−1(m)), p(t, 0,m) = ϕ(m,r̄(t))r(t,m). additionally, we assume that µp, µr and ϕ do not depend on m. integrating the above equations over the age variable a and using the boundary conditions we obtain ∂r ∂t + ∂(g1r) ∂m = −(µr + ϕ(r̄))r + 2e−µpτϕ(r̄(t− τ))λ′(m)r(t− τ,λ(m)). (16) we recall that λ(m) = π2(−τ,h−1(m)) is the maturity of the mother cell at the moment of entering proliferating phase, if the new born daughter cell has maturity m. integrating both sides of (16) over m we obtain r̄′(t) =−(µr+ϕ(r̄))r̄+2e−µpτϕ(r̄(t−τ))r̄(t−τ). (17) biomath 8 (2019), 1905261, http://dx.doi.org/10.11145/j.biomath.2019.05.261 page 9 of 15 http://dx.doi.org/10.11145/j.biomath.2019.05.261 katarzyna pichór, ryszard rudnicki, one and two-phase cell cycle models the following result is proved in [27]. theorem 6.1: assume that equation (17) has a constant solution r̄0 > 0 and r̄0 is globally asymptotically stable. if (µr + ϕ(r̄0)) log(h −1)′(0) < g′1(0) (18) then there exists a stationary solution r0(m) of equation (16) and for every solution r(t,m) of it we have lim t→∞ ∫ |r(t,m) −r0(m)|dm = 0. (19) condition (18) has an interesting biological interpretation. it shows that the stability of the population depends on the dynamics of immature (small) cells. the term (h−1)′(0) describes the relation between the maturity of the mother and daughter cells. if m is the maturity of a small mother cell at the moment of entering the proliferating phase, then the maturity of a new-born daughter cell is m/(h−1)′(0). the term c = µr +ϕ(r̄0) is the rate of leaving of the resting phase (by being lost or by entering the proliferating phase). since g′1(0) is the rate at which small cells mature, condition (18) means that the maturity of a large part of small cells will increase in the next generation. now we present a model from the paper [28]. in this model we assume that the rate of entering proliferating phase ϕ for cell with maturity m depends on the total number of cells with this maturity r(t,m). we consider a simplified version of this model with g1(m) = g1m, g2(m) = g2m, h(m) = hm, where g1,g2,h are positive constants. we have also λ(m) = λm, λ > 0. then equation (16) becomes a special case of the following nonlinear equation: ∂u ∂t +g(x) ∂u ∂x = f(t,u(t,x),u(t−τ,λ(x))), (20) where u = r, x = m. equations of the form (20) were used in description of cellular models in the papers [9], [12], [40]. we assume that the fuctions g : [0, 1] → r, λ: [0, 1] → [0, 1] and f : [0,∞) × r × r → r have continuous derivatives and (a) g(0) = 0, g(x) > 0 for x ∈ ( 0, 1 ], (b) λ(0) = 0, λ(x) < x for x ∈ ( 0, 1 ], (c) there exist continuous functions α1 and α2 such that |f(t,u,v)| ≤ α1(t,v)|u| + α2(t,v). we consider the solution of (20) with the initial condition u(t,x) =ψ(t,x) for (t,x)∈[−τ, 0]×[0, 1]. (21) now we consider the following delay differential equation associated with (20): z′(t) = f(t,z(t),z(t− τ)). (22) the following theorem plays a central role in investigations of equation (20). theorem 6.2: let u(t,x) be a solution of (20). let z(t) be the solution of (22) with the initial condition z(t) = u(t, 0) for t ∈ [−τ, 0]. then for every t0 ≥ 0 and ε > 0 there exist t1 > 0 and another solution ū(t,x) of (20) such that (i) sup{|ū(t,x)−z(t)|: (t,x)∈[−τ,t0]×[0, 1]}<ε, (ii) ū(t,x) = u(t,x) for (t,x) ∈ [t1,∞)×[0, 1]. the proof of this result is given in [28]. from theorem 6.2 the entire strategy of studying of equation (20) becomes clear. namely, if z0(t) is a globally asymptotically stable solution of (22) and u0(x,t) = z0(t) is a locally asymptotically stable solution of (20), then u0(x,t) is globally asymptotically stable solution of (20). thus, rather surprisingly, the question of determining the global stability of a solution of (20) can be reduced to the problem of examining the global stability of the corresponding differential delay equation (22) and the local stability of (20). therefore, in the general case it is sufficient to focus on the global stability of the associated differential delay equation (22), which is itself usually quite difficult, and the local stability of (20), which is often easier. we now turn to considerations of the local stability of the full partial differential equation (20). we assume that the function f does not depend on t. then equation (20) takes the form ∂u ∂t + g(x) ∂u ∂x = f(u,uτ ), (23) where uτ = u(t− τ,λ(x)). biomath 8 (2019), 1905261, http://dx.doi.org/10.11145/j.biomath.2019.05.261 page 10 of 15 http://dx.doi.org/10.11145/j.biomath.2019.05.261 katarzyna pichór, ryszard rudnicki, one and two-phase cell cycle models let ū(x,t) be a given solution of (23) and let a be a subset of c([0, 1]× [−τ, 0]). we say that the solution ū of (23) is exponentially stable on the set a if there exists µ > 0 such that for every ψ ∈ a the solution of the problem (23), (21) satisfies the inequality max{|u(t,x)−ū(t,x)|: x ∈ [0, 1]}≤ce−µt, (24) where c is a constant which depends only on ψ. let aε={ψ : |ψ(t,x)−ū(t,x)|<ε, (t,x)∈[−τ, 0]×[0, 1]}. we say that ū is locally exponentially stable if there exist an ε > 0, µ and c such that condition (24) holds for every solution of the problem (23), (21) with ψ ∈ aε. theorem 6.3: let w be a constant such that f(w,w) = 0 and ∂f ∂u (w,w) < − ∣∣∣∣ ∂f∂uτ (w,w) ∣∣∣∣ . (25) then the solution ū(x,t) ≡ w of (23) is locally exponentially stable. let us summarize the results. consider the associated delay differential equation corresponding to (23): z′(t) = f(z(t),z(t− τ)). (26) let ϕ ∈ c[−τ, 0] and denote by zϕ the solution of (26) satisfying the initial condition zϕ(t) = ϕ(t) for t ∈ [−τ, 0]. let w ∈ r be a constant such that f(w,w) = 0. then w is a stationary solution of (26). the set b(w) = {ϕ ∈ c[−τ, 0] : lim t→∞ zϕ(t) = w} is called the basin of attraction of w. denote by p the projection operator p : c([0, 1] × [−τ, 0]) → c[−τ, 0] given by (pψ)(t) = ψ(0, t) for t ∈ [−τ, 0]. corollary 3: let w ∈ r satisfies (25) and f(w,w) = 0. then equation (23) is globally exponentially stable on the set a = {ψ ∈ c([0, 1] × [−τ, 0]) : pψ ∈ b(w)}. in paper [28] one can find applications of this theory to maturation structured models and to blood production systems. vii. chaos thus far, we have restricted our mathematical results to study such asymptotic properties of the models as the asymptotic stability and sweeping. but some of our models can have more complicated behavior which can be studied using theoretical methods of dynamical systems. now we present some results concerning chaotic and ergodic properties. it is not widely known that solutions of simple linear partial differential equations may behave in a chaotic way. the following equation with the initial condition: ∂u ∂t + x ∂u ∂x = cu, u(0,x) = v(x), c > 0, (27) defines a semiflow on the space x = {v ∈ c[0, 1] : v(0) = 0} given by stv(x) = u(t,x) = ectv(e−tx), which is chaotic, practically in each sense of the meaning of this word. for example, for λ > 0 there exists a gaussian measure with the support x invariant under {st}t≥0 and the system is mixing [44] (see also review papers [46], [51]). this implies the topological chaos: the existence of dense trajectories (topological transitivity) and instability of trajectories. the invariant measure µ for the semiflow {st}t≥0 can be given by the formula µ(a) = p(ξx ∈ a), where ξx = wx2c and wx is the standard wiener process. since equation (27) can describe the evolution of the distribution of maturity in a cellular population, one can prefer to consider the semiflow {st}t≥0 restricted to the set x+ = {v ∈ x : v ≥ 0}. in this case the invariant measure on x+ can be induced by the process ξx = |wx2c| and we still have very strong ergodic and chaotic properties of this semiflow. similar results can be obtained for semiflows generated by equations of the form ∂u ∂t + c(x) ∂u ∂x = f(x,u). (28) biomath 8 (2019), 1905261, http://dx.doi.org/10.11145/j.biomath.2019.05.261 page 11 of 15 http://dx.doi.org/10.11145/j.biomath.2019.05.261 katarzyna pichór, ryszard rudnicki, one and two-phase cell cycle models in structured population models we mainly investigate the long time behavior of densities. although in some models we expect chaotic behavior on the set of densities, it is rather difficult to prove such a result. one of the reasons is that there are no good mathematical tools to investigate such problems. analytic methods of studying chaos in infinite dimensional spaces, based on the paper [10], do not work in this case because these methods are strictly connected with semiflows acting on the whole linear space. now we present one model from the paper [49], which is well biologically motivated, where, by using methods of ergodic theory, we are able to prove chaotic behaviour of a semiflow acting on the set of densities. we consider a population of stem cells. these cells live in the bone marrow and they are precursors of any blood cells. they are subjects of two biological processes: maturation and division. stem cells can be at different levels of morphological development called maturity. the maturity of a cell differs from its age, because we assume that a newly born cell is in the same morphological state as its mother at the point of division. we assume that maturity is a real number x ∈ [0, 1]. the function u(t,x) describes the density distribution function of cells with respect to their maturity. the maturity grows according to the equation x′ = g(x). when one cell reaches the maturity 1 it leaves the bone marrow, then one of cells from the bone marrow splits. this cell is chosen randomly according to the distribution given by the density u(t,x). it follows from the assumptions that a newly born cell has the same maturity as its mother cell and each cell can divide with the same probability (see fig. 5). although our model is rather simple in comparison with other models for erythroid production (e.g. [26], [23]), it is based on the same continuous maturation-proliferation scheme. we neglect here the fact that with the growth of maturity cells pass through consecutive morphological compartments from pluripotential stem cell to erythrocytes. in our model we do not have any exterior regulatory system in which the production of erythrocytes is 0 1x fig. 5. scheme of maturation and division of cells in the bone marrow. stimulated by the hormone erythropoietin and the system tries to keep the number of erythrocytes on a constant level. one can say that chaos appears if the exterior regulatory system does not work (a pathological case). the model is described by a nonlinear semiflow induced by the equation ∂u ∂t + ∂ ∂x (g(x)u) = g(1)u(t, 1)u(t,x) (29) with the initial condition u(0,x) = u0(x), x ∈ [0, 1]. (30) the semiflow is defined on the space of densities. in the paper [49] it was shown that the semiflow generated by the initial problem (29)–(30) posses an invariant measure which is mixing and supported on the whole set of all densities. from this result there follows instability of all trajectories and topological transitivity. the main idea of the proof is to show that the semiflow generated by (29)–(30) is isomorphic to a semiflow generated by (28). then we construct an invariant measure for the second semiflow and transfer it to the initial semiflow. we skip the precise proof here. fig. 6 presents the spatial temporal plot of the solution of (29) with the initial condition u0(x) = x sin 2 (1 x ) . if we change a little the initial condition replacing u0 by ū0(x) = x sin 2 ( 1 x + 0.05 ) we obtain the solution with the plot shown in fig. 7. biomath 8 (2019), 1905261, http://dx.doi.org/10.11145/j.biomath.2019.05.261 page 12 of 15 http://dx.doi.org/10.11145/j.biomath.2019.05.261 katarzyna pichór, ryszard rudnicki, one and two-phase cell cycle models 0 1 2 3 4 u (t ,x ) 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.5 1.0 1.5 2.0 2.5 3.0 t fig. 6. the plot of the solution of (29) with the initial condition u0(x) = x sin2 ( 1 x ) in the time interval [0, 3]. 0 1 2 3 4 u (t ,x ) 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.5 1.0 1.5 2.0 2.5 3.0 t fig. 7. the plot of the solution of (29) with the initial condition ū0(x) = x sin2 ( 1 x+0.05 ) . biomath 8 (2019), 1905261, http://dx.doi.org/10.11145/j.biomath.2019.05.261 page 13 of 15 http://dx.doi.org/10.11145/j.biomath.2019.05.261 katarzyna pichór, ryszard rudnicki, one and two-phase cell cycle models the second example is the bell and anderson model of size structured cellular population given by the equation ∂u ∂t + ∂ ∂x (g(x)u) =−(µ+b)u(t,x)+4bu(t,2x), (31) where x ∈ [0, 1] and we put u(t, 2x) = 0 if 2x > 1. it is well known that if g(2x) 6= 2g(x) at least for one x ∈ [0, 1], then the solutions of (31) have asynchronous exponential growth, i.e., there exist λ ∈ r, positive functions f∗, and a constant c which depends on the initial condition u(0,x) such that e−λtu(t, ·) → cf∗ in l1[0, 1]. having in mind this result, it is difficult to imagine that some versions of this model can be chaotic. it is interesting that if g(x) = ax, then the semiflow generated by the equation (31) is chaotic. the chaotic behaviour of the semiflow generated by this equation was studied using analytic methods by howard [17] and el mourchid et al. 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[60] p. zwoleński, on continuous time version of two-phase cell cycle model of tyrcha, mathematica applicanda 41 (2013), 13–31. biomath 8 (2019), 1905261, http://dx.doi.org/10.11145/j.biomath.2019.05.261 page 15 of 15 http://dx.doi.org/10.11145/j.biomath.2019.05.261 introduction asymptotic properties of stochastic operators and semigroups rubinow-type models bell-anderson-type models two-phase model two-phase population models chaos references www.biomathforum.org/biomath/index.php/biomath original article modelling and analysis of a within-host model of hepatitis b and d co-infections plaire tchinda mouofo∗, jean jules tewa†, samuel bowong‡ ∗ department of mathematics, university of yaounde i, p.o. box 812 yaounde, cameroon tchindaplaire@yahoo.fr †department of mathematics and physics, national advanced school of engineering (polytechnic), university of yaounde i, p.o. box 8390 yaounde, cameroon tewajules@gmail.com ‡department of mathematics and computer science, faculty of science, university of douala, po box 24157 douala, cameroon sbowong@gmail.com received: 17 july 2017, accepted: 21 july 2018, published: 7 august 2018 abstract—the hepatitis delta virus (hdv) is a defect rna virus that requires the presence of the hepatitis b virus (hbv) for cellular infection. a coinfection may result in a more severe acute disease and a higher risk of developing acute liver failure compared with those infected with hbv alone. at the present time, there has been very little to the modeling of hdv. the derivation and analysis of such a mathematical model poses difficulty as it requires the inclusion of (hbv). in this paper, a within-host model for the co-interaction of hdv and hbv is presented and rigorously analyzed. we calculate the basic reproduction number (r0), the disease-free equilibrium, boundary equilibrium, which we define as the existence of one disease along with the complete eradication of the other disease, and the co-infection equilibrium. we determine stability criteria for the disease-free and boundary equilibrium. we also use the optimal control theory to assess the disease control. numerical simulations have been presented to illustrate analytical results. keywords-hepatitis d, hepatitis b, immune system, basic reproduction number, optimal control. ams subject classifications: 34a34, 34d23, 34d40, 92d30 i. introduction hepatitis d is a liver disease caused by the hepatitis d virus (hdv), a defective virus that needs the hepatitis b virus to exist. the hepatitis d virus requires the outer coating of the hepatitis b virus called the surface antigen in order to reproduce itself in a human host. the virus currently copyright: c©2018 mouofo et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: plaire tchinda mouofo, jean jules tewa, samuel bowong, modelling and analysis of a within-host model of hepatitis b and d co-infections, biomath 7 (2018), 1807219, http://dx.doi.org/10.11145/j.biomath.2018.07.219 page 1 of 17 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2018.07.219 p. t. mouofo, j. j. tewa, s. bowong, modelling and analysis of a within-host model of hepatitis ... infects 15 million worldwide, nearly all adults, and it is most common among injecting drug user populations and in countries bordering the mediterranean sea. hdv is transmitted through blood and body fluids, similar to the hepatitis b virus. there are two types of hdv infection, coinfection and super-infection. co-infection occurs when a patient is simultaneously infected with hdv and hbv. the majority of these patients completely recover but there is a higher rate of fulminant hepatic failure and death than with hbv infection alone. super-infection occurs when someone with an existing chronic hbv infection becomes infected with hdv. these patients usually experience a sudden worsening of liver disease. patients with hepatitis b who become chronically infected with hdv experience a very high rate of cirrhosis and end stage liver disease, which makes this super-infection a very dangerous disease. there is no specific treatment for hdv infection. the most common therapeutic approach is based on the administration of interferon-α. however, the clinical response is variable, and in most cases reversible upon interruption of treatment [1], [2]. the concomitant use of antiviral drugs like ribavirin or lamivudine, showed no significant benefits in the treatment of hepatitis delta patients [3], [4]. although these drugs may have some inhibitory effect on hbv replication, they do not suppress hdv replication probably due to the fact that hbsags expression, at least in part, seems not to be affected. the use of mathematical models to study dynamics of virus infections may represent a powerful approach to simulate the course of infection and predict the potential response to different therapies. they have been previously developed for a number of pathologies including hbv and hcv [5], [6], [7], [8], [9], [11], [12], [34]. the humoral immune response is universal and necessary to eliminate or control the disease after viral infection [23]. therefore, several mathematical models have been proposed to describe the virus dynamics with humoral immunity [24], [25], [26], [27], [28], [30], [31], [32]. mostafa et al. introduce an improved hbv model with standard incidence function, cytotoxic t lymphocytes (ctl) immune response, and take into account the effect of the export of precursor ctl cells from the thymus and the role of cytolytic and noncytolytic mechanisms [33]. hattaf et al. [29] study the global stability of a generalized model of a viral dynamic that includes the adaptive immune response, represented by cytotoxic lymphocyte t-cell (ctl-cell ). so, their work does not take in consideration the role of the innate immune response. noura yousfi et al. [38] investigate a new mathematical model that describes the interactions between hepatitis b virus (hbv), liver cells (hepatocytes), and the adaptive immune response. more recently, the numerical simulation of the spread of hdv and hbv in a population was reported [13]. however, a mathematical model to study hdv and hbv dynamics in infected individuals is still lacking. moreover, to our knowledge, none of the existing models take into account the reaction of immune ctl cells. the main interest in studying hbv and hdv model is to understand the long and short term behavior of the dynamics of both diseases and to predict whether the diseases will die out or will persist. in this paper, the dynamical behavior of a cointeraction of hdv and hbv virus model with ctl immune responses is studied. the main objective is in carrying out a detailed qualitative analysis of the resulting model. the existence and the uniqueness results of the solution are discussed. we compute the basic reproduction number. we also investigate the existence of equilibria and study their stability. the model is used to determine the optimal methodology for administering anti-viral medication therapies to fight hbv and hdv infection. in particular, we investigates the fundamental role of chemotherapy treatment in controlling the virus reproduction. a characterization of the optimal control via adjoint variables is also established. we obtain an optibiomath 7 (2018), 1807219, http://dx.doi.org/10.11145/j.biomath.2018.07.219 page 2 of 17 http://dx.doi.org/10.11145/j.biomath.2018.07.219 p. t. mouofo, j. j. tewa, s. bowong, modelling and analysis of a within-host model of hepatitis ... mality system that we seek to solve numerically. it is our view that this study represents the very first modelling work that provides an in-depth analysis of the qualitative dynamics of hbv-hdv co-infection and its control. the paper is organized as follows. in the next section, we propose a new mathematical model for hbv infection alone that takes into account the ctl immune system. to be more realistic, we have assumed that the infection rate is given by the standard incidence function [16]. the model is rigorously analyzed. in section 3, we formulate and analyze a realistic mathematical model for hbv-hdv co-infection, which incorporates the key epidemiological and biological features of each of the two diseases. optimal control approach is applied in section 4, in order to find the best way to fight the co-infection between the two diseases. we end this paper with a brief discussion and conclusion. ii. the hbv model a. the model description we consider the hbv model given by the following differential equations:  ẋ = λ−dx x− β(1 −η)xv x + y ẏ = β(1 −η) xv x + y −ay y −δ y i v̇ = k(1 −ε) y −dv v, i̇ = ρy i + pi −q i2 (1) where x is the number of uninfected liver cells, y is the number of infected liver cells, v is the number of free virus, and i is the number of ctl cells. all the parameters λ, dx, η, β, ay, δ, ε, k, dv, ρ, p, and q are positive. dx, ay and dv are the death rates of uninfected liver cells, the infected cells and free virus, respectively. the constant parameter λ represents the production of the liver cells. β is the contact rate between uninfected cells and free virus. free virus is produced from infected cells at rate ky. infected cells are removed at rate δi by ctl immune responses. the virus-specific ctl cells proliferate at rate ρy by contacts with infected cells. the parameter η is the efficacy of inhibiting new virus infections as a consequence of virus clearance and ε the efficacy of inhibiting viral production from infected cells. of course the behaviour when there is no treatment is obtained by setting η = ε = 0. the parameter p denotes the proliferation rate of immune cells and q the density-dependent rate of immune cells suppression. more precisely, we suppose that the immune cells expand at a net rate p, which encapsulate the positive feedback upon the immune system. the parameter q comes from the fact that we assume a regulatory negative feedback force such as the effect of cell density, inhibitory cytokines or natural apoptosis, which oparates to suppress immune population growth. we suppose in this case that immune population is suppressed at a net rate q which is proportional to the square of its density (qi2). so, the term pi − qi2 can be written as pi ( 1 − ip q ) which express a logistic law for evolution of immune population in the absence of infected cells. b. analysis of the model herein, we present some basic results, such as the positive invariance of model system (1), the boundedness of solutions, the existence of equilibria and and its stability analysis. 1) positivity and boundedness of solutions: the following result guarantees that model system (1) is biologically well behaved and its dynamics is concentrated on a bounded region of r4+. more precisely, the following result holds. theorem 1. let r4+ = {(x,y,v,i) ∈ r4 : x ≥ 0, y ≥ 0,v ≥ 0,i ≥ 0}. then, r4+ is positively invariant under the flow induced by model system (1). moreover, the region ∆ = { (x,y,v,i) ∈ r4 : x + y ≤ λ dx , v ≤ kλ(1 −ε) dxdv , p q ≤ i ≤ p + ρλ dx q } is positively invariant and absorbing with respect to model system (1). biomath 7 (2018), 1807219, http://dx.doi.org/10.11145/j.biomath.2018.07.219 page 3 of 17 http://dx.doi.org/10.11145/j.biomath.2018.07.219 p. t. mouofo, j. j. tewa, s. bowong, modelling and analysis of a within-host model of hepatitis ... proof: no solution of model system (1) with initial conditions (x(0),y(0),v(0),i(0)) ∈ r4+ is negative. in fact, for (x(t),y(t),v(t),i(t)) ∈ r4+, we have ẋ |x=0= λ > 0, ẏ |y=0= (1 − η)βv ≥ 0, v̇ |v=0= (1 − ε)k ≥ 0, i̇ |i=0= 0 ≥ 0, this immediately implies that all solutions of model system (1) with initial condition (x(0),y(0),v(0),i(0)) ∈ r4+ stay in the first quadrant. for the invariance property of ∆, it suffices to show that the vector field, on the boundary, does not point to the exterior. adding the first and second equations of model system(1) yields on the boundary of ∆: d(x + y) dt ∣∣∣∣ x+y= λ dx = λ−dxx−ayy −δyi |x+y= λ dx ≤ (λ−dx(x + y))|x+y= λ dx = 0. similarly, we get dv dt ∣∣∣∣ v= kλ(1−ε) dxdv ≤ kλ(1 −ε) dx −dvv ∣∣∣∣ v= kλ(1−ε) dxdv = 0, di dt ∣∣∣∣ i= p q ≥ (p−qi) i|i= p q = 0, i.e i(t) ≥ p q ∀t ∈ [0, +∞), and di dt ∣∣∣∣ i= p+ ρλ µ q ≤ ( ρλ dx + p−qi ) i ∣∣∣∣ i= p+ ρλ µ q = 0. therefore, solutions starting in ∆ will remain there for t ≥ 0. now, we prove the attractiveness of the trajectories of model system (1). to do so, from model system (1), one has d(x + y) dt ≤ λ−dx(x + y). therefore, lim sup t→∞ (x + y)(t) ≤ λ dx . similarly, since dv dt ≤ kλ(1 −ε) dx −dvv, one has lim sup t→∞ v(t) ≤ kλ(1 −ε) dxdv . concerning the variable i, we have di dt ≤ ( ρλ dx + p ) i −qi2, which implies that i(t) ≤ 1 q p+ ρλ dx + ( 1 i(0) − q p+ ρλ dx ) e − ( p+ ρλ dx ) t . so, i is bounded and hence, ∆ is attracting, that is, all solutions of model system (1) eventually enters ∆. this concludes the proof. 2) basic reproduction number and equilibria: the basic reproduction number is defined as the average number of secondary infections produced by one infected cell during the period of infection when all cells are uninfected. this threshold is obtained at the virus free equilibrium. the virus free equilibrium is obtained by setting v = 0 in all equations of model system (1) at the equilibrium. thus, when v = 0, we have p0 = (x∗, 0, 0, 0) and p1 = (x ∗, 0, 0,i∗) where x∗ = λ dx and i∗ = p q . since i(t) ≥ p q , the virus free equilibrium is p1. we use the method of van den driessche[17] to compute the basic reproduction number. using the notations of van den driessche and watmough[17], for model system (1), we have f = ( 0 β(1 −η) k(1 −ε) 0 ) and v = ( ay + δi ∗ 0 0 dv ) . thus, the basic reproduction number is given by: r0 = k(1 −ε)β(1 −η) dv(ay + δi∗) . (2) biomath 7 (2018), 1807219, http://dx.doi.org/10.11145/j.biomath.2018.07.219 page 4 of 17 http://dx.doi.org/10.11145/j.biomath.2018.07.219 p. t. mouofo, j. j. tewa, s. bowong, modelling and analysis of a within-host model of hepatitis ... from theorem 2 of van den driessche[17], we have the following result. lemma 1. the virus-free equilibrium p1 of the model system (1) is locally asymptotically stable (las) if r0 < 1, and unstable if r0 > 1. we now study the existence of equilibria of model system (1). setting the right-hand sides of model system (1) equals to zero gives λ−dx x− β(1 −η)xv x + y = 0, (3) β(1 −η) xv x + y −ay y −δ y i = 0, (4) k(1 −ε) y −dv v = 0, (5) ρy i + pi −q i2 = 0. (6) model system (1) has always equilibrium p1 = (x∗, 0, 0,i∗) which is the virus free equilibrium and represents the state when the viruses are absent. to find the endemic equilibrium, we look for p2 = (x̄, ȳ, v̄, ī) that represents the state in which both the viruses and ctl cells are present. let γ = dv k(1 −ε) , a = ρdv qk(1 −ε) , b = p q and assume that r0 > 1. from eqs.(5) and (6), one obtains ȳ = γv̄ and ī = av̄ + b. (7) substituting the above expression of ȳ in eq.(4) yields v̄ ( β(1 −η) x̄ x̄ + γv̄ −ayγ −δγ(av̄ + b) ) = 0 (8) since v̄ 6= 0, eq.(8) leads to x̄ = γ2v̄(ay + δb + δav̄) β(1 −η) −γ(ay + δb + δav̄) . (9) since x̄ > 0, one can deduce that v̄ < β(1 −η) −γ(ay + δb) γδa . note that β(1 −η) −γ(ay + δb) = dv(ay + δb) k(1 −ε) [r0 − 1]. thus, x̄ > 0 implies that v ∈]0, ṽ1[, where ṽ1 = dv(ay + δb) γδak(1 −ε) [r0 − 1]. (10) moreover, since x̄ < λ dx , the above relation implies that γ2δadxv̄ 2 + [ dxγ 2(ay + δb) + λγδa ] v̄ −λ [ β(1 −η) −γ(ay + δb) ] < 0. (11) since r0 > 1, the discriminant of (11) is delta = [ dxγ 2(ay + δb) + λγδa ]2 +4λγ2δadx [ β(1 −η) −γ(ay + δb) ] > 0. thus, the condition x̄ < λ dx implies that v̄ ∈]0, ṽ2[ where ṽ2 = − [ dxγ 2(ay + δb) + λγδa ] + √ delta 2γ2δadx (12) we have the following result. lemma 2. let ṽ1 and ṽ2 given in eqs. (10) and (12), respectively. then, ṽ2 < ṽ1 whenever r0 > 1. proof: since r0 > 1, applying some technical manipulation one can show that γdx [ β(1−η)−γ(ay + δb) ] + dxγ 2[ay + δb] > 0 (13) is equivalent to ṽ1 > ṽ2. remark 1. 1) from lemma 2, conditions v̄ ∈ ]0, ṽ1[ and v̄ ∈]0, ṽ2[ can be limited to v̄ ∈ ]0, ṽ2[. thus, when r0 > 1, model system (1) may have an endemic equilibrium p2 = (x̄, ȳ, v̄, ī) with v̄ ∈]0, ṽ2[. 2) we point out that v̄ = ṽ2 implies that x̄ = λ dx . substituting eq. (7) and eqs. (9) into (3), we obtain the following cubic equation in v̄: a3v̄ 3 + a2v̄ 2 + a1v̄ + a0 = 0, (14) biomath 7 (2018), 1807219, http://dx.doi.org/10.11145/j.biomath.2018.07.219 page 5 of 17 http://dx.doi.org/10.11145/j.biomath.2018.07.219 p. t. mouofo, j. j. tewa, s. bowong, modelling and analysis of a within-host model of hepatitis ... where a3 = [ d2δρ qk2(1 −ε)2 ]2 , a2 = d2v ( ay + δ p q ) k2(1 −ε)2 − dxδρd 3 v qk3(1 −ε)3 − δρd3v ( ay + δ p q ) qk3(1 −ε)3 [ r0 − 1 ] , a1 = − λd2vδρ qk2(1 −ε)2 − dxd 2 v ( ay + δ p q ) k2(1 −ε)2 − γdv ( ay + δ p q )2 k(1 −ε) [ r0 − 1 ] , a0 = λdv ( ay + δ p q ) k(1 −ε) [ r0 − 1 ] . the coefficient a3 is always positive. also, if r0 > 1, then a0 > 0 and a1 < 0. using the descartes’ rules of sign, if r0 > 1, the above equation given in (14) has exactly nought or two positive solutions. let us consider the following polynomial p : [0, ṽ2]→r v̄ 7→p(v̄) =a3v̄3 +a2v̄2 +a1v̄+a0. (15) we have • p(0) = a0 > 0 since r0 > 1 • p(ṽ2) = − β(1 −η)k(1 −ε)λṽ2 λk(1 −ε) + dxdvṽ2 < 0 since p(0) > 0 and p(ṽ2) < 0, eq.(14) has only one or three solutions on ]0,v2[. by the descartes’ rules of sign, eq.(14) cannot have three solutions on ]0, ṽ2[. we have proved the following result. theorem 2. if r0 > 1, model system (1) has exactly one endemic equilibrium p2 = (x̄, ȳ, v̄, ī) where x̄, ȳ and ī are defined in eqs. (7) and (9) with v̄ ∈]0, ṽ2[ satisfying eq. (14). remark 2. when r0 < 1, we have x̄ < 0. so, there is no endemic equilibrium. 3) stability of equilibria: here, we study the stability of equilibria. we have the following result about the global stability of the virus free equilibrium. theorem 3. when r0 < 1, then the virus free equilibrium p1 = (x∗, 0, 0,i∗) is globally asymptotically stable in ∆. proof: we consider the following lasallelyapunov candidate function: l(t) = b1y + b2v, (16) where b1 and b2 are positive constants to be determined later. using the fact that x ≤ x∗ and i ≥ i∗, its time derivative along the trajectories of (1) satisfies l̇ = ( b1β(1 −η)x x + y − b2dv ) v +[b2k(1 −ε) − b1(ay + δi)]y, ≤ ( b1β(1 −η)x∗ x∗ + y − b2dv ) v +[b2k(1 −ε) − b1(ay + δi∗)]y. the constants b1 and b2 are chosen such that the coefficient of y are equal to zero, that is, b1 = k(1 −ε) and b2 = ay + δi∗. then, we obtain l̇ ≤ (ay + δi∗)dv(r0 − 1)v. (17) thus, l̇(t) ≤ 0 when r0 < 1. by lasalle’s invariance principle, the largest invariant set in {(x,y,v,i) ∈ r4+, l̇(t) = 0} ⊂ ∆ is reduced to the virus free equilibrium. this proves the global asymptotic stability of p1 = (x∗, 0, 0,i∗) on ∆ [18] (theorem 3.7.11, page 346). this achieves the proof. now, we study the stability of the endemic equilibrium. the jacobian matrix of model system (1) at the endemic equilibrium p2 = (x̄, ȳ, v̄, ī) is j3 =   −dx−u w −v 0 u −w−ay−δī v −δȳ 0 k(1 −ε) −dv 0 0 ρī 0 −qī   , biomath 7 (2018), 1807219, http://dx.doi.org/10.11145/j.biomath.2018.07.219 page 6 of 17 http://dx.doi.org/10.11145/j.biomath.2018.07.219 p. t. mouofo, j. j. tewa, s. bowong, modelling and analysis of a within-host model of hepatitis ... where u = (1 −η)βv̄ȳ (x̄ + ȳ)2 , v = (1 −η)βx̄ x̄ + ȳ and w = (1 −η)βv̄x̄ (x̄ + ȳ)2 . the characteristic equation of j3 is ξ4 + c1ξ 3 + c2ξ 2 + c3ξ + c4 = 0, (18) where c1 = dx + u + w + ay + δ ī + dv + qī, c2 = dxw + dxay + dxδ ī + dxdv + dxqī + uay + uδ ī + udv + uqī + wdv + qīw + qīay + qī 2δ + qīdv + δ ȳρ ī, c3 = uaydv + uδ īdv + dxwdv + dxqīw + dxqīay + dxqī 2δ + dxqīdv + dxδ ȳρ ī + uqīay + uqī 2δ + uqīdv + uδ ȳρī + wdvqī + δ ȳρ īdv c4 = dxwdvqī + dxδ ȳρ īdv + uδ ȳρīdv + uaydvqī + uδ ī 2dvq. it is clear that c1 > 0, c3 > 0 and c4 > 0. after some technical computation, we obtain that c1c2c3 − c23 − c 2 1c4 is positive. hence, we have proved the following result. theorem 4. the endemic equilibrium p2 of model system (1) is locally asymptotically stable if r0 > 1. iii. the hbv-hdv co-infection model a. model construction in this section, we incorporate the hdv infection into the previous model in order to obtain a mathematical models that can describe the dynamics of hdv and hbv co-infection. we add three state variables, namely the hdv viral load, hepatocyte infected with hdv and those coinfected with both hbv and hdv. so, we use the following state variables: • x(t) the number of uninfected cells at time t, • y(t) the number of hbv infected cells at time t, • z(t) the number of hdv infected cells at time t, • w(t) the number of infected cells with both hbv and hdv at time t, • v1(t) the hbv viral load at time t, • v2(t) the hdv viral load at time t, • i(t) the number of ctl cells at time t. we make the following hypothesis: 1) uninfected liver cells (x) can only be infected by hbv virions alone and become y, or by hdv virions alone and become z. 2) infected cells by hbv virions can be superinfected by hdv and become w. 3) infected cells by hdv virions can be superinfected by hbv and become w. putting the above formulations and assumptions together gives the following system of differential equations:  ẋ=λ−dxx− β1(1 −η)xv1 p − β2(1−η)xv2 p , ẏ = β1(1−η)xv1 p − β2(1−η)yv2 p −ayy−δ1yi, ż = β2(1−η)xv2 p − β1(1−η)zv1 p −azz −δ2zi, ẇ= β2(1−η)yv2 p + β1(1−η)zv1 p −aww−δ3wi, v̇1 =k1(1−ε)y+k3(1−ε)w−d1v1, v̇2 =k2(1−ε)w−d2v2, i̇ =ρ1yi+ρ2zi+ρ3wi+pi−qi2, (19) where p = x + y + z + w. (20) susceptible host (healthy hepatocytes) cells are produced at a rate λ, died at a rate dx. β1 and β2 are respectively the contact rates between uninfected cells with free hbv virions, and uninfected cells with free hdv virions. ay, az and aw are respectively death rate of hbv only infected cells, hdv only infected cells and coinfected cells by hdv and hbv. in these equations, all the parameters are positive and we assume that the death rate of uninfected cells is not greater than the death rate of infected cells, that is, min{dx,ay,az,aw} = dx. hbv virions v1 are biomath 7 (2018), 1807219, http://dx.doi.org/10.11145/j.biomath.2018.07.219 page 7 of 17 http://dx.doi.org/10.11145/j.biomath.2018.07.219 p. t. mouofo, j. j. tewa, s. bowong, modelling and analysis of a within-host model of hepatitis ... table i numerical values for the parameters of model system (19) symbols definition value source λ production rate of hepathocyte 252666.6667 day−1 [10] dx normal death rate of healthy liver cells 0.0039 days−1 [15] ay infected cell death rate (hbv) 0.0693 − 0.00693 day−1 [9] az infected cell death rate (hdv) 0.009 day−1 assumed aw infected cell death rate (hbv-hdv) 0.0091 day−1 assumed d1 death rate of free hbv 0.693 day−1 [14] d2 death rate of free hdv 0.6 day−1 assumed β1 infection rate of hbv 3.6 × 10−5 − 1.8 × 10−3 [14] β2 infection rate of hdv 0.002 day−1 assumed k1 hbv production per infected liver cells 200 − 1000 days−1 [8] k2 hdv production per infected liver cells 300 days−1 assumed k3 hbv production per co-infected hbv& hdv cells 300 days−1 assumed δ1 rate of ctl elimination 0.02 day−1 assumed in infected hbv cells δ2 rate of ctl elimination in infected hdv cells 0.02 day−1 assumed δ3 rate of ctl elimination in infected hbv-hdv cells 0.02 day−1 assumed p the proliferation rate of immune cells 0.5 day−1 assumed q density-dependent rate of immune 0.03 day−1 assumed cells suppression ρi hbv-specific ctl stimulation rate 0.02 day−1 assumed produced by hbv only infected cells y and by coinfected cells w. in fact, in order for hdv to successfully complete its replication cycle, a hepatocyte must be coinfected with hbv and hdv. in these coinfected cells, the replication of hbv is suppressed by hdv, although not completely abolished [19], [20]. so, v1 can be produced by w. hbv only infected cells y produce hbv virions particles v1 at a rate k1(1 − ε)y and are killed by the ctl immune response at a rate δ1yi. for the same reason, hdv are killed by the ctl immune response at a rate δ1yi. finally, ctl cells increase at a rates ρ1yi, ρ2zi and ρ3wi as a result of stimulation by the viral antigen of the infected cells. p and q have the same meaning as in the previous model. the parameter values used for numerical simulation are given in table 1. b. mathematical analysis of the model 1) positivity and boundedness of trajectories: we have the following result. theorem 5. the nonegative orthant r7+ is positively invariant for model system (19). moreover, system (19) is pointwise dissipative and the abbiomath 7 (2018), 1807219, http://dx.doi.org/10.11145/j.biomath.2018.07.219 page 8 of 17 http://dx.doi.org/10.11145/j.biomath.2018.07.219 p. t. mouofo, j. j. tewa, s. bowong, modelling and analysis of a within-host model of hepatitis ... sorbing set is given by σ = { (x,y,z,w,v1,v2,i) ∈ r7+ : t ≤ λ dx , v1 ≤ (k1 +k3)(1 −ε)λ dxd1 , v2 ≤ λk2(1−ε) dxd2 , p q ≤ i ≤ ρ0 q } , where t = x+y+z+w and ρ0 = (ρ1+ρ2+ρ3)λ dx +p. the proof is similar to the proof of theorem 1. 2) basic reproduction number and equilibria: the disease-free equilibrium of model system (19) is given by e0 = (x∗, 0, 0, 0, 0, 0,i∗). using the notations in van den driessche and watmough[17] for model system (19), the matrices f and v for the new infection terms and the remaining transfer terms are, respectively, given by f =   0 0 0 (1−η)β1 0 0 0 0 0 (1 −η)β2 0 0 0 0 0 k1(1−ε) 0 k3(1−ε) 0 0 0 0 k2(1−ε) 0 0   and v =   ay +δ1i ∗ 0 0 0 0 0 az +δ2i ∗ 0 0 0 0 0 aw +δ3i ∗ 0 0 0 0 0 d1 0 0 0 0 0 d2   . it then follows that the basic reproduction number is given by r0 = k1(1 −ε)β1(1 −η) d1(ay + δ1i∗) . (21) thus, using theorem 2 of van den driessche and watmough[17], we have the following result. lemma 3. : the virus free equilibrium e0 of model system (19) is locally asymptotically stable (las) if r0 < 1, and unstable if r0 > 1. remark 3. note that the basic reproduction number of the co-infection model of hbv and hdv is the same than the basic the basic reproduction of the hbv model alone. this suggests that to control the co-infection of hbv and hdv within the body of a host, one only needs to control the hbv infection. we now process with the existence of steady states. the steady states of model system (19) satisfy the following equations:  λ−dxx̄− β1(1−η)x̄v̄1 p̄ − β2(1−η)x̄v̄2 p̄ = 0, β1(1−η)x̄v̄1 p̄ − β2(1−η)ȳv̄2 p̄ −ayȳ−δ1ȳī = 0, β2(1−η)x̄v̄2 p̄ − β1(1−η)z̄v̄1 p −azz̄−δ2z̄ī = 0, β2(1−η)ȳv̄2 p̄ + β1(1−η)z̄v̄1 p̄ −aww̄−δ3w̄ī = 0, k1(1 −ε)ȳ + k3(1 −ε)w̄ −d1v̄1 = 0, k2(1 −ε)w̄ −d2v̄2, ρ1ȳī + ρ2z̄ī + ρ3w̄ī + pī −qī2 = 0, (22) where p̄ = x̄ + ȳ + z̄ + w̄. from the last equation of (22), we have (ρ1ȳ + ρ2z̄ + ρ3w̄ + p−qī)ī = 0, (23) which has two possible solutions: ī = 0 and ρ1ȳ+ ρ2z̄ + ρ3w̄ + p−qī = 0. since ī ≥ i∗ and ī 6= 0, one has that ī = 1 q (ρ1ȳ+ρ2z̄+ρ3w̄+p). from the sixth equation of (22), one has w̄ = d2 k2(1 −ε) v̄2. adding the third and fourth equations of (23) and using the expression of w̄ yields z̄ = [ β2(1−η)(x̄+ȳ) (az +δ2ī)p̄ − (aw + δ3ī)d2 k2(1−ε)(az +δ2ī) ] v̄2. (24) substituting eq.(24) into the fourth equation of (22) gives{ β2(1−η)ȳ p̄ + β1(1−η)v̄1 p̄ [ β2(1−η)(x̄+ȳ) (az +δ2ī)p̄ − (aw +δ3ī)d2 k2(1−ε)(az +δ2ī) ]} v̄2 = 0. (25) eq. (25) has two possible solutions. if v̄2 = 0, then from eq. (22) one has that v̄1 = 0 or z̄ = 0. biomath 7 (2018), 1807219, http://dx.doi.org/10.11145/j.biomath.2018.07.219 page 9 of 17 http://dx.doi.org/10.11145/j.biomath.2018.07.219 p. t. mouofo, j. j. tewa, s. bowong, modelling and analysis of a within-host model of hepatitis ... note that v̄1 = 0 leads to the virus free equilibrium e0 = (x∗, 0, 0, 0, 0, 0,i∗). if z̄ = 0, we obtain the hbv-only persistence equilibrium ē = (x̄, ȳ, 0, 0, v̄1, 0, ī) which has biological significance if r0 > 1. if v̄2 6= 0, the model admits a coinfection equilibrium ê = (x̂, ŷ, ẑ, ŵ, v̂1, v̂2, î) where the existence is verified numerically. 3) stability of equilibria: we have the following result. theorem 6. if r0 < 1, then the infection free equilibrium e0 of model system (19) is globally asymptotically stable in σ whenever k3(1−ε)(ay +δ1i∗)+ k1(1−ε)k2(1−ε)β2(1−η) d2 −k1(1−ε)(aw +δ3i∗) ≥ 0. (26) proof: consider the following lasalle function candidate: l(t) = ay + az + aw + bv1 + cv2, (27) where a, b and c are positive constants to be determined later. using the fact that x ≤ x∗ and i ≥ i∗, the derivative of l along the solution of (19) satisfies l̇ = aβ1(1−η)xv1 p + aβ2(1−η)xv2 p −a(ay+δ1i)y −a(az + δ2i)z −a(aw + δ3i)w+bk1(1−ε)y + bk3(1−ε)w−bd1v1 + ck2(1−ε)w−cd2v2 ≤ [ aβ1(1 −η) − bd1 ] v1 + [ bk1(1 −ε) −a(ay + δ1i∗) ] y + [ aβ2(1 −η) − cd2 ] v2 + [ bk3(1−ε)+ck2(1−ε)−a(aw +δ3i∗) ] w. we choose a = k1(1 − ε), b = ay + δ1i∗ and c = k1(1 −ε)β2(1 −η) d2 so that the coefficients of y, v2 and w are equal to zero. in this case, if the condition (26) holds, we obtain l̇ ≤ (ay + δ1i∗) [ r0 − 1 ] v1. thus, if r0 ≤ 1 then l̇ ≤ 0 ∀x, y, z, w, v1, v2, i ≥ 0 and l̇ = 0 if only if (x, y, z, w, v1, v2) = (x∗, 0, 0, 0, 0, 0,i∗). then the globally asymptotically attractivity of e0 follows from lyapunov lasalle invariance principle [18]. this completes the proof. now, we study the stability of the hbv-only persistence equilibrium ē = (x̄, ȳ, 0, 0, v̄1, 0, ī). the jacobian matrix of model system (19) at ē is jē=   −dx−s2 s1 s1 s2 −s1−ay−δ1ī −s1 0 0 −s1−az−δ2ī 0 0 s1 0 k1(1−ε) 0 0 0 0 0 ρ1ī ρ2ī s1 −s3 −s4 0 −s1 s3 −s5 −δ1ȳ 0 0 s4 0 −aw−δ3ī 0 s5 0 k3(1−ε) −d1 0 0 k2(1−ε) 0 −d2 0 ρ3ī 0 0 −qī   , where s1 = β1(1 −η)x̄v̄1 (x̄ + ȳ)2 , s2 = β1(1 −η)ȳv̄1 (x̄ + ȳ)2 , s3 = β1(1 −η)x̄ x̄ + ȳ , s4 = β2(1 −η)x̄ x̄ + ȳ and s5 = β2(1 −η)ȳ x̄ + ȳ . the local stability of ē is governed by the eigenvalues of the jē. hence, conditions for local stability of ē have been derived by applying the routh-hurwitz criterion to the characteristic equation of je. the expresions are complicated and are not presented here, but available from the authors on request. importantly, the set of parameters satisfying these conditions is not empty. figure 1 presents the time evolution of model system (19) when β1 = 0.02 β2 = 0.07, ε = η = 0, k1 = 600 and ρ1 = 0.00002 (so that r0 > 1). all other parameter values are as in table 1. in this case, the co-infection equilibrium is ê(6.5× 104, 1.8 × 104, 490.93, 310.3, 1.572 × 107, 1.55 × 105, 670.9). initial conditions have been chosen to be x(0) = 2 × 107, y(0) = 104, z(0) = 4 × 104, w(0) = 3 × 103, v1(0) = 2 × 103, v2(0) = 1.5 × biomath 7 (2018), 1807219, http://dx.doi.org/10.11145/j.biomath.2018.07.219 page 10 of 17 http://dx.doi.org/10.11145/j.biomath.2018.07.219 p. t. mouofo, j. j. tewa, s. bowong, modelling and analysis of a within-host model of hepatitis ... 0 10 20 30 40 50 60 70 80 90 100 0 0.5 1 1.5 2 2.5 x 10 7 time(days) u n in fe ct e d c e lls (a1) 0 10 20 30 40 50 60 70 80 90 100 0 2 4 6 8 10 12 14 x 10 4 time(days) in fe ct e d c e lls b y h b v (b1) 0 10 20 30 40 50 60 70 80 90 100 0 500 1000 1500 2000 2500 3000 3500 4000 time(days) in fe ct e d c e lls b y h d v (c1) 0 10 20 30 40 50 60 70 80 90 100 0 500 1000 1500 2000 2500 3000 time(days) c o − in fe ct e d c e lls (d1) 0 10 20 30 40 50 60 70 80 90 100 0 2 4 6 8 10 12 x 10 7 time(days) f re e h b v v ir u s (e1) 0 10 20 30 40 50 60 70 80 90 100 0 0.5 1 1.5 2 2.5 3 x 10 5 time(days) f re e h d v v ir u s (f1) 0 10 20 30 40 50 60 70 80 90 100 0 500 1000 1500 2000 2500 time(days) c t l c e lls (g1) fig. 1. time evolution of model system (19) when β1 = 0.02 β2 = 0.07, ε = η = 0, k1 = 600 and ρ1 = 0.00002 (so that r0 > 1). all other parameter values are as in table 1. 103 and i(0) = 4 × 102. from this figure, it is evident that the trajectories of model system (19) converge to the co-infection equilibrium. iv. optimal control of treatment in the hbv-hdv co-infection model this section deals with the problem of optimal control of the co-infection of hbv and hdv. more precisely, we are concerned with the problem of adopting the best strategy of treatment to fight the hbv-hdv co-infection. we seek to search a maximum count of healthy cells with a minimum dose of the administered drugs. hence, if we denote η(t) the first control variable which is efficacy of inhibiting new virus infections as a consequence of virus clearance and ε(t) the second control variable which represents the efficacy of inhibiting viral production from infected cells, model system (19) can be written, to accommodate control actions or chemotherapy treatment, as follows:  ẋ=λ−dxx− β1(1−η(t))xv1 p − β2(1−η(t))xv2 p , ẏ = β1(1−η(t))xv1 p − β2(1−η(t))yv2 p −ayy −δ1yi, ż = β2(1−η(t))xv2 p − β1(1−η(t))zv1 p −azz −δ2zi, ẇ= β2(1−η(t))yv2 p + β1(1−η(t))zv1 p −aww −δ3wi, v̇1 =k1(1−ε(t))y+k3(1−ε(t))w−d1v1, v̇2 =k2(1−ε(t))w−d2v2, i̇ = ρ1yi + ρ2zi + ρ3wi + pi −qi2, (28) biomath 7 (2018), 1807219, http://dx.doi.org/10.11145/j.biomath.2018.07.219 page 11 of 17 http://dx.doi.org/10.11145/j.biomath.2018.07.219 p. t. mouofo, j. j. tewa, s. bowong, modelling and analysis of a within-host model of hepatitis ... with x0, y0, z0, w0, v01 , v 0 and i0 the given initial values of x, y, w, z, v1, v2, i at t0 = 0 respectively. the control functions, η(t) and ε(t), are bounded, lebesgue integrable functions. our objective functional to be maximized is j(η,ε) = ∫ tf 0 [ x(t) − ( a 2 η2 + b 2 ε2 )] dt. (29) in other words, we want to maximize the benefit based on the healthy liver cells count and minimizing the cost based on the percentage effect chemotherapy given (i.e. η and ε). the parameters a and b are positive and represent the weights on the benefit and cost. the goal is to seek an optimal control pair (η∗,ε∗) such that j(η∗,ε∗) = max{j(η,ε) : (η,ε) ∈u}, (30) where u is the control set defined by: u = {u = (η,ε), η,εmeasurable, 0 ≤ η(t) ≤ 1, 0 ≤ ε(t) ≤ 1,∀t ∈ [0, tf ]}. the basic framework of this problem is to prove the existence and the uniqueness of the optimal control and to characterize it. a. analysis of optimal controls the necessary conditions that an optimal pair must satisfy come from pontryagin’s maximum principle [21]. this principle converts (28) (30) into a problem of minimizing pointwise a hamiltonian, h, with respect to η and ε: h = x(t) − ( a 2 η2 + b 2 ε2 ) +λ1 [ λ−dxx− β1(1−η(t))xv1 p − β2(1−η(t))xv2 p ] +λ2 [ β1(1−η(t))xv1 p − β2(1−η(t))yv2 p −ayy −δ1yi ] +λ3 [ β2(1−η(t))xv2 p − β1(1−η(t))zv1 p −azz−δ2zi ] + λ4 [ β2(1−η(t))yv2 p + β1(1−η(t))zv1 p −aww−δ3wi ] +λ5 [k1(1 −ε(t))y + k3(1 −ε(t))w −d1v1] +λ6 [k2(1 −ε(t))w −d2v2] +λ7 [ ρ1yi + ρ2zi + ρ3wi + pi −qi2 ] . (31) by applying pontryagins maximum principle [21] and the existence result for the optimal control pairs from [22], we have the following result. theorem 7. there exists an optimal control pair η∗, ε∗ and corresponding solution x∗, y∗, z∗, w∗, v∗1 , v ∗ 2 and i ∗, such that j(η∗,ε∗) = max u j(η,ε). furthermore, there exist adjoint functions λ1, λ2, λ3, λ4, λ5, λ6, and λ7 such that equation (32) holds with the transversality conditions λi(tf ) = 0, for i = 1, ..., 7 (33) and p = x∗ + y∗ + z∗ + w∗. the following characterization holds: η∗ = min{max{0, r1(t)}, 1}, ε∗ = min{max{0, r2(t)}, 1} (34) where r1(t) = 1 a [ λ1 β1x ∗v∗1 +β2x ∗v∗2 p +λ2 β2y ∗v∗2 −β1x ∗v∗1 p +λ3 β1z ∗v∗1−β2x ∗v∗2 p −λ4 β1z ∗v∗1 + β2y ∗v∗2 p ] r2(t) = −λ5(k1y∗ + k3w∗) −λ6k2w∗ b proof: corollary 4.1 in fleming and rishel [22] gives the existence of an optimal control pair due to the concavity of the integrand of j with respect to (η,ε), a priori boundedness of the state solutions, and the lipschitz property of the state system with respect to the state variables. applying pontryagin’s maximum principle, we obtain biomath 7 (2018), 1807219, http://dx.doi.org/10.11145/j.biomath.2018.07.219 page 12 of 17 http://dx.doi.org/10.11145/j.biomath.2018.07.219 p. t. mouofo, j. j. tewa, s. bowong, modelling and analysis of a within-host model of hepatitis ... λ̇1 = −1 + λ1 ( dx + β1(1 −η)v∗1 (y ∗ + z∗ + w∗) p 2 + β2(1 −η)v∗2 (y ∗ + z∗ + w∗) p 2 ) , −λ2 ( β1(1 −η)v∗1 (y ∗ + z∗ + w∗) p 2 + β2(1 −η)y∗v∗2 p 2 ) −λ3 ( β2(1 −η)v∗2 (y ∗ + z∗ + w∗) p 2 + β1(1 −η)z∗v∗1 p 2 ) −λ4 ( β2(1 −η)y∗v∗2 p 2 + β1(1 −η)z∗v∗1 p 2 ) λ̇2 = −λ1 ( β1(1 −η)x∗v∗1 p 2 + β2(1 −η)x∗v∗2 p 2 ) + λ3 ( β2(1 −η)x∗v∗2 p 2 − β1(1 −η)z∗v∗1 p 2 ) +λ2 ( β1(1 −η)x∗v∗1 p 2 + β2(1 −η)v∗2 (x ∗ + z∗ + w∗) p 2 + ay + δ1i ∗ ) +λ4 ( β1(1 −η)z∗v∗1 p 2 − β2(1 −η)v∗2 (x ∗ + z∗ + w∗) p 2 ) −k1(1 −ε)λ5 −ρ1i∗λ7, λ̇3 = −λ1 ( β1(1 −η)x∗v∗1 p 2 + β2(1 −η)x∗v∗2 p 2 ) + λ2 ( β1(1 −η)x∗v∗1 p 2 − β2(1 −η)y∗v∗2 p 2 ) +λ3 ( β2(1 −η)x∗v∗2 p 2 + β1(1 −η)v∗1 (x ∗ + y∗ + w∗) p 2 + az + δ2i ∗ ) +λ4 ( β2(1 −η)y∗v∗2 p 2 − β1(1 −η)v∗1 (x ∗ + y∗ + w∗) p 2 ) −ρ2i∗λ7, λ̇4 = −λ1 ( β1(1 −η)x∗v∗1 p 2 + β2(1 −η)x∗v∗2 p 2 ) + λ2 ( β1(1 −η)x∗v∗1 p 2 − β2(1 −η)y∗v∗2 p 2 ) +λ3 ( β2(1 −η)x∗v∗2 p 2 − β1(1 −η)z∗v∗1 p 2 ) −k3(1 −ε)λ5 −k2(1 −ε)λ6 −ρ3i∗λ7 +λ4 ( β2(1 −η)y∗v∗2 p 2 + β1(1 −η)z∗v∗1 p 2 + aw + δ3i ∗ ) , (32) λ̇5 = β1(1 −η)x∗ p (λ1 −λ2) + β1(1 −η)z∗ p (λ3 −λ4) + d1λ5, λ̇6 = β2(1 −η)x∗ p (λ1 −λ3) + β2(1 −η)y∗ p (λ2 −λ4) + d2λ6, λ̇7 = δ1y ∗λ2 + δ2z ∗λ3 + δ3w ∗λ4 − (ρ1y∗ + ρ2z∗ + ρ3w∗ + p− 2qi∗)λ7, biomath 7 (2018), 1807219, http://dx.doi.org/10.11145/j.biomath.2018.07.219 page 13 of 17 http://dx.doi.org/10.11145/j.biomath.2018.07.219 p. t. mouofo, j. j. tewa, s. bowong, modelling and analysis of a within-host model of hepatitis ... λ̇1 = − ∂h ∂x , λ̇2 = − ∂h ∂y , ..., λ̇7 = − ∂h ∂i , evaluated at the optimal control pair and corresponding states, which results in the stated adjoint system (32) and (33) [35]. by considering the optimality conditions, ∂h ∂η = 0 and ∂h ∂ε = 0 and solving for η∗, ε∗, subject to the constraints, the characterizations (34) can be derived. to illustrate the characterization of ε∗, we have ∂h ∂ε = −bε−λ5(k1y + k3w) −λ6k2w = 0 at ε∗ on the set {t | 0 < ε∗(t) < 1}. on this set, we have ε∗(t) = −λ5(k1y + k3w) −λ6k2w b . taking into account the bounds on ε, we obtain the characterization of ε in (34). next, we discuss the numerical solutions of the optimality system and the corresponding optimal control pairs, the parameter choices, and the interpretations from various cases. b. numerical results in this section, we study numerically an optimal treatment strategy of our hbv-hdv co-infection model. the optimal treatment strategy is obtained by solving the optimality system, consisting of 14 odes from the state and adjoint equations. an iterative method is used for solving the optimality system. we start to solve the state equations with a guess for the controls over the simulated time using a forward fourth order runge-kutta scheme. because of the transversality conditions (33), the adjoint equations are solved by a backward fourth order runge-kutta scheme using the current iteration solution of the state equations. then, the controls are updated by using a convex combination of the previous controls and the value from the characterizations (34). this process is repeated and iteration is stopped if the values of unknowns at the previous iteration are very close to the ones at the present iteration. in order to evaluate our control strategy, we consider six co-infected patients by hbv and hdv: a1, a2, b1, b2, c1, and c2 divided in three groups such that the first group consists of a1 and a2 and the viral hdv load of patients a1 and a2 are the same and are 1.16×103 copies/ml. the second group are patients b1 and b2 with the same hdv viral loads of 1.16 × 105 copies/ml; the end group are patients c1, and c2 with high levels of hdv viremia which are 1.16 × 107 copies/ml. we suppose that patients a1, b1 and c1 are not under treatment. otherwise, patients a2, b2 and c2 are under our treatment control strategy. we want to evaluate in 100 days the evolution of viral hdv load in every patient. we suppose that the viral load of hbv in those groups are 108 copies/ml. note that some studies report improvements in patients, with interferon-α (ifn) efficacy as high as 90% (η = 0.9) [36]. although lamivudine (lmv) does not have a direct effect on hdv viral production k2, its effect on the viral production of hbv will also have an effect on the hdv viral dynamics. studies such as lewin et al [37] show efficacy levels of therapies based on lmv varying between 90% to 99% (ε > 0.9) for the hbv infection. with this in mind, we consider combination of interferon-α and lamivudine in our simulations. in fig.(2), parameter values are the same as in fig.(1). figure (2) (a1) − (c1) presents the time evolution of viral load of patients of group 1. in this figure, we observe that when there is not strategy of treatment, the disease persists in the host. but, with our strategy of treatment, the disease disappear in the host. to have this result, the control ε(t) is at it maximal value almost all the period of our control strategy. this means that, to control hdv, we need medication with hight efficacy. we have the same observation on the other figures for group 2 and group 3. fig.(3) illustrate the importance of immune system in our model. we observe that without ctl cells, the viral load of patients a1, b1 and c1 increase. otherwise, the viral load of patients a2, b2 and c2 converge to zero if only if ε(t) and η(t) are to their maximal values during all the period of control strategy. so, it is difficult biomath 7 (2018), 1807219, http://dx.doi.org/10.11145/j.biomath.2018.07.219 page 14 of 17 http://dx.doi.org/10.11145/j.biomath.2018.07.219 p. t. mouofo, j. j. tewa, s. bowong, modelling and analysis of a within-host model of hepatitis ... 0 10 20 30 40 50 60 70 80 90 100 0 2 4 6 8 10 12 x 10 4 time (days) f re e h d v v ir u s hdv viral load of patient a 2 with control treatment hdv viral load for patient a 1 without treatment (a1) 0 10 20 30 40 50 60 70 80 90 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time (days) t re a tm e n t co n tr o l ε (t ) fo r p a tie n t a 2 (b1) 0 10 20 30 40 50 60 70 80 90 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 time (days) t re a tm e n t co n tr o l η (t ) o f p a tie n t a 2 (c1) 0 10 20 30 40 50 60 70 80 90 100 0 2 4 6 8 10 12 x 10 4 time (days) f re e h d v v ir u s hdv viral load of patient b 2 with treatment strategy hdv viral load of patient b 1 without treatment (a2) 0 10 20 30 40 50 60 70 80 90 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time (days) t re a tm e n t co n tr o l ε( t) f o r p a tie n t b 2 (b2) 0 10 20 30 40 50 60 70 80 90 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 time (days) t re a tm e n t co n tr o l η (t ) fo r p a tie n t b 2 (c2) 0 10 20 30 40 50 60 70 80 90 100 0 2 4 6 8 10 12 x 10 4 time (days) f re e h d v v ir u s hdv viral load of patient c 2 with control strategy hdv viral load of patient c 1 without treatment (a3) 0 10 20 30 40 50 60 70 80 90 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time (days) t re a tm e n t co n tr o l ε (t ) fo r p a tie n t c 2 (b3) 0 10 20 30 40 50 60 70 80 90 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 time (days) t re a tm e n t co n tr o l η (t ) fo r p a tie n t c 2 (c3) fig. 2. time evolution of viral load of patients a1, b1, c1 without treatment, and patients a2, b2 and c2 with our treatment strategy. to control the disease in this case. this show the importance of immune system in our model. v. discussion and conclusion hepatitis d virus infection has a worldwide distribution. studying the transmission, epidemiology and dynamic virus of hdv infection is an important topic. it is a unique virus for which many open questions remain. for example, hdvspecific treatment protocols still do not exist. the investigation of the inter-related dynamics of chronic hbv and hdv infections are important to understanding how treatment may affect this complex system. to this end, the development of biologically realistic mathematical models is an important tool. the main objective of this paper was to shed light on the interaction between hbv and hdv. a realistic deterministic ode based compartmental model for the transmission of hbv and hdv co-dynamics within the body of a host has been proposed and analyzed. the hbv-only model was qualitatively examined, first of all. the mathematical analysis results show that the basic reproductive number r0 of the cointeraction hbv-hdv model is the same than the basic reproduction number of the hbv model alone. this suggests that the eradication of hdv is conditioned by the eradication of hbv. the epidemiological implication of this is that for biomath 7 (2018), 1807219, http://dx.doi.org/10.11145/j.biomath.2018.07.219 page 15 of 17 http://dx.doi.org/10.11145/j.biomath.2018.07.219 p. t. mouofo, j. j. tewa, s. bowong, modelling and analysis of a within-host model of hepatitis ... 0 20 40 60 80 100 0 1 2 3 4 5 6 7 8 x 10 9 time (days) f re e h d v v ir u s hdv viral load of patient c 2 without immune system reaction hdv viral load of patient c 1 without immune system reaction (d1) 0 20 40 60 80 100 0 1 2 3 4 5 6 7 8 x 10 9 time (days) f re e h d v v ir u s hdv viral load of patient b 2 without immune system reaction hdv viral load of patient b 1 without immune system reaction (d2) 0 20 40 60 80 100 0 1 2 3 4 5 6 7 8 x 10 9 time (days) f re e h d v v ir u s hdv viral load of patient a 2 without immune system reaction hdv viral load of patient a 1 without immune system reaction (d3) fig. 3. time evolution of viral load different groups without ctl cells . the effective eradication and control of hdv, r0 should be less than one. moreover, achieving this may be too costly, because it means that for constant controls, one needs to keep treating for infinite time. therefore, we considered time dependent controls as a way out, to ensure the eradication of the disease in a finite time. in this light, we addressed the optimal control by deriving and analyzing the conditions for optimal eradication of the disease. from our numerical results, we can conclude that immune responses play a significant role in eradication of disease. moreover, to eradicate the disease, it is important to manufacture a drug treatment with hight efficacy which can block the viral production. references [1] hoofnagle j, mullen k, peters m (1987);“ treatment of chronic delta hepatitis with recombinant alpha interferon”. in: rizzetto m, gerin jl, purcell rh, editors. the hepatitis delta virus and its infection. new york: alan r. liss. 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(2015); “a generalized hbv model with diffusion and two delays”. comput. math. appl., 69, 3140. [35] m. i. kamien and n. l. schwarz,(1991); “dynamic optimization: the calculus of variations and optimal control”, north holland, amsterdam. [36] chien rn, (2008); “current therapy for hepatitis c or d or immunodeficiency virus concurrent infection with chronic hepatitis b. hepatol int 2: 296303. [37] lewin sr, ribeiro rm, walters t, lau gk (2001) “analysis of hepatitis b viral load decline under potent therapy: complex decay profiles observed”. hepatology 34: 10121020. [38] noura yousfi khalid hattaf abdessamad tridane, (2011); modeling the adaptive immune response in hbv infection, j. mathematical biology 63 933957. biomath 7 (2018), 1807219, http://dx.doi.org/10.11145/j.biomath.2018.07.219 page 17 of 17 http://dx.doi.org/10.11145/j.biomath.2018.07.219 introduction the hbv model the model description analysis of the model positivity and boundedness of solutions basic reproduction number and equilibria stability of equilibria the hbv-hdv co-infection model model construction mathematical analysis of the model positivity and boundedness of trajectories basic reproduction number and equilibria stability of equilibria optimal control of treatment in the hbv-hdv co-infection model analysis of optimal controls numerical results discussion and conclusion references www.biomathforum.org/biomath/index.php/biomath original article enumerative numerical solution for optimal control using treatment and vaccination for an sis epidemic model vianney mbazumutima∗, christopher thron†, léonard todjihounde‡ ∗department of mathematics institute of mathematics and physical sciences (imsp), porto-novo, bénin vianney.mbazumutima@imsp-uac.org † department of sciences and mathematics texas a & m university-central texas, killeen tx 76549 usa. thron@tamuct.edu ‡ department of mathematics institute of mathematics and physical sciences (imsp), porto-novo, bénin leonardt@imsp-uac.org received: 14 october 2019, accepted: 13 december 2019, published: 18 december 2019 abstract—optimal control problems in mathematical epidemiology are often solved by hamiltonian methods. however, these methods require conditions on the problem to guarantee that they give global solutions. because of the improved computational power of modern computers, numerical approximate solutions that systematically try a large number of possibilities have become practical. in this paper we give an efficient implementation of an enumerative numerical solution method for an optimal control problem, which applies to cases where standard methods cannot guarantee global optimality. we demonstrate the method on a model where vaccination and treatment are used to control the level of prevalence of an infectious disease. we describe the solution algorithm in detail, and verify the method with simulations. we verify that the enumerative numerical method produces solutions that are locally optimal. keywords-epidemic model, vaccination, treatment, optimal control, numerical method, enumerative method, global optimum. i. introduction within the field of epidemiology, extensive research efforts have been devoted to establishing mathematical models that accurately characterize disease dynamics, including the effects of disease controls. numerous mathematical techniques have been developed since the groundbreaking 1926 paper by a. g. m’kendrick [37]. the most basic and most widely used models in epidemiology are multi-compartment models such as sis (susceptible-infected-susceptible), sir copyright: c©2019 mbazumutima et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: vianney mbazumutima, christopher thron, léonard todjihounde, enumerative numerical solution for optimal control using treatment and vaccination for an sis epidemic model, biomath 8 (2019), 1912137, http://dx.doi.org/10.11145/j.biomath.2019.12.137 page 1 of 22 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2019.12.137 v. mbazumutima, c. thron, l. todjihounde, enumerative numerical solution for optimal control using ... (susceptible-infected-recovered) and similar models. the sis model is suitable to model diseases which do not confer immunity. some of these diseases are discussed in [1] and [28]. in the basic sis model, individuals move between two compartments, susceptible and infected [22]. numerous elaborations of the basic model have been developed, such as age-structured epidemic models [15], stochastic models [39], and models with vaccination [23]. important model properties include the basic reproduction number, equilibria and stability characteristics. in mathematical models with controls, finding optimal controls is an important problem with significant practical implications. many general numerical techniques have been developed for this purpose [8]. in [10] it is suggested a new methodology to find solution for an optimal control problems with delays by shooting method which is mixed with continuation on the delay. a formulation on binary indicator functions, direct and simultaneous adaptive collocation approach to optimal control are developed in [9] while in [13] proposes a numerical solution technique for constrained optimal control problems with parameters where an extension penalty function is used to adjust the state, controls, and parameter inequality constraints. in [26] an approach with haar wavelets method is applied for finding the piecewise constant feedback controls for a finitetime linear optimal control problem of a timevarying state-delayed system. a direct method based on hybrid of block-pulse functions and legendre polynomials is discussed in[35] while a direct collocation method is used in [49] to solve numerically optimal control problems. in [50], a sequential quadratic programming is used to solve an optimal control problem which has convex control constraints. [43] studies dynamical tunneling versus fast diffusion for a non-convex hamiltonian and find that dynamical tunneling results at an important quicker rate than classical transport while [14] develops for certain nonconvex hamilitonian-jacobi equations, their homogenization and non-homogenization. an algorithm for solving a non-convex state-dependent hamilton-jacobi partial differential equations is established in [12]. researchers have also been involved in finding solutions to the infectious diseases and techniques to control them. in [40], r. m. neilan and s. lenhart introduce the theory of optimal control applied to systems of ordinary differential equations with an application on seir model while in [33] s. lenhart and j. t. workman give an interesting overview on optimal control applied to biological models. many authors have focused on this topic of optimal control of different diseases. treatment in the sis model under learning have been considered in [34], while in [19], treatment is used in the controlled sis model, but considers different cost structures than the earlier literature. for controlling an epidemic spread, optimal quarantine programs are used in [47], while [16] and [1] consider non-vaccine prevention in the si and sis models respectively. the prevention by strategic individuals in linked sub-populations is analyzed in [45], while [46] considers prevention through social distancing. the prevention and treatment in an sis framework are considered in [17]. in [51], vaccination and treatment are used in an sir setting and simulate optimal paths while a similar approach is discussed in [5]for an hiv type disease. in [29], it was examined the fundamental role of three type of controls, personal protection, treatment and mosquito reduction strategies in controlling malaria. in addition to prevention and treatment strategies and the allocation of resources available to reduce these diseases, researchers have focused in the development of techniques to evaluate which are more effective for approximating the solutions of those epidemic models. in [52], a. zeb et al. studied the seir and employed the multistep generalized differential transform method and compared the results with those obtained by the fourth-order runge-kutta method and nonstandard finite difference method in the case of integer. in [2] f.s. akinboro et al. used differential transformation method and variational iteration biomath 8 (2019), 1912137, http://dx.doi.org/10.11145/j.biomath.2019.12.137 page 2 of 22 http://dx.doi.org/10.11145/j.biomath.2019.12.137 v. mbazumutima, c. thron, l. todjihounde, enumerative numerical solution for optimal control using ... method to obtain the approximate solution of sir model with initial condition. o. j.peter and al.[3] use the differential transform method to study the transmission dynamics of typhoid fever diseases in a population while [38] uses differential transformation method and variational iteration method in finding the approximate solution of ebola model. in [20], it is investigated the application of matrix nonstandard finite difference schemes to obtain numerical solutions of epidemic models while [4] applies non standard finite difference (nsfd) scheme to a modified sir epidemic model with the effect of time delay. laplace-adomian decomposition method (ladm) in [21] is used to study the fractional order childhood disease model and shows that this method provides excellent numerical solutions for nonlinear fractional order models compared to homotopy analysis, homotopy perturbation method, and fourth order runge-kutta. in [7], the homotopy analysis method (ham ) is applied and finds that it is successfully for finding the approximate solution of fractional sir model. in [6] a. j. arenas and al. developed the non standard finite difference scheme with conservation law (nsfdcl) for predictorcorrector type for epidemic models and compared the result with the rungekutta type schemes. v. k. srivastava and al. [48] compare the solution from euler’s and rk4 methods with those obtained by the differential transform method when they studied hiv infection of cd4+ t cells. in this paper, we will investigate the effect of the vaccination and treatment on an sis epidemic model. in this model, neither the pontryagin theorem for local optimality nor the arrow theorem for global optimality applies, so the usual analytical methods for finding locally or globally optimal solutions cannot be used. we develop instead a numerical algorithm that first finds the overall best control from a large class of controls, and then improve it successively until local optimal conditions are satisfied. the paper is organized as follows. in section 2, we recall basic analytical results on optimal control problems that give necessary and sufficient conditions for a globally optimal solution. in section 3, we establish an sis epidemic model under treatment and vaccination controls, and in section 4, we formulate objective functions for the model. in section 5 we discuss the necessary conditions for locally optimal control vaccination and treatment. section 6 describes an enumerative numerical method for finding near-optimal controls, while in section 7, we present simulation results and conclude. ii. some basic results on optimal control problems two of the foundational results in optimal control theory are pontryagin’s maximum principle and the arrow sufficiency theorem. pontryagin’s theorem gives conditions that a locally optimal solution must satisfy; while arrow’s theorem guarantees global optimality of a locally optimal solution. these theorems are stated below. theorem 2.1: (pontryagin’s maximum principle)[33] if u∗(t) and x∗(t) are optimal for the problem max u j[x(t),u(t)], where j[x(t),u(t)] = ∫ tf t0 f(t,x(t),u(t))dt, (1) subject to{ dx dt = g(t,x(t),u(t)), x(t0) = x0, . (2) where the functions f and g are continuously differentiable and x(t) is piecewise differentiable. then there exists a piecewise differentiable adjoint function λ(t) such that h(t,x∗(t),u(t),λ(t))≤h(t,x∗(t),u∗(t),λ(t)), (3) for all controls u at each time t, where the hamiltonian h is given by h(t,x(t),u(t),λ(t)) = f(t,x(t),u(t)) + λ(t)g(t,x(t),u(t)), (4) and{ λ′(t) = −∂h(t,x ∗(t),u∗(t),λ(t)) ∂x , λ(tf ) = 0. (5) biomath 8 (2019), 1912137, http://dx.doi.org/10.11145/j.biomath.2019.12.137 page 3 of 22 http://dx.doi.org/10.11145/j.biomath.2019.12.137 v. mbazumutima, c. thron, l. todjihounde, enumerative numerical solution for optimal control using ... theorem 2.1 is proved in [44]. the following theorem gives sufficient conditions under which local optima are also globally optimal. theorem 2.2: (the arrow theorem)[11] for the optimal control problem (2), the conditions of the maximum principle are sufficient for the global minimization of j[x(t),u(t)] if the maximized hamiltonian function h, defined in(4), is convex in the variable x for all t in the time interval [t0, tf ] for the given λ. theorem 2.2 is proved in [24] and [36]. when the conditions of arrow’s theorem are not satisfied, it may be difficult to guarantee global optimality of a locally optimal solution. in this case, numerical algorithms that widely explore the space of possible controls may be used to avoid converging prematurely to a local optimum that is not globally optimal. the strategy used in this paper is to find the best control from a large class of controls and then improve it successively until local optimal conditions are satisfied. while this strategy does not necessarily give the global optimum, by first doing a preliminary search over a large class of controls it does help to prevent obtaining a suboptimal solution that is only locally optimal. iii. sis model under treatment and vaccination controls the sis (susceptible-infected-susceptible) model is the model where a susceptible individual is sick and when he recovers immediately becomes susceptible again. in this basic model, each individual belongs in one of the following two states: susceptible or infectious. in the literature, many studies have been made to analyze the importance of the use vaccination and treatment on the spread of infectious diseases by using the control theory ([25]; [27]; [31]; [30]; [32]). those optimal controls techniques play the role of limiting the spread of the infectious disease from the concerned population. in this section, we will etablish the controlled sis system . the classical sis epidemic model under vaccination and treatment has four groups or compartments, whose populations are represented by four letters: s, the number of individuals who are healthy but susceptible to the infection; i, the number of individuals who have been contaminated and can spread the infection to susceptibles; t , the number of individuals who have undergone treatment to cure an infection; and v the number of individuals who have been vaccinated when susceptible. we also denote the total population by n, so that n = s +i +v +t . note s,i,v,t may all vary with time, so that all are represented as functions of time. in our model we suppose that individuals enter and leave the population (either by birth/death or immigration/emigration), but the total population remains constant. individuals leave from each group in the same proportion, but all incoming individuals are susceptible. we suppose that susceptibles are infected by direct contact with infected individuals, so that susceptibles’ rate of infection is proportional to the number of infected individuals. we suppose that susceptible individuals are vaccinated at a given rate (which may depend on time), and infected individuals are treated and become no longer infective at a different rate (also possibly time-dependent). finally, we assume that the vaccination is not efficacious at hundred percent, the vaccinated individuals who contact infected individuals may become reinfected at the small rate. these assumptions are represented by the following system of ordinary differential equations:   ds dt = µn −βsi + γi −µs −u1s, di dt = βsi − (µ + γ + u2)i + β�v i, dv dt = u1s −µv −β�v i, dt dt = u2i −µt, n = s + i + t + v, (6) with initial conditions s(0) =s0,i(0) =i0,v (0) =v0,t(0) =t0, (7) biomath 8 (2019), 1912137, http://dx.doi.org/10.11145/j.biomath.2019.12.137 page 4 of 22 http://dx.doi.org/10.11145/j.biomath.2019.12.137 v. mbazumutima, c. thron, l. todjihounde, enumerative numerical solution for optimal control using ... and control conditions 0 ≤ ui(t) ≤ uimax ≤ 1, i = 1, 2. the parameters in (6) have the following significances: µ and γ are the population replacement rate and the recovery rate from infection respectively; β = λ n is the disease transmission coefficient, so that 1/β is the average number of infective contacts per unit time which result in the susceptible individual becoming infected; � is the fraction of vaccinated individuals for whom the vaccine is ineffective; and u1(t) and u2(t) are the proportionate vaccination and treatment control levels, so that for example u1(t)s(t) is the number of susceptible individuals vaccinated per unit time at time t. we may rewrite the system (6) in matrix format. define ~x(t) ≡ (s(t),i(t),v (t),t(t))t , (8) and define the matrix a(~x(t)) as follows: a(~x(t)) = −βi(t)−u1(t)−µ γ 0 0βi(t) −(µ+γ+u2(t)) β�i(t) 0 u1(t) 0 −(µ+β�i(t)) 0 0 u2(t) 0 −µ  , (9) where n = s0 + i0 + v0 + t0. then system (6) can be written as d dt ~x(t) = a(~x(t))~x(t) +   µn 0 0 0   , (10) with initial conditions ~x(0) = (s0,i0,v0,t0). definition 3.1: [18]. metzler matrices are square (real) matrices in which all the off-diagonal elements are non-negative: aij ≥ 0,∀i 6= j. the matrix a is a metzler matrix according to definition (3.1). proposition 3.1: the set γ = {(s,i,v,t) ∈ r4+} is positively invariant for the system (6). proof: whenever xj = 0 and xi ≥ 0 for i 6= j, since a is metzler matrix it follows that dxj/dt ≥ 0. it follows that if xi(0) ≥ 0 ∀i, then none of the xi(t) will change sign for t ≥ 0. therefore, we conclude that the system (6) determined by the matrix(9) preserves invariance of the non-negative cone r4+. iv. objective functions formulation in this section, we establish objective functions in case of nonlinear cost function and the piecewise linear cost function. a. original objective function we suppose a nonlinear cost function in order to take into account various types of costs affecting the susceptible and infected populations. the cost function for system (6) takes the following form: j(u1,u2,s,i) =∫ tf 0 [ f1(u1(t),s(t))+f2(u2(t),i(t)) ] dt+zi(tf ), (11) where f1(u1,s) ={ c′0 if u1 = 0, c0 + c1u1s + c2u 2 1 if 0 < u1 ≤ u1max, (12) and f2(u2,i) ={ d′0 + d2i if u2 = 0, d0 + d1u2i + d2i if 0 < u2 ≤ u2max, (13) the different terms in (11)-(13) are motivated as follows. the functions f1 and f2 represent cost rates associated with vaccination and treatment respectively, while the final additive term in (11) represents costs attributed to latent infections which remain after the treatment period is complete. as far as vaccination cost, we expect a fixed, low-level maintenance cost rate during time periods when no active vaccination efforts are being made: this fixed cost rate is represented by the constant c′0 in (12). when vaccination efforts are being prosecuted, higher fixed costs are incurred, including salaries and facilities: this higher cost level is represented by c0 in (12), where c0 > c′0. we suppose that each vaccination has a fixed biomath 8 (2019), 1912137, http://dx.doi.org/10.11145/j.biomath.2019.12.137 page 5 of 22 http://dx.doi.org/10.11145/j.biomath.2019.12.137 v. mbazumutima, c. thron, l. todjihounde, enumerative numerical solution for optimal control using ... cost c1, which multiplies the total vaccination rate u1s in (12). finally, the quadratic term c2u21 is included to model diminishing returns of higher levels of vaccination effort, including the cost of vaccinating harder-to-reach or less cooperative individuals. as far as treatment cost, as with vaccination we also expect a low-level maintenance cost rate even when no treatments are given, which is modeled by the constant d′0 term in (13). there is also a cost rate per infected individual (which may include both financial cost and quality-of-life cost), expressed by the coefficient d2 in (13). lastly, each patient who receives treatment has a cost rate d2, which multiplies the treatment rate u2i to give the non-fixed cost rate associated with patient care. the values u1max and u2max represent maximum population penetration rates for vaccination and treatment, respectively. for example, a value u1max = 0.05 indicates that at most 5% of the susceptible population can be vaccinated per basic time unit (which is typically taken as days); while a value u2max = 0.1 means that at most 10% of the infected population is treated per day. in summary, the optimization problem is to find the controls (u∗1,u ∗ 2) that minimize the objective function: (u∗1,u ∗ 2) = argmin u1,u2 j(u1,u2,s,i), (14) where (u1,u2) ∈ u such that 0 < u1 < u1max and 0 < u2 < u2max. (15) in this problem, the cost function is not continuously differentiable, and the hamiltonian is not convex. so neither theorem 2.1 nor theorem 2.2 can be applied in this case. b. piecewise linear objective function since we are using an enumerative approach to numerical solution, it is preferred to have a finite number of possible optimal control values. in section v, we will show that the optimal treatment level is either 0 or u2max, given the cost function (13). however, with the cost function (12) there is an infinite number of possible optimal values. for this reason, we consider a modification of the vaccination cost function f1 which closely approximates (12), but which leads to a finite number of optimal vaccination levels (as will be shown in the next section). the modified function is piecewise linear with the following mathematical form: f1(u1,s) = (16)  c′0 if u1 = 0, c0 + c1u1s + c ′ 2u1 + c ′ 3(u1 −u1mid) +, if 0 < u1 ≤ u1max, where the function x+ is the ramp function, x+ = max(x, 0). (17) the constants u1mid,c′2 and c ′ 3 may be chosen to approximate the quadratic term c2u21 in (12). figure 1 shows various possibilities for piecewise linear approximations. given the constant c2, the values u1mid = 0.5, c′2 = 0.5c2 and c ′ 3 = c2 produce a cost function that is an upper bound to the quadratic term c2u21; the values u1mid = 5/9, c′2 = 0.2c2, and c ′ 3 = 2c2 produce an effective lower bound; and the values u1mid = 0.5, c′2 = ( √ 2−1)c2 and c′3 = (3− √ 2)c2 gives an approximation that minimizes the maximum deviation (maximum deviation is equal to 0.043c2). using these different functions, upper and lower bounds on the optimal cost for the original quadratic cost function (12) may be obtained. v. local optimality conditions for the non-autonomous vaccination and treatment control problem a solution ~x(t) with controls u1(t) and u2(t) is locally optimal if small perturbations of the controls u1 and u2 during small time intervals never decrease the cost. this means that there is no way to improve the solution by making slight adjustments to the controls. local optimization is applicable to the global problem in that a globally optimal solution must also be locally optimal. hence local optimality is a necessary condition for global optimality. in this section, we explicitly calculate the effect of local changes in the controls biomath 8 (2019), 1912137, http://dx.doi.org/10.11145/j.biomath.2019.12.137 page 6 of 22 http://dx.doi.org/10.11145/j.biomath.2019.12.137 v. mbazumutima, c. thron, l. todjihounde, enumerative numerical solution for optimal control using ... figure 1. approximation of the quadratic cost function with piecewise linear functions. the linear functions shown give upper and lower bounds to the quadratic cost function, as well as a best approximation that minimizes the maximum deviation between the quadratic cost function and piecewise linear approximation. u1(t) and u2(t) on the cost function. changes in the two controls are considered separately, because in practice they can be varied independently. this gives us a way to identify local optima, which correspond to control levels for which differential changes yield no improvement. in previous sections, we have considered the autonomous problem in which the parameters β,�,γ, and λ and also the costs c0,c′0,d0,d ′ 0,c1,c2,d1,d2 are constants independent of time. in this section we consider the more general non-autonomous problem, in which all parameters can be continuous function of time. the reader should understand that parameters and costs in this section now represent time dependent functions (e.g. β represents β(t)). a. necessary conditions for the optimal control vaccination with simplified cost function first, we will perturb the control u1(s) and calculate the effect on the cost function. given a control u1(t), the perturbed control u′1(s,t) differs from u1(t) by a small amount on an interval of length δ, as follows: u′1(s,t) ={ u1(t) + du1 for s < t < s + δ, u1(t) otherwise, (18) where s is a fixed value between 0 and tf . the system evolution x′(s,t) corresponding to controls u′1(s,t) and u2(t) may be written ~x ′(s,t) ≡ (s′(s,t),i′(s,t),v ′(s,t),t ′(s,t))t , (19) and satisfies the equations ~x′(s, 0) = ~x(0), ∂ ∂t ~x′(s,t) =   a(~x′(t))~x′(s,t) for t < s and t > s + δ, a(~x(t))~x′(s,t)−s′(s,t)du1~∆10 for s ≤ t ≤ s + δ, (20) biomath 8 (2019), 1912137, http://dx.doi.org/10.11145/j.biomath.2019.12.137 page 7 of 22 http://dx.doi.org/10.11145/j.biomath.2019.12.137 v. mbazumutima, c. thron, l. todjihounde, enumerative numerical solution for optimal control using ... where −→ ∆10 ≡ (1, 0,−1, 0)t . (21) system (20) corresponds to system (10) with perturbed control u′1(s,t) on the interval s ≤ t ≤ s + δ. note that ~x′(s,t) = ~x(t) for t ≤ s since ~x′(s, 0) = ~x(0) and ~x′(s,t) and ~x(t) satisfy the same differential equation on [0,s]. in order to compute the new cost associated with the perturbed solution x′(s,t) with controls u′1(s,t) and u2(t), we need to solve (20) for x′(s,t). since equations (20) have different forms on the intervals s < t < s + δ and t > s + δ, we will treat these two intervals separately. first we find the solution on s ≤ t ≤ s + δ. for t > s, we may define: (∆1s(s,t), ∆1i(s,t), ∆1v (s,t), ∆1t (s,t)) ≡ 1 a [ s(t) −s′(s,t),i(t) − i′(s,t), v (t) −v ′(s,t),t(t) −t ′(s,t) ] , (22) where a = s(s)du1δ. (23) we also use the notation: ~∆1(s,t)≡(∆1s(s,t),∆1i(s,t),∆1v(s,t), ∆1t(s,t))t. (24) note that ~∆1(s,s) = ~x(s) −~x′(s,s) = ~0. for simplicity, in the remaining discussion we will write the functions s,s′,i,i′,v,v ′,t,t ′, ∆1, ∆1, ∆2 without arguments (e.g. we write i′(s,t) as i′). from (10) and (20) we have for interval s < t < s + δ,  ds dt =µn −βsi + γi −µs −u1s, ds′ dt =µn−βs′i′+γi′−µs′−u1s′−s′du1. (25) using the notations (23) and (24) for the system (25), we obtain for s < t < s + δ ds dt − ds′ dt =−aβ(i∆1s +s∆1i)+aγ∆1i −aµ∆1s−au1∆1s +s′du1 +o(a2). (26) using similar computations, we may obtain corresponding equations for infected, vaccinated, and treated compartments on the interval s < t < s+δ: di dt − di′ dt =aβ(∆1si+∆1is)−(µ+γ+u2)a∆1i + β�(a∆1iv + a∆1v )i + o(a2). (27) dv dt − dv ′ dt =au1∆1s−aµ∆1v −aβ�(∆1v i+∆1iv )−s′du1 +o(a2). (28) dt dt − dt ′ dt = au2∆1i −aµ∆1t . (29) equations (26)-(29) may be rewritten in matrix form: ∂ ∂t [ ~∆1(s,t) ] =(−βi−u1−µ −βs+γ 0 0 βi −(µ+γ+u2)+βs+β�v β�i 0 u1 −β�v −µ−β�i 0 0 u2 0 −µ ) ~∆1(s,t) + ~∆10 δ + o(a). (30) using (30), we have on t ∈ [s,s + δ] ∂~∆1(s,t) ∂t =ã(~x(t))~∆1(s,t)+ 1 δ −→ ∆10 +o(a), (31) where ã(~x(t)) = a(~x(t)) + ( 0 −βs 0 0 0 β(s+�v ) 0 0 0 −β�v 0 0 0 0 0 0 ) . (32) we also note ~∆1(s,s) = ~0. to lowest order in δ, the solution of (31) gives ~∆1(s,s + δ) ≈ ~∆10. on the interval t ∈ [s+δ,tf ], ~∆1(s,t) is a solution to ∂~∆1(s,t) ∂t = ã(~x(t))~∆1(s,t) (33) we are now ready to compute the difference between cost functions for x′ and x. we denote these cost functions by j′ and j respectively. for simplicity, we first consider the case where c′0 = c0 and d′0 = d0. from (11),(13) and (16), we obtain the sequence of equations (34)-(36), where h(x) in (36) denotes biomath 8 (2019), 1912137, http://dx.doi.org/10.11145/j.biomath.2019.12.137 page 8 of 22 http://dx.doi.org/10.11145/j.biomath.2019.12.137 v. mbazumutima, c. thron, l. todjihounde, enumerative numerical solution for optimal control using ... j′ = ∫ s 0 [ [c0 + c1u1(t)s(t) + c ′ 2u1 + c ′ 3(u1 −u1mid) +] + [d0 + d1u2(t)i(t) + d2i(t)] ] dt + ∫ s+δ s [ c0 + c1(u1(t) + du1)s ′(s,t) + c′2(u1(t) + du1) + c ′ 3((u1 + du1) −u1mid) + + d0 + (d1u2(t) + d2)i ′(s,t) ] dt + ∫ tf s+δ [ [c0 +c1u1(t)s ′(s,t)+c′2u1 +c ′ 3(u1−u1mid) +]+[d0 +(d1u2(t)+d2)i ′(s,t)] ] dt + i′(s,tf ); (34) j = ∫ s 0 [ [c0 + c1u1(t)s(t) + c ′ 2u1 + c ′ 3(u1 −u1mid) +] + [d0 + (d1u2(t) + d2)i(t)] ] dt + ∫ s+δ s [ [c0 + c1u1(t)s(t) + c ′ 2u1 + c ′ 3(u1 −u1mid) +] + [d0 + (d1u2(t) + d2)i(t)] ] dt + ∫ tf s+δ [ [c0 + c1u1(t)s(t) + c ′ 2u1 + c ′ 3(u1 −u1mid) +] + [d0 + (d1u2(t) + d2)i(t)] ] dt + i(tf ); (35) j′−j du1 = ∫ s+δ s [ −s(s)δ (c1u1(t)∆1s(s,t)+[d1u2(t)+d2]∆1i(s,t))+c1s′1(s,t)+c ′ 2 +c ′ 3h(u1−u1mid) ] dt −s(s)δ [∫ tf s [c1u1(t)∆1s(s,t) + (d1u2(t) + d2)∆1i(s,t)] ] dt−s(s)δ∆1i(s,tf ). (36) the heaviside (step) function (note the heaviside function is the derivative of the ramp function). taking the limit as du1 → 0 for small δ we obtain ∂j ∂u1 =δ [ c1s+c ′ 2 +c ′ 3h (u1(s)−u1mid)−ψ1 ] , (37) where ψ1≡s(s) [∫ tf s [ c1u1(t)∆1s(s,t) + (d1u2(t)+d2)∆1i(s,t) ] dt+∆1i(s,tf ) ] . (38) in this case, the local optimum conditions on vaccination control u1(s) depend on 3 cases: (a) ψ1 ≤ c1s + c′2: then ∂j ∂u1(s) ≥ 0 and j(u1(s)) is minimized when u1(s) = 0. (b) c1s + c′2 ≤ ψ1 ≤ c1s + c ′ 2 + c ′ 3: in this case, it is necessary to check whether j(u1mid) < j(0). using (37), we obtain j(u1mid) −j(0+) = δu1mid[c1s + c′2 − ψ1] and j(0+) −j(0) = (c0 − c′0)δ. so, j(u1mid) < j(0) is equivalent to ψ1 > (c0 − c′0)/u1mid + c ′ 2 + c1s. if j(u1mid) − j(0) < 0 then u1(s) = u1mid is locally optimal, otherwise u1(s) = 0 is locally optimal. (c) ψ1 ≥ c1s+c′2+c ′ 3: in this case, it is necessary to check whether j(u1max) < j(0). from biomath 8 (2019), 1912137, http://dx.doi.org/10.11145/j.biomath.2019.12.137 page 9 of 22 http://dx.doi.org/10.11145/j.biomath.2019.12.137 v. mbazumutima, c. thron, l. todjihounde, enumerative numerical solution for optimal control using ... (37), we have j(u1max)−j(0+) = δ[u1max(c′2 − ψ1) +c′3(u1max −u1mid)] and j(0+) −j(0) = δ(c0 − c′0). so j(umid) < j(0) is equivalent to ψ1 > c1s + c ′ 2 + c ′ 3(1−u1mid/u1max) + δ(c0−c ′ 0). if j(u1max) −j(0) < 0 then u1(s) = u1max is optimal otherwise if j(u1max) −j(0) > 0 then u1(s) = 0 is optimal. b. necessary conditions for the optimal control treatment to find the necessary conditions for the optimal control treatment function u2(t), we will perturb the control u2(s) and calculate the effect on the cost function. the perturbed control u′2 which differs from u2 by a small amount on an interval of length δ is u′2(s,t) = { u2(t) + du2 for s ≤ t ≤ s + δ, u2(t) otherwise . (39) for this purpose, we define ~x ′(s,t) ≡ (s′(s,t),i′(s,t),v ′(s,t),t ′(s,t))t , (40) which satisfies the equations: ~x ′(s, 0) = ~x(0), ∂ ∂t ~x ′(s,t) ={ a(~x(t))~x ′(s,t) for t < s or t > s + δ, a(~x(t))~x ′(s,t)−i′(s,t)du2~∆20 for s≤t≤s+δ, (41) where −→ ∆20 ≡ (0,−1, 0, 1)t . (42) system (41) corresponds to system (10) with perturbed control u′2(s,t) on the interval s ≤ t ≤ s + δ. note that ~x′(s,t) = ~x(t) for t ≤ s since ~x′(s, 0) = ~x(0) and ~x′(s,t) and ~x(t) satisfy the same differential equation on [0,s]. in order to compute the new cost associated with the perturbed solution x′(s,t) with controls u′2(s,t) and u1(t), we need to solve (20) for x′(s,t). since equations (20) have different forms on the intervals s < t < s + δ and t > s + δ, we will solve for x′(s,t) on these two intervals separately. for s ≤ t ≤ s + δ, we define : (∆2s(s,t), ∆2i(s,t), ∆2v (s,t), ∆2t (s,t)) ≡ 1 b [s(t)−s′(s,t),i(t)−i′(s,t),v (t)−v ′(s,t), t(t) −t ′(s,t)], (43) where b ≡ i(s)du2δ. the following notation will be also used: ~∆2(s,t)≡(∆2s(s,t), ∆2i(s,t), ∆2v(s,t), ∆2t(s,t)) t . (44) note that ~∆2(s,s) = ~0. following the same arguments used from (25) to (30), to first order for s < t < s + δ, we obtain ∂~∆2(s,t) ∂t = ã(~x(t))~∆2(s,t) + 1 δ −→ ∆20, (45) and for t > s + δ ∂~∆2(s,t) ∂t = ã(~x(t))~∆2(s,t). (46) to lowest order in b, the solution of (45) gives ~∆2(s,s + δ) ≈ ~∆20, (47) and ~∆2(s,t) is solution to  ∂~∆2(s,t) ∂t = ã(~x(t))~∆2(s,t) + 1 δ ~∆20 for t ∈ (s,s + δ), ∂~∆2(s,t) ∂t = ã(~x(t))~∆2(s,t) for t ∈ [0,s] ∪ [s + δ,∞). by the same argument which has been used for (34) and (35), we have the series of equations (48)(50), where b ≡ i(s)du2δ in (50). biomath 8 (2019), 1912137, http://dx.doi.org/10.11145/j.biomath.2019.12.137 page 10 of 22 http://dx.doi.org/10.11145/j.biomath.2019.12.137 v. mbazumutima, c. thron, l. todjihounde, enumerative numerical solution for optimal control using ... j = ∫ s 0 [ [c0 + c1u1(t)s(t) + c ′ 2u1(t) + c ′ 3(u1 −u1mid) +] + [d0 + (d1u2(t) + d2)i(t)] ] dt + ∫ s+δ s [ [c0 + c1u1(t)s(t) + c ′ 2u1(t) + c ′ 3(u1 −u1mid) +] + [d0 + (d1u2(t) + d2)i(t)] ] dt + ∫ tf s+δ [ [c0 + c1u1(t)s(t) + c ′ 2u1(t) + c ′ 3(u1 −u1mid) +] + [d0 + (d1u2(t) + d2)i(t)] ] dt + i(tf ); (48) j′ = ∫ s 0 [ [c0 + c1u1(t)s(t) + c ′ 2u1(t) + c ′ 3(u1 −u1mid) +] + [d0 + d1u2(t)i(t) + d2i(t)] ] dt + ∫ s+δ s [ [c0 + c1u1(t)s ′(s,t) + c′2u1(t) + c ′ 3(u1 −u1mid) +] + [d0 + d1(u2(t) + du2)i ′(s,t) + d2i ′(s,t)] ] dt + ∫ tf s+δ [ [c0 + c1u1(t)s ′(s,t) + c′2u1(t) + c ′ 3(u1 −u1mid) +] + [d0 + d1u2(t)i ′(s,t) + d2i ′(s,t)] ] dt + i′(s,tf ); (49) j′ −j du2 = ∫ s+δ s [ −s(s)δ (c1u1(t)∆2s(s,t) + [d1u2(t) + d2]∆2i(s,t)) + d1i′(s,t) ] dt − i(s)δ [∫ tf s+δ [(d1u2(t) + d2)∆2i(s,t) + c1u1(t)∆2s(s,t)]dt + ∆2i(s,tf ) ] . (50) taking limits as before, we obtain 1 δ ∂j ∂u2(s) = i(s)(d1 − ψ2), (51) where ψ2≡ ∫ tf s [ (d1u2(t)+d2)∆2i(s,t) + c1u1(t)∆2s(s,t) ] dt −∆2i (s,tf ). (52) for this case, there is no effect from the control u2(s) on ∂j ∂u2(s) · the local optimum conditions on u2(s) depends 3 cases: (a) d1 > ψ2 =⇒ ∂j∂u2(s) > 0, then j(u2(s)) is minimized when u2(s) = 0, (b) d1 = ψ2 =⇒ ∂j∂u2(s) = 0, then u2(s) has no effect on j(u2(s)), (c) d1 < ψ2 =⇒ ∂j∂u2(s) < 0, then j(u2(s)) is minimized at u2 when u2(s) = u2max. c. summary of necessary conditions for optimal controls the following theorem summarizes the result of the previous discussion: theorem 5.1: suppose u∗1 and u ∗ 2 are locally optimal controls for the system (6) with objective function (11),where c0 > c′0 and d0 > d ′ 0. let us consider ψ1(s) and ψ2(s) given by the expressions (38) and (52) respectively. then (a) if ψ1(s) < 0 or s(s) = 0 then u1(s) = 0 is locally optimal, biomath 8 (2019), 1912137, http://dx.doi.org/10.11145/j.biomath.2019.12.137 page 11 of 22 http://dx.doi.org/10.11145/j.biomath.2019.12.137 v. mbazumutima, c. thron, l. todjihounde, enumerative numerical solution for optimal control using ... (b) if c1s + c′2 ≤ ψ1 ≤ c1s + c ′ 2 + c ′ 3 and ψ1 > (c0−c′0)/u1mid + c1s + c ′ 2 and s(s) > 0 then u∗1(s) = u1mid is locally optimal, (c) if c1s + c′2 ≤ ψ1 ≤ c1s + c ′ 2 + c ′ 3 and ψ1 < (c0 −c′0)/u1mid + c1s + c ′ 2 then u ∗ 1(s) = 0 is locally optimal, (d) if ψ1 ≥ c1s + c′2 + c ′ 3 and ψ1 > c1s + c ′ 2 + c′3(1−u1mid/u1max)+δ(c0−c ′ 0) and s(s) > 0 then u∗1(s) = u ∗ 1max is locally optimal, (e) if ψ1 ≥ c1s + c′2 + c ′ 3 and ψ1 < c1s + c ′ 2 + c′3(1−u1mid/u1max)+δ(c0−c ′ 0) then u ∗ 1(s) = 0 is locally optimal, (f) if ψ2 < d1 or i(s) = 0 then u∗2(s) = 0 is locally optimal, (g) if ψ2 > d1 or i(s) > 0 and ψ2 > (d0−d′0) i(s)u2max + d1 then u∗2(s) = u2max is locally optimal, (h) if ψ2 > d1 or i(s) > 0 and ψ2 = (d0−d′0) i(s)u2max + d1 then u∗2(s) = 0 or u ∗ 2(s) = u2max is locally optimal, (i) if ψ2 > d1, i(s) > 0 and ψ2 < (d0−d′0) i(s)u2max +d1 then u∗2(s) = 0 is locally opimal, (j) if i(s) = 0 then the value of u2(s) has no effect on the solution to system (6) or on the value of j. corollary 5.1: for the system (6) with objective function (11), given any locally optimal control (u∗1,u ∗ 2), then for any 0 ≤ s ≤ tf , we have u∗1 ∈{0,u1mid,u1max} and u∗2 ∈{0,u2max} . proof: follows immediately from theorem 5.1. corollary 5.1 implies that at any given time instant, there are only 6 possible optimal control pairs. so, given that an optimal control is constant on a specified set n of intervals, then there are 6n of possible optimal controls. this fact is based to the discussion in the next section. vi. efficient numerical method for finding near-optimal controls theorem 5.1 gives conditions for local optimality, which does not necessary imply global optimality. many algorithms employ a process which converges to a locally optimal solution given a starting point. if the starting point is close enough to a globally optimal solution, these algorithms will converge to a global optimum. thus, it is important to identify near-optimal solutions. one way on doing this is to find optimal solutions from a large class of controls that are representative of the different possibilities. first, consider the class of control strategies that are constant in intervals of length t/n where t is the total time of the system and n is a positive integer. we also consider strategies that are restricted to the optimal values specified in theorem 5.1. then there are 6n strategies of which meet these conditions. if n is small enough, the best solution from this class can be found by simply evaluating the cost for all 6n strategies. this limits on the size of n for practical computation. in order to increase n, we may make further assumptions. we expect the vaccination level u1mid to occur as the system is transitioning from no control to full control. in order to reduce computation time, we consider only the two extreme vaccination strategies: 0,u1max. this leads to 4n strategies that are constant on the n intervals. on each interval 0 ≤ k ≤ n − 1, there are four strategy options: (u1k,u2k) = {(0, 0), (u1max, 0), (0,u2max), (u1max,u2max)}. these options may be indexed as follows: (u1k,u2k)∈(0, 0) ⇐⇒ an−k−1 = 0, (u1k,u2k) = (u1max, 0) ⇐⇒ an−k−1 = 1, (u1k,u2k) = (0,u2max) ⇐⇒ an−k−1 = 2, (u1k,u2k) = (u1max,u2max) ⇐⇒ an−k−1 = 3, where an−k−1 is the index of the control on interval k with k = 0, · · · ,n−1. then (a0, · · · ,an−1) completely specifies the control. we associate this control with the index an−k−14 n−k−1 + an−k−24 n−k−2 + · · · + a0. schematically, we have the figure 2. for more explanation of the numerical method used, we consider the flowchart in figure 3. biomath 8 (2019), 1912137, http://dx.doi.org/10.11145/j.biomath.2019.12.137 page 12 of 22 http://dx.doi.org/10.11145/j.biomath.2019.12.137 v. mbazumutima, c. thron, l. todjihounde, enumerative numerical solution for optimal control using ... figure 2. strategies computed on kth interval. figure 3. algorithm’s flowchart to find near-optimal strategy an enumerative algorithm for finding a nearoptimal strategy from among these strategies proceeds as follows: algorithm 1 calculates the optialgorithm 1 calculation of optimal piecewise constant control strategy part i. 1: n = number of intervals 2: cbest = 1e100 3: for j in 0, · · · , 4n − 1 do 4: generate strategy sj = (a (j) 0 , · · · ,a (j) n−1) associated with index j 5: evaluate cj = cost of strategy sj 6: if cj < cbest then 7: cbest ← cj 8: sbest ← sj 9: end if 10: end for 11: return cbest,sbest mal piece-wise constant control strategy by computing the costs of all 4n possible strategies. this is computationally expensive. it is possible to greatly reduce costs by reusing cost computations between strategies as follows. suppose s1 and s2 are two strategies which agree on the first k intervals. this means that the cost contributions from first k intervals are the same for both strategies. if these interval costs are known for s1, then it is not necessary to recompute them for s2. thus in the calculation for s2, it is only necessary to compute the costs from the last n−k intervals. it may be shown that on average, only 2 intervals’ costs must be recomputed, instead of n intervals as in algorithm 1. it follows that the total amount of computation required is reduced by a multiplicative factor of 2/n. a pseudo-code for the improved algorithm is shown in algorithm 2. after part i, we consider on flowchart part ii.a, where the intervals are divided in half to obtain 2n intervals. as shown in figure 4, we keep the u20, · · · ,u2n−1 the same. we recompute u′10, · · · ,u ′ 12n−1 by evaluating the 2n strategies and choosing the optimum. the resulting strategy has vaccination constant on 22n intervals and treatment constant on n intervals. we denote the best vaccination strategy obtained from the previous procedure by (u∗10, . . . ,u ∗ 12n−1). after this, the next step shown biomath 8 (2019), 1912137, http://dx.doi.org/10.11145/j.biomath.2019.12.137 page 13 of 22 http://dx.doi.org/10.11145/j.biomath.2019.12.137 v. mbazumutima, c. thron, l. todjihounde, enumerative numerical solution for optimal control using ... algorithm 2 improved version of algorithm 1. 1: n = number of intervals 2: j = 0 3: generate strategy s0 = (a (0) 0 , · · · ,a (0) n−1) 4: compute cost for each interval c0(0), · · · ,c0(n − 1) 5: cbest = c0(0) + · · · + c0(n − 1) 6: sbest = s0 7: for j = 1, · · · , 4n − 1 do 8: compute largest ` such that a(j)` 6= a (j−1) ` 9: cj(0) = cj(k),k = 0 . . .`− 1 10: recompute cj(`), · · · ,cj(n − 1) 11: if ∑ cj(k) < cbest then 12: cbest ← ∑ cj 13: sbest ← (a (j) 0 , · · · ,a (j) n−1) 14: end if 15: end for 16: return cbest,sbest figure 4. schema of flowchart part ii.a. in flowchart part ii.b is to divide the intervals of constant treatment in half and recompute u′20, · · · ,u ′ 22n−1 keeping the vaccination control u∗10, · · · ,u ∗ 12n−1 fixed. the resulting strategy is represented in figure 5. figure 5. schema of flowchart part ii.b. at the end of part ii, the resulting strategy is constant on 2n intervals of equal length. so in part iii, the purpose is to improve the solution by adjusting the sizes of active treatment and vaccination intervals. each iteration of this part iii changes each active treatment or vaccination interval by at most 1. also, this part iii works by considering all strategies that agree with the previous best strategy except at the endpoints of the active treatment or vaccination intervals. figures 6 and 7 show how different vaccination and treatment controls are tried which differ from the previous best solution only where active treatment intervals begin or end. note that the dashed line in figures 6 and 7 represents the previous optimal vaccination and previous optimal treatment respectively. part iii of the flowchart figure 3 has the following pseudocode: algorithm 3 pseudocode for part iii of algorithm. 1: s0 = (a (0) 0 , · · · ,a (0) t/dt−1) is the previous best strategy 2: identify change points in vaccination strategy. n1 = number of change points. 3: j = 0 4: time steps before and after changes =n1 5: identify change points in treatment strategy. n2 = number of change points 6: compute the costs on all 2n intervals for previous optimal strategy c0(0), · · · ,c0(2n−1) 7: sbest = s0 8: for j = 1, · · · , 3n1 2n2 − 1 do do 9: compute next candidate strategy snew = a (j) 1 , . . .a (j) t/dt 10: compute the first ` such that a(j)` 6= a (j−1) ` 11: cj(0) = cj(k),k = 0 . . .`− 1 12: recompute cj(`), · · · ,cj(n − 1) 13: ∑ cj(k) ← cnew 14: if ∑ cj(k) < cbest then 15: cbest ← cnew 16: sbest ← snew 17: end if 18: end for 19: return cbest,sbest biomath 8 (2019), 1912137, http://dx.doi.org/10.11145/j.biomath.2019.12.137 page 14 of 22 http://dx.doi.org/10.11145/j.biomath.2019.12.137 v. mbazumutima, c. thron, l. todjihounde, enumerative numerical solution for optimal control using ... figure 6. vaccination strategy modification in stage iii of algorithm. figure 7. treatment strategy modification in stage iii of algorithm. vii. numerical simulations and discussion a. numerical simulations in this section, we use this algorithm described in section vi to solve the control problem given by equations (11),(12) and (16). table i, shows the baseline parameters for our simulations. in our analysis, we used the baseline configuration described in table i and varied one cost parameter at a time. figures 8 14 show the results for the parameters c0,c1,c2,d0,d1,d2 and z respectively. in each set of three sub-figures, the first two sub-figures give the optimal vaccination and treatment controls found by the algorithm. for all solutions found by the algorithm, local optimality was numerically verified. the third subfigure shows the process of convergence during the algorithm. in each graph, the thin solid line shows the default configuration, and the other two biomath 8 (2019), 1912137, http://dx.doi.org/10.11145/j.biomath.2019.12.137 page 15 of 22 http://dx.doi.org/10.11145/j.biomath.2019.12.137 v. mbazumutima, c. thron, l. todjihounde, enumerative numerical solution for optimal control using ... table i baseline parameters used in simulation parameters interpretation values units source c1 cost coefficient of u1s 10 dimensionless assumed c2 cost coefficient of u21 10 5 dimensionless assumed d1 cost coefficient of u2i 40 dimensionless assumed d2 cost coefficient of i 5 dimensionless assumed z final cost coefficient of i 50 dimensionless assumed c′2 approximative constant of c2 when 0 < u1 ≤ u1mid 0 dimensionless assumed c′3 approximative constant of c2 when u1 ≤ u1mid ≤ u1max 0 dimensionless assumed ncint number of intervals on which control is constant 3 dimensionless assumed β disease transmission coefficient 8×10−5 day−1 assumed γ recovery rate 0.65 ∈ [0.25,1.5] day−1 [42] � small rate infection of vaccinated individuals 0.0001 day−1 assumed µ rate of replacement including both birth/death and immigration/emigration 0.004 day−1 [41] u1max maximum population rate for vaccination 0.05 day−1 assumed u2max maximum population rate for treatment 0.1 day−1 assumed tf final time 100 days assumed dt time increment 0.1 day assumed n population size 10000 humans assumed lines give configurations with different values of the chosen parameter. the labels i.a, ii.a,b and iii on each third subfigure correspond to different parts of the algorithm described in the flowchart in figure 3. b. discussion figures 8, (9) and (10) show that increasing c0,c1 or c2 (which are fixed cost of vaccination and cost per vaccination respectively) reduces the active vaccination interval and increases the active treatment interval. for figures (10) shows that no matter how much the quadratic vaccination cost c2 is increased, the vaccination time interval is reduced while the active treatment time interval is increasing until to a certain maximum level less than 20 . figure 11 shows that increasing the value of d0, increases a little the vaccination interval, while the treatment interval decreases to zero. in figure 12, when the value of d1 is increased, it follows that the vaccination interval is increased slightly while the treatment interval decreases. figure 13 shows that no matter how much d2 is increased, both vaccination and treatment intervals don’t change much. figure 14 shows that by increasing the value of z, the vaccination interval increases a little while the treatment interval is reduced. the rightmost sub-figure in each set of figures shows the rates of convergence, and the costs of the solutions found by the algorithm. the algorithm takes 3 to 20 iterations to converge, and increasing any cost parameter produces increased final cost. typically initial large decreases in cost which represents the cost improvement from algorithm i followed by algorithm iia. usually little improvement iteration of part iii modify each treatment or vaccination interval by stimulating variable reduction in the cost. the algorithm some times iii brings rapid improvement followed by slower. during the rapid phase, both controls are being adjusted. the rapid improvement phase ends when one control has reached its optimal configuration and the control continues to adjust. in most cases, execution time to compute each optimal control was a minute or less on an intel r©core(tm) i3-2328m cpu at 2.20ghz, 2200 mhz, 2 cores, 4 logical processors with 4g ram. conclusions in this paper, we introduced an sis epidemic model under vaccination and treatment controls. we formulated an objective function and simplified it for numerical computations: the simplified objective function can configured to give an upper bound, lower bound, or best estimate for the cost. biomath 8 (2019), 1912137, http://dx.doi.org/10.11145/j.biomath.2019.12.137 page 16 of 22 http://dx.doi.org/10.11145/j.biomath.2019.12.137 v. mbazumutima, c. thron, l. todjihounde, enumerative numerical solution for optimal control using ... figure 8. optimal vaccination, treatment controls and strategy cost for different values of c0. figure 9. optimal vaccination, treatment controls and strategy cost for different values of c1. we determined necessary conditions for optimal control treatment and necessary conditions for optimal control vaccination with simplified cost function. we established an algorithm to optimize the strategy cost. this algorithm has been improved to reduce the execution time to find the best strategy cost. we also verified that the final strategy obtained by the algorithm in simulation biomath 8 (2019), 1912137, http://dx.doi.org/10.11145/j.biomath.2019.12.137 page 17 of 22 http://dx.doi.org/10.11145/j.biomath.2019.12.137 v. mbazumutima, c. thron, l. todjihounde, enumerative numerical solution for optimal control using ... figure 10. optimal vaccination, treatment controls and strategy cost for different values of c2. figure 11. optimal vaccination, treatment controls and strategy cost for different values of d0. satisfies the local optimum conditions given in theorem 5.1 . although, this does not guarantee global optimality, the fact that we have tried a large diversity of strategies in the algorithm makes it plausible that the final strategy is indeed the global optimum. finally, some numerical simulations are presented to illustrate the performance of this algorithm. biomath 8 (2019), 1912137, http://dx.doi.org/10.11145/j.biomath.2019.12.137 page 18 of 22 http://dx.doi.org/10.11145/j.biomath.2019.12.137 v. mbazumutima, c. thron, l. todjihounde, enumerative numerical solution for optimal control using ... figure 12. optimal vaccination, treatment controls and strategy cost for different values of d1. figure 13. optimal vaccination, treatment controls and strategy cost for different values of d2. acknowledgements the support of this research through the german academic exchange service (daad) and the anonymous reviewers are hereby acknowledged. biomath 8 (2019), 1912137, 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[52] anwar zeb, madad khan, gul zaman, shaher momani, and vedat suat ertürk. comparison of numerical methods of the seir epidemic model of fractional order. zeitschrift für naturforschung a, 69(1-2):81–89, 2014. biomath 8 (2019), 1912137, http://dx.doi.org/10.11145/j.biomath.2019.12.137 page 22 of 22 http://dx.doi.org/10.11145/j.biomath.2019.12.137 introduction some basic results on optimal control problems sis model under treatment and vaccination controls objective functions formulation original objective function piecewise linear objective function local optimality conditions for the non-autonomous vaccination and treatment control problem necessary conditions for the optimal control vaccination with simplified cost function necessary conditions for the optimal control treatment summary of necessary conditions for optimal controls efficient numerical method for finding near-optimal controls numerical simulations and discussion numerical simulations discussion references references mathematical modelling of hiv-hcv co-infection dynamics in presence of hiv therapy biomath https://biomath.math.bas.bg/biomath/index.php/biomath b f biomath forum original article mathematical modelling of hiv-hcv co-infection dynamics in presence of hiv therapy edison mayanja1,2,∗, livingstone s. luboobi3, juma kasozi1, rebecca n. nsubuga4 1department of mathematics, makerere university, kampala, uganda edisonmayanja.phd@gmail.com 0000-0002-8077-3083 kasozi@cns.mak.ac.ug 0000-0002-0941-9604 2department of economics and statistics, kabale university, kabale, uganda 3independent researcher, c/o department of mathematics, makerere university, kampala, uganda luboobi@gmail.com 0000-0003-0438-9845 4independent researcher, kampala, uganda nrebeccansubuga@gmail.com 0000-0001-8527-6222 received: november 27, 2021, accepted: july 15, 2022, published: august 11, 2022 abstract: in this work, we formulated and analysed a deterministic model to study the hiv-hcv co-infection dynamics in presence of hiv therapy. the hcv chronic stage was split into two periods: the period before and the period after onset of cirrhosis. this was done because the hcv chronic stage of infection is long, asymptomatic and infectious. the effective reproduction numbers, one of our outcome measures, were computed using the next generation matrix method. numerical simulations were performed to support the analytical results from the model. the different parameters in the model were subjected to a sensitivity analysis to determine their relative importance on the hiv-hcv co-infection dynamics. the results indicated that both hiv and hcv infections enhance each other; and in the long run, increasing the rates at which people are put on hiv treatment reduces the prevalence of hcv in the community; however, it increases the prevalence of hiv. therefore, there should be increased safer sexual behaviour campaigns among individuals on hiv treatment. keywords: hiv/aids, hcv, co-infection, reproduction number, sensitivity analysis, therapy i. introduction the human immunodeficiency virus (hiv) and hepatitis c virus (hcv) are a global challenge. these two viruses have a considerable impact on the global morbidity and mortality. hiv is a virus that weakens the immune system by destroying the cd4+ t-cells and hence making it harder for the body to fight off other infections. hepatitis c infection is a liver disease caused by hepatitis c virus (hcv). globally, chronic hcv infection is the leading cause of chronic liver disease, and is associated with cirrhosis, hepatocellular carcinoma, and liver failure related mortality [1]. both hcv and hiv are blood borne viruses caught through exposure to hcv and hiv infected blood, respectively; having common routes of transmission [2, 3], namely by: injection drug use, sexual contact, mother to child transmission during pregnancy or birth, blood and blood products transfusion, organ transplants from infected copyright: © 2022 edison mayanja, livingstone s. luboobi, juma kasozi, rebecca n. nsubuga. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. *corresponding author citation: edison mayanja, livingstone s. luboobi, juma kasozi, rebecca n. nsubuga, mathematical modelling of hiv-hcv co-infection dynamics in presence of hiv therapy, biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 1/15 https://biomath.math.bas.bg/biomath/index.php/biomath mailto:edisonmayanja.phd@gmail.com https://orcid.org/0000-0002-8077-3083 mailto:kasozi@cns.mak.ac.ug https://orcid.org/0000-0002-0941-9604 mailto:luboobi@gmail.com https://orcid.org/0000-0003-0438-9845 mailto:nrebeccansubuga@gmail.com https://orcid.org/0000-0001-8527-6222 https://creativecommons.org/licenses/by/4.0/ https://doi.org/10.55630/j.biomath.2022.07.158 mayanja et al, mathematical modelling of hiv-hcv co-infection dynamics in presence of hiv therapy donors, and exposure to blood by health care professionals [3]. several treatment options of hcv exist, however, infection with hcv takes long to manifest unlike infection with hiv. therefore, most of the individuals infected with hcv may not be aware that they are infected until many years later [4], and thus would not seek for treatment. screening, diagnosis and treatment of hcv infected individuals remains a global challenge. globally, in most countries there is: lack of prioritization and hence lack of dedicated budgets or programmes for hepatitis c [5]. according to the global hepatitis report of 2017 [6], it was revealed that globally, in 2015 of the 71 million people living with hcv: only 20% had been tested for hcv and knew their status (africa had the lowest proportion diagnosed of 6%); 7% started treatment; and cumulatively, 7.7% had ever received hcv treatment; most of these treatments were older, less effective interferon-based regimens. unlike in industrialized countries, where it is recommended that all hiv infected patients should be screened for hcv on entry into the healthcare system; in most resource limited countries like uganda there is no mandatory hcv screening even among hiv infected individuals who are under health care. this is due to the high hcv screening and treatment costs [7, 8]. however, hcv screening is only recommended when investigating whether hcv is the possible cause of liver disease among hiv infected patients [7, 9]. this is still the situation in uganda [9]. hiv and hcv share transmission routes, as such hiv-hcv coinfection is common [10]. in 2015, it was estimated that, globally close to 71 million people were chronically infected with hcv; and of the 36.7 million people that were living with hiv, 2.3 million (6.3%) had been co-infected with hcv [6]. coinfection with hiv completely changes the dynamics of hcv infection. hiv infection reduces the chance of spontaneous clearance of acute hcv [11], with 80% of the acute hcv infected individuals developing chronic hcv infection [10]. there is rapid progression to cirrhosis and higher rates of liver failure in hiv-hcv co-infected individuals than they are in individuals who are infected with hcv only [11,12]. cirrhosis has been observed to occur 12 to 16 years earlier in persons coinfected with hcv and hiv [10] compared to the 20 to 30 years in individuals infected with only hcv [7]. in this era of antiretroviral therapy (art) and in efforts to achieve the 95-95-95 (hiv testing, treatment, and viral suppression) target, numbers of aids-related deaths have greatly reduced, hence making hiv infection a chronic illness among individuals infected with only hiv. however, co-infection with hcv complicates the management of hiv by increasing the risk of death among hiv infected individuals. for persons living with hiv and co-infected with hcv, liverrelated morbidity and mortality has become the leading cause of non-aids-related morbidity and deaths [13]. furthermore, hiv infected individuals on highly active antiretroviral therapy (haart) and co-infected with hcv, have slower cd4+ t-cells recovery than those infected with only hiv [14]. over the years, mathematical models have been applied in the modelling of hiv and its common coinfections such as, tuberculosis, malaria, and hepatitis viruses to understand the dynamics of hiv and its co-infections. for example, [2, 3, 11, 12, 15] and [16] developed mathematical models to study the dynamics of hiv-hcv co-infection. some of these models have considered treatment for both hiv and hcv. these models have either considered hcv infection in stages, namely: acute and chronic or simply hcv infection without considering stages of infection. however, the chronic stage of hcv infection needs to be given a special attention because is very long, asymptomatic and infectious. recently, mayanja et al. (2020) [17], formulated and analysed a deterministic model to study the hiv and hcv co-infection dynamics in absence of therapy; but with the hcv chronic stage split into two stages i.e. before and after onset of cirrhosis and its complications. findings revealed that, in the long run, the number of individuals co-infected with hiv and latent hcv was far greater than that in any other class of individuals. the dynamics of hiv-hcv co-infection in absence of therapy were dominated by hiv. in this work, we extend the work in [17] by introducing hiv treatment and model the dynamics of hiv-hcv coinfection in presence of hiv treatment. though hiv and hcv may share other transmission routes, in this work we only considered sexual transmission among sexually active individuals as done in [17]. ii. model formulation a. description of the hiv-hcv co-infection dynamics we categorized the chronic hcv stage as in [17], that is: latent and advanced hcv, where latent hcv is characterized by a long infectious period during which infection is undiagnosable because there are no or mild symptoms for a long time; and advanced hcv characterized by onset of cirrhosis and its related complications. the total population is divided into eleven distinct compartments, defined as follows: the biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 2/15 https://doi.org/10.55630/j.biomath.2022.07.158 mayanja et al, mathematical modelling of hiv-hcv co-infection dynamics in presence of hiv therapy susceptible, s(t); infected with hiv without aids symptoms and not on hiv treatment, ih(t); infected with hiv without aids symptoms and are on hiv treatment, it (t); individuals who have developed aids symptoms, ã(t); acutely hcv-infected, ia(t); latent hcv, il(t); advanced hcv, b̃(t); co-infected with hiv and acute hcv, not on hiv treatment, iha(t); coinfected with hiv and acute hcv, on hiv treatment, ita(t); co-infected with hiv and latent hcv, not on hiv treatment, ihl(t); and those co-infected with hiv and latent hcv, on hiv treatment, itl(t). the assumptions regarding the transmission of hiv and hcv are as follows: susceptibles are sexually active individuals at age a and above (mean age of sexual debut in uganda is 16 years [18], however, 18 years is the legal age of marriage); hcv or hiv transmission is through sexual acts; for simplicity, we assume that hiv and hcv cannot be transmitted simultaneously; individuals infected with hiv are put on hiv treatment immediately they are diagnosed; individuals in the aids class do not revert to earlier classes despite successful treatment; aids cases on hiv treatment adhere never to have new sexual partners otherwise practice safe sex if they must, whereas those not on hiv treatment have undergone counselling and do not engage in unprotected sexual activities, similarly, for individuals in the advanced hcv class don’t spread diseases. we suppose that there is a constant recruitment rate λ into the susceptible class through sexual maturing; and a constant natural mortality at a per capita rate µ in all classes. susceptible individuals are infected with hcv at per capita rate πc, which is given by: πc = c̃βc[ia + il + ρ(iha + ita + ihl + itl)] n , (1) where c̃ is the average number of sexual partners acquired per year; βc is the hcv transmission probability per sexual contact; n is the total active population; and ρ > 1 is the enhancement factor for increased risk of being infected with hcv by a dually infected individual. susceptible individuals are infected with hiv at per capita rate πh, which is given by: πh = c̃βh[ih + r3it + ω(iha + r3ita + ihl + r3itl)] n , (2) where βh is the hiv transmission probability per sexual contact; r3 < 1 is a reduction parameter catering for the reduced risk of being infected with hiv by an hiv infected individual on hiv treatment; and ω > 1 is the enhancement factor for increased risk of being infected with hiv by a dually infected individual. individuals co-infected with hiv and hcv have higher viral loads of hiv and hcv as compared to those mono infected. this may increase their risk of transmission of each of the viruses [19]. both ω and ρ model the fact that co-infected individuals are more infectious than their counterparts who are mono infected [12]. when susceptible individuals are infected with hiv, they enter the class of hiv infected not on treatment, ih(t). individuals in the class ih(t), once detected are put on hiv treatment at a rate δ1 to enter the class it (t). individuals in it (t) class, progress to aids class (ã) at a rate r2α, where r2 < 1 is a reduction parameter catering for the reduced risk of progression to aids due to hiv treatment. individuals in class ih(t), who are not on hiv treatment, progress to aids class at a rate α. apart from natural death, individuals in aids class have an additional aids-induced death at a per capita rate σã. susceptible individuals once infected with hcv enter the class of acute hcv infected, ia(t). some of the individuals in ia(t) class, clear acute hcv spontaneously at a rate τ and become susceptible to hcv again. others who fail to spontaneously clear, progress to latent hcv class, il(t), at a rate γ. then, individuals from il(t) progress to advanced hcv class, b̃(t), at a rate φ. apart from natural death, individuals in class b̃(t) have an additional advanced hcv-induced death at a per capita rate σb̃. since hiv weakens the immune system, this leaves the body more vulnerable to other infections and illnesses. hence, individuals infected with hiv are at higher risk of contracting hcv than their counterparts without hiv [20]. hiv also increases hcv rna, thus making sexually active hiv infected individuals at an increased risk of sexual transmission of hcv [21]. due to hiv and hcv having similar ways of transmission, it means that individuals who are infected with hiv are at a high risk of being infected with hcv and vice versa. therefore, we included amplification parameters, ki=1,2 > 1, to cater for the increased risk of getting infected with hcv for those individuals who are already infected with hiv. amplification parameters, qi=1,2 > 1, were included to cater for the increased risk of getting infected with hiv for those individuals who are already infected with hcv [12]. a sexual encounter between individuals in classes ih(t) and ia(t), is likely to result into a co-infection with hiv and acute hcv, where: individuals who are infected with hiv only and not on hiv treatment, ih(t), get infected with acute hcv at a rate k1πc to enter the class of those individuals co-infected with biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 3/15 https://doi.org/10.55630/j.biomath.2022.07.158 mayanja et al, mathematical modelling of hiv-hcv co-infection dynamics in presence of hiv therapy table i parameter values used in the numerical simulations of hiv-hcv co-infection model. parameter description value source βh hiv transmission probability 0.0217 yr −1 assumed βc hcv transmission probability 0.05 yr−1 [15] c̃ average number of sexual partners acquired 4∗ yr−1 [17] θ rate of progression from iha(t) to ihl(t) 0.52 yr −1 [17] α rate of progression from hiv to aids 0.068∗ yr−1 assumed φ rate of progression from il to b̃ class 0.095 ∗ yr−1 [17] µ per capita natural mortality rate 0.0158 yr−1 [17] γ rate of progression from ia to il class 2 yr −1 [17] λ recruitment rate 602095 yr−1 [17] k1,k2 amplification factor for individuals in ih and it classes, respectively 1.001 [12] q1,q2 amplification factor for individuals in ia and il classes, respectively 1.0001 [12] ω enhancement factor for increased risk of being infected with hiv by a co-infected individual 1.0002 [15] ρ enhancement factor for increased risk of being infected with hcv by a coinfected individual 1.0002 [12] τ rate of spontaneous clearance of acute hcv 0.27 yr−1 [15] r1 reduction factor for risk of acute hcv spontaneous clearance in presence of co-infection 0.25 [11] r2 reduction factor for progressing from it to ã 0.2 assumed r3 reduction factor of risk of being infected by an hiv infected individual, on hiv treatment 0.5 assumed δi=1,2,3 rates at which individuals who are infected with hiv are identified and put on hiv treatment 0.12 yr−1 assumed ϕ rate of progression from it a to it l class 0.52 yr −1 [17] parameter values with * have: c̃ ∈ [1, 4], α ∈ [0.066, 0.1], φ ∈ [0.095, 0.1]; yr represents year. hiv and acute hcv but not on hiv treatment, iha(t); whereas those who are infected with acute hcv, ia(t), become co-infected with hiv at a rate q1πh. in addition, some of the individuals who are in class iha(t) can spontaneously clear acute hcv at a rate r1τ and return to the ih(t) class. these individuals are susceptible to hcv infection again. due to the fact that the probability of spontaneous clearance of the hcv virus is reduced in case of co-infection [11], a reduction parameter r1 < 1 was introduced to cater for the reduced risk of spontaneous clearance of acute hcv due to the coinfection of acute hcv and hiv. some of the individuals in class iha(t) who fail to spontaneously clear acute hcv, are detected to be coinfected with hiv and are immediately put on hiv treatment at a rate δ3 to enter the class of dually infected with hiv and acute hcv but on hiv treatment, ita(t); whereas those undetected to be co-infected with hiv, progress to dually infected with latent hcv and hiv but not on hiv treatment, ihl(t), at a rate θ. when individuals in classes ih(t) and il(t) sexually interact, individuals who are in the class ih(t) are likely to become co-infected with acute hcv at a rate k1πc to enter iha(t) class whereas individuals in the class il(t) are likely to become co-infected with hiv at a rate q2πh to enter ihl(t) class. when individuals in classes it (t) and ia(t) sexually interact, individuals who are in the class it (t) are likely to become co-infected with acute hcv at a rate k2πc to enter class ita(t); whereas individuals in the class ia(t) are likely to become co-infected with hiv at a rate q1πh to enter class iha(t). when individuals in classes it (t) and il(t) sexually interact, individuals who are in the class it (t) are likely to become co-infected with acute hcv at a rate k2πc to enter class ita(t) whereas individuals in the class il(t) are likely to become coinfected with hiv at a rate q2πh to enter class ihl(t). individuals who are dually infected with latent hcv and hiv but not on hiv treatment, ihl(t), are detected and put on hiv treatment at a rate δ2 to enter the class itl(t). some of the individuals in class ita(t) spontaneously clear acute hcv at a rate r1τ and return to it (t) whereas those who fail to spontaneously clear acute hcv, progress to class itl(t) at a rate of ϕ. as we summarize the description of the hiv-hcv co-infection dynamics, the parameters presented in the description of hiv-hcv co-infection dynamics in subsection ii-a are summarized in table i. biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 4/15 https://doi.org/10.55630/j.biomath.2022.07.158 mayanja et al, mathematical modelling of hiv-hcv co-infection dynamics in presence of hiv therapy b. compartmental model for the hiv-hcv co-infection dynamics the hiv-hcv co-infection dynamics are presented in a compartmental diagram (figure 1). c. mathematical model from the compartmental diagram in figure 1, the associated mathematical model is shown as a system of differential equations (3a)–(3k): ds dt = λ + τia −µs −πcs −πhs, (3a) dih dt = πhs + r1τiha −k1πcih −δ1ih −µih −αih, (3b) dit dt = δ1ih + r1τita −k2πcit −r2αit −µit , (3c) dia dt = πcs − τia −q1πhia −γia −µia, (3d) dil dt = γia −q2πhil −φil −µil, (3e) diha dt = k1πcih + q1πhia −r1τiha −θiha − δ3iha −µiha, (3f) dihl dt = θiha + q2πhil −δ2ihl −µihl, (3g) ditl dt = δ2ihl + ϕita −µitl, (3h) dita dt = k2πcit + δ3iha −ϕita −µita −r1τita, (3i) dã dt = r2αit + αih −σãã−µã, (3j) db̃ dt = φil −σb̃b̃ −µb̃, (3k) where πc and πh are as defined in (1) and (2), respectively. the initial values of the variables of the system (3a)–(3k) are as follows: s(0) > 0,ih(0) ≥ 0,it (0) ≥ 0, ã(0) ≥ 0,ia(0) ≥ 0,il(0) ≥ 0, b̃(0) ≥ 0,iha(0) ≥ 0,ihl(0) ≥ 0, itl(0) ≥ 0 and ita(0) ≥ 0. since we assume that due to diminished immunity, aids cases, ã(t), and advanced hcv infected cases, b̃(t), do not engage in sexual activity. as a result, equations (3a)–(3i) are independent of ã(t) and b̃(t). therefore, compartments ã and b̃ do not feed into any other compartments. thus, the total active population at time t, n(t), is given by (4): n(t) = s(t) + ih(t) + it (t) + ia(t) + il(t) + iha(t) + ihl(t) + itl(t) + ita(t). (4) iii. model analysis and results a. positivity and boundedness of solutions of hiv-hcv co-infection model it is important to establish whether system (3a)–(3i) is well posed and biologically meaningful. a study of the non-negativity and boundedness properties of the solutions of the hiv-hcv co-infection model (3a)–(3i) is made in this subsection. lemma 1. the solutions s(t), ih(t), it (t), ia(t), il(t), iha(t), ihl(t), itl(t), ita(t) of the system (3a)– (3i) are non-negative for t ≥ 0. proof: let the initial values of the variables of the system (3a)–(3i) be non-negative with s(0) > 0. we prove that the solution component of s(t) remains positive. assume that there exists a first time t1 : s(t1) = 0, s ′ (t1) < 0 and s(t) > 0,ih(t) > 0,it (t) > 0,ia(t) > 0,il(t) > 0,iha(t) > 0,ihl(t) > 0,itl(t) > 0, ita(t) > 0 for 0 < t < t1. from (3a) of the system, we have: ds(t1) dt = λ + τia(t1) > 0, which is a contradiction and consequently s(t) remains positive. the non-negativity of the other variables can similarly be proved. therefore, the solutions of the system are nonnegative for t ≥ 0. lemma 2 (invariant region). the region ω = { (s(t),ih(t),it (t),ia(t),il(t),iha(t),ihl(t), itl(t),ita(t)) ∈ r9+ : n(t) ≤ max { n0, λ µ }} , is positively invariant and attracting with respect to the model. proof: let (s(t),ih(t),it (t),ia(t),il(t),iha(t),ihl(t), itl(t),ita(t)) ∈ r9+ be any solution of the system with non-negative initial condition given by (s(0),ih(0),it (0),ia(0),il(0),iha(0),ihl(0), itl(0),ita(0)). adding equations (3a)–(3i), gives: dn dt = λ −µn −αih(t) −r2αit −φil. (5) biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 5/15 https://doi.org/10.55630/j.biomath.2022.07.158 mayanja et al, mathematical modelling of hiv-hcv co-infection dynamics in presence of hiv therapy 𝑰𝒉 𝑰𝑻 𝑰𝒉𝒍 𝑰𝒉𝒂 𝑩 𝑰𝒍 𝑰𝒂 𝑺 𝜇𝐼ℎ 𝜋ℎ𝑆 𝜎𝐵 𝐵 ∅𝐼𝑙 𝛾𝐼𝑎 𝜇𝑩 𝜇𝐼𝑙 𝜇𝐼𝑎 𝜗𝐼ℎ𝑎 𝛿2𝐼ℎ𝑙 𝜇𝑆 ∧ (𝜎𝐴 + 𝜇)𝑨 𝛿1𝐼ℎ 𝑨 𝑰𝑻𝒂 𝑰𝑻𝒍 𝜇𝐼𝑇 𝛼𝐼ℎ 𝜇𝐼ℎ𝑙 𝜇𝐼𝑇𝑎 𝑟2𝛼𝐼𝑇 fig. 1. compartmental diagram for hiv-hcv co-infection dynamics. solid arrows indicate movement from one compartment to another whereas dashed connections indicate the interaction between the connected compartments. for ih(t),it (t),il(t) ≥ 0 for t ≥ 0, (5) reduces to: dn(t) dt ≤ λ −µn, (6) from which n(t) ≤ λ µ + ( n0 − λ µ ) e−µt (7) where n0 ≥ 0 is the initial total population size. two observations are made: i: if n0 > λ µ , then (7) implies n(t) ≤ n0 for all values of t. ii: if n0 < λ µ , then (7) implies n(t) ≤ λ µ for all values of t. therefore, n(t) ≤ max{n0, λµ} for all values of t ≥ 0. hence, every feasible solution of the model system that starts in the region ω remains in the region for all values of t. thus, the region ω is biologically feasible and positively invariant. therefore, the model is epidemiologically and mathematically well posed. before analysing the dynamics of the hiv-hcv coinfection model (3a)–(3i), it is instructive to first analyse the hiv-only and hcv-only submodels. this is done in the subsections iii-b, iii-c and iii-d. biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 6/15 https://doi.org/10.55630/j.biomath.2022.07.158 mayanja et al, mathematical modelling of hiv-hcv co-infection dynamics in presence of hiv therapy b. the hiv-only submodel we set ia(t) = il(t) = iha(t) = ihl(t) = itl(t) = ita(t) = 0 in system (3a)–(3i), thus the hiv-only submodel is as in (8a)–(8c): dshiv dt = λ −πhshiv −µshiv , (8a) dih dt = πhshiv −αih − δ1ih −µih, (8b) dit dt = δ1ih −r2αit −µit , (8c) where πh = c̃βh[ih + r3it ] nhiv (9) and nhiv = shiv + ih + it . (10) 1) the hiv-free equilibrium and effective reproduction number: the hiv-free equilibrium point, ε0hiv , for system (8a)–(8c) is obtained by setting the righthand side of system (8a)–(8c) equal to zero, and hence is found to be: ε0hiv = (s 0 hiv ,i 0 h,i 0 t ) = ( λ µ , 0, 0). the effective reproduction number is defined as the spectral radius of the next generation matrix [23]. as interpreted in [12], it is the “expected number of secondary infections produced by a single infectious individual during his/her entire infectious period in the presence of an intervention strategy”. using the next generation matrix method [23], we obtain the jacobian matrices of new hiv infections, fhiv , and for the rate of transfer into and out of compartment i by all other processes, vhiv , evaluated at hiv-free equilibrium as: fhiv = [ c̃βh c̃βhr3 0 0 ] and vhiv = [ (δ1 + α + µ) 0 −δ1 (r2α + µ) ] . the effective reproduction number of the hiv-only submodel, rhiv , is given by the spectral radius of the next generation matrix, fhiv v −1 hiv . that is, rhiv = c̃βh (δ1 + α + µ) ( 1 + r3δ1 (r2α + µ) ) . (11) expressing (11) as: rhiv = c̃βh (r2α + µ) ( r2α + µ + r3δ1 δ1 + µ + α ) = c̃βh (r2α + µ) f(δ1), (12) in which f(δ1) = ( r2α + µ + r3δ1 δ1 + µ + α ) = i + r3δ1 g + δ1 , (13) where i = r2α + µ and g = α + µ. now, lim δ1→+∞ f(δ1) = r3 and lim δ1→0 f(δ1) = i g . (14) from (12) and (14), we deduce that, varying δ1 alone while other parameters are kept fixed, rhiv is bounded, that is: rhiv (δ1) ≤ c̃βh (r2α + µ) max { i g ,r3 } . (15) using theorem 2 in [23], the following result follows. theorem 1. the hiv-free equilibrium ε0hiv is locally asymptotically stable if rhiv < 1 and unstable otherwise. we can establish the following results using, for instance, the techniques in [17]: theorem 2. the hiv-free equilibrium ε0hiv is globally-asymptotically stable whenever rhiv < 1. theorem 3. the hiv-only submodel has a unique endemic equilibrium if and only if rhiv > 1. the global stability property of the endemic equilibrium of the hiv-only submodel can be investigated using the techniques in [17]. c. the hcv-only submodel to obtain the hcv-only submodel, we set ih(t) = it (t) = iha(t) = ihl(t) = itl(t) = ita(t) = 0 in (3a)–(3i). thus, we obtain: dshcv dt = λ + τia −πcshcv −µshcv , (16a) dia dt = πcshcv −γia − τia −µia, (16b) dil dt = γia −µil −φil, (16c) where πc = c̃βc[ia + il] nhcv and nhcv = shcv + ia + il. (17) basic reproduction number: the hcv-only submodel has a basic reproduction number, rhcv , as derived in [17], is given by: rhcv = c̃βc (γ + τ + µ) ( 1 + γ (µ + φ) ) . (18) biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 7/15 https://doi.org/10.55630/j.biomath.2022.07.158 mayanja et al, mathematical modelling of hiv-hcv co-infection dynamics in presence of hiv therapy in [17], it is deduced that, keeping other parameters constant and varying γ alone, rhcv is bounded. that is: rhcv (γ) ≤ c̃βc (µ + φ) max {d e , 1 } , (19) where d = µ + φ and e = µ + τ. existence and stability of hcv-free and hcv endemic equilibria: the existence of hcv-free and hcv endemic equilibria and their stability is as studied in [17]. d. the disease-free equilibrium and effective reproduction number for the hiv-hcv co-infection model the disease-free equilibrium of the hiv-hcv coinfection model (3a)–(3i) is given by: ε0 = (s0f,i 0f h ,i 0f t ,i 0f a ,i 0f l ,i 0f ha,i 0f hl ,i 0f tl ,i 0f ta) = (λ µ , 0, 0, 0, 0, 0, 0, 0, 0 ) . using the next generation matrix method [23] on model equations (3a)–(3i), we obtain jacobian of new infections matrix at disease free-equilibrium f̃ as:  c̃βh c̃βhr3 0 0 n1 n1 n1r3 n1r3 0 0 0 0 0 0 0 0 0 0 c̃βc c̃βc c̃βcρ c̃βcρ c̃βcρ c̃βcρ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   where n1 = c̃βhω; and the jacobian matrix for the rate of transfer from one compartment to another by all other processes at disease-free equilibrium v as: v =   a1 0 0 0 −r1τ 0 0 0 −δ1 a2 0 0 0 0 0 −r1τ 0 0 a3 0 0 0 0 0 0 0 −γ a4 0 0 0 0 0 0 0 0 a5 0 0 0 0 0 0 0 −θ a6 0 0 0 0 0 0 0 −δ2 a7 −ϕ 0 0 0 0 −δ3 0 0 a8   where a1 = (δ1 + α + µ), a2 = (r2α + µ), a3 = (γ + τ + µ), a4 = (φ + µ), a5 = (r1τ + θ + δ3 + µ), a6 = (δ2 + µ), a7 = µ, a8 = (ϕ + µ + r1τ). the effective reproduction number, r0, for the hivhcv co-infection model is the maximum of eigenvalues of the next generation matrix f̃v −1: λ1 = c̃βh (δ1 + µ + α) ( 1 + r3δ1 (r2α + µ) ) , λ2 = c̃βc (γ + τ + µ) ( 1 + γ (µ + φ) ) , and λ3 = λ4 = λ5 = λ6 = λ7 = λ8 = 0. that is, r0 = max { c̃βh (δ1 + µ + α) ( 1 + r3δ1 (r2α + µ) ) , c̃βc (γ + τ + µ) ( 1 + γ (µ + φ) ) , 0, 0, 0, 0, 0, 0 } . (20) thus r0 = max{rhiv ,rhcv} (21) where rhiv and rhcv are the reproduction numbers of hiv-only and hcv-only submodels as indicated in equations (11) and (18), respectively. from (21), it is deduced that the disease with the bigger effective reproduction number will dominate the dynamics of the hiv-hcv co-infection. using theorem 2 in [23], the following result follows. theorem 4. the disease-free equilibrium ε0 of the hiv-hcv co-infection model is locally asymptotically stable if r0 < 1 and unstable otherwise. 1) global stability of disease-free equilibrium for hiv-hcv co-infection model: to study the global behaviour of system (3a)–(3i), we use the theorem in [24] as shown in appendix a. re-writing model (3a)–(3i) in the form of equation (a.1) and using the same notation as used in [24], we have: x = (s), z = (ih,it ,ia,il,iha,ihl,itl,ita), f(x, 0) = [λ −µs], and a =   g1 h1 0 0 h2 c̃βhω ωh1 ωh1 δ1 g2 0 0 0 0 0 r1τ 0 0 g3 c̃βc c̃βcρ c̃βcρ c̃βcρ c̃βcρ 0 0 γ g4 0 0 0 0 0 0 0 0 g5 0 0 0 0 0 0 0 θ g6 0 0 0 0 0 0 0 δ2 g7 ϕ 0 0 0 0 δ3 0 0 g8   biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 8/15 https://doi.org/10.55630/j.biomath.2022.07.158 mayanja et al, mathematical modelling of hiv-hcv co-infection dynamics in presence of hiv therapy where g1 = c̃βh − (δ1 + α + µ), g2 = −(r2α + µ), g3 = c̃βc − (γ + τ + µ), g4 = −(φ + µ), g5 = −(r1τ + θ + µ + δ3), g6 = −(δ2 + µ), g7 = −µ, g8 = −(ϕ + µ + r1τ), h1 = c̃βhr3, h2 = c̃βhω + r1τ. ĝ defined as ĝ(x,z) = az−g(x,z) is given by: ĝ(x,z) =   ĝ1(x,z) ĝ2(x,z) ĝ3(x,z) ĝ4(x,z) ĝ5(x,z) ĝ6(x,z) ĝ7(x,z) ĝ8(x,z)   =   c̃βh(1 − sn )(ih + r3it + ωiha + ωr3ita +ωihl + ωr3itl) + k1πcih k2πcit c̃βc(1 − sn )(ia + il + ρiha + ρita + ρihl +ρitl) + q1πhia q2πhil −k1πcih −q1πhia −q2πhil 0 −k2πcit   . it can be seen that since ĝ5(x,z) < 0, ĝ6(x,z) < 0, and ĝ8(x,z) < 0, then ĝ(x,z) < 0. this implies that the second condition (h2) in theorem by [24] is not fulfilled. thus, the disease-free equilibrium of system (3a)–(3i) may not be globally asymptotically stable for r0 < 1. 2) hiv-hcv co-infection endemic equilibrium: establishment of the expressions for the endemic equilibrium for the hiv-hcv co-infection model (3a)– (3i) analytically is laborious, therefore, its existence and stability could be numerically investigated as, for instance, done in [17], by varying the initial values of the variables to determine whether they would level off to the same non-zero values in the long run, irrespective of the different initial values of the variables. 3) impact of hiv infection on hcv infection and vice versa: the impact of hcv infection on hiv infection and vice versa is analysed by expressing the reproduction number of one infection in terms of the other. the impact of hiv infection on hcv infection is analysed by expressing the reproduction number of hcv infection in terms of that of hiv infection, that is, rhcv = rhiv βc(δ1 + α + µ)(r2α + µ)(µ + φ + γ) βh(r2α + µ + r3δ1)(γ + τ + µ)(µ + φ) . (22) now, taking the partial derivative of rhcv in (22) with respect to rhiv , gives: ∂rhcv ∂rhiv = βc(δ1 + α + µ)(r2α + µ)(µ + φ + γ) βh(r2α + µ + r3δ1)(γ + τ + µ)(µ + φ) . (23) since, according to (23), ∂rhcv ∂rhiv > 0, epidemiologically this is an implication that an increase in hiv cases would result in an increase in hcv cases in the population. that is, hiv prevalence enhance hcv infections in the population. thus, hiv control has a positive effect in controlling the hcv transmission dynamics. similarly, it can be shown that ∂rhiv ∂rhcv = βh(γ + τ + µ)(µ + φ)(r2α + µ + r3δ1) βc(µ + φ + γ)(δ1 + α + µ)(r2α + µ) . (24) since, from (24), ∂rhiv ∂rhcv > 0, this implies that an increase in hcv cases would result in an increase in hiv cases in the population. that is, hcv prevalence enhance hiv infections in the population. thus, hiv and hcv infections enhance each other. this is in concurrence with the findings in [2] and [12]. e. sensitivity analysis 1) derivation of parameter values: in uganda’s efforts to achieve the 90-90-90 targets, in 2018, of the 1.4 million ugandans that were living with hiv, 84% knew their hiv status, 72% were on treatment and 64% were virally suppressed [22]. in this work, the rate at which hiv infected individuals are identified and put on hiv treatment has been assumed to be 0.12, that is: δi=1,2,3 = 0.12. we make a further assumption that there is no difference in the duration in acute hcv stage between individuals who are dually infected with hiv and on hiv treatment, and those not on hiv treatment. hence, θ = ϕ = 0.52. some of the parameter values are cited from the respective studies with literature similar to this work, and others have been assumed only to illustrate numerical results. all the parameter input values are summarised in table i indicating the sources of parameter values. 2) computation of r0: substituting for the parameter values in table i in (11) and (18), we obtain rhiv = 1.295 and rhcv = 1.667. from (21), r0 = max{1.295, 1.667} = 1.667 which is rhcv . biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 9/15 https://doi.org/10.55630/j.biomath.2022.07.158 mayanja et al, mathematical modelling of hiv-hcv co-infection dynamics in presence of hiv therapy hence, the dynamics of hiv-hcv co-infection in presence of hiv therapy is dominated by hcv. introduction of hiv treatment changes the dynamics of hiv-hcv co-infection. for the hiv-hcv co-infection dynamics in absence of treatment, the dynamics were dominated by hiv [17]. therefore, there is need to investigate the dynamics of hiv-hcv co-infection in presence of therapies for both infections. 3) computation of sensitivity indices of the effective reproduction numbers with respect to the parameters of the hiv-hcv co-infection model: the sensitivity analysis of the effective reproduction number of the hivhcv co-infection model, r0, to each of the parameter values has been performed using the normalized forward sensitivity index method [25]. this helps to determine the relative contribution of each parameter on r0 such that appropriate intervention strategies can be taken. the normalized forward sensitivity index of re with respect to parameter, x, is defined as the ratio of the relative change in re to the relative change in parameter, x, that is: rrex = ∂re ∂x × x re . (25) sensitivity analysis of r0 to each of the parameter values has been computed separately for rhiv and rhcv , since r0 = max{rhiv ,rhcv}. sensitivity indices of rhiv and rhcv have been calculated analytically using formulas rrhivx = ∂rhiv ∂x × x rhiv (26) and rrhcvx = ∂rhcv ∂x × x rhcv , (27) respectively. table ii presents sensitivity indices of both rhiv and rhcv . table ii sensitivity indices of rhiv and rhcv with respect to parameters. parameter index of rhiv parameter index of rhcv βh +1.0000 βc +1.0000 c̃ +1.0000 c̃ +1.0000 r2 −0.3105 φ −0.8123 r3 +0.6712 µ −0.14201 α −0.6442 τ −0.1181 µ −0.4382 γ +0.0725 δ1 +0.0823 the interpretation of the sensitivity indices presented in table ii is as follows: for a parameter with a negative index, it implies that the corresponding basic reproduction number decreases (increases) with an increase (decrease) in the value of that parameter while keeping the values of other parameters fixed. more still, a positive index implies that the corresponding basic reproduction number increases (decreases) with an increase (decrease) in the value of that parameter. for example, increasing (decreasing) the value of average number of sexual partners acquired per year, c̃, by 10% while other parameter values are kept fixed, increases (decreases) the value of rhiv by 10%. in addition, a 10% increase (decrease) in the value of the rate of progression of individuals infected with hiv to aids, α, while other parameter values are kept fixed, decreases (increases) the value of rhiv by 6.4%. similarly, the sensitivity indices of other parameters can be interpreted. from table ii, we deduce that endemicity of hiv infection increases when the values of βh, c̃, δ1, and r3 are increased and or those of r2, α, and µ are decreased. the most sensitive parameters in hiv infection are c̃ and βh (which are equally sensitive) followed by r3, α and r2. therefore, interventions should target and concentrate on reducing the values of c̃ and βh. parameters c̃, βh, δ1, r2, r3 and α are related in a way that: an increase in the rate at which individuals who are infected with only hiv are identified and put on hiv treatment, δ1, reduces the value of reduction factor of hiv individuals on treatment progressing to aids, r2, hence increasing the duration in the hiv stage (reduction in α). an increase in δ1 results into increased r3 which in the long run leads to increased hiv infections. this is because hiv infected individuals on treatment have a prolonged life span, look healthy and can easily get many sexual partners as they would wish like any hiv negative individual. they have a prolonged time of infecting other individuals with hiv. therefore, for reduced hiv infections, hiv infected individuals on hiv treatment need to be sensitised on how to live positively (for example, having safe sex using condoms and having few sexual partners). the sensitivity indices of rhcv in table ii are similar to those in [17], since the hcv-only submodel and the corresponding parameter values in this work are the same as those in [17]. thus, from table ii, we deduce that endemicity of hcv infection increases when the values of βc, c̃, and γ are increased and or those of φ, τ, and µ are decreased. the most sensitive parameters in hcv infection are c̃ and βc (which are equally sensitive) followed by φ. in subsection iii-e2, it is revealed that the dynamics of hiv-hcv co-infection biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 10/15 https://doi.org/10.55630/j.biomath.2022.07.158 mayanja et al, mathematical modelling of hiv-hcv co-infection dynamics in presence of hiv therapy in presence of hiv therapy is dominated by hcv. therefore, r0 will be more sensitive to βc, c̃, and φ just like rhcv . from sensitivity analysis, we deduce that, βh (or βc) and c̃ are equally likely to increase hiv (or hcv) infections. increment in the values of these parameters is leading other parameters in increasing the hiv (or hcv) infection. this is in agreement with the findings in [17] in which they investigated the dynamics of hiv-hcv co-infection in absence of treatment for the two infections. therefore, for reduced hiv (or hcv) infections: individuals need to greatly reduce the rate of sexual partner acquisition, c̃; and transmit-ability probabilities βh (or βc); that is, by having safe sex which doesn’t expose them to infected blood, like using condoms as recommended in [17]. iv. numerical simulations we use the ode45 solver in matlab to perform a numerical simulation of the hiv-hcv co-infection model using parameter values presented in table i. the initial values for the variables are set as follows: in uganda, the total population in 2014 was 34, 634, 650 of which 49.2% was aged 15-64 years [26]. in this study, the population aged 15-64 years is taken to be sexually active. this implies that sexually active population, p = 49.2 100 × 34, 634, 650 = 17, 040, 248. in 2015, the hiv prevalence in uganda was 6.2% [22]. this implies that sexually active population that was living with hiv in 2015 = 6.2 100 × 17, 040, 248 = 1, 056, 496. in uganda, hcv prevalence is 2.7% [1]. this implies that sexually active population infected with hcv= 2.7 100 × 17, 040, 248 = 460, 087. sexually active population co-infected with hiv and hcv= 2.7 100 × 1, 056, 496 = 28, 526. in 2018, 72% of hiv infected ugandans were on hiv treatment [22]. in [3, 4] and [12] it is revealed that approximately 85% of the people infected with acute hcv develop chronic hcv. basing on these facts, the following initial values for the variables are derived: it (0) = 740, 138,ih(0) = 287, 832,itl(0) = 17, 458,ita(0) = 3, 081,il(0) = 366, 827,ia(0) = 64, 734,ihl(0) = 6, 789 and iha(0) = 1, 198. therefore, s(0) = p − [it (0) + ih(0) + il(0) + ia(0) + itl(0) + ita(0) + iha(0) + ihl(0)] = 15, 552, 191. in figures 2, 3, 4, and 5, values for the rates at which individuals who are infected with only hiv, dually infected with hiv and latent hcv, and dually infected with hiv and acute hcv are identified and put on hiv treatment, δ1, δ2, and δ3, respectively, are varied to investigate the effect of increasing hiv treatment on 0 5 10 15 20 0.7 0.75 0.8 0.85 0.9 0.95 t re a tm e n t p ro p o rt io n time (years) δ 1 =δ 2 =δ 3 =0.12 δ 1 =δ 2 =δ 3 =0.24 δ 1 =δ 2 =δ 3 =0.36 δ 1 =δ 2 =δ 3 =0.48 δ 1 =δ 2 =δ 3 =0.60 fig. 2. proportion on hiv treatment under varying rates of initiation on hiv treatment. the number of individuals co-infected with hiv and hcv. starting with δ1 = δ2 = δ3 = 0.12, these values were doubled, tripled, multiplied four times, and lastly multiplied five times. figure 2 shows that increasing the rate at which individuals are identified and put on hiv treatment, in the long run increases the proportion of individuals on hiv treatment. a country with such increasing treatment proportions, will nearly achieve the second “95” of the 95-95-95 targets. figure 3 shows the effect of increasing the values of δ1, δ2, and δ3 on hcv and hiv prevalences. figure 3(b) is a magnification of figure 3(a). from figures 3(a) and 3(b), it is revealed that increasing the values of δ1, δ2, and δ3, eventually, leads to the decrease in the prevalence of hcv. this implies that hiv control has a positive effect in controlling the hcv transmission dynamics as mentioned in subsection iii-d3. hiv treatment boosts the immunity of the body to fight off opportunistic infections such as hcv. figure 3(c) reveals that increasing the rate at which individuals are identified and put on hiv treatment, in the long run increases the prevalence of hiv. this is due to hiv treatment improving the health of these people and prolonging their life span. they get many sexual partners and infect them with hiv. this is in concurrence with the findings in [27] and [28] in which it is inferred that treatment of hiv/aids patients would prolong the patients’ lives, and if the treatment does not reduce the infectiousness of such people, they become key agents in the spread of the hiv/aids. therefore, for reduced hiv infections, hiv positively biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 11/15 https://doi.org/10.55630/j.biomath.2022.07.158 mayanja et al, mathematical modelling of hiv-hcv co-infection dynamics in presence of hiv therapy infected people need continuous sensitisation on how to live positively as mentioned in this work in subsection iii-e3. figure 4(a) shows that increasing the rate at which people are identified and put on hiv treatment, decreases the number of individuals who are co-infected with hiv and acute hcv not on hiv treatment in the long run. on the other hand, increasing the rate at which people are identified and put on hiv treatment, increases the number of individuals who are co-infected with hiv and acute hcv on treatment in the long run as shown in figure 4(b). figure 5(a) reveals that with increasing values of δ1, δ2, and δ3, in the long, there are decreasing numbers of people co-infected with hiv and latent hcv not on hiv treatment, however, the population co-infected with hiv and latent hcv on hiv treatment increases as shown in figure 5(b). figures 6 and 7 show the effect of varying amplification parameters catering for increased risk of getting infected with hcv (or hiv) for those individuals who are already infected with hiv (or hcv), k1 and k2 (or q1 and q2), on the size of individuals co-infected with hiv and hcv. figure 6(a) shows that when the value of k1, is increased, eventually, there will be an increase in the number of individuals infected with hiv but on treatment becoming co-infected with acute hcv. more still, figure 6(b) shows that when the value of k2, is increased, in the long run, there will be an increase in the number of individuals infected with hiv but on treatment becoming co-infected with acute hcv. figure 7(a) reveals that when the value of q1 is increased, eventually, there will be an increase in the number of individuals acutely infected with hcv getting co-infected with hiv. in addition, figure 7(b) shows that when the value of q2 is increased, there will be an increase in the number of individuals latently infected with hcv becoming co-infected with hiv, in the long run. therefore, figures 7 and 6 confirm that individuals who are already infected with hiv (or hcv) are at an increased risk of being infected with hcv (or hiv). this is in agreement with the literature in [20] and [12] which reveals that individuals who are already infected with hiv have an increased risk of contracting hcv and vice versa. v. conclusion building on the earlier work in [17], we have added the aspect of hiv treatment. thus, we formulated and analysed a mathematical model for the hiv-hcv co-infection dynamics in presence of hiv therapy. 0 2 4 6 8 10 12 14 16 18 20 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 h c v p re va le n ce time (years) δ 1 =δ 2 =δ 3 =0.12 δ 1 =δ 2 =δ 3 =0.24 δ 1 =δ 2 =δ 3 =0.36 δ 1 =δ 2 =δ 3 =0.48 δ 1 =δ 2 =δ 3 =0.60 (a) hcv prevalence against time 18.8 19 19.2 19.4 19.6 19.8 20 0.09 0.091 0.092 0.093 0.094 0.095 0.096 0.097 0.098 h c v p re va le n ce time (years) δ 1 =δ 2 =δ 3 =0.12 δ 1 =δ 2 =δ 3 =0.24 δ 1 =δ 2 =δ 3 =0.36 δ 1 =δ 2 =δ 3 =0.48 δ 1 =δ 2 =δ 3 =0.60 (b) magnified graph of hcv prevalence against time 0 2 4 6 8 10 12 14 16 18 20 0.0595 0.06 0.0605 0.061 0.0615 0.062 0.0625 0.063 0.0635 0.064 h iv p re va le n ce time (years) δ 1 =δ 2 =δ 3 =0.12 δ 1 =δ 2 =δ 3 =0.24 δ 1 =δ 2 =δ 3 =0.36 δ 1 =δ 2 =δ 3 =0.48 δ 1 =δ 2 =δ 3 =0.60 (c) hiv prevalence against time fig. 3. simulation results showing hiv-hcv co-infection dynamics under varying rates of initiation on hiv treatment. biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 12/15 https://doi.org/10.55630/j.biomath.2022.07.158 mayanja et al, mathematical modelling of hiv-hcv co-infection dynamics in presence of hiv therapy 0 2 4 6 8 10 12 14 16 18 20 0 1000 2000 3000 4000 5000 6000 7000 8000 time (years) h iv − a cu te h c v c o − in fe ct e d n o t o n t re a tm e n t δ 1 =δ 2 =δ 3 =0.12 δ 1 =δ 2 =δ 3 =0.24 δ 1 =δ 2 =δ 3 =0.36 δ 1 =δ 2 =δ 3 =0.48 δ 1 =δ 2 =δ 3 =0.60 (a) population co-infected with hiv and acute hcv not on hiv treatment against time 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 3 3.5 x 10 4 time (years) h iv − a cu te h c v c o − in fe ct e d o n t re a tm e n t δ 1 =δ 2 =δ 3 =0.12 δ 1 =δ 2 =δ 3 =0.24 δ 1 =δ 2 =δ 3 =0.36 δ 1 =δ 2 =δ 3 =0.48 δ 1 =δ 2 =δ 3 =0.60 (b) population co-infected with hiv and acute hcv on hiv treatment against time fig. 4. hiv-acute hcv co-infection dynamics under varying rates of initiation on hiv treatment. analytical analysis revealed that both hiv and hcv infections enhance each other. the different parameters in the hiv-hcv co-infection model were subjected to a sensitivity analysis and it was deduced that hiv (or hcv) transmission probability per sexual contact and average number of sexual partners acquired per year are not only equally likely to result into increased hiv (or hcv) infections, but also increment in the values of these parameters is leading other parameters in increasing the hiv (or hcv) infections, just like the case in the work in [17] when there was no treatment for both infections. through numerical simulations it was revealed that in the long run, increasing the rates 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10 4 time (years) h iv − la te n t h c v c o − in fe ct e d n o t o n t re a tm e n t δ 1 =δ 2 =δ 3 =0.12 δ 1 =δ 2 =δ 3 =0.24 δ 1 =δ 2 =δ 3 =0.36 δ 1 =δ 2 =δ 3 =0.48 δ 1 =δ 2 =δ 3 =0.60 (a) population co-infected with hiv and latent hcv not on hiv treatment against time 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 x 10 5 time (years) h iv − la te n t h c v c o − in fe ct e d o n t re a tm e n t δ 1 =δ 2 =δ 3 =0.12 δ 1 =δ 2 =δ 3 =0.24 δ 1 =δ 2 =δ 3 =0.36 δ 1 =δ 2 =δ 3 =0.48 δ 1 =δ 2 =δ 3 =0.60 (b) population co-infected with hiv and latent hcv on hiv treatment against time fig. 5. hiv-latent hcv co-infection dynamics under varying rates of initiation on hiv treatment. at which people are put on hiv treatment reduces the prevalence of hcv in the community, however, it increases the prevalence of hiv. we recommend that there should be increased safer sexual behaviour campaigns among individuals on hiv treatment. this work can be extended by including treatment for hcv in the model. acknowledgments we are very grateful to the financial support extended by sida phase-iv bilateral program with makerere university [2015-2020, project 316] “capacity building in mathematics and its applications”. biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 13/15 https://doi.org/10.55630/j.biomath.2022.07.158 mayanja et al, mathematical modelling of hiv-hcv co-infection dynamics in presence of hiv therapy 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 3 3.5 x 10 4 time (years) h iv − a cu te h c v c o − in fe ct e d n o t o n t re a tm e n t k 1 =1.001 k 1 =2.001 k 1 =3.001 k 1 =4.001 k 1 =5.001 (a) iha under varying values of k1 0 2 4 6 8 10 12 14 16 18 20 0 1 2 3 4 5 6 7 8 9 10 x 10 4 time (years) h iv − a cu te h c v c o − in fe ct e d o n t re a tm e n t k 2 =1.001 k 2 =2.001 k 2 =3.001 k 2 =4.001 k 2 =5.001 (b) it a under varying values of k2 fig. 6. hiv-hcv co-infection dynamics under varying values of amplification parameters for the risk of getting co-infected with hcv for hiv infected individuals, k1 and k2. appendix a global stability conditions for the disease-free equilibrium when r0 < 1 according to castillo-chavez et al. [24], for the system written in the form:  dx dt = f(x,z), dz dt = g(x,z), g(x, 0) = 0, (a.1) where the components of x ∈ rm denotes the number of uninfected individuals and the components of z ∈ rn denotes the number of infected individuals. 0 2 4 6 8 10 12 14 16 18 20 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 time (years) h iv − a cu te h c v c o − in fe ct e d n o t o n t re a tm e n t q 1 =1.0001 q 1 =2.0001 q 1 =3.0001 q 1 =4.0001 q 1 =5.0001 (a) iha under varying values of q1 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 x 10 4 time (years) h iv − la te n t h c v c o − in fe ct e d n o t o n t re a tm e n t q 2 =1.0001 q 2 =2.0001 q 2 =3.0001 q 2 =4.0001 q 2 =5.0001 (b) ihl under varying values of q2 fig. 7. hiv-hcv co-infection dynamics under varying values of amplification parameters for the risk of getting co-infected with hiv for hcv infected individuals, q1 and q2. let u0 = (x∗, 0) denote the disease-free equilibrium of this system. the fixed point u0 = (x∗, 0) is a globally asymptotically stable equilibrium of system (a.1) provided that r0 < 1 and the following conditions (h1) and (h2) are satisfied: (h1): for dx dt = f(x, 0), x∗ is globally asymptotically stable, (h2): g(x,z) = az−ĝ(x,z), ĝ(x,z) ≥ 0 for (x,z) ∈ ω, where a = dzg(x∗, 0) is an m–matrix (the off diagonal elements of a are non-negative) and ω is the region where the model makes biological sense. if system (a.1) satisfies conditions (h1) and (h2), then the theorem below holds: biomath 11 (2022), 2207158, https://doi.org/10.55630/j.biomath.2022.07.158 14/15 https://doi.org/10.55630/j.biomath.2022.07.158 mayanja et al, mathematical modelling of hiv-hcv co-infection dynamics in presence of hiv therapy theorem ([24]). the fixed point u0 = (x∗, 0) is a globally asymptotic stable equilibrium of system (a.1) provided r0 < 1 (locally asymptotically stable) and that conditions (h1) and (h2) are satisfied. references [1] m.w. sonderup, m. afihene, r. ally, b. apica, y. awuku y, l. cunha, et al., “hepatitis c in sub-saharan africa: the current status and recommendations for achieving elimination by 2030”, the lancet gastroenterology & hepatology, 2(12):910-919, 2017, doi: 10.1016/s2468-1253(17)30249-2. 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https://www.researchgate.net/profile/dennis-muriithi/publication/283072306_mathematical_analysis_of_a_comprehensive_hiv_aids_model_treatment_versus_vaccination/links/5628b57a08ae518e347c6083/mathematical-analysis-of-a-comprehensive-hiv-aids-model-treatment-versus-vaccination.pdf https://doi.org/10.55630/j.biomath.2022.07.158 introduction model formulation description of the hiv-hcv co-infection dynamics compartmental model for the hiv-hcv co-infection dynamics mathematical model model analysis and results positivity and boundedness of solutions of hiv-hcv co-infection model the hiv-only submodel the hiv-free equilibrium and effective reproduction number the hcv-only submodel the disease-free equilibrium and effective reproduction number for the hiv-hcv co-infection model global stability of disease-free equilibrium for hiv-hcv co-infection model hiv-hcv co-infection endemic equilibrium impact of hiv infection on hcv infection and vice versa sensitivity analysis derivation of parameter values computation of r0 computation of sensitivity indices of the effective reproduction numbers with respect to the parameters of the hiv-hcv co-infection model numerical simulations conclusion appendix a: global stability conditions for the disease-free equilibrium when r0<1 references www.biomathforum.org/biomath/index.php/biomath original article model-based control strategies for anaerobic digestion processes neli dimitrova∗, mikhail krastanov†∗ ∗institute of mathematics and informatics bulgarian academy of sciences sofia, bulgaria †faculty of mathematics and informatics sofia university “st kl. ohridski” nelid@math.bas.bg, krastanov@fmi.uni-sofia.bg received: 28 february 2019, accepted: 12 july 2019, published: 17 august 2019 abstract—in this paper we consider a fourdimensional bioreactor model, describing an anaerobic wastewater treatment with methane production. different control strategies for stabilizing the dynamics are presented and discussed. a general and practice-oriented bounded open-loop control is proposed, aimed to steer the model solutions towards an a priori given set in the phase plane. keywords-continuous bioreactor; dynamical nonlinear model; global asymptotic stabilization; feedback control; open-loop control; model uncertainty i. introduction anaerobic digestion (ad) is a biological process in which organic degradable material is converted into biogas by microorganisms [7], [8]. recently, ad has been evaluated as one of the most promising processes for waste recovery, environmental protection and bioenergy production. the biogas is a mixture of gases composed of methane, carbon dioxide, nitrogen, oxygen, hydrogen sulphide and traces of other gases. the biogas is classified as a renewable energy, which can be used in gas engines to produce electricity and heat energies. storing biogas prevents greenhouse gas emissions from entering the atmosphere. some estimates from 1997 [23] report that recovery of organic wastes and industrial effluents could reduce 20% of the global warming effect on the planet. at laboratory or industrial scales, the ad process occurs inside an anaerobic digester (reactor), where degradation of organic material holds by plenty of anaerobic microorganisms. the growth of the microorganisms (bacteria, yeasts, etc.) proceeds by consumption of appropriate nutrients (substrates) involving carbon, nitrogen, oxygen, etc., under favorable conditions (temperature, ph, etc.). the mass of the living organisms (or cells) copyright: c©2019 dimitrova et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: neli dimitrova, mikhail krastanov, model-based control strategies for anaerobic digestion processes, biomath 8 (2019), 1907127, http://dx.doi.org/10.11145/j.biomath.2019.07.127 page 1 of 17 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2019.07.127 neli dimitrova, mikhail krastanov, model-based control strategies for anaerobic digestion processes is called biomass. the number, behavior and interaction of the included organisms pose a challenge to the specialist in the field and have been extensively investigated in the literature. the reactor configuration and environmental conditions (retention time, temperature, feedstock, stirring, etc.) influence the dynamics and composition of the different groups of bacteria responsible for the organic material degradation [7]. according to the user objectives and the nature of the biodegradable waste (solid wastes or released in wastewater), different technologies can be used. we mention below some commonly cited reactors in the literature [7], [8]. batch reactor is a reactor without inflow nor outflow. the digester is filled with the biodegradable materials and left until the substrate has been degraded. then the digester is emptied and a new cycle can start again. fed-bach reactor (also called semi-continuous or fed/sequencing batch reactor) is a reactor without outflow. the process is cyclic, the digester is filled gradually according to the progress of the reaction in order to ensure optimal growth conditions. at the end of the digestion phase, decantation allows to separate the liquid phase and suspended biomass. continuous bioreactor: the tank is continuously fed at a constant rate and the digestat (the material that remains after the ad process) is evacuated by a mechanical action. depending on the contact between the substrate and biomass or the feeding mode, the continuous bioreactors fall into several categories. among them we mention the continuously stirred tank reactor, where the outflow is equal to the rate of inflow and a continuous mixing ensures the medium homogeneity. independently of the chosen reactor type, a key parameter in biogas plants is the dilution rate. it is proportional to the speed of the input mechanisms which feeds the reactor with substrate. to avoid wash out of bacteria, the dilution rate is always constrained, i. e. u ∈ [umin,umax] with umin > 0 (cf. [8]). the ad process follows several phases: hydrolysis is the step where polymers (macromolecules) are hydrolysed to monomers (simple organic matter); the speed of degradation depends on the substrate type (glucide, proteins, lipids, cellulose, etc.). acidogenesis, where monomers are degraded to volatile fatty acids (vfa) and alcohol. acetogenesis, performed by acitogenic bacteria which transform the vfa into acetic acid, hydrogen (h2) and carbon dioxide (co2). the responsible bacteria for this step produce h2 and can be inhibited by an excess of h2 concentration in the digester, that’s why they live fixed to the methanogenic bacteria which consume the hydrogen. methanogenesis, where methanogenic bacteria reduce the specific substrate into methane. a good management and control of the ad process can be achieved via validated mathematical models—an area, which is extensively studied in recent years. the proposed models are specific to a couple of criteria such as the waste nature and its composition, the used technology, collected data and its quality, the possible experiments and changes in the operating conditions. in general, a dynamical model accounts for the time evolution of substartes and biomasses, and is based on ordinary differential equations representing mass balance within the process. the mathematical modeling of ad processes has a long history. in 1968, andrews [5] modelled the methanogenic fermentation by only the final step methanogenesis. in 1973, graef and andrews [24] included the acidogenesis step in the description of fermentation. later on, other researchers, hill and barth (1977, [28]), boone and bryant (1980, [12]), eastman and ferguson (1981, [20]) added a hydrolysis step to their description, and modelled a three-step process. the interested reader is referred to [32] for more details about other models. over time the models were extended depending on the different substrates (wastewater, sludge, etc.). in 2002, a group of experts in the ad processes (iwa task group for mathematical modbiomath 8 (2019), 1907127, http://dx.doi.org/10.11145/j.biomath.2019.07.127 page 2 of 17 http://dx.doi.org/10.11145/j.biomath.2019.07.127 neli dimitrova, mikhail krastanov, model-based control strategies for anaerobic digestion processes elling of anaerobic digestion processes) developed a standard model for the ad process, called adm1 (anaerobic digestion model no 1) [9]. in order to make the adm1 a standard platform for ad simulation, it has been decided to generalize the composition of waste, so it is measured by an unified unit chemical oxygen demand (cod) and the process is supposed to occur in a continuously stirred tank reactor. adm1 is described by 32 ordinary differential equations and its parameters are collected from different applications. many modifications, adaptations and variations of the adm1 have been done later, cf. [43], [44], [47], [48] and the references therein. the adm1 and its variations are complex models suitable for process knowledge and simulation, but not appropriate for process control and software sensors design, because they require a plenty of input parameters which are difficult to obtain. to overcome the adm1 complexity, simpler models based on mass balance equations [8] have been developed, more suitable to support monitoring or control strategies. such a model (called am1), including two reactions (acidogenesis and methanogenesis) is proposed by bernard et al [11], and turns out to approximate efficiently the adm1 for modeling anaerobic wastewater treatment. this model will be investigated in the present paper. the management and control of the ad process require a good information about the internal state of the system. biological processes are known to be highly unstable due to the specific behavior of the system itself or to the presence of disturbances. thus, an obvious need for an efficient control and monitoring of such systems arises. a summary and review of different sensoring approaches can be found e. g. in [31], [33], [42], [49]. the majority of sensors intended to measure the process key variables often require complex equipment and careful maintenance, so that the plant costs may climb quickly, which is not desirable from industrial standpoint. therefore, it is crucial to find a methodology which allows cost-effective and easily adopted to practice monitoring of ad plants. such methodology is the development of efficient software sensors, also called observers. the observers are auxiliary dynamical systems that provide information on the unmeasurable state variables of the system by using its mathematical model and its input and output signals (the measurable variables of the system), see e. g. [2], [3], [4], [8], [26], [40]. in biological processes observers are mainly used in on-line estimations for control purposes. in addition to the modeling and observer design, the ad control in biogas plants is gaining an increased importance. the main reason is the significant growth of bioenergy markets. moreover, due to the climate-energy package of the european commission, the produced biogas must be rich in methane to fulfil the environmental standards [51]. controlling the ad processes is a delicate problem due to the high complexity and strong instability of the ecosystem inside the reactor [26]. several factors are to be handled like e. g. the highly nonlinear behavior of the system itself, load disturbances, system uncertainties, constraints on manipulated and state variables and the limited online measurements information [39]. moreover, the ad process involves living organisms which are very sensitive to the operating conditions and may be washed out or inhibited due to an accidental toxic feeding, leading, in the worst case, to a stop of the digester. the control design varies with the application objectives. usually, in biogas plants, the controller is designed to satisfy one specific criterion, either economical (maximizing methane production) or ecological (minimizing cod concentration of the effluent) or stability (vfa or dissolved hydrogen) criteria [21], [39]. the controller type depends on many factors such as the accuracy of the monitoring, knowledge of the system and availability and complexity of the considered model. among the classical controllers are the proportional-integral (pi) controller, the proportional integral-differential (pid) controller, the adaptive pid and the cascade pi controls; all they have been recognized as a good alternative biomath 8 (2019), 1907127, http://dx.doi.org/10.11145/j.biomath.2019.07.127 page 3 of 17 http://dx.doi.org/10.11145/j.biomath.2019.07.127 neli dimitrova, mikhail krastanov, model-based control strategies for anaerobic digestion processes for the regulation of ad plants (cf. [22], [40] and the references therein). various advanced control approaches like expert systems (rule-based and fuzzy-logic-based systems, neural networks, etc.) have been recently developed for ad control. an other type of model-based control designed for the ad processes, is the linearizing control [8]. the latter is based on a nonlinear model, aimed to achieve linear closed-loop dynamics. a drawback of this method is that it relies of full knowledge of the system parameters. later on, an adaptive linearizing control has been proposed [35], which ensures global asymptotic stability of the closed-loop system. when intervals of the model uncertainties are known a priori, robust linearizing control based on interval observers has been proposed in [1], [41] and [45]. other recently developed approaches for controlling ad processes are based on differential geometry [29], [38]. sliding mode approaches have been also used to control anaerobic continuous bioreactors. further, the nonlinear adaptive control law, which is robust with respect to unknown kinetic rates has been proposed for the global stabilization of ad processes [35]. extremum seeking control (esc) is another technique to handle dynamic optimization problems. the goal of esc is to find the operating set-point, a priori unknown, such that a performance function reaches its extremum value. the classical extremum seeking approach [36], [37], [50] is designed in the form of a block diagram (scheme) that is implemented on the bioreactor to tune the dilution rate of the open-loop system towards a set-point, where an optimal value of the output is achieved. the main limitation in applying this approach is that the dynamics should be open-loop stable. otherwise, a local controller is necessary to stabilize the system around the optimal operating point. extremum seeking numerical techniques are developed in recent years to overcome the above drawbacks. based on a mathematical model, this new approach splits the extremum seeking problem into two steps: global dynamics stabilization and application of a numerical optimization method. in [46] this approach is applied to the classical two-dimensional model of methane fermentation. for further information about instrumentation and control of ad processes we refer the reader to the excellent review [30]. the present paper is organized as follows. section ii presents the dynamic model of the anaerobic wastewater treatment process and gives an overview of authors’ results on global stabilizability of the model dynamics using different control strategies. section iii reports on a new result, concerning a general and practically oriented stabilization approach by means of an arbitrary measurable bounded control function. ii. basic properties and global stabilizability of the model dynamics we consider the mass balance model am1 of anaerobic wastewater treatment in a continuous bioreactor, described by the following nonlinear system of ordinary differential equations (cf. e. g. [6], [11], [25], [27], [34]): ds1 dt = u(si1 −s1) −k1µ1(s1)x1 dx1 dt = (µ1(s1) −αu)x1 ds2 dt = u(si2−s2)+k2µ1(s1)x1−k3µ2(s2)x2 dx2 dt = (µ2(s2) −αu)x2. (1) the definition of the state variables s1, s2 and x1, x2 as well as of the model parameters is given in table i. all coefficients are assumed to be positive. the parameter α ∈ (0, 1) represents the proportion of bacteria that are affected by the dilution; α = 0 and α = 1 correspond to a fed-batch reactor and to a continuously stirred tank reactor, respectively (cf. [2], [6], [11], [25]). the input substrate concentrations si1 and s i 2 are assumed to be constant. the dilution rate u is considered as a control (manipulated) input. the model describes two steps of the ad process: biomath 8 (2019), 1907127, http://dx.doi.org/10.11145/j.biomath.2019.07.127 page 4 of 17 http://dx.doi.org/10.11145/j.biomath.2019.07.127 neli dimitrova, mikhail krastanov, model-based control strategies for anaerobic digestion processes table i definition of the model phase variables and parameters s1 concentration of chemical oxygen demand (cod) [g/l] s2 concentration of volatile fatty acids (vfa) [mmol/l] x1 concentration of acidogenic bacteria [g/l] x2 concentration of methanogenic bacteria [g/l] u dilution rate [day−1] si1 influent concentration s1 [g/l] si2 influent concentration s2 [mmol/l] k1 yield coefficient for cod degradation [g cod/(g x1)] k2 yield coefficient for vfa production [mmol vfa/(g x1)] k3 yield coefficient for vfa consumption [mmol vfa/(g x2)] k4 coefficient [l2/g] q methane gas flow rate m1 maximum acidogenic biomass growth rate [day−1] m2 maximum methanogenic biomass growth rate [day−1] ks1 saturation parameter associated with s1 [g cod/l] ks2 saturation parameter associated with s2 [mmol vfa/l] ki inhibition constant associated with s2 [(mmol vfa/l)1/2] (i) acidogenesis, where the organic substrate s1 is degraded into volatile fatty acids (vfa) (s2) by acidogenic bacteria (x1); (ii) methanogenesis, where vfa (s2) are degraded by methanogenic bacteria (x2) into methane ch4. the methane solubility is very low, therefore the methane produced by the second step is not stored in the liquid phase, thus the output methane flow rate q = qch4 is written in the form q = k4µ2(s2)x2. (2) the model dynamics can be described schematically by the following biological reaction pathways: acidogenesis: k1s1 r1(·)−→ x1 + k2s2 methanogenesis: k3s2 r2(·)−→ x2 + k4q, where rj(·), j = 1, 2, are the reaction rates, which are given by rj(·) = µj(·)xj. the functions µ1(s1) and µ2(s2) model the specific growth rates of the microorganisms. usually, the model is studied using the following particular expressions of µ1 and µ2 (cf. [2], [6], [11], [25], [27]) µ1(s1) = m1s1 ks1 + s1 (monod law) µ2(s2) = m2s2 ks2 +s2 + ( s2 ki )2 (haldane law), (3) where m1, ks1 , m2, ks2 and ki are positive coefficients (see table i). the most crucial problem in investigating ad models is the formulation of reasonable analytic expressions for µ1(s1) and µ2(s2). in our theoretical studies on the model (1) we do not assume to know explicit expressions for µ1 and µ2, we only impose the following general assumption on the latter. assumption a1. the function µj(sj) is defined for sj ∈ [0, +∞), µj(0) = 0, µj(sj) > 0 for sj > 0; µj(sj) is continuously differentiable and bounded for all sj ∈ [0, +∞), j = 1, 2. the model (1) exhibits very rich dynamics. considering u as a bifurcation parameter, the system possesses different types of bifurcations of the equilibrium points, most of them leading to washout of the biomass and thus to process break-down [10], [14]. biomath 8 (2019), 1907127, http://dx.doi.org/10.11145/j.biomath.2019.07.127 page 5 of 17 http://dx.doi.org/10.11145/j.biomath.2019.07.127 neli dimitrova, mikhail krastanov, model-based control strategies for anaerobic digestion processes define s = k2 k1 s1 + s2, s i = k2 k1 si1 + s i 2. (4) the quantity s is called biological oxygen demand (bod) and represents the biological equivalent of cod, i. e. the biological equivalent of the total amount of organic substrate in the digester. for the practical application it is worth to note that bod is online measurable and is used as a depollution factor in wastewater treatment, see e. g. [2], [6], [11], [13] and the references therein. define the set ω0 = { (s1,x1,s2,x2) ∈ r4 : s1 > 0, x1 > 0, s2 > 0, x2 > 0} . (5) the basic properties of the model solutions are summarized in the next lemma, which extends assertions that are given in [25], [53] and [54]. lemma 1. let assumption a1 be fulfilled. then for each point p0 = (s01,x 0 1,s 0 2,x 0 2) ∈ ω0 the corresponding solution (s1(t),x1(t),s2(t),x2(t)) of (1) with (s1(0),x1(0),s2(0),x2(0)) = p0 is defined for each t > 0. moreover, for each ε > 0 there exists tε such that for each t > tε the following inequalities hold true: si −ε < s(t) + k3x2(t) < si α + ε, si1 −ε < s1(t) + k1x1(t) < si1 α + ε, s1(t) < s i 1 and x1(t) ≥ ε. below we present some authors’ results related to global stabilizability of the model dynamics by means of different control strategies and satisfying different criteria—the ecological criterion in subsections a and b or the economical criterion in subsection c. a. global stabilizability via input control here we investigate the global stabilizability of the dynamics (1) using the classical approach, where the dilution rate u is considered as a control variable. more precisely, we show that for any admissible value of u there exists a nontrivial (with positive components) equilibrium point, which is globally asymptotically stable for the system. although the manipulated input u is the most exploited variable for control purposes, to the authors’ knowledge there is no rigorous proof so far in the literature for global stabilizability of this model. assume that the control variable u varies in the interval u ∈ (0,ub), where ub ≤ 1 α min { µ1(s i 1),µ2(s i 2) } ≤ umax. let for some ū ∈ (0,ub) the following assumption holds true: assumption a2. there exist points s1(ū) = s̄1 ∈( 0,si1 ) and s2(ū) = s̄2 ∈ ( 0,si2 ) , such that the following equalities hold true ū = 1 α µ1(s̄1) = 1 α µ2 (s̄2) . assumption a2 is called regulability [25] of the system: it ensures the existence of a nontrivial equilibrium of the model (1), corresponding to the chosen value of the dilution rate u. determine the points s̄1 and s̄2 according to assumption a2 and compute further x1(ū) = x̄1 = si1 − s̄1 αk1 , x2(ū) = x̄2 = si2 − s̄2 + αk2x̄1 αk3 . then the point p(ū) = p̄ = (s̄1, x̄1, s̄2, x̄2) is an equilibrium point of the system (1). in practical applications, the chosen equilibrium point is also called an operating or a reference point. let s and si be determined according to (4). the next assumption is assumption a3. the following inequalities hold true: (i) µj(sj) < µj(s̄j) for sj ∈ (0, s̄j),j = 1, 2; (ii) µ1(s1) > µ1(s̄1) for s1 ∈ (s̄1,si1); (iii) µ2(s2) > µ2(s̄2) for s2 ∈ (s̄2,si). biomath 8 (2019), 1907127, http://dx.doi.org/10.11145/j.biomath.2019.07.127 page 6 of 17 http://dx.doi.org/10.11145/j.biomath.2019.07.127 neli dimitrova, mikhail krastanov, model-based control strategies for anaerobic digestion processes assumption a3 is technical. it is always fulfilled when the functions µj(·), j = 1, 2, are monotone increasing (like the monod specific growth rate). if µj(·) is not monotone increasing (like e. g. the haldane law) then ū has to be chosen in a proper way, such that assumption a3 will be satisfied. for biological evidence s2 satisfies the inequality s2 ≤ si2. the requirement s2 < s i in assumption a3(iii) is motivated by the fact that si can be considered as the worst-case upper bound of the concentration s2 due to imbalance between acidogenesis and methanogenesis, leading to acidification (x2 = 0) in the bioreactor (cf. [27]). let ū ∈ (0, ub) be chosen according to assumptions a2 and a3. denote by σ1 the system obtained from (1) by substituting the control variable u by ū. then the following theorem 1 reports on the global stability of the equilibrium point p̄. theorem 1. (cf. [18]) let the assumptions a1, a2 and a3 be fulfilled and let p0 = (s01,x 0 1,s 0 2,x 0 2) be an arbitrary point from the set ω0. then the solution of σ1 starting from the point p0 converges asymptotically towards p̄. a drawback of this control approach is that it requires exact and full knowledge of the system states and precise tuning of u to achieve the global stabilizability. this drawback will be overcome by applying an output feedback control. b. global stabilizability via output feedback control this subsection is devoted to global stabilizability of the dynamics (1) by means of a feedback control and in the presence of model uncertainties. the proposed state feedback is of the form u ≡ κ(s2,x2) = βk4µ2(s2)x2, where β is a properly chosen positive parameter. the proposed feedback law is strongly related to the output (2). the parameter β provides some degrees of freedom in choosing an admissible reference (equilibrium) point, usually determined by ecological rules. this fact is also exploited in designing an extremum seeking algorithm for maximizing the methane production in real time (cf. subsection c below). consider the dynamical system (1) in the state space ζ = (s1,x1,s2,x2). using the definition of si from (4) we make the following assumption: assumption a4. lower bounds si− and k−4 for the values of si and k4 respectively, as well as an upper bound k+3 for the value of k3 are known. define the following feedback control law: κ(ζ) = β k4 µ2(s2) x2 (6) with β ∈ ( k+3 si− ·k−4 , +∞ ) . the feedback control law κ(·) can be written in the form κ(·) = β ·q(·), where q is the methane output (2). for the practical applications it is worth to note that q is on-line measurable, so this holds true for the feedback control κ(·) as well. denote by σ2 the closed-loop system obtained from (1) by substituting the control variable u by the feedback κ(ζ) from (6). choose some β ∈ ( k+3 si− ·k−4 , +∞ ) and let ξ̄ = si − k3 βk4 ; obviously, ξ̄ belongs to the interval (0,si). the next assumption is similar to the regulability assumption a2. assumption a5. there exists a point s̄1 such that µ1(s̄1) = µ2 ( ξ̄ − k2 k1 s̄1 ) > 0, s̄1 ∈ ( 0,si1 ) . find s̄1 according to assumption a5 and define s̄2 = ξ̄ − k2 k1 s̄1, x̄1 = si1 − s̄1 αk1 , x̄2 = 1 αβk4 . it is straightforward to see that the point p̄β = (s̄1, x̄1, s̄2, x̄2) is an equilibrium point of σ2. we shall show below that the feedback law (6) stabilizes asymptotically the closed-loop system towards p̄β (cf. [16]). biomath 8 (2019), 1907127, http://dx.doi.org/10.11145/j.biomath.2019.07.127 page 7 of 17 http://dx.doi.org/10.11145/j.biomath.2019.07.127 neli dimitrova, mikhail krastanov, model-based control strategies for anaerobic digestion processes let ω0 be defined according to (5). using the definitions of s and si from (4), we define the sets ω1 = {(s1,x1,s2,x2) : s1 + k1x1 ≤ si1/α, s + k3x2 ≤ s i/α } , ω2 = {( s1,x1, ξ̄ − k2 k1 s1, x̄2 ) : 0 < s1 < k1 k2 ξ̄, x1 > 0 } , ω = ω0 ∩ ω1. assumption a6. let the inequality µ′1(s̄1+θ(s1−s̄1))+ k2 k1 ·µ′2 ( s̄2−θ k2 k1 (s1−s̄1) ) >0 be satisfied for each s1, belonging to the projection of the set ω ∩ ω2 on the s1-axis and for each θ ∈ [0, 1]. assumption a6 is technical. it is always fulfilled when the functions µj(·), j = 1, 2, are monotone increasing. if µj(·) is not monotone increasing then the set-point ξ̄ (or equivalently the value for β) has to be chosen in a proper way in order to satisfy assumption a6. theorem 2. (cf. [16]) let the assumptions a1, a4, a5 and a6 be satisfied. let us fix an arbitrary number β ∈ ( k+3 si− ·k−4 , +∞ ) and let p̄β = (s̄1, x̄1, s̄2, x̄2) be the corresponding equilibrium point. then the feedback control law κ(·) defined by (6) stabilizes asymptotically the control system σ2 to the point p̄β for each starting point ζ0 = (s01,x 0 1,s 0 2,x 0 2) ∈ ω0. c. extremum seeking control consider the equation (2) describing the process output, i. e. the methane production. a modelbased numerical extremum seeking algorithm is proposed in authors’ publications [14]–[18] to steer and stabilize the dynamics (1) towards a steady state, where maximum methane flow rate qmax is achieved. for that purpose the function q is computed on the set of all equilibrium points, parameterized with respect to: (i) u in the case of the input control (theorem 1), (ii) β in the case of the output feedback law (theorem 2). denote the so obtained function by q(q), where q ∈ (q−,q+) denotes one of u or β, and q− > 0, q+ > 0 are the corresponding bounds according to theorem 1 or theorem 2. the function q(q) is called input-output static characteristic of the model. assume that q(q), q ∈ (q−,q+), is strongly unimodal, i. e. there exists a unique point qmax ∈ (q−,q+) where q(q) takes a maximum, qmax = q(qmax), the function strictly increases in the interval (q−,qmax) and strictly decreases in (qmax,q +). denote by e(q) the equilibrium point parameterized on q and let e(qmax) be the steady state where qmax is achieved. the goal is to stabilize in real time the systems σ1 and σ2 towards this (a priori unknown) equilibrium point e(qmax) and therefore to the maximum methane flow rate qmax. this is realized by applying a numerical model-based extremum seeking algorithm. the main idea of the algorithm is the following: a sequence of points q1,q2, . . . ,qn, . . . from the interval (q−,q+) is constructed, each qj being in the form qj = qj−1 ± hj, hj > 0, and such that {qj} tends to qmax; theorems 1 and 2 guarantee that the dynamics is globally asymptotically stabilizable towards the equilibrium e(qj), j = 1, 2, . . .. then by computing and comparing the values q(q1),q(q2), . . . ,q(qn), . . ., the desired equilibrium point e(qmax) and thus qmax are achieved. in the computer implementation the algorithm is carried out in two stages. in the first stage, “rough” intervals [q] and [q] are found which enclose qmax and qmax respectively; in the second stage, the interval [q] is refined using an elimination procedure based on the golden mean value (or fibonacci search) strategy. the second stage produces the final intervals [qmax] = [q−max,q + max] and [qmax] such that qmax ∈ [qmax], qmax ∈ [qmax] and q+max − q−max ≤ �, where the tolerance � > 0 is specified by the user. iii. open–loop control stabilization the previous section ii was devoted to global stabilization of the dynamics (1) towards a previously chosen equilibrium (operating) point. this biomath 8 (2019), 1907127, http://dx.doi.org/10.11145/j.biomath.2019.07.127 page 8 of 17 http://dx.doi.org/10.11145/j.biomath.2019.07.127 neli dimitrova, mikhail krastanov, model-based control strategies for anaerobic digestion processes section proposes a different approach for stabilizing the model solutions. instead to an equilibrium point, the goal here is to steer the model solutions so that the bod values s fall onto an interval [s−,s+], given a priori by ecological rules, and remain there for all time. this is achieved by suitably constructed bounded open-loop control. assumption a7. let the functions µj(sj) are strictly increasing in the intervals (0,sij), j = 1, 2. assumption a7 is technical. it is always fulfilled when the functions µj(sj), j = 1, 2, are presented by the monod specific growth rate. let s and si be defined according to (4), i. e. s = k2 k1 s1 + s2, s i = k2 k1 si1 + s i 2. let us choose an operating (reference) point s∗, s∗ ∈ (0,si). assumption a8 (regulability). there exists a point s∗1 such that µ1(s ∗ 1) = µ2 ( s∗ − k2 k1 s∗1 ) > 0, s∗1 ∈ ( 0,si1 ) . find s∗1 according to assumption a8 and determine further s∗2 = s ∗ − k2 k1 s∗1, x ∗ 1 = si1 −s ∗ 1 αk1 , x∗2 = si2 −s ∗ 2 + αk2x ∗ 1 αk3 = si −s∗ αk3 . it is straightforward to see that the point ζ∗ = (s∗1,x ∗ 1,s ∗ 2,x ∗ 2) is an equilibrium point of system (1). one can directly check that the equilibrium points of (1) satisfy the equalities (cf. the straight line l2 in fig. 1) s1 + αk1x1 = s i 1, s + αk3x2 = s i. denote u∗ = 1 α µ1(s ∗ 1) = 1 α µ2(s ∗ 2). practically, ecological norms prescribe an admissible interval [s−,s+] for the bod values s. we choose an arbitrary interval [s−,s+] contained in the interior of [s−,s+], i. e. [s−,s+] ⊂ (s−,s+). assumption a9. the positive reals s−1 , s + 1 , s − 2 , s+2 , u − and u+ satisfy the relations s− = k2 k1 s−1 + s − 2 , s + = k2 k1 s+1 + s + 2 , αu− = µ1(s − 1 ) = µ2(s − 2 ), αu+ = µ1(s + 1 ) = µ2(s + 2 ), 0 < s−1 < s ∗ 1 < s + 1 < s i 1, 0 < s−2 < s ∗ 2 < s + 2 < s i 2, u− < u∗ < u+, and each point of the interval [u−,u+] is an admissible value for the dilution rate u. the imposed boundedness of u in assumption a9 is not restrictive. practically, the dilution rate is associated with the speed of the feeding pump of the bioreactor and so, there are always a lower bound u− and an upper bound u+ for u (see [25] for more details). it follows from (4) and assumption a9 that 0 < s− < s∗ < s+ < si. consider the following open-loop system σ3 ds1 dt = χ(t)(si1 −s1) −k1 µ1(s1) x1 (7) dx1 dt = (µ1(s1) −αχ(t)) x1 (8) ds2 dt = χ(t)(si2 −s2) −k2µ1(s1)x1 −k3µ2(s2)x2 (9) dx2 dt = (µ2(s2) −αχ(t))x2, (10) obtained from system (1) after replacing u by an arbitrary bounded measurable control function χ(t) such that χ(t) ∈ [u−,u+] for each t ≥ 0. (11) consider first the subsystem (7)–(8), which equations do not depend on s2 and x2. let s11 and s 2 1 be arbitrary real numbers, satisfying the inequalities 0 < s11 < s − 1 < s + 1 < s 2 1 < s i 1. biomath 8 (2019), 1907127, http://dx.doi.org/10.11145/j.biomath.2019.07.127 page 9 of 17 http://dx.doi.org/10.11145/j.biomath.2019.07.127 neli dimitrova, mikhail krastanov, model-based control strategies for anaerobic digestion processes fig. 1. the parallelogram l(s1,s2) = abcd, a = (s1,x12), b = (s 1, x̃12), c = (s 2,x22), d = (s 2, x̃22); the parallelogram l(s−,s+) = efgh, e = (s−,x−2 ), f = (s −, x̃−2 ), g = (s +,x+2 ), h = (s +, x̃+2 ); the lines l1 : s + k3x2 = s i, l2 : s + αk3x2 = s i, l3 : s + k3x2 = si/α. denote by l(s11,s 2 1) the parallelogram determined by the points (s11,x 1 1), (s 1 1, x̃ 1 1), (s 2 1,x 2 1) and (s21, x̃ 2 1), where the reals x 1 1, x̃ 1 1, x 2 1 and x̃ 2 1 are the solutions of s11 + αk1x 1 1 −s i 1 = 0, s 2 1 + αk1x 2 1 −s i 1 = 0, s11 + k1x̃ 1 1 = s 2 1 + k1x 2 1, s 1 1 + k1x 1 1 = s 2 1 + k1x̃ 2 1. analogously, a parallelogram l(s−1 ,s + 1 ) can be defined. for reader’s convenience we note that l(s11,s 2 1) and l(s − 1 ,s + 1 ) are similar to the parallelograms on fig. 1 with s1 and x1 instead of s and x2 respectively. the following assertion holds true for the openloop subsystem (7)–(8). theorem 3. (cf. [19]) let the restriction of assumptions a1, a7, a8 and a9, concerning s1, x1 and µ1(s1) be fulfilled. then for each point ζ01 = (s 0 1,x 0 1) ∈ {(s1,x1) : s1 > 0,x1 > 0} and for each measurable function χ(t) ∈ [u−,u+], t ≥ 0, the corresponding solution ϕ1(t,ζ 0 1 ) = (s1(t),x1(t)) of (7)–(8) with s1(0) = s01 and x1(0) = x 0 1 is well defined for all t ∈ [0, +∞) and tends to the parallelogram l(s−1 ,s + 1 ) as t → +∞. below we shall prove a similar result related to the whole open-loop system (7)–(10). let us choose arbitrary real numbers s1 and s2, satisfying the inequalities 0 < s1 < s− < s+ < s2 < si. (12) denote by l(s1,s2) the parallelogram determined by the points (s1,x12), (s 1, x̃12), (s 2,x22) and (s2, x̃22), where the reals x 1 2, x̃ 1 2, x 2 2 and x̃ 2 2 are solutions of the equations s1 + αk3x 1 2 −s i = 0, s2 + αk3x 2 2 −s i = 0, s1 + k3x̃ 1 2 = s 2 + k3x 2 2, s1 + k3x 1 2 = s 2 + k3x̃ 2 2. (13) the parallelograms l(s1,s2) and l(s−,s+) are visualized on fig. 1. biomath 8 (2019), 1907127, http://dx.doi.org/10.11145/j.biomath.2019.07.127 page 10 of 17 http://dx.doi.org/10.11145/j.biomath.2019.07.127 neli dimitrova, mikhail krastanov, model-based control strategies for anaerobic digestion processes let the set ω0 be defined according to (5) and ζ0 = (s01,s 0 2,x 0 1,x 0 2) ∈ ω0 be an arbitrary point. denote by ϕ(t,ζ0) = (s̄1(t), s̄2(t), x̄1(t), x̄2(t)), t ≥ 0, the corresponding solution of the open-loop system σ3 (i. e. of (7)–(10)) with (s̄1(0), s̄2(0), x̄1(0), x̄2(0)) = ζ 0. according to lemma 1, ϕ(t,ζ0) is bounded and well defined for each t ∈ [0,∞). if we set s̄(t) := k2 k1 s̄1(t) + s̄2(t) and s0 = s(0) = k2 k1 s̄1(0) + s̄2(0), then one can verify that (s̄(t), x̄2(t)), t ≥ 0, satisfy the following openloop system starting from the point (s0,x02): ds dt = u(si −s) −k3 µ2(s2) x2 dx2 dt = (µ2(s2) −αu) x2, (14) where χ is defined by (11). with s̃ > 0, x̃2 > 0 and δ > 0 we define b(s̃, x̃2; δ) :={(s,x2) : |s−s̃| ≤ δ, |x2−x̃2| ≤ δ}. proposition 1. let the assumptions a1, a7, a8 and a9 be fulfilled, and χ : [0, +∞) → [u−,u+] be the measurable function defined by (11). then for each point (s̃, x̃2) from the boundary of l(s1,s2) there exists δ > 0 such that if (s̄(τ), x̄2(τ)) ∈ b(s̃, x̃2; δ) for some sufficiently large τ ≥ 0, then there exists t > τ so that the point (s̄(t), x̄2(t)) belongs to the interior of the set l(s1,s2) \l(s−,s+). proof: let us fix an arbitrary point (s̃, x̃2) from the boundary of l(s1,s2) and let τ0 ≥ 0 be a sufficiently large positive number so that s̄1(τ) ∈ ( s−1 −η,s + 1 + η ) (15) for each τ ≥ τ0, where η := k1 3k2 min { s− −s1,s2 −s+ } > 0. (16) we define z := ( s x2 ) , f(z,u,s2) := ( −k3µ2(s2)x2 + u(sin −s) (µ2(s2) −αu) x2 ) . let r > 0 be sufficiently small, so that b(s̃, x̃2; r) ∩l(s−,s+) = ∅. we set l = max{‖f ′z(z,u,s2)‖ : z ∈ b(s̃, x̃2; r), u ∈ [u−,u+], s2 ∈ [ms2,m s 2 ]}, where [ms2,m s 2 ] is an interval containing the values of s̄2(t) for t ≥ 0, f ′z(z,u,s2) is the jacobian of f with respect to z calculated at the point (z,u,s2), and ‖ ·‖ is the euclidean norm. let τ be an arbitrary number in (τ0, +∞). without loss of generality we may find t0 > τ such that t0 − τ > 0 is so small that the solution z̃(·) := (s̃(·), x̃2(·)) of (14), starting from the point z̃ = (s̃, x̃2)t at the moment of time τ is well defined on the interval [τ,t0] and z̃(t) ∈ b(s̃, x̃2; r/2). let us choose δ0 > 0 to be sufficiently small, so that for each point z ∈ b(s̃, x̃2; δ0) the solution of σ3, starting from the point z at the moment of time τ is well defined on [τ,t0] and the following inequality holds true 3el(t0−τ)δ0 < r. (17) let z be an arbitrary point from b(s̃, x̃2; δ0) and z(t) be the value at the moment of time t of the solution of (14) starting from the point z at the moment of time τ. we set t1 := sup{t ∈ [τ,t0] : z(t) ∈ b(s̃, x̃2; r)}. clearly t1 > τ. moreover, for each t ∈ (τ,t1] we have that ‖z(t)−z̃(t)‖≤‖z+ ∫ t τ f(z(ξ),χ(ξ), s̄2(ξ))dξ−z̃ − ∫ t τ f(z̃(ξ),χ(ξ), s̄2(ξ))dξ‖ ≤ ‖z − z̃‖ + ∫ t τ ‖f(z(ξ),χ(ξ), s̄2(ξ))−f(z̃(ξ),χ(ξ), s̄2(ξ))‖dξ ≤‖z − z̃‖ + ∫ t τ l‖z(ξ) − z̃(ξ)‖dξ. applying the gronwall inequality we obtain ‖z(t) − z̃(t)‖≤ el(t−τ)‖z − z̃‖≤ el(t0−τ)δ0 < r 3 for each t ∈ (τ,t1] (18) biomath 8 (2019), 1907127, http://dx.doi.org/10.11145/j.biomath.2019.07.127 page 11 of 17 http://dx.doi.org/10.11145/j.biomath.2019.07.127 neli dimitrova, mikhail krastanov, model-based control strategies for anaerobic digestion processes according to the choice of δ0 from (17), and hence ‖z(t)‖ ≤ ‖z̃(t)‖ + ‖z(t) − z̃(t)‖. this means that z(t) ∈ b(s̃, x̃; r/2) + b(0; r/3) = b(s̃, x̃; 5r/6) ⊂ b(s̃, x̃; r). from this inclusion, applied for t = t1, we obtain that t1 = t0. assume first that (s̃, x̃2) = (s̃(τ), x̃2(τ)) = (s1,x12), i. e. (s̃, x̃2) coincides with the point a in fig. 1. without loss of generality, we may think that δ0 ∈ ( 0, 1 2 min{s− −s1,s2 −s+} ) . let us assume that (s̄(τ), x̄2(τ)) ∈ b(s̃, x̃2; δ0). using the relations (15) and (16), we obtain that s̄2(τ) = s̄(τ) − k2 k1 s̄1(τ) ≤ s̃ + δ0 − k2 k1 ( s−1 − k1 3k2 (s− −s1) ) = s1 − k2 k1 s−1 + s− −s1 3 + δ0 = s− − (s− −s1) − k2 k1 s−1 + s− −s1 3 + δ0 = s− − k2 k1 s−1 − (s − −s1) + s− −s1 3 + δ0 = s−2 − 2(s− −s1) 3 + δ0 < s−2 − s− −s1 6 < s−2 . then for each t ∈ ( τ, min { τ + s− −s1 12ms2 ,t0 }) (19) we have that s̄2(t) = s̄2(τ) + ∫ t τ ˙̄s2(σ)dσ < s−2 − s− −s1 6 + (t − τ)ms2 < s−2 − s− −s1 12 (20) for each t ∈ (τ,t ] . we set κ := µ2(s − 2 ) −µ2 ( s−2 − s− −s1 12 ) > 0. then, using (20) and assumption a9, we obtain that d dt x̃2(t) = x̃2(t)(µ2(s̄2(t)) −αχ(t)) ≤ x̃2(t)(µ2(s̄2(t)) −µ2(s−2 )) ≤−κx̃2(t) < 0, and hence x̃2(t) < e −κ(t−τ)x̃2(τ) = e −κ(t−τ)x12 < x 1 2 (21) for each t ∈ (τ,t ]. the equality (s̃(τ), x̃2(τ)) = (s1,x12) implies s̃(τ) + k3x̃2(τ) − d = 0, where d := si + (1 −α)k3x12. using the presentation d dt (s̃(t) + k3x̃2(t) −d) = −χ(t)(s̃(t) + k3x̃2(t) −d) + (1 −α)k3χ(t)(x̃2(t) −x12), we obtain that s̃(t) + k3x̃2(t) −d = − ∫ t τ e ∫ ζ t χ(ξ)dξ(1−α)k3χ(ζ)(x12−x̃2(ζ))dζ< 0 (22) for each t ∈ (τ,t ]. as it was shown above, we have that for each t ∈ (τ,t ], z̃(t) ∈ b(s̃, x̃; r/2) and b(s̃, x̃2; r/2)∩ l(s−,s+) = ∅. moreover, the estimate (20) implies that µ2(s̄2(t)) < µ2(s − 2 ) −κ. taking into account this inequality, (21) and (13), we obtain that ds̃(t) dt = −k3µ2(s̄2(t))x̃2(t) + χ(t)(si − s̃(t)) > −k3(µ2(s−2 )−κ)e −κ(t−τ)x12 +u −(si−s1) > −k3µ2(s−2 )x 1 2 + u −(si −s1) = 0 for each t ∈ (τ,t ]. then for each t satisfying (19) the inequality s̃(t) > s1 holds true for each t ∈ (τ,t ]. so, we obtain that s̃(t) ∈ (s1,s−) for each t ∈ (τ,t ]. (23) also, without loss of generality, we may think that l(s1,s2) ∩b(s1,x12; r) = {(s,x2) : s ≥ s1,s+k3x2 ≤ d}∩b(s1,x12; r). biomath 8 (2019), 1907127, http://dx.doi.org/10.11145/j.biomath.2019.07.127 page 12 of 17 http://dx.doi.org/10.11145/j.biomath.2019.07.127 neli dimitrova, mikhail krastanov, model-based control strategies for anaerobic digestion processes taking into account (22) and (23), we obtain from here that the point z̃(t) = (s̃(t), x̃2(t)) belongs to the interior of the set l(s1,s2) \ l(s−,s+). hence there exists γ > 0 so that b(s̃(t), x̃2(t); γ) is a subset of the interior of l(s1,s2) \l(s−,s+). next we choose an arbitrary δ from the interval( 0,γe−l(t−τ) ) and assume that for some τ > τ0 the point (s̄(τ), x̄2(τ)) ∈ b(s̃, x̃2; δ) (note that the real δ > 0 does not depend on τ and on t , but on the difference t−τ). having fixed δ and τ, we fix an arbitrary t according to (19). for this choice of the real t we apply (18) with δ instead of δ0 and obtain that the point (s̄(t), x̄2(t)) belongs to b(s̃(t), x̃2(t); γ), and hence it belongs to the interior of the set l(s1,s2) \l(s−,s+). the remainder cases concerning the location of the point (s̃, x̃2) on the boundary of the parallelogram l(s1,s2) can be considered in analogous way. this completes the proof of proposition 1. corollary 1. let the assumptions a1, a7, a8 and a9 be fulfilled, and χ : [0, +∞) → [u−,u+] be the measurable function defined by (11). then for each point (s̃, x̃2) from the boundary of l(s1,s2) there exists δ > 0 such that if (s̄(τ), x̄2(τ) ∈ b(s̃, x̃2; δ) for some sufficiently large τ ≥ 0, then there exists t > τ so that (s̃(t), x̃2(t)), t ∈ (τ,t ], belongs to the interior of the set l(s1,s2) \ l(s−,s+), where (s̃(t), x̃2(t)) denotes the solution at the time moment t of (9)–(10) with initial condition s̃(τ) = s̃ and x̃2(τ) = x̃2. theorem 4. let the assumptions a1, a7, a8 and a9 be fulfilled. let ζ0 = (s01,s 0 2,x 0 1,x 0 2) ∈ ω0 be an arbitrary point and ϕ(t,ζ0) = (s̄1(t), s̄2(t), x̄1(t), x̄2(t)), t ≥ 0, be the corresponding solution of the open-loop system σ3 (i. e. of (7)–(10)) with ϕ(0,ζ0) = ζ0, where the measurable function χ is defined according to (11). then (s̄1(t), x̄1(t)) and (s̄(t), x̄2(t)) tend to the parallelograms l(s−1 ,s + 1 ) and l(s −,s+) respectively, as t → +∞. proof: let ζ0 = (s01,s 0 2,x 0 1,x 0 2) ∈ ω0 be an arbitrarily fixed point and ϕ(t,ζ0) = (s̄1(t), s̄2(t), x̄1(t), x̄2(t)), t ≥ 0, be the corresponding solution of the open-loop system σ3. theorem 3 implies that (s̄1(t), x̄1(t)) tends to the parallelogram l(s−1 ,s + 1 ) as t → +∞. one can directly check that (s̄(t), x̄2(t)) is a trajectory of (9)–(10) starting from the point (s0,x02), where s0 := k2 k1 s01 + s 0 2. we consider the following two cases: case 1. s(t) + k3x2(t) ≥ si α for each t ≥ 0 (cf. fig. 1). according to lemma 1 we have that for each ε > 0 there exists t1 > 0 such that si α ≤ s(t) + k3x2(t) < si α + ε for each t ≥ t1. hence s(t) + k3x2(t) → si/α as t →∞. let (s(tk),x2(tk)) → (ŝ, x̂2) for some sequence tk → ∞ as k → ∞. clearly, ŝ + k3x̂2 = si/α holds true. without loss of generality we may think that s̄1(tk) → ŝ1 ≥ s−1 and s̄2(tk) → ŝ2 ≥ 0 as tk → +∞. because s(tk) = k2 k1 s1(tk) + s2(tk), we obtain that ŝ = k2 k1 ŝ1 + ŝ2 ≥ k2 k1 s−1 > 0. then the relations ŝ > 0 and ŝ + k3x̂2 = si α imply ŝ+αk3x̂2 = si α −(1−α)k3x̂2 > si α −(1−α) si α = si. hence there exists δ > 0 such that each point (s,x2) satisfying the inequalities |s− ŝ| < δ and |x2 − x̂2| < δ satisfies the inequality s + αk3x2 > si, too. we set m = max { (|u(si−s)−k3µ2(s2)x2|, |x2(µ2(s2)−αu)|) : u∈[u−,u+], |s−ŝ|≤δ, |x2−x̂2|≤δ,s2∈[0,ms2 ]} biomath 8 (2019), 1907127, http://dx.doi.org/10.11145/j.biomath.2019.07.127 page 13 of 17 http://dx.doi.org/10.11145/j.biomath.2019.07.127 neli dimitrova, mikhail krastanov, model-based control strategies for anaerobic digestion processes and m= min { s+αk3x2−si : |s−ŝ|≤δ, |x2−x̂2|≤δ } . according to the choice of δ > 0, we have that m > 0. next we choose the positive reals τ and ε so small that the following inequalities hold true: 2τm < δ and 2ε < τmu−. (24) denote q(t) := s(t) + k3x2(t)− si α and choose tk so large that |s(tk)−ŝ| < δ 2 , |x2(tk)−x̂2| < δ 2 , 0 < q(tk) < ε. we set τ̄ := sup{σ̃ ∈ [0,τ] : |s(tk + σ) − ŝ| < δ, |x2(tk + σ) − x̂2| < δ for each σ ∈ [0, σ̃]}. clearly τ̄ > 0. then, using (24) we obtain that for each σ̃ ∈ [0, τ̄) |s(tk + σ̃) − ŝ| = ∣∣∣∣s(tk) + ∫ tk+σ̃ tk ṡ(ξ)dξ − ŝ ∣∣∣∣ ≤ |s(tk) − ŝ|+∫ tk+σ̃ tk |χ(ξ)(si−s(ξ))−k3µ2(s2(ξ))x2(ξ)|dξ ≤ δ 2 + τm < δ and |x(tk +σ̃)−x̂2| = ∣∣∣∣x2(tk)+ ∫ tk+σ̃ tk ẋ2(ξ)dξ−x̂2 ∣∣∣∣ ≤ |x2(tk) − x̂2| + ∫ tk+σ̃ tk |µ2(s2(ξ))x2(ξ) −αχ(ξ)x2(ξ)|dξ ≤ δ 2 + τm < δ. these inequalities imply that τ̄ = τ, and then for each τ̃ ∈ [0,τ] we have |s(tk + τ̃) − ŝ| < δ and |x2(tk + τ̃) − x̂2| < δ. on the other hand we have that q(tk + τ) = q(tk) + ∫ tk+τ tk q̇(ξ)dξ = s(tk) + k3x2(tk) − si α − ∫ tk+τ tk χ(ξ)(s(ξ) + αk3x2(ξ) −si)dξ < ε−mτu− < −ε. this contradicts the assumption that q(t) = s(t) + k3x2(t)−si/α ≥ 0 for each t ≥ 0 and shows that case 1 is impossible. case 2. s(t) + k3x2(t) ≤ si for each t ≥ 0. similarly to the previous case 1, one can easily show that this case 2 is also impossible. since the previous two cases 1 and 2 are impossible, there exists t1 > 0 so that si < s(t1) + k3x2(t1) < si α and 0 < s(t1) < s i. if (s(t1),x2(t1)) ∈ l(s−,s+), we are done. if (s(t1),x2(t1)) 6∈ l(s−,s+) then there exists a parallelogram l(s1,s2) determined by (12) and (13) such that the point (s(t1),x2(t1)) belongs to the boundary of l(s1,s2), and l(s−,s+) is contained in the interior of l(s1,s2). let l+ be the set of all ω-limit points of the trajectory (s̄(t), x̄2(t)) as t →∞, i. e. (s̄, x̄2) ∈ l+ iff there exists a sequence tk tending to infinity as k → ∞ and such that (s̄(tk), x̄2(tk)) tends to (s̄, x̄2) as k → ∞ (cf. [52]). according to proposition 1, the set l+ is a nonempty compact subset of the parallelogram l(s1,s2). let us assume the existence of a point (s̄, x̄2) ∈ l+ such that the distance between this point and the set l(s−,s+) is strictly positive. we denote by l(s̄1, s̄2) a parallelogram such that the point (s̄, x̄2) belongs to its boundary and l(s−,s+) is contained in the interior of l(s̄1, s̄2). now, applying proposition 1 to the point (s̄, x̄2), the parallelogram l(s̄1, s̄2) and the function χ from (11), we obtain that there exists a neighborhood b(s̄, x̄2; δ) of the point (s̄, x̄2) such that if (s̄(τ̄), x̄2(τ̄) ∈ b(s̄, x̄2; δ) for some sufficiently large τ̄ ≥ 0, then there exists t > τ̄ so that the point (s̄(t), x̄2(t)) belongs to the interior of the set l(s̄1, s̄2) \l(s−,s+). because (s̄, x̄2) is a ω-limit point, there exists a moment of time τ > τ̄ so that (s̄(τ), x̄2(τ)) ∈ b(s̄, x̄2; δ). according to proposition 1, there exists t > τ such that (s̄(t), x̄2(t)) belongs to the interior of the set l(s̄1, s̄2) \l(s−,s+). denote by l(ŝ1, ŝ2) a parallelogram containing the point (s̄(t), x̄2(t)) on its boundary, containbiomath 8 (2019), 1907127, http://dx.doi.org/10.11145/j.biomath.2019.07.127 page 14 of 17 http://dx.doi.org/10.11145/j.biomath.2019.07.127 neli dimitrova, mikhail krastanov, model-based control strategies for anaerobic digestion processes ing l(s−,s+) in its interior and contained in the interior of l(s̄1, s̄2). according to corollary 1, the solution (s̄(t), x̄2(t)), t ≥ t , will remain in l(ŝ1, ŝ2). but this is impossible, because (s̄, x̄2) is a ω-limit point. this contradiction completes the proof of theorem 4. iv. conclusion we investigate a four-dimensional nonlinear dynamic system, which models anaerobic degradation of soluble organic wastes in a continuous bioreactor with methane production. the main result of the paper (section iii) is the construction of a general and practically oriented approach for stabilizing the model. the aim is to ensure in finite time the values of bod (denoted by s) to fall onto a given interval [s−,s+], determined by known ecological norms, and to remain there for all time. the approach is based on suitably constructed open-loop control by means of an arbitrary bounded measurable function. the openloop control approach can serve as a valuable tool for stability investigations using concrete control functions, in particular in the presence of time delays. this new technique can also be considered as a generalization of the previous authors’ results, presented in section ii, which are related to global stabilizability of the model dynamics towards an equilibrium (operating) point using different control approaches. acknowledgements the work of the first author has been accomplished with the financial support of the ministry of education and science by grant no do1161/28.08.2018 for the national geo-information centre – part of the bulgarian national roadmap for research infrastructure. the work of the second author has been partially supported by the sofia university “st. kl. ohridski” under grant no 80-10-20/09.04.2019 and by the bulgarian science fund under grant no kp-06-h22/4/04.12.2018. references [1] v. alcaraz-gonzález, j. harmand, a. rapaport, j.p. steyer, v. gonzález-alvarez and c. pelayo-ortiz, robust interval-based siso regulation under maximum uncertainty conditions in an anaerobic digester, in: intelligent control, 2001 (isic’01), proc. ieee intern. symposium, 240–245, ieee, 2001. 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[54] h. xia, g. s. k. wolkowicz and l. wang, transient oscillations induced by delayed growth response in the chemostat, j. math. biol., 50, 489–530, 2005. biomath 8 (2019), 1907127, http://dx.doi.org/10.11145/j.biomath.2019.07.127 page 17 of 17 http://dx.doi.org/10.11145/j.biomath.2019.07.127 introduction basic properties and global stabilizability of the model dynamics global stabilizability via input control global stabilizability via output feedback control extremum seeking control open–loop control stabilization conclusion references www.biomathforum.org/biomath/index.php/biomath original article the gompertz model revisited and modified using reaction networks: mathematical analysis svetoslav m. markov institute of mathematics and informatics bulgarian academy of sciences smarkov@bio.bas.bg the paper is dedicated to prof. dr. roumen anguelov on the occasion of his 65th birthday. received: 22 august 2021, accepted: 2 october 2021, published: 4 october 2021 abstract— in the present work we discuss the usage of the framework of chemical reaction networks for the construction of dynamical models and their mathematical analysis. to this end, the process of construction of reaction-network-based models via mass action kinetics is introduced and illustrated on several familiar examples, such as the exponential (radioactive) decay, the logistic and the gompertz models. our final goal is to modify the reaction network of the classic gompertz model in a natural way using certain features of the exponential decay and the logistic models. the growth function of the obtained new gompertz-type hybrid model possesses an additional degree of freedom (one more rate parameter) and is thus more flexible when applied to numerical simulation of measurement and experimental data sets. more specifically, the ordinate (height) of the inflection point of the new generalized gompertz model can vary in the interval (0, 1/e], whereas the respective height of the classic gompertz model is fixed at 1/e (assuming the height of the upper asymptote is one). it is shown that the new model is a generalization of both the classic gompertz model and the one-step exponential decay model. historically the gompertz function has been first used for statistical/insurance purposes, much later this function has been applied to simulate biological growth data sets coming from various fields of science, the reaction network approach explains and unifies the two approaches. keywords-systems of ordinary differential equations, reaction networks, chemical reaction networks, evolutionary growth-decay models, relative growth rate, exponential (radioactive) decay, logarithmic change rate, logistic model, gompertz model. i. introduction: reaction networks and evolutionary growth-decay models the present study is devoted to mathematical models induced by chemical reaction networks describing evolutionary changes of biologcopyright: © 2021 markov. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: svetoslav m. markov, the gompertz model revisited and modified using reaction networks: mathematical analysis, biomath 10 (2021), 2110023, http://dx.doi.org/10.11145/j.biomath.2021.10.023 page 1 of 21 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2021.10.023 s m markov, the gompertz model revisited and modified using reaction networks: mathematical ... ical species. we are interested in smooth changes that are described either as monotone growth, or monotone decay, in certain (large) time intervals, oscillation processes will not be considered in this work. we briefly call such changes, resp. functions, growth-decay changes/functions. the exponential radioactive decay, the gompertz and the logistic models are three textbook examples, where we can consider the state variables involved as species that react between each others or are catalysts of certain reaction(s). the main purpose of this work is to propose a new modification of the classic gompertz model possessing more flexibility and functionalities. on the way to achieve this goal we offer a brief introduction into the method of chemical reaction networks, which turns to be essential in the construction of new meaningful mathematical models, in particular when it comes to biological growth and decay processes. in the preliminary part of this work we present several examples demonstrating the role of the chemical reaction network theory (crnt) in tracing the characteristics of elementary familiar mathematical models for the numerical simulation of complex phenomenological (biological) processes. the example with the gompertz model demonstrates the need of a detailed knowledge on the basics of elementary reaction networks. biological growth-decay functions, describing evolutionary processes, are often presented in the literature by means of explicit algebraic expressions. such a presentation offers little or no information on the physico-chemical mechanism of the studied process. more information in this direction is provided when the growth-decay functions are defined as solutions to systems of ordinary differential equations. in the latter case we may look for a possible (chemical) reaction network, which implies the particular dynamical system via mass action kinetics [20]. if such a network does exist, we say that the differential system has a realization (formulation) in terms of a (chemical) reaction network [6]. models formulated in terms of reaction networks offer additional knowledge for the particular biological process, possibly leading to further modifications and improvements of the particular model. an essential step towards the generalizion of purely chemical reaction network, involving just particular chemical substances towards a more generalized chemical objects, such as enzymes and substrates, seem to be done first by the prominent scientist victor henri, who proposed the enzyme kinetic reaction network, see example 7 below. later on scientists working in fields, such as population dynamics and epidemiological modelling, began to realize that many of their models can be based on reaction networks, wherein the chemical substances are considered as more generalized biological objects often denoted as species [6]. the logistic model is an instructive example of a growth-decay model possessing a chemical reaction network [6], [13], [21]. the reaction network presentation enables an easy identification of the logistic model as a constituent part of other (more complex) growth-decay models and to expect similar behaviour of the growth and decay functions, such as sigmoidal growth and exponential decay. section 2 is intended for readers who are not familiar with the chemical reaction network theory (crnt) and its application to mathematical modelling in biology. on several examples we give a brief introduction of method of “translation” of a reaction network into a system of ordinary differential equations (ode’s) using the simple “mass action kinetic” principle. such a translation turns the reaction network into an unique mathematical problem for the time evolution of the masses (concentrations , densities) of the species. readers who are interested in the implementation of the more involved “power law kinetic” postulates may consult some textbooks on crnt [8], [20]. in section 3 we consider the classical gompertz model from the perspective of crnt. for this purpose we formulate the gompertz model in terms of a reaction network. readers already equipped with the technique of translation will be able to easily biomath 10 (2021), 2110023, http://dx.doi.org/10.11145/j.biomath.2021.10.023 page 2 of 21 http://dx.doi.org/10.11145/j.biomath.2021.10.023 s m markov, the gompertz model revisited and modified using reaction networks: mathematical ... trace the relation between the “chemical” reaction network and the classical ode’s involved. this approach allows us to enlighten certain characteristics of the gompertz growth function and the closely related gompetz decay function involved (known as mortality law). in section 4 we propose a new modification of the reaction network of the classic gompertz model obtaining thus an original gompertz-like model possessing one more degree of freedom. the proposed new model is mathematically analysed in the spirit of the reaction network approach. it is shown that the model is a generalization of both the classic gompertz model and the one-step radioactive exponential decay model, forming thus a hybrid between these two familiar models. ii. preliminaries: reaction networks and their translation into ode’s we briefly recall some features of growth-decay models based on reaction networks as well as some appropriate terminology and notation. a. reaction networks. canonical forms. systems of chemical reactions, briefly: reaction networks, are symbolically presented as systems of elementary reactions of the following canonical form: s + q k−→ p + r, (1) showing that one, two or more species on the left side of the reaction arrow, called reactants or reagents (in this example species s and q), react, and, as result of the reaction, one, two or more species, named products (here p and r), are produced. note that the arrow should point to the right and the sign “+” has different meaning when placed on the left or on the right side of the reaction arrow: on the left side the “+” means reaction between the enlisted reactants, whereas on the right no reaction is assumed between the product speciess; if such a reaction exists, then it should be described by a separate elementary reaction. just one species on each side of the arrow is also possible, in fact, as we shall see below, a reaction of the form s k−→ p is a basic one. note that a presentation, such as s k−→ p k1−→ q, is not canonical, the corresponding canonical presentation for this reaction network is: s k−→ p, p k1−→ q. as another example, the often used non-standard presentation of a reverse reaction network: s k−→←− k1 p has the following canonical form: s k−→ p, p k1−→ s. as one more example of a non-canonical familiar reaction network let us mention the enzyme kinetic reaction scheme between an enzyme e with a single active site and a substrate s, forming an enzyme-substrate complex c, which then yields product p , known as henri-michaelis-menten reaction [7]: s + e k1−→←− k−1 c k2−→ p + e. in canonical form the above reaction network should be presented as: s + e k1−→ c, c k−1−→ s + e, c k2−→ p + e. as we shall see below canonical forms are useful for an easier transform of a reaction network into a system of differential equations via mass action principle. notation. all species (reactants and products) partaking in a reaction are denoted by uppercase letters, such as p,q,r,s,x,y . a positive number called “rate parameter” is written over the reaction arrow and indicates the velocity of the reaction. the reactants on the left side of a reaction arrow either decay or remain constant, whereas the product species on the right side of the arrow are growing or constant. in some cases a species may appear two or more times at one side of the arrow, such as a + a, briefly denoted as 2a. biomath 10 (2021), 2110023, http://dx.doi.org/10.11145/j.biomath.2021.10.023 page 3 of 21 http://dx.doi.org/10.11145/j.biomath.2021.10.023 s m markov, the gompertz model revisited and modified using reaction networks: mathematical ... for example, in the reaction (1) the reactants s,q on the left side decay with time, whereas the product species p,r on the right side are growing. in a biological context we can say that in this case the product species grow at the expenses of the reactant species, or that the growing species consume the species on the left side of the arrow (as food resources). in population dynamics (some of) the species on the left side could be considered as “parent” species that give birth (reproduce) into the outcome species on the right side of the arrow. catalyst species. one more special case should be mentioned, namely when a species x appears on both sides of the arrow, e.g. s +x k−→ p +x. in this case species x does not change in time, it is called a catalyst. catalyst species enable a reaction to perform, e. g. in this example without x, the reaction s k−→ p cannot practically happen. when studying biological growth/decay processes, it is important to identify the species with catalytic action, the catalysts. in this work catalysts will be usually denoted by some of the letters x,y,z. once again, by definition a catalyst species appears on both sides of a reaction arrow: on the left side as a reactant and on the right side as a product. note also that some species may partake as catalysts in a particular reaction, but can also be involved as reagents in other reaction(s) as part of the same network; there they may change (grow or decay). in such situations a catalyst species may change as result of its total participation in a particular reaction network [20]. in the “logistic” reaction s + x k−→ 2x species x is a catalyst which catalyses the reaction s k−→ x, that is x catalyzes its own production (growing). such species are called auto-catalysts. as a (total) result of the logistic reaction, the catalyst species x is growing in time. b. differential systems induced by reaction networks via mass action kinetics law of mass action: the rate of a reaction is proportional to the (mathematical) product of the concentrations of the reactants. using mass action kinetics principle every particular reaction network can be uniquely transformed (translated) into a system of ordinary differential equations (ode’s), briefly: system of rate equations, or dynamical system [20], [23]. such a transformation (“translation”) is performed in the following way. firstly, we assume homogenous distribution of the species involved, say p,q,s, in a certain volume/area/compartment. then the quantitative (numeric) values assigned to species p,q,s, such as masses, concentrations, densities, number of entities (individuals, molecules, cells, etc.), are considered as smooth functions of time denoted respectively: p = p(t),q = q(t),s = s(t), so that their derivatives, resp. p′ = dp(t)/dt,q′ = dq(t)/dt,s′ = ds(t)/dt, exist up to a certain order. functions p,q,s are briefly called concentrations or masses, and the first derivatives of the masses p′,q′,s′ are called rates of change (growth or decay or both) of the respective species. under these assumptions, the mass action principle says that the rate of change of each species is proportional to the product of the masses of the reacting species, thereby the coefficient of proportionality is negative if the species decays (which is the case when it appears on the left hand-side of the reaction arrow) and is positive if the species grows (when appearing on the right hand-side of the arrow). the proportionality coefficient is called the rate parameter and is usually written over the reaction arrow. this procedure is performed for every elementary reaction of the system of reactions, that is the reaction network. the “translation” of a reaction network into a system of ode’s via mass action kinetics is illustrated on the following examples. example 1. consider the reaction network s + r k−→ p, k > 0. (2) under mass action kinetics principle, reaction (2) biomath 10 (2021), 2110023, http://dx.doi.org/10.11145/j.biomath.2021.10.023 page 4 of 21 http://dx.doi.org/10.11145/j.biomath.2021.10.023 s m markov, the gompertz model revisited and modified using reaction networks: mathematical ... induces a system of three differential equations, one for each of the three species, also known as reaction equations or rate equations: s′ = −ksr, r′ = −ksr, p′ = ksr, k > 0. (3) system (3) demonstrates the application of the mass action law when two reacting species s,r produce a new species p . in this reaction species s,r decay, whereas species p grows; so the signs of the rate parameters for s and r are negative, and the sign of the rate parameter for p is positive. the rates (of change) of all three species are proportional to the product of the masses of the reacting species s and r, so the absolute values of all rates are of the form ksr, with k > 0. dynamical system (3) implies the identities s′+ r′ = 0, s′+p′ = 0, resp. s+r = c1 = const, s+p = c2 = const. such identities are often known as conservation relations (laws). example 2. consider the reaction network: s + x k−→ p + x, k > 0. (4) the induced system of odes is: s′ = −ksx, x′ = 0, p′ = ksx, k > 0. (5) note that equation x′ = 0 is obtained as x′ = −ksx + ksx = 0. species x is a catalyst. the masses of species s and p satisfy the identity s+ p = const. example 3. consider the following reaction network involving two reactions and three species: s + x k−→ p + x, x α−→ p, (6) wherein k > 0,α > 0. the first reaction s + x k−→ p + x of reaction network (6) does not cause changes in catalyst species x, whereas the second reaction x α−→ p causes exponential decay of x. the other declining species is s; species p is growing. the induced dynamical system is: s′ = −ksx, x′ = −αx, p′ = ksx + αx. (7) note that the rate p′ of species p is obtained as the sum ksx + αx of the rates of p from the two reactions involving p . note also that the catalyst species x changes (decays) due to reaction x′ = −αx. dynamical system (7) induces the identity p + x + s = const . example 4. for the henri-michaelis-menten reaction network s + e k1−→←− k−1 c k2−→ p + e a correct translation produces the following system of ode’s for the concentrations s,e,c,p of the resp. species s,e,c,p : s′ = −k1es + k−1c, e′ = −k1es + (k−1 + k2)c, c′ = k1es− (k−1 + k2)c, p′ = k2c. in this last example the “most difficult” rates formulation seem to be the rates e′ = de/dt and c′ = dc/dt as they correspond to three distinct reaction arrows: e.g. for c′ one incoming in species c and two outgoing arrows from c. the example also demonstrates one more practically useful property of reaction networks: they are more obvious than the resp. systems of ode’s and more easy to memorize. the above examples illustrate the process of translation of a chemical reaction networks into systems of ordinary differential equations. they also illustrate the derivation of an identity relation connecting the state variables in the system of ode’s. we are going next to illustrate how the induced dynamical systems can be further mathematically analysed. c. growth-decay models based on reaction networks below we present three case studies of familiar growth/decay functions generated by reaction networks. the goal is to demonstrates the use of the reaction networks methodology in several aspects: i) parallel to the growth function other useful functions appear (such as decay and wavelike functions) that should be analysed; ii) the original reaction networks offers meaningful interpretations of the resulting model solutions; iii) in biomath 10 (2021), 2110023, http://dx.doi.org/10.11145/j.biomath.2021.10.023 page 5 of 21 http://dx.doi.org/10.11145/j.biomath.2021.10.023 s m markov, the gompertz model revisited and modified using reaction networks: mathematical ... the process of modelling and numerical simulation of particular data sets the modeler can modify the basic reaction network by introducing meaningful changes in (some of) the elementary reactions. this latter possibility will be demonstrated in section 4, where we modify the classic gompertz model. case study 1. consider the reaction (network): s k−→ p, k > 0, (8) known in chemistry as a “first-order (fo)” reaction and in nuclear physics as “one-step exponential radioactive decay (1-serd)”. this elementary reaction is known under several additional names due to its application to various processes such as radioactive nuclear decay, fluid dynamics, enzyme kinetics, marine ecology, physico-chemistry, etc. by definition, a first-order reaction proceeds at a rate that depends linearly on only one reactant concentration. indeed, reaction (8) induces the following dynamical system for the change rates of the concentrations s = s(t), p = p(t) of species s,p : s′ = −ks, p′ = ks, k > 0. (9) dynamical system (9) illustrates how the expression “product of masses (concentrations)” should be interpreted in the definition of the mass action principle when just one species appears on the left hand-side of the reaction arrow. in such a situation the “product” consists of only one state variable, in the case of system (9)—concentration s. system (9) implies the relation s′ + p′ = 0, which after integration gives the identity (conservation) relation s + p = c = const. (10) identity (10) says that at any time moment variable p gains as muuch as s loses. in certain real life situations this could be interpreted either as: i) species p consumes species s as a food resource, or: ii) compartment s migrates (flows, transforms) into compartment x. thus reaction network (8) exhibits a specific mechanism for the time evolution of the two species s and p . species p grows for the expense of species s, which proportionally decays. when equipped with initial conditions s(0) = s0 > 0, p(0) = p0 ≥ 0, (11) such that s0 + p0 = c, dynamical system (9) turns into an initial value problem (ivp) (9)–(11) and relation (10) becomes s + p = s0 + p0 = c, hence s = s0 + p0 −p = c−p. let us briefly analyse the ivp for odes (9)– (11). substituting s = c − p in equation p′ = ks we obtain an autonomous ordinary differential equation for the growth function p of the form: p′ = k(c−p), (12) with initial condition p(0) = p0. the differential equations (9), (12) for functions s and p under initial conditions (11) admit explicit solutions as functions of t ≥ 0. the solution for s is the familiar first-order exponential (radioactive) decay: s(t) = s0e −kt. (13) function s has an asymptote s(∞) = 0, that is concentration s vanishes at infinity. species with such a property are known as “limiting reagents” in chemistry. the solution for p can be obtained using identity (10) when substituting s by c−p in (13): p(t) = c−s = c− (c−p0)e−kt, c = s0 + p0. (14) function p solving (14) has an upper asymptote p(∞) = c; it is known as exponential (also saturation) growth model. to obtain the absolute change rates of functions s,p we can differentiate expressions (13), (14), or insert the expression for s,p in the resp. differential equations. for the absolute decay rate of s we obtain: s′ = −ks0e−kt. (15) biomath 10 (2021), 2110023, http://dx.doi.org/10.11145/j.biomath.2021.10.023 page 6 of 21 http://dx.doi.org/10.11145/j.biomath.2021.10.023 s m markov, the gompertz model revisited and modified using reaction networks: mathematical ... for the absolute growth rate of function p we have p′ = (c−s)′ = ks0e−kt = k(c−p0)e−kt, c = s0 + p0. (16) the logarithmic (relative) decay rate rs of function s is constant, namely: rs = (ln s) ′ = s′ s = −ks s = −k. (17) the logarithmic (relative) growth rate rp of p can be obtained as follows: rp = (ln p) ′ = p′ p = ks c−s = k ( c−p p ) = k ( (c−p0)e−kt c−(c−p0)e−kt ) . (18) the second derivatives of functions s,p are: s′′ = (−ks)′ = k2s > 0, (19) resp.: p′′ = k(c−p)′ = −kp′ = −k2(c−p0)e−kt < 0. (20) relations s′′ > 0 and p′′ < 0 show that function s is convex, whereas function p is concave for all t ≥ 0. remarks. 1. in numerical simulation studies the upper asymptote c of the growth function p, also known as environmental capacity, is usually set to one: p(∞) = c = 1. 2. reaction (8) is the first chain-link of a multi-step exponential radioactive decay chain. in the field of radioactive decay and some population studies one is only interested in the decay process and ignores the evolution of the growth species. in such situations one often presents reaction (8) in the form s k−→ ø, k > 0. the symbol ø indicates that the reaction equation for the growth species in dynamical system (9), in our case equation p′ = ks, is suppressed. case study 2. this example is an extension of the previous reaction network (8). an exponential mechanism involving two sequential first order steps in the transformation of three species s,p,q is given in the reaction network: s k1−→ p, p k2−→ q, (21) where k1,k2 are positive rate parameters. (as already mentioned, reaction network (21) is often written in the concise non-canonical form s k1−→ p k2−→ q,). in nuclear physics reaction (21) is known as rwo-step exponential radioactive decay (2-serd). denoting the concentrations (densities, masses) of species s,p,q as functions of time t by s = s(t),p = p(t),q = q(t) and their derivatives resp. by s′,p′,q′, we arrive at the following dynamical system: s′ = −k1s, p′ = k1s−k2p, q′ = k2p. (22) dynamical system (22) induces the following conservation identity: s + p + q = c = const. (23) system (22) shows that s′ < 0 and q′ > 0, so function s decays, whereas function q grows. it can be proved that function p first increases until a certain time moment t∗ and then decreases in [t∗,∞). such functions are called unimodal, their graphs are wave-like; such functions will also be considered as growth-decay functions. in chemistry, species like p , having zero concentration at the beginning and at the end of the process, are called “intermediate”. a detailed discussion of reaction network (21) and the solutions s,p,q to system (22) are given in [5]. for the solution to general n-step exponential radioactive decay system of differential equations the reader may consult [4]. case study 3. let us discuss the familiar logistic model as induced by a reaction network. the logistic (verhulst) growth function is originally introduced in [28] as the solution of a differential equation of the form x′ = kx(c−x). the solution of this equation is a sigmoidal growth function x = x(t), t ∈ r. one usually ignores the related decay function which is implicitly involved in the right-hand side of the differential equation as c−x. biomath 10 (2021), 2110023, http://dx.doi.org/10.11145/j.biomath.2021.10.023 page 7 of 21 http://dx.doi.org/10.11145/j.biomath.2021.10.023 s m markov, the gompertz model revisited and modified using reaction networks: mathematical ... in contrast, the growth-decay presentation of the logistic model based on reaction networks involves simultaneously the two functions—growing and decaying—as a single tuple (pair). the logistic growth-decay pair is generated by the following reaction network involving two reacting species s,x: s + x k−→ 2x, (24) wherein k is a positive rate parameter. as already mentioned, the symbol 2x in (24) is an abbreviation for x + x. reaction network (24) shows that s is a decaying species, and x is a growing species that catalyses the 1-serd reaction s k−→ x, hence, x is an auto-catalyst species. under the assumption of mass action kinetics reaction network (24) induces the following dynamical system of two differential reaction equations for the masses (concentrations, densities) s = s(t), x = x(t) of species s,x, resp.: s′ = −ksx, x′ = kxs, k > 0. (25) due to s′+x′ = 0, after integration, system (25) induces the conservation identity relation: s + x = const = c. (26) assume initial value conditions s(0) = s0 > 0, x(0) = x0 > 0, (27) satisfying relation (26), so that s0 + x0 = c. (28) the initial value problem (25)–(27) implies the following autonomous differential equations for the growth function x and the decay function s: x′ = kx(c−x), x(0) = x0, s′ = −ks(c−s), s(0) = s0 = c−x0. (29) differential equations (29) show that function x is monotonically increasing and bounded in r+ with values the interval [x0,c), where the value c = s0 + x0 is known as (environmental) carrying capacity. more specifically, function x approaches asymptotically c: x(∞) = x∞ = c. function s is monotonically decreases approaching zero: s(∞) = s∞ = 0. as traditionally accepted in the literature, we shall assume c = 1, thus relation (26) becomes s + x = 1. (30) equations (29) posses explicit algebraic solutions for t ∈ r. to find solution x we have to solve: x′ x(1 −x) = k, which can be written as x′ x + x′ 1 −x = k. (31) integrating (31) we obtain ln x− ln (1 −x) = kt + ln c, or x 1 −x = cekt, c = x0 1 −x0 , which can be presented as x = cekt 1 + cekt = x0 (1 −x0)e−kt + x0 . (32) for the boundary values of x at t = 0, t = ∞, expression (32) gives resp. x(0) = x0, x(∞) = 1, as expected. using expression (32) for the growth function x, the decay function s is readily obtained from identity relation (30) as follows: s = 1 −x = (1−x0)e −kt (1−x0)e−kt+x0 = s0e −kt s0e−kt+(1−s0) = s0 s0+(1−s0)ekt . (33) absolute and logarithmic (relative) change rates. to obtain the absolute rate of change of the growing species x, also called absolute growth rate (agr), we can differentiate expression (32). biomath 10 (2021), 2110023, http://dx.doi.org/10.11145/j.biomath.2021.10.023 page 8 of 21 http://dx.doi.org/10.11145/j.biomath.2021.10.023 s m markov, the gompertz model revisited and modified using reaction networks: mathematical ... alternatively, we can substitute the obtained expressions for s and x in the equation x′ = ksx from (29) to obtain: x′ = k s0e −kt s0e−kt+(1−s0) · x0 (1−x0)e−kt+x0 = kx0(1−x0)e−kt [(1−x0)e−kt+x0]2 . (34) for the boundary values of function x′ we have x′(0) = kx0s0 = kx0(1 −x0), x′(∞) = 0. the logarithmic change rate of function x, known also as relative growth rate (rgr), is defined as: rx = (lnx) ′ = d(lnx)/dt = x′ x . (35) the rgr (35) of x can be obtained by substituting expression (33) for function s in differential equation x′/x = ks to get: rx = x′ x = ks = k(1 −x0)e−kt (1 −x0)e−kt + x0 . (36) for the boundary values of function rx = x′/x we have rx(0) = ks0 = k(1 −x0), rx(∞) = 0. to obtain the absolute change (decay) rate of the species s, we can proceed as follows: s′ = (1 −x)′ = −x′ = − kx0(1−x0)e −kt [(1−x0)e−kt+x0]2 = − ks0(1−s0)e −kt [s0e−kt+(1−s0)]2 . (37) the boundary values of function s′ are s′(0) = −ks0(1 −s0), s′(∞) = 0. the logarithmic change rate (relative decay rate) of species s is: rs = (lnx) ′ = d(lnx)/dt = s ′ s = −kx = −k x0 (1−x0)e−kt+x0 = − k(1−s0) s0e−kt+(1−s0) . (38) for the boundary values of the relative decay rate we have rs(0) = −ks0, rs(∞) = 0. inflection point of the growth function. to look for inflection points of growth function x we need an expression for function x′′ = x′′(t): x′′ = (x′)′ = (kxs)′ = k(x′s + s′x) = k ((kxs)s + (−kxs)x) = k2xs(s−x). (39) expression (39) reduces the solution of equation x′′(t∗) = 0 for an inflection point t∗ to equation s(t∗) = x(t∗), (40) showing that the values of the decay function s and the growth function x at t∗ are identical. the two equations (40): s(t∗) = x(t∗), and s(t∗) + x(t∗) = 1, due to (30), imply s(t∗) = x(t∗) = 1/2. thus we have: x(t∗) = x0 (1 −x0)e−kt ∗ + x0 = 1 2 , (41) equivalently e−kt ∗ = x0 1 −x0 = x0 s0 , (42) or t∗ = − 1 k ln x0 1 −x0 = ln( x0 (1 −x0) )− 1 k . (43) expression (43) for t∗ shows that for t∗ > 0, that is for the existence of inflection of the growth function, it is necessary that the logarithm in (43) is positive, that is x0 (1 −x0) = x0 s0 < 1, that is x0 < s0, hence x0 < 1/2. consequently, when x0 ≥ 1/2 growth function x has no inflection. in this case we have x′′(t) < 0 for all t ≥ 0, hence growth function x is concave on r+. in the special case x0 = s0 = 1/2, we obtain the simple expressions: x = 1/(1 + e−kt), s = e−kt/(1 + e−kt) = 1/(1 + ekt). lag time (phase). let us find an expression for the slope of function x at the inflection point. using (34) and (42), we obtain x′(t∗) = kx0(1−x0)e−kt ∗ [(1−x0)e−kt∗+x0]2 = kx0 2 (x0+x0) 2 = k 4 . (44) biomath 10 (2021), 2110023, http://dx.doi.org/10.11145/j.biomath.2021.10.023 page 9 of 21 http://dx.doi.org/10.11145/j.biomath.2021.10.023 s m markov, the gompertz model revisited and modified using reaction networks: mathematical ... using (44), we can compute the lag time y, which satisfies the relation x(t∗) y = x′(t∗) = k 4 , hence y = 2/k. applications of the logistic and the one-step exponential radioactive decay (1-serd) models. the logistic model finds numerous applications. a popular application is the lotka-volterra “prey-predator” model in population dynamics. denoting the prey species by s and the predator by x, in its simplest form the lotka-volterra model can be written as a reaction network [6]: s + x k−→ 2x, s ν−→ 2s, x µ −→ ø, (45) wherein k,ν,µ are positive rate parameters. reaction network (45) induces the dynamical system: s′ = −ksx + νs, x′ = ksx−µx. (46) the logistic reaction s + x k−→ 2x describes the natural reproduction of the predator population. reaction x µ −→ ø represents the mortality of the predator. reaction s ν−→ 2s describes the natural reproduction of the prey population. the last two reactions vary in different versions of the lotka-volterra (45), however, the logistic reaction remains usually the same. another familiar application of the logistic model is the epidemiological si model, where s stays for susceptible population and i for infective one: s + i k−→ 2i. (47) as we see, the basic epidemiological reaction network (47) coincides with the logistic reaction (24). again, the epidemiological si model (47) is the backbone of various modifications, such as the popular sir model, where r means “removed” (or “recovered”) population: s + i k−→ 2i, i ν−→ r, (48) where k > 0, ν > 0 are positive rate parameters. as a further extension to (48), the “vital” sir model includes additionally newborn (b) and dead (d) population compartments; in the simple case of equal birth and death rates the reaction-networkformulation of the vital sir model reads: s + i k−→ 2i, i ν−→ r, d µ −→ b, (49) where k > 0, ν > 0, µ > 0 are positive rate parameters. the last reaction: d µ −→ b looks somewhat strange; however, it describes adequately the situation in stable populations. models (45), (48), (49) demonstrate an useful property of the reaction-network-formulation of models. namely, such a formulation allows a modeller to construct easily various combinations of existing familiar elementary models with already established characteristics. we shall demonstrate this property in section 4 with a modification of the gompertz model implementing in it certain features of the 1-serd and the logistic models. the two models considered next in the present work—the classic and modified gompertz models—can serve as further examples for our proposed methodology of treating growth-decay models induced by reaction networks. iii. the classic gompertz model revisited from the perspective of reaction networks theory the gompertz growth function has been initially designed for insurance purposes [9], and later used more generally as a modelling growth function in life sciences, like the logistic one [33]. it is usually presented as the explicit algebraic solution x = x(t) to an autonomous differential equation of the form: x′ = νx ln(1/x), ν > 0. in the sequel we deduce the gompertz function x = x(t) starting from a reaction network using the terminology of crnt. this allows us to obtain and analyse the gompertz growth function together with the related decay function, giving biomath 10 (2021), 2110023, http://dx.doi.org/10.11145/j.biomath.2021.10.023 page 10 of 21 http://dx.doi.org/10.11145/j.biomath.2021.10.023 s m markov, the gompertz model revisited and modified using reaction networks: mathematical ... us a general view on the gompertzian growthdecay process, as well as a meaningful physicochemical interpretation of the state variables and rate parameters involved. the reaction network approach explanes and unifies the two approaches. consider the following reaction network involving three species s,x,q and two reactions: s ν−→ q, s + x k−→ 2x + s, (50) wherein ν,k are positive rate parameters [5], [21]. remarks. reaction s + x k−→ 2x + s of network (50) says that both species x and s act as catalysts. more specifically species x catalyses the reaction s−→x + s, turning it into reaction: s + x−→x + x + s. so, x is a growing species autocatalysing itself. on the other hand, species s is also a catalyst in this reaction, it catalyses the reaction: x−→x + x. as result in this reaction, species s does not change in time; however, globally s changes (declines) as result of the first-order exponential decay reaction s−→q. the latter reaction shows that s flows (migrates) into species q, that is, outside the system of the two compartments of our interest (s,x). as mentioned, species q can be replaced by the symbol ø: s ν−→ ø, meaning that we shall ignore the time evolution of species q. assuming homogeneity, denoting the massrelated numerical characteristics (such as concentrations, masses, densities, etc.) of species s,x, resp., by lowercase letters s,x, reaction network (50) induces the following system of two ode’s for the state variables s = s(t), x = x(t), t ∈ r+ = [0,∞) [21]: s′ = −νs, x′ = kxs, (51) where ν > 0, k > 0 are rate parameters. system (51) belongs to the class of biochemical systems (s-systems), cf. [25], [27] [26], [29], [31]. from system (51) we see that function s satisfies the uncoupled autonomous first order differential equation: s′ = −νs, ν > 0. as mentioned in section 2, case study 1., solution s = s(t) is given by: s(t) = s0e −νt, t ∈ r+, (52) for any initial value s(0) = s0 > 0. hence, function s is monotone decreasing, exponentially approaching zero at t −→∞. proposition 1. let functions s = s(t), x = x(t) satisfy the system of ode’s (51) for t ∈ r+. then the following identity relation holds true in r+: γs + ln x = ln x(∞), γ = k/ν. (53) wherein x(∞) = x∞ = x(t)|t−→∞ is the ordinate of the horizontal asymptote of growth function x. proof: dynamical system (51) implies the identity: x′ = kxs = −kx(s′/ν), which can be written as: γs′ + x′/x = 0, γ = k/ν. (54) the integration of (54) yields γs + ln x = const = c. this equation shows that, while function s decreases with time, function x increases, however, the latter increase is bounded by the constant c in the equation. the constant c has an important geometric meaning. indeed, boundary values s(∞),x(∞) satisfy identity (53), so that γs(∞) + ln x(∞) = c, γ = k/ν. (55) using that for t −→ ∞ function s = s0e−νt approaches zero for any positive s0, ν, symbolically s(∞) = s∞ = s|t−→∞ = 0, expression (52) implies ln x(∞) = c. this proves identity (53). the asymptote of the growth function. the constant c = ln x(∞) from identity (53) determines the value of the horizontal asymptote x = x(∞) of growth function x(t). as traditionally done in the literature on gompertz model, we fix the value for the asymptote as x(∞) = x|t−→∞ = 1. this choice of the asymptote leads to the value of c in expression (53) as c = ln x(∞) = ln 1 = 0. (56) biomath 10 (2021), 2110023, http://dx.doi.org/10.11145/j.biomath.2021.10.023 page 11 of 21 http://dx.doi.org/10.11145/j.biomath.2021.10.023 s m markov, the gompertz model revisited and modified using reaction networks: mathematical ... fixing c = 0, identity (53) becomes: γs + ln x = 0, γ = k/ν. (57) in what follows we shall consider the solution to (51) as an ordered 2-tuple (pair) (s,x), satisfying identity (57) in r+. under the choice c = 0 relation (57) guarantees that the growth solution x approaches the asymptote x = 1 at t −→∞. initial value problem. we shall next consider the system of ode’s (51) as initial value problem involving an initial tuple (s(0),x(0)) = (s0,x0) for the solutions. identity (57) is satisfied by solution (s,x) for all t ≥ 0 including t = 0 and t = ∞. hence, when considering system (51) as an initial value problem, we shall naturally assume that the initial tuple (s0,x0) satisfies identity (57), i.e.: γs0 + ln x0 = 0, γ = k/ν. (58) relation (58) restricts the range of x0 in the interval x0 ∈ (0, 1). indeed, if x0 ≥ 1, then (58) implies s0 ≤ 0, which makes no practical sense. so, the choice c = 0 scales the total evolution of the (monotonically increasing) growth function x in the range x ∈ [x0, 1). in contrast, the monotonically decreasing decay function s ranges in the interval (0,s0], thereby the value s0 can be greater than one, s0 > 1. identity (57) implies the following practically useful relations: ks + ν ln x = 0, (59) or, equivalently, using notation δ = 1/γ = ν/k: s = −δ ln x = ln x−δ, x = e−γs, (60) in particular. at t = 0; s0 = −δ ln x0 = ln x0−δ, x0 = e −γs0, (61) to be used in the calculations to follow. in particular, for s0 = 1 we need to have, according to (61): x0 = e −γ. we now formulate the following: proposition 2. let initial value pair (s0,x0) satisfy 0 < x0 < 1, s0 = −δ ln x0, δ = ν/k, (62) then: 1) solution (s,x) to initial value problem (51)– (62) satisfies in r+ = [0,∞) relation (57): γs + ln x = 0; 2) solution x to system (51) satisfies the autonomous differential equation: x′ = νx(− ln x); (63) 3) solution x to initial value problem (51)–(62), can be presented in the form x = x0 e−νt. (64) proof: 1) using initial values (62) the integration of relation (54) under the choice c = 0 yields (57) together with x∞ = 1. 2) substituting (59): ks = −ν ln x, in differential equation x′ = kxs yields: x′ = kxs = x(ks) = x(−ν ln x) = νx(− ln x), = νx ln(1/x), (65) which is the familiar gompertz differential equation (63). 3) using expressions (60), (52), solution x = x(t) can be obtained from relation (57), as follows: ln x = −γs = −γ(s0e−νt) = (−γs0)e−νt = (ln x0)e−νt = ln x0 e−νt, (66) resp. for x we obtain (64). this proves the proposition. using part 3 of proposition 2 we can present the solution tuple (s,x) to initial value problem (51)–(62) in the form (s,x) = ( s0e −νt, x0 e−νt ) . (67) biomath 10 (2021), 2110023, http://dx.doi.org/10.11145/j.biomath.2021.10.023 page 12 of 21 http://dx.doi.org/10.11145/j.biomath.2021.10.023 s m markov, the gompertz model revisited and modified using reaction networks: mathematical ... change rates. to obtain an explicit algebraic expression for the absolute growth rate of species x we write: x′ = kxs = (ks)x = (−ν ln x)x = −νln x0 e−νt exp(ln x0 e−νt), (68) which is positive due to ln x0 < 0. for the boundary values of function x′ we have: x′(0) = kx0s0 = x0(−ν ln x0) > 0, x′(∞) = kx(∞)s(∞) = 0. for the logarithmic (relative) growth rate rx = rx(t) of gompertz growth function x we obtain: rx = (ln x) ′ = x′/x = −ν ln x0 e−νt = ln x0−νe −νt . (69) for the boundary values of rx = x′/x we have: rx(0) = −ν ln x0 e0 = ln(1/x0)ν, rx(∞) = −ν ln x∞ e−∞ = 0. (70) to obtain the absolute change (decay) rate of species s we write s′ = −νs = −νs0e−νt. (71) the boundary values of function s′ are s′(0) = −νs0, s′(∞) = 0. the logarithmic (relative) change rate rs = rs(t) of species s is constant (and so are the boundary values of rs): rs = s′ s = −ν. (72) inflection points. consider next the existence of a possible inflection point t∗ for the gompertz growth function x. for the second derivative x′′ = x′′(t) of growth function x we have: x′′ = (x′)′ = (kxs)′ = k(x′s + s′x) = k ((kxs)s + (−νs)x) = kxs(ks−ν) = k2xs(s−ν/k). (73) expression (73) reduces the solution of equation x′′(t∗) = 0 for t∗ to equation s(t∗) = ν/k = δ, (74) showing that the value of the decay function s at the inflection point t∗ is equal to the rate parameter ratio δ = ν/k. using (52), equation (74) reads: s(t∗) = s0e −νt∗ = δ, hence e−νt ∗ = δ/s0 = 1/(γs0), (75) thus we obtain t∗ = (1/ν) ln(γs0) = ln(γs0) 1 ν . (76) expressed via x0, the inflection time moment t∗ can be obtained when substituting γs0 in (76) by ln(1/x0): t∗ = ln(ln 1 x0 ) 1 ν . (77) to compute the value x(t∗) we can use relations (74): x(t∗) = e−γs(t ∗) = e−γδ = e−1 = 1/e. (78) expression (76) implies: in order to have t∗ > 0, that is to exist an inflection point for growth function x on r+, it is necessary relation 1 < γs0 to take place. in terms of x0 this reads (using γs0 = − ln x0): 1 < ln(1/x0), equivalently: x0 < 1/e. (79) the condition for existence of inflection point s(t∗) = δ < s0 is equivalent to 0 < x0 < 1/e = x(t∗). lag time (phase). to compute the so-called lag time interval of growth function x for the classic gompertz model, we need the slope of function x at the inflection point, that is x′(t∗). denote the intersection of the tangent line to the graph of x through the inflection point t∗,x(t∗) with the abscissa and the asymptote x = x∞ = 1, resp. by (ta, 0) and (tb, 1). the length of interval [ta, t∗] on the abscissa is the lag time, whereas the length of the interval [t∗, tb] is the log time. biomath 10 (2021), 2110023, http://dx.doi.org/10.11145/j.biomath.2021.10.023 page 13 of 21 http://dx.doi.org/10.11145/j.biomath.2021.10.023 s m markov, the gompertz model revisited and modified using reaction networks: mathematical ... substituting x(t∗) = 1/e, resp. ln x′(t∗) = −1, in expression x′ = ksx = −νx ln x, we obtain x′(t∗) = ν/e. (80) the lag time l = t∗ − ta is equal to the ratio l = x(t∗)/x′(t∗) = 1/ν, (81) observing the triangle below its vertex point (t∗,x(t∗)). we summarize the results obtained on the expressions for solution rates, inflection points and lag/log times in the following proposition 3. 1. solution tuple (s,x) to gompertz initial value problem (51)–(62) is characterized by the following properties: 1a. solution (s,x) is given by (67): (s,x) = ( s0e −νt, x0 e−νt ) , thereby γs = − ln x, in particular: γs0 = − ln x0, γ = k/ν = 1/δ. the boundary values of gompertz growth/decay functions s,x are: s(0) = ln x0 −δ, s(∞) = 0; x(0) = e−γs0 , x(∞) = 1. 1b. the absolute change rates of species s,x are given by expressions (71), (68): s′ = −νs0e−νt; x′ = −ν ln x0 e−νt x0e −νt . the boundary values of functions s′, x′, are: s′(0) = −νs0, s′(∞) = 0; x′(0) = −νx0 ln x0 > 0, x′(∞) = 0. 1c. the logarithmic change rates of functions s,x are given by (72), (69): rs = (ln s) ′ = s ′ s = −ν, rx = (ln x) ′ = x′/x = ln x0 −νe−νt. for the boundary values of the logarithmic change rates of gompertz growth/decay functions s,x we have: rs(0) = rs(∞) = −ν; rx(0) = ln(1/x0) ν, rx(∞) == 0. 2a. the inflection point t∗ of gompertz growth function x is given by (77): t∗ = ln ( ln 1 x0 ) 1 ν . the values of the growth/decay functions at inflection point t∗ are, cf. (74), (78): s(t∗) = δ = ν/k, x(t∗) = e−1. for the existence of inflection point in [0,∞), the necessary and sufficient condition is s0 > δ, resp.: 0 < x0 < 1/e. 2b. the lag time l is given by the ratio (81): l = x(t∗)/x′(t∗) = 1/ν. remarks on the logistic and gompertz models. 1) the inflection point of the growing species x in the gompertz model is lower than those in the logistic model, 1/e < 1/2. as a consequence, the gompertzian growth curve has a shorter lag time, resp. longer lag (ageing, mortality) time, than the logistic growth curve. in both models the growing species x reproduces by a doubling mechanism, being constrained by species s which declines with time until vanishing. the inhibiting decay mechanism of species s is different in the two growth-decay models. in the logistic case species s is consumed by x as nutritional (food) resource (s charges x); thereby x is the solely species using s. in contrast, in the gompertz model species s serves as a catalyst for x; thereby s charges some “other” species as well. the catalytic vs. the resource-charging role of species s turns out to be decisive in the distinction of the two models. 2) both the logistic and the gompertz models make use of just one rate parameter, which is not so obvious in the gompertz model. the parameter k in the gompertz model participates only in the identity relation and its role there is to determine the value of s0, resp. the limit value (one) of the upper asymptote of the growth function. without loss of generality the parameter k can be set to one, se e.g. [32]. the decisive role of the rate parameter ν is noticed by many authors, biomath 10 (2021), 2110023, http://dx.doi.org/10.11145/j.biomath.2021.10.023 page 14 of 21 http://dx.doi.org/10.11145/j.biomath.2021.10.023 s m markov, the gompertz model revisited and modified using reaction networks: mathematical ... using for ν names such as “relative maturity rate”, “mortality rate”, etc. iv. a new model based on a modified (hybrid) gompertz-like reaction network in this section we propose and mathematically analyse a growth-decay model induced by a reaction network that is close to gompertz network (50) but borrows some features of the one-step exponential decay 1-serd model. consider the following reaction network involving two species s and x: s ν−→ x, s + x k−→ 2x + s, (82) wherein ν,k are positive rate parameters. denoting the mass-related quantitative (numerical) characteristics of species s,x, resp. by s,x, under the assumption of mass action kinetics, reaction network (82) induces the following dynamical system of two reaction equations: s′ = −νs, x′ = kxs + νs, (83) where ν,k are positive rate parameters. proposition 4. if functions s = s(t), x = x(t) satisfy the system of ode’s (83) on r+ = [0,∞), then the following identity relation holds true on r+: γs + ln(x + 1/γ) = ln(x∞ + 1/γ), (84) wherein γ = k/ν and x∞ = x(∞) = x(t)|t−→∞. proof: system (83) implies the relation: s′ ν + x′ kx + ν = 0, or γs′ + x′ x + ν/k = 0. after integration, the above relation leads to the following identity γs + ln(x + 1/γ) = const = c, γ = k/ν. (85) as in the classic gompertz model, solution s to system (83) satisfies the autonomous ordinary differential equation s′ = −νs, with solution (13) (or (52)). hence function s is monotone decreasing, approaching zero: s(∞) = s∞ = 0. passing to limit t −→ ∞ in identity (85), using s(∞) = 0, we obtain const = c = ln(x(∞) + 1/γ), hence (84). identity (84) suggests that while function s monotonically decays, function x monotonically grows remaining bounded from above by x(∞), so the line x = x(∞) is a horizontal asymptote for the growth function x = x(t). we have the freedom to choose the boundary value x(∞) for x at t = ∞; so, as done traditionally, we set x∞ = x(∞) = 1. using boundary values s∞ = 0, x∞ = 1, we obtain relation (84) in the form γs + ln(x + 1/γ) = ln(1 + 1/γ), or, using notation δ == 1/γ = ν/k: γs + ln(x + δ) = ln(1 + δ), equivalently γs + ln x + δ 1 + δ = 0. (86) remark. introducing the “deviated” growth function xδ: xδ = x + δ 1 + δ , in relation (86), we obtain γs + ln xδ = 0, which formally matches the corresponding identity (57) for the classic gompertz model: γs + ln x = 0. this similarity takes place for a number of results to follow. in fact it is possible to rewrite most of the classical gompertz results from section 3 replacing function x by xδ, then performing a reverse transformation: x = xδ(1 + δ) −δ. from relation (86) we can obtain expressions for s in terms of x and for x in terms of s. here are given some practically useful relations: s = δ ln 1 + δ x + δ = ln ( x + δ 1 + δ )−δ , (87) biomath 10 (2021), 2110023, http://dx.doi.org/10.11145/j.biomath.2021.10.023 page 15 of 21 http://dx.doi.org/10.11145/j.biomath.2021.10.023 s m markov, the gompertz model revisited and modified using reaction networks: mathematical ... γs = ln ( 1 + δ x + δ ) , (88) eγs = 1 + δ x + δ , (89) x = (1 + δ)e−γs −δ. (90) in the special case t = 0 we have the following relations for the initial pair (s0,x0), assuring the limit condition x∞ = 1: s0 = ln ( 1 + δ x0 + δ )δ = ln ( x0 + δ 1 + δ )−δ ; (91) γs0 = ln ( 1 + δ x0 + δ ) ; (92) eγs0 = 1 + δ x0 + δ ; (93) x0 = (1 + δ)e −γs0 −δ. (94) substituting s from (87) in the differential equation for growth function x in dynamical system (83), leads to the following autonomous differential equation: x′ = kxs + νs = k(x + δ)s = k(x + δ) ln ( 1+δ x+δ )δ . (95) to deduce an explicit solution for growth function x, we first use relation (89) to write: 1 + δ x + δ = eγs = eγs0e −νt = (eγs0 ) e−νt . (96) we then substitute the term eγs0 in (96), using the expression (93), to get 1 + δ x + δ = (eγs0 ) e−νt = ( 1 + δ x0 + δ )e−νt . (97) relation (97) implies an explicit expression for growth function x = x(t): x(t) = (1 + δ) ( x0 + δ 1 + δ )e−νt −δ. (98) based on the above considerations, we formulate the following proposition 5. let initial value tuple (s0,x0) be such that 0 < x0 < 1, s0 = ln ( 1+δ x0+δ )δ > 0, δ = 1/γ = ν/k, (99) then i) solution (s,x) to initial value problem (83)– (99) satisfy on r+ = [0,∞) relation (84); in particular relations (87), (90): s = ln ( x + δ 1 + δ )−δ ; x = (1 + δ)e−γs −δ. ii) the growth function x satisfies the autonomous ordinary differential equation (95); x′ = k(x + δ) ln ( 1 + δ x + δ )δ ; iii) the solution x to equation (95) with initial value x(0) = x0, resp. system (83)–(99) can be presented in the explicit form (98): x(t) = (1 + δ) ( x0 + δ 1 + δ )e−νt −δ. change rates. to obtain an explicit algebraic expression for the absolute growth rate of species x we use expressions (95) and (98) to obtain: x′ = k(x + δ) ln ( 1+δ x+δ )δ = k(1 + δ) ( x0+δ 1+δ )e−νt ln ( 1+δ x0+δ )e−νt . (100) for the boundary values of function x′ we have: x′(0) = kx0s0 = x0(−ν ln x0) > 0 x′(∞) = kx(∞)s(∞) = 0. (101) biomath 10 (2021), 2110023, http://dx.doi.org/10.11145/j.biomath.2021.10.023 page 16 of 21 http://dx.doi.org/10.11145/j.biomath.2021.10.023 s m markov, the gompertz model revisited and modified using reaction networks: mathematical ... for the relative (logarithmic) growth rate rx = rx(t) of growth function x we obtain: rx = (ln x) ′ = x′/x = −ν ln x0 e−νt = ln x0−νe −νt . (102) for the boundary values of rx = x′/x we have: rx(0) = −ν ln x0 e0 = ln(1/x0)ν, rx(∞) = −ν ln x∞ e−∞ = 0. (103) the expressions for the absolute and logarithmic change (decay) rates of decay species s are the same as those for the classic gompertz model, cf. (71), (72). inflection points. to calculate the inflection points of the growth function x (if any) we need to obtain an expression for the second derivative x′′ of x: x′′ = (x′) ′ = (ksx + νs)′ = (ksx)′ + (νs)′ = k(s′x + sx′) + νs′ = k(−νsx + s(ksx + νs)) −ν2s = ks(−νx + ksx + νs−ν2/k) = ks (ks(x + ν/k) −ν(x + ν/k)) = ks(x + δ)(ks−ν) = k2s(x + δ)(s−δ). (104) according to expression (104) equation x′′(t) = 0 is equivalent to equation s(t)−δ = 0, or s(t) = δ. let time instant t∗ solve the latter equations, then s0 > s(t ∗) = δ (105) is a necessary condition for the existence of an inflection point. indeed, if (105): s0 > δ, then the monotone decreasing function s(t) equals to δ at time instant t∗: s(t∗) = δ. in other words, for s0 > δ then there exists time moment t∗, such that the pair t∗,x(t∗) is an inflection point for growth function x, such that s(t∗) = δ, resp. x′′(t∗) = 0. expression (104) reduces the solution of equation x′′(t∗) = 0 for t∗ to equation s(t∗) = ν/k = δ, (106) saying that the value of the decay function s at inflection time instant t∗ is equal to the rate parameter ratio δ = ν/k. using (52), we have s(t∗) = s0e−νt ∗ = δ, hence e−νt ∗ = δ/s0 = 1/(γs0), (107) thus we obtain t∗ = (1/ν) ln(γs0) = ln(γs0) 1 ν . (108) let us now “translate” formulae (105), (108) in terms of growth function x. expressed via x0, the inflection time instant (108) can be obtained when substituting γs0 in (108) by ln((1 + δ)/(x0 + δ)): t∗ = ln(ln 1 + δ x0 + δ ) 1 ν . (109) knowing the s-value s(t∗) = δ, we compute the x-value x(t∗) using expression (90): x(t∗) = (1 + δ)e−γs(t ∗) −δ = (1 + δ)e−γδ −δ = (1 + δ)e−1 −δ = (1 − (e− 1)δ)/e, thus finally we have: x(t∗) = 1 − (e− 1)δ e . (110) from (110) we obtain a necessary and sufficient condition for the existence of inflection: 1 e > x(t∗) = 1 − (e− 1)δ e > x0 > 0. (111) relation (111) implies a necessary condition for the existence of inflection: x(t∗) > 0, namely: δ < 1 e− 1 = e ≈ 0.58197671, (112) resp. ν < e k. (113) using (111) we obtain a second necessary condition for the existence of inflection: 1/e > x(t∗) > x0, namely: 1 − (e− 1)δ > ex0, (114) biomath 10 (2021), 2110023, http://dx.doi.org/10.11145/j.biomath.2021.10.023 page 17 of 21 http://dx.doi.org/10.11145/j.biomath.2021.10.023 s m markov, the gompertz model revisited and modified using reaction networks: mathematical ... resp. δ < 1 −ex0 e− 1 = e(1 −ex0), (115) resp. ν < e(1 −ex0)k. (116) practically, the necessary and sufficient condition (111) for the existence of inflection point can be tested by verifying the two necessary conditions (112), (115), resp. (113), (116). inequality (112) implies the following restriction on the initial value x0 for existence of inflection point: x0 < 1/e. (117) remarks. i) restriction (117) says that for the existence of inflection initial value x0 should be below the inflection value for the classical gompertz model, i.e. 1/e. ii) note that in the classic gompertz case the growth function may have no inflection only when x0 > 1/e. in contrast, the hybrid gompertz growth function may have no inflection for any initial values x0 ∈ (0, 1), even for initial values satisfying (117). iii) depending on the values of the rate parameters ν,k, the inflection point can be arbitrarily close to the classic gompertz value 1/e, as well as to initial value x0 no matter how small x0 is. in the latter case the inflection point can be arbitrarily close to zero (providing x0 itself is sufficiently small). this possibility makes the shape of the graph of x extremely flexible, which makes a considerable difference with the classic gompertz case. under suitable choice of the initial conditions and rate parameters the hybrid model can be close to the one-step exponential decal model. iv) inequality (112) implies δ < 1 −ex0 e− 1 < 1 e− 1 . (118) the “rough” inequality (118) can be used when x0 is close to 0. lag time (lag phase). to compute the lag time of growth function x for the hybrid gompertz model, we need the value of slope of function x at inflection time moment t∗, that is x′(t∗). denote the intersection of the tangent line through the inflection point (t∗,x(t∗)) with the abscissa and the asymptote x = x∞, resp. by (ta, 0) and (tb, 1). the width (length) of interval [ta, t∗] is by definition the lag time. to compute the slope x′(x∗) of growth function x at inflection time moment t∗, we substitute the value: x∗ = x(t∗) from (110), resp. x∗ + δ = (1 + δ)/e, in the expression for the slope x′ to obtain: x′(t∗) = k(x∗ + δ) ln ( 1+δ x∗+δ )δ = kδ e (1 −δ) ln ( e1+δ 1+δ ) = ν e (1 −δ). (119) as in the classical gompertz case, we define the lag time (phase) l as the length of the segment on the abscissa between inflection moment t∗ and the intersection point of the abscissa and the tangent with slope x′(t∗). hence, for the lag time l we obtain: l = x∗ x′(t∗) = 1 ν ( 1 − eδ 1 + δ ) . (120) we summarize the obtained results as follows. proposition 6. 1. solution pair (s,x) to initial value hybrid gompertz problem (83), (s(0) = s0,x(0) = x0), is characterized by the following properties: 1a. the absolute change rate of species x is given by: (100): x′ = k(1 + δ) ( x0 + δ 1 + δ )e−νt ln ( 1 + δ x0 + δ )e−νt . 1b. for the boundary values of function x′ we have expression (101): x′(0) = kx0s0 = x0(−ν ln x0) > 0 x′(∞) = kx(∞)s(∞) = 0. 1c. the logarithmic change rate rx = rx(t) of the hybrid gompertz growth function x is given by expression (102): rx = (ln x) ′ = x′/x = −ν ln x0 e−νt. biomath 10 (2021), 2110023, http://dx.doi.org/10.11145/j.biomath.2021.10.023 page 18 of 21 http://dx.doi.org/10.11145/j.biomath.2021.10.023 s m markov, the gompertz model revisited and modified using reaction networks: mathematical ... for the boundary values of the logarithmic change rates of the hybrid gompertz growth function x we have (103): rx(0) = −ν ln x0 e0 = ln(1/x0)ν, rx(∞) = −ν ln x∞ e−∞ = 0. (121) 2a. the inflection time moment t∗ of the hybrid gompertz growth function x is given by (109): t∗ = ln(ln 1 + δ x0 + δ ) 1 ν . the values of the growth/decay functions at inflection point t∗ are (74): s(t∗) = δ = ν/k, resp. (110): x(t∗) = 1 − (e− 1)δ e . the slope of the tangent line through the inflection point is given by (119): x′(t∗) = ν e (1 −δ). for the existence of inflection point in [0,∞), the necessary and sufficient condition is (111): 1 e > x(t∗) = 1 − (e− 1)δ e > x0 > 0. 2b. the lag time l is given by the ratio (120): l = x∗ x′(t∗) = 1 ν ( 1 − eδ 1 + δ ) . finally, the following proposition holds true: proposition 7. the hybrid gompertz function (98) is a generalization of the classical gompertz function (64). proof: the classical gompertz function (64) is obtained from the hybrid gompertz function (98) for the special case k −→ ∞, resp. δ = ν/k −→ 0, while keeping the rate parameter ν fixed. v. concluding remarks biological growth functions are usually presented in the mathematical literature by means of their explicit expressions or as solutions to differential equations [11]–[18]. however, biological growth models are usually related to decay processes/functions, which becomes especially transparent when the models are based on reaction equations. using chemical reaction network theory, one can easily observe close relations between various growth/decay processes, as well as between existing growth-decay models, e.g. classes of biochemical systems [25]–[27], [29]–[31]. in the present work we propose an elementary introduction in the reaction network approach based on mass action kinetics. to this end we discuss in some detail several familiar examples, such as the oneand two-step exponential (radioactive) decay, the logistic and the gompertz models. we focus on the simultaneous analysis of the growth and the decay functions using the identity relation between the two functions naturally induced by the reaction equations. the power of the reaction network approach is fully revealed in section 3 when applied to the analysis of the classical gompertz model. there we propose a revision of the model based on the reaction network inducing the original gompertz model, which we call “gompertz reaction network” in honour of the author of the well-known growth model and his seminal paper [9]. our final goal in this work is the modification of the gompertz reaction network in a natural way, using fully the dynamical features of the one-step (first-order) exponential decay reaction. in this way we obtain a hybrid of the one-step exponential and the classic gompertz model in a natural way, performing a small modification in the gompertz reaction network. the growth function of the obtained new hybrid gompertzlike model possesses one additional degree of freedom (one more rate parameter) and is thus more flexible when applied to modelling and numerical simulation of measurement and experimental biomath 10 (2021), 2110023, http://dx.doi.org/10.11145/j.biomath.2021.10.023 page 19 of 21 http://dx.doi.org/10.11145/j.biomath.2021.10.023 s m markov, the gompertz model revisited and modified using reaction networks: mathematical ... data sets. more specifically, the ordinate (height) of the inflection point of the hybrid model can largely vary, whereas the resp. height of the classic gompertz model is fixed (at 1/e). the presented generalization of the gompertz model possesses some common features with the richards model in direction of improved flexibility when simulating measurement data sets [24], [34]. acknowledgments. this paper is supported by the national scientific program “information and communication technologies for a single digital market in science, education and security (ictinses)”, contract no. do1–205/23.11.2018, financed by the ministry of education and science in bulgaria. the author is indebted to his colleagues the professors r. anguelov, m. krastanov, n. kyurkchiev, for their constant encouragements and useful suggestions. special thanks also go to dr. a. vassilev for his expert numerical simulations (to be published jointly in a forthcoming article). references [1] anguelov, r., borisov m., iliev a., kyurkchiev n., s. markov, on the chemical meaning of some growth models possessing gompertzian-type property. math. meth. appl. sci. 2017, 1-2, https://doi.org/10.1002/mma.4539 [2] asadi, m., di crescenzo, a., sajadi, f.a. et al. a generalized gompertz growth model with applications and related birth-death processes. ricerche mat (2020). https://doi.org/10.1007/s11587-020-00548-y [3] bajzer, z., vuk-pavlovic, s., new dimensions in gompertzian growth. journal of theoretical medicine 1997; 2:307–315. 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[34] zwietering, m. h., jongenburger, i., rombout, f. m., k. van’t riet, modeling of the bacterial growth curve, applied and environmental microbiology 56 (6): 1875– 1881, 1990. doi:10.1128/aem.56.6.1875-1881.1990 biomath 10 (2021), 2110023, http://dx.doi.org/10.11145/j.biomath.2021.10.023 page 21 of 21 http://dx.doi.org/10.11145/j.biomath.2021.10.023 introduction: reaction networks and evolutionary growth-decay models preliminaries: reaction networks and their translation into ode's reaction networks. differential systems induced by reaction networks via mass action kinetics growth-decay models based on reaction networks the classic gompertz model revisited from the perspective of reaction networks theory a new model based on a modified (hybrid) gompertz-like reaction network concluding remarks references www.biomathforum.org/biomath/index.php/biomath original article age-structured delayed sipcv epidemic model of hpv and cervical cancer cells dynamics i. numerical method vitalii v. akimenko∗, fajar adi-kusumo† ∗institute of biology and medicine,, taras shevchenko national university of kyiv,ukraine vitaliiakm@gmail.com †faculty of mathematics and natural sciences universitas gadjah mada, yogyakarta, indonesia. f adikusumo@ugm.ac.id received: 29 april 2021, accepted: 2 october 2021, published: 6 december 2021 abstract— the numerical method for simulation dynamics of nonlinear epidemic model of agestructured sub-populations of susceptible, infectious, precancerous and cancer cells and unstructured population of human papilloma virus (hpv) is developed (sipcv model). cell population dynamics is described by the initial-boundary value problem for the delayed semi-linear hyperbolic equations with ageand time-dependent coefficients and hpv dynamics is described by the initial problem for nonlinear delayed ode. the model considers two time-delay parameters: the time between viral entry into a target susceptible cell and the production of new virus particles, and duration of the first stage of delayed immune response to hpv population growing. using the method of characteristics and method of steps we obtain the exact solution of the sipcv epidemic model in the form of explicit recurrent formulae. the numerical method designed for this solution and used the trapezoidal rule for integrals in recurrent formulae has a second order of accuracy. numerical experiments with vanished mesh spacing illustrate the second order of accuracy of numerical solution with respect to the benchmark solution and show the dynamical regimes of cellhpv population with the different phase portraits. keywordssipcv epidemic model; agestructured model; hpv. numerical epidemiology; method of characteristics; i. introduction human papilloma virus (hpv) is the most common sexually transmitted infection that can cause dysplasia (precancerous tissue) and cervical cancer [27], [36], [45]. the importance of this problem in medicine motivated extensive studied of hpv-epidemic models over the last decades. the problem is considered at two levels: (i) social level (trans-mission of hpv between people and copyright: © 2021 akimenko et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: vitalii v. akimenko, fajar adi-kusumo, age-structured delayed sipcv epidemic model of hpv and cervical cancer cells dynamics i. numerical method, biomath 10 (2021), 2110027, http://dx.doi.org/10.11145/j.biomath.2021.10.027 page 1 of 23 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2021.10.027 v v akimenko, f adi-kusumo, age-structured delayed sipcv epidemic model of hpv and cervical ... effectiveness of hpv vaccination of population) [13], [43], and (ii) molecular or tissue level (cellhpv population dynamics) [17], [48], [54]. early epidemic models of the second level (ii) are based on the systems of nonlinear ode and consider the dynamics of several compartments-subclasses of cells and virus. since viruses are non-living things which do not replicate their population dynamics can be efficiently described by the unstructured models. but such models are not efficient enough for the cell populations because they neglect the life history of biological cell as living organ-ism and provide only restricted description of system in many applications. the need for integration of cell life-histories with population dynamics motivates the development of age-structured, and more generally, physiologically structured, population models in cell population dynamics and tumor growth modelling [18], [21], [29], [32], [35], [38], [39], [40], [41], [42], [44], [47], [49], [50]. the model considered in this paper is based on the epidemic models studied in [3], [48], and includes the age-structured models of susceptible, infective, precancerous, cancer cells populations and unstructured model of human papilloma virus population (sipcv epidemic model). we use the l. hayflick limit theory [20] for modelling proliferation in all cell subpopulations. biological cell assumed to divide into mother and daughter cells with different developmental potential [23]. the model considers life history of mother cells: birth, maturing up to the age when they can proliferate, limited number of divisions at the reproductive age when it gives birth to several daughter cells, aging up to the final reproductive age and death. the features of new model are: (i) death rates of infected, precancerous and cancerous cells do not depend from the hpv abundance since the immune response of biological organism is tolerant with respect to its own cells [27], [30], [36], [45]; (ii) death rate of hpv is densitydependent function due to the immune response on the virus population growing [27], [36], [45]; (iii) interaction strength between susceptible and hpv is a product of the lotka-voltera incidence rate and result in the growth of infective cells [3]; (iv) infective cells partially move to the precancerous subclass and partially apoptose when viruses leave infectious cells and ready to infect new susceptible cells [31], [48]; (v) precancerous cells move to the cancer subclass with the non-linear densitydependent saturated rate [48]; (vi) two time-delay parameters describe the time between viral entry into a target susceptible cell and the production of new virus particles [26] and a duration of the first stage of delayed immune response to hpv population growing [27], [36], [45]. development of population dynamics models of mathematical epidemiology necessarily leads to the development of methods of numerical epidemiology in simulation of cell-virus interaction dynamics. there are two main approaches to numerical simulation of physiologically structured models of population dynamics. the first one uses the difference-finite or element-finite approximation of the boundary-initial value problem for the transport equation [1], [7], [10], [19], [34], [44], [51], [52]. this approach is preferred when solving the problems with complex equations, complex domain of definition or/and non-local boundary conditions for which it is difficult or impossible to obtain an exact solution. on the other hand, only biologically correct monotone and conservative difference schemes are applicable for the transport and reaction-diffusion equations in practice. since in work [33] it was proved that the physically correct monotone linear difference scheme of the second order of approximation and higher for the transport or reaction-diffusion equations does not exist, one has to design of the new special nonlinear monotone difference schemes of the second order of approximation and higher for these types of equations [7], [10], [19]. the second approach is based on method of characteristics of the theory of hyperbolic equations of the first order which reduces the pde transport equation to the ode of the first order and allows for applying the corresponding well-known numerical methods to this equation. this approach includes the escalator boxcar train (ebt) method [25], method of charbiomath 10 (2021), 2110027, http://dx.doi.org/10.11145/j.biomath.2021.10.027 page 2 of 23 http://dx.doi.org/10.11145/j.biomath.2021.10.027 v v akimenko, f adi-kusumo, age-structured delayed sipcv epidemic model of hpv and cervical ... acteristics [1], [3], [5], and different combinations and variations of them [2], [14], [15], [16], [37], [52], [56]. numerical methods of this type provide the biologically correct numerical solution of the second and higher order of approximation in most cases. but despite the importance of study and development of numerical epidemiology nowadays there are no works with numerical implementations of such methods for the physiologically structured epidemic models so far. using the method of characteristics [3], [4], [5], [6], [8], [9], [11], [24], [53], [55], [58] and method of steps from the theory of delayed differential equations [12], [28], [46], [57], [59], we obtain an exact solution of the sipcv epidemic model. this solution is given in form of the recurrent formulae (like in works [3], [4], [8], [55]) in which the densities of all subpopulations are defined through the integrals from solution taken at previous instance of time. for this exact solution we create the numerical method of the second order of accuracy using the trapezoidal rule [22] for approximation of all integrals in the recur-rent formulae. the exact solution of the age-structured delayed sipcv epidemic system provides us the advantages in developing of the more fast and accurate numerical method in comparison with the other methods used for the ageand sizestructured models [1], [5], [25]. evaluation of the convergence rate of the approximate solution to the exact one is considered in the next paper, the second part of our work. for illustration of accuracy of the designed numerical method we evaluate in experiments the residual of deviation of numerical solution from the benchmark solution of sipcv model obtained for the special particular coefficients of equations and initial values. simulations show that the relative numerical error converges pointwise to zero with h → 0 (where h is a mesh spacing) by the quadratic low that illustrates and confirms the second order of accuracy of numerical solution. these results are in good agreement with the evaluations of numerical errors obtained earlier in experiments [8] with the numerical method designed for the nonlinear age-structured models of population dynamics by the same approach. numerical experiments with model parameters of the system reveal two types of the asymptotically stable dynamical regimes-non-oscillating and oscillating dynamics of population when the quantity of cells and hpv converge to the non-trivial asymptotically stable states of the system. these dynamical regimes correspond to the localization of dysplasia (precancer cells) and cancer tumor in biological tissue without metastases. overall, the numerical method obtained in this paper provides the reliable and accurate theoretical instrument for simulation and study of age-structured sipcv epidemic models. ii. model we consider a sipcv epidemic model that consists of susceptible (noninfected), in-fectious (without significant changing of morphology, cin i and cin ii stages [30], [36]), precancerous (with changed by virus morphology dysplasia, but is differentiable yet, cin iii stage [30], [36]), cancer (nondifferentiable) cells and human papilloma virus (hpv) that moves freely between cells. the age-specific densities of susceptible, infectious, pre-cancerous and cancer cells are denoted as s(a,t), i(a,t), p(a,t) and c(a,t). the dynamics of cell subclasses (subpopulations) is described by the nonlinear age-structured model with ageand time-dependent death rates of susceptible ds(a,t), infectious dq(a,t), precancerous dr(a,t) and cancer cells dc(a,t) with maximum lifespan ad, an age reproductive window of non-cancer cells [ar,am] and cancer cells [ac,ag], ac < ar, ag < am. we assume that adaptive behavior of the hpv makes the immune response of biological organism (both t-killers cells and humoral immunity) tolerant with respect to infectious and precancerous cells and noneffective with respect to cancer cells that is their death rates does not depend from the hpv abundance [27], [30], [36]. organism recognizes cancer cells as its own and does not at-tempt to destroy them. since viruses are not living things and cannot reproduce (multiply) until they enter a living cell, we use the ode model biomath 10 (2021), 2110027, http://dx.doi.org/10.11145/j.biomath.2021.10.027 page 3 of 23 http://dx.doi.org/10.11145/j.biomath.2021.10.027 v v akimenko, f adi-kusumo, age-structured delayed sipcv epidemic model of hpv and cervical ... instead of age-structured model for describing the dynamics of hpv quantity v (t). the model considers the time-dependent recruitment rate of viruses λ(t), their density-dependent death rate dv(v (t − σ)) with delay parameter σ the duration of the first stage of delayed immune response to hpv population growing [36]. the interaction strength be-tween susceptible and hpv is a product of the lotka-voltera incidence rate α(a − θ,t − θ)v (t − θ)s(a,t) (α(a,t) is a time and age dependent infection rate, θ is a delay parameter (θ < σ) time between viral entry into a target cell and the production of new virus particles [26]), and result in the growth of infectious cells, with partial move of them to the precancerous subclass with rate δ(a,t)i(a,t) (δ(a,t) is a time and age dependent progression rate from infectious to precancerous cells dysplasia). model considers also the partial apoptosis of infectious and precancerous cells with rate n(t) ad∫ 0 (d∗q(a,t)i(a,t) + d ∗ r(a,t)p(a,t))da, (where n(t) is a time dependent mean number of virions produced by one cell, d∗q(a,t) and d∗r(a,t) are time and age dependent death rates of infectious and precancerous cells as a result of virus replication), when viruses leave destroyed cells and ready to infect new susceptible cells. we assume that 0 < d∗q(a,t) < dq(a,t), 0 < d∗r(a,t) < dr(a,t), that is some of infectious and precancerous cells may heal themselves or, at least, can slow down the replication of the hpv within themselves and die when they reach the maximum age or as a result of exposure to some external factors. cancer cells cannot be defined in absolute terms and usually they are recognized only by the abnormal rapid proliferation (ac < ar, ag < am) when cells do not have time for maturation (”nondifferentiable” cells). cancer cells differ from the normal ones in their lack of response to normal fertility control mechanism [27], [30], [36], [45]. that is why precancerous cells turn to the cancer in our model only through the proliferation when a precancerous cell may divide into two new cancer ones. fertility rate of precancerous cells turned to the cancer is proportional to the density-dependent saturated rate µ(nr(t)) = ρnr(t) 1+wnr(t) [48], where the quantity of precancerous cells of re-productive age and older is nr(t) = ad∫ ar p(a,t)da, ρ ∈ (0, 1] is a progression rate from precancerous to cancerous cells, w ≥ 1 is a coefficient of satura-tion, 0 ≤ µ(nr) < ρw−1 for nr ≥ 0. these assump-tions lead to the following age-structured epidemic model in domain q = {(a,t) |a ∈ (0,ad), t ∈ (0,t)}: ∂s(a,t) ∂t + ∂s(a,t) ∂a = −ds(a,t)s(a,t) −α(a−θ,t−θ)v (t−θ)s(a,t), (1) ∂i(a,t) ∂t + ∂i(a,t) ∂a =−(dq(a,t)+δ(a,t))i(a,t) + α(a−θ,t−θ)v (t−θ)s(a,t), (2) ∂p(a,t) ∂t + ∂p(a,t) ∂a = −dr(a,t)p(a,t) + δ(a,t)i(a,t), (3) ∂c(a,t) ∂t + ∂c(a,t) ∂a = −dc(a,t)c(a,t), (4) ∂v (t) ∂t = λ(t) −dv(v (t−σ))v (t) + n(t) ad∫ 0 (d∗q(a,t)i(a,t)+d ∗ r(a,t)p(a,t))da (5) equations (1) (5) are completed by the boundary conditions and initial values: s(0, t) = am∫ ar βs(a,t)s(a,t)da (6) i(0, t) = am∫ ar βq(a,t)i(a,t)da (7) p(0, t) = (1−µ(nr(t)) am∫ ar βr(a,t)p(a,t)da (8) biomath 10 (2021), 2110027, http://dx.doi.org/10.11145/j.biomath.2021.10.027 page 4 of 23 http://dx.doi.org/10.11145/j.biomath.2021.10.027 v v akimenko, f adi-kusumo, age-structured delayed sipcv epidemic model of hpv and cervical ... c(0, t) = ag∫ ac βc(a,t)c(a,t)da + µ(nr(t)) am∫ ar βr(a,t)p(a,t)da (9) s(a, 0) =ϕ(a),i(a,0) =p(a,0) =c(a,0) = 0 (10) v (t) = v0(t), t ∈ [−σ, 0], (11) where the birth (fertility) rates of the susceptible, infectious, precancerous and cancer cells are βs(a,t), βq(a,t), βp(a,t) and βc(a,t), respectively; ϕ(a) is an initial density of susceptible cells, v0(t) is an initial value of hpv quantity. function α(a−θ,t−θ) is a rate of infection which takes into account infection of susceptible cells over the age θ (survived after incubation period) at the unit of time: α(a−θ,t−θ) = { 0, if a < θ α0(a,t−θ), if a ≥ θ, (12) where α0(a,t) is an auxiliary rate of infection defined for a ∈ [θ,ad], t ∈ [−θ,t−θ]. we impose the following restrictions on the density-dependent hpv death rate and cells’ birth and death rates [27], [30], [31], [36], [45]: δ(a,t),dc(a,t),dr(a,t),dq(a,t),ds(a,t)>0, (13) βs(a,t),βq(a,t),βr(a,t),βc(a,t) > 0, (14) α0(a,t) ≥ 0, λ(t) ≥ 0,n(t) > 0, (15) dv(v ) > 0, ∂dv(v ) dv ≥ 0 for v > 0, (16) ϕ(a) ≥ 0, ad∫ 0 ϕ(a)da > 0,v0(t) ≥ 0 (17) the positiveness of derivative of densitydependent death rate of hpv in (15) means that increasing of hpv quantity changes the characteristics of intracellular space that result in the organism immune response through the activation of cell immunity (t-killers) and humoral immunity (b-lymphocytes) that leads to the elimination of viruses (i.e. monotone increasing of their death rate). we assume that all coefficients and initial values of system (1)-(11) are twice continuously differentiable functions and have the private derivatives of the second order by all their arguments. iii. susceptible cell population dynamics since the time delay θ in equation (1) provides the formal linearization, we can apply the method of steps [3], [28], from the theory of linear delayed ode (1), (6), (10) and the method of characteristic [3], [4], [8], [21], [53], [55] from the theory of line-ar hyperbolic equations of the first order to the initial-boundary problem for the quasi-linear transport equation. without loss of generality time cut is covered by the set of consequent time periods [tk−1, tk] (tk = kad, k = 1, ...,k, t0 = 0, tk = t), and we define the following sets (fig. 1): q (1) k = {(a,t)|t ∈ [(k − 1)ad,a + (k − 1)ad), a ∈ [0,ad]} (18) q (2 ) k = {(a,t)|t ∈ [a + (k − 1)ad,kad], a ∈ [0,ad]} where q = k⋃ k=1 ( q (1) k ⋃ q (2) k ) . we define also the auxiliary set of age intervals: ω(k) = {[−a(k)l ,−a (k) l+1]|a (k) l = lar + (k − 1)ad, l = 0, ...,l− 1,a(k)l = am + (k − 1)ad, a (k) l+1 = kad} (19) l = { [am/ar] + 1, if am/ar − [am/ar] > 0, am/ar, if am/ar − [am/ar] = 0, (20) where k = 1, ...,k, and [a] is an integer part of real number a. using new characteristic variable v = a − t, and time t we reduce the problem (1), (6), (10) to cauchy problem for the linear homogeneous delayed ode: ∂s ∂t = −ds(v + t,t)s(v + t,t) −α(v + t−θ,t−θ)v (t−θ)s(v + t,t) (21) s(v, 0) = ϕ(v). (22) biomath 10 (2021), 2110027, http://dx.doi.org/10.11145/j.biomath.2021.10.027 page 5 of 23 http://dx.doi.org/10.11145/j.biomath.2021.10.027 v v akimenko, f adi-kusumo, age-structured delayed sipcv epidemic model of hpv and cervical ... fig. 1. splitting of domain q̄ and the age intervals[ a (1) l ,a (1) l+1 ] , for l = 0, ...,l. renewal boundary condition (6) completes the problem (19), (20). in original variables, solution of problem (21), (22), has a form (k = 1, ...,k): s(a,t) =   f (k−1) 1 (a−t)w1(a−t,tk−1, t), if (a,t) ∈ q(1 )k f (k) 1 (a−t)w1(a−t,t−a,t), if (a,t) ∈ q(2 )k , (23) w1(v,tk,t) =e − t∫ tk (ds(v+ξ,ξ)+α(v+ξ−θ,ξ−θ)v (ξ−θ))dξ (24) functions f(k)1 (a−t) are defined from the initial (10) and boundary (6) conditions: f (0) 1 (v) = ϕ(v),v ∈ [0,ad] (25) f (k) 1 (v) = φ (k) 1l (v),v ∈ [ −a(k)l ,−a (k) l−1 ] , (26) l = 1, ...,l + 1, where the set of auxiliary functions φ(k)1l (v) is defined as: φ (k) 11 (u) = am+u∫ ar+u βs(v −u,−u) f (k−1) 1 (v) w1(v,tk−1,−u)dv (27) u ∈ [ −a(k)1 ,−a (k) 0 ] , φ (k) 1l (u) = −a(k)l−2∫ ar+u βs(v−u,−u)φ (k) 1(l−1)(v)w1(v,−v,−u)dv + l−3∑ j=0 −a(k)j∫ −a(k)j+1 βs(v−u,−u)φ (k) 1(j+1) (v)w1(v,−v,−u)dv + am+u∫ −a(k)0 βs(v−u,−u)f (k−1) 1 (v)w1(v,tk−1,−u)dv, (28) u ∈ [ −a(k)l ,−a (k) l−1 ] , l = 2, ...,l, φ (k) 1(l+1) (u) = am+u∫ ar+u βs(v −u,−u) f (k) 1 (v) w1(v,−v,−u)dv (29) u ∈ [ −a(k)l+1,−a (k) l ] . two parts of the solution (23) have to be linked continuously, that is s(a,t) ∈ c(q) at the points of characteristics a = t − tk−1 in directions a = const, . by analogy with [3], [4], [8], we can write the compatibility (continuity) condition of solution s(a,t) ∈ c(q) in the form: ϕ(0) = am∫ ar βs(a, 0)ϕ(a)da (30) where the second term in the right side of eq. (28) should be omitted for l = 2, function f(k)1 (v) in eq. (29) has been already defined on the previous steps because its argument v ∈ [ar + u,am + u] biomath 10 (2021), 2110027, http://dx.doi.org/10.11145/j.biomath.2021.10.027 page 6 of 23 http://dx.doi.org/10.11145/j.biomath.2021.10.027 v v akimenko, f adi-kusumo, age-structured delayed sipcv epidemic model of hpv and cervical ... and v > u. the schematic diagram of definition of the functional sequence φ(1)1l (v) for the example l = 4 is given in fig. 2. iv. infectious cell population dynamics using the same approach as in the previous section we reduce the problem (2), (7), (10), to the cauchy problem for linear nonhomogeneous delayed ode: ∂i ∂t =−d̃q|(v+t,t)i(v+t,t) (31) + α(v+t−θ,t−θ)v (t−θ))s(v+t,t) i(v, 0) = 0, (32) where d̃q(v,t) = dq(v,t) + δ(v,t), function v (t) is taken from the previous instant t − θ, function s(v + t,t) is taken from the previous section. solution of problem (31), (32) has a form (k = 1, ...,k): i(a,t) =   f (k−1) 2 (a− t)w2(a− t,tk−1, t) +z (k−1) 2 (a− t,tk−1, t), if (a,t) ∈ q(1)k , f (k) 2 (a− t)w2(a− t,t−a,t) +z (k) 2 (a− t,t−a,t), if (a,t) ∈ q(2)k , (33) w2(v,tk, t) = exp  − t∫ tk d̃q(v + ξ,ξ)dξ   , (34) z (k) 2 (v,tk, t) = t∫ tk w2(v,ξ,t)α(v + ξ −θ,ξ −θ) v (ξ −θ)s(v + ξ,ξ)dξ. (35) functions f(k)2 (v) are defined from the initial (10), and boundary (7) conditions: f (0) 2 (v) = 0, v ∈ [0,ad], (36) f (k) 2 (v) = φ (k) 2l (v), v ∈ [ −a(k)l ,−a (k) l−1 ] , (37) l = 1, ...,l + 1, where the set of auxiliary functions φ(k)2l (v) is defined as: φ (k) 21 (u)= am+u∫ ar+u βq(v−u,−u)(f (k−1) 2 (v)w2(v,tk−1,−u) + z (k−1) 2 (v,tk−1,−u))dv, (38) u ∈ [ −a(k)1 ,−a (k) 0 ] , φ (k) 2l (u)= −a(k)l−2∫ ar+u βq(v−u,−u)(φ (k) 2(l−1)(v)w2(v,−v,−u) + z (k) 2 (v,−v,−u))dv + l−3∑ j=0 −a(k)j∫ −a(k)j+1 βq(v −u,−u)(φ (k) 2(j+1) (v) w2(v,−v,−u) + z (k) 2 (v,−v,−u))dv + am+u∫ −a(k)0 βq(v −u,−u)(f (k−1) 2 (v) w2(v,tk−1,−u)+z (k−1) 2 (v,tk−1,−u))dv, (39) u ∈ [ −a(k)l ,−a (k) l−1 ] , l = 2, ...,l, φ (k) 2(l+1) (u)= am+u∫ ar+u βq(v−u,−u)(f (k) 2 (v)w2(v,−v,−u) + z (k) 2 (v,−v,−u))dv (40) u ∈ [ −a(k)l+1,−a (k) l ] , where the second term in the right side of equation (39) should be omitted for l = 2, function f(k)2 (v) in eq. (40) has been already defined on the previous steps be-cause its argument v ∈ [ar + u,am + u] and v > u (see example in fig.2). since i(a,t) has the trivial initial value, two parts of the solution (33) satisfy the compatibility (continuity) condition of solution, i(a,t) ∈ c(q). biomath 10 (2021), 2110027, http://dx.doi.org/10.11145/j.biomath.2021.10.027 page 7 of 23 http://dx.doi.org/10.11145/j.biomath.2021.10.027 v v akimenko, f adi-kusumo, age-structured delayed sipcv epidemic model of hpv and cervical ... fig. 2. schematic diagram of definition of the sequence φ(1)1l (v), l = 1, ...,l + 1, l = 4. v. precancerous cell population dynamics the problem (3), (8), (10) is reduced to the cauchy problem for nonlinear non-homogeneous ode ∂p ∂t = −dr(v + t,t)p(v + t,t) + δ(v + t,t)i(v + t,t) (41) p(v, 0) = 0, (42) where i(v+t,t) is taken from the previous section. function µ(v+t−σ,np (t)) will be defined below in the recurrent formulae for the travelling wave solution. solution of problem (41), (42) has a form (k = 1, ...,k): p(a,t) = p(v,t) =   f (k−1) 3 (a− t)w3(a− t,tk−1, t) +z (k−1) 3 (a− t,tk−1, t), if (a,t) ∈ q(1 )k , f (k) 3 (a− t)w3(a− t,t−a,t) +z (k) 3 (a− t,t−a,t), if (a,t) ∈ q(2 )k , , (43) w3(v,tk, t) = exp  − t∫ tk dr(v + ξ,ξ)dξ   , (44) z (k) 3 (v,tk−1, t) = t∫ tk−1 w3(v,ξ,t)δ(v + ξ,ξ) i(v + ξ,ξ)dξ (45) functions f(k)3 (v) and np (ξ) are defined from the initial (10) and boundary (8) conditions: f (0) 3 (v) = 0,v ∈ [0,ad] (46) f (k) 3 (v) = φ (k) 3l (v),v ∈ [ −a(k)l ,−a (k) l−1 ] , (47) l = 1, ...,l + 1, where the auxiliary functions φ(k)3l (v) are defined as: φ (k) 31 (u) = (1 −µ(nr(−u))) am+u∫ ar+u βr(v −u,−u) ( f (k−1) 3 (v)w3(v,tk−1,−u) +z (k−1) 3 (v,tk−1,−u) ) dv, (48) u ∈ [ −a(k)1 ,−a (k) 0 ] biomath 10 (2021), 2110027, http://dx.doi.org/10.11145/j.biomath.2021.10.027 page 8 of 23 http://dx.doi.org/10.11145/j.biomath.2021.10.027 v v akimenko, f adi-kusumo, age-structured delayed sipcv epidemic model of hpv and cervical ... np(−u) = ad+u∫ ar+u ( f (k−1) 3 (v)w3(v,tk−1,−u) +z (k−1) 3 (v,tk−1,−u) ) dv,u∈[−a(k)1 ,−a (k) 0 ], (49) φ (k) 3l (u) = (1 −µ(nr(−u)))( −a(k)l−2∫ ar+u βr(v−u,−u) ( φ (k) 3(l−1)(v)w3(v,−v,−u) ) +z (k) 3 (v,−v,−u) ) dv l−3∑ j=0 −a(k)j∫ −a(k)j+1 βr(v −u,−u) ( φ (k) 3(j+1) (v) w3(v,−v,−u) +z (k) 3 (v,−v,−u) ) dv + am+u∫ −a(k)0 βr(v −u,−u) ( f (k−1) 3 (v) w3(v,tk−1,−u) +z (k−1) 3 (v,tk−1,−u) ) dv ) u ∈ [ −a(k)l ,−a (k) l−1 ] , l = 2, ...,l (50) np(−u) = −a(k)l−2∫ ar+u ( φ (k) 3(l−1)(v)w3(v,−v,−u) +z (k) 3 (v,−v,−u) ) dv + l−3∑ j=0 −a(k)j∫ −a(k)j+1 ( φ (k) 3(j+1) (v)w3(v,−v,−u) +z (k) 3 (v,−v,−u) ) dv + ad+u∫ −a(k)0 ( f (k−1) 3 (v)w3(v,tk−1,−u) +z (k−1) 3 (v,tk−1,−u) ) dv, (51) u ∈ [ −a(k)l ,−a (k) l−1 ] , l = 2, ...,l. φ (k) 3(l+1) (u) = 2(1 −µ(np(−u))) am+u∫ ar+u βr(v−u,−u)f (k) 3 (v)w3(v,−v,−u) +z (k) 3 (v,−v,−u) ) dv, (52) u ∈ [ −a(k)l+1,−a (k) l ] , np(−u) = ad+u∫ ar+u ( f (k) 3 (v)w3(v,−v,−u) +z (k) 3 (v,−v,−u) ) dv (53) u ∈ [ −a(k)l+1,−a (k) l ] , where the second term in the right side of eqs. (50), (51) should be omitted for l = 2, function f (k) 3 (v) in eqs. (52), (53) has been already defined on the previous steps because its argument v ∈ [ar + u,am + u] and v > u (see example in fig.2). because the initial value of p(a,t) is trivial, solution (43) satisfies the compatibility (continuity) condition of solution, p(a,t) ∈ c(q). vi. cancer cell population dynamics since cancer cells have a specific age reproductive window [ac,ak] differed from the one of noncancer cells, we introduce a new auxiliary set of age intervals for the problem (4), (9), (10): ω̃(k) = {[ −ã(k)l ,−ã (k) l+1 ]∣∣∣ã(k)l = lac + (k − 1)ad, l = 0, ...,l− 1, ã(k)l = ag + (k − 1)ad, ã (k) l+1 = kad} (54) l̃= { [ag/ac]+1, if ag/ac−[ag/ac]>0, ag/ac, if ag/ac − [ag/ac] = 0, (55) biomath 10 (2021), 2110027, http://dx.doi.org/10.11145/j.biomath.2021.10.027 page 9 of 23 http://dx.doi.org/10.11145/j.biomath.2021.10.027 v v akimenko, f adi-kusumo, age-structured delayed sipcv epidemic model of hpv and cervical ... where k = 1, ...,k, and [a] is an integer part of real number a. in new characteristic variable v = a−t = const problem (4), (9), (10) is reduced to the cauchy problem for linear homogeneous ode: ∂c ∂t = −dc(v + t,t) c(v + t,t) (56) c(v, 0) = 0, (57) where function µ(np (t − σ)) is taken from the previous instant of time, p(a,t) is taken from the previous section. solution of problem (56), (57) has a form (k = 1, ...,k): (a,t) = (v,t) (58) =   f (k−1) 4 (a− t)w4(a−t,tk−1, t), if (a,t) ∈ q(1 )k , f (k) 4 (a−t)w4(a−t,t−a,t), if (a,t) ∈ q(2 )k , (59) w4(v,tk, t) = exp  − t∫ tk dc(v + ξ,ξ)dξ   . (60) functions f(k)4 (v) are defined from the initial (10) and boundary (9) conditions: f (0) 4 (v) = 0, v ∈ [0,ad] (61) f (k) 4 (v) = φ (k) 4l (v), v ∈ [ −ã(k)l ,−ã (k) l−1 ] , l = 1, ..., l̃ + 1, (62) where the auxiliary functions φ(k)4l (v) are defined as: φ (k) 41 (u) = ag+u∫ a+u βc(v −u,−u)f (k−1) 4 (v) w4(v,tk−1,−u)dv + µ(np(−u)) × am+u∫ ar+u βp(v −u,−u)p(v −u,−u)dv, (63) u ∈ [ −a(k)1 ,−a (k) 0 ] , φ (k) 4l (u) = −ã(k)l−2∫ a+u β(v −u,−u)φ(k) 4(l−1)(v) w4(v,−v,−u)dv+ l−3∑ j=0 −ã(k)j∫ −ã(k)j+1 β(v−u,−u) × φ(k) 4(j+1) (v)w4(v,−v,−u)dv + ag+u∫ −ã(k)0 βc(v −u,−u)f (k−1) 4 (v) w4(v,tk−1,−u)dv + µ(np(−u)) am+u∫ ar+u βp(v −u,−u)p(v −u,−u)dv, (64) u ∈ [ −ã(k)l ,−ã (k) l−1 ] , l = 2, ..., l̃, φ (k) 4(l̃+1) (u) = ag+u∫ ac+u βc(v −u,−u) f (k) 4 (v) w4(v,−v,−u)dv + µ(np(−u)) × am+u∫ ar+u βp(v−u,−u)p(v−u,−u)dv, (65) u ∈ [ −ã(k) l̃+1 ,−ã(k) l̃ ] , where the second term in the right side of equation (62) should be omitted for l = 2, function f (k) 4 (v) in eq. (63) has been already defined on the previous steps be-cause its argument v ∈ [ac + u,ag + u] and v > u (see example in fig.2). because the initial value of c(a,t) is trivial, solution (57) satis-fies the compatibility (continuity) condition of solution, c(a,t) ∈ c(q). vii. hpv population dynamics nonlinear death rate dv(v ) of cauchy problem for nonlinear delayed ode (5), (11), satisfies biomath 10 (2021), 2110027, http://dx.doi.org/10.11145/j.biomath.2021.10.027 page 10 of 23 http://dx.doi.org/10.11145/j.biomath.2021.10.027 v v akimenko, f adi-kusumo, age-structured delayed sipcv epidemic model of hpv and cervical ... restrictions (16) and, hence, it is a locally lipschitz continuous function. hence, the unique smooth solution of problem (5), (11) exists and can be ob-tained by the method of steps [28]: v (t) = v (0) exp  − t∫ 0 dv(v (ξ −σ))dξ   + t∫ 0 exp  − t∫ η dv(v (ξ −σ))dξ  (λ(η) + n(η) ad∫ 0 (d∗q(a,η)i(a−η,η) + d ∗ p(a,η) p(a−η,η))da) dη, (66) where functions i(a,t) and p(a,t) are taken from the previous sections. overall, the results obtained above are summarized in the theorem 1. theorem 1. let the coefficients of sipcv epidemic system (1)-(11) satisfy conditions (12)-(17) and initial value ϕ(a) satisfies the compatibility condition (30), then the unique continuous solution of problem (1)-(11), p(a,t) ∈ c(q̄), c(a,t) ∈ c(q̄), v (t) ∈ c([0,t]) exists and can be obtained by explicit recurrent formulae (23)-(29), (33)-(40), (43)-(53), (58)-(65),, (66), respectively. viii. numerical implementations the numerical age-specific densities sji , i j i , p j i , c j i of susceptible, infectious, precancerous, cancer cells, respectively, are de-fined in the nodes of uniform grid ω̄h = {(ai, tj) |ai = ih,i = 1, ...,n,tj = jh, j = 1, ...,m,n = ad/h,m = t/h} . the mesh spacing h, the same for the variables a and t, is chosen so that 0 < h ≤ min(θ,ac), θ/h − [θ/h] = 0, ac/h − [ac/h] = 0, ar/h − [ar/h] = 0, ag/h− [ag/h] = 0, am/h− [am/h] = 0, ad/h− [ad/h] = 0. in this case each point a (k) l ∈ ω (k) (eq. (19)) and ã (k) l ∈ ω̃ (k) (eq. (54)) coincides always with some point tj. the numerical hpv quantity vj is defined at the points tj. we introduce the special iand mindexes: ic = ac/h, ir = ar/h, ig = ag/h, im = am/h, mθ = t/θ. numerical density of susceptible cells sji is defined from eq. (23): s0i = ϕ(ai), i = 0, ...,n,j = 0, (67) for k = 1: s j i =   ( f (0) 1 ) i−j (w01) j i−j, 0 , if 1 ≤ j < i ≤ n,( f (1) 1 ) j−i (w1) j j−i, j−i , if 0 ≤ i ≤ j ≤ n, (68) for k > 1: s j i =   ( f (k−1) 1 ) j−(k−2)n−i (w1) j j−i(k−1)n , if 1 ≤ (j − (k − 1)n) < i,( f (k) 1 ) j−(k−1)n−i (w1) j j−i ,j−i , if i ≤ (j − (k − 1)n) ≤ n, (69) where ( f (k) 1 ) l , (w01) j i,p, and (w1) j i,p are given:( f (0) 1 ) l = ϕ(al), l = 0, ...,n, (70)( f (k) 1 ) l = tr ( g (k) 1 ,e (k) 1 , ir, im, l ) , (71) k ≥ 1, l = 0, ...,n, ( g (k) 1 ) l,i =   βs(ai, tl) ( f (0) 1 ) i−l (w01) l i−l,0 , if k = 1, βs(ai, tl+(k−1)n ) ( f (k−1) 1 ) l−i+n (w1) l+(k−1)n l+(k−1)n−i, (k−1)n , if k > 1, (72) ( e (k) 1 ) l,i = βs(ai, tl+(k−1)n ) ( f (k) 1 ) l−i (w1) l+(k−1)n l+(k−1)n−i,l+(k−1)n−i , (73) k = 1, ...,k, biomath 10 (2021), 2110027, http://dx.doi.org/10.11145/j.biomath.2021.10.027 page 11 of 23 http://dx.doi.org/10.11145/j.biomath.2021.10.027 v v akimenko, f adi-kusumo, age-structured delayed sipcv epidemic model of hpv and cervical ... (w01) j i,0 = exp ( −0.5h ( (f1) 0 i + ( f (0) 1 )j i+j ) − h j−1∑ l=1 (f1) l i+l ) , (74) (w1) j i,p=   1, if j = p, exp ( −0.5h ( (f1) p p−i+(f1) j j−i ) −h j−1∑ l=p+1 (f1) l l−i ) , if j > p, (75) (f1) j i = ds(ai, tj) + α(ai −θ,tj −θ)ṽ (tj −θ), (76) where we use the trapezoidal rule [22] for approximating the integrals in eqs. (27)-(29) with the order of approximation o(h2): tr(x,y,ir,im,l) =   h ( 0.5 ( (x)l,ir+(x)l,im ) + im−1∑ i=ir+1 (x)l,i ) , if 0 ≤ l ≤ ir, h ( 0.5 ( (x)l,ir+(x)l,l ) + l−1∑ i=ir+1 (x)l,i ) +h ( 0.5 ( (y)l,l+(y)l,im ) + im−1∑ i=l+1 (y)l,i ) , if ir < l < im, h ( 0.5 ( (y)l,ir +(y)l,im ) + im−1∑ i=ir+1 (y)l,i ) , if im ≤ l ≤ n, (77) the smooth function ṽ (tj − θ) is obtained from the initial value v0(t) and numerical virus density v j by parabolic interpolating procedure [22]. we imply here that sums like j∑ i=l fi in all equations are omitted if j < l. the numerical function ( f (k) 1 ) l depends from the numerical function ( e (k) 1 ) li (eqs. (71), (73)) which uses the value of( f (k) 1 ) i−l obtained at the previous steps (index l > i − l in eq.(72)). the numerical density of infectious cells i1i is given: i0i = 0, i = 0, ...,n, j = 0, (78) for k = 1: i j i =   ( z (0) 2 )j i−j ,0 if 1 ≤ j < i ≤ n,( f (1) 2 ) j−i (w2) j j−i ,j−i+ ( z (1) 2 )j j−i,j−i , if 0 ≤ i ≤ j ≤ n, (79) for k > 1: i j i =   ( f (k−1) 2 ) j−(k−2)n−i (w2) j j−i,(k−1)n + ( z (k−1) 2 )j j−i,(k−1)n , if 1 ≤ (j − (k − 1)n) < i,( f (k) 2 ) j−(k−1)n−i (w2) j j−i,j−i + ( z (k) 2 )j j−i,j−i , if i ≤ (j − (k − 1)n) ≤ n, (80) where ( f (k) 2 ) i , (w2) j i,p and ( z (k) 2 )j i,p are given: ( f (0) 2 ) l = 0, l = 0, ...,n, (81)( f (k) 2 ) l = tr ( g (k) 2 ,e (k) 2 , ir, im, l ) , (82) k ≥ 1, l = 0, ...,n, ( g (k) 2 ) l,i =   βq(ai, tl) ( z (0) 2 )l i−l,0 , if k = 1, βq(ai, tl+(k−1)n ) (( f (k−1) 2 ) l−i+n (w2) l+(k−1)n l+(k−1)n−i,(k−1)n + ( z (k−1) 2 )l+(k−1)n l+(k−1)n−i,(k−1)n ) , if k > 1, (83) biomath 10 (2021), 2110027, http://dx.doi.org/10.11145/j.biomath.2021.10.027 page 12 of 23 http://dx.doi.org/10.11145/j.biomath.2021.10.027 v v akimenko, f adi-kusumo, age-structured delayed sipcv epidemic model of hpv and cervical ...( e (k) 2 ) l,i = βq(ai, tl+(k−1)n ) (( f (k) 2 ) l−i (w2) l+(k−1)n l+(k−1)n−i,l+(k−1)n−i + ( z (k) 2 )l+(k−1)n l+(k−1)n−i,l+(k−1)n−i ) , (84) k = 1, ...,k, (w02) j i,0 = exp ( −0.5h ( d̃q(ai, t0) + d̃q(ai+j, tj) ) −h j−1∑ l=1 d̃q(ai+l, tl) ) (85) (w2) j i,p=   1, if j = p, exp ( −0.5h ( d̃q(ap−i, tp) +d̃q(aj−i, tj) ) −h j−1∑ l=p+1 d̃q(al−i, tl) ) , if j > p, (86) ( z (0) 2 )j i,0 = 0.5h ( (f02) j i,0 + (f02) j i,j ) + h j−1∑ l=1 (f02) j i,l, i = j, ...,n, (87) ( z (k) 2 )j i,p =   0, if j = p, 0.5h ( (f2) j i,p + (f2) j i,j ) +h j−1∑ l=p+1 (f2) j i,l, i = 1, ...,n, if j > p, (88) (f02) j i,p= (w02) j i,pα(ai+p−θ,tp−θ)ṽ (tp−θ)s p i+p, (89) (f2) j i,p= (w2) j i,pα(ap−i−θ,tp−θ)ṽ (tp−θ)s p p−i, (90) where in eqs.(82), (86), (88), we use the trapezoidal rule [22] for approximating the inte-grals with the order of approximation o(h2). the numerical density of precancerous cells pji is given: p0i = 0, i = 0, ...,n, j = 0, (91) for k = 1: p j i =   ( z (0) 3 )j i−j,0 , if 1 ≤ j < i ≤ n,( f (1) 3 ) j−i (w3) j j−i,j−i+ ( z (1) 3 )j j−i,j−i , if 0 ≤ i ≤ j ≤ n, (92) p j i =   ( f (k−1) 3 ) j−(k−2)n−i (w3) j j−i, (k−1)n + ( z (k−1) 3 )j j−i,(k−1)n , if 1 ≤ (j − (k − 1)n) < i,( f (k) 3 ) j−(k−1)n−i (w3) j j−i,j−i + ( z (k) 3 )j j−i,j−i , if i ≤ (j − (k − 1)n) ≤ n, (93) where ( f (k) 3 ) i , (w3) j i,p and ( z (k) 3 )j i,p are given: ( f (0) 3 ) l = 0, l = 0, ...,n. (94)( f (k) 3 ) l = (1 −µ(k)l )tr ( g (k) 3 ,e (k) 3 , ir, im, l ) , k ≥ 1, l = 1, ...,n, (95) µ (k) l = tr ( g̃ (k) 3 , ẽ (k) 3 , ir, id, l ) ,k ≥ 1 (96) ( g (k) 3 ) l,i =   βr(ai, tl) ( z (0) 3 )l i−l,0 , if k = 1, βr(ai, tl+(k−1)n ) (( f (k−1) 3 ) l−i+n (w3) l+(k−1)n l+(k−1)n−i,(k−1)n + ( z (k−1) 3 )l+(k−1)n l+(k−1)n−i,(k−1)n ) if k > 1, (97) ( e (k) 3 ) l,i = βr(ai, tl+(k−1)n ) (( f (k) 3 ) l−i (w3) l+(k−1)n l+(k−1)n−i,l+(k−1)n−i + ( z (k) 3 )l+(k−1)n l+(k−1)n−i,l+(k−1)n−i ) , (98) biomath 10 (2021), 2110027, http://dx.doi.org/10.11145/j.biomath.2021.10.027 page 13 of 23 http://dx.doi.org/10.11145/j.biomath.2021.10.027 v v akimenko, f adi-kusumo, age-structured delayed sipcv epidemic model of hpv and cervical ... ( g (k) 3 ) l,i =   ( z (0) 3 )l i−l,0 , if k = 1,( f (k−1) 3 ) l−i+n (w3) l+(k−1)n l+(k−1)n−i,(k−1)n + ( z (k−1) 3 )l+(k−1)n l+(k−1)n−i,(k−1)n , if k > 1, (99) ( ẽ (k) 3 ) l,i = ( f (k) 3 ) l−i (w3) l+(k−1)n l+(k−1)n−i,l+(k−1)n−i + ( z (k) 3 )l+(k−1)n l+(k−1)n−i,l+(k−1)n−i , (100) (w03) j i,0 = exp (−0.5h (dr(ai, t0) + dr(ai+j, tj)) −h j−1∑ l=1 dr(ai+l, tl) ) , (101) (w3) j i,p =   1, if j = p, exp (−0.5h (dr(ap−i, tp) +dr(aj−i, tj)) −h j−1∑ l=p+1 dr(al−i, tl) ) , if j > p, (102) ( z (0) 3 )j i,0 = 0.5h ( (f03) j i,0 + (f03) j i,j ) + h j−1∑ l=1 (f03) j i,l, i = j, ...,n, (103) ( z (k) 3 )j i,p =   0, if j = p, 0.5h ( (f3) j i,p + (f3) j i,j ) +h j−1∑ l=p+1 (f3) j i,l, i = 1, ...,n, if j > p, (104) (f03) j i,p = (w03) j i,p δ(ai+p, tp)i p i+p, (105) (f3) j i,p = (w3) j i,p δ(ap−i, tp)i p p−i, (106) where we use in eqs.(95), (96), (101) (104) the trapezoidal rule [22] for approximating the integrals with accuracy o(h2). the numerical density of cancer cells cji is given: c0i = 0, i = 0, ...,n,j = 0, (107) for k = 1: c j i =   0, if 1 ≤ j < i ≤ n,( f (1) 4 ) j−i (w4) j j−i,j−i , if 0 ≤ i ≤ j ≤ n, (108) for k > 1: c j i =   ( f (k−1) 4 ) j−(k−2)n−i (w4) j j−i, (k−1)n , if 1 ≤ (j − (k − 1)n) < i,( f (k) 4 ) j−(k−1)n−i (w4) j j−i,j−i , if i ≤ (j − (k − 1)n) ≤ n, (109) where ( f (k) 4 ) i and (w4) j i,p are given:( f (0) 4 ) l = 0, l = 0, ...,n, (110) ( f (k) 4 ) l = tr ( g (k) 4 ,e (k) 4 , ic, in, l ) + µ (k) l h( p l+(k−1)n ir βr(ar, tl+(k−1)n ) + p l+(k−1)n im ×βr(am, tl+(k−1)n ) +2 im−1∑ p=ir+1 p l+(k−1)np βr(ap, tl+(k−1)n )   , k ≥ 1, l = 1, ...,n, (111) ( g (k) 4 ) l,i =   0, if k = 1, βc(ai, tl+(k−1)n ) ( f (k−1) 4 ) l−i+n (w4) l+(k−1)n l+(k−1)n−i,(k−1)n , if k > 1, (112) ( e (k) 4 ) l,i = βc(ai, tl+(k−1)n ) ( f (k) 4 ) l−i (w4) l+(k−1)n l+(k−1)n−i,l+(k−1)n−i (113) biomath 10 (2021), 2110027, http://dx.doi.org/10.11145/j.biomath.2021.10.027 page 14 of 23 http://dx.doi.org/10.11145/j.biomath.2021.10.027 v v akimenko, f adi-kusumo, age-structured delayed sipcv epidemic model of hpv and cervical ... (w4) j i,p=   1, if j = p, exp (−0.5h (dc(ap−i, tp) +dc(aj−i, tj)) −h j−1∑ l=p+1 dc(al−i, tl) ) , if j > p, (114) where we use in eqs. (111), (114) the trapezoidal rule [22] for approximating the integrals with accuracy o(h2). the exact solution (66) of initial problem (5), (11) for hpv density can be written in a more convenient form for numerical implementation: v (t) = v (t−h) exp  − t∫ t−h dv(v (ξ −σ))dξ   + t∫ t−h exp  − t∫ η dv(v (ξ −σ))dξ  (λ(η)+ n(η) ad∫ 0 (d∗q(a,η)i(a,η)+d ∗ r(a,η)p(a,η))da ) dη. (115) using the trapezoidal rule [22] for approximating the integrals in eq. (115) yields the numerical density of virus population: v j = v j−1 exp ( −0.5h ( dv(ṽ (tj −σ)) +dv(ṽ (tj−1 −σ)) )) + 0.5h ( (g5) j + (g5) j−1 exp ( −0.5h ( dv(ṽ (tj −σ)) +dv(ṽ (tj−1 −σ)) ))) , (116) (g5) j = λ(tj) + n(tj) ( 0.5h((e5) j 0 + (e5) j id ) +h id−1∑ i=1 (e5) j i ) , (117) (e5) j i = d ∗ q(ai, tj)i j i + d ∗ r(ai, tj)p j i . (118) the above results lead to the following theorem. theorem 2. let the conditions of the theorem 1 are hold. the numerical method for the recurrent formulae (23)-(30), (33)-(40), (43)(53), (58)-(66) exact solution of the system (1)-(11), developed for the uniform grid ω̄h = {(ai, tj) |ai = ih,i = 1, ...,n,tj = jh, j = 1, ...,m,n = ad/h,m = t/h} is given by eqs. (67)-(118). the method is based on the trapezoid rules and approximates the exact solution of the system (1)-(11) with accuracy ψ = o(h2). along with the study of the order of approximation of numerical method it is im-portant to obtain the estimations of the velocity of conversation of numerical solution to exact one. this question is studied in the next, second part of our paper. in the next section we consider a series of numerical experiments which illustrate the theoretical results ob-tained in theorems 1 and 2. ix. numerical experiments the theoretical results obtained in theorem 2 are illustrated by the numerical experiments. in the first group of experiments we study the residual as a measure of deviation of obtained numerical solution from some benchmark solution of system (1)-(11) taken from [8], [37], [56]. it is easy to verify by direct substitution that functions s(a,t) = i(a,t) = p(a,t) = c(a,t) = exp(−εa)(1 + exp(−t))−1, (119) v (t) = (2σ + t) 0.5 , (120) are exact solution of the system (1)-(11) with the following particular coefficients and initial values: βs(a,t)=βq(a,t) =ε(exp(−εar)−exp(−εam))−1 = const, (121) µ(t) = ρ (ε (1 + exp(−t)) (exp(−εar) −exp(−εad)) −1 + w ) , (122) βr(t)=ε(1−µ(t))−1(exp(−εar)−exp(−εam))−1, (123) βc(t)= (ε−µ(t)βr(t) (exp(−εar)−exp(−εam))) ×(exp(−εac)−exp(−εag))−1, (124) biomath 10 (2021), 2110027, http://dx.doi.org/10.11145/j.biomath.2021.10.027 page 15 of 23 http://dx.doi.org/10.11145/j.biomath.2021.10.027 v v akimenko, f adi-kusumo, age-structured delayed sipcv epidemic model of hpv and cervical ... ds(t) =ε−α(1+exp(−t))−0.5−(1 + exp(t))−1, (125) dq(t) =ε + α(1 + exp(−t))−0.5 −δ − (1 + exp(t))−1, (126) dr(t) = ε + δ − (1 + exp(t))−1, (127) d∗q(t) = 0.2dq(t), (128) d∗r(t) = 0.2dr(t), (129) dc(t) = ε− (1 + exp(t))−1, (130) dv(v (t−σ)) = 0.05(1 + v (t−σ))0.5, (131) λ(t) = 0.5(2σ + t) −0.5 + 0.05 ( 1 + (σ + t) 0.5 )0.5 (2σ + t) 0.5 −nε−1 ( d∗q(t) + d ∗ r(t) ) ×(1−exp(−εad)) (1+exp(−t)) −1 , (132) s(a, 0) = i(a, 0) = p(a, 0) = c(a, 0) = 0.5 exp(−εa), (133) v0(t) = (2σ + t) 0.5 , t ∈ [−σ, 0]. (134) exact solutions (119) satisfy the continuity condition (30) and conditions: i(0, 0) = am∫ ar βs(a, 0)i(a, 0)da, (135) p(0, 0) = (1 −µ(nr(0)) am∫ ar βr(a, 0)p(a, 0)da, (136) c(0, 0) = ag∫ ac βc(a, 0)c(a, 0)da + µ(nr(0)) am∫ ar βr(a, 0)p(a, 0)da. (137) the set of constants used in experiments is given in table i. our goal here is to pro-vide a detailed description of the numerical experiments and obtained numerical results which will help the potential users to evaluate the numerical method of characteristics and increase the chances of its implementation in simulation of epidemic dynamics in various applications. table i set of constants constant value constant value ar 0.3 σ 0.2 am 0.9 n 1.5 ac 0.1 α 0.01 ag 0.4 δ 0.01 ad 1 ρ 0.1 t 10 w 0.5 θ 0.1 ε 4 the time of simulation in all experiments equals to 10 cell’s lifespan t = 10ad. deviation of the numerical solution y ji = (s j i ,i j i ,p j i ,c j i ) from the benchmark one y (ai, tj) = (s(ai, tj),i(ai, tj),p(ai, tj),c(ai, tj)) and v j from v (tj) with x = h/ad → 0 (dimensionless parameter) is measured by the two dimensionless norms of functional spaces h and c [8], [51]: δ (1) h (y ) = m∑ j=0 n∑ i=0 ∣∣∣y ji −y (ai, tj)∣∣∣  m∑ j=0 n∑ i=0 ∣∣∣y ji ∣∣∣  −1, (138) δ (1) c (y ) = max 0≤i≤n 0≤j≤m ∣∣∣y ji −y (ai, tj)∣∣∣   max 0≤i≤n 0≤j≤m ∣∣∣y ji ∣∣∣  −1, (139) δ (2) h (v ) = m∑ j=0 ∣∣v j −v (tj)∣∣   m∑ j=0 ∣∣v j∣∣  −1, (140) δ (2) c (v ) = max 0≤j≤m ∣∣v j −v (tj)∣∣( max 0≤j≤m ∣∣v j∣∣)−1, (141) all program projects were created in microsoft visual c 2019, platform microsoft .net framework and were launched on a pc (cpu i5 of 8th generation 1.6 ghz, ram 16 gb). the results biomath 10 (2021), 2110027, http://dx.doi.org/10.11145/j.biomath.2021.10.027 page 16 of 23 http://dx.doi.org/10.11145/j.biomath.2021.10.027 v v akimenko, f adi-kusumo, age-structured delayed sipcv epidemic model of hpv and cervical ... of numerical experiments are presented in table 2 and are shown in fig. 3 ( � the values of δ(1)h (s), δ (1) c (s) from table ii depending from x = h/ad, the graphs of corresponding regressions y(x) (with denoted equations)), fig. 4 (δ(1)h (i), δ (1) c (i) and y(x)), fig. 5 (δ(1)h (p), δ (1) c (p) and y(x)), fig. 6 (δ(1)h (c), δ (1) c (c) and y(x)) and fig. 7 (δ (2) h (v ), δ (2) c (v ) and y(x)). the session time of simulation ts (sec) (diagnostic session time of microsoft visual c 2019) depending from parameter x in each numerical experiment is shown in table iii. table ii the values of δ(1)h (y ), δ (1) c (y ), y j i = (s j i ,i j i ,p j i ,c j i ), and δ (2) h (v ), δ (2) c (v ) for the different values of x. x 0.005 0.01 0.025 0.05 δ (1) h (s) 0.00038 0.0015 0.0094 0.0371 δ (1) c (s) 0.00074 0.0023 0.0184 0.0716 δ (1) h (i) 0.00035 0.0014 0.0088 0.0347 δ (1) c (i) 0.00066 0.0026 0.0164 0.0639 δ (1) h (p) 0.00035 0.0014 0.0086 0.0342 δ (1) c (p) 0.00065 0.0026 0.0161 0.0629 δ (1) h (c) 0.00077 0.0031 0.0193 0.0757 δ (1) c (c) 0.0015 0.0054 0.0368 0.1397 δ (1) h (v ) 0.00021 0.0008 0.0051 0.0210 δ (1) c (v ) 0.00036 0.0014 0.0091 0.0382 table iii the session time of simulation ts in seconds (diagnostic session time of mi-crosoft visual c 2019), depending from the x. x 0.005 0.01 0.025 0.05 ts (sec) 651 52 4 2 the equations of regressions in figs. 3 7 are built automatically in ms excel at the points of δ(1)h (y ), δ (1) c (y ), δ (2) h (v ), δ (2) c (v ) depending from the values of x. they exhibit the quadratic (parabolic) low of convergence δ(1)h (y ) → 0, δ (1) c (y ) → 0, (y j i = (s j i ,i j i ,p j i ,c j i )) and δ (2) h (v ) → 0, δ (2) c (v ) → 0 with x → 0 in all experiments that illustrates and confirms the second fig. 3. � values of δ(1)h (s), δ (1) c (s), graphs of regressions y(x), x = h/ad. fig. 4. � values of δ(1)h (i), δ (1) c (i), graphs of regressions y(x), x = h/ad. fig. 5. � values of δ(1)h (p), δ (1) c (p), graphs of regressions y(x), x = h/ad. order of accuracy of numerical solution. the most numerical error among all 5 subclasses is obtained for the cancer cell population (fig. 6, table ii δ(1)h (c), δ (1) c (c)). the large deviation (> 3%) of numerical cancer cell population density from the benchmark solution for x ≥ 0.025 can be biomath 10 (2021), 2110027, http://dx.doi.org/10.11145/j.biomath.2021.10.027 page 17 of 23 http://dx.doi.org/10.11145/j.biomath.2021.10.027 v v akimenko, f adi-kusumo, age-structured delayed sipcv epidemic model of hpv and cervical ... fig. 6. � values of δ(1)h (c), δ (1) c (c), graphs of regressions y(x), x = h/ad. fig. 7. � values of δ(1)h (v ), δ (1) c (v ), graphs of regressions y(x), x = h/ad. explained by the complexity of nonlinear boundary condition (9) which includes the integral from the numerical precancerous cell population density com-puted with some numerical error. nevertheless, we observe the significand decreasing of δ (1) h (c), δ (1) c (c) for x ≤ 0, 01 with follow-ing convergence of them to 0. besides the accuracy of numerical method, we consider also the valuable in practice parameter session time of simulation ts given in table iii. since the refine of difference grid leads necessarily to increasing of ts, numerical experiments with benchmark solution provide the optimal range of mesh spacing h for which the appropriate accuracy of numerical solution can be reachable for an acceptable time of simulation. according to the data of table iii the critical point of mech spacing after which the time of simulation begins to grow rapidly is h = 0.01 (or x = 0.01). on the other hand, the results of fig. 8. graphs of ns(t) for non-oscillating dynamics for 3 different initial values ϕ(a). table ii show that for x = 0.01 the deviation of numerical solution from the benchmark one (relative numerical error) is less or equal than 0, 5% for all subclasses of population that corresponds to the acceptable accuracy of simulation in most biological systems. from tables ii and iii, it follows that the optimal value of the dimensionless parameter is x = h/ad = 0.01 that corresponds to the mesh spacing h = 0.01. this result is in good agreement with the recommended value of time spacing obtained in the numerical experiments for nonlinear age-structured model of population dynamics in [8]. in the second group of numerical experiments we study the asymptotically stable dynamical regimes of population. simulations reveal two types of asymptotically stable dynamics: nonoscillating and oscillating convergence of all subpopulation quantities to the nontrivial equilibrium states. the dynamical regime of the first type non-oscillating dynamics is shown in figs. 8 12. in particular, in fig. 8 it is shown the graphs of susceptible cell quantity dynamics for three different initial values ϕ(a). the corresponding graphs of infected cell quantity, precancerous cell quantity, cancer cell quantity and hpv quantity dynamics are shown in figs. 9, 10, 11, 12, respectively. it should be noted that the non-oscillating dynamics of dysplasia (precancerous cells) and cancer growth shown in figs. 10 and 11 by dotted lines is a most realistic type of tumor tissue biomath 10 (2021), 2110027, http://dx.doi.org/10.11145/j.biomath.2021.10.027 page 18 of 23 http://dx.doi.org/10.11145/j.biomath.2021.10.027 v v akimenko, f adi-kusumo, age-structured delayed sipcv epidemic model of hpv and cervical ... fig. 9. graphs of ni (t) for non-oscillating dynamics for 3 different initial values ϕ(a). fig. 10. graphs of np (t) for non-oscillating dynamics for 3 different initial values ϕ(a). fig. 11. graphs of nc (t) for non-oscillating dynamics for 3 different initial values ϕ(a). growth. the second type of dynamical regimes oscillating dynamics is shown in figs. 13 17. in fig. 13, it is shown the susceptible cell quantity dynamics for two different initial values ϕ(a). the corresponding graphs of infected cell quantity, fig. 12. graphs of v (t) for non-oscillating dynamics for 3 different initial values ϕ(a). fig. 13. of ns(t) for oscillating dynamics for 2 different initial values ϕ(a). precancerous cell quantity, cancer cell quantity and hpv quantity dynamics are shown in figs. 14, 15, 16, 17, respectively. asymptotically stable regimes of sipcv model shown in figs. 8 12 and 13 17 demonstrate the localization of dysplasia (precancerous cells) and cancer in biological tissue without metastases. thus, the results of simulations exhibit that the numerical method designed for the age-structured sipcv epidemic model can be applied for numerical study and modelling of cell-hpv population dynamics with high accuracy. x. conclusion in this study we consider a new epidemic model of age-structured sub-populations of susceptible, infectious, precancerous and cancer cells and unstructured population of hu-man papilloma virus (hpv) (sipcv epidemic model). the model biomath 10 (2021), 2110027, http://dx.doi.org/10.11145/j.biomath.2021.10.027 page 19 of 23 http://dx.doi.org/10.11145/j.biomath.2021.10.027 v v akimenko, f adi-kusumo, age-structured delayed sipcv epidemic model of hpv and cervical ... fig. 14. of ni (t) for oscillating dynamics for 2 different initial values ϕ(a). fig. 15. of np (t) for oscillating dynamics for 2 different initial values ϕ(a). fig. 16. of nc (t) for oscillating dynamics for 2 different initial values ϕ(a). is based on the competi-tive system of initialboundary value problems for delayed semi-linear transport equations with integral boundary conditions and initial problem for delayed nonlinear ode. for this system we obtained the exact solution in the form of recurrent formulae in which fig. 17. of v (t) for oscillating dynamics for 2 different initial values ϕ(a). the den-sities of all subpopulations depend from the integrals from solution taken at previous instance of time. the main ideas and method of derivation of such exact solutions are taken from works [3], [8]. the new result obtained in this work is the numerical implementation of recurrent formulae for exact solution and development of the numerical method of the second order of approximation for simulation of susceptible, infectious, precancerous, can-cer cells and human papilloma virus population dynamics. since evaluation of the accuracy of numerical solution and session time of simulation are essential to successful use of numerical method in applications, we estimated the difference between computed solution and benchmark solution of model and session time on the refined difference grid. numeri-cal experiments showed that the relative numerical error of solution may be reduced up to 0.1% for the very small value of mesh spacing parameter h = 0.005 (that is 0.5% of ad) but for large value of session time of simulation. we recommend to use in applications the optimal value of mesh spacing h = 0.01 (1% of ad) that pro-vides the small value of relative numerical error (less than 0,5%) for acceptable session time. this recommendation is in a good agreement with the results of numerical experiments obtained in [8] for nonlinear age-structured model of population dynamics. the numerical experiments with model parameters revealed two types of asymptotically stable dynamical regimes of spicv population biomath 10 (2021), 2110027, http://dx.doi.org/10.11145/j.biomath.2021.10.027 page 20 of 23 http://dx.doi.org/10.11145/j.biomath.2021.10.027 v v akimenko, f adi-kusumo, age-structured delayed sipcv epidemic model of hpv and cervical ... non-oscillating and oscillating convergence of solution to the positive steady states. the nonoscillating type of spicv population dynamics corresponds to the observed in practice dynamics of tumor growth-localization of dysplasia (precancerous cells) and cancer in biological tissue without metastases. overall, development of agestructured sipcv epidemic model, derivation of its ex-act solution and design of corresponding numerical methods provide the theoretical instrument for study the dynamics of susceptible, infected, precancerous, cancerous cells and viruses populations that help us better understand the features of human papilloma virus infectious disease. acknowledgment we would like to acknowledge department of mathematics, universitas gadjah mada for the 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[59] j. zhou, l. song, j. wei, mixed types of waves in a discrete diffusive epidemic mod-el with nonlinear incidence and time delay, j. differential equations 268 (8) (2020) 4491-4524. biomath 10 (2021), 2110027, http://dx.doi.org/10.11145/j.biomath.2021.10.027 page 23 of 23 http://dx.doi.org/10.11145/j.biomath.2021.10.027 introduction model susceptible cell population dynamics infectious cell population dynamics precancerous cell population dynamics cancer cell population dynamics hpv population dynamics numerical implementations numerical experiments conclusion references www.biomathforum.org/biomath/index.php/biomath original article a tribute to the use of minimalistic spatially-implicit models of savanna vegetation dynamics to address broad spatial scales in spite of scarce data ivric valaire yatat djeumena,b, alexis tchuinté tamenc,∗, yves dumonta,d,e, pierre couteronc,f auniversity of pretoria, department of mathematics and applied mathematics, pretoria, south africa buniversity of yaoundé 1, national advanced school of engineering, yaoundé, cameroon cird, umr amap, lmi dycofac, yaoundé, cameroon alexis.tchuinte@yahoo.fr; yatat.valaire@gmail.com; ivric.yatatdjeumen@up.ac.za dcirad, umr amap, pretoria, south africa eamap, université de montpellier, cirad, cnrs, inra, ird, montpellier, france yves.dumont@cirad.fr; yves.dumont@up.ac.za f amap, ird, cirad, cnrs, inra, université de montpellier, montpellier, france pierre.couteron@ird.fr ∗the first two authors contributed equally received: 5 september 2018, accepted: 16 december 2018, published: 20 december 2018 abstract—the savanna biome encompasses a variety of vegetation physiognomies that traduce complex dynamical responses of plants to the rainfall gradients leading from tropical forests to hot deserts. such responses are shaped by interactions between woody and grassy plants that can be either direct, disturbance-mediated or both. there has been increasing evidence that several vegetation physiognomies, sometimes highly contrasted, may durably coexist under similar rainfall conditions suggesting multi-stability or at least not abrupt transitions. these fascinating questions have triggered burgeoning modelling efforts which have, however, not yet delivered an integrated picture liable to furnish sensible predictions of potential vegetation at broad scales. in this paper, we will recall the key ecological processes and resulting vegetation dynamics that models should take into account. we will also present the main modelling options present in the literature and advocate the use of minimalistic models, capturing only the essential processes while retaining sufficient mathematical tractability and restricting themselves to a minimal set of parameters assessable from the overall literature. keywords-biogeography; rainfall; fires; ordinary differential equations; impulsive differential equation; tree-grass interactions; multi-stability. copyright: © 2018 yatat djeumen et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: ivric valaire yatat djeumen, alexis tchuinté tamen, yves dumont, pierre couteron, a tribute to the use of minimalistic spatially-implicit models of savanna vegetation dynamics to address broad spatial scales in spite of scarce data, biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 1 of 29 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2018.12.167 i.v. yatat djeumen, a. tchuinté tamen, y. dumont, p. couteron, a tribute to the use of minimalistic ... i. introduction savannas have been identified by biogeographers as a biome corresponding to warm mean annual temperatures (> 20°c) and a broad range of intermediate mean annual rainfall (100–2000 mm.yr−1) (sarmiento (1984) [76], youta happi (1998) [114], abbadie et al. (2006) [1], lehmann et al. (2011) [58]). such climatic context predominates along rainfall gradients leading from subequatorial wet climates to hot arid climates. the wider definition to which we refer here tends to integrate climatic variant allowing for nearly desert vegetation or on the contrary seasonal tropical forests. savannas display specific interplays of natural constraints that prevent or at least impede closure of woody cover and ensuing suppression of light-demanding herbs and grasses. a central, albeit non-exclusive cause for this is the ’ideal fire climate’ (trollope (2011) [98]) that characterizes tropical regions with seasonal droughts alternating with warm and wet rainy seasons producing high herbaceous biomass that once dried-up becomes highly ignitable and fuels fires (frost et al. (1986) [39], thonicke et al. (2001) [93], govender et al. (2006) [44]). high frequency of lightning storms which is a characteristic of africa (abbadie et al. (2006) [1], trollope (2011) [98]) also contributes to make it the ”fire continent” even though present fire regimes mostly rely on human-made ignitions (archibald et al. (2009) [7], govender et al. (2006) [44], trollope (2011) [98]). dynamics of vegetation within the savanna biome has long interested ecologists as it clearly departs from the classical post-disturbance succession pathways that are expected to rapidly bring back closed canopy forest, as observed in most of temperate and wet tropical climatic zones (bond et al. (2005) [22]). the last decades have witnessed burgeoning efforts of modelling as to account for the possibly long-lasting coexistence of grassy and woody components and try to predict potential shifts from two-phased vegetation physiognomies. these efforts have, however, not yet delivered an integrated picture liable to furnish at broad scales (i.e., for fractions of continents) sensible predictions of possible vegetation dynamics. such a big picture is nevertheless desirable for figuring out the future of vegetation in the face of climate and anthropic change scenarios (mayaux et al. (2004) [60], bond et al. (2005) [22], archibald et al. (2009) [7], accatino et al. (2010) [4], favier et al. (2012) [36]). it is also necessary for applications to territories devoid of reference data and longterm observation sites, as it is the case for most of tropical africa. the objectives of the present contribution are fourfold. it first aims at recalling and synthetizing the main array of facts about ecological processes and resulting vegetation dynamics that models should aim to capture and render (see section ii). second, in order to claim genericity, we synthetize the main modelling options present in the literature, and put emphasis on minimalistic models, capturing only essential processes while retaining sufficient mathematical tractability and restricting themselves to a minimal set of assessable parameters (see section iii). thirdly, on this basis, we argue that such models have now become more comprehensive, and useful for meaningful predictions (see section iv). finally, we discuss how those models may now help guiding data collection for improved calibration and testing of dynamical hypotheses (see section v). ii. a brief review on space-implicit tree-grass interactions modelling a. tree-grass coexistence and possible alternative stable states over very large tropical territories, field observers have documented long lasting coexistence of notable levels of grass and woody biomass (backéus (1992) [9]). the most frequently reported form of coexistence is observed locally through vegetation physiognomies that associate fairly continuous grassy cover and more or less scattered populations of trees and shrubs of varying clumping levels. this is referred to as savanna physiognomy (see figure 1). such vegetation types mixing both lifeforms are manifold and progressively merge in space or through time biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 2 of 29 http://dx.doi.org/10.11145/j.biomath.2018.12.167 i.v. yatat djeumen, a. tchuinté tamen, y. dumont, p. couteron, a tribute to the use of minimalistic ... without clear-cut boundaries (torello-raventos et al. (2013) [95]). another modality of long lasting association between herbaceous and woody lifeforms occur at landscape scale under the form of mosaics featuring forests (usually closed canopy ones) and open savannas or grasslands (e.g. figure 1; bond and parr (2010) [21]). in those landscapes that pertain to moist-wet climates, normally seen as favourable to forests, the mosaics appear highly contrasted and among the most ”emblematic vegetation transitions” in the world (oliveras and malhi (2016) [72]): outside the closed forest, woody vegetation is of low biomass and the dominant physiognomies relate mainly to grassland. moreover, boundaries between forest and grassland are generally sharp (hoffmann et al. 2012 [48], cunisanchez et al. (2016) [27]). our interpretations of those various physiognomies are limited by the length of the observation windows we can rely on for distinguishing trends against fluctuations. for field observations, this window length barely extend over some decades and this only for a very small number of sites where invaluable data have been gathered. at the scale of extensive territories, representativeness of those sites remains yet an open question. remote sensing is progressively broadening our observational means. but the best nowadays space-borne sensors for estimating woody cover (buccini and hanan (2007) [26]) or biomass (mermoz et al. (2014) [64], bouvet et al. (2018) [24]) are recent and do not allow tracking changes far back. moreover, the accuracy of those estimations, notably for woody cover is limited, due to the difficulty to separate grass vs. tree in signal responses in mixed stands. this is particularly true regarding long diachronic series that mainly feature optical images of insufficient spatial resolution. apart from blatant changes, e.g., forest encroachment or recession, mitchard and flintrop (2013) [68], subtle evolution of the grass-tree balance in mixed physiognomies are still beyond reach. remote sensing, however, recently brought two interesting contributions to the savanna debate. first, broad scale assessment of woody cover at regional (central africa, favier et al. (2012) [36]) to continental/global scales (hirota et al. (2011) [47]) clearly showed that contrasted levels of cover can coexist under the same ranges of climatic conditions, making the existence of multi-stable states at least plausible. second, in both central and west africa, comparison between ancient air photographs from the 50s and satellite images from the 80-90s frequently evidenced a progress of forest over savannas/grasslands in landscape featuring contrasted mosaics of the type exemplified in figure 1 (youta happi (1998) [114], mitchard et al. (2011) [69]). even though there is still no conclusive evidence that alternative stable states may exist within the savanna biome, models should be able to account for them as plausible outcomes of tree-grass interactions. the same applies to savanna physiognomies locally associating trees and grasses that may be seen as either stable or transient twophase states. since those mixed physiognomies are observable at broad scale, there is no reason to a priori rule out that some observed mixtures may be stable under their local environmental context. indeed, hypothesis testing is a fundamental role of models, though this use is not so widespread in ecology. and to this aim, the wider the array of reasonable predictions the more relevant is the model. b. lines of thoughts most authors agree on the fact that soil water budget, herbivory (i.e. grazing and/or browsing) and fires are the principal factors influencing growth of woody and herbaceous plants and their dynamical interactions (scholes and archer (1997) [79], higgins et al. (2000) [45], scholes (2003) [78], van langevelde et al. (2003) [101], bond et al. (2005) [22], bond (2008) [17], abbadie et al. (2006) [1], accatino et al. (2010) [4], staver and levin (2012) [84], baudena et al. (2010) [11], (2014) [12], jeffery et al. (2014) [51]). authors however diverge on the relative importance of those factors in shaping dynamical outcomes of tree-grass interactions. this is not surprising considering the broad extent of the savanna biome biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 3 of 29 http://dx.doi.org/10.11145/j.biomath.2018.12.167 i.v. yatat djeumen, a. tchuinté tamen, y. dumont, p. couteron, a tribute to the use of minimalistic ... figure 1: landscape-scale mosaic between dense forest and herbaceous savanna (grassland) observed in central cameroon (ayos). brownpink tree crowns indicate marshy forests in talwegs. note the weak congruence between topography and the occurrences of forest vs. grassland. airborne photo from n. barbier, june 2017. and the variety of both environmental conditions and anthropogenic pressures that apply therein. a factor appearing pervasive in a given context is not systematically due to prevail elsewhere. one group of authors has been insisting on direct interactions among or between plant-types (i.e. tree-tree or tree-grass) such as competition for light or for soil limiting resources (often moisture via root systems) (e.g. scholes and archer (1997) [79], scholes (2003) [78]). it is obvious that the tree– grass interaction is highly asymmetric: trees have a strong competitive effect on grasses, but grasses have a weak competitive effect on mature trees, although they may have a strong effect on saplings that have not grown above the grass layer (scholes (2003) [78], figure 2-a). another group of authors has been emphasizing that woody vegetation would be likely to reach a closed canopy situation and suppress grasses in the absence of recurrent disturbances induced by fires or browsers (or both sources) that delay or block the build-up of woody biomass by destroying the aerial part of seedlings and saplings (e.g. bond et al. (2005) [22], bond (2008) [17], staver and levin (2012) [84], baudena et al. (2010) [11], (2014) [12], jeffery et al. (2014) [51]; figure 2b & -c). browsers impact, though undoubtedly pervasive in certain situations (mcnaughton and georgiadis (1986) [62], scholes and walker (1993) [80], van langevelde et al. (2003) [101], holdo et al. (2009) [49]) is not systematic across the savanna biome and the generality of the disturbance hypothesis relies mainly on fire. indeed, experimental fire suppression systematically leads to the thickening-up of the woody vegetation and to the development of dense woodlands or thickets. for sufficient annual rainfall, shifts toward close canopy forests are also observed (bond et al. (2005) [22], jeffery et al. (2014) [51]). literature may sometimes overemphasize the distinction between ’interaction’ (between plant types for limited resource) and ’disturbance’ hypotheses (see scholes and archer (1997) [79]) as to make them appear as alternatives, though they are by no means mutually exclusive. it is widely acknowledged that to have notable impact on vegetation, fire disturbance requests sufficient intensity through enough dry grass biomass as main source of fuel. under a certain level of grass biomass, owing to insufficient rainfall or intense grazing, fires tend to spread difficultly and, where occurring, have modest impacts on woody plants. logically, most authors tend now to distinguish disturbance-limited (i.e., under moist climate) vs. water limited (i.e., arid) savannas (e.g. bond et al. (2003) [20]). inter-tree competition shapes the second type (sankaran et al. (2005) [75]), while asymmetric and fire-meditated treebiomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 4 of 29 http://dx.doi.org/10.11145/j.biomath.2018.12.167 i.v. yatat djeumen, a. tchuinté tamen, y. dumont, p. couteron, a tribute to the use of minimalistic ... figure 2: three facets of woody plant resprouting just after fire and rainfall onset in the humid savannas of the sanagha basin (cameroon, central africa). note that tufts of perennial grasses did also systematically resprout. seedling struggling in a middle of a grass tuft a). seedling resprouting after topkill either at ground b) or stem c) level. photos: pierre couteron (march 2018). grass interactions is central to the first one. but less clear-cut situations obviously occur under intermediate rainfall (diouf et al. (2012) [31]) or because of modulation by edaphic conditions, grazing and anthropogenic pressures. grazing may lead savannas toward physiognomies and functioning looking less fire-prone, i.e. more ”arid-like”, than expected from the only climate features as an emergent consequence of dynamical amplification of external forcing. iii. main published modelling options the questions raised by observed or putative dynamics within the savanna biome have triggered an increasing interest in terms of modelling. pioneering works (walker et al. (1981) [103], walker and noy-meir (1982) [104]) first used systems of ordinary differential equations (ode) to address the particular case of arid, fire-immune savannas in which excessive grazing fosters bush encroachment (skarpe (1990) [82]). this line of modelling featured grass and woody biomasses as state variables and aimed at explicitly depicting their interactions in relation to soil moisture dynamics. as such, it became a paradigm for ’interaction models’ involving a limited resource, but the central assumption of soil niche partitioning between the two plant forms called walter’s (1971) hypothesis [105] has been ever since hotly debated and is obviously not verified in all ecological contexts where savannas, dry thickets or grasslands are observable. another line of ode-based modelling built on the application to savannas of the initial concept of asymmetric competition of (tilman (1994) [94]) through a simple framework that allows considering both direct and disturbance-mediated plant interactions. tilman’s framework reinterpretation (see accatino et al. (2010) [4]), de michele et al. (2011) [28] used two states variables, namely cover-fractions of grass (g) and tree (t ) assumed exclusive and summing between zero and one. it modelled their interacting dynamics in a system of two ode.   dt dt = ctt(1 −t) −δtt, dg dt = cgg(1 −t −g) − cttg−δgg, (1) where, t and g are dimensionless and denote the fractions of sites occupied by tree and grass biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 5 of 29 http://dx.doi.org/10.11145/j.biomath.2018.12.167 i.v. yatat djeumen, a. tchuinté tamen, y. dumont, p. couteron, a tribute to the use of minimalistic ... respectively. ct and cg are the colonisation rates of tree and grass respectively. δt and δg represent the mortality rates of tree and grass respectively. in the sequel, we refer to system (1) as tilman’s model. logistic growth of the inferior competitor (grasses plus herbs) is bounded and depressed by the cover of the superior competitor (woody plants) which logistic growth is not directly affected by grasses (asymmetric competition). in system (1) there is no fire-mediated retroaction of g on t . this was however introduced by subsequent authors (van langevelde et al. (2003) [101], beckage et al. (2009) [14], accatino et al. (2010) [4], beckage et al. (2011) [13], de michele et al. (2011) [28]) via a linear function of g. thus, explicitly including the impact of fire on t in tilman’s model, the first equation of (1) becomes: dt dt = ctt(1 −t) −δtt −δffω(g)t, (2) where, δf represents the trees vulnerability to fire, f is the fire frequency (inversely proportional to fire return time period) and ω(g) is a function of grass biomass that represents the fire impact. through ω(g), there is thus indirect, fire-mediated negative feed-back of grass cover onto tree cover that counterbalance direct, tree-grass asymmetric interactions. a larger array of models (see tables i and ii) took a leaf from the previous modelling framework (system 1 and equation 2). main sources of variations between models were: (1) nature of the equations and temporal treatment of fire disturbance (time-continuous forcing, i.e. ode vs. time-discrete or impulsive occurrences); (2) nature of the function expressing grass-fire feedback on trees (linear vs. nonlinear); (3) integration of herbivory in addition to fire; (4) facultative explicit treatment of water availability through models with one (and sometimes more) additional state variables expressing water resource in interaction with vegetation variables. we will refer to such models as ’ecohydrological’ (see table i), among which is system (3) proposed by accatino et al. (2010) [4] that features a first equation devoted to the dynamics of a soil moisture variable (s)   ds dt = p w1 (1−s)−εs(1−t−g)−τtst −τgsg, dt dt =ctst(1−t)−δtt−δfftω(g), dg dt =cgsg(1−t−g)−ctstg−δgg−fg, (3) where p w1 (per year) represents the rainfall rate normalized with respect to root zone capacity, ε (per year) is the evaporation, τt and τg (per year), are water uptake parameters for tree and grass respectively. ct , cg, δt , δg, δf and f are defined as in system (1) and equation (2). note that setting ω(g) = 0 in the second equation of (3) makes tilman’s model (1) analogous to the system coupling the second and the third equations of (3). moreover, if the s variable is held constant, the main difference between systems (1) and (3) is that accatino et al. (2010) [4] considered ω(g) = g (i.e., impact of fire on trees as a linear function of grass biomass) while in tilman’s model, this function is equal to zero (no impact of fire). taking ω(g) as any increasing function of the grass cover, provides a more general expression of the fire impact on trees. without loss of generality, we referred to holling type functions (holling (1959) [50], augier et al. (2010) [8], tewa et al. (2013) [92], see equation (4) for generic ones). the general form of ω(g) reads as ω(g) = gθ gθ + αθ , (4) where, g in tons per hectare (t.ha−1) is grass biomass, α is the value takes by g when fire intensity is half its maximum, and the integer θ determines the steepness of the sigmoid. nonlinear response was retained by some other authors (scheiter and higgins (2007) [77], higgins et al. (2010) [46], staver et al. (2011) [83], touboul et biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 6 of 29 http://dx.doi.org/10.11145/j.biomath.2018.12.167 i.v. yatat djeumen, a. tchuinté tamen, y. dumont, p. couteron, a tribute to the use of minimalistic ... al. (2018) [96]). accatino and de michele (2016) [3] also introduced a piecewise linear function of grass biomass (indeed qualitatively mimicking extreme non-linearity) in their non-equilibrium model (nem) as the probability of the occurrence of fire. considering a nonlinear (sigmoidal) shape for ω(g) allows for the existence of up to three treegrass coexistence (i.e. savanna) equilibria, while two of them may be simultaneously stable (i.e. bistability) and forest-savanna-grassland tristability is reachable (yatat et al. (2014) [112], (2018) [109], tchuinté tamen et al. (2014) [91], (2017b) [88]). conversely, we proved that for linear ω(g) functions, tristability is unreachable (yatat et al. (2018) [109], tchuinté tamen et al. (2018) [88]). as pointed out in yatat et al. (2018) [109], possible tristability is in good agreement with results of favier et al. (2012) [36] obtained along a general climatic transect over central african (latitude in the range of 3 – 4◦ north). these results concerned a very large range of woody cover (wc) variations (from very low values approaching grassland physiognomies to nearly 80% cover (i.e. forest) through wc of 40%, i.e. savanna) which suggests grassland/savanna/forest tristability as at least plausible. ode models have been criticized on the basis that they only predict abrupt transitions between two alternative stable states (accatino and de michele (2016) [3]) that are deemed unrealistic. however, tristability of equilibria as well as bistability of two savanna equilibria suggests that shifts from one stable state to another may be less spectacular than hypothesized from previous models and that models may render more complex pathways of vegetation changes (see yatat et al. (2018) [109] for further discussion). in our earlier works (yatat et al. (2014) [112], (2017) [110], (2018) [109], tchuinté tamen et al. (2014) [91], (2016) [89], (2017) [90]), we chose to use above-ground biomasses instead of covers as state variables in contrast to accatino et al. (2010) [4] and most of the models which have been proposed on the subject (reinterpretation of tilman (1994) [94], baudena et al. (2010) [11], staver et al.(2011) [83], de michele et al. (2011) [28], synodinos et al. (2015) [85]) that considered cover fractions. modelling biomasses help accounting from the fact that plant types are not mutually exclusive at a given point in space since field studies suggested that grass often develop under tree crowns (belsky et al. (1989) [16], belsky (1994) [15], weltzin and coughenour (1990) [107], abbadie et al. (2006) [1], dohn et al. (2012) [33], moustakas et al. (2013) [70]). moreover, biomasses directly refer to the cycle of carbon and can be assessed from radar remote sensing in savanna ecosystems that correspond to woody biomasses below the saturating level of the backscatter l-band radar signal (mermoz et al. (2015) [65], bouvet et al. (2018) [24]). the minimal configuration of the published models (tables i & ii) featured only two vegetation state variables (e.g., van langevelde et al. (2003) [101], tchuinté tamen et al. (2014) [91], (2016) [89], (2017) [90], (2017b) [88], synodinos et al. (2018) [86]) as in tilman’s (1994) [94] initial framework, but several models distinguished size classes within the woody component of vegetation (e.g. favier et al. (2004) [37], baudena et al. (2010) [11], staver et al. (2011) [83], yatat et al. (2014) [112], (2017) [110], touboul et al. (2018) [96]). some models used more than two size classes through matrix population models (e.g. accatino et al. (2013) [2], (2016) [5] and references therein). simpler models used only two sizerelated variables in addition to grass and simply account for the asymmetric nature of tree-grass interactions as discussed previously (see also scholes (2003) [78], yatat et al. (2014) [112], (2017) [110]). other models separate large trees, having top buds above the flame zone and therefore facing limited risks of topkill, from smaller trees and shrubs which have high probability of having their aerial systems destroyed (beckage et al. (2009) [14], staver et al. (2011) [83], yatat et al. (2014) [112], (2017) [110]). this distinction stems from field observations (trollope (1984) [97], trollope and trollope (1996) [99]) that evidence rapid decline of percent topkill with tree height (see biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 7 of 29 http://dx.doi.org/10.11145/j.biomath.2018.12.167 i.v. yatat djeumen, a. tchuinté tamen, y. dumont, p. couteron, a tribute to the use of minimalistic ... table i: comparison of several models of tree–grass dynamics with respect to some modelling options. walter’s hypothesis refers to the differences in root depth of herbaceous and woody vegetation in water seeking while ecohydrological frameworks stand for models that consider additional state variables expressing water resource in interaction with vegetation variables. from tchuinté tamen (2017) [87] and yatat (2018) [113]. the symbol ∗ means that we refer to system (1) state tree herbivory walter’s ecohydroauthors variables state variables perturbation hypothesis logical cover biomass all sizes lumped size-structured applied frameworks walker et al. (1981) [103] x x x x tilman (1994)∗ [94] x x higgins et al. (2000) [45] x x x van langevelde et al. (2003)[101] x x x x d’odorico et al. (2006) [32] x x beckage et al. (2009) [14] x x baudena et al. (2010) [11] x x higgins et al. (2010) [46] x x x accatino et al. (2010) [4] x x x de michele et al. (2011) [28] x x x x staver et al. (2011) [83] x x beckage et al. (2011) [13] x x yu and d’odorico (2014) [115] x x x x touboul et al. (2018) [96] x x synodinos et al. (2018) [86] x x x x yatat et al. (2014) [112] x x x tchuinté tamen et al. (2014)[91] x x x tchuinté tamen et al. (2016)[89] x x x tchuinté tamen et al. (2017)[90] x x x yatat et al. (2017) [110] x x x yatat et al. (2018) [109] x x x tchuinté tamen et al. (2018)[88] x x x x also figure 3). ode models featuring two woody variables in addition to grass proved analytically tractable (beckage et al. (2009) [14], staver et al. (2011) [83], yatat et al. (2014) [112], (2017) [110]) as long as other complexities were not introduced. a strong objection against ode models is that fire is not a forcing that continuously removes a small fraction of biomass through time as per the previous ode equation systems. instead, fire actually suppresses a substantial fraction of biomass at once through punctual outbreaks that shape ecosystem aspect and immediate post-fire functioning (figure 4). this principle was implemented biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 8 of 29 http://dx.doi.org/10.11145/j.biomath.2018.12.167 i.v. yatat djeumen, a. tchuinté tamen, y. dumont, p. couteron, a tribute to the use of minimalistic ... table ii: summary of the characteristics of tree–grass interactions models with respect to fire modelling options (continued table i). from tchuinté tamen (2017) [87] and yatat (2018) [113]. some references (unticked) do not model fire. the symbol ∗ means that we refer to system (1). fire perturbation impact of fire authors timetimetimetimelinear sigmoidal continuous stochastic discrete impulsive forms walker et al. (1981) [103] tilman (1994)∗ [94] higgins et al. (2000) [45] x x van langevelde et al. (2003) [101] x x d’odorico et al. (2006) [32] x beckage et al. (2009) [14] x x baudena et al. (2010) [11] x x higgins et al. (2010) [46] x x accatino et al. (2010) [4] x x de michele et al. (2011) [28] x x staver et al. (2011) [83] x x beckage et al. (2011) [13] x x yu and d’odoricco (2014) [115] x x touboul et al. (2018) [96] x x x synodinos et al. (2018) [86] x x yatat et al. (2014) [112] x x tchuinté tamen et al. (2014) [91] x x tchuinté tamen et al. (2016) [89] x x tchuinté tamen et al. (2017) [90] x x yatat et al. (2017) [110] x x yatat et al. (2018) [109] x x tchuinté tamen et al. (2018) [88] x x x via time-discrete recurrence equation models by scheiter and higgins (2007) [77], higgins (2010) [46]. but another framework of impulsive differential equations (ide) also proved relevant to gain realism regarding nature and consequences of fire while keeping a high level of analytical tractability thanks to the ode modelling of inter-fires vegetation dynamics (yatat et al. (2017) [110], tchuinté tamen et al. (2016) [89], (2017) [90]). iv. reaching sensible predictions from minimal models a. a seminal ”big picture” at biogeographic scale the model from accatino et al. (2010) [4] was pioneering and inspiring in that it first ventures into generically predicting vegetation physiognomies (in terms of percent covers of woody vs. herbaceous plants) over the entire savanna biome. the authors used bifurcation analysis (accatino biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 9 of 29 http://dx.doi.org/10.11145/j.biomath.2018.12.167 i.v. yatat djeumen, a. tchuinté tamen, y. dumont, p. couteron, a tribute to the use of minimalistic ... 0 10 20 30 40 50 60 70 80 90 100 height classes − m t o p k il l − % 0 − 0.50 0.51 − 1.00 1.01 − 1.50 1.51 − 2.00 2.01 − 2.50 2.51 − 3.00 3.01 − 3.50 3.51 − 4.00 4.01 − 4.50 4.51 − 5.00 4 6 8 13 16 29 48 70 87 92 (a) 0 10 20 30 40 50 60 70 80 90 100 height − m t o p k il l − % 0.5 1 2 3 4 5 9 27 29 43 49 69 75 89 91 9999 59 (b) figure 3: illustration of the effect of height on the frequency of topkill of individual trees subjected to fires in the kruger national park (panel (a)) and in the central highlands of kenya (panel (b)). in the panel (b), continuous bars denote head fire while black-dash bars represent back fire (reproduced from trollope and trollope (2010) [100], © trollope and trollope (2010) [100] ). figure 4: aspects of two nearby savannas both located close to a forest boundary in the sanagha basin, cameroon (central africa) depending on recent fire occurrence (left) or not (right). photos were taken the same day. on the left, note the general resprouting of both the herbaceous stratum and topkilled woody plants. photos: pierre couteron, march 2018. biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 10 of 29 http://dx.doi.org/10.11145/j.biomath.2018.12.167 i.v. yatat djeumen, a. tchuinté tamen, y. dumont, p. couteron, a tribute to the use of minimalistic ... et al. (2010) [4], tchuinté tamen et al. (2018) [88], see also figure 7) based on two important parameters of strong intuitive meaning, namely mean total annual rainfall and fire frequency. they thereby achieved delineation of domains in which main physiognomies (i.e., grassland, savanna, forest) can be expected as stable. bistability situations (forest-grassland and forest-savanna) were also highlighted for sufficiently high fire frequencies. for low fire frequencies, a sensible gradient of increasing woody cover with increasing annual precipitation was found. but one may note that only dense woodlands (i.e. still two-phase vegetation) were obtained even for the highest rainfall range, while forest stricto -sensu (mono-phase, with no tall, light-demanding savanna grasses in the understory) is widely observed for the corresponding ranges of precipitations. moreover, transitions between vegetation types in relation to fire frequencies proved tricky. indeed, in the high rainfall range, increasing fire frequency leads from the aforementioned woodlands to forest. in the intermediate rainfall range, increasing fire frequency leads from savanna mono-stability to forest-savanna bistability (see figure 7). analogously, for fairly low rainfall, grassland stability shifts to grassland-forest bistability. hence, all over the rainfall gradient it looks as if sufficient frequency of fire were a necessary condition to reach forest (bi)-stability. this is contradicted by empirical knowledge according to which frequent fires are known to jeopardize or at least delay woody biomass build-up, but never favour it (bond et al. (2005) [22], archibald et al. (2009) [7], bond and parr (2010) [21]). where did this critical problem come from? most of subsequent papers barely evoked the question. a large share of them investigated different modelling options, often more complex and/or less tractable; or they assumed particular biogeographic conditions. in a further contribution, accatino and de michele (2016) [3] argued about intrinsic limitations of ode-based modelling. they also put forward that there is no evidence according to which observed vegetation physiognomies may be close to a stable equilibrium point. it is in fact undisputable that climate is likely to vary through time, and there is no guaranty that woody vegetation can track such variation with enough celerity. they also underline as questionable the assumption according to which the parameter f of fire frequency should be treated as a constant forcing, independently of vegetation characteristics. all these arguments brought them to propose a ’non-equilibrium model’, based on stochastic difference equations, as alternative to the timecontinuous model of accatino et al. (2010) [4] referred to as ’equilibrium-model’ (em). in their non-equilibrium (nem) model, fire occurrence is a stochastic event all the more likely to occur in a given dry season that ignitable dry grass biomass abundantly built-up in the foregone rainy seasons. accatino and de michele (2016) [3] compared the predictions of their (em) vs. (nem) models. they argued that separation of fire-immune vs. fire-prone savannas as an indirect consequence of rainfall is an emergent property with their nem while it is artificially induced by the choice of the f parameter with em. they also pointed out that when considering high rainfall, the nem is able to reproduce the ”bimodality” of woody cover extent observed in remote sensing studies. but in fact, their nem was not a straightforward time-discrete analogue of the ode based em of accatino et al. (2010) [4], since it features several novelties. therefore, the differences they reported between em and nem are not a simple consequence of time-continuous fire forcing vs. time-discrete fire occurrences. in fact, several aspects altogether contribute to the more satisfactory results obtained with the nem by accatino and de michele (2016) [3]. we will subsequently illustrate the fact that predictions that are qualitatively satisfactory can be obtained by directly improving the ode based ’em’ framework, notably regarding the firemediated feedback of grass onto tree dynamics. b. fire frequency, grass biomass and fire impact the way in which fire impact is modelled in the previous equation systems (1 and 3) is obviously a crucial question. as stated by scholes (2003) biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 11 of 29 http://dx.doi.org/10.11145/j.biomath.2018.12.167 i.v. yatat djeumen, a. tchuinté tamen, y. dumont, p. couteron, a tribute to the use of minimalistic ... 0 2 4 6 8 10 0 2 4 6 8 10 grass biomass (t.ha−1) f ir e i m p a c t (w (g )) holling type i (a) 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 grass biomass (t.ha−1) f ir e i m p a c t (w (g )) holling type ii holling type iii (b) figure 5: graphical representation of the function shapes w(g). the fire impact term in the odes is given by λft fω(g). table iii: functions involving ω(g). functions models (ode) coexistence equilibria ω(g) = 0 tilman (1994) [94] one savanna monostability ω(g) = g accatino et al. (2010) [4] 2 savannas van langevelde et al. (2003) [101] bistability holling type ii tchuinté tamen et al. (2014) [91] 2 savannas bistability holling type iii staver et al. (2011) [83] 3 savannas tchuinté tamen et al. (2014) [91] bistability yatat et al. (2014) [112] 3 savannas bistability & tristabilitytchuinté tamen et al. (2018) [88] [78], modelling savanna dynamics in fire-prone contexts actually requires introducing an ”equation predicting the effect of grass biomass, via fire intensity, on tree biomass”. in fact, non-linearity in ω(g) may be justified on various, non-exclusive grounds, since what is important is to properly model, as a whole, the causal chain that leads from grass abundance and ignition regime to woody biomass suppression. as steps in this chain we may identify: (i) fire frequency to be seen as an external forcing upon the tree-grass system (think about a targeted fire regime in a managed area such as a ranch or a protected area); (ii) actual yearly fire probability (or frequency) of occurrence in any arbitrary small piece of land once (i) has been set; (iii) fire potential impact (intensity and flame height) on woody vegetation; (iv) fire actual impact that also depends on features intrinsic to woody vegetation (see below section iv-c). fire intensity which is strictly speaking a quantity of energy released (bond and keeley (2005) biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 12 of 29 http://dx.doi.org/10.11145/j.biomath.2018.12.167 i.v. yatat djeumen, a. tchuinté tamen, y. dumont, p. couteron, a tribute to the use of minimalistic ... figure 6: example of spatial heterogeneity of fire propagation in an altitude mosaic of forests and low biomass grasslands in cameroon. here local community hunters tend to set fire every year at landscape scale (i.e. f = 1) as to flush small game from spots of dense grass cover. but all the area does not burn every year because fire actually do not propagate everywhere (from p. couteron, february 2017, mount cameroon national park). [18]) appears empirically as a fairly linear function of grass biomass. but impact on trees also depends on flame height which is reported as increasing exponentially with observed quantity of dry grass biomass (scheiter and higgins (2007) [77], staver et al. (2011) [83], synodinos et al. (2018) [86]), though one may suppose some levelling off for maximal grass biomass (and height) values. in earlier works (yatat et al. (2014) [112], (2017) [110], (2018) [109], tchuinté tamen et al. (2014) [91], (2016)[89], (2017) [90]), we systematically assumed fire impact on woody vegetation as a non-linear, increasing bounded function of grass biomass (yatat et al. (2014) [112], (2017) [110], (2018) [109], tchuinté tamen et al. (2014) [91], (2016) [89], (2017) [90], (2018) [88]), w(g). in our modelling, the f parameter was kept as constant multiplier of w(g), but we interpret it as a man-induced ”targeted” fire frequency (as for instance in a fire management plan), which will not translate into actual frequency of fires of notable impact as long as grass biomass is not of sufficient quantity. with this interpretation, the actual fire regime may substantially differ from the targeted one, as frequently observed in the field. and, whatever f values, any hypothetical piece of land will actually be fire-prone only if other forcing factors (climate, herbivory, etc.) allow for sufficient grass biomass. most previous modelling papers including those from our group did not elaborate much regarding the successive steps involved in the grass-fire feedback. distinction between (i) and (ii) may appear subtle and to our knowledge was never emphasized before. it directly results from space-implicit savanna modelling. fire regime, which is nowadays overwhelmingly maninduced (govender et al (2006) [44], archibald (2009) [7]) is a forcing at landscape scale since people do not go and set fire in every piece of land. they instead count on fire propagation that depends on abundance and spatial evenness of dry grass. in presence of low and unevenly distributed grassy fuel, fire will barely propagate leaving a large share of the area unburnt. this makes the difference between steps (i) and (ii), as frequently observed in the field (see diouf et al. (2012) [31]; figure 6). the response of percent area burnt to grass abundance is likely to be sharply nonlinear, as suggested by the impressive results reported by mcnaughton (1992) [61] at the scale of the entire serengeti complex in tanzania. in this remarkable study, local fire frequency dwindled over a decade biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 13 of 29 http://dx.doi.org/10.11145/j.biomath.2018.12.167 i.v. yatat djeumen, a. tchuinté tamen, y. dumont, p. couteron, a tribute to the use of minimalistic ... following grass biomass suppression by soaring herbivore populations, while the ignition regime by people dwelling around the park likely remained more or less the same. in fact, since we are here dealing with mean-field models the ω(g) function is also due to embody the difficult spreading of fire in presence of fuel of overall low quantity keeping in mind natural spatial variability of grass biomass (figure 6). non-linearity of ω(g) seems therefore a necessary feature for adequately capturing the fire-mediated grass-tree feedback. c. tree survival to be relevant, the most parsimonious models featuring just grass and tree state variables must overcome the limitation pointed out in sub-section iv-a for the precursor model of accatino et al. (2010) [4]. all things being equal, any increase in fire frequency should never increase the woody component of vegetation. fire, if any, is expected to be of no substantial consequence over the driest stretch of the rainfall gradient while for the moister part, it is widely observed that extending the average time between successive fires (decreasing frequency) favours the building up of woody vegetation. accounting for that proved to be a challenge for minimal two-variable models that do not distinguish between fire sensitive and fire insensitive woody fractions. non-linearity of the ω(g) function, though important proved not sufficient to overcome this problem. tchuinté tamen et al. (2017) [90] further introduced a second non-linear decreasing function, which directly expresses that high woody biomasses, corresponding to tall trees proportionately experience far less firerelated losses than low woody biomasses relating to seedlings, saplings and shrubs (see figure 2). we hence proposed the following function to denote the fire-induced tree/shrub mortality: ϑ(t) = λminft + (λ max ft −λ min ft )e −pt , (5) where λminft (in yr −1) is minimal loss of tree biomass due to fire in systems with a very large tree biomass, while λmaxft (in yr −1) is maximal loss of tree/shrub biomass due to fire in open vegetation (e.g. for an isolated small woody individual having its crown within the flame zone), p (in t−1.ha) is proportional to the inverse of biomass suffering an intermediate level of mortality. this general form was suggested by experimental observations showing dwindling rate of topkill with increasing tree height, since tall trees are likely to have their upper parts above the flame zone, even for high grass biomass (trollope and trollope (2010) [100]). notice that taking into account a nonlinear and decreasing function of tree biomass is a way to bypass introducing size classes as to keep the model minimal and retain mathematical tractability (see inspiring examples in meron et al. (2004) [67], lefever et al. (2009) [55]). the addition of the ϑ(t) function was indeed decisive in ensuring that a two-equation ode system provides predictions that qualitatively agree with the general ecological knowledge about the role of fire return period. on this basis, tchuinté tamen et al. (2017) [90] designed a model that also implemented punctual fire events through impulsive differential equations. but predictions of the ode model itself were already satisfactory. d. relating to water resource several savanna models have explicitly modelled soil water resources, via a dedicated equation as in system (3). but the soil moisture dynamics is very rapid compared to change in vegetation. soil moisture variations linked to a given rainfall event are damped within a few days (barbier et al. (2008) [10]), while vegetation growth proceeds over months for grasses and even years for woody plants. it therefore makes sense to consider vegetation dynamics in relation to a level of soil water resource that is approximately constant for a given level of total annual rainfall or, ideally, water deficit (rainfall minus evapotranspiration). martinez-garcia et al. (2014) [59] reached similar conclusions for a partial differential equation model of non-local plant-plant interactions for water. this justifies letting parameters in the vegetation dynamics equations directly depend on biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 14 of 29 http://dx.doi.org/10.11145/j.biomath.2018.12.167 i.v. yatat djeumen, a. tchuinté tamen, y. dumont, p. couteron, a tribute to the use of minimalistic ... climatic parameters. this principle sustains the model expressed through the following system:   dg dt = rg(w)g ( 1 − g kg(w) ) −δgg −ηtgtg−λfgfg, dt dt = rt (w)t ( 1 − t kt (w) ) −δtt −fϑ(t)ω(g)t, g(0) =g0,t(0) = t0, (6) where, • g and t are grass and tree biomasses respectively; • w is the mean annual precipitation (in millimeters per year, mm.yr−1); • rg(w) = γgw bg + w and rt (w) = γtw bt + w are annual productions of grass and tree biomasses respectively, where γg and γt (in yr−1) express maximal growths of grass and tree biomasses respectively, half saturations bg and bt (in mm.yr−1) determine how quickly growth increase with water availability; • kg(w) = cg 1 + dge−agw , and kt (w) = ct 1 + dte−at w are carrying capacities of grass and tree respectively, where cg and ct (in t.ha−1) denote the maximum values of the grass and tree biomasses, ag and at (mm−1yr) control the steepness of the curves of kg and kt respectively, and dg and dt control the location of their inflection points; • δg and δt respectively express the rates of grass and tree biomasses loss by herbivores (grazing and/or browsing) or by human action; • ηtg denotes the asymmetric influence of trees on grass for light (shading) and resources (water, nutrients) which relate to competitive or facilitative influences; • λfg is the specific loss of grass biomass due to fire; • f = 1 τ is the fire frequency, where τ is the fire return period. submitting model (6) to bifurcation analysis provides figure 7-(b) that is to be compared to figure 7-(a) from accatino et al. (2010) [4], which has been reobtained using matcont (see govaerts (2000) [42], dhooge et al. (2003) [29], govaerts et al. (2007) [43] and references therein). both figures 7-(a) and 7-(b) are sensible regarding low fire frequencies for which increasing mar leads to a sequence of physiognomies of increasing woody biomass (i.e., grassland, savanna, forest). but in figure 7-(b), the improvement resulting from introducing ω(g) and ϑ(t) is apparent when increasing fire frequency (f) at different levels of the rainfall gradient. for high mar values, the expected physiognomy shifts from monostable forest to forest-grassland bistability. indeed, in presence of high mar, it is known that grasslands are due to be encroached by forest under fire prevention or even just because of decreasing fire frequencies (jeffery et al. (2014) [51]). our model accords with field observations in that a high fire frequency is indeed a necessary condition to perpetuate the grassland (or savannas of low woody biomass) physiognomies. moreover, largescale observations of bimodality between high and very low woody cover situations (hirota et al. (2011) [47], favier et al. (2012) [36]) can be accounted for by the forest-grassland bistability, though the converse is not necessarily true. in fact bimodality may stem from either transient situations or topographical heterogeneity and does not automatically implies bistability. for low to intermediate mar values, say 600 − 1000 mm, fire is known to be less pervasive, though field observations or experiment results depict woody vegetation thickening in case of fire frequency decrease (brookman-amissah (1980) [25]). the model is able to render such thickening as a shift from the grassland to the savanna stability domain. the model also predicts forest-savanna or savanna-grassland bistability, and even tristability thereof for restricted domains in the mar-fire frequency plane that were situated around 1000 biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 15 of 29 http://dx.doi.org/10.11145/j.biomath.2018.12.167 i.v. yatat djeumen, a. tchuinté tamen, y. dumont, p. couteron, a tribute to the use of minimalistic ... mm of mar. this value is of course dependent on parameter values used for computations underlying figure 7-(b). refined calibrations relating to a specific regional context may displace the thresholds. notably, the parameter expressing the influence of woody biomass on grasses proved to be influential on the thresholds between vegetation states (tchuinté tamen et al. (2018) [88]). e. impulsive time-periodic occurrences of fire events in previous models, the traditional timecontinuous fire forcing formalism is often used. however, it is questionable to model fire as a permanent forcing that continuously removes fractions of fire sensitive biomass all over the year. indeed, several months and even years can pass between two successive fires, such that fire may be considered as an instantaneous perturbation of the savanna ecosystem (yatat et al. (2017) [110], tchuinté tamen (2016) [89], (2017) [90]; see also table iv, page 18). several recent papers have proposed to model fires as stochastic events while keeping the continuous-time differential equation framework (baudena et al. (2010) [11], beckage et al. (2011) [13], klimasara and tyran-kamińska (2018) [54], synodinos et al. (2018) [86]) or using a time-discrete model (higgins et al. (2000) [45], accatino and de michele (2013) [2], accatino et al. (2016) [5]). however, a drawback of most of the aforementioned recent time-discrete stochastic models (higgins et al. (2000) [45], baudena et al. (2010) [11], beckage et al. (2011) [13]) is that they barely lend themselves to analytical approaches. based on table iv, page 18, we further consider in our group (yatat et al. (2017) [110], (2018) [109], yatat and dumont (2018) [111], tchuinté tamen (2016) [89], (2017) [90], (2018) [88]) impulsive time-periodic fire events which is an approximation that keeps the potential of analytical investigation as large as possible while modelling discrete fires. an impulsive differential equations system can be used to express fire through impulsive periodic occurrences (e.g. system (7), below):   dg dt = rg(w)g ( 1− g kg(w) ) −δgg −ηtgtg, dt dt = rt (w)t ( 1− t kt (w) ) −δtt, t 6= nτ, ∆g(nτ) = −λfgg(nτ), ∆t(nτ) = −ϑ(t(nτ))ω(g(nτ))t(nτ), n = 1, 2, ...,nf, (7) where, • for π ∈{g,t}, ∆π(nτ) = π(nτ+) −π(nτ) and π(nτ+) = lim θ→0+ π(nτ + θ); • τ = 1 f is the period between two consecutive fires; • nf is a countable number of fire occurrences; • nτ, n = 1, 2, ...,nf , are called moments of impulsive effects of fire, and satisfy 0 ≤ τ < 2τ < ... < nfτ. properties of models (6) and (7) have been analysed in tchuinté tamen et al. (2018) [88]. below we provide some numerical simulations as to illustrate the bifurcations between the stable domains delineated in figure 7-b as a consequence of increasing the fire frequency for two particular values of mar, i.e. w = 946 mm.y−1 and w = 1003 mm.y−1. for each of the two mar values, we compare the consequences of increasing f with the ode (system (6)) and the ide (system (7)) frameworks, by comparing figure 8 against figure 9 and figure 10 against figure 11. for the lower mar situation, ode and ide frameworks qualitatively agree and show the bifurcation from monostable savanna to monostable grassland through an intermediate bistable situation (figure 8-b; figure 9-b). for the higher mar case, both frameworks also show the transition from monostable forest to forest-grassland bistability through tristability involving savanna. qualitatively, the biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 16 of 29 http://dx.doi.org/10.11145/j.biomath.2018.12.167 i.v. yatat djeumen, a. tchuinté tamen, y. dumont, p. couteron, a tribute to the use of minimalistic ... (a) (b) figure 7: bifurcation diagrams using matcont. (a) accatino et al. (2010) [4] model re-implementation. (b) implementation of system (6). single red, green and black symbols (rectangles in panel (a), dots in panel (b)) stand for grassland, forest and desert respectively. twinned red and green symbols stand for savanna (coexistence state). size of the symbols qualitatively denote grass and tree cover fractions in panel (a) and biomass levels in panel (b). parameters used to compute figure 7-(a) are from accatino et al (2010) [4] (see also table v in appendix). the parameter values used in 7-(b) are from tchuinté tamen et al. (2018) [88] (see also table vi in appendix). (color in the online version). biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 17 of 29 http://dx.doi.org/10.11145/j.biomath.2018.12.167 i.v. yatat djeumen, a. tchuinté tamen, y. dumont, p. couteron, a tribute to the use of minimalistic ... table iv: average fire period (τ, in yr) ranges of values found in literature with respect of the mean annual rainfall (mar). ranges of values of mar are from yatat et al. (2017) [110] and tchuinté tamen et al. (2016) [89] ranges references low mar: 5 – 50 frost and robertson (1985) [40] mar ≤ 650 mm.yr−1 4 – 8 trollope (1984) [97] intermediate mar: 3 – 5 february et al. (2013) [38] 650 mm.yr−1 ≤ mar ≤ 1100 mm.yr−1 5 – 7 van wilgen et al. (2004) [102] high mar: 0.5 – 3 jeffery et al. (2014) [51] mar ≥ 1100 mm.yr−1 0.5 – 2 bond and keeley (2005) [18] accatino et al. (2010) [4] 1 – 5 abbadie et al. (2006) [1] 1 menaut and cesar (1979) [63] gignoux et al. (2009) [41] predictions of the two frameworks agree about the predicted sequence of vegetation physiognomies when increasing the fire frequency. however, the ide model systematically predicted bifurcations for lower values of f than for the ode. this indicates that shifting to the conceptually more satisfactory ide framework will introduce specificities in forthcoming stages concerning refined calibration and comparison with real-world observations. v. discussion and prospects in the present paper we emphasize that ecologists did probably not yet exploit all the potential of simple ode systems for modelling vegetation dynamics in the savanna biome to which most seasonal tropical ecosystems pertain. we showed that reasonable, non-trivial predictions can be obtained in reference to hypothetical situations directly relating to rainfall and fire frequency gradients. application to specific contexts and locations would request refined calibration for the parameters of the generic minimalistic model. but it may also invite to better address specific processes deemed influential in a particular situation under study. this would mean complexifying the model to match a specific piece of reality. though this is actually a natural and sensible trend in science, parsimony is an opposing principle that tells us to keep complexification under control. a meaningful, balanced modelling approach should restrict to what we strictly need to account for a well-defined array of empirical facts in a particular situation. on the empirical side, the ongoing development of remote-sensing techniques and derived products is providing avenues to better depict the spatiotemporal variation of environmental factors, such as rainfall (e.g. via chelsa, karger et al. (2017) [52]), topography (via the srtm, farr et al. (2007) [35], published at increased spatial resolution by nasa in 2013) or fires (http://modis-fire.umd.edu; e.g. archibald et al. (2009) [7], diouf et al. (2012) [31]). this also applies to the monitoring of some vegetation variables, though disentangling effects on most remotely-sensed signals from grasses vs. trees in mixed savanna physiognomies is still challenging. improving and diversifying sources of remote sensing information will obviously help sorting out relevant predictions from unrealistic ones and refine the benchmarking of models. but most of the parameters expressing vegetation dynamics will remain out of reach of remotelysensed assessment and will remain dependent on field information. an increased effort of field data biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 18 of 29 http://modis-fire.umd.edu http://dx.doi.org/10.11145/j.biomath.2018.12.167 i.v. yatat djeumen, a. tchuinté tamen, y. dumont, p. couteron, a tribute to the use of minimalistic ... 0 5 10 15 20 grass biomass (t/ha) 0 10 20 30 40 50 60 70 80 90 100 t re e b io m a s s ( t/ h a ) (w=946, f=0.4) (a) 0 5 10 15 20 grass biomass (t/ha) 0 10 20 30 40 50 60 70 80 90 100 t re e b io m a s s ( t/ h a ) (w=946, f=0.51) (b) 0 5 10 15 20 grass biomass (t/ha) 0 10 20 30 40 50 60 70 80 90 100 t re e b io m a s s ( t/ h a ) (w=946, f=0.7) (c) figure 8: illustration of a bifurcation due to f with the ode model of system (6) for a constant mar value of w=946 mm.y−1. when the fire frequency f increases, the system shifts from a savanna monostable (see panel (a)) to a grassland monostable (see panel (c)) passing through a bistability between savanna and grassland (see panel (b)). (a) (b) (c) figure 9: illustration of a bifurcation due to fire frequency f with the impulsive model (7) for a constant mar value of w=946 mm.y−1. this figure is based on same parameters values as figure 8 but with the impulsive ide framework savanna-grassland bistability is already observable for f = 0.4 (panel (a)) and give way to monostable grassland for f=0.51 (panel (b)). these values are to be compared to f = 0.25 and f = 0.7, respectively when using the ode model (system (6)). 0 5 10 15 20 grass biomass (t/ha) 0 20 40 60 80 100 120 140 160 180 200 t re e b io m a s s ( t/ h a ) (w=1008, f=0.4) (a) 0 5 10 15 20 grass biomass (t/ha) 0 20 40 60 80 100 120 140 160 180 200 t re e b io m a s s ( t/ h a ) (w=1008, f=0.5) (b) 0 5 10 15 20 grass biomass (t/ha) 0 20 40 60 80 100 120 140 160 180 200 t re e b io m a s s ( t/ h a ) (w=1008, f=0.56) (c) figure 10: numerical simulations with the ode model (6) illustrating a bifurcation induced by increasing the fire frequency (f) in presence of a constant mar (w) value of 1008 mmy−1. vegetation shifts from monostable forest (left) to savanna-forest bistability (center) to grassland-savanna-forest tristability (right). biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 19 of 29 http://dx.doi.org/10.11145/j.biomath.2018.12.167 i.v. yatat djeumen, a. tchuinté tamen, y. dumont, p. couteron, a tribute to the use of minimalistic ... 0 5 10 15 20 grass biomass (t/ha) 0 20 40 60 80 100 120 140 160 180 200 t re e b io m a s s ( t/ h a ) (w=1008, f=0.3) (a) (b) (c) (d) figure 11: numerical simulations with the ide model (7) illustrating a bifurcation induced by increasing the fire frequency (f) in presence of a constant mar (w) value of 1008 mm−1. this figure is analogous to figure 10 obtained with the ode model. note that with the ode model, there is still tristability for f = 0.56 while forest is still monostable for f = 0.4. collection is obviously desirable but insufficient means for research in most tropical countries is enduring reality. that strong data limitation is alas probably here to stay finally pleads for parsimony in modelling. it also underlines the importance for modelling to be sufficiently convincing and accessible to ecologists as to guide data acquisition and orient scarce resources towards assessing parameters proven as the most influential by sensibility analyses. on the modelling side, within the class of models that distinguished size classes for the woody component of vegetation, some used more than two size classes through matrix population models (e.g. accatino et al. (2013) [2], (2016) [5]). however, such models remain generally simulationbased and usually involve a fairly large number of parameters. thus, it is not easy/possible to assess how model parameter variations may influence the model outcomes (yatat et al. (2018) [109]). ode biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 20 of 29 http://dx.doi.org/10.11145/j.biomath.2018.12.167 i.v. yatat djeumen, a. tchuinté tamen, y. dumont, p. couteron, a tribute to the use of minimalistic ... and ide models featuring two woody variables, in addition to grass, proved analytically tractable (beckage et al. (2009) [14], staver et al. (2011) [83], yatat et al. (2014) [112], (2017) [110]) as long as other complexities were not introduced. for example, yatat et al. (2014) [112] (resp. yatat et al. (2017) [110]) studied a ode-like (resp. ide-like) tree-grass interactions model where in addition to grass they considered two classes of woody plants: fire-sensitive like seedlings and fire insensitive. but, based on recent publications of our group, we found that even with models that feature only one state variable for woody component, meaningful results are obtained and, to some extent, are qualitatively similar to those obtained with models that used two size-related variables for woody component (tchuinté tamen et al. (2014) [91], (2016) [89], (2017) [90], (2018) [88], yatat et al. (2018) [109]). the ide framework is an obvious improvement that expresses a reasonable trade-off between increased realism and decreased analytical tractability. within the framework of ide, a further step in that direction could be considering patterns of fire occurrences featuring stochastic components instead of the deterministic periodic regime we used (yatat et al. (2017) [110], (2018) [109], tchuinté tamen et al. (2016) [89], (2017) [90], (2018) [88]). but one may note here that periodicity of fire outbreaks is not that unrealistic in the context of subequatorial humid savannas for which fires can only occur at the end of dry seasons which are of short duration. here the annual climatic cycle strongly defines the temporal window for fires while in less humid savannas fire is simultaneously less frequent and liable to occur all over extensive dry seasons (diouf et al. (2012) [31]). we believe that while mathematical tractability or theoretical study of a model is not an absolute requirement or is not always possible, it remains nevertheless desirable at least for two reasons. first, it can appear as a kind of guarantee that numerical simulations displayed by the model are not the result of some numerical artifacts. in other words, the choice or the construction of a suitable algorithm to solve a given (complex) model strongly relies on its qualitative study or, when this study is not possible, on the analysis of some sub-models, that can be mathematically tractable. nowadays, there are more and more works that point out some spurious behaviors that may appear when using some ’classical’ schemes for model simulations (see for example anguelov et al. (2012) [6]). second, any theoretical analysis may provide useful informations about the role of some particular parameters in the dynamics of the system. we have here focused on spatially-implicit models because we believe that such models have still important insights to provide and also because spatially-explicit models are far more demanding in terms of parametrization and more difficult to study theoretically. substantial efforts to design and run spatial models of savannas have however been made during the last decade (borgogno et al. (2009) [23]). most of them relied on individual-based models such as cellular automata. at this step of the discussion, it seems meaningful to point out that there are some authors who pleaded for a mutualistic or complementary relationship between mathematical tractabilitybased and simulation-based formalisms (omohundro (1984) [73], wolfram (1985) [108], weimar (1997) [106], narbel (2006) [71], dietrich et al. (2014) [30], dumont et al. (2018) [34]) and we also agree with that. as an illustration, it may be more difficult to achieve a very deep theoretical analysis of a partial differential equations model when taking into account spatial heterogeneity while it seems more easy to handle it when using for example an individual-based model or cellular automaton formalism. independently, partial differential equations (pde) have also widely been used to account for the genesis of vegetation regular spatial patterns (bare soil vs. dense shrubby cover) in the particular context of arid savannas (lefever and lejeune (1997) [56], klausmeier (1999) [53], sherratt (2005) [81], borgogno et al. (2009) [23], lefever et al. (2009) [55], meron (2011) [66], lefever and turner (2012) [57]). in biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 21 of 29 http://dx.doi.org/10.11145/j.biomath.2018.12.167 i.v. yatat djeumen, a. tchuinté tamen, y. dumont, p. couteron, a tribute to the use of minimalistic ... this line, there was a recent attempt by yatat et al. (2018) [109] to account for the dynamics of forestsavanna boundaries by introducing local diffusion operators for herbaceous and woody biomasses in a parsimonious space-implicit model, with timecontinuous fire events forcing, closely related to the one of tchuinté tamen et al. (2017) [90]. the analysis of this model shows that there exists monostable or bistable travelling solutions, related to the boundary movements in the forestgrassland mosaic. and the authors showed that depending on fire-return time as well as difference in diffusion coefficients of woody and herbaceous vegetation, fire events are able to greatly slow down or even stop the progression of forest in the humid part of the savanna biome. this kind of results, obtained from theoretical analysis, are of great interest for practical needs or management policies. however, as an improvement of yatat et al. (2018) [109], there are also some ongoing works that aim to deal with existence of travelling waves for system of impulsive partial differential equations (ipde) that model savanna dynamics (e.g. yatat and dumont (2018) [111]). this type of modelling is of great interest considering that the forest-savanna ecotone is the most widespread in the tropics and that forests have been encroaching during the last decades in many humid savannas of west and central africa, and to a lesser extent in other regions of the world (oliveras and malhi (2016) [72]). on the other hand, forest encroachment, which is of great consequence for the global carbon balance of terrestrial ecosystems proved heterogeneous in space and time (oliveras and malhi (2016) [72]). hence, modelling should help better understand the hierarchy of processes and forces accounting for such heterogeneity. this makes spatially-explicit modelling desirable. but, interpretation and calibration of local, diffusion operators as used in yatat et al. (2018) [109] cannot rely on much empirical knowledge in plant ecology. and one may note that this also applies to colonization rates between adjacent cells that are central to cellular automata models. in the case of pdes, modelling non-local plant-plant interaction processes (e.g. through interaction kernels) is mathematically more challenging, though attempts in martinez-garcia et al. (2014) [59] and in lefever et al. (2009) [55] or lefever and turner (2012) [57] nevertheless provide sources of inspiration. a recent line of criticism questions the relevance of reasoning in reference to equilibrium states. there is indeed no compelling evidence that physiognomies presently observable correspond or even are close to predictable equilibria determined by current environmental conditions. in fact, parameters reflecting environmental variables, notably climate are due to fluctuate or even change through time. and vegetation, especially its woody component may be unable to keep pace with such variations and rather track them with delay thereby remaining distant from any equilibrium state. long-lasting consequences of past climate periods probably still mark present vegetation. for instance, in central africa, concomitant to, drier period occurred some centuries ago (europe’s ”little ice age”, 500-200 years bp), which probably provoked forest cover recession and fragmentation (oslisly et al. (2013) [74]). the trend of widespread forest boundaries displacement within savannas, as observed during the last decades may be a delayed recovery after this past drier period that is progressing at slow and unequal pace owing to the counteracting influence of fires. increased co2 availability that favours c3 woody plants against c4 savanna grasses (bond and midgley (2000) [19]) may also reinforce this trend. vi. conclusion minimal savanna models using ode systems have been criticized from different standpoints. the first one was the poor realism of the overall picture made by the predicted stable equilibria. the present paper shows that some unrealistic predictions are not a direct drawback of the ode framework, but rather derive from inadequate modelling of the crucial fire-mediated negative feedback of grassy biomass onto woody vegetation. using nonlinear functions such as ω(g) and ϑ(t) (as in equations 5) is not only justified by the biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 22 of 29 http://dx.doi.org/10.11145/j.biomath.2018.12.167 i.v. yatat djeumen, a. tchuinté tamen, y. dumont, p. couteron, a tribute to the use of minimalistic ... nature of the mechanisms at play, but also proved sufficient to get a meaningful ”big picture” of vegetation physiognomies predicted as stable for varying mean annual rainfall and fire frequency (as in figure 7-(b)). another argument against ode models is that they predict too contrasted stable states meaning that shifts between them would appear as more catastrophic than actually observed. but some strong contrasts such as landscape mosaics of forest and grassland are indeed observable in the field (see figure 1). moreover equilibria that have attraction domains ”adjacent” in figure 7 do not systematically show contrasted biomass values. transitions may actually be progressive in terms of state variables. indeed, at low fire frequency, the transition from savanna to forest along the mar gradient corresponds to a continuous increase of tree biomass with concomitant decrease of grass biomass. the same applies to the transition from savanna to grassland via increased fire frequency (see figures 8 and 9). here, the shape of the nonlinear functions embodying the grass-fire feedback matters as shown in previous works (yatat et al. (2014) [112], tchuinté tamen et al. (2014) [91]). strongly nonlinear shapes (e.g. holling functions of higher order see table iii) allow for multiple coexistence equilibria (i.e. multiple savanna physiognomies) of different woody biomass values, which may be seen as ”stepping stones” between grassland and forest. thus, the ode framework does not automatically imply abrupt changes between equilibria of very contrasted biomass values. on the other hand, it is undisputable that modelling fire as an external forcing continuously suppressing small amount of biomass through time is not satisfactory. models based on punctual fires impacting large shares of biomasses are more relevant. this is implemented in time-discrete stochastic models which are however of limited analytical tractability. impulsive differential equation systems are a good compromise since they permit to model time-discrete fire impact while keeping ode for modelling vegetation growth. as such, they remain analytically tractable to a large extent while being more realistic. in this paper, we show that minimal savanna models are able to provide a wide array of meaningful and relevant predictions of savanna dynamics while retaining sufficient mathematical tractability and restricting themselves to a minimal set of parameters assessable from the overall literature. moreover, simplicity is overarching whatever the level of tractability. with a simple model, simulations can claim a thorough exploring of all parameters space. conversely, it is difficult to be sure that sufficient exploration of model behaviors has been carried out for overcomplicated models which tend to flourish in ecology. moreover, because there is naturally substantial uncertainty for many parameter values, it is difficult to conclude whether results are due to the ranges taken for parameters or to the structure of the model itself. therefore, using complex models, it becomes even more illusory to test hypotheses, while this is one of the fundamental roles of modelling. acknowledgements a. tchuinté tamen and i.v. yatat djeumen are grateful to the french national institute for research for sustainable development (ird), the french agricultural research centre for international development (cirad), the french institute for research in computer science and automation (inria, epi masaie), the department of cooperation and cultural action (scac) of the french embassy in yaoundé, the international center for pure and applied mathematics (cimpa), the international laboratory for computer sciences and applied mathematics (lirima), the annual international conference on mathematical methods and models in biosciences (biomath 2016 & 2017), for their financial supports during the preparation of their ph.d theses. i.v. yatat djeumen also acknowledges the support of the sarchi chair in mathematical models and methods in bioengineering and biosciences (university of pretoria, south africa). references [1] l. abbadie, j. gignoux, x. le roux, and m. lepage. lamto: structure, functioning, and dynamics of a savanna ecosystem. springer, 2006. biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 23 of 29 http://dx.doi.org/10.11145/j.biomath.2018.12.167 i.v. yatat djeumen, a. tchuinté tamen, y. dumont, p. couteron, a tribute to the use of minimalistic ... 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(2010) [4] (here see model (3)). w1, − ε, yr−1 ct , yr−1 cg, yr−1 γt , yr−1 γg, yr−1 δt , yr−1 δg, yr−1 δf , − 345 20 30 10 2 180 0.04 2.8 0.35 table vi: parameter values used to get figure 7-(b), page 17. from tchuinté tamen et al. (2018) [88] (here see model (6)). cg, t.ha−1 ct , t.ha−1 bg, mm.yr−1 bt , mm.yr−1 ag, yr−1 at , yr−1 20 450 501 1192 0.0029 0.0045 dg, − dt , − γg, yr−1 γt , yr−1 δg, yr−1 δt , yr−1 14.73 106.7 2.5 1 0.01 0.1 λfg, − λminft , − λ max ft , − p, t −1ha α, t.ha−1 ηt g, ha.t−1yr−1 0.3 0.05 0.7 0.01 1 0.01 biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 page 29 of 29 http://dx.doi.org/10.11145/j.biomath.2018.12.167 introduction a brief review on space-implicit tree-grass interactions modelling tree-grass coexistence and possible alternative stable states lines of thoughts main published modelling options reaching sensible predictions from minimal models a seminal "big picture" at biogeographic scale fire frequency, grass biomass and fire impact tree survival relating to water resource impulsive time-periodic occurrences of fire events discussion and prospects conclusion references www.biomathforum.org/biomath/index.php/biomath original article covid-19 changing the face of the world. can sub-sahara africa cope? mandidayingeyi h machingauta1,2, bwalya lungu3, edward m lungu1 1 department of mathematics and statistical sciences botswana international university of science and technology palapye, botswana hellen.machingauta@studentmail.biust.ac.bw, lungue@biust.ac.bw 2department of surveying and geomatics midlands state university, gweru, zimbabwe pfupajenamh@staff.msu.ac.zw 3department of food science university of california,davis, ca 95616, usa blungu@ucdavis.edu received: 7 september 2020, accepted: 11 march 2021, published: 29 march 2021 abstract— we formulate a mathematical model for the spread of the coronavirus which incorporates adherence to disease prevention. the major results of this study are: first, we determined optimal infection coefficients such that high levels of coronavirus transmission are prevented. secondly, we have found that there exists several optimal pairs of removal rates, from the general population of asymptomatic and symptomatic infectives respectively that can protect hospital bed capacity and flatten the hospital admission curve. of the many optimal strategies, this study recommends the pair that yields the least number of coronavirus related deaths. the results for south africa, which is better placed than the other sub-sahara african countries, show that failure to address hygiene and adherence issues will preclude the existence of an optimal strategy and could result in a more severe epidemic than the italian covid-19 epidemic. relaxing lockdown measures to allow individuals to attend to vital needs such as food replenishment increases household and community infection rates and the severity of the overall infection. keywords-covid-19; hospital bed capacity; removal rates; optimal strategies i. introduction the coronavirus pandemic has disrupted global economies and health systems in unprecedented ways. as of 20 january 2021 there were 96 715 656 recorded cases with 2 068 062 deaths copyright: © 2021 machingauta et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: mandidayingeyi h machingauta, bwalya lungu, edward m lungu, covid-19 changing the face of the world. can sub-sahara africa cope?, biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 page 1 of 23 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2021.03.117 m h machingauta, b lungu, e m lungu, covid-19 changing the face of the world. can sub-sahara ... and 69 410 634 recoveries, [34]. the coronavirus currently infecting humans for the first time started in china in november 2019 in the city of wuhan. the virus is being referred to as sars-cov-2 to differentiate it from other coronaviruses, [33]. according to the centers for disease control (cdc) the virus displays symptoms within 2 − 14 days of exposure that may include fever, body aches, dry cough, fatigue, chills, headache, sputum production, myalgias, anorexia, sore throat, shortness of breath, loss of appetite and loss of smell. most of these symptoms are similar to the symptoms of diseases caused by other pathogens such as bacteria (e.g tuberculosis, cholera etc), viruses (e.g influenza, hiv, sars, mers) and parasites (e.g malaria). in its severe form covid-19 induces severe pneumonia and has led to high rates of death in about 1 − 5% of those infected [31]. however, some people infected with the virus may show little to no symptoms and are classified as asymptomatic carriers that are still able to transmit the virus to other individuals. these individuals play a major role in transmission as they are silent spreaders and it is challenging to determine the rate at which they spread the disease. there are currently no vaccines or treatments available for covid-19, but patients suffering from severe symptoms are usually hospitalized with median hospital stays of 10 to 13 days [9], [31], [37]. most of the countries experiencing high numbers of covid-19 cases are developed countries including the united states of america (usa), italy, spain, united kingdom (uk), france, germany, russia, brazil and china. most of these developed countries have excellent health facilities. however, the covid-19 epidemic in italy, spain, france and the usa has demonstrated that the current medical facilities were not designed to serve the populations during a pandemic. moreover, the democratic system of governance which we have all cherished has been seriously challenged during these times when some form of authoritarian governance was required in order to enforce prevention measures such as social distancing, masks, lockdown etc [6], [21], [27], to control virus transmission. the current western health systems performed well during the initial stages of the coronavirus disease progression but have since been challenged due to the acute rise in infection rates. various countries have had to make decisions over who is offered or not offered a bed in an intensive care unit (icu). the decisions were based on hospital bed capacity to avoid hospital overload and were dependent on early testing and isolating those who test positive as was the case in south korea, germany and china [16]. the epidemic in these developed countries has had very serious effects on the overall infrastructure and livelihoods of the people living there, and we believe that the effects on the sub-sahara africa region could very likely overwhelm this region where the economies are very weak and their icu capacity compared to population sizes averages only 9 beds per 10 000 inhabitants [32]. the current economic landscape will likely impact the ability of the various health systems in the long term to supply personal protective equipment (ppe) required by the frontline workers as they take care of infected patients. the main mode of transmission of the virus is through person to person transmission and this can happen through droplets, airborne transmission, surface transmission and the fecal oral route. the required measures to stop the spread of covid-19 include social distancing, face mask requirements for out of home activities, staying home if one feels sick or is nursing a cold. good hygiene practices such as frequent washing of hands or not touching the face area especially the mouth, nose and eyes, good sanitary practices that include frequent sanitation of surfaces. drastic measures have included mandatory shelter-ins or lockdowns. while these types of measures are not new and have been shown to work, they will likely present a different set of problems for sub-sahara africa. the problems sub-sahara africa will face will be compounded but not limited to the following: the low standards of good sanitary and hygiene measures may present an environment that biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 page 2 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 m h machingauta, b lungu, e m lungu, covid-19 changing the face of the world. can sub-sahara ... supports virus spread. good sanitary and hygiene practices have been encouraged and used in previous epidemics for thousands of years. the availability of clean water and poor hygiene (sanitary) will likely fuel the sub-sahara coronavirus epidemic as has been the case for other ongoing epidemics such as the ebola and cholera. the ebola [17] and cholera epidemics which remain unresolved will compete with the coronavirus for health facilities and workers. the current health facilities are already inadequate, for example the number of hospital beds 9 per 10 000 people [32], will prove inadequate based on the higher hospitalization rates reported in italy, france and spain for individuals infected with the coronavirus [24]. the average family in the poorest country in sub-sahara africa lives on less than $1 per day and most of what they earn goes towards food and other necessities. most families will not be able to afford the cost of soap and sanitizers to maintain the high levels of hygiene required to control or stop the spread of covid-19 [26]. as a result of inadequate water supplies and lack of sanitizers/soaps a large number of the population may not practice good sanitation and hygiene practices. in addition, public bathrooms if available, do not provide soaps and sanitizers as they are not considered an essential commodity. therefore, it will be challenging to adequately implement the frequent hygienic hand washing and sanitation of environmental surfaces. social distancing and isolation of suspected/confirmed covid-19 cases is essential to stop virus transmission and spread. however, the facilities required to effectively isolate infected individuals or individuals suspected to have been exposed to the virus are non-existent in sub-sahara africa. for example, most infected individuals or those exposed to the virus will not have the luxury of a separate bedroom with separate sanitary facilities in their homes. it is common to find 8 people living in a small two bed-roomed house, therefore it will be impossible to institute social distancing and isolation practices in the event of suspected exposure or infection. as part of social distancing, shelter-in or lockdown procedures, limitations have been placed on our movements. we can go to the grocery store to purchase food and other essential products. however, there are limits to how many grocery trips we can make and this means that families will have to buy food in bulk and store it at home. most of the food we eat such as fresh vegetables, fruits and meats are perishable and thus require refrigeration in order to extend their shelf life. most people in sub-sahara africa do not have refrigerators and even those who have refrigerators are subjected to frequent power interruption and can only buy enough supplies for a few days. this means that frequent visits to the shops and open markets is unavoidable. food supply, harvesting etc in most countries will be challenged as lockdowns are implemented to slowdown social interaction and consequently disease spread. we may start to see disruptions in the supply chain, a dip in demand for commodities, loss of income as a result of layoffs and cash flow issues. we have already observed the impact to the food and commodity distribution system. in the most impacted developed countries such as the usa, it has not been uncommon since early march to find empty shelves where commodities such as sanitizers, disinfectants, soaps, toilet paper, paper towels, rice and canned goods used to be found. as a result of the early panic, the demand of these goods exceeded the supply and shortages resulted. such scenarios would likely become the norm in sub-sahara africa. developed countries such as canada, australia, germany and the uk are paying their citizens to stay at home. the state of our economies in sub-sahara africa will not allow us to pay salaries and ensure every member of society has adequate supplies. people will have to calculate the risk of sure starvation if they obey the lockdown measures or risk going out and contracting covid-19 because they have to make a living. the world health organization (who) recently released guidance on the use of face masks, as a result countries are instituting rules requiring biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 page 3 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 m h machingauta, b lungu, e m lungu, covid-19 changing the face of the world. can sub-sahara ... citizens to wear face masks from cloth material. in sub-saharan countries, this has brought up questions of affordability of face masks and most of the population is waiting for government handouts of face masks. while it will be challenging to implement face mask wearing overall, it will be even more challenging to ensure face masks are washed prior to reuse and that people understand that face masks are not a 100% effective barrier. therefore, people need to be educated on how the face masks should be worn and removed to avoid virus transmission and/or cross contamination as cloth masks are being removed. in addition face masks need to be used with all the hygiene and sanitary practices to control this virus. in light of all this, education of the masses is critical in the control of covid-19 and needs to be a part of the complete covid-19 mitigation framework. african culture will play a significant role during this pandemic as is the case for ebola and cholera epidemics. for example, most relatives of infected individuals will be unlikely to comply fully with the directives of social distancing and isolation. adherence to precautionary measures in public places such as shops and open markets that are visited by people from different economic, educational and healthy backgrounds will be challenged (markets in nigeria, kenya, tanzania, burundi etc). behavioral norms are the most difficult to break. for example, young people are taught to cough into their fisted hands but now it is recommended to use disposable tissues that are a luxury to many or to cough into the elbow rather than hands [33]. the african culture is based on the use of hands, for example to greet, which makes it difficult to adhere to the new recommendation. though the washing of hands during ebola and cholera epidemics has been recommended, many people still walk out of the toilet without washing hands. it will be difficult for people during this pandemic to remember to wash their hands and wash them in the proper manner too [33]. ubuntu (you are, because i am) is one of the cornerstones of african culture where a sense of community is still being practiced to a large extent. social distancing becomes a challenge as africans are very social people. for example, if one is unable to till land on their own, one can invite people in the village to help with the tilling whilst they provide beer and food for the helpers. social gatherings such as funerals and weddings are held in high esteem. limiting the number of people who can attend to limit social interactions is considered a taboo. in a typical household people go to different jobs and engage in different activities throughout the day and the question that comes uppermost is can social distancing be implemented in such situations? sub-sahara african governments do not have the capacity to construct overflow hospital bed capacity. this study will address the following objectives: first, to determine the total number of individuals likely to be infected with covid-19. secondly, to find possible hospitalization strategies that would not overload hospital bed capacity and the number of infected individuals who would need to be safely isolated. finally, to find a strategy which yields the lowest number of deaths. ii. do people learn from previous epidemics: ebola to date africa has registered very few cases of covid-19 infections and deaths, even though the cases are rising. one wonders whether frequent exposure of the african population to epidemics has prepared it to prevention protocols. we give examples from ebola in the democratic republic of congo (drc) and the sudan, where prior exposure to the epidemic reduced the disease caseload. the data for various epidemics in the drc and the sudan presented in figures 1(a) and 1(b) shows a very high peak of total infections for the first epidemic and declining number of total infected during subsequent epidemics. the data suggests that the populations in the two countries have learnt how to manage an epidemic (ebola) which requires high standards of hygiene and social distancing. levy et al, [15] have shown that the severity of an outbreak is linked to the level biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 page 4 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 m h machingauta, b lungu, e m lungu, covid-19 changing the face of the world. can sub-sahara ... (a) 1976 1979 2004 year 0 50 100 150 200 250 300 n u m b e r o f e b o la c a s e s (b) 1976 1995 2007 2008 2012 2014 year 0 50 100 150 200 250 300 350 n u m b e r o f e b o la c a s e s fig. 1: ebola outbreaks plots for (a) sudan and (b) the democratic republic of congo. 0 5 10 15 20 25 30 35 40 45 50 week 0 20 40 60 80 100 120 140 n u m b e r o f n e w e b o la c a s e s (a) 0 5 10 15 20 25 30 35 40 45 50 week 0 10 20 30 40 50 60 70 80 90 100 n u m b e r o f n e w e b o la d e a th s (b) fig. 2: democratic republic of congo plots for (a) new ebola cases (b) new ebola deaths. of prior knowledge and education of the general population as well as preparedness of health care facilities. using data from the democratic republic of congo (drc) for new ebola cases and new ebola deaths, we notice a trend of rising to a peak and then declining, indicating two phases (i) naivety to the virus early in the infection and (ii) experience towards adherence to prevention protocols (figures 2(a) and 2(b)). ebola and covid-19 management are similar in many ways. they both rely on very high standards of hygiene, avoiding hand shaking and low density occupancy in residential homes. the difficulty one encounters when modeling ebola or covid-19 is modeling the infections arising from poor hygiene, social distancing etc. from figures 1 and 2, we suggest modeling infections from poor hygiene and lack of social distancing by a function fitted to the data and depicted in figure 3. this function represents high infections due to poor understanding of the prevention measures early into the epidemic, but as the infection progresses individuals who apply the knowledge from previous epidemics adapt to avoid infections as was the case for ebola. the function in figure 3 mimics the process observed for ebola and is adopted in this study. iii. quality of the lockdown since sars-2 is a respiratory disease, we want to incorporate the effects of household transmission due to an infected individual i, in household biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 page 5 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 m h machingauta, b lungu, e m lungu, covid-19 changing the face of the world. can sub-sahara ... 0 200 400 600 800 1000 1200 time (days) 0 5 10 15 20 25 30 35 40 45 50 e n v ir o n m e n t, ( 1 = 0 .6 5 , 2 = 0 .3 3 ) fig. 3: time course for the environment h at time t during the lockdown. the rate of exposure from a member of the family j to i is given by: νh = ηg × ψh × ψi,inf (1) where ψh depends on household size and ψi,inf represents asymptomatic or symptomatic status of the infection. an individual in the household h is also exposed to community transmission because a member of the household has to leave the house to transact with the community, for example to buy food or to go to work (if they are front-line workers) during the lockdown. this movement is inevitable in poverty stricken africa where food insecurity is common and food must be sourced daily. the rate of community exposure is given by: ch = εg × fg(t) × ψi,age age group, (2) where εg is the baseline exposure from the community, fg is a time dependent curve that modifies the community rate of exposure over time and ψi,age accounts for disease susceptibility depending on age. the rate of exposure of individual i in household h in which a member goes out to transact with the community is given by: λ = si,g(t)  mi,h(t) ∑ j 6=i νh + ch   (3) (3) has been used to moderate transmission of respiratory synctial virus which is similar to sars2, [13]. the approach described in (1), (2) and (3) is described in detail in [13]. we have used the data from [13] to incorporate the quality of the lockdown and to explain why the number of covid-19 cases exploded after the lockdown. iv. model description we develop a simple model in the context of the sub-sahara africa environment which consists of a class of individuals, s, who are susceptible to the disease, a class of individuals, e, who have been exposed to the disease, a class of asymptomatic individuals, ias. these are individuals who are not showing symptoms but are transmitting the virus. a class of symptomatic infectives, is, a class of individuals who require hospitalization or to be isolated, h, and a class of recovered individuals, r. we note that ias and is can infect s directly and indirectly by contaminating the environment φ. for simplicity of notation, let x(t) = (x1(t),x2(t),x3(t),x4(t),x5(t),x6(t)), = (s(t),e(t),ias(t),is(t),h(t),r(t)), x7(t) = φ(t), n(t) =x1(t)+x2(t)+x3(t)+x4(t)+x5(t)+x6(t). we consider the following model: dx1 dt = π − (β1x3 + β2x4)x1 n −β3x7x1 −µx1, (4) dx2 dt = (β1x3 + β2x4)x1 n + β3x7x1 −(α1 + µ)x2, (5) dx3 dt = α1x2 − (γ1 + α2 + µ + κ1)x3, (6) biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 page 6 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 m h machingauta, b lungu, e m lungu, covid-19 changing the face of the world. can sub-sahara ... dx4 dt = α2x3 − (γ2 + δ1 + κ2)x4, (7) dx5 dt = γ1x3 + γ2x4 − (δ2 + κ3)x5, (8) dx6 dt = κ1x3 + κ2x4 + κ3x5 −µx6, (9) where (π, µ) =   (0, 0) during lockdown 6= (0, 0) no lockdown, inter-zonal movement allowed. (10) for short lockdown periods one can assume that µ = 0. this assumption is not valid for long lockdowns. the quality of the environment is described by the equation: dx7 dt = α3(x3 + x4) n −µ1x7. (11) the parameter α3 measures the rate of contaminating the environment, that is, the rate at which the amount of pathogens are released into the environment by both asymptomatic and symptomatic infectives. equation (11) is also supported by berge et al. [2] for their model on ebola. the model flow diagram is given below: fig. 4: model flow diagram where θ = (β1x3+β2x4)x1 n + β3x7x1. equation (4) describes the rate of change in the susceptible population, x1. the first term in (4) represents recruitment of people into the susceptible class through movement of people from different zones when this is allowed. this term is zero during a lockdown but may be nonzero if inter-zone movements are allowed. the second term represents loss due to infection of susceptible individuals by asymptomatic infectives, x3, at the rate β1 and symptomatic infectives, x4, at the rate β2. the third term represents indirect infections due to an unclean environment at the rate β3. as time has progressed, it has become necessary to account for loss due to other causes such as natural death at the rate µ. there has been confusion over the number of covid deaths and it has become pathologically necessary to ascertain that a covid infected individual actually died of covid complications, [22], [30]. however, µ was ignored during early stages of the pandemic and every covid infected individual was assumed to have died of covid. equation (5) describes the rate of change of the exposed class, x2. the first two terms represent gain from infection of susceptible individuals and the third term represents loss due to sero-conversion to the asymptomatic state at the rate α1 and natural death at a constant rate µ, respectively. the same comment regarding µ in equation (4) applies in this case. equation (6) describes the rate of change of the asymptomatic infected class, x3. the first term represents gain from exposed individuals who are converting to sero-positive status without exhibiting symptoms. the second term represents loss due to conversion to symptomatic state and hospitalization at the rates α2 and γ1, respectively. κ1 represents losses from this class due to recovery and µ represents natural death as explained in (10). the first term on the right hand side of equation (7) represents gain from the asymptomatic state. the second term represents loss due to hospitalization or isolation and a blanket term representing loss due to both natural death and disease induced death at the rates γ2 and δ1 = (µ + δx4 ), respectively. loss from this class due to recovery is assumed to occur at a constant rate κ2. equation (8) describes the rate of change of the hospitalized and isolated class. the first two terms represent gain from testing and contact tracing of biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 page 7 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 m h machingauta, b lungu, e m lungu, covid-19 changing the face of the world. can sub-sahara ... both asymptomatic and symptomatic individuals. the third term represents loss due to blanket death at the rate δ2 = (µ + δx5 ) and recovery at the rate κ3. equation (9) describes the rate of change of the recovered individuals. the first three terms represent gain from recovery of asymptomatic, symptomatic and hospitalized cases. the recovered class loses people through natural death at a rate µ for a long pandemic. equation (11) describes the rate of contaminating the environment due to the release of pathogens into the environment by symptomatic and asymptomatic infectives at the rate α3. the second term represents cleaning of the environment (naturally or due to interventions) at the rate µ1. v. model analysis a. positivity of solutions denote by <6+ the points x(t) = (x1(t), x2(t), x3(t), x4(t), x5(t), x6(t)) in <6 with positive coordinates and consider the system (4) − (9) with initial values x0 = (x01, x 0 2, x 0 3, x 0 4, x 0 5, x 0 6) ∈ < 6 +. we can state the following lemma: lemma 1. if x0i ≥ 0, i = 1, ..., 6 then xi(t) ≥ 0 for t > 0, i = 1, ..., 6. proof: first, we want to show that 0 ≤ x7(t) ≤ α3µ1 where the lower bound signifies a clean environment and the upper bound signifies a covid-19 contaminated environment. from (11), we have dx7 dt = α3(x3 + x4) n −µ1x7 ≤ α3 −µ1x7. the solution for a totally contaminated covid-19 environment is given by x7(t) ≤ x7(0) e−µ1 t + α3 µ1 (1 −e−µ1 t). as t → ∞, x7(t) ≤ α3 µ1 . for a clean environment, (11) becomes dx7 dt = α3 (x3 + x4) n −µ1x7 ≥ −µ1x7 x7(t) = x0 e −µ1t ≥ 0. hence, we have 0 ≤ x7(t) ≤ α3 µ1 . from equation (4), we have dx1 dt = π − (β1x3 + β2x4)x1 n −β3x7x1 −µx1 ≥ π − ( β1 + β2 + β3 α3 µ1 ) x1 ≥ −wx1, w = ( β1 + β2 + β3 α3 µ1 ) . the solution is x1(t) ≥ x01 e −wt ≥ 0, ∀t ≥ 0. similarly, we can show that for i = 2, ..., 6 xi(t) ≥ 0, i = 2, ..., 6. this completes the proof. b. invariance the total population, n(t), at time t is given by dn dt = π −µn −δx4x4 −δx5x5 ≤ π −µn. by gronwall inequality, it is easy to show that 0 ≤ n(t) ≤ π µ . for the existence of a unique bounded solution, we infer that any solution of the system (4) − (11) is non negative and bounded in ω ={ (x1,x2,x3,x4,x5,x6) ∈<6+ : n ≤ π µ , x7 ≤ α3µ1 } . biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 page 8 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 m h machingauta, b lungu, e m lungu, covid-19 changing the face of the world. can sub-sahara ... c. disease free equilibrium (dfe) 1) disease free equilibrium for π = 0: during a strict lockdown, we can take π = 0. the system (4) − (11) has two equilibrium points, the disease eradication point ζ0 = (0, 0, 0, 0, 0, 0, 0) and the disease endemic point ζ1 = (x ∗ 1,x ∗ 2,x ∗ 3,x ∗ 4,x ∗ 5,x ∗ 6,x ∗ 7). to establish the stability of ζ0, we use the hartmann-grobmann theorem which roughly states that near an equilibrium point, the dynamics of the original (nonlinear) system are the same as those for the linearized system. the linearized system for (4) − (11) is given by: dy(t) dt = ay(t), where y(t) = (x1(t),x2(t),x3(t),x4(t),x5(t),x6(t),x7(t)) t, a =   0 0 0 0 0 0 0 0 −α1 0 0 0 0 0 0 α1 −b11 0 0 0 0 0 α2 −c11 0 0 0 0 0 γ1 γ2 −d11 0 0 0 0 κ1 κ2 κ3 0 0 0 0 α3 α3 0 0 −µ1   and b11 = (γ1 + α2 + κ1) c11 = (γ2 + κ2 + δ1) d11 = (δ2 + κ3) the eigenvalues of |a−λi| are given by (0, 0,−α1,−b11,−c11,−d11,−µ1). the reproduction number r0, is given by the largest spectral radius of the matrix |a−λi| . in this instance the possible values of r0 are max{α1, b11, c11, d11, µ1}. since λi < 0, i = 1, 2, ..., 7, the system (4)−(11) is stable and tends to ζ0 as t →∞. provided the population maintains the covid control protocols and keeps r0 below 1, the disease will fail to establish in the population. 2) disease free equilibrium for π 6= 0: when inter-zone movements are allowed, that is, π 6= 0, the system (4)−(11) has a disease free equilibrium point ζ0 = (πµ, 0, 0, 0, 0, 0, 0). we use the technique by van den driessche and watmough [29] to find the model reproduction number. the matrix for new infections is given by f whilst the matrix for other transitions is given by v, where f =   (β1 x3 +β2 x4)x1 n + β3 x7 x1 0 0 0   , v =   ax2 bx3 −α1 x2 cx4 −α2 x3 µ1 x7 − α3 (x3+x4) n   and a = (α + µ) b = (γ1 + α2 + κ1 + µ) c = (γ2 + κ2 + δ1). the jacobean matrices f of f and v of v are computed at the point ζ0 with respect to the infected classes (x2, x3, x4, x7). thus, the basic reproduction number r0, given by the spectral radius of the matrix fv−1 is the maximum of the moduli of the eigenvalues of that matrix fv−1 given by, r0 = rx3 + rx4 + rx7, where, rx3 = α1 β1 ab rx4 = α1 α2 β2 abc rx7 = α1 α3 β3 (c + α2) abcµ1 . based on r0 we can state the following theorem: theorem 2. the dfe point, ζ0, is locally asymptotically stable if r0 < 1 and unstable if r0 > 1. biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 page 9 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 m h machingauta, b lungu, e m lungu, covid-19 changing the face of the world. can sub-sahara ... remark 1: r0 can be greater than 1 even if rx3 < 1, rx4 < 1, rx7 < 1. it is necessary for each sub-reproduction number rx3 , rx4 and rx7 to be less than 1 and their sum to be less than 1 for the disease to fail to establish in the population. remark 2: unlike the case π = 0, now the adherence to the covid protocols by each infected subpopulation, xi, i = 3, 4, is more strict (adherence to social distancing, mask wearing etc) and the hygiene measures must be more strictly observed for the stability of ζ0 to be achieved. 3) global stability of the disease free equilibrium: theorem 3. the dfe point, ζ0, of system (4) − (11) is globally asymptotically stable for r0 < 1. proof. to prove theorem 3, we use kamgangsallet stability theorem, [10]. let z = (z1, x2) with z1 = (x1, x6) ∈ <2 and z2 = (x2, x3, x4, x5, x7) ∈ <5. in terms of z1 and z2 system (4) − (11) can be written as: ż1 = a1 (z)(z1 −z∗1 ) + a12(z) z2 ż2 = a2(z) (z2) where z∗1 = ( π µ , 0), with a1(z) = [ −µ 0 0 −µ ] , a12(z) = [ 0 −β1x1 n −β2x1 n 0 −β3x1 0 κ1 κ2 κ3 0 ] and a2(z) =   −a β1x1 n β2x1 n 0 β3x1 α1 −b 0 0 0 0 α2 −c 0 0 0 γ1 γ2 −d 0 0 α3 n α3 n 0 −µ1   , where d = (δ2+κ3). we want to show that the five sufficient conditions of kamgang-sallet theorem in [10] are satisfied as follows: (i) the system (4)−(11) is a dynamical system on ω, as defined and shown in section v . (ii) the eigenvalues of a1(z) are real and negative, thus the system ż1 = a1(z) (z1−z∗1 )+ a12(z) z2 is globally asymptotically stable at the equilibrium z∗1 . (iii) the matrix a2(z) is a metzler matrix, i.e. a matrix such that off diagonal terms are non negative and is irreducible for any given z ∈ ω. (iv) there exists a matrix ā2, which is an upper bound for the set m = a2(z) : z ∈ ω. indeed, ā2 =   −a β1 β2 0 β3x∗1 α1 −b 0 0 0 0 α2 −c 0 0 0 γ1 γ2 −d 0 0 α3 n∗ α3 n∗ 0 −µ1   is an upper bound for m. (v) for r0 < 1 , λ is the eigenvalue of ā2, α(ā2) = max{re(λ) : λ}≤ 0 to check condition (v) we will use the following lemma which is a characterization of metzler stable matrices: lemma 4. let m be a square matrix written in block form d = [ a b c d ] with a and d being square matrices. m is metzler stable if and only if matrices a and d −ca−1b are metzler stable. using lemma 4, matrix ā2 can be expressed in the form of matrix m with: a =  −a β1 β2α1 −b 0 0 α2 −c   , b =  0 β3x∗10 0 0 0   c = [ 0 γ1 γ2 0 α3 n∗ α3 n∗ ] , d = [ −d 0 0 −µ1 ] . clearly a, is a stable metzler matrix and after computations we obtain d − ca−1b is a stable biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 page 10 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 m h machingauta, b lungu, e m lungu, covid-19 changing the face of the world. can sub-sahara ... metzler matrix if and only if rmax0 = α1 β1 ab + α1 α2 β2 abc + α1 α3 β3 (c + α2) abcµ1 ≤ 1. d. endemic equilibrium point (eep) solving for the system of equations (4) − (11) by equating the rhs to zero, we find the coordinates of the eep given by ζ1 = (x∗1, x ∗ 2, x ∗ 3, x ∗ 4, x ∗ 5, x ∗ 6, x ∗ 7), where x∗1 = abcn∗µ1 α1 a33 (12) x∗2 = a22 (r0 − 1) aα1 a33 (13) x∗3 = a22 (r0 − 1) aba33 (14) x∗4 = α2 a22 (r0 − 1) abca33 (15) x∗5 = a22 a44 (r0 − 1) abcda33 (16) x∗6 = κ3 a22 a44 a55 (r0 − 1) abcdµa33 (17) x∗7 = α3(α2 + c)a22(r0 − 1) abcn∗µ1 a33 (18) and a22 = abcn ∗µµ1 a33 = (c + α2) α3 β3 + (cβ1 + α2 β2) µ1 a44 = (α2 γ2 + cγ1) a55 = (cdκ1 + dκ2 α2). the coordinates (13) − (18) exist if and only if r0 > 1. 1) global stability of the endemic equilibrium point: the global stability of the eep is explored by proving the following theorem: theorem 5. if r0 > 1 then the eep given by ζ1 is globally asymptotically stable in the region ω. proof. following the work of [5], we construct a lyapunov function l of the type: l(xi) = 7∑ i=1 ( xi −x∗i −x ∗ i ln xi x∗i ) . differentiating l with respect to xi gives: dl dt = 7∑ i=1 ( xi −x∗i xi ) dxi dt . substituting for dxi dt , i = 1, ..., 7, we get dl dt = ( x1 −x∗1 x1 ) [ π − (β1x3 + β2x4)x1 n −β3x7x1 −µx1 ] + ( x2 −x∗2 x2 ) [ (β1x3 + β2x4)x1 n + β3x7x1 −ax2 ] + ( x3 −x∗3 x3 ) [α1x2 − bx3] + ( x4 −x∗4 x4 ) [α2x3 − cx4] + ( x5 −x∗5 x5 ) [γ1x3 + γ2x4 −dx5] + ( x6 −x∗6 x6 ) [κ1x3 + κ2x4 + κ3x5 −µx6] + ( x7 −x∗7 x7 )[ α3(x3 + x4) n −µ1x7 ] = a33 −a22, biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 page 11 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 m h machingauta, b lungu, e m lungu, covid-19 changing the face of the world. can sub-sahara ... where a22 = ( x1−x∗1 x1 )2[ (β1x3 +β2x4) n +β3x7 +µ ] + [ x∗1 x1 π+ (β1x3 +β2x4)x ∗ 1 n +β3x7x ∗ 1 +µx ∗ 1 ] + ( x2 −x∗2 x2 )2 a + ax∗2 + x∗2 x2 [ β1x3 + β2x4)x1 n + β3x7x1 ] + ( x3 −x∗3 x3 )2 b + bx∗3 + x∗3 x3 α1x2 + ( x4 −x∗4 x4 )2 c + cx∗4 + x∗4 x4 α2x3 + ( x5−x∗5 x5 )2 d+dx∗5 + x∗5 x5 (γ1x3 +γ2x4) + ( x6 −x∗6 x6 )2 µ + µx∗6 + x∗6 x6 (κ1x3 + κ2x4 + κ3x5) + ( x7−x∗7 x7 )2 µ1 +µ1x ∗ 7 + x∗7 x7 α3(x3 +x4) n and a33 = π + x∗ 2 1 x1 [ (β1x3 + β2x4) n + β3x7 + µ ] + [ (β1x3 + β2x4) n + β3x7 + µ ] + x∗ 2 2 x2 a + α1x2 + x∗ 2 3 x3 b + α2x3 + x∗ 2 4 x4 c +γ1x3 + γ2x4 + x∗ 2 5 x5 d + κ1x3 + κ2x4 +κ3x5 + x∗ 2 6 x6 µ+ α3(x3 + x4) n + x∗ 2 7 x7 µ1. since all the parameters used in the system (4) − (11) are non negative we have dl dt ≤ 0 for a33 ≤ a22 and dldt = 0 if and only if x1 = x ∗ 1,x2 = x∗2,x3 = x ∗ 3,x4 = x ∗ 4,x5 = x ∗ 5,x6 = x ∗ 6,x7 = x ∗ 7. thus by la-salle’s invariance principle, the eep is globally asymptotically stable. table i: numerical values for the parameters of the italian case parameter value/range source β1 0.492 [8] β2 1.30 [20] β3 0 estimate δ1 0.015 [19] δ2 4827 53578 [33] α1 1 3.21 [20] α2 1 2.27 [20] α3 0 estimate γ1 [0, 0.6] estimate γ2 [0, 0.6] estimate κ1 1 6 [8] κ2 1 10 [8] κ3 1 14 [8] µ 0.00003032 [35] µ1 0 estimate π µ× 400 000 [11] vi. numerical simulations we want to find optimal isolation rates γ1 and γ2, using matlab programs, under which the number of infected individuals will not overwhelm the hospital capacity, h. we performed numerical simulations using data from the italian coronavirus epidemic from 31st january to the 15th of may 2020 to demonstrate that our model can accurately reproduce the recorded data on numbers of infected and dead individuals. the parameter values used for numerical simulations are given in table i. 1) sensitivity analysis: five parameters, β3, α3, γ1, γ2 and µ1 in table i have been estimated based on the number of hospitalized cases in the lombardy region of northern italy. we have analyzed how sensitive the reproduction number is to the changes in these five parameters (figure biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 page 12 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 m h machingauta, b lungu, e m lungu, covid-19 changing the face of the world. can sub-sahara ... prcc p a r a m e t e r fig. 5: a diagram showing the sensitivity of r0 to various model parameters. 5). the parameters α3 and β3 are important and provide insight into understanding how the subsahara africa situation will differ from the italian or in general the situation in western countries, where we assumed α3 = 0. a. italy lombardy is a region with a population of approximately 10 million. by an iterative procedure, since we know the number of people infected with the coronavirus from 31st january to 22nd march 2020, we have estimated that the number of people susceptible to infection by the virus, through their failure to observe prevention measures such as self quarantine, social distancing etc at the start of the coronavirus epidemic, was about 400 000. as at 31st january 2020, the number who were potentially exposed to the infection is estimated to be around 200 individuals. on the 31st january 2020, italy recorded its first 3 cases of individuals infected with the coronavirus. based on this initial data, we present examples of strategies by fixing the rate of isolating symptomatic infectives and then finding the corresponding rate of isolating asymptomatic infectives which ensures that the combined number of infectives does not exceed the hospital bed capacity and vice versa. the examples discussed here are not unique but are typical of other scenarios and provide insight into the following: (i) how the infection curve can be flattened to ensure that the hospital bed capacity is not exceeded. (ii) for non optimal cases, where hospital bed capacity is exceeded, to quantify the number of infectives for each non optimal case which must be safely isolated outside hospital facilities. according to [23], italy has 12.5 beds per 100 000 individuals of intensive care unit (icu) or critical care beds (ccb) beds and 3.18 beds per 1 000 individuals ordinary hospital beds. for lombardy, this data (combining both ordinary and icu hospital beds) gives 33 182 beds. figure 6 presents an example for a fixed rate γ2 = 0.33 of isolating symptomatic infectives. we find that the optimal rate of securing asymptomatic infectives when there is a lockdown, π = 0, should be γ1 = 0.46. converting these rates to time, we see that asymptomatic infectives should be identified, hospitalized in a time almost 1.5 times faster than the time of hospitalizing symptomatic infectives. currently, every nation is experiencing a lack of materials and equipment to conduct tests. this rate of hospitalizing asymptomatic infectives would be difficult to achieve as there has been a shortage of testing materials, and most governments have decided they would only test individuals who present with covid-19 symptoms. moreover, the time it is taking to obtain test results 2−3 days is slowing down the testing significantly. for this strategy, we have determined the number of infected people that should be safely secured for varying values of γ1 at peak infection. for γ1 = 0, the number in excess of hospital bed capacity that must be secured is 22 408, for γ1 = 0.3, the number in excess of hospital bed capacity that must be secured is 13 407. for the non optimal cases, if the hospital overflow bed capacity is increased by 50% (as was done in most western countries) the only case that would have accommobiomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 page 13 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 m h machingauta, b lungu, e m lungu, covid-19 changing the face of the world. can sub-sahara ... 0 50 100 150 200 250 time (days) 0 1 2 3 4 5 6 h o s p it a li s e d , ( 2 = 0 .3 3 ) 10 4 hospital bed capacity=33182 1 =0, ~=0 1 =0.3 1 =0.49 1 =0, =0 1 =0.3 1 =0.46 data1 fig. 6: population level plots for hospitalized individuals for varying values of γ1 dated the overflow is (γ1,γ2) = (0.3, 0.33). figure 6 shows that when inter zonal movement is allowed, π 6= 0, the number of people needing hospitalization or to be safely isolated increases. the optimum removal rates with inter-zonal movement are given by (γ∗1,γ ∗ 2 ) = (0.49, 0.33), implying that asymptomatic infectives must be isolated in a time 1.6 times faster than when there is no inter zonal movement. the optimal solutions in figure 6 show that the peak hospitalization capacity occurs earlier when there is a lockdown (π = 0) than when there is inter-zonal movement (π 6= 0). figure 7 presents a strategy where the rate of hospitalizing symptomatic infectives is fixed at γ2 = 0.6. the optimal rate for γ1 which flattens the hospital admission curve is found to be γ1 = 0.16. in other words, almost one third of the effort must be devoted to testing, contact tracing and hospitalizing asymptomatic infectives. for this strategy the situation regarding the non optimal cases is as follows: for γ1 = 0 the number needed to be secured is 10 097, which is too high to accommodate. using optimization techniques to find γ∗1 and γ ∗ 2 we have found and demonstrated in figure 8 that the optimum rates are (γ∗1,γ ∗ 2 ) = (0.6, 0.26). this optimum pair implies that the effort devoted to 0 50 100 150 200 250 time (days) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 h o s p it a li s e d , ( 2 = 0 .6 ) 10 4 hospital bed capacity=33182 1 =0 1 =0.16 data1 fig. 7: population level plots for hospitalized individuals for varying values of γ1 0 50 100 150 200 250 time (days) 0 1 2 3 4 5 6 h o s p it a li s e d , ( 1 = 0 .6 ) 10 4 hospital bed capacity=33182 2 =0.1 2 =0.26 data1 fig. 8: population level plots for hospitalized individuals for varying values of γ2 testing, contact tracing and isolating symptomatic infectives must be 2.3 times higher than that of isolating asymptomatic infectives. figure 9 shows that for γ1 = 0.3 the optimum pair is (γ∗1,γ ∗ 2 ) = (0.3, 0.45). this strategy targets to remove symptomatic infectives faster than the asymptomatic infectives. this strategy removes symptomatic infectives in a time at least one and half times faster than the time of isolating biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 page 14 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 m h machingauta, b lungu, e m lungu, covid-19 changing the face of the world. can sub-sahara ... table ii: comparison of deaths at different optimal cases strategy γ1 γ2 30 days 60 days 90 days 105 days 105 days δ = 0.01 δ = 0.03 δ = 0.02 δ = 0.015 δ = 0.09 lockdown no lockdown 1 0.49 0.33 210 10 126 32 108 31 433 120 200 (29) (11 591) (27 682) (31 368) 2 0.16 0.6 226 12 861 33 994 31 202 115 690 (29) (11 591) (27 682) (31 368) 3 0.6 0.26 200 8 989 30 734 31 162 120 640 (29) (11 591) (27 682) (31 368) 4 0.3 0.45 217 11 179 32 771 31 135 117 600 (29) (11 591) (27 682) (31 368) 0 50 100 150 200 250 time (days) 0 1 2 3 4 5 6 h o s p it a li s e d , ( 1 = 0 .3 ) 10 4 hospital bed capacity=33182 2 =0.1 2 =0.3 2 =0.45 data1 fig. 9: population level plots for hospitalized individuals for varying values of γ2 asymptomatic infectives. to illustrate the impact of non adherence to prevention measures, such as social distancing, not wearing masks etc, on the optimal case given in figure 9, we considered how the case β3 6= 0 for italy would have altered the conclusions presented in figures 6 to 9. figure 10 shows that if we vary the parameter α3 the hospital bed capacity for lombardy in italy would have been exceeded for any α3 ≥ 0.01. this suggests that high standards of hygiene are key to controlling covid-19 infections. we can see from the examples of the four strategies illustrated in figures 6, 7, 8, 9 that there is no unique way of flattening the curve in order to 0 50 100 150 200 250 time (days) 0 2 4 6 8 10 12 14 h o s p it a li s e d , ( 1 = 0 .3 , 2 = 0 .4 5 ) 10 4 hospital bed capacity=33182 3 =0 3 =0.05 3 =0.1 data1 fig. 10: population level plots for hospitalized individuals for varying values of α3 protect the hospital bed capacity. the question we address is how does one choose the best strategy among the optimal strategies? table ii gives the number of deaths resulting from the four strategies above and compares each strategy with the actual number of recorded deaths. we conclude that the best strategy is one which reduces the number of deaths. in this case (table ii) any strategy that isolates the symptomatic infectives faster is preferable as it is more economical. b. optimal control 1) optimal control without incorporating household and community exposure: to study measures that reduce disease transmission, biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 page 15 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 m h machingauta, b lungu, e m lungu, covid-19 changing the face of the world. can sub-sahara ... such as lockdown or isolation of infectives, we introduced two controls, u1 and u2, on the infection coefficients β1 and β2 as shown below: dx1 dt = π−(1−u1) β1x3x1 n −(1−u2) β2x4x1 n −β3x7x1 −µx1 (19) dx2 dt = (1 −u1) β1x3x1 n + (1 −u2) β2x4x1 n +β3x7x1 − (α1 + µ)x2 (20) dx3 dt = α1x2 − (γ1 + α2 + µ + κ1)x3 (21) dx4 dt = α2x3 − (γ2 + δ1 + κ2)x4 (22) dx5 dt = γ1x3 + γ2x4 − (δ2 + κ3)x5 (23) dx6 dt = κ1x3 + κ2x4 + κ3x5 −µx6 (24) and dx7 dt = α3(x3 + x4) n −µ1x7 (25) we want to minimize the objective functional given by j(u1,u2) = ∫ tt 0 a1x2 + a2x3 + a3x4 + 1 2 a4u 2 1 + 1 2 a5u 2 2. (26) the goal is to find a set of controls that minimize the number of susceptible individuals who come into contact with infected individuals. let u∗1, u ∗ 2 be the optimal controls. the problem is to find j(u∗1,u ∗ 2) = minj(u1,u2) , (u1,u2) ∈ u, (27) subject to system (19) − (25), where u is the set of measure functions defined from [0, tt ] to [0, 1]. the optimality conditions are given by, u∗1 = min { max [ 0,d1 ( β1x3x1 a4n )] ,u1max } , u∗2 = min { max [ 0,d1 ( β2x4x1 a5n )] ,v2max } , where d1 = (λx2 −λx1 ). calculation for u∗1 and u ∗ 2 is based on pontryagin’s maximum principle (see [14]) for a detailed description. 2) optimal control incorporating household and community exposure.: to study the impact that household and community exposure has on disease transmission control measures, we modify equations (19) − (25) by adding an additional term λ given in (3) which captures household and community exposure of an individual. the modified equations are given by: dx1 dt = π − (1 −u1 + λ) β1x3x1 n −(1 −u2 + λ) β2x4x1 n −β3x7x1 −µx1 (28) dx2 dt = (1 −u1 + λ) β1x3x1 n +(1 −u2 + λ) β2x4x1 n + β3x7x1 −(α1 + µ)x2 (29) dx3 dt = α1x2 − (γ1 + α2 + µ + κ1)x3 (30) dx4 dt = α2x3 − (γ2 + δ1 + κ2)x4 (31) dx5 dt = γ1x3 + γ2x4 − (δ2 + κ3)x5 (32) dx6 dt = κ1x3 + κ2x4 + κ3x5 −µx6 (33) and dx7 dt = α3(x3 + x4) n −µ1x7 (34) where λ is given by (3). the optimality conditions that we want to satisfy are the same as those for equations (19)−(25). we consider the non optimal strategy in figure 6 using equations (19) − (25) and equations (28) − (34) for γ1 = 0.3 and γ2 = 0.33 where u1 = 0 and u2 = 0. if we use the controls u∗1 ≥ 0.2, u ∗ 2 ≥ 0.2 and (γ1, γ2) = (0.3, 0.33) we obtain figures 11(a) − 11(e). it is clear from figure 11(e) that household and community disease transmission would increase the number hospitalized though the numbers would still be below the hospital bed capacity for optimal values of γ1 and γ2. the number of susceptible individuals (figure 11(a)) would decline due to increased infection rates. biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 page 16 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 m h machingauta, b lungu, e m lungu, covid-19 changing the face of the world. can sub-sahara ... 0 50 100 150 time (days) 0.5 1 1.5 2 2.5 3 3.5 4 s u sc e p ti b le s 10 5 (a) without controls with controls including household and community exposure with controls excluding household and community exposure 0 50 100 150 time (days) 0 0.5 1 1.5 2 2.5 3 3.5 4 e x p o se d 10 4 (b) without controls with controls including household and community exposure with controls excluding household and community exposure 0 50 100 150 time (days) 0 2000 4000 6000 8000 10000 12000 14000 a sy m p to m a ti c i n fe c ti v e s (c) without controls with controls including household and community exposure with controls excluding household and community exposure 0 50 100 150 time (days) 0 2000 4000 6000 8000 10000 12000 s y m p to m a ti c i n fe c ti v e s (d) without controls with controls including household and community exposure with controls excluding household and community exposure 0 50 100 150 time (days) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 h o sp it a li z e d 10 4 (e) hospital bed capacity=33182 without controls with controls including household and community exposure with controls excluding household and community exposure data1 fig. 11: a comparison of state variables with and without controls c. sub-sahara africa: south africa as an example. to provide insight into the sub-sahara africa outlook, simulations were done using data from south africa, one of the countries in this region with the highest number of coronavirus cases and with the best medical facilities. it is our view that if south africa cannot cope then most, if not all, countries in sub-sahara africa would not cope. on the 4th of march 2020, south africa recorded its first case of the coronavirus. as of the 1st of april 2020 the highest case counts of coronavirus had been reported in 3 provinces namely gauteng, kwazulu natal and the western cape. the 3 provinces had 1 380 reported coronavirus cases distributed as follows: 645 in gauteng, 326 in kwazulu natal and 186 in the western cape. these 3 provinces have a total population of 33 309 473 people. the number of people infected with the coronavirus from the 4th of march to the 1st april 2020 is known, we have estimated that the number of people susceptible to infection by biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 page 17 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 m h machingauta, b lungu, e m lungu, covid-19 changing the face of the world. can sub-sahara ... table iii: numerical values for the parameters of the south african case parameter value/range source β1 0.492 [8] β2 1.30 [20] β3 0.001006082 [11] δ1 0.015 [19] δ2 5 1380 [33] α1 1 3.21 [20] α2 1 2.27 [20] α3 5.3346 × 10−6 estimate γ1 [0, 0.77] estimate γ2 [0, 0.33] estimate κ1 1 6 [8] κ2 1 10 [8] κ3 1 14 [8] µ 0.00004290 [36] µ1 0.00274 [11] π µ × 1 200 000 [11] the virus through failure to self quarantine, self isolate, observe social distancing etc was about 1 200 000. at that time the number who were potentially exposed to the infection is estimated to be about 1 000 individuals. except for β3, π, α3, µ, µ1 and δ2, we use the parameters in table 1 for italy. this is justified on the basis that italian family bonds are similar to sub-sahara africa. the values used for this simulation are given in table 3. according to [4], gauteng province, kwazulu natal province and the western cape province combined have a hospital bed capacity of 62 787. this number includes both icu and ordinary hospital beds. figure 12 compares real time epidemic curves for italy, spain, the united kingdom and south africa for the first 40 days of the epidemic. each country implemented the lockdown strategy at different stages of infection. italy introduced the lockdown on the 9th of march 2020 when the total number of individuals infected with covid19 was 9 172. spain introduced the lockdown on the 15th of march 2020 when the total number of those infected was 7 798. the united kingdom introduced the lockdown on the 23rd of march 2020 when the number of individuals infected was 6 650. south africa introduced the lockdown on the 27th of march 2020 when the number infected was only 1 170. figure 12 shows different infection trends for the four countries. it is obvious that south africa which introduced the lockdown early enough displays an epidemic which rises at a gentle rate. we consider the problem of finding the hospitalization rates γ1 and γ2 for which the hospital bed capacity of 62 787 would suffice. fig. 12: total covid cases, uk, spain, italy, south africa 1) scenario with no lockdown measures: figure 13 shows that for a fixed rate γ2 = 0.33 of symptomatic infectives, the least rate of isolating asymptomatic infectives should be γ1 = 0.77 for φ = 0 (implying that the population must observe all the prevention measures ,social distancing, mask wearing etc) but that for φ 6= 0 the hospital bed capacity is never sufficient. biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 page 18 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 m h machingauta, b lungu, e m lungu, covid-19 changing the face of the world. can sub-sahara ... 0 50 100 150 200 250 time (days) 0 0.5 1 1.5 2 2.5 h o s p it a li s e d , ( 2 = 0 .3 3 ) 10 5 hospital bed capacity=62787 1 =0 1 =0.3 1 =0.6 1 =0.77 data1 fig. 13: population level plots for hospitalized individuals for varying values of γ1 for the strategies in figure 13, the numbers of people that must be isolated are given in table 4. table 4 presents cases when the hospital bed capacity, denoted by h, is increased by 50% as was done in italy, spain and the state of new york, usa. the overflow is given by (hp −h × 1.5). we can see that no pair of isolating infectives γ1 and γ2 would accommodate the number of infected individuals in the 3 provinces of south africa within hospital facilities. the overflow capacity would be too large for any non-optimal pair γ1 and γ2 and φ 6= 0, to safely isolate the overflow infected population. the only feasible solution is hospitalizing at the optimal rates (γ∗1,γ ∗ 2 ) = (0.77, 0.33) with φ = 0. this option requires isolating asymptomatic infectives faster than symptomatic infectives, a strategy which requires perfect hygiene and sanitary measures. this requirement cannot be met in sub-sahara africa given the state of the economies. to illustrate the impact of non adherence to disease prevention measures on the optimal case in figure 13, we varied the parameter α3. figure 14 shows how the optimal solution in figure 13 is altered by varying α3. the hospital bed capacity is exceeded for any case α3 6= 0. for the case φ 6= 0, no matter how small φ is, no optimal strategy exists for the three provinces of south africa. it is unrealistic to expect perfect adherence to social distancing, wearing of masks and lockdown measures, in a region where the population survives on less than $1 per day and there are no food banks. 0 50 100 150 200 250 time (days) 0 1 2 3 4 5 6 h o s p it a li s e d , ( 2 = 0 .3 3 ) 10 5 hospital bed capacity=62787 3 =0 3 =0.05 3 =0.1 3 =0.2 data1 fig. 14: population level plots for hospitalized individuals for varying values of α3 2) effect of early lockdown: south africa introduced the lockdown very early. the strategy discussed in section c.1 is therefore not relevant. hence, we consider the modified system with control given in (19) − (25) and (28) − (34). we consider hypothetically how the lockdown could have altered a non optimal strategy in figure 13. we chose γ1 = 0.6 and γ2 = 0.33 (u∗1 ≥ 0.1, u∗2 ≥ 0.1) to illustrate this example (see figures 15(a) − 15(e)). the number of deaths for a model with controls for south africa for different optimal cases of γ1 and γ2 is given in table v . table v compares the actual number of recorded deaths, the controlled number of deaths and the uncontrolled number of deaths (no lockdown). it is clear that the lockdown was very effective. vii. conclusion the analysis has revealed that there are several strategies that can flatten the infection curve and protect hospital bed capacity. however, we have biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 page 19 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 m h machingauta, b lungu, e m lungu, covid-19 changing the face of the world. can sub-sahara ... table iv: number of people who must be hospitalized. γ1 γ2 h h × 1.5 h peak (hp ) oveflow 0 0.33 62 787 94 180 246 174 151 994 0.3 0.33 62 787 94 180 207 654 113 474 0.6 0.33 62 787 94 180 112 952 18 772 0 50 100 150 time (days) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 e x p o se d 10 4 (b) without controls with controls including household and community exposure with controls excluding household and community exposure 0 50 100 150 time (days) 0 2000 4000 6000 8000 10000 12000 a sy m p to m a ti c i n fe c ti v e s (c) without controls with controls including household and community exposure with controls excluding household and community exposure 0 50 100 150 time (days) 0 2000 4000 6000 8000 10000 12000 s y m p to m a ti c i n fe c ti v e s (d) without controls with controls including household and community exposure with controls excluding household and community exposure 0 50 100 150 time (days) 0 2 4 6 8 10 12 h o sp it a li z e d 10 4 (e) hospital bed capacity=62787 without controls with controls including household and community exposure with controls excluding household and community exposure data1 fig. 15: a comparison of state variables with and without controls found that of all the strategies, the preferred strategy is either that which removes asymptomatic infectives much faster than the symptomatic infectives or the strategy that removes symptomatic infectives faster than asymptomatic infectives as either of these strategies gives the least number of deaths. the optimal control analysis suggests that if italy had introduced the lockdown much earlier the number of deaths would have been reduced biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 page 20 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 m h machingauta, b lungu, e m lungu, covid-19 changing the face of the world. can sub-sahara ... table v: comparison of deaths at different optimal cases γ1 γ2 30 days 60 days 72 days 90 days δ = 0.003 δ = 0.01 δ = 0.01 δ = 0.01 0.77 0.33 no lockdown 178 3 358 5 930 12 394 lockdown 7 151 211 327 actual (5) (123) (238) (705) 0.4 0.6 no lockdown 187 3 805 6 834 14 287 lockdown 7 153 216 339 actual (5) (123) (238) (705) significantly and the hospital bed capacity would have been protected. south africa introduced the lockdown early. this is evident from the flatness of the infection curve (figure 12). it is obvious from fig.12 that early lockdown slowed down the number of infections. the question is why despite this measure the number of infected individuals has kept on rising. the number of infections in fact rose rapidly after the lockdown. it is clear from figure 15 that the lockdown was ineffective because the individuals in various households came out to mingle with individuals from other households at community places such as shops, markets etc. the lockdown did not have the desired effect. from figure 15 it is evident that if household and community transmission had been avoided due to higher levels of adherence to control measures the situation would have resulted in 33% fewer asymptomatic infectives, 33% fewer symptomatic infectives and 62% less hospitalizations. the difficulty for south africa, and indeed any sub-sahara african country, will be how to ensure the populations living in high density areas observe social distancing and good hygiene practices. it is easier in developed countries with social security arrangements to maintain disease prevention measures. in sub-sahara africa where the majority of the people live a subsistence life and are self employed, it will be hard to enforce the measures employed in developed countries. declarations of competing interest: none. this research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. we thank the simons foundation (us) through the research and graduate studies in mathematics (rgsma) project at the botswana international university of science and technology (biust) and the department of mathematics and statistical sciences of biust for their support. references [1] africa news (2020). south africa coronavirus: us gives 1,000 ventilators; cases pass 11,000. retrieved from https://www.africanews.com/2020/05/12/south-africacoronavirus-daily-updates-on-covid-19/ on 13 may 2020. 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(2020). risk factors associated with acute respiratory distress syndrome and death in patients with coronavirus disease 2019. pneumonia in wuhan, china jama intern med. 2020 biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 page 23 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 introduction do people learn from previous epidemics: ebola quality of the lockdown model description model analysis positivity of solutions invariance disease free equilibrium (dfe) disease free equilibrium for =0 disease free equilibrium for =0 global stability of the disease free equilibrium endemic equilibrium point (eep) global stability of the endemic equilibrium point numerical simulations sensitivity analysis italy optimal control optimal control without incorporating household and community exposure optimal control incorporating household and community exposure. sub-sahara africa: south africa as an example. scenario with no lockdown measures effect of early lockdown conclusion references original article biomath 3 (2014), 1410061, 1–7 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum descriptor-based fitting of lysophosphatidic acid receptor 3 antagonists into a single predictive mathematical model olaposi idowu omotuyi, hiroshi ueda department of pharmacology and therapeutic innovation university graduate school of biomedical sciences, 852-8521 nagasaki, japan email: bbis11r104@cc.nagasaki-u.ac.jp received: 6 june 2013, accepted: 6 october 2014, published: 16 october 2014 abstract—sixty six diverse compounds previously reported as lysophosphatidic acid receptor (lpa3) inhibitors have been used to derive a mathematical model based on partial least square (pls) clustering of 41 molecular descriptors and pic50 values. the preand postcross-validated correlation coefficient (r2) is 0.94462 (rmse=0.21390) and 0.74745 (rmse=0.49055) respectively. bivariate contingency analysis tools implemented in moe was used to prune the descriptors and refit the equations at a descriptor-pic50 correlation coefficient of 0.8 cutoff. a new equation was derived with r2 and rmse values estimated at 0.88074 and 0.31388 respectively. both equations correctly predicted the 95% of the pic50 values of the test dataset. principal component analysis (pca) was also used to reduce the dimension and linearly transform the raw data; 8 principal components sufficiently account for more than 98% of the variance of the dataset. the numerical model derived here may be adapted for screening chemical database for lpa3 antagonism. keywords-upscaling; lpa3; lpa3 antagonists; mathematical model; pca; molecular descriptors i. introduction quantitative structure activity relationship (qsar) allows statistical analysis of experimental data and building of predictive mathematical models from the dataset. the numerical models built using this approach has been successfully implemented in screening of large database of chemical compounds for hit-compound detection [1]. in the presence of experimental dataset [2], the success of qsar depends on two key factors: array of descriptors that optimally represent the structural parameters required for molecular interaction or reactions [3] and an appropriate statistical learning and validation algorithms [4]. in practice, physical properties descriptors (1d-descriptor), pharmacophore descriptors (2d-descriptors) and geometrical descriptors (3d-descriptors, often requires prior knowledge of target protein binding-pocket) are the most commonly used descriptor types for qsar modeling [5,6,7]. we seek to answer a single question here, what combination of citation: o. omotuyi, h. ueda, descriptor-based fitting of lysophosphatidic acid receptor 3 antagonists into a single predictive mathematical model, biomath 3 (2014), 1410061, http://dx.doi.org/10.11145/j.biomath.2014.10.061 page 1 of 7 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2014.10.061 o. omotuyi et al., descriptor-based fitting of lysophosphatidic acid receptor 3 ... molecular predictors would numerically and accurately predict the experimental antagonist activities of lpa3 inhibitors? when answered, the mathematical relationship derived from the descriptors will enable screening of chemical databases for compounds exhibiting lpa3 antagonism required for the treatment of diseased conditions such as ovarian cancer [8] and neuropathic pain [9] with lpa3 etiology. ii. statistical basis of qsar modeling using partial least square method the qsar/pls modeling equations and algorithms have been well described in moe documentations [10]. given m molecules of a training dataset, suppose that each of the molecules is described by an n-vector of descriptors xi = (xi1, ..., xin), for one of the molecules denoted as i. let yi be a representation of the experimental result (pic50) for a molecule i. a linear model for y (the experimental result) is given by eq. (1) [11]. y = a0 + a t x , (1) where a0 is a scalar, and at is a n-vector. if each molecule has an importance weight (nonnegative) w representing the relative probability that the associated molecule will be encountered, and that the sum of all the weights are designated as w . the mean square error is given as eq. (2) [12]. msea0,a = 1 w m∑ i=1 [yi −(y = a0 +at xi)]2 . (2) differentiating mse with respect to the parameters satisfying the normal eqs (3,4,5,6 &7) solvable by matrix diagonalization: a0 = y0 − at xi , (3) y0 = 1 w m∑ i=1 [wiyi] , (4) x0 = 1 w m∑ i=1 [wixi] , (5) sa = b = 1 w m∑ i=1 [wiyi(xi − x0)] , (6) s = 1 w m∑ i=1 [wi(xi − x0)(xi − x0)t ] . (7) starting from the normal equations above, an estimate of a can be computed if columns of the weight matrix (ga) (eq. (8)) is obtained through gram-schmidt orthogonalization [13] of the vectors generated by krylov sequence b, sb, s2b, ..., sa−1b [14]. the ath pls coefficient vector is then estimated using eq. (9). ga = (gi, g2, . . . , ga) . (8) a = ga(g t asga) −1gtab . (9) noting that gi is the column vectors of length n and a is the degree of the pls fit; an integer less than or equals n. moe [10] descriptor calculator was used to generate the numerical representations (a aro, asa, asa h, a hyd, slogp, slogp vsa0, slogp vsa1, slogp vsa2, slogp vsa3, slogp vsa4, slogp vsa5, slogp vsa6, slogp vsa7, slogp vsa8, slogp vsa9, smr vsa0, smr vsa1, smr vsa2, smr vsa3, smr vsa4, smr vsa5, smr vsa6, smr vsa7, a acc, kier1, kier2, kier3, kiera1, kiera2, kiera3, kierflex, chi0, chi0v, chi0v c, chi0 c, chi1, chi1v, chi1v c, chi1 c, chiral, chiral u) of the 66 (supplementary fig. 1) randomly selected lpa3 antagonists retrieved from the european institute of bioinformatics dataset (https://www.ebi.ac.uk/chembl/) representing our training dataset (chembl3250). using the pls method as described above, eq. (10) was generated relating the descriptors to the pic50 with a correlation coefficient (r2) 0.94462 (rmse = 0.21390) (fig. 1, blue biomath 3 (2014), 1410061, http://dx.doi.org/10.11145/j.biomath.2014.10.061 page 2 of 7 http://dx.doi.org/10.11145/j.biomath.2014.10.061 o. omotuyi et al., descriptor-based fitting of lysophosphatidic acid receptor 3 ... circles and line); when cross validated, r2 was estimated as 0.74745 (rmse = 0.49055). fig. 1: scatter plot of the experimental pic50 vs. pic50-predictions of eq. (10) (blue) and eq. (12) (green). pic50 = 3.57363 − 0.25353 · a aro − 0.00361 · asa + 0.23510 · a hyd + 0.05890 · slogp − 0.02287 · slogp v sa0 +0.00032 · slogp v sa1 + 0.03125 · slogp v sa2 −0.02059 · slogp v sa3 + 0.02954 · slogp v sa4 +0.07226 · slogp v sa5 + 0.02879 · slogp v sa6 +0.04687 · slogp v sa7 + 0.03836 · slogp v sa8 +0.06880 · slogp v sa9 + 0.04912 · smr v sa0 +0.02536 · smr v sa1 + 0.08743 · smr v sa2 +0.00289 · smr v sa3 − 0.01524 · smr v sa4 +0.04694 · smr v sa5 + 0.09067 · smr v sa6 − 0.01442 · smr v sa7 + 0.18393 · a acc − 0.77650 · kier1 − 0.43968 · kier2 − 0.30735 · kier3 − 0.43752 · kiera1 − 0.03578 · kiera2 + 0.76916 · kiera3 − 0.09573 · kierflex + 0.00332 · chi0 + 0.55223 · chi0v + 0.13554 · chi0v c − 0.16530 · chi0 c + 0.59498 · chi1 + 0.05911 · chi1v − 0.93262 · chi1v c − 1.22808 · chi1 c − 0.16986 · chiral − 0.56204 · chiral u. (10) fig. 2: bar chart representations of the residual (experimental pic50-predicted pic50 values of the test dataset. only 1 out of tested compounds (compound 23, see supplementary fig. 2 for structural details) showed > 1.0 pic50 unit (indication of wrong prediction). noting that root mean square error (rmse) is the square root of mse function (eq. (2)) at a given parameter value and the correlation coefficient (r2) is 1-mse/yvar with values raging between 0 and 1 (0= no fit, 1 is perfect fit and yvar is the sample variance of the yi values). the predictive suitability of our equation was tested on 23 compounds (supplementary fig. 2) with experimentally determined ic50 for lpa3 antagonism. if we assume that residual value above 1.0 pic50 unit represents poor fitting. our data (fig. 3) suggest that eq. (10) accurately predicted 22 of the 23 test compounds. iii. descriptor contingency analysis to determine the level of significance of each of the descriptors to the overall equation and we performed contingency analysis. the data presented here provides a window of decision on whether pruning of the descriptor set is required. in moe [10], qsar-contingency tool performs a bivariate contingency analysis for each descriptor and the experimental activity value and produces a table of correlation coefficients (eq. (11)) for each descriptor given that x represents a randomly selected molecular descriptor and y is a randomly selected activity value for a randomly selected sample m, v ar(x) and v ar(y ), then the covariance of biomath 3 (2014), 1410061, http://dx.doi.org/10.11145/j.biomath.2014.10.061 page 3 of 7 http://dx.doi.org/10.11145/j.biomath.2014.10.061 o. omotuyi et al., descriptor-based fitting of lysophosphatidic acid receptor 3 ... the random variables x and y is defined to be cov(x, y ) = e(xy ) − e(x)e(y ) [10, 15]. r2 = [e(xy ) − e(x)e(y )]2 v ar(x)v ar(y ) . (11) given that the values of r2 ranges from 0 to 1, and 1 represents a perfectly linear correlation, we therefore proposed that only descriptors r2 values ≥ 0.8 are useful and that the descriptors outside this range can be pruned. our data suggest that 31 out of the original 41 descriptors have r2 values ≥ 0.8 (fig. 3, supplementary table 1). with the exclusion of the descriptors with unsatisfactory coefficient, qsar is re-calculated using the residual set of descriptors. new numerical relationship was generated (eq. (12)) with r2 (0.88074) and rmse values (0.31388). the scatter plot of the predicted pic50 and the experimental values for the new eq. (12) is given in fig. 1 (green circles and line). lpic50 = 2.23199 − 0.00516xasa − 0.00516xasa h − 0.48596xa hyd − 0.33917xslogp −0.05298xslogpv sa0 − 0.03967xslogpv sa1 −0.02243xslogpv sa2 + 0.01681xslogpv sa7 + 0.02107xslogpv sa9 −0.00757xsmrv sa0 − 0.00087xsmrv sa1 − 0.00089xsmrv sa3 −0.01173xsmrv sa4 + 0.00955xsmrv sa5 − 0.01412xsmrv sa6 − 0.02508xsmrv sa7 − 0.26771xkier1 + 0.15306xkier20.56650xkier3 − 0.30504xkiera2 + 0.98837xkiera3 − 0.28849xkierflex + 0.48535xchi0 + 0.90693xchi0v + 0.10234xchi0vc + 0.24407xchi0c + 0.66154xchi1 + 0.36006xchi1v − 1.03589xchi1vc − 0.62474xchi1c − 0.36725xaaro . (12) when this equation was used for predicting the fig. 3: bar chart representations of descriptorexperimental pic50 correlation coefficient. only 31 out of 41 descriptors lie above 0.8 coefficient cutoff. fig. 4: the 3d plot of the first three principal components. each point represents a compound in the training dataset and each colour represents a distinct cluster of pic50 values. pic50 values of the test set, only one compound lies above the 1.0 pic50 unit cutoff (data not shown). thus, eq. (12) is less bulky and as accurate as eq. (10) in predicting lpa3 antagonism. iv. principal component analysis of equation we sought to further study the dataset descriptors along the principle components through the reduction of the dimensionality and linear transformation of the raw data [13]. given the initial 66 training dataset compounds (represented as m) and for one of the compounds say i its descriptors are represented by n-vector of real numbers xi = biomath 3 (2014), 1410061, http://dx.doi.org/10.11145/j.biomath.2014.10.061 page 4 of 7 http://dx.doi.org/10.11145/j.biomath.2014.10.061 o. omotuyi et al., descriptor-based fitting of lysophosphatidic acid receptor 3 ... (xi1, ..., xin), where n = 1 − 31, new eq. (12). assuming that each molecule i has an associated importance weight wi, (non-negative, real number) and that the weights is relative probability that the associated molecule xi will be encountered (adding up to 1); if w denotes the sum of all the weights then, the eigenvalues and eigenvectors for the final data are estimable from the raw data using eq. (1). if s is a symmetric, semi-definite sample covariance matrix, s can be diagonalized such that s = qt ddq (q is orthogonal, d is diagonal-sorted in descending order from top left to bottom right) [13, 14]. e(x) ≈ x = x0 = 1 w m∑ i=1 [wixi] (13) cov(x) ≈ s = 1 w m∑ i=1 [wixix t i − xx t ]. (14) the effect of the each of the principal components (eigenvectors) on the condition and the variance shows that nine (8) principal components sufficiently accounts for more than 98% of the variance in the dataset [15]. the 3d-scatter plot of the first three principal components (pca1, pca2 and pca3) with respect to pic50 values is shown in fig. (4); each point in the plot corresponds to a dataset molecule colored according to clustered pic50 values. v. conclusion given the good mathematical correlation between the set of descriptors and lpa3 antagonism, it is not unusual to propose that the equation is prejudiced for those set of compounds with highly related descriptor properties and therefore may not be a universal formula for lpa3 antagonist screening. that said, it will however capture the compounds with structural properties found within the dataset accurately and therefore may be piped as into ligand-based screening protocol for more successful hit-compound identification. acknowledgment this work was supported by platform for drug discovery, informatics, and structural life science from the ministry of education, culture, sports, science and technology, japan. appendix supplementary  table  1.0  showing  correlation  coefficient  of  each  descriptor s/n desciptors corr. coefficient 1 slogp_vsa6 0.57623 2 chiral_u 0.65734 3 slogp_vsa4 0.66609 4 slogp_vsa5 0.6996 5 chiral 0.72218 6 smr_vsa2 0.78566 7 slogp_vsa8 0.78621 8 a_acc 0.78922 9 slogp_vsa3 0.79094 10 kiera1 0.79264 11 a_aro 0.80122 12 slogp_vsa9 0.80481 13 slogp_vsa1 0.80575 14 chi0_c 0.806 15 chi1v 0.80603 16 kierflex 0.80836 17 chi1v_c 0.81041 18 kiera3 0.81376 19 slogp_vsa2 0.81493 20 smr_vsa7 0.81623 21 asa 0.81908 22 asa_h 0.81908 23 chi0v 0.82223 24 chi0v_c 0.82394 25 smr_vsa4 0.82512 26 chi1_c 0.82535 27 kiera2 0.82725 28 chi0 0.82827 29 slogp_vsa7 0.82933 30 smr_vsa5 0.82941 31 kier2 0.83257 32 slogp_vsa0 0.83519 33 kier1 0.83644 34 smr_vsa1 0.83839 35 chi1 0.84721 36 smr_vsa6 0.84762 37 smr_vsa3 0.8525 38 kier3 0.85924 39 slogp 0.86886 40 smr_vsa0 0.87545 41 a_hyd 0.88264 scatter plot of the experimental pic50 vs. pic50predictions of eq.(10) (blue) and eq. (12). biomath 3 (2014), 1410061, http://dx.doi.org/10.11145/j.biomath.2014.10.061 page 5 of 7 http://dx.doi.org/10.11145/j.biomath.2014.10.061 o. omotuyi et al., descriptor-based fitting of lysophosphatidic acid receptor 3 ... 1 o n o p o o o o 2 o n o p o o o o 3 o o n n n f cl 4 o n o p o o o o 5 o p s o o 6 o o o p s o o o o 7 o o n n n o o 8 o n o p o o o o 9 o p o o o 10 o o n n n o o 11 o p s o o 12 o o n o o o 13 o n o p o o o o 14 p o o o 15 o o o n oo n o o o 16 o o o n o o n o o + 17 o n o p o o o o o 18 o n o p o o o o 19 o p s o o 20 o o o p o o o p o o o oo 21 o p s o o 22 o o o p so o o o 23 o o n nn n o o f f f 1 o o o nn o o o o o pic50: 7.5229 $pred: 7.5301 $res: -0.0072 2 o o o nn o o o o o pic50: 7.5229 $pred: 7.5301 $res: -0.0072 3 o p o o o pic50: 6.4157 $pred: 6.4861 $res: -0.0705 4 o o n n n n o o pic50: 4.7305 $pred: 4.7125 $res: 0.0179 5 o o o p oo o o o pic50: 6.6840 $pred: 6.8724 $res: -0.1883 6 n oo p s o o no pic50: 6.5200 $pred: 6.4167 $res: 0.1033 7 o n o p o o o o o pic50: 6.1925 $pred: 6.1022 $res: 0.0902 8 o on nn n o o f f f pic50: 5.1637 $pred: 5.0749 $res: 0.0888 9 o o o n oo n o o o pic50: 5.9101 $pred: 5.9231 $res: -0.0131 10 o o o n oo n o o o pic50: 5.9101 $pred: 5.9231 $res: -0.0131 11 o p o o o pic50: 6.4157 $pred: 6.3266 $res: 0.0890 12 o n o p o o o o pic50: 5.0000 $pred: 5.0393 $res: -0.0393 13 o o n n n o o f pic50: 5.2366 $pred: 5.4671 $res: -0.2305 14 o p o o o pic50: 6.3747 $pred: 6.6310 $res: -0.2563 15 n oo p o o o no pic50: 6.0292 $pred: 6.1727 $res: -0.1435 16 o o n n n o o pic50: 4.5229 $pred: 4.5776 $res: -0.0547 17 o o p s o o pic50: 5.6308 $pred: 5.6943 $res: -0.0635 18 n oo p s o o no pic50: 6.6003 $pred: 6.4167 $res: 0.1836 19 o n o p o o o o pic50: 5.1649 $pred: 5.1271 $res: 0.0379 20 o n o p o o o o pic50: 5.1904 $pred: 5.5739 $res: -0.3834 lpa3 inihibitors: training set for qsar modeling 21 o o n n n n o n s o o pic50: 5.6364 $pred: 5.7584 $res: -0.1220 22 o o n n n cl pic50: 4.5229 $pred: 4.4398 $res: 0.0830 23 o o n n n pic50: 4.5229 $pred: 4.4673 $res: 0.0556 24 o p s o o pic50: 6.9136 $pred: 6.9600 $res: -0.0463 25 o o o p o o o o o pic50: 6.8447 $pred: 6.6932 $res: 0.1515 26 o n o p o o o o n pic50: 6.0269 $pred: 6.4243 $res: -0.3974 27 o o p s o o pic50: 5.8962 $pred: 5.8462 $res: 0.0500 28 o o n n n n o o pic50: 4.6588 $pred: 4.2947 $res: 0.3641 29 o n o p oo o o n pic50: 5.0334 $pred: 5.2622 $res: -0.2288 30 o n o p o o o o pic50: 5.1931 $pred: 5.3161 $res: -0.1230 31 o p s o o pic50: 7.5528 $pred: 7.3033 $res: 0.2495 32 o p s o o pic50: 6.4685 $pred: 6.8160 $res: -0.3474 33 o o f p o o o pic50: 5.1124 $pred: 4.9684 $res: 0.1439 34 o o n n n o o f pic50: 5.0830 $pred: 4.9372 $res: 0.1458 35 o p o o o pic50: 6.1135 $pred: 6.1509 $res: -0.0374 36 o n o p o o o o pic50: 5.1593 $pred: 5.3161 $res: -0.1568 37 o n o p o o o o pic50: 5.0000 $pred: 4.7783 $res: 0.2217 38 o o o n o o n o o + pic50: 6.1238 $pred: 6.1151 $res: 0.0087 39 o p s o o pic50: 7.5686 $pred: 7.1552 $res: 0.4134 40 o o n n n o o f f f pic50: 6.3206 $pred: 6.3223 $res: -0.0018 lpa3 inihibitors: training set for qsar modeling 41 n o n sn o o no o o n o o o o + + pic50: 7.6198 $pred: 7.6400 $res: -0.0202 42 n o n sn o o no o o n o o o o + + pic50: 7.6198 $pred: 7.6400 $res: -0.0202 43 o p o o o pic50: 6.0809 $pred: 6.3361 $res: -0.2551 44 o o n n n o o pic50: 5.0788 $pred: 4.8447 $res: 0.2341 45 o o o p oo o o o pic50: 7.0706 $pred: 6.8724 $res: 0.1982 46 o p s o o pic50: 7.5528 $pred: 7.0811 $res: 0.4717 47 p o o o pic50: 6.1844 $pred: 6.2144 $res: -0.0299 48 o p o o o pic50: 6.9872 $pred: 6.7202 $res: 0.2669 49 o n o p o o o o o pic50: 5.1415 $pred: 5.0991 $res: 0.0424 50 o o p o o o o pic50: 6.8447 $pred: 6.6623 $res: 0.1824 51 o p o o o pic50: 7.0177 $pred: 6.7353 $res: 0.2824 52 o o n n n o o pic50: 4.5229 $pred: 4.7671 $res: -0.2443 53 o o n o o o pic50: 5.5240 $pred: 5.5696 $res: -0.0456 54 o p s o o pic50: 6.7905 $pred: 7.2006 $res: -0.4101 55 o n o p oo o o pic50: 5.6364 $pred: 5.6100 $res: 0.0264 56 o n o p oo o o n pic50: 5.5800 $pred: 5.2622 $res: 0.3179 57 n oo p o o o no pic50: 6.3830 $pred: 6.1727 $res: 0.2103 58 o on n n n on so o f f f pic50: 7.1871 $pred: 7.1826 $res: 0.0045 59 o o n n n f cl pic50: 4.5229 $pred: 5.0926 $res: -0.5698 60 o o o p s o o o o pic50: 6.7352 $pred: 6.9852 $res: -0.2500 lpa3 inihibitors: training set for qsar modeling biomath 3 (2014), 1410061, http://dx.doi.org/10.11145/j.biomath.2014.10.061 page 6 of 7 http://dx.doi.org/10.11145/j.biomath.2014.10.061 o. omotuyi et al., descriptor-based fitting of lysophosphatidic acid receptor 3 ... 61 o n o p o o o o pic50: 5.9259 $pred: 5.7730 $res: 0.1529 62 o o p s o o o pic50: 6.7352 $pred: 7.0541 $res: -0.3189 63 o n o p o o o oo pic50: 5.2541 $pred: 5.5111 $res: -0.2570 64 o n o p o o o o n pic50: 6.7570 $pred: 6.4243 $res: 0.3327 65 p o o o pic50: 5.9208 $pred: 6.0663 $res: -0.1455 66 o o n n o s o ocl pic50: 6.5214 $pred: 6.2442 $res: 0.2772 lpa3 inihibitors: training set for qsar 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[11] m.j. wichura, ”the coordinate-free approach to linear models”. cambridge series in statistical and probabilistic mathematics. cambridge: cambridge university press. pp. xiv+199. isbn 978-0-521-86842-6. 2006. mr 2283455 [12] d. wackerly, w. scheaffer. ”mathematical statistics with applications” (7 ed.). belmont, ca, usa: thomson higher education. isbn 0-49538508-5. 2008 biomath 3 (2014), 1410061, http://dx.doi.org/10.11145/j.biomath.2014.10.061 page 7 of 7 http://dx.doi.org/10.1016/j.ejmech.2012.10.035 http://dx.doi.org/10.1002/9783527628766 http://dx.doi.org/10.1016/j.bbalip.2012.08.014 http://dx.doi.org/10.11145/j.biomath.2014.10.061 introduction statistical basis of qsar modeling using partial least square method descriptor contingency analysis principal component analysis of equation conclusion references original article biomath 1 (2012), 1210045, 1–7 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum dynamics of an sis model on homogeneous networks with delayed reduction of contact numbers maoxing liu∗† and gergely röst† ∗ department of mathematics north university of china, taiyuan, shanxi, p. r. china, 030051 email: liumaoxing@126.com † bolyai institute university of szeged, szeged, hungary email: rost@math.u-szeged.hu received: 12 july 2012, accepted: 4 october 2012, published: 27 december 2012 abstract—during infectious disease outbreaks, people may modify their contact patterns after realizing the risk of infection. in this paper, we assume that individuals make the decision of reducing a fraction of their links when the density of infected individuals exceeds some threshold, but the decision is made with some delay. under such assumption, we study the dynamics of a delayed sis epidemic model on homogenous networks. by theoretical analysis and simulations, we conclude that the density of infected individuals periodically oscillate for some range of the basic reproduction number. our results indicate that information delays can have important effects on the dynamics of infectious diseases. keywords-epidemic model; networks; delay; stability; oscillation i. introduction a great deal of research in mathematical epidemiology has been done for complex networks during the past years [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. these works focus mostly on two different complex networks: the watts-strogatz model and the barabási-albert model. the first model is a relatively homogeneous network exhibiting small-world properties [1], and the second one is a typical example of a scalefree network [2]. on these networks, the sis model is generally used to study the spread of infections via connections between the nodes of the networks. in disease transmission models, time delay plays an important role in many epidemiological mechanisms. various delayed epidemic models have been extensively studied (see, for example, refs. [13], [14], [15], [16], [17], [18], [19]). recently there have been some results about delay epidemic models on complex networks. in the paper [20], the authors present a modified sis model with the effect of time delay in the transmission on small-world and scale-free networks. they found that the presence of the delay may enhance outbreaks and increase the prevalence of infectious diseases in these networks. whereas in [21], the authors consider a delayed sir epidemic model on uncorrelated complex network and addressed the effect of time lag on the shape and number of epidemic waves. they showed that even when the transmission rate is below the critical threshold, a larger delay can cause that the disease takes off while the force of infection is not increased. otherwise, a large delay can cause multiple waves with larger amplitudes in the second and subsequent waves. in fact, on one hand, during infectious disease outbreaks, individuals may reduce their activities after receiving information about the risk of infection. for citation: m. liu , g. röst, dynamics of an sis model on homogeneous networks with delayed reduction of contact numbers, biomath 1 (2012), 1210045, http://dx.doi.org/10.11145/j.biomath.2012.10.045 page 1 of 7 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.10.045 m. liu et al., dynamics of an sis model on homogeneous networks with delayed reduction of contact numbers example, people will reduce the time that they go out, students will not attend school, and so on, and such information on the ongoing epidemics may impact the dynamics itself [22], as the contact network changes [23]. social distancing can be used as a control measure as well. on the other hand, due to the latent period of diseases, the concealment of infected individuals, or the time needed for collecting and analyzing epidemic data, the exact number of infected individuals is hard to be known in real time. based on the above facts, in this paper we assume that individuals adjust their connections according to some information delay concerning the actual disease prevalence. for simplicity, we work with the assumption that individuals uniformly and randomly reduce the number of their connections by some factor whenever the density of infected individuals exceeds a given threshold. we study the effects of time-delayed information on the decision-making of individuals, and then we study the dynamics of the spread of the disease under such circumstances. by theoretical analysis and simulations, we conclude that the disease will be eradicated under some conditions, or tend to an endemic state, but it can also oscillate in a periodic pattern when the structure of networks and the time delay are properly given. all these behaviors are completely characterized by the basic reproduction number. the paper is organized as follows. in the following section, we set up a delayed sis model on homogenous networks and analyse the existence of equilibria. we also give some results on stability, permanence and oscillation of the disease in section 2. in section 3, some numerical simulations illustrating the key points of the theoretical analysis are given. at the last section we offer a discussion of our results. ii. model and analysis a. the model in this section, we consider a susceptible-infectedsusceptible (sis) model on homogenous networks. in the sis model, infectious (i) individuals contaminate their susceptible (s) neighbors with some transmission rate. meanwhile, infected individuals recover at some rate and return to the susceptible state again. by using the mean-field approach on homogenous networks, in [6] the authors arrived to the following epidemic model: di(t) dt = −µi(t) + β〈k〉i(t)(1 − i(t)). (1) here i(t) ∈ [0, 1] denotes the density of infected nodes at time t. the first term considers infected nodes recovering with rate µ. the second term of the right-hand side of eq. (1) represents the newly infected nodes. this is proportional to the transmission rate β, the number of links emanating from each node 〈k〉, and the probability that a given link points to a healthy node, which is 1 − i(t). here µ, β, 〈k〉 are positive constants. we suppose that individuals will reduce their links according to the information they learn on the disease spread. if the disease is not widespread, people remain in contact with others as usual. with the increasing number of the infected individuals, people reduce their activities and temporarily terminate some of their links. we assume this is governed by the following function: h(i) = { 1, i ≤ p, q, i > p, fig. 1. the graph of h(i). where 0 < p, q < 1. as shown in fig. 1, when i ≤ p the number of links of individuals are the same as usual 〈k〉; when i > p (i.e. the density of infectious nodes exceeds the threshold p), the links of individuals are reduced to a lower level q〈k〉. by assuming a time delay τ > 0 in making this reduction, we obtain the following epidemic model with discontinuous right hand side: di(t) dt = −µi(t) + β〈k〉h(i(t − τ ))i(t)(1 − i(t)). (2) for the sake of simplicity, we rescale time by ĩ(t) = i(µ−1t), then writing the equation for ĩ(t) and dropping the tilde to use the notation i(t) for the variable in the biomath 1 (2012), 1210045, http://dx.doi.org/10.11145/j.biomath.2012.10.045 page 2 of 7 http://dx.doi.org/10.11145/j.biomath.2012.10.045 m. liu et al., dynamics of an sis model on homogeneous networks with delayed reduction of contact numbers rescaled time, eq. (2) is transformed into di(t) dt = −i(t) + r0h(i(t − τ ))i(t)(1 − i(t)), (3) where r0 = β〈k〉 µ is the basic reproduction number, which expresses the number of secondary infections generated by a single infected node in a fully susceptible homogenous network. eq. (3) is a scalar delay differential equation, which are known to be able to exhibit complicated behaviour if the nonlinearity is nonmonotone [24]. the technical complications due to the discontinuity of the right hand side can be avoided if by a solution of eq. (3) we mean a continuous function i(t) satisfying i(t) = i(t−τ )+ ∫ t t−τ i(s)(r0h(i(s−τ ))(1−i(s))−1)ds for all t ≥ τ . throughout the paper by i′(t) we mean the right derivative when i(t − τ ) = p, this will not cause any confusion. given the interpretation of the model, we only consider solutions i(t) ∈ [0, 1]. clearly solutions from this interval remain in [0, 1] for all future time. for the existence of the equilibria, we have the following statement. proposition 2.1. (a) the disease free equilibrium i∗0 = 0 always exists, (b) if 1 < r0 ≤ 11−p , eq. (3) has a positive equilibrium i∗1 , (c) if r0 > 1 q(1−p) , eq. (3) has a different positive equilibrium i∗2 , (d) if r0 < 1 or 11−p < r0 < 1 q(1−p) , there is no positive equilibrium. proof. to obtain the equilibria of eq. (3), we set i(t) ≡ i(t − τ ) ≡ i∗, and let the right hand side of eq. (3) be zero, so r0h(i ∗)i∗(1 − i∗) = i∗. (4) first we have i∗ = 0, which corresponds to the disease free equilibrium. if i∗ 6= 0, then from eq. (4) we get r0h(i ∗)(1 − i∗) = 1. (5) this does not have a positive solution for r0 < 1. if r0 > 1, we distinguish two cases. when i∗ ≤ p, we have i∗ = 1 − 1 r0 (let’s denote it by i∗1 ), and when i∗ > p, we have i∗ = 1 − 1 qr0 (denoted by i∗2 ). to satisfy i∗1 ≤ p, we obtain 1− 1 r0 ≤ p, which is r0 ≤ 11−p . similarly, i∗2 > p is equivalent to r0 > 1 q(1−p) . proposition 2.2. if r0 < 1, then i0 is globally asymptotically stable. if r0 > 1, then i0 is unstable. proof. if r0 < 1, the statement easily follows from the comparison di(t) dt ≤ i(t)[r0 − 1]. (6) as di(t) dt = i(t)[r0 − 1] (7) is the linear variational equation around zero, the disease free equilibrium is unstable if r0 > 1. b. permanence for the permanence of eq. (3), we find the following result. theorem 2.3: if r0 > 1, system (3) is permanent. more precisely, 1 − 1 qr0 ≤ lim inf t→∞ i(t) ≤ lim sup t→∞ i(t) ≤ 1 − 1 r0 . (8) proof: first we show i∞ = lim inf t→∞ i(t) ≥ 1 − 1 qr0 . suppose that 0 < i(t1) ≤ 1− 1qr0 for some t1. thus, by the definition of h, (1 − i(t1))r0h(i(t1 − τ )) ≥ 1 and from eq. (3) we have i′(t1) = i(t1) ( − 1 + r0h(i(t1 − τ ))(1 − i(t1)) ) ≥ 0, that is to say, the solution at t1 is increasing, and since by proposition 2.1. a positive equilibrium less than 1 − 1 qr0 can not exist, we obtain i∞ ≥ 1 − 1 qr0 . (9) next we show i∞ = lim sup t→∞ i(t) ≤ 1 − 1 r0 . using similar argument, supposing i(t2) ≥ 1 − 1r0 for some t2, we obtain (1 − i(t2))r0h(i(t2 − τ ) ≤ 1, and from eq. (3) we have i′(t2) = i(t2) ( − 1 + r0h(i(t2 − τ ))(1 − i(t2)) ) ≤ 0. the solution at t2 is decreasing, and by the nonexistence of equilibrium greater than 1 − 1 r0 , we obtain i∞ ≤ 1 − 1 r0 . (10) the graph showing the relation of i∗ and r0 is depicted in fig.2. from fig.2, we can also see the effect of the structure of the network. by increasing the average degree 〈k〉, r0 is also increasing, and the equilibrium will change from zero to nonzero, then to oscillation, then back to a nonzero equilibrium again; thus system (3) will experience different dynamical behaviors. biomath 1 (2012), 1210045, http://dx.doi.org/10.11145/j.biomath.2012.10.045 page 3 of 7 http://dx.doi.org/10.11145/j.biomath.2012.10.045 m. liu et al., dynamics of an sis model on homogeneous networks with delayed reduction of contact numbers fig. 2. the relation of i∗ and r0, here a = 1 1−p , b = 1 q(1−p) . the solutions are oscillating when a < r0 < b. c. stability and oscillation for the stability of the positive equilibria and oscillations of solutions, we have the following results. proposition 2.4: for eq. (3), the following claims are true: (a) if 1 < r0 < 11−p , the positive equilibrium i ∗ 1 is locally asymptotically stable. (b) if r0 > 1 q(1−p) , the positive equilibrium i ∗ 2 is locally asymptotically stable. proof: (a) if 1 < r0 < 11−p , eq. (3) has a positive equilibrium i∗1 = 1− 1 r0 < p, thus we can choose a small neighborhood of i∗1 such that i(t) < p as long as the solution is in this neighborhood. thus, restricted to this neighborhood h ≡ 1 and eq. (3) becomes an ordinary differential equation, and we can easily obtain that the positive equilibrium i∗1 is locally asymptotically stable. using the same argument, we can prove (b). theorem 2.5: if 1 1−p < r0 < 1 q(1−p) , then all solutions of eq. (3) oscillate around p. proof: let 1 1−p < r0 < 1 q(1−p) , and suppose that a solution i(t) of eq. (3) does not oscillate around p, then there must exist t1 such that i(t) ≤ p, for t ≥ t1 or a t2 such that i(t) ≥ p, for t ≥ t2. in the following we show that none of them is possible. if i(t) ≤ p for t ≥ t1, we have i′(t) = −i(t) + r0i(t)(1 − i(t)) for t ≥ t1 + τ . in this case the solution i(t) converges to 1 − 1 r0 > p, which is a contradiction. similarly, if i(t) > p for t larger than some t3, we have i′(t) = −i(t) + qr0i(t)(1 − i(t)) for t ≥ t2 + τ . in this case the solution i(t) converges to 1 − 1 qr0 < p, this is also a contradiction. we finish by demonstrating that if i(t) ≥ p for all t > t2, then there is a t3 such that i(t) > p for all t > t3. if that is not true, then we can find arbitrarily large tk such that i(tk) = p. if i(tk − τ ) > p, then i′ is continuous at tk and i′(tk) = 0 must hold, but then h(i(tk −τ )) = q and so i′(tk) = p(r0q(1 − p) − 1) 6= 0. to exclude the remaining case when i(tk − τ ) = p, we impose the additional assumption that on some interval of length τ our solution takes the value p at most finitely many times, then this property of i(t) will be inherited to any interval [t − τ, t] for future t by the integral representation. by integration we also obtain i(tk) i(tk − τ ) = exp [∫ tk tk−τ r0h(i(s − τ ))(1 − i(s)) − 1ds ] , which implies that the integral in the exponent should be zero. however, r0h(i(s − τ ))(1 − i(s)) − 1 ≤ 0 whenever i(s − τ ) > p and h(i(s − τ )) = q. by our assumption the equality i(s − τ ) = p holds on a set of measure zero, thus the integral is negative, which is a contradiction. iii. simulations in this section, we discuss some examples and simulations. our purpose is to illustrate the sharpness of the results of the previous section. here we set initial data as constant functions. first we set p = 0.8, q = 0.6 and demonstrate the stability of the zero equilibrium, as shown in fig. 3 with r0 = 0.8. in the case r0 = 2, as shown in fig.4, the positive equilibrium i∗1 is asymptotically stable. while for r0 = 3.6, p = 0.5, q = 0.4 and τ = 1.2, as shown in fig.5, solutions of eq.(3) are oscillatory. in the case r0 = 2.7, p = 0.1, q = 0.6, as shown in fig.6, the positive equilibrium i∗2 is asymptotically stable. if 1 1−p < r0 < 1 q(1−p) , then all solutions of eq. (3) oscillate around p. in this case, to illustrate the effect of the time delay, we consider distinct values of the delay while other parameters are fixed. in the first case the delay is small, depicted in fig. 7 , with τ = 0.3, and the amplitude of the solution around p is apparently small. biomath 1 (2012), 1210045, http://dx.doi.org/10.11145/j.biomath.2012.10.045 page 4 of 7 http://dx.doi.org/10.11145/j.biomath.2012.10.045 m. liu et al., dynamics of an sis model on homogeneous networks with delayed reduction of contact numbers fig. 3. the time evolutions of the density of infection with i(0) = 0.1 and 0.8, r0 = 0.8, p = 0.8, q = 0.6, and τ = 0. fig. 4. the time evolutions of the density of infection with i(0) = 0.1 and 0.8, r0 = 2, p = 0.8, q = 0.6 and τ = 0. in the second case, if the delay is larger than in the first case (τ = 3.6), we observe increased amplitude in fig. 8. however, for all delays, the solutions always oscillate between 1 − 1 r0 and 1 − 1 qr0 . these values are represented by the straight lines in fig. 7 and fig. 8, and one can see that the bounds are rather sharp for large delays. iv. conclusion in this paper, we studied a delayed model for an sis epidemic process in a population of individuals on a homogenous network. we assumed that individuals temporarily reduce the number of their links by a factor fig. 5. the time evolutions of the density of infection with i(0) = 0.8,r0 = 3.6, p = 0.5, q = 0.4 and τ = 1.2. fig. 6. the time evolutions of the density of infection with i(0) = 0.1 and 0.8, r0 = 2.7, p = 0.1, q = 0.6 and τ = 0. q when the density of infections exceeds the threshold number p, but this modification in the contact pattern is done with some delay τ . when the basic reproduction number is smaller than one, the disease will be eradicated. for reproduction numbers larger than one, we showed that the disease persists in the population. if the endemic state is lower than the threshold, then the reduction of contacts will never be triggered and the solution converges to an endemic equilibrium i∗1 . if there is an endemic state even with the reduction of contacts which is at higher level than the threshold, then the terminated links remain inactive for all future time and biomath 1 (2012), 1210045, http://dx.doi.org/10.11145/j.biomath.2012.10.045 page 5 of 7 http://dx.doi.org/10.11145/j.biomath.2012.10.045 m. liu et al., dynamics of an sis model on homogeneous networks with delayed reduction of contact numbers fig. 7. the time evolutions of the density of infection with i(0) = 0.8, r0 = 3.6, p = 0.5, q = 0.4 and τ = 0.3. fig. 8. the time evolutions of the density of infection with i(0) = 0.8, r0 = 3.6, p = 0.5, q = 0.4 and τ = 3.6. the solution converges to a different endemic state i∗2 . there is an interesting intermediate situation though for a range of basic reproduction numbers, when the reduction is triggered, but with such reduced transmission the endemic state is below the threshold, thus the links will be reactivated again, which helps the disease to spread more, thus triggering the reduction and so on, forming an interesting periodic oscillatory pattern. the time delay has significance in determining the characteristics of this oscillation: longer delay leads to larger amplitudes. our results indicate that the structure of the network (which influences the reproduction number), the threshold type reduction in contacts and the delayed decision in reduction interestingly interplay on influencing the spreading dynamics of infectious diseases. acknowledgment the authors would like to thank the support of national sciences foundation of china (10901145), top young academic leaders of higher learning institutions of shanxi, european research council starting investigator grant nr. 259559, hungarian scientific research fund otka k75517, and bolyai scholarship of hungarian academy of sciences. references [1] d. j. watts and s. h. strogatz, “collective dynamics of ’smallworld’ networks,” nature, vol. 393, pp. 440–442, 1998. http://dx.doi.org/10.1038/30918 [2] a. l. barabasi and r. albert, “emergence of scaling in random networks”, science, vol. 286, pp. 509–512, 1999. http://dx.doi.org/10.1126/science.286.5439.509 [3] r. m. may and a. l. lloyd, “infection dynamics on scale-free networks”, phys. rev. e, vol. 64, 066112, 2001. 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http://dx.doi.org/10.11145/j.biomath.2012.10.045 introduction model and analysis the model permanence stability and oscillation simulations conclusion references bifurcation analysis of a mathematical model of microalgae growth under the influence of sunlight biomath https://biomath.math.bas.bg/biomath/index.php/biomath b f biomath forum original article bifurcation analysis of a mathematical model of microalgae growth under the influence of sunlight lingga sanjaya putra mahardhika, fajar adi-kusumo∗, dwi ertiningsih department of mathematics, faculty of mathematics and natural sciences, universitas gadjah mada, yogyakarta, indonesia linggasanjaya2018@mail.ugm.ac.id 0000-0001-5063-2848 f adikusumo@ugm.ac.id 0000-0002-1643-4466 dwi ertiningsih@ugm.ac.id 0000-0002-4529-8440 received: july 9, 2022, accepted: january 30, 2023, published: april 28, 2023 abstract: in this paper is considered a microalgae growth model under the influence of sunlight. the model is a two-dimensional system of the first order ordinary differential equations (ode) with a ten-dimensional parameter space. for this model, we study the existence of equilibrium points and their stability, and determine a bifurcation of the system when the value of some parameters is varied. the lambert ω function is used to calculate equilibrium points and apply the linearization technique to provide their stabilities. by varying the value of some parameters numerically, we found a transcritical bifurcation of the system and show stability regions of the equilibrium points in parameter diagrams. the bifurcation shows that the microalgae have a minimum sustainable nutrition supply and have a minimum light intensity that plays an important role for survival in a chemostat which has a certain depth. the results can be used to design a chemostat in optimizing the growth of microalgae. keywords: microalgae growth model, quota cell, parameter diagram, bifurcation i. introduction at the moment, fossil fuels still generate about 80% of the demand of global energy. this demand increases along with the increase in population, because each individual needs a means of transportation to carry out activities and move to other places. however, the extensive use of fossil fuels plays an important role for global climate change, environmental pollution, and problems in health [1]. most scientists are looking for new types of energy. the most interesting renewable energy that is expected to have an important role in the future global energy structure is biofuels. biodiesel is one of the biofuels that is recognized as an ideal carrier of renewable energy which has the potential to become a primary energy source. there are several candidates of plants for biodiesel production, but most of them grow slowly, contain little vegetable oil for biodiesel, and require large areas of land to be grown. therefore, they are considered inefficient for biodiesel production. microalgae is a microorganism that has an ability to convert solar energy to chemical energy through the fixation of carbon dioxide (co2). the growth of microalgae is relatively fast, so it can be considered as a valuable source for making biodiesel [1]. microalgae can form their energy by photosynthesis to support their life needs. it carries out photosynthesis with the help of sunlight and absorbs carbon dioxide and nutrients around it to form the various substances it needs. the result of photosynthesis is glucose that can be stored in cells as vegetable fats. copyright: © 2023 lingga sanjaya putra mahardhika, fajar adi-kusumo, dwi ertiningsih. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. *corresponding author citation: lingga sanjaya putra mahardhika, fajar adi-kusumo, dwi ertiningsih, bifurcation analysis of a mathematical model of microalgae growth under the influence of sunlight, biomath 12 (2023), 2301307, https://doi.org/10.55630/j.biomath.2023.01.307 1/9 https://biomath.math.bas.bg/biomath/index.php/biomath mailto:linggasanjaya2018@mail.ugm.ac.id https://orcid.org/0000-0001-5063-2848 mailto:f_adikusumo@ugm.ac.id https://orcid.org/0000-0002-1643-4466 mailto:dwi_ertiningsih@ugm.ac.id https://orcid.org/0000-0002-4529-8440 https://creativecommons.org/licenses/by/4.0/ https://doi.org/10.55630/j.biomath.2023.01.307 mahardhika et al., bifurcation analysis of mathematical model of microalgae growth under the influence of sunlight biofuel is mostly made from vegetable oil or animal fats [2]. therefore, microalgae can play an important role to reduce carbon dioxide levels in the atmosphere and become a candidate for renewable energy sources to replace fossil fuels [3]. furthermore, the potential of biofuel products from the microalgae is expected to be abundant [4]. the theoretical investigation of microbial growth and nutrient dynamics in chemostats was pioneered by j. monod [5]. the model proposed that extracellular nutrient concentration is a determinant of cellular growth. however, microorganisms have nutrient reserves in cells that are used for growth and development, which causes the growth rate of microorganisms depend on the nutrient reserves in cells. thus the monod model is a weak growth predictor. furthermore, m.r. droop [6], purposed a new empirical model as an update of the monod model which is known as the cell quota model with a specific growth rate. droop’s model is a function of cell quota (intracellular density) of limiting nutrients. the monod and droop models are the basis of the general microorganism growth model. in fact, microalgae not only absorb nutrients but also perform photosynthesis. the growth model of microalgae is mostly based on the droop model and then modified with the addition of the sunlight factor. however, the model is limited by the conditions of low nutrient levels. since light intensity which transmits a medium can be explained with lambert-beer law [7], the maximum growth of microalgae under influence of light can be formulated by lambert-beer law. furthermore, the microalgae growth model – this is a new derivation after the lambert-beer law is considered – contains the exponential term. there are several problems in the mathematical models of microalgae growth. one of which is the model with the influence of sunlight that contains an exponential form, so it is quite difficult to carry out stability analysis and bifurcation analysis on the model. our model is motivated by the one in [5], where the mortality rate of the system in [5] represents chemostat dilution. however, based on the fact that microalgae have a natural mortality rate at room temperature, see [8], we add the microalgae natural mortality rate in our system. since this model contains an exponential term, we apply the lambert ω function to determine equilibrium points (see [9]). furthermore, in the fourth section, we analyze the stability of the equilibrium point by linearization technique (see [10, 11]) to provide the stability of the equilibrium points. since the stability conditions depend on the parameters, then we can divide the parameter space into several regions. lastly, the occurrence of bifurcation is investigated by analyzing those regions numerically (see the bifurcation theory in [12–14]) and are established bifurcation parameters with respect to four varying parameters. ii. microalgae growth model a. microalgae microalgae are autotrophs that produce their food through the process of photosynthesis [15]. photosynthesis is the process of converting inorganic compounds co2 and h2o into glucose under sunlight. solar energy is used in this process [16, 17]. nutrients in cell quotas depend on nutrients in the environment, while microalgae growth depends on photosynthesis and nutrient levels in cell quotas. microalgae require large amounts of macro and micro nutrients for their growth. there is a linear relationship between nutrient density and biomass, because microalgae absorb nutrients from the environment and then store them in the cell quotas [1]. microalgae assimilate various organic carbon and inorganic sources as nutrients (such as glucose, acetate, nitrogen, and phosphor) for their growth [18]. light is used by microalgae to break down co2 into glucose in the process of photosynthesis. to make an efficient microalgae culture for microalgae growth, it is necessary to optimize the intensity of light that enters the culture. in indoor and outdoor microalgae cultivation systems, light source and light intensity are important factors that affect the phototrophic growth performance of microalgae [1]. light can be transmitted by a medium that absorbs the light intensity. the light intensity that is absorbed depends on the concentration of liquid and the thickness of the medium [19]. microalgae have a level of turbidity in the water so that they can block the light intensity. thus, if the microalgae content in the chemostat is denser, less light can enter. so, less light can be used by microalgae. several models in the literature have been developed to explain microalgae growth under influence of light, especially in monocultures [20]. b. microalgae growth models under influence of sunlight we propose a new microalgae growth model under the sunlight influence which is motivated by the one in [5]. in our model, we consider the microalgae natural mortality rate at room temperature m, not due to harvesting d. this could explain more the natural biomath 12 (2023), 2301307, https://doi.org/10.55630/j.biomath.2023.01.307 2/9 https://doi.org/10.55630/j.biomath.2023.01.307 mahardhika et al., bifurcation analysis of mathematical model of microalgae growth under the influence of sunlight behavior of microalgae at room temperature. the model is as follows: da(t) dt = a(t) ( ln(t)βki0e −(σ+ka(t))z ln(t) + qβki0e−(σ+ka(t))z −m ) (1) dn(t) dt = d ( nc −n(t) ) − ln(t)a(t), (2) where a(t),n(t),σ,k,i0,nc and z represent the microalgae content, nutrient content, turbidity of the water, the coefficient of turbidity due to the presence of microalgae concentration, light intensity, stock of nutrient, and depth of the medium respectively. note: nc, a(t), q, and n(t) were normalized by the density of water (1 g cm−3) to obtain relative density (n.d. – non-dimensional) in the model (see [5]). the chemostat contains n(t) (in g cm−3) extracellular nutrients for microalgae. microalgae a(t) (in g cm−3) in the chemostat is always given new nutrients nc in the form of dilution d to replace the old nutrients in the chemostat. furthermore, the new nutrients enter the chemostat with the rate dnc and the old nutrients are pushed out through the pipe with the rate dn(t). according to lambert-beer law, the light intensity can be absorbed by microalgae at the rate ka(t)z. the thickness of the water causes the light intensity to decrease with a decreasing rate σz. by using lambertbeer law equation, the light intensity can be formulated as i = i0e−(σ+ka(t))z. furthermore, microalgae need sunlight to carry out photosynthesis and then the products of photosynthesis are used to grow. we assume the maximum microalgae growth rate has a linear relationship with the light intensity. the relationship between the maximum microalgae growth rate and the light intensity can be formulated as µmax = βki, where β represents a constant the ratio between the maximum microalgae growth rate and the light intensity. microalgae absorb nutrients from outside the cells into the quota cell at the rate ln(t). according to the droop quota cell model, the nutrients in the quota cells help the microalgae to grow. then the nutrient in the quota cell can be formulated q = ln(t) µmax + q. by using the droop quota cell model, the microalgae growth rate can be formulated as µ = µmax(1 − qq), where q and µ represent the minimum quota cell for microalgae to grow and the growth rate of microalgae, respectively. furthermore, the microalgae content is increased by µa(t). theorem 1 guarantees a non-negative model solution. next, theorem 2 guarantees that the model solution is bounded. theorem 1. given a(0) and n(0) are non-negative. if all parameters in the model are positive, then a(t) and n(t) are non-negative. proof: will be proved by contradiction. assume that the solution a(t) and n(t) always decrease and then will be negative. let say that as follows. for 0 < t′ < t0, then a(t′) > 0 and n(t′) > 0. at the time t0, the variables are a(t0) = 0 and n(t0) = 0, and its derivatives are a′(t0) < 0 and n′(t0) < 0. substituting t0 into equation (1) and (2), then we get the following: da(t0) dt = a(t0) ( ln(t0)βki0e −(σ+ka(t0))z ln(t0) + qβki0e−(σ+ka(t0))z −m ) = 0, dn(t0) dt = d ( nc −n(t0) ) − ln(t0)a(t0) = dnc > 0. so that, there is a contradiction. hence, the solution a(t) and n(t) will not be negative. theorem 2. given a(0) and n(0) are non-negative. if all parameters in the model are positive, then the solution a(t) and n(t) are bounded. proof: will be proved by contradiction. assume that the solution a(t) and n(t) are unbounded. hence, a(t) and n(t) always increase. for n(t) will be: n(t) > nc. let say that as follows. for 0 < t′ < t0, at the time t′: n(t′) < nc. at the time t0, the variables are: n(t0) = nc. because the solution n(t) always increase, then dn(t0) dt > 0. substituting t0 into equation (2), then we get the following: dn(t0) dt = d (nc −n(t0)) − ln(t0)a(t0) = −lnca(t0) < 0. so that, there is a contradiction. hence, the solution n(t) is bounded. furthermore, n(t) will not be greater than nc. biomath 12 (2023), 2301307, https://doi.org/10.55630/j.biomath.2023.01.307 3/9 https://doi.org/10.55630/j.biomath.2023.01.307 mahardhika et al., bifurcation analysis of mathematical model of microalgae growth under the influence of sunlight define r1 = lβki0, r2 = qβki0. now for the solution a(t), considered below: da(t) dt = a(t) ( ln(t)βki0e −(σ+ka(t))z ln(t) + qβki0e−(σ+ka(t))z −m ) = a(t) ( n(t)r1 ln(t)e(σ+ka(t))z + r2 −m ) = a(t) ( n(t) ( r1 −mle(σ+ka(t))z ) −mr2 h1 ) , for h1 = ln(t)e(σ+ka(t))z + r2. since a(t) always increase, then da(t)/dt > 0 for all t > 0. it means, n(t) ( r1 −mle(σ+ka(t))z ) −mr2 h1 > 0, for all t > 0. since h1 > 0, then: n(t) ( r1 −mle(σ+ka(t))z ) > mr2. since n(t) and mr2 are non-negative, then we get as follows: r1 −mle(σ+ka(t))z > 0. furthermore, note that a(t) always increase. let say that as follows. for 0 < ts < t̄, such that at the time ts we have: r1 −mle(σ+ka(ts))z > 0. because the solution a(t) always increase, at the time t̄ we get: r1 −mle(σ+ka(t̄))z = 0, and da(t̄)/dt > 0. next step, substituting t̄ into equation (1), then we get: da(t̄) dt = a(t̄) n(t̄) ( r1 −mle(σ+ka(t̄))z ) −mr2 h1 = −a(t̄) mr2 h1 < 0. so that there is a contradiction. hence, the solution a(t) is bounded. iii. determine equilibrium points in this section we will discuss about how to determine equilibrium points of the model. the lambert ω function is used to determine the equilibrium point. equilibrium conditions are da(t)/dt = 0 and dn(t)/dt = 0, so that the following equation is obtained: 0 = a(t) ( ln(t)βki0e −(σ+ka(t))z ln(t) + qβki0e−(σ+ka(t))z −m ) , (3) 0 = d (nc −n(t)) − ln(t)a(t). (4) from the equation (3) we divide it into two cases, they are a = 0 or a 6= 0. for a = 0 obtained below: 0 = d (nc −n(t)) − ln(t)a(t) ⇒ n(t) = nc. so, the first equilibrium point is (a∗1,n ∗ 1 ) = (0,nc). for a 6= 0 in the nutritional equation dn(t)/dt = 0, it is obtained as follows: 0 = d (nc −n(t)) − ln(t)a(t) ⇒ n(t) = dnc d + la(t) . (5) furthermore, because a 6= 0, in order to satisfy the equilibrium condition, it is obtained as follows: n(t) ( r1 −mle(σ+ka(t))z ) −mr2 = 0. (6) substituting equation (5) into equation (6) yields: dnc ( r1 −mle(σ+ka(t))z ) −mr2(d + la(t)) d + la(t) = 0. (7) prior to further discussion, note the following coefficients. recall that r1 = lβki0, r2 = qβki0. define a = mldnce σz, b = lmr2, c = dmr2 −dncr1, and d = kz. so that to meet the equilibrium conditions, (7) yields: aeda(t) + ba(t) + c = 0. (8) furthermore, equation (8) is an exponential equation, using the lambert ω function (for example see [21]), the root of the equation is as follows: a∗ = − bω ( ade−cd/b b ) + cd bd , (9) where ω ( ade−cd/b b ) is the lambert ω function that is executed on ade −cd/b b . by substituting equation (9) into equation (5), we get the following: n∗ = dncbd dbd− l ( bω(ade −cd/b b ) + cd ). (10) so, the second equilibrium point (a∗2,n ∗ 2 ) is:( − bω(ade −cd/b b ) + cd bd , dncbd dbd− l ( bω(ade −cd/b b ) + cd ) ) . furthermore, the existence of these two equilibrium points will be investigated. in this model, the equilibrium point exist, if a∗,n∗ ≥ 0. lemma 3. if all parameters in the model are positive, then the equilibrium point (a∗1,n ∗ 1 ) = (0,nc) exists. biomath 12 (2023), 2301307, https://doi.org/10.55630/j.biomath.2023.01.307 4/9 https://doi.org/10.55630/j.biomath.2023.01.307 mahardhika et al., bifurcation analysis of mathematical model of microalgae growth under the influence of sunlight proof: it is known that all parameters are greater than 0, then nc > 0. since a∗1 = 0 and n ∗ 1 = nc > 0, the equilibrium point (a∗1,n ∗ 1 ) = (0,nc) exists. lemma 4. if all parameters in the model are positive and a ≤ −c, then the equilibrium point (a∗2,n∗2 ) =( − bω( ade−cd/b b )+cd db , dnc d+la∗2 ) exists. proof: first, we will prove that a∗2,n ∗ 2 ∈ r. it is known that all parameters are positive, then a = mldnce σz, b = lmr2, and d = kz are positive. so, it yields: ade−cd/b b > 0 > − 1 e . let y = ade −cd/b b . according to the nature of the lambert ω function, since y > −1 e , then ω(y) ∈ r. so that a∗2,n ∗ 2 ∈ r. next, we will prove a∗2,n ∗ 2 ≥ 0. it is known that a ≤−c, then because b,d > 0, it is obtained as follows: a ≤−c ⇒ ade−cd/b b ≤− cde−cd/b b . let x = −cd b , then y = ade x b . let g = xex, as follows: g = xex = − cde−cd/b b . therefore, y ≤ g. also, we have x = ω(g). since 0 < a ≤ −c, then y and g are ascending (increasing) functions. then we apply lambert ω function as follows: ω(y) ≤ ω(g) ⇒ ω (ade−cd/b b ) ≤− cd b ⇒− bω ( ade−cd/b b ) + cd b ≥ 0 ⇒−d bω ( ade−cd/b b ) + cd bd ≥ 0 ⇒ da∗2 ≥ 0. since d > 0, then a∗2 ≥ 0. furthermore, because all parameters are positive, then n∗2 = ncd d+la∗2 ≥ 0. therefore, the second equilibrium point exists. iv. stability analysis in this section, we will discuss the stability of the equilibrium points that had been obtained. further on we use a = a(t) and n = n(t) for brevity. define da/dt = g1(a,n) and dn/dt = g2(a,n). note the following: ∂ ∂a g1(a,n) = −ar1kzlne(σ+ka)z (lne(σ+ka)z + r2)2 + r1n lne(σ+ka)z + r2 −m ∂ ∂n g1(a,n) = ar1r2 (lne(σ+ka)z + r2)2 ∂ ∂a g2(a,n) = −ln ∂ ∂n g2(a,n) = −d − la. by using the above equations, the jacobian matrix is obtained as follows, j(a,n) =   ∂∂ag1(a,n) ∂∂n g1(a,n) ∂ ∂a g2(a,n) ∂ ∂n g2(a,n)   . theorem 5. given all of the parameters are positive. if ∂g1(a∗1,n ∗ 1 )/∂a < 0 ⇒ r1nc lnceσz+r2 < m, then the first equilibrium point (a∗1,n ∗ 1 ) is stable. proof: note that ∂ ∂n g1(a ∗ 1,n ∗ 1 ) = 0. substituting the first equilibrium point into the jacobian matrix, it yields: j(a∗1,n ∗ 1 ) =   ∂∂ag1(a∗1,n∗1 ) 0 ∂ ∂a g2(a ∗ 1,n ∗ 1 ) ∂ ∂n g2(a ∗ 1,n ∗ 1 )   . according to the jacobian matrix, the polynomial characteristic is as follows: |j(a∗1,n ∗ 1 ) −λi| = 0( ∂ ∂a g1(a ∗ 1,n ∗ 1 ) −λ )( ∂ ∂n g2(a ∗ 1,n ∗ 1 ) −λ ) = 0. furthermore, the eigenvalues of the jacobian matrix at the first equilibrium point are obtained: λ∗1 = ∂ ∂a g1(a ∗ 1,n ∗ 1 ) = r1nc lnceσz + r2 −m λ∗2 = ∂ ∂n g2(a ∗ 1,n ∗ 1 ) = −d. because all parameters are positive and r1nc lnceσz+r2 < m ⇒ r1nc lnceσz+r2 − m < 0, then λ∗1 < 0 and λ∗2 < 0. so, the first equilibrium point is stable. theorem 6. given all of the parameters are positive. if ∂g1(a∗2,n ∗ 2 )/∂a < 0, then the second (a ∗ 2,n ∗ 2 ) equilibrium point is stable. biomath 12 (2023), 2301307, https://doi.org/10.55630/j.biomath.2023.01.307 5/9 https://doi.org/10.55630/j.biomath.2023.01.307 mahardhika et al., bifurcation analysis of mathematical model of microalgae growth under the influence of sunlight proof: by substituting the second equilibrium point into the jacobian matrix, it yields: j(a∗2,n ∗ 2 ) =   ∂∂ag1(a∗2,n∗2 ) ∂∂n g1(a∗2,n∗2 ) ∂ ∂a g2(a ∗ 2,n ∗ 2 ) ∂ ∂n g2(a ∗ 2,n ∗ 2 )   . let: j(a∗2,n ∗ 2 ) =  t1 t2 t3 t4   . furthermore, the polynomial characteristic is |j(a∗2,n ∗ 2 ) −λi| = 0, λ2 − (t1 + t4)λ + t1t4 − t2t3 = 0. so, the eigenvalues are obtained: λ∗∗1,2 = t1 + t4 ± √ (t1 + t4)2 − 4(t1t4 − t2t3) 2 . note that all parameters are positive. then: t1 < 0, t2 > 0, t3 < 0, t4 < 0, since by the theorem we have t1 < 0. therefore the real part is negative: re(λ∗∗1,2) < 0. so, the second equilibrium point is stable. v. bifurcation analysis in the previous section, the equilibrium point stability conditions have been obtained. in this section, we will discuss bifurcation analysis of the model (1) and (2). since the equilibrium point stability conditions are a function of parameters, then these functions divide the parameter space into several regions. furthermore, the stability areas are formed on the parameter space. first, we investigate the bifurcation for the varied nc and d parameters. to find the bifurcation line, at least one of the eigenvalues in theorem 5 should be 0. let: λ∗1 = ∂ ∂a g1(a ∗ 1,n ∗ 1 ) = r1nc lnceσz + r2 −m = 0, λ∗2 = ∂ ∂n g2(a ∗ 1,n ∗ 1 ) = −d 6= 0. it means that a = −c. hence, we get the first equilibrium point a∗1 = 0 and n ∗ 1 = nc. since the other parameter values have been determined in table i, then we get the bifurcation line as function of parameters: b(nc,d) = r1nc lnceσz + r2 −m. the graph of the function b(nc,d) = 0 can be seen in figure 1. this function divides the parameter space into two regions, i.e., d1 and d2. in the d1 region for (nc,d) = (0.048, 1) and d2 region for (nc,d) = (0.378266, 1), the phase portrait can be seen in figure 2 and figure 3 respectively. in both figures, the red dot is the first equilibrium point (a∗1,n ∗ 1 ), while the blue dot is the second equilibrium point (a∗2,n ∗ 2 ). in the region d1 for the parameter value nc = 0.048 and d = 1, the solution a(t) converges at 0 while the solution n(t) converges at 0.048. so, the solution (a(t),n(t)) is stable at the first equilibrium point (a∗1,n ∗ 1 ) = (0, 0.048), as can be seen in figure 2. note that in the phase portrait, the second equilibrium point (a∗2,n ∗ 2 ) = (−1.865, 0.077) is not only non-existent but also unstable. so, the microalgae population tends to be 0 g cm−3 and nutrition tends to be 0.048 g cm−3 as t →∞. in the region d2 for the parameter value nc = 0.3783 and d = 1, the solution a(t) converges at 9.1807 while the solution n(t) converges at 0.1334. so, the solution (a(t),n(t)) is stable at the second equilibrium point (a∗2,n ∗ 2 ) = (9.1807, 0.1334), as can be seen in figure 3. note that in the phase portrait, the first equilibrium point (a∗1,n ∗ 1 ) = (0, 0.3783) is unstable. so, the microalgae population tends to be 9.1807 g cm−3 and nutrition tends to be 0.1334 g cm−3 as t →∞. based on the bifurcation analysis for the varied nc and d parameters, it shows that there is a shift in the stability of the equilibrium point. in the region d1, the first equilibrium point is stable while the second equilibrium point is unstable. in the region d2, the second equilibrium point is stable while the first equilibrium point is unstable, thus a transcritical bifurcation occurs. in the next discussion, we investigate the bifurcation for the varied i0 and z parameters. to find the bifurcation line, at least one of the eigenvalues in theorem 6 should be 0. let: λ∗∗1 = t1 + t4 + √ (t1 + t4)2 − 4(t1t4 − t2t3) 2 6= 0, λ∗∗2 = t1 + t4 − √ (t1 + t4)2 − 4(t1t4 − t2t3) 2 = 0. we easily derive: λ∗∗2 = t1t4 − t2t3 = alnr1r2 + a(d + la)r1kzlne (σ+ka)z (lne(σ+ka)z + r2)2 − r1n(d + la) lne(σ+ka)z + r2 + m(d + la) = 0. it means that a < −c. hence, we get the second equilibrium point (a∗2,n ∗ 2 ) defined earlier in section iii. biomath 12 (2023), 2301307, https://doi.org/10.55630/j.biomath.2023.01.307 6/9 https://doi.org/10.55630/j.biomath.2023.01.307 mahardhika et al., bifurcation analysis of mathematical model of microalgae growth under the influence of sunlight table i: parameter table with variations nc and d. parameter description value reference d dilution (0,1]day−1 assumption m microalgae natural death rate 0.3day−1 [8] k algae turbidity 0.032cm−1 [5] β proportional constant 10j−1 cm3 [5] σ water turbidity 0.1cm−1 [5] z chemostat depth 8cm assumption l nutrition uptake rate 0.2day−1 [5] i0 sunlight intensity 50j cm−1 day−1 [5] nc stock nutrition (0,1]g cm−3 assumption q minimum quota cell for algae grow 0.05g cm−3 [5] table ii: parameter table with variations i0 and z. parameter definition value reference d dilution 0.24day−1 [5] m microalgae natural death rate 0.3day−1 [8] k algae turbidity 0.032cm−1 [5] β proportional constant 10j−1 cm3 [5] σ water turbidity 0.1cm−1 [5] z chemostat depth (0,100]cm assumption l nutrition uptake rate 0.2day−1 [5] i0 sunlight intensity (0,100]j cm−1 day−1 assumption nc stock nutrition 1g cm−3 assumption q minimum quota cell for algae grow 0.05g cm−3 [5] since the other parameter values have been determined in table ii, then we get the bifurcation line as function of parameters: b(i0,z) = λ ∗∗ 2 . the graph of the function b(i0,z) = 0 can be seen in figure 4. this function divides the parameter space into two regions, i.e., d1 and d2. in the d1 region for (i0,z) = (88, 50) and d2 region for (i0,z) = (88, 20), the portrait phase can be seen in figure 5 and figure 6 respectively. in both figures, the red dot is the first equilibrium point (a∗1,n ∗ 1 ) while the blue dot is the second equilibrium point (a∗2,n ∗ 2 ). in the region d1 for the parameter value i0 = 88 and z = 50, the solution a(t) converges at 0 while the solution n(t) converges at 1. so, the solution (a(t),n(t)) is stable at the first equilibrium point (a∗1,n ∗ 1 ) = (0, 1), as can be seen in figure 5. note that in the phase portrait, the second equilibrium point (a∗2,n ∗ 2 ) = (−0.3216, 1.3661) is not only non-existent but also unstable. so, the microalgae population tends to be 0 g cm−3 and nutrition tends to be 1 g cm−3 as t →∞. in the region d2 for the parameter value i0 = 88 and z = 20, the solution a(t) converges at 3.437 while the solution n(t) converges at 0.259. so, the solution (a(t),n(t)) is stable at the second equilibrium point (a∗2,n ∗ 2 ) = (3.437, 0.258), as can be seen in figure 6. note that in the phase portrait, the first equilibrium point (a∗1,n ∗ 1 ) = (0, 1) is unstable. so, the microalgae population tends to be 3.437 g cm−3 and nutrition tends to be 0.258 g cm−3 as t →∞. based on the bifurcation analysis for the varied i0 and z parameters, it shows that there is a shift in the stability of the equilibrium point. in the d1 area, the first equilibrium point is stable while the second equilibrium point is unstable. for the d2 region, the second equilibrium point is stable while the first equilibrium point is unstable, thus a transcritical bifurcation occurs. vi. conclusion according to the bifurcation analysis in the previous section, the transcritical bifurcation occurred for the varied nc and d parameters. in the same way, the transcritical bifurcation occurred for the varied i0 and z parameters. furthermore, for the varied nc and d parameters, the occurrence of transcritical bifurcations indicated that microalgae have the minimum supply of nutrition to survive. in the other words, microalgae can survive, if nutrition supply is more than the minimum biomath 12 (2023), 2301307, https://doi.org/10.55630/j.biomath.2023.01.307 7/9 https://doi.org/10.55630/j.biomath.2023.01.307 mahardhika et al., bifurcation analysis of mathematical model of microalgae growth under the influence of sunlight 0.0 0.2 0.4 0.6 0.8 nc 0.0 0.2 0.4 0.6 0.8 d d1 d2 parameter diagram fig. 1: parameter diagram (nc,d). −2.0 −1.5 −1.0 −0.5 0.0 a −0.4 −0.2 0.0 0.2 0.4 n phase portrait fig. 2: phase portrait in region d1 for nc = 0.048, d = 1. 0 2 4 6 8 10 a −0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 n phase portrait fig. 3: phase portrait in region d2 for nc = 0.3783, d = 1. 0 10 20 30 40 50 60 70 i0 0 10 20 30 40 50 60 70 z d2 d1 parameter diagram fig. 4: parameter diagram (i0,z). −0.4 −0.2 0.0 0.2 0.4 a 0.8 1.0 1.2 1.4 1.6 n phase portrait fig. 5: phase portrait in region d1 for i0 = 88, z = 50. −0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 a 0.0 0.2 0.4 0.6 0.8 1.0 n phase portrait fig. 6: phase portrait in region d2 for i0 = 88, z = 20. biomath 12 (2023), 2301307, https://doi.org/10.55630/j.biomath.2023.01.307 8/9 https://doi.org/10.55630/j.biomath.2023.01.307 mahardhika et al., bifurcation analysis of mathematical model of microalgae growth under the influence of sunlight supply of nutrition required for their survival. in this study, microalgae can be sustained, if the supply of nutrition is greater than 0.0967 g cm−3. meanwhile, for the varied i0 and z parameter, the occurrence of transcritical bifurcation can be interpreted as follows. microalgae that live in chemostat at depth z requires sufficient light intensity to survive. in other words, microalgae can be sustained in the chemostat at a constant light intensity i0, if the chemostat has a depth no more than a certain value. thus, microalgae have the minimum light intensity required for their survival in a chemostat at a certain depth. vii. acknowledgements the authors would like to thank the mathematics department of gadjah mada university and mr. hardi who have encouraged the authors to write this research paper. references [1] c.-y. chen, k.-l. yeh, r. aisyah, d.-j. lee, j.-s. chang, cultivation, photobioreactor design and harvesting of microalgae for biodiesel production: a critical review, bioresource technology, 102(1):71–81, 2011. 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determine equilibrium points stability analysis bifurcation analysis conclusion acknowledgements references www.biomathforum.org/biomath/index.php/biomath original article mathematical modeling and analysis of covid-19 infection spreads with restricted optimal treatment of disease incidence d. pal1,∗, d. ghosh2, p.k. santra3, g.s. mahapatra2 1chandrahati dilip kumar high school (h.s.) chandrahati 712504, west bengal, india 2department of mathematics, national institute of technology puducherry karaikal-609609, india 3maulana abul kalam azad university of technology kolkata-700064, india ∗correspondence email: pal.debkumar@gmail.com received: 13 august 2020, accepted: 14 june 2021, published: 17 july 2021 abstract— this paper presents the current situation and how to minimize its effect in india through a mathematical model of infectious coronavirus disease (covid-19). this model consists of six compartments to population classes consisting of susceptible, exposed, home quarantined, government quarantined, infected individuals in treatment, and recovered class. the basic reproduction number is calculated, and the stabilities of the proposed model at the disease-free equilibrium and endemic equilibrium are observed. the next crucial treatment control of the covid-19 epidemic model is presented in india’s situation. an objective function is considered by incorporating the optimal infected individuals and the cost of necessary treatment. finally, optimal control is achieved that minimizes our anticipated objective function. numerical observations are presented utilizing matlab software to demonstrate the consistency of present-day representation from a realistic standpoint. keywords-novel coronavirus; sehgir model; basic reproduction number; stability; optimal control i. introduction recently, the coronavirus disease has turned out to be a pandemic over almost the whole world. the basic indication of this infection is ordinary fever, cough, and breathing problems. this virus also showed the capability to produce serious health problems among a specific group of individuals, including the aged populace as well as patients with cardiovascular disease and diabetes [1]. however, a clear picture of the nature of this copyright: © 2021 pal et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: d pal, d ghosh, p k santra, g s mahapatra, mathematical modeling and analysis of covid-19 infection spreads with restricted optimal treatment of disease incidence, biomath 10 (2021), 2106147, http://dx.doi.org/10.11145/j.biomath.2021.06.147 page 1 of 20 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2021.06.147 d pal, d ghosh, p k santra, g s mahapatra, mathematical modeling and analysis of covid-19 ... epidemiology is still being explained [2]. this virus transmits from human to human as a result, covid-19 disease spread across the globe, and the total number of active covid-19 cases increases day by day ([3]-[10]). in particular, india has become the stiffest affected country with covid-19 endemic [11] due to its very high population densities. the figure for positive covid-19 infection started to increase from 4th march 2020. as of 8th may 2020, a total of 59690 confirmed covid cases, together with 17887 recovered and 1986 deaths in india [12]. different precaution measures ([13]-[16]) have been taken by the indian government to maintain social distance [17] among the huge numbers of the population in india. there is no specific medicine for covid-19 infection to date. therefore, doctors recommended different treatments via medication to covid-19 patients depending on their symptoms. the therapy and vaccine yet to get, spread of coved 19 diseases can be restricted via appropriate precautionary measures like quarantined mechanisms ([18]-[20]), individual safeguard from the infected individual by using social distancing [21], etc. as the covid-19 virus spread very quickly throughout the world, so various mathematical models depending on the pandemic outbreak ([22]-[32], [33], [34], [35]) have been performed already. wu et al. [36] studied the dynamics behind the spread of covid19 virus world-wise using seir model. read et al. [37] developed a covid-19 seir model based on poisson-distributed daily time augmentations. paul et al. [38] presented a mathematical model on covid-19 incorporating the different safety strategies to protect the citizens from the virus. sardar et al. [39] proposed a mathematical model to identify the lockdown effect of the spreading of covid-19 disease in india. pal et al. [40] explored a covid-19 based seqir model to understand india’s disease situation. this paper introduces a six-compartmental covid-19 infection model by separating the total populace into six classes, purposely susceptible, exposed, home quarantined, government quarantined, infected individuals in treatment as well as recovered class. we introduce treatment control in the model to assimilate realistic and biologically significant in the pandemic situation. a brief description of the necessary and sufficient conditions for the existence of multi-objective optimal control is provided in section 2. the model derivation and preliminaries are explained in section 3. the basic properties of our proposed model structure are discussed in section 4. in section 5, we introduce the concept of the basic reproduction number (r0) [41]. next, we deal with disease-free equilibrium (dfe) (e0) and endemic equilibrium (e1) points of the system. it is clear that covid19 infection is not only community health trouble [42] but also a tremendous societal and monetary shock on the developing countries. therefore, it is an essential concern to control ([43]-[46]) the spread of covid-19 infection in india by adopting several optimal control policies. in section 6, we have formulated the covid-19 epidemic model with control treatment. this section provides us a procedure to find optimal control [47] u(t) that increases the recovery rate as well as minimizes the cost associated with the treatment. analytical results are obtained in the previous sections are numerically verified in section 7 with the help of realistic values of the model parameters using matlab. lastly, a general conclusion about our proposed model structure is provided in section 8. ii. multi-objective optimal control suppose x(t) ∈ x ⊂ rn represents the state variables of a system and u(t) ∈ u ⊂ rm represents the control variables at time t, with t0 ≤ t ≤ tf . an optimal control problem consists of finding a piecewise continuous control u(t) and the associated state x (t) that optimizes a cost function j [u(t),x (t)]. the majority of mathematical models that use the optimal control theory rely on pontryagin’s maximum principle, a first-order condition for finding the optimal solution. theorem 1. (pontryagin’s maximum principle [48]) if u∗ (t) and x∗(t) are optimal for the problem max u j [u(t),x (t)] , (1) biomath 10 (2021), 2106147, http://dx.doi.org/10.11145/j.biomath.2021.06.147 page 2 of 20 http://dx.doi.org/10.11145/j.biomath.2021.06.147 d pal, d ghosh, p k santra, g s mahapatra, mathematical modeling and analysis of covid-19 ... where j [u(t),x (t)] = max u ∫ tf t0 f (t,x (t) ,u (t)) dt, subject to { dx dt = g (t,x (t) ,u (t)) , x (t0) = x0, then there exists a piecewise differentiable adjoint variable λ (t) such that h(t,x∗(t),u(t),λ(t))≤h(t,x∗(t),u∗(t),λ(t))) for all controls u at each time t, where the hamiltonian h is given by h(t,x(t),u(t),λ(t)) = f (t,x (t) ,u (t)) + λg (t,x (t) ,u (t)) (2) and{ λ ′ (t) = − ∂h (t,x∗ (t) ,u∗ (t) ,λ (t)) ∂x , λ (tf ) = 0. while pontryagin’s maximum principle gives the necessary conditions for the existence of an optimal solution, the following theorem provides sufficient conditions. theorem 2. (arrow sufficiency theorem [49]) for the optimal control problem (1), the conditions of the maximum principle are sufficient for the global minimization of j [u(t),x (t)], if the minimized hamiltonian function h, defined in (2), is convex in the variable x for all t in the time interval [t0, tf ] for a given λ. one of the major side effects of vaccination/treatment is the creation of drug resistant virus/bacteria which eventually leads to drug failure (due to ineffectiveness of the vaccine/treatment). optimal control has been used to curb the creation of drug resistant virus/bacteria or drug failure (at the same time reducing the cost of treatment or vaccination) by imposing a condition that monitors the global effect of the vaccination/treatment program. hence if x(t) represents the group of individuals to be vaccinated/treated and u(t) ∈ u represents the control on vaccination/treatment, where the control set u is given by u ={u(t) :v0≤ u(t)≤v1, lebesgue measurable}, then, the following objective functions are to be minimized simultaneously: i1 (u) = ∫ tf t0 x (t) dt and i2 (u) = ∫ tf t0 um (t) dt, for m > 0, and the optimal solution can be represented as min u {i1 (u) ,i2 (u)} . in general, there does not exist a feasible solution that minimizes both objective functions simultaneously. hence, the pareto optimality concept is used to simultaneously find optimal control u∗ that minimizes both objective functions. iii. derivation and preliminaries of covid-19 model this section develops a mathematical model of covid-19 transmission with the subsequent suppositions: the underlying human population is split up into six mutually exclusive compartments, namely, susceptible (s), exposed (infected but not yet infectious) (e), home quarantined population (population were exposed to the virus but viewing light symptoms of coronavirus disease and stay at home isolation) (h), government quarantined population (population was infective in symptomatic phase, i.e., showing symptoms of coronavirus disease and stay at government observation places for isolation) (g), infected (i), and recovered class (r) (infectious people who have cleared or recovered from coronavirus infection). therefore, the total human population n(t) = s(t) + e(t) + h(t) + g(t) + i(t) + r(t). this model involves certain assumptions which consist of the following: (i) the susceptible population (s) comprises individuals who have not yet been infected by covid-19, but may be infected through biomath 10 (2021), 2106147, http://dx.doi.org/10.11145/j.biomath.2021.06.147 page 3 of 20 http://dx.doi.org/10.11145/j.biomath.2021.06.147 d pal, d ghosh, p k santra, g s mahapatra, mathematical modeling and analysis of covid-19 ... contact with both types of home quarantined (h) and government quarantined (g) people. (ii) the exposed population (e) comprises individuals infected with covid-19 infection but not infectious. (iii) the infective population in home quarantined phase (h) comprises individuals who have covid-19 infection with light symptoms (but capable of infecting) and quarantined at home for isolation. (iv) the infective population in the symptomatic phase (g) comprises individuals who have developed covid-19 infection with complications and various symptoms but their test report yet not come positive and are quarantined by the government facility for isolation. (v) the infected population (i), whose covid 19 test is positive clinically and stayed at hospital for treatment (incapable of infecting others). the infected individuals coming from home and government quarantined compartments if their test report comes positive. (vi) the recovered class (r) consists of those who become healed from the disease by treatment or quarantined program. (vii) the susceptible individuals become infected by adequate contact with infective individuals in the asymptomatic phase (home quarantined) and symptomatic phase (government quarantined), and enter into the exposed class. the susceptible population is also decreased due to natural death. (viii) the exposed population is entered into the home quarantined, government quarantined, and infected population, respectively. the said population is also diminished due to natural death. (ix) one part of home quarantined individuals enters into the infected population, and the other becomes recovered. this population is also decreased by natural death. (x) one part of the government quarantined individuals enters into the infected population, and the other becomes recovered. this individual is also decreased by natural death. (xi) one part of the infected population enters into the recovered class. other individuals are decreased due to infection and natural death. (xii) home quarantined (asymptomatic), government quarantined (symptomatic), and the infected population recover from the coronavirus disease and enters into the recovered class. the recovered population diminishes by natural death. the parameters of the covid-19 model are presented as follows: λ : the recruitment rate of susceptible from embedding population. α1 : the coefficient of transmission rate from home quarantined to susceptible individuals, and the expression gives the transmission rate: α1h(t)s(t). α2 : the coefficient of transmission rate from government quarantined population to susceptible individuals, and the transmission rate is: α2g(t)s(t). β1 : the fraction of exposed individuals that will start to show light symptoms of covid-19 (but remains capable of infecting others) and move to the class h. β2 : the rate at which the exposed individuals become infected by covid 19 infection and move to the class i. β3 : the fraction of the exposed individuals that will start to show symptoms of infection and move to the class g. γ2 : the rate of home quarantined individuals eventually show disease symptoms and move to class i. γ1 : the recovery rate of the home quarantined population h. σ2 : the rate at which government quarantined individuals eventually show disease symptoms and move to class i. σ1 : the recovery rate of the government quarantined population g. biomath 10 (2021), 2106147, http://dx.doi.org/10.11145/j.biomath.2021.06.147 page 4 of 20 http://dx.doi.org/10.11145/j.biomath.2021.06.147 d pal, d ghosh, p k santra, g s mahapatra, mathematical modeling and analysis of covid-19 ... d2 : the disease-related death rate of infective population in the infected phase i � : the recovery rate of the infected population i. d1 : the natural death rate of all human epidemiological classes. in our proposed covid-19 model, s(t), e(t), h(t), g(t), i(t), and r(t) denote the numbers of susceptible, exposed, home quarantined, government quarantined, infected, and recovered, respectively. through the contact between susceptible and home quarantined populations, a part of the susceptible population, i.e., α1h(t)s(t), becomes infected and enters into exposed class. similarly, through the contact between susceptible and government quarantined populations, a part of the susceptible people, i.e., α2g(t)s(t), becomes infected and enters into the exposed category. the fraction of the home quarantined population γ2 will start to show symptoms of and move to the class i. another portion of the home quarantined population γ1 is recovered from infection due to treatment or quarantined process and move to the recovered class r. similarly, a fraction of the government quarantined community σ2 will start to show symptoms of covid 19 and move to the class i. other portion of the home quarantined population σ1 is recovered from infection due to treatment or quarantined process and move to the recovered class r. a fraction of infected individuals � is recovered from infection through treatment in hospital and move to recovered class r. another fraction d2 of the infected individuals is diminished due to the disease-related death rate of the infective population. from every class, a part of the inhabitants is reduced at the natural death rate d1. we diagrammatically represent the flow of individuals from one class to another in fig. 1. therefore, our proposed mathematical model of the covid-19 infection is presented through the fig. 1. pictorial representation of proposed covid 19 model for indian scenario following set of non-linear differential equation ds dt = λ − (α1h + α2g) s −d1s, (3) de dt = (α1h + α2g) s − (β1 + β2 + β3 + d1) e, dh dt = β1e −γ1h −γ2h −d1h, dg dt = β3e −σ1g−σ2g−d1g, di dt = β2e + γ2h + σ2g−d1i −d2i − �i, dr dt = �i −d1r + γ1h + σ1g; with initial conditions: s(0) > 0,e(0) ≥ 0,h(0) ≥ 0, g(0) ≥ 0,i(0) ≥ 0,r(0) ≥ 0. (4) the sehgir model formulation (3) can be biomath 10 (2021), 2106147, http://dx.doi.org/10.11145/j.biomath.2021.06.147 page 5 of 20 http://dx.doi.org/10.11145/j.biomath.2021.06.147 d pal, d ghosh, p k santra, g s mahapatra, mathematical modeling and analysis of covid-19 ... rewritten as: ds dt = λ − (α1h + α2g) s −d1s, (5) de dt = (α1h + α2g) s −ae, dh dt = β1e −bh, dg dt = β3e −cg, di dt = β2e + γ2h + σ2g−di, dr dt = γ1h + σ1g + �i −d1r; where a = β1 + β2 + β3 + d1, b = γ1 + γ2 + d1, c = σ1 + σ2 + d1, d = d1 + d2 + �, with initial conditions (4). iv. fundamental properties a. positivity of the solutions theorem 3. each solution of the proposed system (5) under conditions (4) satisfy s(t) > 0, e(t) ≥ 0, h(t) ≥ 0, g(t) ≥ 0, i(t) ≥ 0, r(t) ≥ 0 for all values of t ≥ 0. proof: the first equation of the system (5), can be written ds dt = λ − (α1h + α2g) s −d1s = λ −ψs; where ψ = (α1h + α2g) − d1. thereafter by integration, we obtain the following expression s(t)=s(0) exp ( − ∫ t 0 ψ(s)ds ) +λ exp ( − ∫ t 0 ψ(s)ds )∫ t 0 e ∫ t 0 ψ(v)dvds>0. hence s(t) is non-negative for all t. from the next equation of (5), we get, de dt ≥−ae. this equation provides e(t) ≥ e(0) exp(−at) ≥ 0. also, from the remaining equations and with the help of initial conditions, we obtain h(t) ≥ h(0) exp(−bt) ≥ 0, g(t) ≥ g(0) exp(−ct) ≥ 0, i(t) ≥ i(0) exp(−dt) ≥ 0, as well as r(t) ≥ r(0) exp(−d1t) ≥ 0. so, it is observed that s(t) > 0, e(t) ≥ 0, h(t) ≥ 0, g(t) ≥ 0, i(t) ≥ 0, r(t) ≥ 0 for all values of t ≥ 0. hence the theorem. b. invariant region theorem 4. the feasible region γ defined by γ = { (s,e,h,g,i,r) ∈ r6+ : 0 < n ≤ λ η } , where η = min{d1,d1 + d2} is positively invariant for the system (3). proof: let ((s(0),e(0),h(0),g(0),i(0),r(0)) ∈ γ. adding the equations of the system (3) we obtain dn dt = λ−d1s−d1e−d1h−d1h−d1g−(d1+d2)i−d1r. therefore, dn dt +ηn = λ−(d1−η)s−(d1−η)e−d1h −(d1−η)h−(d1−η)g−(d1 +d2−η)i (6) −(d1−η)r ≤ λ, where η = min{d1,d1 + d2}. the solution n(t) of the differential equation (6) has the following property, 0 < n(t) ≤ n(0) exp(−ηt) + λ η (1 − exp(−ηt)), where n(0) represents the sum of the initial values of the variables. as t → ∞, we have 0 < n(t) ≤ λ η . also, if n(0) ≤ λ η then n(t) ≤ λ η biomath 10 (2021), 2106147, http://dx.doi.org/10.11145/j.biomath.2021.06.147 page 6 of 20 http://dx.doi.org/10.11145/j.biomath.2021.06.147 d pal, d ghosh, p k santra, g s mahapatra, mathematical modeling and analysis of covid-19 ... for all t. this means λ η is the upper bound of n. on the other hand, if n(0) > λ η implies n(t) will decrease to λ η . this means that if n(0) > λ η , the solution (s(t),e(t),h(t),g(t),i(t),r(t)) enters γ or approaches it asymptotically. hence it is positively invariant under the flow induced by the systems (3) and (4). thus in γ the mathematical model (3) with initial conditions (4) is well-posed epidemiologically. hence it is sufficient to study the dynamics of the model in γ. v. existence of equilibrium and stability analysis in this section, we will study the existence and stability behavior of the system (3) at various equilibrium points. the equilibrium points of the system (2.1) are: (i) disease-free equilibrium (dfe): e0 ( λ d1 , 0, 0, 0, 0, 0 ) , (ii) endemic equilibrium: e1(s ∗,e∗,h∗,g∗,i∗,r∗). a. the basic reproduction number the basic reproduction number (brn) ([50][53]) of the system (3) will be obtained by the next-generation matrix method [54]. let z = (e(t),h(t),g(t),i(t),s(t),r(t))t , the proposed covid-19 system (3) can be written in the following form: dz dt = z(z) −υ(z); where z(z) =   (α1h + α2g) s 0 0 0 0 0   , υ(z) =   ae −β1e + bh −β3e + cg −(β2e + γ2h + σ2g) + di −(γ1h + σ1g + �i) + d1r −λ + (α1h + α2g) s + d1s   . the jacobian matrices of z(z) and υ(z) at the dfe e0 are as follows, respectively: dz(e0) =  f4×4 0 00 0 0 0 0 0   , dυ(e0) =  v4×4 0 00 0 0 0 0 0   , where f =   0 λα1 d1 λα2 d1 0 0 0 0 0 0 0 0 0 0 0 0 0   , v =   a 0 0 0 −β1 b 0 0 −β3 0 c 0 −β2 −γ2 −σ2 d   . following [54], r0 = ρ ( fv −1 ) where ρ is the spectral radius of the next-generation matrix (fv −1). thus, from the model (3), we have the following expression for brn r0 : r0 = λ d1 1 abc [α1β1c + α2β3b]. notice that λ d1 is the number of susceptibles at the dfe. b. existence of endemic equilibrium e1(s ∗,e∗,h∗,g∗,i∗,r∗) in this section, we will analyze the existence of a non-trivial endemic equilibrium e1(s ∗,e∗,h∗,g∗,i∗,r∗) of the system (3). to find the endemic equilibrium of the system (3), we consider the following: s(t) > 0,e(t) > 0,h(t) > 0, g(t) > 0,i(t) > 0,r(t) > 0 biomath 10 (2021), 2106147, http://dx.doi.org/10.11145/j.biomath.2021.06.147 page 7 of 20 http://dx.doi.org/10.11145/j.biomath.2021.06.147 d pal, d ghosh, p k santra, g s mahapatra, mathematical modeling and analysis of covid-19 ... and ds dt = 0, de dt = 0, dh dt = 0, dg dt = 0, dr dt = 0. (7) from the third, fourth, fifth, and sixth equation of (7) we obtain h∗ = β1e ∗ b , g∗ = β3e ∗ c , i∗ = e∗ d [β2 + γ2β1 b + β3σ2 c ], r∗ = e∗ d1 [ � d {β2 + γ2β1 b + β3σ2 c }+ γ1β1 b + β3σ1 c ] . now from de dt = 0 and using the values of h∗and g∗, we get, s∗ = λ d1r0 > 0. again, putting the value of s∗ in the first equation of (7) we gain, e∗ = λ a [ 1 − 1 r0 ] . hence, e∗ has a unique positive solution iff r0 > 1. summarizing the above discussions, we arrive at the following result. theorem 5. the system (3) has a dfe e0( λ d1 , 0, 0, 0, 0, 0), which exists for all parameter values. if r0 > 1 the system (3) also admits a unique endemic equilibrium e1(s ∗,e∗,h∗,g∗,i∗,r∗). c. asymptotic behavior for the stability of dfe e0( λd1 , 0, 0, 0, 0, 0) we consider the theorems given below theorem 6. the dfe e0 of the system (3) is locally asymptotically stable if r0 < 1. proof: see appendix a. theorem 7. the diseases free equilibrium (dfe) e0( λ d1 , 0, 0, 0, 0, 0)is globally asymptotically stable (gas) in r6+ for the system (3) if r0 < 1 and becomes unstable if r0 > 1. proof: we rewrite the system (3) as dx dt = f(x,v ), dv dt = g(x,v ), g(x, 0) = 0, where x = (s,r) ∈ r2 (the number of uninfected individuals compartments), v = (e,h,g,i) ∈ r4 (the number of infected individuals compartments), and e0( λd1 , 0, 0, 0, 0, 0) is the dfe of the system (3). the global stability of the dfe is guaranteed if the following two conditions are satisfied: (i) for dx dt = f(x, 0), x∗ is globally asymptotically stable in r2. (ii) g(x,v ) = bv − ĝ(x,v ), ĝ(x,v ) ≥ 0 for (x,v ) ∈ ω, where b = dv g(x∗, 0) is a metzler matrix, and ω is the positively invariant set to the model (3). following castillo-chavez et al. [55], we check for aforementioned conditions. for system (3), f(x, 0) = [ λ −d1s 0 ] , b =   −a λα1 d1 λα2 d1 0 β1 −b 0 0 β3 0 −c 0 β2 γ2 σ2 −d   and ĝ(x,v ) =   ( λ d1 −s)(α1h + α2g) 0 0 0   . clearly, ĝ(x,v ) ≥ 0 (using theorem 2), whenever the state variables are inside ω (the positively invariant set of the model (3)). again, it is clear that x∗ = ( λ d1 , 0)t is a globally asymptotically stable equilibrium of the system dx dt = f(x, 0). hence, the theorem follows. theorem 8. the endemic equilibrium point e1(s ∗,e∗,h∗,g∗,i∗,r∗) of the system (3) is locally asymptotically stable if r0 > 1, b1b2 − b3 > 0 and b1b2b3 −b21b4 −b 2 3 > 0 proof: see appendix b. biomath 10 (2021), 2106147, http://dx.doi.org/10.11145/j.biomath.2021.06.147 page 8 of 20 http://dx.doi.org/10.11145/j.biomath.2021.06.147 d pal, d ghosh, p k santra, g s mahapatra, mathematical modeling and analysis of covid-19 ... vi. proposed covid-19 model with control in this section, the primary focus is to set up an optimal control problem of the epidemic model (3). in the present situation of the covid19 outbreak, it is highly essential to construct an optimal control problem so that the total amount of drug is minimized. here we take one control variable u(t) on the recovery rate of the infectious individuals in the infected phase with treatment in the hospital. therefore, our epidemic model with one control and the same initial conditions (4) becomes: ds dt = λ − (α1h(t) + α2g(t)) s(t) −d1s(t), de dt = (α1h + α2g) s − (β1 +β2 +β3 +d1) e, dh dt =β1e −γ1h −γ2h −d1h, dg dt =β3e −σ1g−σ2g−d1g, di dt =β2e(t) + γ2h(t) + σ2g(t) (8) −(d1 + d2)i(t) −u(t)i(t), dr dt =u(t)i(t) + γ1h(t) + σ1g(t) −d1r(t). the control function u(t), 0 ≤ u(t) ≤ 1 represents the fraction of the infected individuals who are identified and will be treated. when u(t) is close to 1 then the treatment failure is low, but the implementation cost is high. for the model (8), the single-objective cost functional to be minimized is given by the objective functional ([56]-[59]) j(u(t)) = ∫ tf 0 [g1i + 1 2 g2u 2]dt; (9) with g1 > 0 and g2 > 0, where we want to minimize the infectious group i while also keeping the cost of treatment u(t) low. the term g1i represents the cost of infection, while the term 1 2 g2u 2 represents the cost of treatment. the goal is to find an optimal control, u∗, such that j (u∗) = min{j(u) : u ∈ u} , (10) where u ={u : u is lebesgue measurable, 0 ≤ u ≤ 1, t ∈ [0, tf ] } (11) applying the pontryagins maximum principle, we have the following result( s ∗ ,e ∗ ,h ∗ ,g ∗ ,i ∗ ,r ∗ ) of the system (8), that minimizes j(u) over u. theorem 9. there exists an optimal control u∗and corresponding solutions ( s ∗ ,e ∗ ,h ∗ ,g ∗ ,i ∗ ,r ∗ ) of the system (8), that minimizes j(u) over u. furthermore, there exist adjoint functions λi(t), i = 1, 2, 3, 4, 5, 6, such that dλ1 dt = (λ1−λ2)(α1h+α2g)+λ1d1, dλ2 dt = (λ2−λ3)β1 +(λ2−λ5)β2 +(λ2−λ4)β3 +λ2d1, dλ3 dt = (λ1−λ2)α1s+(λ3−λ6)γ1 +(λ3−λ5)γ2 +λ3d1, dλ4 dt = (λ1−λ2)α2s+(λ4−λ6)σ1 +(λ4−λ5)σ2 +λ4d1, dλ5 dt = (λ5−λ6)u+(d1 + d2)λ5−g1, dλ6 dt =d1λ6; with transversality conditions λi(tf ) = 0, i = 1, 2, 3, 4, 5, 6, and the control u∗ satisfies the optimality condition u∗ = min{max{0, (λ5 −λ6)i ∗ g2 }, 1}. proof: the hamiltonian is defined as follows: ĥ =g1i+ 1 2 g2u 2 +λ1[λ−(α1h+α2g) s−d1s] +λ2[(α1h + α2g) s −ae] (12) +λ3[β1e −bh] + λ4[β3e −cg] +λ5[β2e + γ2h + σ2g− (d1 + d2)i −ui] +λ6[ui + γ1h + σ1g−d1r], biomath 10 (2021), 2106147, http://dx.doi.org/10.11145/j.biomath.2021.06.147 page 9 of 20 http://dx.doi.org/10.11145/j.biomath.2021.06.147 d pal, d ghosh, p k santra, g s mahapatra, mathematical modeling and analysis of covid-19 ... where λi (i = 1, 2, 3, 4, 5, 6) are the adjoint functions to be determined. the form of the adjoint equations and transversality conditions are expected results from pontryagin’s maximum principle [60]. the adjoint system can be obtained as follows: dλ1 dt = − ∂ĥ ∂s = (λ1−λ2) (α1h+α2g)+λ1d1, dλ2 dt = − ∂ĥ ∂e = (λ2 −λ3)β1 + (λ2 −λ5)β2 +(λ2 −λ4)β3 + λ2d1, dλ3 dt = − ∂ĥ ∂h = (λ1 −λ2)α1s + (λ3 −λ6)γ1 +(λ3 −λ5)γ2 + λ3d1, dλ4 dt = − ∂ĥ ∂g = (λ1 −λ2)α2s + (λ4 −λ6)σ1 +(λ4 −λ5)σ2 + λ4d1, dλ5 dt = − ∂ĥ ∂i = (λ5−λ6)u+(d1 +d2)λ5−g1, dλ6 dt = − ∂ĥ ∂r = d1λ6. (13) the transversality conditions (or boundary conditions) are λi(tf ) = 0, i = 1, 2, 3, 4, 5, 6. (14) by the optimality condition, at u = u∗(t) we have ∂ĥ ∂u = g2u ∗ − (λ5 −λ6)i ∗ = 0 ⇒ u∗(t) = (λ5−λ6)i ∗ g2 . (15) by using the bounds for the control u(t), we get u∗ =   (λ5−λ6)i ∗ g2 , if 0 ≤ (λ5−λ6)i ∗ g2 ≤ 1. 0, if (λ5−λ6)i ∗ g2 ≤ 0. 1, if (λ5−λ6)i ∗ g2 ≥ 1. in compact notation: u∗ = min { max { 0, (λ5 −λ6)i ∗ g2 } , 1 } . (16) using (16), we obtain the following optimality system: ds dt = λ − (α1h + α2g) s −d1s, (17) de dt = (α1h + α2g) s −ae, dh dt =β1e −bh, dg dt =β3e −cg, di dt =β2e + γ2h + σ2g− (d1 + d2)i −min { max { 0, (λ5 −λ6)i ∗ g2 } , 1 } i, dr dt = min { max { 0, (λ5 −λ6)i ∗ g2 } , 1 } i +γ1h + σ1g−d1r, dλ1 dt = (λ1 −λ2) (α1h + α2g) + λ1d1, dλ2 dt = (λ2 −λ3)β1 + (λ2 −λ5)β2 +(λ2 −λ4)β3 + λ2d1, dλ3 dt = (λ1 −λ2)α1s + (λ3 −λ6)γ1 +(λ3 −λ5)γ2 + λ3d1, dλ4 dt = (λ1 −λ2)α2s + (λ4 −λ6)σ1 +(λ4 −λ5)σ2 + λ4d1, dλ5 dt = (λ5−λ6) min{max{0, (λ5−λ6)i ∗ g2 }, 1} +(d1 + d2)λ5 −g1, dλ6 dt =d1λ6; subject to the following conditions: s(0) > 0, e(0) ≥ 0, h(0) ≥ 0, g(0) ≥ 0, i(0) ≥ 0, r(0) ≥ 0 and λ1(tf ) = 0, λ2(tf ) = 0, λ3(tf ) = 0, λ4(tf ) = 0, λ5(tf ) = 0, λ6(tf ) = 0. this completes the proof. biomath 10 (2021), 2106147, http://dx.doi.org/10.11145/j.biomath.2021.06.147 page 10 of 20 http://dx.doi.org/10.11145/j.biomath.2021.06.147 d pal, d ghosh, p k santra, g s mahapatra, mathematical modeling and analysis of covid-19 ... 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 −10 0 0.5 1 1.5 x 10 −10 0 0.5 1 1.5 2 2.5 3 3.5 α 1 α 2 r e p ro d u c ti o n n u m b e r( r 0 ) fig. 2. sensitivity of r0 to α1 and α2, rest of the parameters are based on table 1 vii. numerical simulations the current section presents some computer simulations to assess the proposed model’s applicability for the covid-19 scenario. the simulation is carried out based on available data of pandemic infection in india. also, these numerical simulation is very much crucial from a practical viewpoint. estimating the parameters of the model for india, we have studied the proposed covid-19 system. the main objective is to study the effects of two quarantined population parameters α1 and α2, to show the impact of these parameters on the pandemic curve through the graphical presentation. by changing the values of the mentioned parameters, we observe the infected population’s behavior for 60 days from 2nd april for their particular base values. table 1 and table 2 give the values of the model parameters and initial population density, respectively. based on table 1, the brn is r0 = 3.0909, which is much greater than one. hence the infection spread so quickly in india. therefore, it needs to take the right policy to reduce the value of r0 much less to 1. for the proposed model, a graphical presentation of r0 to α1and α2 is given in figure 2. table i model parameters for covid-19 system parameters values (unit) data source λ 50000 day−1 [61] α1 2 × 10−10 day−1 estimated α2 1 × 10−10 day−1 estimated β1 0.4 day−1 assumed β2 1 × 10−6 day−1 assumed β3 0.05 day−1 assumed γ1 0.15 day−1 estimated γ2 0.0028 day−1 estimated σ1 0.15 day−1 estimated σ2 0.002 day−1 estimated � 0.06 day−1 estimated d1 2 × 10−5 day−1 [61] d2 0.001 day−1 estimated table ii preliminary population density for covid-19 model s(0) e(0) h(0) g(0) i(0) r(0) 12×108 2×105 2×105 5×104 1649 5×104 in figure 3, the ’red’ curve presents an infected individual for this proposed model, and the bar diagram is the actual infected individual as per our available data. figure 3 depicts that the actual infected individual almost coincides with our biomath 10 (2021), 2106147, http://dx.doi.org/10.11145/j.biomath.2021.06.147 page 11 of 20 http://dx.doi.org/10.11145/j.biomath.2021.06.147 d pal, d ghosh, p k santra, g s mahapatra, mathematical modeling and analysis of covid-19 ... 1/4 10/4 20 /4 30/4 10/5 20/5 29/5 0 2 4 6 8 10 x 10 4 time p o p u la ti o n i fig. 3. time series of infected population with parameter values and initial conditions from table 1 and 2 during 1/4/2020 to 29/5/2020. 0 50 100 150 200 250 300 350 400 450 500 0 2 4 6 8 10 12 14 16 18 x 10 5 time p o p u la ti o n i α 1 =0.0000000002 α 1 =0.00000000019 α 1 =0.00000000018 fig. 4. time series of the infected population with α2 = 1 × 10−10 for different values of α1 and other input values taken from table 1 and 2. proposed model curve from 1st april to 29th may 2020. therefore, the proposed covid-19 model is best fitted to the current situation of india. for fixed α2 if we gradually decrease α1, the infected individuals is also reduces steadily, which is presented via figure 4. therefore, practically if we strictly follow the home quarantined restriction, then naturally α1 decrease, and also the pick of the infected individual reduces. it is also observed from figure 4 that for α1 = 2 × 10−10, the pick of the disease reached almost after 160 days from 1st april 2020, and the height number of infected cases around 1700000. for α1 = 1.9 × 10−10 and α1 = 1.8 × 10−10 the pick of infection reached almost after 175 and 220 days, respectively from 1st april 2020, and the corresponding height number of infected cases may be around 1200000 and 800000, respectively. again if we fixed α1 at 2×10−10 and the values of α2 gradually increase, then the infected number of individuals is also gradually increasing, which biomath 10 (2021), 2106147, http://dx.doi.org/10.11145/j.biomath.2021.06.147 page 12 of 20 http://dx.doi.org/10.11145/j.biomath.2021.06.147 d pal, d ghosh, p k santra, g s mahapatra, mathematical modeling and analysis of covid-19 ... 0 50 100 150 200 250 300 350 0 0.5 1 1.5 2 2.5 3 3.5 x 10 6 time p o p u la ti o n i α 2 =0.0000000001 α 2 =0.0000000002 α 2 =0.0000000003 fig. 5. time series of infected population with parameter values and initial conditions from table 1 and 2 during 1/4/2020 to 29/5/2020. 0 20 40 60 80 100 120 140 160 180 200 0 1000 2000 3000 4000 5000 6000 7000 8000 time p o p u la ti o n i α 1 =0.00000000001 α 1 =0.00000000005 α 1 =0.0000000001 fig. 6. time series of infected population with α1 = α2 using data from table 1, 2 starting from 1st april to 29th may 2020. is depicted in figure 5. this situation arises as we increase α2, then the government quarantined technique is slackly applied to the population. in this case α2 = 3×10−10, the pick of infection reached almost 125 days after 1st april 2020, and the total number of highest infected individuals will be around 3000000. also, we making α1 = α2 = 2 × 10−10, i.e., if government take a policy to convert all home quarantined individuals into government quarantined. in that case, the value of r0 = 1.6368 < 3.0909 is less than the previous value of r0. therefore, infected individuals are automatic decreases, which are depicted in figure 6. figure 6 shows that if the government takes said policy, then the maximum number of infected is biomath 10 (2021), 2106147, http://dx.doi.org/10.11145/j.biomath.2021.06.147 page 13 of 20 http://dx.doi.org/10.11145/j.biomath.2021.06.147 d pal, d ghosh, p k santra, g s mahapatra, mathematical modeling and analysis of covid-19 ... 30/4 20/5 10/6 30/6 20/7 7/8 0 0.2 0.4 0.6 0.8 1 time u fig. 7. the graph of u with respect to time t based on table 1, 2 and g1 = 0.005 and g2 = 1000 starting from 30/4/2020 to 7/8/2020. 30/4 20/5 10/6 30/6 20/7 7/8 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 5 time p o p u la ti o n i fig. 8. the control diagram for the infected population (i) using data given in table 1 along with g1 = 0.005 and g2 = 1000 from 30/4/2020 to 7/8/2020. restricted to about 8000. unfortunately, to arrange this type of quarantined system is not possible for government since india is a country with large populations. therefore, the government has to think above other possible ways to restrict covid19 infection. therefore, this paper provides a way to restrict the infection by optimal control policy. we try to recover the infected patients by using the minimum drug. the present section explores the idea to solve the control problem numerically and will interpret the findings graphically. the boundary value problem in this paper estranged boundary conditions ranges between t = 0 to t = tf . the optimality problem is solved intended for 100 days. actually, given time tf = 100 represent the period at which the given treatment is stopped. table iii initial densities for the optimal control problem (17) s(0) e(0) h(0) g(0) i(0) r(0) 11.76×108 2×106 5×106 5×105 89987 16×106 the collocation method is the best technique to solve two-point bvps numerically. the current optimization problem solves numerically using matlab for our control problem based on table 1 and table 3. here, we choose weight constants g1 = 0.005 and g2 = 1000, respectively in the objective function given in (17). figure 7 represents the optimal control graphs for treatment control u. it shows that treatment control is very much necessary when the disease prevails. also, this control function minimizes the cost function j. biomath 10 (2021), 2106147, http://dx.doi.org/10.11145/j.biomath.2021.06.147 page 14 of 20 http://dx.doi.org/10.11145/j.biomath.2021.06.147 d pal, d ghosh, p k santra, g s mahapatra, mathematical modeling and analysis of covid-19 ... 0 10 20 30 40 50 60 70 80 90 100 0 1 2 3 4 5 6 7 8 x 10 6 time p o p u la ti o n i fig. 9. diagram for the infected populace (i) without control using data from tables 1, 2 and g1 = 0.005, g2 = 1000 from 30/4/2020 to 7/8/2020. 0 10 20 30 40 50 60 70 80 90 100 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10 8 time p o p u la ti o n r fig. 10. the control diagram of the recovered population (r) based on tables 1, 2 and g1 = 0.005, g2 = 1000 from 30/4/2020 to 7/8/2020. the graphs of the infected population (i) and recovered population (r) with treatment control and without treatment control with respect to time t are presented in figure 8 to figure 11, respectively. from these figures, we can predict that treatment control is exceptionally efficient in reducing covid infection. therefore, control acquiesces the best result to control the covid-19 epidemic outbreak. viii. discussions and conclusions this paper explores the idea of a sixcompartmental covid-19 infection model fitted for the india scenario. the present model has exhibited the effects of different precautions proposed by the administration to control india’s infectious disease. this study has also presented the impact of home and government quarantined technique biomath 10 (2021), 2106147, http://dx.doi.org/10.11145/j.biomath.2021.06.147 page 15 of 20 http://dx.doi.org/10.11145/j.biomath.2021.06.147 d pal, d ghosh, p k santra, g s mahapatra, mathematical modeling and analysis of covid-19 ... 0 10 20 30 40 50 60 70 80 90 100 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10 8 time p o p u la ti o n r fig. 11. diagram of the recovered population (r) without of control based on table 1, 2 and g1 = 0.005, g2 = 1000 during 30th april to 7th august 2020. on the covid-19 epidemiological model via a nonlinear differential equation system. the dynamical behavior of our proposed model is presented at dfe and endemic equilibrium points. we have calculated the bnr and verify that it behaves a crucial role in predicting the stability nature of all possible equilibrium points and the existence of the disease soon. also, a sensitivity analysis of r0 is carried out to α1 and α2 through figure 2. this analysis has shown that the parameters α1 and α2 are vital to restrict the spread of infection. the most crucial part related to public health importance is that this paper has built up a suitable optimal control problem to reduce the number of infected individuals. though the infection may be controlled by reducing the parameters’ values α1 and α2, it is not a long term solution to restrict the spread of the disease. therefore, we deem the treatment of infected individuals by medicine as a control to diminish the spread of covid-19 infection. in this paper, we include a quadratic control to quantify this goal. to minimize the objective functional (9), the control function u(t) is considered. this study also numerically verified the theoretical analysis by using matlab software to validate scientific findings through plot comparative figures of infected populations with different values of α1 and α2. we have observed from figure 4 for α1 = 2 × 10−10 and α1 = 1 × 10−10 that the pick of infection will be attained in the mid of september 2020 and around 1700000 may be affected in that time. again if the administration has taken the policy to cover all the populations under the government quarantined process (which is impossible for a country like india), i.e., α1 = α2, then figure 6 show that a maximum number of infected individuals are around 80000, the infection may be eradicated within october 2020. the proposed optimal control strategies are beneficial to reduce the number of infected populations, which is presented through comparative figure 8 to figure 11. it may also be concluded from these figures that the only treatment of an infected individual by medicine may not be the possible way to die out the disease from india and the globe. therefore, vaccination should be necessary as early as possible to protect the world from the covid-19 endemic. our mathematical model on the covid-19 epidemic diseases gives some consequences of public health policies. many of our proposed model biomath 10 (2021), 2106147, http://dx.doi.org/10.11145/j.biomath.2021.06.147 page 16 of 20 http://dx.doi.org/10.11145/j.biomath.2021.06.147 d pal, d ghosh, p k santra, g s mahapatra, mathematical modeling and analysis of covid-19 ... parameters are assumed or estimated, but it depends on many factors; these parameters may be considered as fuzzy or stochastic rather than deterministic. consequently, it may include fuzzy or stochastic differential equations in the proposed model for future work consideration. the progress of treating covid-19 disease by different medicines in a cost-effective way is the main objective of health administrators, policy-makers, and scientists until a vaccine is discovered. the present paper gives a little effort to reach this objective to restrict covid-19 infection. acknowledgments: the authors would like to express their gratitude to the editor hristo v kojouharov and referees for their encouragement and 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[62] m. kot, elements of mathematical ecology, cambridge university press, cambridge, 2001. appendix a. proof of theorem 4 the variational matrix of system (3) at dfe e0 is given by me0 =   −d1 0 −α1λd1 − α2λ d1 0 0 0 −a α1λ d1 α2λ d1 0 0 0 β1 −b 0 0 0 0 β3 0 −c 0 0 0 β2 γ2 σ2 −d 0 0 0 γ1 σ1 ∈ −d1   therefore, eigenvalues of the characteristic equation of me0 are −d1,−d1 and −d and and the solution of the cubic equation, p (λ) ≡ λ3 + a1λ2 + a2λ + a3 = 0 (a.1) where a1 = (a + b + c), a2 = (ab + ac)(1−r0)+bc + ( α1β1c b + α2β3b c ) λ d1 , a3 =abc(1 −r0). now, it is easily noted that, a1 > 0, a3 > 0 if r0 < 1. after some simplifications, we get a1a2−a3 = (a+b)ab [ (1−r0)+ λα2β3 d1ac ] +(a + c)ac [ (1 −r0) + λα1β1 d1ab ] +bc(b + c) + 2abc here, we can notice that, if r0 < 1 then a1a2 − a3 > 0 if r0 < 1. therefore, by the routh– hurwitz routh–hurwitz criterion [62] it follows that p (λ) = 0 has negative real roots if r0 < 1, i.e., the system (3) at dfe e0 when r0 < 1.this completes the proof. appendix b. proof of theorem 6 the variational matrix of system (3) at e1(s ∗,e∗,h∗,g∗,i∗,r∗) is given by, me1 =   b11 0 b13 b14 0 0 b21 b22 b23 b24 0 0 0 b32 b33 0 0 0 0 b42 0 b44 0 0 0 b52 b53 b54 b55 0 0 0 b63 b64 b65 b66   where, b11 = −d1r0, b13 = −α1s∗, b14 = −α2s∗, b21 = d1(r0−1), b22 = −a, b23 = α1s∗, b24 = α2s ∗, b32 = β1, b33 = −b, b42 = β3, b44 = −c, b52 = β2, b53 = γ2, b54 = σ2, b55 = −d, b63 = γ1, b64 = σ1, b65 =∈, b66 = −d1. therefore, eigenvalues of the characteristic equation of me1 are −d, −d1and the solution of the equation, q (λ) ≡ λ4 +b1λ3 +b2λ2 +b3λ+b4 = 0 (b.1) where b1 = a + b + c + d1r0, b2 = λ d1r0 [ α1β1c b + α2β3b c ] +(a + b + c)d1r0 + bc, b3 = (α1β1 + α2β3)d1(r0 − 1) + bcd1r0 + λα1β1c b + λα2β3b c b4 = abcd1(r0 − 1). biomath 10 (2021), 2106147, http://dx.doi.org/10.11145/j.biomath.2021.06.147 page 19 of 20 http://dx.doi.org/10.11145/j.biomath.2021.06.147 d pal, d ghosh, p k santra, g s mahapatra, mathematical modeling and analysis of covid-19 ... now, it is easily noted that bi > 0 (i = 1, 2, 3) and b4 > 0 if r0 > 1. by the routh–hurwitz criterion [62], it follows that q (λ) = 0 has negative real roots if bi > 0 for i = 1, 2, 3, 4, d1 = b1 > 0, d2 = ∣∣∣∣b1 b31 b2 ∣∣∣∣ = b1b2 −b3 > 0, d3 = ∣∣∣∣∣∣ b1 b3 0 1 b2 b4 0 b1 b3 ∣∣∣∣∣∣ = b1b2b3 −b21b4 −b 2 3 > 0. therefore the system (3) shows local asymptotic stability at e1 when r0 > 1, b1b2 −b3 > 0 and b1b2b3 − b21b4 − b 2 3 > 0. this completes the proof. biomath 10 (2021), 2106147, http://dx.doi.org/10.11145/j.biomath.2021.06.147 page 20 of 20 http://dx.doi.org/10.11145/j.biomath.2021.06.147 introduction multi-objective optimal control derivation and preliminaries of covid-19 model fundamental properties positivity of the solutions invariant region existence of equilibrium and stability analysis the basic reproduction number existence of endemic equilibrium e1(s,e,h,g,i,r) asymptotic behavior proposed covid-19 model with control numerical simulations discussions and conclusions references original article biomath 1 (2012), 1209022, 1–6 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum a state-efficient zebra-like implementation of synchronization algorithms for 2d rectangular cellular arrays hiroshi umeo∗ and akira nomura∗ ∗ univ. of osaka electro-communication neyagawa-shi, hatsu-cho, 18–8, osaka, 572–8530, japan emails: umeo@cyt.osakac.ac.jp, nomura@cyt.osakac.ac.jp received: 15 july 2012, accepted: 2 september 2012, published: 12 october 2012 abstract—the firing squad synchronization problem on cellular automata has been studied extensively for more than forty years, and a rich variety of synchronization algorithms have been proposed for not only one-dimensional (1d) but two-dimensional (2d) arrays. in the present paper, we propose a simple and state-efficient mapping scheme: zebra-like mapping for implementing 2d synchronization algorithms for rectangular arrays. the zebra-like mapping we propose embeds two types of configurations alternately onto a 2d array like a zebra-like pattern, one configuration is a synchronization configuration of 1d arrays and the other is a stationary configuration which keeps its state unchanged until the final synchronization. it is shown that the mapping gives us a smallest, known at present, implementation of 2d fssp algorithms for rectangular arrays. the implementation itself has a nice property that the correctness of the constructed transition rule set is clear and transparent. it is shown that there exists a ninestate 2d cellular automaton that can synchronize any (m x n) rectangle in (m+n+max(m,n)-3) steps. keywords-cellular automaton; fssp; firing squad synchronization problem i. introduction we study a synchronization problem that gives a finite-state protocol for synchronizing large scale cellular automata. the synchronization in cellular automata has been known as the firing squad synchronization problem (fssp, for short) which was originally proposed by j. myhill in moore [1964] to synchronize all/some parts of a self-reproducing cellular automaton. the problem has been studied extensively for more than forty years. 1 2 3 n 1 2 3 ... ... ... c11 c12 c13 c1n c21 c22 c23 c31 c32 c2n c33 c3n 4 c14 c24 c34 m ... cm1 cm2 cm3 cmncm4 .. . .. . .. . .. . .. . ... fig. 1. a 2d rectangle cellular automaton. in the present paper, we propose a simple and stateefficient mapping scheme: zebra-like mapping for implementing 2d synchronization algorithms. the zebra-like mapping we propose embeds two types of configurations alternately onto a 2d array like a zebra-like pattern, one configuration is a synchronization configuration of 1d arrays and the other is a stationary configuration which keeps its state unchanged until the final synchronization. the mapping gives us a smallest, known at present, implementation of 2d fssp algorithms. not only the number of states in the implementation is smaller, but the correctness of the constructed transition function with citation: h. umeo, a. nomura, a state-efficient zebra-like implementation of synchronization algorithms for 2d rectangular cellular arrays, biomath 1 (2012), 1209022, http://dx.doi.org/10.11145/j.biomath.2012.09.022 page 1 of 6 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.09.022 h. umeo et al., a state-efficient zebra-like implementation of synchronization algorithms... 2561 rules is clear and transparent. ii. firing squad synchronization problem a. fssp on 2d arrays figure 1 shows a finite two-dimensional (2d) rectangular array consisting of m × n cells. each cell is an identical (except the border cells) finite-state automaton. the array operates in a lock-step mode in such a way that the next state of each cell (except border cells) is determined by both its own present state and the present states of its north, south, east and west neighbors. thus, we assume the von neumann-type four nearest neighbors. all cells (soldiers), except the north-west corner cell (general), are initially in the quiescent state at time t = 0 with the property that the next state of a quiescent cell with quiescent neighbors is the quiescent state again. at time t = 0, the north-west corner cell c11 is in the fire-when-ready state, which is the initiation signal for synchronizing the array. the firing squad synchronization problem is to determine a description (state set and next-state function) for cells that ensures all cells enter the fire state at exactly the same time and for the first time. the tricky part of the problem is that the same kind of soldier having a fixed number of states must be synchronized, regardless of the size m × n of the array. the set of states and next state function must be independent of m and n. the problem was first solved by j. mccarthy and m. minsky who presented a 3n-step algorithm for 1d cellular array of length n. in 1962, the first optimumtime, i.e. (2n − 2)-step, synchronization algorithm was presented by goto [1962], with each cell having several thousands of states. mazoyer [1987] developed a sixstate synchronization algorithm which, at present, is the algorithm having the fewest states for 1d arrays. on the other hand, a rich variety of synchronization algorithms for 2d rectangular arrays has been proposed. the first optimum-time rectangle synchronization algorithm was proposed by beyer [1969] and shinahr [1974], independently. concerning the time optimality of the 2d rectangle synchronization algorithms, the following theorems have been shown. theorem 1beyer [1969], shinahr [1974] there exists no cellular automaton that can synchronize any 2d rectangle array of size m×n in less than m + m + max(m, n)−3 steps, where the general is located at one corner of the array. theorem 2shinahr [1974] there exists a 28-state cellular automaton that can synchronize any 2d rectangle array of size m×n at exactly m+m+max(m, n)−3 optimum steps, where the general is located at one corner of the array. b. optimum-time l-shaped mapping algorithm 1 2 3 4 n 1 2 3 4 m m 1 2 3 4 nm 1 2 3 4 m 3 4 nm 3 4 m nm m 4 nm 4 m 2 3 4 nm 2 3 4 m 1/1 1/11/1 1/15 1/7 1/3 1/31/3 1/1 1/3 1/7 1/1 t = 2m-2 time t = 0 1/1 t = n -1 t = m-1 1/11/1 m m gm n t = 2n+m -3 n fig. 2. l-shaped decomposition of an m × n rectangle cellular automaton and a space-time diagram for the l-shaped base synchronization algorithm. a black circle • in a shaded small square represents a general on each li and a wake-up signal for the synchronization generated by the general is indicated by a horizontal and vertical arrow. the first optimum-time synchronization algorithm developed by beyer [1969] and shinahr [1974] for rectangle arrays operates as follows: we assume that an initial general is located on c11 on a rectangular array of size m×n. by dividing the entire rectangle array of size m×n into min(m, n) rotated l-shaped 1d arrays, shown in fig. 2, one treats the rectangle synchronization problem biomath 1 (2012), 1209022, http://dx.doi.org/10.11145/j.biomath.2012.09.022 page 2 of 6 http://dx.doi.org/10.11145/j.biomath.2012.09.022 h. umeo et al., a state-efficient zebra-like implementation of synchronization algorithms... as min(m, n) independent 1d generalized synchronizations with the general located at the bending cell of the l-shaped array. all cells on each l-shaped array fall into a pre-specified synchronization state, simultaneously. we denote the ith (from outside) l-shaped array by li and its horizontal and vertical segment is denoted by lhi and lvi , 1 ≤ i ≤ min(m, n), respectively. see fig. 2. concerning the synchronization of li, it can be easily seen that each general is generated at the cell cii at time t = 3i − 3, and the general initiates the horizontal (row) and vertical (column) synchronizations on lhi and lvi , via a 1d generalized optimum-time synchronization algorithm which can synchronize arrays of length ` with a general on kth cell from left end (from bottom in the l-shaped decomposition) in ` − 2 + max(k, ` − k + 1) optimum steps, where 1 ≤ k ≤ `. note that the length of li is `i = m + n − 2i + 1 and the general is on ki = (m − i + 1)th cell from left (bottom) end. for any i, 1 ≤ i ≤ min(m, n), all cells on li can be synchronized at time t = 3i−3+`i−2+max(ki, `i−ki +1) = m+n+ i−4+max(m−i+1, n−i+1) = m+n+max(m, n)−3. thus, the rectangle array of size m × n can be synchronized at time t = m + n + max(m, n) − 3 in optimumsteps. in fig. 2 (top), each general is represented by a black circle • in a shaded square and a wake-up signal for the synchronization generated by the general is indicated by a horizontal and vertical arrow. the algorithm itself is very simple and now we are going to discuss its implementation in terms of a 2d cellular automaton. the question is: how many states are required for its realization? let q be a set of internal states for the 1d optimumtime generalized synchronization algorithm which is embedded onto a 2d array as a base algorithm. when we implement the algorithm on rectangle arrays based on the scheme above, we usually have to add a direction information to each state in order to simulate the embedded synchronization operations on each horizontal and vertical segment. thus, approximately, 2 | q | −1 states are usually required for its independent row and column synchronization operations in order to avoid state mixing. only a firing state is shared by the two areas. shinahr [1974] gave a 28-state implementation based on the idea above. c. zebra-like mapping on square arrays the proposed mapping for square arrays is basically based on the rotated l-shaped mapping scheme presented in the previous section, however, the mapping onto square arrays consists of two types of configurations: one is a one-cell smaller synchronized configuration and the other is a filled-in configuration with a stationary state. the stationary state remains unchanged once filled-in by the time before the final synchronization. each configuration is mapped alternatively onto an l-shaped array in a zebra fashion. the mapping is referred to as zebra-like mapping. a key idea of the small-state implementation is: • alternative mapping of two types of configurations: a stationary layer separates two consecutive synchronization layers and it allows us to use the same state set for the vertical and horizontal synchronization on each layer, helping us to construct a small-state transition rule set for the synchronization layers. • a one-cell smaller synchronization configuration embedded: a one-cell smaller synchronization configuration than the classical l-shaped mapping (shinahr [1974]) is embedded, where we can save synchronization time by two steps. • a shared pre-firing state: a single state x is shared between an initial general state of the square synchronizer, the stationary state in the stationary layer, and a pre-firing state of the embedded 1d synchronization algorithm used. the state x itself acts as a pre-firing state of the square synchronizer to be constructed. • a simple condition for final synchronization: any cell in state x, except cn,n, enters the final synchronization state at the next step if all its neighbors are in state x or the boundary state of the square. the cell cn,n enters the synchronization state if and only if its north and west cells are in state x and its east and south cells are in the boundary state. a cell in state x that is adjacent to the cell cn,n is also an exception. these are the only conditions that make cells fire. in our construction we take the mazoyer’s 6-state 1d synchronization rule as an embedded synchronization algorithm. the set of the 6-states is {g, q, a, b, c, x}, where g is a general, q is a quiescent, and x is a firing state, respectively. the other three states a, b and c are auxiliary states, respectively. the seven-state square synchronizer that we construct has the following state set: {g, q, a, b, c, x, f}, where f is a newly introduced firing state, x is a biomath 1 (2012), 1209022, http://dx.doi.org/10.11145/j.biomath.2012.09.022 page 3 of 6 http://dx.doi.org/10.11145/j.biomath.2012.09.022 h. umeo et al., a state-efficient zebra-like implementation of synchronization algorithms... t = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x q q q q q q q q q q q q 2 q q q q q q q q q q q q q 3 q q q q q q q q q q q q q 4 q q q q q q q q q q q q q 5 q q q q q q q q q q q q q 6 q q q q q q q q q q q q q 7 q q q q q q q q q q q q q 8 q q q q q q q q q q q q q 9 q q q q q q q q q q q q q 10 q q q q q q q q q q q q q 11 q q q q q q q q q q q q q 12 q q q q q q q q q q q q q 13 q q q q q q q q q q q q q t = 1 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x g q q q q q q q q q q q 2 g q q q q q q q q q q q q 3 q q q q q q q q q q q q q 4 q q q q q q q q q q q q q 5 q q q q q q q q q q q q q 6 q q q q q q q q q q q q q 7 q q q q q q q q q q q q q 8 q q q q q q q q q q q q q 9 q q q q q q q q q q q q q 10 q q q q q q q q q q q q q 11 q q q q q q q q q q q q q 12 q q q q q q q q q q q q q 13 q q q q q q q q q q q q q t = 2 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x a c q q q q q q q q q q 2 a x q q q q q q q q q q q 3 c q q q q q q q q q q q q 4 q q q q q q q q q q q q q 5 q q q q q q q q q q q q q 6 q q q q q q q q q q q q q 7 q q q q q q q q q q q q q 8 q q q q q q q q q q q q q 9 q q q q q q q q q q q q q 10 q q q q q q q q q q q q q 11 q q q q q q q q q q q q q 12 q q q q q q q q q q q q q 13 q q q q q q q q q q q q q t = 3 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x g b a q q q q q q q q q 2 g x g q q q q q q q q q q 3 b g q q q q q q q q q q q 4 a q q q q q q q q q q q q 5 q q q q q q q q q q q q q 6 q q q q q q q q q q q q q 7 q q q q q q q q q q q q q 8 q q q q q q q q q q q q q 9 q q q q q q q q q q q q q 10 q q q q q q q q q q q q q 11 q q q q q q q q q q q q q 12 q q q q q q q q q q q q q 13 q q q q q q q q q q q q q t = 4 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x g c g g q q q q q q q q 2 g x x c q q q q q q q q q 3 c x x q q q q q q q q q q 4 g c q q q q q q q q q q q 5 g q q q q q q q q q q q q 6 q q q q q q q q q q q q q 7 q q q q q q q q q q q q q 8 q q q q q q q q q q q q q 9 q q q q q q q q q q q q q 10 q q q q q q q q q q q q q 11 q q q q q q q q q q q q q 12 q q q q q q q q q q q q q 13 q q q q q q q q q q q q q t = 5 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x g b a b c q q q q q q q 2 g x x x c q q q q q q q q 3 b x x g q q q q q q q q q 4 a x g q q q q q q q q q q 5 b c q q q q q q q q q q q 6 c q q q q q q q q q q q q 7 q q q q q q q q q q q q q 8 q q q q q q q q q q q q q 9 q q q q q q q q q q q q q 10 q q q q q q q q q q q q q 11 q q q q q q q q q q q q q 12 q q q q q q q q q q q q q 13 q q q q q q q q q q q q q t = 6 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x g c g q c a q q q q q q 2 g x x x c a q q q q q q q 3 c x x a c q q q q q q q q 4 g x a x q q q q q q q q q 5 q c c q q q q q q q q q q 6 c a q q q q q q q q q q q 7 a q q q q q q q q q q q q 8 q q q q q q q q q q q q q 9 q q q q q q q q q q q q q 10 q q q q q q q q q q q q q 11 q q q q q q q q q q q q q 12 q q q q q q q q q q q q q 13 q q q q q q q q q q q q q t = 7 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x g b a q a a g q q q q q 2 g x x x x x b q q q q q q 3 b x x g b a q q q q q q q 4 a x g x g q q q q q q q q 5 q x b g q q q q q q q q q 6 a x a q q q q q q q q q q 7 a b q q q q q q q q q q q 8 g q q q q q q q q q q q q 9 q q q q q q q q q q q q q 10 q q q q q q q q q q q q q 11 q q q q q q q q q q q q q 12 q q q q q q q q q q q q q 13 q q q q q q q q q q q q q t = 8 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x g c g q a b b c q q q q 2 g x x x x x x c q q q q q 3 c x x g c g g q q q q q q 4 g x g x x c q q q q q q q 5 q x c x x q q q q q q q q 6 a x g c q q q q q q q q q 7 b x g q q q q q q q q q q 8 b c q q q q q q q q q q q 9 c q q q q q q q q q q q q 10 q q q q q q q q q q q q q 11 q q q q q q q q q q q q q 12 q q q q q q q q q q q q q 13 q q q q q q q q q q q q q t = 9 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x g b a q q b c c a q q q 2 g x x x x x x c a q q q q 3 b x x g b a b c q q q q q 4 a x g x x x c q q q q q q 5 q x b x x g q q q q q q q 6 q x a x g q q q q q q q q 7 b x b c q q q q q q q q q 8 c c c q q q q q q q q q q 9 c a q q q q q q q q q q q 10 a q q q q q q q q q q q q 11 q q q q q q q q q q q q q 12 q q q q q q q q q q q q q 13 q q q q q q q q q q q q q t = 10 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x g c g g q q c a a g q q 2 g x x x x x x x x b q q q 3 c x x g c g q c a q q q q 4 g x g x x x c a q q q q q 5 g x c x x a c q q q q q q 6 q x g x a x q q q q q q q 7 q x q c c q q q q q q q q 8 c x c a q q q q q q q q q 9 a x a q q q q q q q q q q 10 a b q q q q q q q q q q q 11 g q q q q q q q q q q q q 12 q q q q q q q q q q q q q 13 q q q q q q q q q q q q q t = 11 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x g b a b c q a a b b c q 2 g x x x x x x x x x c q q 3 b x x g b a q a a g q q q 4 a x g x x x x x b q q q q 5 b x b x x g b a q q q q q 6 c x a x g x g q q q q q q 7 q x q x b g q q q q q q q 8 a x a x a q q q q q q q q 9 a x a b q q q q q q q q q 10 b x g q q q q q q q q q q 11 b c q q q q q q q q q q q 12 c q q q q q q q q q q q q 13 q q q q q q q q q q q q q t = 12 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x g c g q c q a b b c c g 2 g x x x x x x x x x c a q 3 c x x g c g q a b b c q q 4 g x g x x x x x x c q q q 5 q x c x x g c g g q q q q 6 c x g x g x x c q q q q q 7 q x q x c x x q q q q q q 8 a x a x g c q q q q q q q 9 b x b x g q q q q q q q q 10 b x b c q q q q q q q q q 11 c c c q q q q q q q q q q 12 c a q q q q q q q q q q q 13 g q q q q q q q q q q q q t = 13 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x g b a q c q q b c c b a 2 g x x x x x x x x x x x x 3 b x x g b a q q b c c a q 4 a x g x x x x x x c a q q 5 q x b x x g b a b c q q q 6 c x a x g x x x c q q q q 7 q x q x b x x g q q q q q 8 q x q x a x g q q q q q q 9 b x b x b c q q q q q q q 10 c x c c c q q q q q q q q 11 c x c a q q q q q q q q q 12 b x a q q q q q q q q q q 13 a x q q q q q q q q q q q t = 14 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x g c g q c a q q c b a c 2 g x x x x x x x x x x x x 3 c x x g c g g q q c a a c 4 g x g x x x x x x x x b q 5 q x c x x g c g q c a q q 6 c x g x g x x x c a q q q 7 a x g x c x x a c q q q q 8 q x q x g x a x q q q q q 9 q x q x q c c q q q q q q 10 c x c x c a q q q q q q q 11 b x a x a q q q q q q q q 12 a x a b q q q q q q q q q 13 c x c q q q q q q q q q q t = 15 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x g b a q a a g q g a c b 2 g x x x x x x x x x x x x 3 b x x g b a b c q a a c b 4 a x g x x x x x x x x x x 5 q x b x x g b a q a a g q 6 a x a x g x x x x x b q q 7 a x b x b x x g b a q q q 8 g x c x a x g x g q q q q 9 q x q x q x b g q q q q q 10 g x a x a x a q q q q q q 11 a x a x a b q q q q q q q 12 c x c x g q q q q q q q q 13 b x b x q q q q q q q q q t = 16 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x g c g q a b b a g c b q 2 g x x x x x x x x x x x x 3 c x x g c g q c q a c b q 4 g x g x x x x x x x x x x 5 q x c x x g c g q a b b a 6 a x g x g x x x x x x c q 7 b x q x c x x g c g g q q 8 b x c x g x g x x c q q q 9 a x q x q x c x x q q q q 10 g x a x a x g c q q q q q 11 c x c x b x g q q q q q q 12 b x b x b c q q q q q q q 13 q x q x a q q q q q q q q t = 17 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x g b a q q b a c g b q q 2 g x x x x x x x x x x x x 3 b x x g b a q c q g b q q 4 a x g x x x x x x x x x x 5 q x b x x g b a q q b a c 6 q x a x g x x x x x x c q 7 b x q x b x x g b a b c q 8 a x c x a x g x x x c q q 9 c x q x q x b x x g q q q 10 g x g x q x a x g q q q q 11 b x b x b x b c q q q q q 12 q x q x a c c q q q q q q 13 q x q x c q q q q q q q q t = 18 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x g c g g q g c b g c q q 2 g x x x x x x x x x x x x 3 c x x g c g q c g g c q q 4 g x g x x x x x x x x x x 5 g x c x x g c g g q g c b 6 q x g x g x x x x x x x x 7 g x q x c x x g c g q c g 8 c x c x g x g x x x c a q 9 b x g x g x c x x a c q q 10 g x g x q x g x a x q q q 11 c x c x g x q c c q q q q 12 q x q x c x c a q q q q q 13 q x q x b x g q q q q q q t = 19 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x g b a b a g b q g b a q 2 g x x x x x x x x x x x x 3 b x x g b a q g a g b a q 4 a x g x x x x x x x x x x 5 b x b x x g b a b a g b q 6 a x a x g x x x x x x x x 7 g x q x b x x g b a q g a 8 b x g x a x g x x x x x x 9 q x a x b x b x x g b a q 10 g x g x a x a x g x g q q 11 b x b x g x q x b g q q q 12 a x a x b x g x a q q q q 13 q x q x q x a x q q q q q t = 20 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x g c g b c g c q g c g c 2 g x x x x x x x x x x x x 3 c x x g c g c g c g c g c 4 g x g x x x x x x x x x x 5 b x c x x g c g b c g c q 6 c x g x g x x x x x x x x 7 g x c x c x x g c g c g c 8 c x g x g x g x x x x x x 9 q x c x b x c x x g c g c 10 g x g x c x g x g x x c q 11 c x c x g x c x c x x q q 12 g x g x c x g x g c q q q 13 c x c x q x c x c q q q q t = 21 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x g b g b g g b g g b g b 2 g x x x x x x x x x x x x 3 b x x g b g b g b g b g b 4 g x g x x x x x x x x x x 5 b x b x x g b g b g g b g 6 g x g x g x x x x x x x x 7 g x b x b x x g b g b g b 8 b x g x g x g x x x x x x 9 g x b x b x b x x g b g b 10 g x g x g x g x g x x x x 11 b x b x g x b x b x x g q 12 g x g x b x g x g x g q q 13 b x b x g x b x b x q q q t = 22 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x g g g g g g g g g g g g 2 g x x x x x x x x x x x x 3 g x x g g g g g g g g g g 4 g x g x x x x x x x x x x 5 g x g x x g g g g g g g g 6 g x g x g x x x x x x x x 7 g x g x g x x g g g g g g 8 g x g x g x g x x x x x x 9 g x g x g x g x x g g g g 10 g x g x g x g x g x x x x 11 g x g x g x g x g x x a a 12 g x g x g x g x g x a x q 13 g x g x g x g x g x a q q t = 23 1 2 3 4 5 6 7 8 9 10 11 12 13 1 x x x x x x x x x x x x x 2 x x x x x x x x x x x x x 3 x x x x x x x x x x x x x 4 x x x x x x x x x x x x x 5 x x x x x x x x x x x x x 6 x x x x x x x x x x x x x 7 x x x x x x x x x x x x x 8 x x x x x x x x x x x x x 9 x x x x x x x x x x x x x 10 x x x x x x x x x x x x x 11 x x x x x x x x x x x x x 12 x x x x x x x x x x x x x 13 x x x x x x x x x x x x q t = 24 1 2 3 4 5 6 7 8 9 10 11 12 13 1 f f f f f f f f f f f f f 2 f f f f f f f f f f f f f 3 f f f f f f f f f f f f f 4 f f f f f f f f f f f f f 5 f f f f f f f f f f f f f 6 f f f f f f f f f f f f f 7 f f f f f f f f f f f f f 8 f f f f f f f f f f f f f 9 f f f f f f f f f f f f f 10 f f f f f f f f f f f f f 11 f f f f f f f f f f f f f 12 f f f f f f f f f f f f f 13 f f f f f f f f f f f f f fig. 3. snapshots of the optimum-time synchronization process on a 13 × 13 square array. general, and q is a quiescent state, respectively. the state g is the general state of the embedded synchronization. those states a, b and c are also auxiliary states, respectively. the transition rule set is constructed in such a way that: the initial general on c1,1 in state x generates a new general in state g on the cell c1,2 and c2,1 at time t = 1. the general in state g initiates a synchronization for the following cells {c1,2, c1,3, ..., c1,n} and {c2,1, c3,1, ..., cn,1}, each of length n − 1. note that the length of the array where optimum-time synchronization operations are embedded is shorter by one than the usual embedding in section 2. the cells on the segments are constructed to operate so that they simulate the mazoyer’s optimum-time synchronization operations. all cells on the two horizontal and vertical segments of length n − 1 enter the pre-firing state x at time t = 1 + 2(n − 1) − 2 = 2n − 3. in this way, the first l1 acts as a synchronization layer. at time t = 2, the cell c2,2 takes the state x and it extends an x-arm (a cell segment in state x) in the right and lower direction, respectively, towards the cells {c2,3, c2,4, ..., c2,n} and {c3,2, c4,2, ..., cn,2}, respectively, each of length n−2. every cell once entered in state x remains unchanged by the time before it meets a local condition for the synchronization given later. at time t = 2 + n − 2 = n, the filled-in operation with the stationary state x on the second layer is finished. in this way, the second l2 acts as a stationary layer. concerning the embedding on the odd ith layer, the cell ci,i takes the stationary state x time t = 2i − 2 and generates a new general in state g on the cell ci,i+1 and ci+1,i at time t = 2i − 1. the general in state g initiates a synchronization for the following cells {ci,i+1, ci,i+2, ..., ci,n} and {ci+1,i, ci+2,i, ..., cn,i}, each of length n − i. all cells on the two horizontal and vertical segments of length n − i enter the pre-firing state x at time t = 2i − 1 + 2(n − i) − 2 = 2n − 3. in this way, for odd i, the ith li acts as a synchronization layer. as for the even ith layer, at time t = 2i − 2, the cell ci,i takes the state x and it extends the x-arm in the right and lower direction, respectively, towards the cells {ci,i+1, ci,i+2, ..., ci,n} and {ci+1,i, ci+2,i, ..., cn,i}, each of length n − i. every cell once entered in state x remains unchanged by the time before synchronization. at time t = 2i − 2 + n − i = n + i − 2, the filled-in operation on the ith layer for even i is finished. at time t = 2n − 3, all of the cells, except cn,n, on the square of size n × n enter the state x, which is a pre-firing state. thus we have seen: theorem 3umeo and kubo [2010] there exists a sevenstate 2d ca that can synchronize any n×n square array in 2n − 2 steps. figure 3 shows some snapshots of the synchronization process operating in optimum-steps on a 13 × 13 square array. iii. zebra-like mapping on rectangle arrays in this section we give three implementations for rectangle synchronization algorithms. as is shown in fig. 2, a 1d generalized fssp algorithm is mapped on an l-shaped 1d array, where the cells on the horizontal and vertical segments have to cooperate with each other. thus, in contrast to the square implementation, two independent, small-size synchronization configurations cannot be implemented on the horizontal and vertical segment on a single synchronization layer in the rectangle case. all the implementations given are variants of the zebra-like mapping. the first ten-state implementation is a straightforward implementation of the zebralike mapping, which yields a non-optimum algorithm. the second 11-state implementation is a variant of the zebra-like mapping where the first synchronization layer l1 and the thereafter layers li, i ≥ 3 take a different set of synchronization rule set. the third one is a ninestate implementation which regards the marking symbol biomath 1 (2012), 1209022, http://dx.doi.org/10.11145/j.biomath.2012.09.022 page 4 of 6 http://dx.doi.org/10.11145/j.biomath.2012.09.022 h. umeo et al., a state-efficient zebra-like implementation of synchronization algorithms... t = 0 1 2 3 4 5 6 7 8 9 1 g q q q q q q q q 2 q q q q q q q q q 3 q q q q q q q q q 4 q q q q q q q q q 5 q q q q q q q q q 6 q q q q q q q q q 7 q q q q q q q q q 8 q q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 1 1 2 3 4 5 6 7 8 9 1 q ] q q q q q q q 2 [ q q q q q q q q 3 q q q q q q q q q 4 q q q q q q q q q 5 q q q q q q q q q 6 q q q q q q q q q 7 q q q q q q q q q 8 q q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 2 1 2 3 4 5 6 7 8 9 1 q a ] q q q q q q 2 b tx q q q q q q q 3 [ q q q q q q q q 4 q q q q q q q q q 5 q q q q q q q q q 6 q q q q q q q q q 7 q q q q q q q q q 8 q q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 3 1 2 3 4 5 6 7 8 9 1 q a q ] q q q q q 2 b x tx q q q q q q 3 q tx q q q q q q q 4 [ q q q q q q q q 5 q q q q q q q q q 6 q q q q q q q q q 7 q q q q q q q q q 8 q q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 4 1 2 3 4 5 6 7 8 9 1 q q a a ] q q q q 2 q x x tx q q q q q 3 b x q q q q q q q 4 b tx q q q q q q q 5 [ q q q q q q q q 6 q q q q q q q q q 7 q q q q q q q q q 8 q q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 5 1 2 3 4 5 6 7 8 9 1 q a a a q ] q q q 2 b x x x tx q q q q 3 b x tx q q q q q q 4 b x q q q q q q q 5 q tx q q q q q q q 6 [ q q q q q q q q 7 q q q q q q q q q 8 q q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 6 1 2 3 4 5 6 7 8 9 1 q a a q a a ] q q 2 b x x x x tx q q q 3 b x g q q q q q q 4 q x q q q q q q q 5 b x q q q q q q q 6 b tx q q q q q q q 7 [ q q q q q q q q 8 q q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 7 1 2 3 4 5 6 7 8 9 1 q a q a a a q ] q 2 b x x x x x tx q q 3 q x q ] q q q q q 4 b x [ q q q q q q 5 b x q q q q q q q 6 b x q q q q q q q 7 q tx q q q q q q q 8 [ q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 8 1 2 3 4 5 6 7 8 9 1 q q a a a q a a c 2 q x x x x x x tx q 3 b x q a ] q q q q 4 b x b tx q q q q q 5 b x [ q q q q q q 6 q x q q q q q q q 7 b x q q q q q q q 8 b tx q q q q q q q 9 [ q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 9 1 2 3 4 5 6 7 8 9 1 q a a a q a a [ c 2 b x x x x x x x tx 3 b x q a q ] q q q 4 b x b x tx q q q q 5 q x q tx q q q q q 6 b x [ q q q q q q 7 b x q q q q q q q 8 b x q q q q q q q 9 q tx q q q q q q q 10 [ q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 10 1 2 3 4 5 6 7 8 9 1 q a a q a a [ ] c 2 b x x x x x x x x 3 b x q q a a ] q q 4 q x q x x tx q q q 5 b x b x q q q q q 6 b x b tx q q q q q 7 b x [ q q q q q q 8 q x q q q q q q q 9 b x q q q q q q q 10 b tx q q q q q q q 11 [ q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 11 1 2 3 4 5 6 7 8 9 1 q a q a a [ b [ c 2 b x x x x x x x x 3 q x tx a a a q ] q 4 b x b x x x tx q q 5 b x b x tx q q q q 6 b x b x q q q q q 7 q x q tx q q q q q 8 b x [ q q q q q q 9 b x q q q q q q q 10 b x q q q q q q q 11 q tx q q q q q q q 12 [ q q q q q q q q 13 q q q q q q q q q t = 12 1 2 3 4 5 6 7 8 9 1 q q a a [ q b [ c 2 q x x x x x x x x 3 b x q a a q a a c 4 b x b x x x x tx q 5 b x b x g q q q q 6 q x q x q q q q q 7 b x b x q q q q q 8 b x b tx q q q q q 9 b x [ q q q q q q 10 q x q q q q q q q 11 b x q q q q q q q 12 b tx q q q q q q q 13 c q q q q q q q q t = 13 1 2 3 4 5 6 7 8 9 1 q a a [ b b g [ c 2 b x x x x x x x x 3 b x q a q a a [ c 4 b x b x x x x x tx 5 q x q x q ] q q q 6 b x b x [ q q q q 7 b x b x q q q q q 8 b x b x q q q q q 9 q x q tx q q q q q 10 b x [ q q q q q q 11 b x q q q q q q q 12 ] x q q q q q q q 13 c tx q q q q q q q t = 14 1 2 3 4 5 6 7 8 9 1 q a [ q b b g [ c 2 b x x x x x x x x 3 b x q q a a [ ] c 4 q x q x x x x x x 5 b x b x q a ] q q 6 b x b x b tx q q q 7 b x b x [ q q q q 8 q x q x q q q q q 9 b x b x q q q q q 10 b x b tx q q q q q 11 ] x [ q q q q q q 12 [ x q q q q q q q 13 c x q q q q q q q t = 15 1 2 3 4 5 6 7 8 9 1 q [ b b q b g [ c 2 b x x x x x x x x 3 q x tx a a [ b [ c 4 b x b x x x x x x 5 b x b x q a q ] q 6 b x b x b x tx q q 7 q x q x q tx q q q 8 b x b x [ q q q q 9 b x b x q q q q q 10 ] x b x q q q q q 11 a x q tx q q q q q 12 ] x [ q q q q q q 13 c x q q q q q q q t = 16 1 2 3 4 5 6 7 8 9 1 [ q b b b [ g [ c 2 q x x x x x x x x 3 b x q a [ q b [ c 4 b x b x x x x x x 5 b x b x q q a a c 6 q x q x q x x tx q 7 b x b x b x q q q 8 b x b x b tx q q q 9 ] x b x [ q q q q 10 q x q x q q q q q 11 a x b x q q q q q 12 ] x b tx q q q q q 13 c x c q q q q q q t = 17 1 2 3 4 5 6 7 8 9 1 b b q b b [ b [ c 2 [ x x x x x x x x 3 b x q [ b b g [ c 4 b x b x x x x x x 5 q x q x tx a a [ c 6 b x b x b x x x tx 7 b x b x b x tx q q 8 ] x b x b x q q q 9 a x q x q tx q q q 10 a x b x [ q q q q 11 g x b x q q q q q 12 ] x ] x q q q q q 13 c x c tx q q q q q t = 18 1 2 3 4 5 6 7 8 9 1 b b b q b [ b [ c 2 [ x x x x x x x x 3 b x [ q b b g [ c 4 q x q x x x x x x 5 b x b x q a [ ] c 6 b x b x b x x x x 7 ] x b x b x g q q 8 q x q x q x q q q 9 a x b x b x q q q 10 a x b x b tx q q q 11 g x ] x [ q q q q 12 ] x [ x q q q q q 13 c x c x q q q q q t = 19 1 2 3 4 5 6 7 8 9 1 b b b b g [ b [ c 2 [ x x x x x x x x 3 g x b b q b g [ c 4 b x [ x x x x x x 5 b x b x q [ b [ c 6 ] x b x b x x x x 7 a x q x q x q ] q 8 a x b x b x [ q q 9 q x b x b x q q q 10 a x ] x b x q q q 11 g x a x q tx q q q 12 ] x ] x [ q q q q 13 c x c x q q q q q t = 20 1 2 3 4 5 6 7 8 9 1 b b b b g q b [ c 2 q x x x x x x x x 3 g x b b b [ g [ c 4 b x [ x x x x x x 5 ] x b x [ q b [ c 6 q x q x q x x x x 7 a x b x b x q a c 8 a x b x b x b tx q 9 a x ] x b x [ q q 10 ] x q x q x q q q 11 g x a x b x q q q 12 ] x ] x b tx q q q 13 c x c x c q q q q t = 21 1 2 3 4 5 6 7 8 9 1 q b b b g b g [ c 2 b x x x x x x x x 3 g x b b b [ b [ c 4 ] x [ x x x x x x 5 a x g x b b g [ c 6 a x b x [ x x x x 7 q x b x b x q [ c 8 a x ] x b x b x tx 9 a x a x q x q tx q 10 ] x a x b x [ q q 11 a x g x b x q q q 12 ] x ] x ] x q q q 13 c x c x c tx q q q t = 22 1 2 3 4 5 6 7 8 9 1 b q b b g b g [ c 2 b x x x x x x x x 3 c x b b b [ b [ c 4 q x q x x x x x x 5 a x g x b b g [ c 6 a x b x [ x x x x 7 a x ] x b x [ ] c 8 q x q x q x q x x 9 a x a x b x b x q 10 ] x a x b x b tx q 11 a x g x ] x [ q q 12 ] x ] x [ x q q q 13 c x c x c x q q q t = 23 1 2 3 4 5 6 7 8 9 1 b b q b g b g [ c 2 ] x x x x x x x x 3 c x q b b [ b [ c 4 [ x b x x x x x x 5 q x g x b b g [ c 6 a x ] x [ x x x x 7 a x a x g x b [ c 8 a x a x b x [ x x 9 g x q x b x b x tx 10 ] x a x ] x b x q 11 a x g x a x q tx q 12 ] x ] x ] x [ q q 13 c x c x c x q q q t = 24 1 2 3 4 5 6 7 8 9 1 ] b b [ g b g [ c 2 [ x x x x x x x x 3 c x b q b [ b [ c 4 ] x b x x x x x x 5 [ x c x b b g [ c 6 q x q x q x x x x 7 a x a x g x b [ c 8 a x a x b x [ x x 9 g x a x ] x b x g 10 q x ] x q x q x q 11 a x g x a x b x q 12 ] x ] x ] x b tx q 13 c x c x c x c q q t = 25 1 2 3 4 5 6 7 8 9 1 a ] b [ b b g [ c 2 ] x x x x x x x x 3 c x b b g [ b [ c 4 [ x ] x x x x x x 5 b x c x q b g [ c 6 [ x [ x b x x x x 7 q x q x g x b [ c 8 a x a x ] x [ x x 9 g x a x a x g x c 10 a x ] x a x b x [ 11 g x a x g x b x q 12 ] x ] x ] x ] x q 13 c x c x c x c tx q t = 26 1 2 3 4 5 6 7 8 9 1 a q ] [ b b g [ c 2 ] x x x x x x x x 3 c x ] b g q b [ c 4 [ x [ x x x x x x 5 b x c x b [ g [ c 6 q x ] x b x x x x 7 [ x [ x c x b [ c 8 ] x q x q x q x x 9 g x a x a x g x c 10 a x ] x a x b x ] 11 g x a x g x ] x [ 12 ] x ] x ] x [ x q 13 c x c x c x c x q t = 27 1 2 3 4 5 6 7 8 9 1 g a a c b b g [ c 2 ] x x x x x x x x 3 c x a ] g b g [ c 4 [ x ] x x x x x x 5 g x c x b [ b [ c 6 b x [ x ] x x x x 7 b x b x c x g [ c 8 c x [ x [ x b x x 9 a x g x q x g x c 10 a x ] x a x ] x [ 11 g x a x g x a x b 12 ] x ] x ] x ] x [ 13 c x c x c x c x q t = 28 1 2 3 4 5 6 7 8 9 1 g a [ c ] b g [ c 2 ] x x x x x x x x 3 c x a q c b g [ c 4 [ x ] x x x x x x 5 g x c x ] [ b [ c 6 b x [ x [ x x x x 7 ] x b x c x g [ c 8 c x q x ] x b x x 9 [ x c x [ x c x c 10 a x q x ] x q x [ 11 g x a x g x a x b 12 ] x ] x ] x ] x q 13 c x c x c x c x c t = 29 1 2 3 4 5 6 7 8 9 1 g [ ] c [ ] g [ c 2 ] x x x x x x x x 3 c x g [ c ] g [ c 4 [ x ] x x x x x x 5 g x c x g c b [ c 6 ] x [ x ] x x x x 7 [ x g x c x g [ c 8 c x ] x [ x ] x x 9 ] x c x g x c x c 10 [ x [ x c x [ x [ 11 g x g x a x g x g 12 ] x ] x ] x ] x ] 13 c x c x c x c x ] t = 30 1 2 3 4 5 6 7 8 9 1 c ] [ c ] [ c [ c 2 ] x x x x x x x x 3 c x c [ c ] c [ c 4 [ x ] x x x x x x 5 c x c x [ c ] [ c 6 [ x [ x ] x x x x 7 ] x c x c x c [ c 8 c x ] x [ x ] x x 9 [ x c x ] x c x c 10 ] x [ x c x [ x [ 11 c x c x [ x c x c 12 ] x ] x ] x ] x q 13 c x c x c x c x [ t = 31 1 2 3 4 5 6 7 8 9 1 c c c c c c c c c 2 c x x x x x x x x 3 c x c c c c c c c 4 c x c x x x x x x 5 c x c x c c c c c 6 c x c x c x x x x 7 c x c x c x c c c 8 c x c x c x c x x 9 c x c x c x c x c 10 c x c x c x c x c 11 c x c x c x c x c 12 c x c x c x c x c 13 c x c x c x c x c t = 32 1 2 3 4 5 6 7 8 9 1 x x x x x x x x x 2 x x x x x x x x x 3 x x x x x x x x x 4 x x x x x x x x x 5 x x x x x x x x x 6 x x x x x x x x x 7 x x x x x x x x x 8 x x x x x x x x x 9 x x x x x x x x x 10 x x x x x x x x x 11 x x x x x x x x x 12 x x x x x x x x x 13 x x x x x x x x x t = 33 1 2 3 4 5 6 7 8 9 1 f f f f f f f f f 2 f f f f f f f f f 3 f f f f f f f f f 4 f f f f f f f f f 5 f f f f f f f f f 6 f f f f f f f f f 7 f f f f f f f f f 8 f f f f f f f f f 9 f f f f f f f f f 10 f f f f f f f f f 11 f f f f f f f f f 12 f f f f f f f f f 13 f f f f f f f f f fig. 4. snapshots of the non-optimum-time ten-state synchronization process on a 13 × 9 rectangular array. used in the recursive division as the pre-firing state, making the algorithm work in optimum-steps. those three implementations are stated in theorems 4, 5 and 6. their proofs are omitted due to the limited space available. some snapshots of the synchronization processes in those three implementations are given in figures 4, 5, and 6. theorem 4 there exists a ten-state 2d ca that can synchronize any m × n rectangle arrays in m + n + max(m, n) − 2 non-optimum steps. theorem 5 there exists an eleven-state 2d ca that can synchronize any m × n rectangle arrays in m + n + max(m, n) − 3 optimum steps. theorem 6 there exists a nine-state 2d ca that can synchronize any m × n rectangle arrays in m + n + max(m, n) − 3 optimum steps. t = 0 1 2 3 4 5 6 7 8 9 1 g q q q q q q q q 2 q q q q q q q q q 3 q q q q q q q q q 4 q q q q q q q q q 5 q q q q q q q q q 6 q q q q q q q q q 7 q q q q q q q q q 8 q q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 1 1 2 3 4 5 6 7 8 9 1 q gx q q q q q q q 2 [ q q q q q q q q 3 q q q q q q q q q 4 q q q q q q q q q 5 q q q q q q q q q 6 q q q q q q q q q 7 q q q q q q q q q 8 q q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 2 1 2 3 4 5 6 7 8 9 1 q a ] q q q q q q 2 b gx q q q q q q q 3 [ q q q q q q q q 4 q q q q q q q q q 5 q q q q q q q q q 6 q q q q q q q q q 7 q q q q q q q q q 8 q q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 3 1 2 3 4 5 6 7 8 9 1 q a q ] q q q q q 2 b x gx q q q q q q 3 q gx q q q q q q q 4 [ q q q q q q q q 5 q q q q q q q q q 6 q q q q q q q q q 7 q q q q q q q q q 8 q q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 4 1 2 3 4 5 6 7 8 9 1 q q a a ] q q q q 2 q x x gx q q q q q 3 b x tx q q q q q q 4 b gx q q q q q q q 5 [ q q q q q q q q 6 q q q q q q q q q 7 q q q q q q q q q 8 q q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 5 1 2 3 4 5 6 7 8 9 1 q a a a q ] q q q 2 b x x x gx q q q q 3 b x g q q q q q q 4 b x q q q q q q q 5 q gx q q q q q q q 6 [ q q q q q q q q 7 q q q q q q q q q 8 q q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 6 1 2 3 4 5 6 7 8 9 1 q a a q a a ] q q 2 b x x x x gx q q q 3 b x q ] q q q q q 4 q x [ q q q q q q 5 b x q q q q q q q 6 b gx q q q q q q q 7 [ q q q q q q q q 8 q q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 7 1 2 3 4 5 6 7 8 9 1 q a q a a a q ] q 2 b x x x x x gx q q 3 q x q a ] q q q q 4 b x b tx q q q q q 5 b x [ q q q q q q 6 b x q q q q q q q 7 q gx q q q q q q q 8 [ q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 8 1 2 3 4 5 6 7 8 9 1 q q a a a q a a gx 2 q x x x x x x gx q 3 b x q a q ] q q q 4 b x b x tx q q q q 5 b x q tx q q q q q 6 q x [ q q q q q q 7 b x q q q q q q q 8 b gx q q q q q q q 9 [ q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 9 1 2 3 4 5 6 7 8 9 1 q a a a q a a [ gx 2 b x x x x x x x tx 3 b x q q a a ] q q 4 b x q x x tx q q q 5 q x b x q q q q q 6 b x b tx q q q q q 7 b x [ q q q q q q 8 b x q q q q q q q 9 q gx q q q q q q q 10 [ q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 10 1 2 3 4 5 6 7 8 9 1 q a a q a a [ ] gx 2 b x x x x x x x x 3 b x tx a a a q ] q 4 q x b x x x tx q q 5 b x b x tx q q q q 6 b x b x q q q q q 7 b x q tx q q q q q 8 q x [ q q q q q q 9 b x q q q q q q q 10 b gx q q q q q q q 11 [ q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 11 1 2 3 4 5 6 7 8 9 1 q a q a a [ b [ gx 2 b x x x x x x x x 3 q x q a a q a a c 4 b x b x x x x tx q 5 b x b x g q q q q 6 b x q x q q q q q 7 q x b x q q q q q 8 b x b tx q q q q q 9 b x [ q q q q q q 10 b x q q q q q q q 11 q gx q q q q q q q 12 [ q q q q q q q q 13 q q q q q q q q q t = 12 1 2 3 4 5 6 7 8 9 1 q q a a [ q b [ gx 2 q x x x x x x x x 3 b x q a q a a [ c 4 b x b x x x x x tx 5 b x q x q ] q q q 6 q x b x [ q q q q 7 b x b x q q q q q 8 b x b x q q q q q 9 b x q tx q q q q q 10 q x [ q q q q q q 11 b x q q q q q q q 12 b gx q q q q q q q 13 gx q q q q q q q q t = 13 1 2 3 4 5 6 7 8 9 1 q a a [ b b g [ gx 2 b x x x x x x x x 3 b x q q a a [ ] c 4 b x q x x x x x x 5 q x b x q a ] q q 6 b x b x b tx q q q 7 b x b x [ q q q q 8 b x q x q q q q q 9 q x b x q q q q q 10 b x b tx q q q q q 11 b x [ q q q q q q 12 ] x q q q q q q q 13 gx tx q q q q q q q t = 14 1 2 3 4 5 6 7 8 9 1 q a [ q b b g [ gx 2 b x x x x x x x x 3 b x tx a a [ b [ c 4 q x b x x x x x x 5 b x b x q a q ] q 6 b x b x b x tx q q 7 b x q x q tx q q q 8 q x b x [ q q q q 9 b x b x q q q q q 10 b x b x q q q q q 11 ] x q tx q q q q q 12 [ x [ q q q q q q 13 gx x q q q q q q q t = 15 1 2 3 4 5 6 7 8 9 1 q [ b b q b g [ gx 2 b x x x x x x x x 3 q x q a [ q b [ c 4 b x b x x x x x x 5 b x b x q q a a c 6 b x q x q x x tx q 7 q x b x b x q q q 8 b x b x b tx q q q 9 b x b x [ q q q q 10 ] x q x q q q q q 11 a x b x q q q q q 12 ] x b tx q q q q q 13 gx x c q q q q q q t = 16 1 2 3 4 5 6 7 8 9 1 [ q b b b [ g [ gx 2 q x x x x x x x x 3 b x q [ b b g [ c 4 b x b x x x x x x 5 b x q x tx a a [ c 6 q x b x b x x x tx 7 b x b x b x tx q q 8 b x b x b x q q q 9 ] x q x q tx q q q 10 q x b x [ q q q q 11 a x b x q q q q q 12 ] x ] x q q q q q 13 gx x c tx q q q q q t = 17 1 2 3 4 5 6 7 8 9 1 b b q b b [ b [ gx 2 [ x x x x x x x x 3 b x [ q b b g [ c 4 b x q x x x x x x 5 q x b x q a [ ] c 6 b x b x b x x x x 7 b x b x b x g q q 8 ] x q x q x q q q 9 a x b x b x q q q 10 a x b x b tx q q q 11 g x ] x [ q q q q 12 ] x [ x q q q q q 13 gx x c x q q q q q t = 18 1 2 3 4 5 6 7 8 9 1 b b b q b [ b [ gx 2 [ x x x x x x x x 3 b x b b q b g [ c 4 q x [ x x x x x x 5 b x b x q [ b [ c 6 b x b x b x x x x 7 ] x q x q x q ] q 8 q x b x b x [ q q 9 a x b x b x q q q 10 a x ] x b x q q q 11 g x a x q tx q q q 12 ] x ] x [ q q q q 13 gx x c x q q q q q t = 19 1 2 3 4 5 6 7 8 9 1 b b b b g [ b [ gx 2 [ x x x x x x x x 3 g x b b b [ g [ c 4 b x [ x x x x x x 5 b x b x [ q b [ c 6 ] x q x q x x x x 7 a x b x b x q a c 8 a x b x b x b tx q 9 q x ] x b x [ q q 10 a x q x q x q q q 11 g x a x b x q q q 12 ] x ] x b tx q q q 13 gx x c x c q q q q t = 20 1 2 3 4 5 6 7 8 9 1 b b b b g q b [ gx 2 q x x x x x x x x 3 g x b b b [ b [ c 4 b x [ x x x x x x 5 ] x g x b b g [ c 6 q x b x [ x x x x 7 a x b x b x q [ c 8 a x ] x b x b x tx 9 a x a x q x q tx q 10 ] x a x b x [ q q 11 g x g x b x q q q 12 ] x ] x ] x q q q 13 gx x c x c tx q q q t = 21 1 2 3 4 5 6 7 8 9 1 q b b b g b g [ gx 2 b x x x x x x x x 3 g x b b b [ b [ c 4 ] x q x x x x x x 5 a x g x b b g [ c 6 a x b x [ x x x x 7 q x ] x b x [ ] c 8 a x q x q x q x x 9 a x a x b x b x q 10 ] x a x b x b tx q 11 a x g x ] x [ q q 12 ] x ] x [ x q q q 13 gx x c x c x q q q t = 22 1 2 3 4 5 6 7 8 9 1 b q b b g b g [ gx 2 b x x x x x x x x 3 gx x q b b [ b [ c 4 q x b x x x x x x 5 a x g x b b g [ c 6 a x ] x [ x x x x 7 a x a x g x b [ c 8 q x a x b x [ x x 9 a x q x b x b x tx 10 ] x a x ] x b x q 11 a x g x a x q tx q 12 ] x ] x ] x [ q q 13 gx x c x c x q q q t = 23 1 2 3 4 5 6 7 8 9 1 b b q b g b g [ gx 2 ] x x x x x x x x 3 gx x b q b [ b [ c 4 [ x b x x x x x x 5 q x c x b b g [ c 6 a x q x q x x x x 7 a x a x g x b [ c 8 a x a x b x [ x x 9 g x a x ] x b x g 10 ] x ] x q x q x q 11 a x g x a x b x q 12 ] x ] x ] x b tx q 13 gx x c x c x c q q t = 24 1 2 3 4 5 6 7 8 9 1 ] b b [ g b g [ gx 2 [ x x x x x x x x 3 gx x b b g [ b [ c 4 ] x ] x x x x x x 5 [ x c x q b g [ c 6 q x [ x b x x x x 7 a x q x g x b [ c 8 a x a x ] x [ x x 9 g x a x a x g x c 10 q x ] x a x b x [ 11 a x a x g x b x q 12 ] x ] x ] x ] x q 13 gx x c x c x c tx q t = 25 1 2 3 4 5 6 7 8 9 1 a ] b [ b b g [ gx 2 ] x x x x x x x x 3 gx x ] b g q b [ c 4 [ x [ x x x x x x 5 b x c x b [ g [ c 6 [ x ] x b x x x x 7 q x [ x c x b [ c 8 a x q x q x q x x 9 g x a x a x g x c 10 a x ] x a x b x ] 11 g x a x g x ] x [ 12 ] x ] x ] x [ x q 13 gx x c x c x c x q t = 26 1 2 3 4 5 6 7 8 9 1 a q ] [ b b g [ gx 2 ] x x x x x x x x 3 gx x a ] g b g [ c 4 [ x ] x x x x x x 5 b x c x b [ b [ c 6 q x [ x ] x x x x 7 [ x b x c x g [ c 8 ] x [ x [ x b x x 9 g x g x q x g x c 10 a x ] x a x ] x [ 11 g x a x g x a x b 12 ] x ] x ] x ] x [ 13 gx x c x c x c x q t = 27 1 2 3 4 5 6 7 8 9 1 g a a gx b b g [ gx 2 ] x x x x x x x x 3 gx x a q c b g [ c 4 [ x ] x x x x x x 5 g x c x ] [ b [ c 6 b x [ x [ x x x x 7 b x b x c x g [ c 8 gx x q x ] x b x x 9 a x c x [ x c x c 10 a x q x ] x q x [ 11 g x a x g x a x b 12 ] x ] x ] x ] x q 13 gx x c x c x c x c t = 28 1 2 3 4 5 6 7 8 9 1 g a [ gx ] b g [ gx 2 ] x x x x x x x x 3 gx x g [ c ] g [ c 4 [ x ] x x x x x x 5 g x c x g c b [ c 6 b x [ x ] x x x x 7 ] x g x c x g [ c 8 gx x ] x [ x ] x x 9 [ x c x g x c x c 10 a x [ x c x [ x [ 11 g x g x a x g x g 12 ] x ] x ] x ] x ] 13 gx x c x c x c x ] t = 29 1 2 3 4 5 6 7 8 9 1 g [ ] gx [ ] g [ gx 2 ] x x x x x x x x 3 gx x c [ c ] c [ c 4 [ x ] x x x x x x 5 g x c x [ c ] [ c 6 ] x [ x ] x x x x 7 [ x c x c x c [ c 8 gx x ] x [ x ] x x 9 ] x c x ] x c x c 10 [ x [ x c x [ x [ 11 g x c x [ x c x c 12 ] x ] x ] x ] x q 13 gx x c x c x c x [ t = 30 1 2 3 4 5 6 7 8 9 1 gx ] [ gx ] [ gx [ gx 2 ] x x x x x x x x 3 gx x c c c c c c c 4 [ x c x x x x x x 5 gx x c x c c c c c 6 [ x c x c x x x x 7 ] x c x c x c c c 8 gx x c x c x c x x 9 [ x c x c x c x c 10 ] x c x c x c x c 11 gx x c x c x c x c 12 ] x c x c x c x c 13 gx x c x c x c x c t = 31 1 2 3 4 5 6 7 8 9 1 gx gx gx gx gx gx gx gx gx 2 gx x x x x x x x x 3 gx x x x x x x x x 4 gx x x x x x x x x 5 gx x x x x x x x x 6 gx x x x x x x x x 7 gx x x x x x x x x 8 gx x x x x x x x x 9 gx x x x x x x x x 10 gx x x x x x x x x 11 gx x x x x x x x x 12 gx x x x x x x x x 13 gx x x x x x x x x t = 32 1 2 3 4 5 6 7 8 9 1 f f f f f f f f f 2 f f f f f f f f f 3 f f f f f f f f f 4 f f f f f f f f f 5 f f f f f f f f f 6 f f f f f f f f f 7 f f f f f f f f f 8 f f f f f f f f f 9 f f f f f f f f f 10 f f f f f f f f f 11 f f f f f f f f f 12 f f f f f f f f f 13 f f f f f f f f f fig. 5. snapshots of the optimum-time eleven-state synchronization process on a 13 × 9 array. table i a list of implementations for 2d rectangle fssp algorithms. implementations # of # of time notes states rules complexity beyer [1969] — — optimum rectangle shinahr [1974] 28 — optimum rectangle umeo, maeda and 6 1718 nonrectangle fujiwara [2002] optimum umeo and kubo [2010] 7 787 optimum square theorem 4 10 1629 nonrectangle (this paper) optimum theorem 5 11 4044 optimum rectangle (this paper) theorem 6 9 2561 optimum rectangle (this paper) iv. conclusion we have proposed a nine-state optimum-time synchronization algorithm that can synchronize any rectangle arrays of size m × n with a general at one corner in m + n + max(m, n)−3 steps. the algorithm is based on a new, simple zebra-like mapping scheme which embeds biomath 1 (2012), 1209022, http://dx.doi.org/10.11145/j.biomath.2012.09.022 page 5 of 6 http://dx.doi.org/10.11145/j.biomath.2012.09.022 h. umeo et al., a state-efficient zebra-like implementation of synchronization algorithms... t = 0 1 2 3 4 5 6 7 8 9 1 g q q q q q q q q 2 q q q q q q q q q 3 q q q q q q q q q 4 q q q q q q q q q 5 q q q q q q q q q 6 q q q q q q q q q 7 q q q q q q q q q 8 q q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 1 1 2 3 4 5 6 7 8 9 1 q x q q q q q q q 2 [ q q q q q q q q 3 q q q q q q q q q 4 q q q q q q q q q 5 q q q q q q q q q 6 q q q q q q q q q 7 q q q q q q q q q 8 q q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 2 1 2 3 4 5 6 7 8 9 1 q a ] q q q q q q 2 b x q q q q q q q 3 [ q q q q q q q q 4 q q q q q q q q q 5 q q q q q q q q q 6 q q q q q q q q q 7 q q q q q q q q q 8 q q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 3 1 2 3 4 5 6 7 8 9 1 q a q ] q q q q q 2 b x c q q q q q q 3 q x q q q q q q q 4 [ q q q q q q q q 5 q q q q q q q q q 6 q q q q q q q q q 7 q q q q q q q q q 8 q q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 4 1 2 3 4 5 6 7 8 9 1 q q a a ] q q q q 2 q x x x q q q q q 3 b x x q q q q q q 4 b x q q q q q q q 5 [ q q q q q q q q 6 q q q q q q q q q 7 q q q q q q q q q 8 q q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 5 1 2 3 4 5 6 7 8 9 1 q a a a q ] q q q 2 b x x x c q q q q 3 b x g q q q q q q 4 b x q q q q q q q 5 q x q q q q q q q 6 [ q q q q q q q q 7 q q q q q q q q q 8 q q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 6 1 2 3 4 5 6 7 8 9 1 q a a q a a ] q q 2 b x x x x x q q q 3 b x q ] q q q q q 4 q x [ q q q q q q 5 b x q q q q q q q 6 b x q q q q q q q 7 [ q q q q q q q q 8 q q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 7 1 2 3 4 5 6 7 8 9 1 q a q a a a q ] q 2 b x x x x x c q q 3 q x q a ] q q q q 4 b x b g q q q q q 5 b x [ q q q q q q 6 b x q q q q q q q 7 q x q q q q q q q 8 [ q q q q q q q q 9 q q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 8 1 2 3 4 5 6 7 8 9 1 q q a a a q a a c 2 q x x x x x x x q 3 b x q a q ] q q q 4 b x b x g q q q q 5 b x q g q q q q q 6 q x [ q q q q q q 7 b x q q q q q q q 8 b x q q q q q q q 9 [ q q q q q q q q 10 q q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 9 1 2 3 4 5 6 7 8 9 1 q a a a q a a [ x 2 b x x x x x x x a 3 b x q q a a ] q q 4 b x q x c g q q q 5 q x b c q q q q q 6 b x b g q q q q q 7 b x [ q q q q q q 8 b x q q q q q q q 9 q x q q q q q q q 10 [ q q q q q q q q 11 q q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 10 1 2 3 4 5 6 7 8 9 1 q a a q a a [ ] x 2 b x x x x x x x x 3 b x q a a a q ] q 4 q x b x x x g q q 5 b x b x x q q q q 6 b x b x q q q q q 7 b x q g q q q q q 8 q x [ q q q q q q 9 b x q q q q q q q 10 b x q q q q q q q 11 [ q q q q q q q q 12 q q q q q q q q q 13 q q q q q q q q q t = 11 1 2 3 4 5 6 7 8 9 1 q a q a a [ b [ x 2 b x x x x x x x x 3 q x q a a q a a c 4 b x b x x x c g q 5 b x b x g q q q q 6 b x q x q q q q q 7 q x b c q q q q q 8 b x b g q q q q q 9 b x [ q q q q q q 10 b x q q q q q q q 11 q x q q q q q q q 12 [ q q q q q q q q 13 q q q q q q q q q t = 12 1 2 3 4 5 6 7 8 9 1 q q a a [ q b [ x 2 q x x x x x x x x 3 b x q a q a a [ c 4 b x b x x x x x b 5 b x q x q ] q q q 6 q x b x [ q q q q 7 b x b x q q q q q 8 b x b x q q q q q 9 b x q g q q q q q 10 q x [ q q q q q q 11 b x q q q q q q q 12 b x q q q q q q q 13 c q q q q q q q q t = 13 1 2 3 4 5 6 7 8 9 1 q a a [ b b g [ x 2 b x x x x x x x x 3 b x q q a a [ ] c 4 b x q x x x x x x 5 q x b x q a ] q q 6 b x b x b g q q q 7 b x b x [ q q q q 8 b x q x q q q q q 9 q x b c q q q q q 10 b x b g q q q q q 11 b x [ q q q q q q 12 ] x q q q q q q q 13 x b q q q q q q q t = 14 1 2 3 4 5 6 7 8 9 1 q a [ q b b g [ x 2 b x x x x x x x x 3 b x q a a [ b [ c 4 q x b x x x x x x 5 b x b x q a q ] q 6 b x b x b x g q q 7 b x q x q g q q q 8 q x b x [ q q q q 9 b x b x q q q q q 10 b x b x q q q q q 11 ] x q g q q q q q 12 [ x [ q q q q q q 13 x x q q q q q q q t = 15 1 2 3 4 5 6 7 8 9 1 q [ b b q b g [ x 2 b x x x x x x x x 3 q x q a [ q b [ c 4 b x b x x x x x x 5 b x b x q q a a c 6 b x q x q x c g q 7 q x b x b c q q q 8 b x b x b g q q q 9 b x b x [ q q q q 10 ] x q x q q q q q 11 a x b c q q q q q 12 ] x b g q q q q q 13 x x c q q q q q q t = 16 1 2 3 4 5 6 7 8 9 1 [ q b b b [ g [ x 2 q x x x x x x x x 3 b x q [ b b g [ c 4 b x b x x x x x x 5 b x q x q a a [ c 6 q x b x b x x x b 7 b x b x b x x q q 8 b x b x b x q q q 9 ] x q x q g q q q 10 q x b x [ q q q q 11 a x b x q q q q q 12 ] x ] x q q q q q 13 x x c a q q q q q t = 17 1 2 3 4 5 6 7 8 9 1 b b q b b [ b [ x 2 [ x x x x x x x x 3 b x [ q b b g [ c 4 b x q x x x x x x 5 q x b x q a [ ] c 6 b x b x b x x x x 7 b x b x b x g q q 8 ] x q x q x q q q 9 a x b x b c q q q 10 a x b x b g q q q 11 g x ] x [ q q q q 12 ] x [ x q q q q q 13 x x c x q q q q q t = 18 1 2 3 4 5 6 7 8 9 1 b b b q b [ b [ x 2 [ x x x x x x x x 3 b x b b q b g [ c 4 q x [ x x x x x x 5 b x b x q [ b [ c 6 b x b x b x x x x 7 ] x q x q x q ] q 8 q x b x b x [ q q 9 a x b x b x q q q 10 a x ] x b x q q q 11 g x a x q g q q q 12 ] x ] x [ q q q q 13 x x c x q q q q q t = 19 1 2 3 4 5 6 7 8 9 1 b b b b g [ b [ x 2 [ x x x x x x x x 3 g x b b b [ g [ c 4 b x [ x x x x x x 5 b x b x [ q b [ c 6 ] x q x q x x x x 7 a x b x b x q a c 8 a x b x b x b g q 9 q x ] x b x [ q q 10 a x q x q x q q q 11 g x a x b c q q q 12 ] x ] x b g q q q 13 x x c x c q q q q t = 20 1 2 3 4 5 6 7 8 9 1 b b b b g q b [ x 2 q x x x x x x x x 3 g x b b b [ b [ c 4 b x [ x x x x x x 5 ] x g x b b g [ c 6 q x b x [ x x x x 7 a x b x b x q [ c 8 a x ] x b x b x b 9 a x a x q x q g q 10 ] x a x b x [ q q 11 g x g x b x q q q 12 ] x ] x ] x q q q 13 x x c x c a q q q t = 21 1 2 3 4 5 6 7 8 9 1 q b b b g b g [ x 2 b x x x x x x x x 3 g x b b b [ b [ c 4 ] x q x x x x x x 5 a x g x b b g [ c 6 a x b x [ x x x x 7 q x ] x b x [ ] c 8 a x q x q x q g x 9 a x a x b x b c q 10 ] x a x b x b g q 11 a x g x ] x [ q q 12 ] x ] x [ x q q q 13 x x c x c x q q q t = 22 1 2 3 4 5 6 7 8 9 1 b q b b g b g [ x 2 b x x x x x x x x 3 x x q b b [ b [ c 4 q x b x x x x x x 5 a x g x b b g [ c 6 a x ] x [ x x x x 7 a x a x g x b [ c 8 q x a x b x [ x x 9 a x q x b x b [ q 10 ] x a x ] x b x q 11 a x g x a x q g q 12 ] x ] x ] x [ q q 13 x x c x c x q q q t = 23 1 2 3 4 5 6 7 8 9 1 b b q b g b g [ x 2 ] x x x x x x x x 3 x x b q b [ b [ c 4 [ x b x x x x x x 5 q x c x b b g [ c 6 a x q x q x x x x 7 a x a x g x b [ c 8 a x a x b x [ x x 9 g x a x ] x b x g 10 ] x ] x q x q x q 11 a x g x a x b c q 12 ] x ] x ] x b g q 13 x x c x c x c q q t = 24 1 2 3 4 5 6 7 8 9 1 ] b b [ g b g [ x 2 [ x x x x x x x x 3 x x b b g [ b [ c 4 ] x ] x x x x x x 5 [ x c x q b g [ c 6 q x [ x b x x x x 7 a x q x g x b [ c 8 a x a x ] x [ x x 9 g x a x a x g x c 10 q x ] x a x b x [ 11 a x a x g x b x q 12 ] x ] x ] x ] x q 13 x x c x c x c a q t = 25 1 2 3 4 5 6 7 8 9 1 a ] b [ b b g [ x 2 ] x x x x x x x x 3 x x ] b g q b [ c 4 [ x [ x x x x x x 5 b x c x b [ g [ c 6 [ x ] x b x x x x 7 q x [ x c x b [ c 8 a x q x q x q x x 9 g x a x a x g x c 10 a x ] x a x b x ] 11 g x a x g x ] x [ 12 ] x ] x ] x [ x q 13 x x c x c x c x q t = 26 1 2 3 4 5 6 7 8 9 1 a q ] [ b b g [ x 2 ] x x x x x x x x 3 x x a ] g b g [ c 4 [ x ] x x x x x x 5 b x c x b [ b [ c 6 q x [ x ] x x x x 7 [ x b x c x g [ c 8 ] x [ x [ x b x x 9 g x g x q x g x c 10 a x ] x a x ] x [ 11 g x a x g x a x b 12 ] x ] x ] x ] x [ 13 x x c x c x c x q t = 27 1 2 3 4 5 6 7 8 9 1 g a a x b b g [ x 2 ] x x x x x x x x 3 x x a q c b g [ c 4 [ x ] x x x x x x 5 g x c x ] [ b [ c 6 b x [ x [ x x x x 7 b x b x c x g [ c 8 x x q x ] x b x x 9 a x c x [ x c x c 10 a x q x ] x q x [ 11 g x a x g x a x b 12 ] x ] x ] x ] x q 13 x x c x c x c x c t = 28 1 2 3 4 5 6 7 8 9 1 g a [ x ] b g [ x 2 ] x x x x x x x x 3 x x g [ c ] g [ c 4 [ x ] x x x x x x 5 g x c x g c b [ c 6 b x [ x ] x x x x 7 ] x g x c x g [ c 8 x x ] x [ x ] x x 9 [ x c x g x c x c 10 a x [ x c x [ x [ 11 g x g x a x g x g 12 ] x ] x ] x ] x ] 13 x x c x c x c x ] t = 29 1 2 3 4 5 6 7 8 9 1 g [ ] x [ ] g [ x 2 ] x x x x x x x x 3 x x c [ c ] c [ c 4 [ x ] x x x x x x 5 g x c x [ c ] [ c 6 ] x [ x ] x x x x 7 [ x c x c x c [ c 8 x x ] x [ x ] x x 9 ] x c x ] x c x c 10 [ x [ x c x [ x [ 11 g x c x [ x c x c 12 ] x ] x ] x ] x q 13 x x c x c x c x [ t = 30 1 2 3 4 5 6 7 8 9 1 x ] [ x ] [ x [ x 2 ] x x x x x x x x 3 x x c c c c c c c 4 [ x c x x x x x x 5 x x c x c c c c c 6 [ x c x c x x x x 7 ] x c x c x c c c 8 x x c x c x c x x 9 [ x c x c x c x c 10 ] x c x c x c x c 11 x x c x c x c x c 12 ] x c x c x c x c 13 x x c x c x c x c t = 31 1 2 3 4 5 6 7 8 9 1 x x x x x x x x x 2 x x x x x x x x x 3 x x x x x x x x x 4 x x x x x x x x x 5 x x x x x x x x x 6 x x x x x x x x x 7 x x x x x x x x x 8 x x x x x x x x x 9 x x x x x x x x x 10 x x x x x x x x x 11 x x x x x x x x x 12 x x x x x x x x x 13 x x x x x x x x x t = 32 1 2 3 4 5 6 7 8 9 1 f f f f f f f f f 2 f f f f f f f f f 3 f f f f f f f f f 4 f f f f f f f f f 5 f f f f f f f f f 6 f f f f f f f f f 7 f f f f f f f f f 8 f f f f f f f f f 9 f f f f f f f f f 10 f f f f f f f f f 11 f f f f f f f f f 12 f f f f f f f f f 13 f f f f f f f f f fig. 6. snapshots of the optimum-time nine-state synchronization process on a 13 × 9 array. 1d synchronization operations onto rectangle arrays. the nine-state implementation described in terms of state transition table is a smallest, known at present, realization of time-optimum rectangle synchronizer. the embedding scheme developed in this paper would be useful for state-efficient implementation of multi-dimensional synchronization algorithms. references [1] r. balzer, “an 8-state minimal time solution to the firing squad synchronization problem”, information and control 10, 22–42 (1967). http://dx.doi.org/10.1016/s0019-9958(67)90032-0 [2] w. t. beyer, “recognition of topological invariants by iterative arrays”, ph.d. thesis, mit, pp. 144 (1969). [3] h. d. gerken, “über synchronisations — probleme bei zellularautomaten”, diplomarbeit, institut für theoretische informatik, technische universität braunschweig, pp. 50 (1987). [4] e. goto, “a minimal time solution of the firing squad problem”, dittoed course notes for applied mathematics 298, harvard university, 52–59 (1962). [5] j. mazoyer, “a six-state minimal time solution to the firing squad synchronization problem”, theoretical computer science 50, 183–238 (1987). http://dx.doi.org/10.1016/0304-3975(87)90124-1 [6] e. f. moore, “the firing squad synchronization problem”, in sequential machines, selected papers, (e. f. moore, ed.), addison-wesley, reading ma, 213–214 (1964). [7] h. schmid, “synchronisationsprobleme für zelluläre automaten mit mehreren generälen”, diplomarbeit, universität karsruhe, (2003). [8] h. schmid and t. worsch, “the firing squad synchronization problem with many generals for one-dimensional ca”, proc. of ifip world congress, 111–124 (2004). [9] i. shinahr, “twoand three-dimensional firing squad synchronization problems”, information and control 24, 163–180 (1974). http://dx.doi.org/10.1016/s0019-9958(74)80055-0 [10] h. szwerinski, “time-optimum solution of the firing-squadsynchronization-problem for n-dimensional rectangles with the general at an arbitrary position”, theoretical computer science 19, 305–320 (1982). http://dx.doi.org/10.1016/0304-3975(82)90040-8 [11] h. umeo, “firing squad synchronization algorithms for twodimensional cellular automata”, journal of cellular automata 4, 1–20 (2008). [12] h. umeo, “firing squad synchronization problem in cellular automata”, in: encyclopedia of complexity and system science, r. a. meyers (ed.), springer, vol.4, 3537–3574 (2009). [13] h. umeo, m. hisaoka, and s. akiguchi, “twelve-state optimum-time synchronization algorithm for two-dimensional rectangular cellular arrays”, proc. of 4th international conference on unconventional computing: uc 2005, lncs 3699, 214–223 (2005). [14] h. umeo, n. kamikawa, k. nishioka, and s. akiguchi, “generalized firing squad synchronization protocols for onedimensional cellular automata — a survey”, acta physica polonica b, proceedings supplement 3, 267–289 (2010). [15] umeo and kubo, “a seven-state time-optimum square synchronizer.”, proc. of the 9th international conference on cellular automata for research and industry, lncs 6350, springerverlag, 219–230 (2010). [16] h. umeo, m. maeda and n. fujiwara, “an efficient mapping scheme for embedding any one-dimensional firing squad synchronization algorithm onto two-dimensional arrays”, proc. of the 5th international conference on cellular automata for research and industry, lncs 2493, springer-verlag, 69–81 (2002). [17] h. umeo, m. maeda, m. hisaoka and m. teraoka, “a stateefficient mapping scheme for designing two-dimensional firing squad synchronization algorithms”, fundamenta informaticae, 74 no. 4, 603–623 (2006). [18] h. umeo and h. uchino, “a new time-optimum synchronization algorithm for two-dimensional cellular arrays”, proc. of intern. conf. on computer aided systems theory, eurocast 2007, (a. quesada-arenciba, j. c. rodriguez, r. moreno-diaz jr., r. moreno-diaz (eds.)), 213–216, (2007). [19] h. umeo, t. yamawaki, n. shimizu and h. uchino, “modeling and simulation of global synchronization processes for largescale-of two-dimensional cellular arrays”, proc. of intern. conf. on modeling and simulation, ams 2007, 139–144 (2007). biomath 1 (2012), 1209022, http://dx.doi.org/10.11145/j.biomath.2012.09.022 page 6 of 6 http://dx.doi.org/10.1016/s0019-9958(67)90032-0 http://dx.doi.org/10.1016/0304-3975(87)90124-1 http://dx.doi.org/10.1016/s0019-9958(74)80055-0 http://dx.doi.org/10.1016/0304-3975(82)90040-8 http://dx.doi.org/10.11145/j.biomath.2012.09.022 introduction firing squad synchronization problem fssp on 2d arrays optimum-time l-shaped mapping algorithm zebra-like mapping on square arrays zebra-like mapping on rectangle arrays conclusion references www.biomathforum.org/biomath/index.php/biomath covid-19 research communications (editorial) the goal of the series is to provide a platform for rapid communication and exchange of ideas concerning the covid-19 epidemic. it is new and unlike the known virus-induced diseases. there is a significant research effort, including mathematical modelling, to understand the characteristics of the virus sars-cov-2 (severe acute respiratory syndrome coronavirus 2) and the epidemiological dynamics of covid-19, the disease caused by it. due to their novelty, the research is often likely to produce results only on specific aspects of the disease, provide just partial answers to research questions, or collect evidence for formulating hypothesis yet to be tested. we believe, however, that the significance of the pandemic for the human population makes it essential to share even such partial results as soon as they are available to facilitate the advancement of the research on this disease. while eventually, a more comprehensive picture of both the virus and the disease will emerge, even incomplete but timely and scientifically-based information will help the authorities to make sound decisions on the course of action during the epidemic. for the series, we invite publications on any aspect of the covid-19 epidemic. specifically, the series aims to cover • the biological research, providing an understanding of the relevant structures and causal relationships in the epidemiological environment, which can facilitate mathematical or statistical modelling, • mathematical models of the structures, causal interactions and epidemiological data, and their analysis, • mathematical models and analysis of the socio-economic aspects of the pandemic, • any new mathematical methods, applicable to the study of any of the mentioned topics. all submissions to the series will be prioritised for a fast peer-review. hristo kojouharov section editor copyright: c© 2020 kojouharov. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: hristo kojouharov, covid-19 research communications (editorial), biomath 9 (2020), 2005047, http://dx.doi.org/10.11145/j.biomath.2020.05.047 page 1 of 1 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2020.05.047 original article biomath 1 (2012), 1209251, 1–5 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum viral dynamic model of antiretroviral therapy including the integrase inhibitor raltegravir in patients with hiv-1 dimitra bon∗, christoph stephan†, oliver t. keppler‡ and eva herrmann∗ ∗ institute of biostatistic and mathematical modeling department of medicine, goethe-university, frankfurt main, germany emails: bon@med.uni-frankfurt.de, herrmann@med.uni-frankfurt.de † hiv center goethe university hospital and faculty of medicine, frankfurt main, germany email: christoph.stephan@mail.hivcenter.de ‡ institute of medical virology goethe university hospital and faculty of medicine, frankfurt main, germany email: oliver.keppler@kgu.de received: 15 july 2012, accepted: 25 september 2012, published: 13 october 2012 abstract —antiviral combination therapies consisting of reverse transcriptase inhibitors, protease inhibitors and an integrase inhibitor, have been developed to suppress hiv below the limit of detection[3]. we introduce a mathematical model for the effect of different combination treatment regimens on the dynamics of hiv rna and cd4 t-cell counts [1,7,8,9]. we will especially focus on modelling the treatment effect of the integrase inhibitorraltegravir [11]. the model consists of a system of ordinary differential equations and the parameters were chosen or estimated in order to agree with clinical data of a recent clinical trial [13]. all the numerical simulations were calculated with matlab. keywords-hiv; mathematical modelling; integrase inhibitor treatment i. in t ro d u ct i o n human immunodeficiency virus (hiv ) is a member of the retrovirus family. it can cause acquired immunodeficiency syndrome (aids) in which the immune system fails, allowing opportunistic infections and cancers to thrive. hiv infection is considered pandemic with approximately 34 million people been infected globally (who-unaids 2010)[15]. a. hiv-1 infection in absence of antiretroviral treatment the course of hiv-1 infection consists of three phases. first phase is the acute phase, it can last several weeks and it is a period of rapid viral replication. the second phase, the chronic phase, can last up to 9-10 years and it is mostly asymptomatic. in this phase an equilibrium is achieved between viral replication and the immune response. the last phase, aids, can last a couple of years and the immune system can no longer control the viral replication.[4 (p.6-10)] b. replication cycle and antiretroviral treatment (fig.1) hiv infects cells in the human immune system such as t cells and in particular the cd4+ t-cells. understanding the hiv replication cycle and how and where the antiretroviral drugs works, helps us to construct our mathematical model. therefore, we focus on antiviral treatment during the asymptomatic phase of hiv-1 infection. the process begins when the virus binds to the surface of the cd4 cell. then rna enzymes enter the citation: d. bon, c. stephan, o. keppler, e. herrmann, viral dynamic model of antiretroviral therapy including the integrase inhibitor raltegravir in patients with hiv-1, biomath 1 (2012), 1209251, http://dx.doi.org/10.11145/j.biomath.2012.09.251 page 1 of 5 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.09.251 d. bon et al., viral dynamic model of antiretroviral therapy... fig. 1. replication cycle-treatment. cytoplasm. reverse transcriptase make a dna copy of the viral rna and with the help from integrase, the viral dna integrates into the host dna. the proviral dna transcribes into rna and then rna is translated into protein. viral rna and proteins assembled into viral particles and with the help of protease the new virus buds out.[4 (p.26-33),14] the management of hiv/aids typically includes a combination of antiretroviral drugs and the approach is known as highly active antiretroviral therapy (haart). the three most important classes of antiretroviral drugs are: reverse transcriptase inhibitors (rti) interfere with reverse transcription. protease inhibitors(pi) interfere with the protease enzyme that hiv uses to produce infectious viral particles. integrase inhibitors (ini) block integrase, the enzyme hiv uses to integrate genetic material of the virus into its target host cell. [4 (p.60-94,105-108)] ii. mo d e l li ng t he dy n ami c s o f hiv we begin with the basic model of untreated chronic viral infection[7,8,9]. the model considers three compartments, the uninfected (target) cells (t), the infected cells (i) and the the free virus (v ). target cells are produced at a constant rate s and die at rate d t . free virus infects target cells in order to produce infected cells at rate k v t and dies at rate c v . infected cells die at rate δi and produce new virus at rate nδi . the equations that describe the basic model are: d t /d t = s − k v t − d t d i /d t = k v t −δi d v /d t = nδi − c v where • s is the source term for target cells • d is the loss rate of target cells • k is the infection rate • c is the loss rate of free virus • δ is the loss rate of infected cells • n is the number of virion production. note that we do not model latently infected cell compartments as the models which are discussed in the following mainly aim to model relatively only limited treatment durations. a. models of viral dynamics under rti and pi treatment patients infected with hiv-1 were usually treated with rti’s or pi’s or a combination of the two. in order to analyse the effect of antiretroviral treatment the model equations become [8,9]: d t /d t = s − (1 −ε2)k vi t − d t d i /d t = (1 −ε2)k vi t −δi d vi /d t = (1 −ε1)nδi − c vi d vn i /d t = ε1nδi − c vn i where • ε1 and ε2 are the efficacies of pi and rti (0 ≤ ε1,2 ≤ 1). • vi and vn i are the infectious and non-infectious virus, respectively, where v = vi + vn i is the viral load. fig. 2. model of viral infection under pi and rti treatment. biomath 1 (2012), 1209251, http://dx.doi.org/10.11145/j.biomath.2012.09.251 page 2 of 5 http://dx.doi.org/10.11145/j.biomath.2012.09.251 d. bon et al., viral dynamic model of antiretroviral therapy... b. models of viral dynamics under raltegravir treatment since 2007, when raltegravir was approved by the fda, we can find in the literature mathematical models that describe the dynamics of hiv-1 under treatment with raltegravir. in sedaghat a. r. et al. [10,11] it is suggested a model with three states, the uninfected, the early stage infection and the late stages infection. murray j. m.[5,6] suggests two models, in the first one includes a compartment of long-lived infected cells and in the second includes a pool of cells with full-length integrated hiv dna. what we suggest, is an extension of the basic model (fig. 3) resulting in a simpler model to assess antiretroviral treatment efficacy. it may be less suited for long-term prediction. consider instead of one compartment of infected cells, two, one with infected cells that can produce virus (i1) and one that because of raltegravir is unable to produce virus (i2). fig. 3. model of viral infection under combination treatment. thus our system of equations becomes: d t /d t = s − (1 −ε2)k vi t − d t d i1/d t = (1 −ε2)(1 −ε3)k vi t −δi1 d i2/d t = (1 −ε2)ε3 k vi t − mδi2 d vi /d t = (1 −ε1)nδi1 − c vi d vn i /d t = ε1nδi1 − c vn i where m is the enhancement loss rate because of raltegravir and ε3 the efficacy of raltegravir. iii. fi t t i n g ex a mp l e we consider an example of one patient of a recent study with patients infected with hiv-1 [13]. this patient was treated with 2 rti’s and rltegravir. in order to verify our model we fitted both measurements of plasma hiv rna concentrations and cd4 t-cell counts with our model (fig.5). all the calculations were performed with matlab. we solve our ode system with ode-23s solver for stiff equations, we calculate the likelihood function and for the minimization process we used the fminsearch, which uses the nelder-mead algorithm. we set the m ,n parameters fix and estimate c , d , δ, ε1, ε2, ε3 (fig.4). fig. 4. fix and estimated parameters for our patient. note that we have calculated s and k as: s=449.592 cells/day, k=2.52 × 10−4 fig. 5. patient treated with raltegravir and 2 rti’s iv. ex te n d e d mo d e l as it is now also possible to quantify unintegrated dna [13], we decided to extend our model. we know that the unintegrated dna enter the nucleus, either integrates with the host dna or binds with itself and biomath 1 (2012), 1209251, http://dx.doi.org/10.11145/j.biomath.2012.09.251 page 3 of 5 http://dx.doi.org/10.11145/j.biomath.2012.09.251 d. bon et al., viral dynamic model of antiretroviral therapy... forms 1-ltr (long terminal repeat) and 2-ltr circles (fig.6)[2,12]. thus we introduce a new compartment (i0) of unintegrated dna (fig.7). fig. 6. replication cycle-ltr circles. fig. 7. extended model of viral infection under combination treatment. the ode system that describes our model (fig.7) is d t /d t = s − (1 −ε2)k t vi − d t d i0/d t = (1 −ε2)k t vi − (a + b)i0 d i1/d t = b(1 −ε3)i0 −δi1 d i2/d t = (a + bε3)i0 − mδi2 d vi /d t = (1 −ε1)nδi1 − c vi d vn i /d t = ε1nδi1 − c vn i where • a is the production rate of 2-ltr circles • b is the production rate of integrated dna (before treatment b > a) • i0 is the compartment of unintegrated dna (cdna) • i1 is the compartment of the 2-ltr circles • i2 is the compartment of integrated dna fitting results of clinical data will be described in an upcoming clinical paper. preliminary results on nonfinal data are promising. v. co n clus i o n s mathematical modelling of hiv-1 dynamics can be a really useful tool for the treatment of hiv-1 as they allow an efficient and early assessment of treatment efficacy. we can focus on viral clearance and cd4 tcell death rates in which the medical and modelling community show interest, as well as the treatment efficacy parameters. therefore we can also compare different treatment groups in order to see which treatment may be preferable. overall, our model is relatively concise but may be less suited for longterm predictions. previous models [5,6,10,11] include a compartment of latently infected cells which makes them more complex and estimating treatment efficacy will be more difficult. nevertheless, such models are better for fitting and predicting long-term clinical response. overall we believe that the proposed model can be a tool to analyze data from clinical trials. re f e r e nc e s [1] s. bonhoffer, r.m. may, g.m. shaw, m.a. nowak, virus dynamics and drug therapy, proc. natl. acad. sci. usa 49 6971–6976, 1997. http://dx.doi.org/10.1073/pnas.94.13.6971 [2] m. j. buzon et al., hiv-1 replication and immune dynamics are affected by raltegravir intensification of haart-suppressed subjects, nature medicine 16 2010. [3] e. de clercq, strategies in the design of antiviral drugs, nature reviews drug discovery 1 13–25, 2002. http://dx.doi.org/10.1038/nrd703 [4] c. hoffmann, j. k. rockstroh, hiv 2011, medizin fokus verlag, 2011. http://hivbook.com/ [5] j.m. murray, s. emery, a.d. kelleher, m. law, j. chen, d.j. hazuda, b-y t. nguyen, h. teppler and d.a. cooper, antiretroviral therapy with the integrase inhibitor raltegravir alters decay kinetics of hiv, significantly reducing the second phase, aids 21 2315–2321, 2007. http://dx.doi.org/10.1097/qad.0b013e3282f12377 [6] j.m. murray, hiv dynamics and integrase inhibitors, antiviral chemistry and chemotherapy 19 157–164, 2002. [7] m. a. nowak and r. m. may, virus dynamics-mathematical principles of immunology and virology, oxford university press, 2000. [8] a. s. perelson, p. w. nelson, mathematical analysis of hiv-1 dynamics in vivo, siam review 41:1 3–44, 1999. biomath 1 (2012), 1209251, http://dx.doi.org/10.11145/j.biomath.2012.09.251 page 4 of 5 http://dx.doi.org/10.1073/pnas.94.13.6971 http://dx.doi.org/10.1038/nrd703 http://hivbook.com/ http://dx.doi.org/10.1097/qad.0b013e3282f12377 http://dx.doi.org/10.11145/j.biomath.2012.09.251 d. bon et al., viral dynamic model of antiretroviral therapy... [9] a. s. perelson, modelling viral and immune system dynamics, nature reviews immunology 2 28–36 january 2002. http://dx.doi.org/10.1038/nri700 [10] a. r. sedaghat, j.b. dinoso, l. shen, c.o. wilke, r.f. siliciano,c.o. wilke, decay dynamics of hiv-1 depend on the inhibited stages of the viral life cycle, pnas vol. 105, no. 12 4832–4837, 2008. http://dx.doi.org/10.1073/pnas.0711372105 [11] a. r. sedaghat, r.f. siliciano, c.o. wilke, constraints on the dominant mechanism for hiv viral dynamics in patients on raltegravir, antiviral therapy 14 263–271, 2009. [12] r. d. sloan, m.a. wainberg, the role of unintegrated dna in hiv infection, retrovirology 8:52 2011. http://dx.doi.org/10.1186/1742-4690-8-52 [13] c. stephan, h.m. baldauf, a. haberl, m. bickel, e. herrmann, a. berger, m. stuermer, c. goffinet, l. kaderali, o. keppler, impact of raltegravir on hiv-1 rna and dna species following initiation of antiretroviral therapy croi 2012, 19th conference on retroviruses and opportunistic infections. [14] national institute of health-national institute of allergy and infectious diseases, hiv replication cycle steps in the hiv replication cycle, 2012. http://www.niaid.nih.gov/topics/ hivaids/understanding/biology/pages/hivreplicationcycle. aspx [15] world health organization-unaids, global summary of the hiv/aids epidemic, 2010. http://www.who.int/hiv/data/2011_epi_core_en.png biomath 1 (2012), 1209251, http://dx.doi.org/10.11145/j.biomath.2012.09.251 page 5 of 5 http://dx.doi.org/10.1038/nri700 http://dx.doi.org/10.1073/pnas.0711372105 http://dx.doi.org/10.1186/1742-4690-8-52 http://www.niaid.nih.gov/topics/hivaids/understanding/biology /pages/hivreplicationcycle.aspx http://www.niaid.nih.gov/topics/hivaids/understanding/biology /pages/hivreplicationcycle.aspx http://www.niaid.nih.gov/topics/hivaids/understanding/biology /pages/hivreplicationcycle.aspx http://www.who.int/hiv/data/2011_epi_core_en.png http://dx.doi.org/10.11145/j.biomath.2012.09.251 introduction hiv-1 infection replication cycle and antiretroviral treatment (fig.1) modelling the dynamics of hiv models of viral dynamics under rti and pi treatment models of viral dynamics under raltegravir treatment fitting example extended model conclusions references www.biomathforum.org/biomath/index.php/biomath original article a rewinding model for replicons with dna-links abdul adheem mohamad1, tsukasa yashiro2 1mathematics section, department of mathematical and physical sciences college of arts and sciences, university of nizwa mohamad@unizwa.edu.om 2independent mathematical institute miyota, kitasaku, nagano, japan t-yashiro@dokusuken.com received: 9 october 2019, accepted: 4 january 2020, published: 23 february 2020 abstract—a double strand dna has a double helical structure and it is modeled by a thin long twisted ribbon fixed at the both ends. a dna-link is a topological model of such a dna segment in the nuclear of a eukaryotic cell. in the cell cycle, the dna is replicated and distributed into new cells. the complicated replication process follows the semi-conservative scheme in which each backbone string is preserved in the replicated dna. this is interpreted in terms of splitting process of the dnalink. in order to split the dna-link, unknotting operations are required. this paper presents a recursive unknotting operations, which efficiently reduce the number of twistings. keywords-dna, replication, link, topological model, replicon mathematics subject classification (2010) 92b99 i. introduction it is known that dna is a polymer consisting of a set of base pairs and sugar-phosphate backbones (watson-click model (1953), see [2] [11]). the backbones form a double-helix structure with opposite directions induced from the ordered pair of 5′ and 3′. the 360◦-rotation of the helical strings is joined by about 10.5 base pairs. therefore, it is natural that the double strand dna (ds-dna) is modeled by a long, thin strip twisted around the centre curve of the strip (see figure 1, also [2][11]). the boundary curves of the strip form a double helix which correspond to the backbones of dna [2][11]. a full twist of the ds-dna is interpreted as the 360◦-rotation of the strip about the centre curve (see figure 1). the replication of dna has been studied since the helical double strand structure was discovered (see [1][3][4][10][12]). in the interphase of the copyright: c© 2020 mohamad et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: abdul adheem mohamad, tsukasa yashiro, a rewinding model for replicons with dna-links, biomath 9 (2020), 2001047, http://dx.doi.org/10.11145/j.biomath.2020.01.047 page 1 of 9 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2020.01.047 abdul adheem mohamad, tsukasa yashiro, a rewinding model for replicons with dna-links 10.5bp γ0 γ0 fig. 1. the dna double helix is modeled by a long thin strip. (b) (c)(a) fig. 2. (a) the linear dna forms a set of loops attached to the nuclear matrix. (b) each loop is anchored at the nuclear matrix (nm). (c) the anchored loop is modeled by twisted strings fixed at their ends of the boundary of a 3-dimensional ball, called a dna-link. eukaryotic cell cycle, there is a substructure of nucleus called the nuclear matrix (nm) (see [8][9]). it is believed that the ds-dna is organized as a set of loop-shaped structures, called replicons. each replicon has a specific site in which the replication starts. this is called a replication origin or simply, origin. it is observed that two ends of the replicon (loop) are anchored at nm [8][9][14]. thus it forms a pair of twisted strings fixed at the ends of the boundary of a 3-dimensional ball, which is a two-component tangle (see [6] for a definition). (see figure 2 (c)); in this paper, we call this a dna-link (see section iii for the definition). in the dna replication process, the sequence of base pairs along each backbone acts as a template to reproduce the base pairs in the synthesized dna. this means that each backbone with base pairs is preserved in the synthesized dna. this is called the semi-conservative scheme, we interpreted this scheme in terms of dna-link in lemma iii.1. as the replication process follows the semiconservative scheme, the dna-link must be split at the end. topologically, it should be done by unknotting operations. biologically, it is believed that enzymes topoisomerases topoia, topoib and topoii are responsible for this operation. our model uses only topoia. this paper is organised as such: in section ii knots, links and linking numbers are introduced. in section iii, dna-link is defined and the semiconservative scheme is interpreted in terms of dna-link (lemma iii.1). in order to rewind the ds-dna it is natural to require the minimal energy and maximal length of rewound segment. we obtained theorem iii.1. section iv describes a topological model for rewinding replicon. our model is equipped with a recursive unknotting operations. this model can reduce the twisting number of biomath 9 (2020), 2001047, http://dx.doi.org/10.11145/j.biomath.2020.01.047 page 2 of 9 http://dx.doi.org/10.11145/j.biomath.2020.01.047 abdul adheem mohamad, tsukasa yashiro, a rewinding model for replicons with dna-links fig. 3. link diagrams of a usual link (left) and linearly very long double helix (right). both are admitted as 2-component links. r3r2r1 fig. 4. three types of reidemeister moves. r2 r1 fig. 5. the left diagram is deformed into the right by r2. the middle diagram is deformed into the very right by r1. the dna-link efficiently. the recursive unknotting operation resolve the unsoloved problem in [5]. ii. knots and links a link is a disjoint union of circles embedded in r3; k1, k2, . . . ,kn denoted by l(k1,k2, . . . ,kn). each ki i = 1, 2, . . . ,n is called a component of the link l and the link l is called an n-component link. a 1-component link is called a knot. a knot is trivial if there is an embedded disc in r3 bounded by the knot. a link is trivial if the link consists of mutually split trivial link components. a link diagram dl of a link l is a projected image of the orthogonal projection (x1,x2,x3) 7→ (x1,x2) with crossing information at the crossings (see [6] for details). there are three types of local moves on link diagrams depicted in figure 4, called reidemeister moves. two links l1 and l2 are equivalent if the diagram dl1 is deformed into the diagram dl2 by a finite sequence of three types of reidemeister moves shown in figure 4 [6] example ii.1. the diagrams in figure 5 are equivalent. the left diagram is deformed into the right by applying r2 move. a. split link let a and b be compact disjoint subsets of r3. the pair {a,b} is said to be split if there exists an embedded sphere s in 3-space such that s bounds one of the subsets from the other. a 2-component link is called a split link if the components are split. a link l is said to be oriented if each component is oriented. a link diagram is the biomath 9 (2020), 2001047, http://dx.doi.org/10.11145/j.biomath.2020.01.047 page 3 of 9 http://dx.doi.org/10.11145/j.biomath.2020.01.047 abdul adheem mohamad, tsukasa yashiro, a rewinding model for replicons with dna-links image of l in the plane r2 under the orthogonal projection (x1,x2,x3) 7→ (x1,x2) with crossing information; that is, at a crossing formed by two short arcs, one arc is upper and the other lower (see figure 3). a link diagram of a link l is denoted by dl. b. linking number let l be an oriented link and let dl be a link diagram of l. at a crossing point of dl, there are two types of crossings formed by short subarcs of dl; positive and negative crossings (see figure 6). +1 −1 k2 k1 k2k1 fig. 6. crossings with signs let l(k1,k2) be an oriented dna-link with distinct link components k1 and k2. let c(dl) be the set of crossings of the diagram dl. the linking number is defined by lk(k1,k2) = 1 2 ∑ c∈c(dl) ε(c)d(c) where c is a crossing of the link diagram, ε(c) is the sign ±1 according to the diagrams in figure 6, and also d(c) =   1 if the crossing c consists of distinct components, 0 otherwise. let k be a knot and let dk be a knot diagram. the total sum of signs w(dk): w(dk) = ∑ c∈c(dk) ε(c) is called a writhe of k. note ii.1. the linking number does not depend on the choice of the diagram of l (see [6]). if a 2-component link is split, then the linking number between the components is of course zero but the converse is not always true (see [6]). the writhe depends on the choice of diagram (see [6]). c. unknotting operations there is an operation to exchange the over arc and the under arc, called an unknotting operation (see figure 7). for every non-trivial knot k, it is fig. 7. unknotting operation in mathematical topology. modified into a trivial knot by applying a finite number of unknotting operations (see proposition 4.4.1 in [6]). iii. dna as a 2-component link for a linear dna, the length of the double strands is much longer than its diameter and also in the eukaryotic cells, the ends of a linear segment of dna are fixed at the nuclear matrix. therefore, it is possible to view the double strand dna as a special 2-component link. a. dna-link a ds-dna can be modeled as the boundary components {s1,s2} of a long thin twisted strip with the centre curve γ. we write this as l = l(s1,s2; γ), where s1 and s2 represent the single strands of the dna. we call l a dna-link. let l0 = l0(s1,s2; γ0) a dna-link. after the dna is replicated and distributed into two daughter cells, there are two identical dna-links representing daughter dnas, l1 = l1(s ′ 1, s̄1,γ1), (1) l2 = l2(s ′ 2, s̄2,γ2), (2) where s′1 and s ′ 2 are single strands (templates) inherited from the original dna. s̄1 and s̄2 represent synthesized single strands from s′1 and s ′ 2 respectively and γ1 and γ2 are centre curves of the strips for l1 and l2 respectively. biomath 9 (2020), 2001047, http://dx.doi.org/10.11145/j.biomath.2020.01.047 page 4 of 9 http://dx.doi.org/10.11145/j.biomath.2020.01.047 abdul adheem mohamad, tsukasa yashiro, a rewinding model for replicons with dna-links from this observation, the authors described in [5] that the semi-conservative scheme is interpreted in terms of dna-links as: lemma iii.1 ([5]). the semi-conservative scheme is interpreted in terms of links: a deformation of the dna-link l0(s1,s2; γ) into a split link {s′1,s ′ 2}, where s ′ 1 and s ′ 2 are inherited single strands from s1 and s2 respectively. the complicated replication process can be interpreted in terms of dna-links. as it is shown by lemma iii.1, the semi-conservative scheme is described as follows: the dna-link l0 is deformed into the split 2-component link {s′1,s ′ 2}, where s′i is obtained from si (i = 1, 2) by applying unknotting operations to l0. γ0 γ1 γ2 s1 s2 s′1 s′2 fig. 8. a topological model of the semi-conservative scheme. white proved in [13] the following formula of the linking number lk(s1,s2): lemma iii.2 (white [13]). for a dna modeled by the dna-link l(s1,s2; γ), the following formula of the linking number holds: lk(s1,s2) = tw(s1,s2) + wr(γ), (3) where tw(s1,s2) is the number of full-twists of the curves {s1,s2} along the centre curve γ and wr(γ) is the writhe of γ. lemma iii.3. suppose that a dna-link l(s1,s2; γ) has unknotted (trivial) γ. then lk(s1,s2) = 0 if and only if l(s1,s2; γ) is split. proof: if l(s1,s2; γ) is split, then lk(s1,s2) = 0. suppose lk(s1,s2) = 0. consider a diagram dl of l. since γ is trivial, we can modify γ into a trivial circle γ′ so that wr(γ′) = 0. let l(s′1,s ′ 2; γ) be the modified link. note that this modification does not change the linking number. then lk(s′1,s ′ 2) = tw(s1,s2) + wr(γ) = tw(s′1,s ′ 2) = 0 the diagram dl is depicted in figure 9. since the writhe wr(γ′) = 0, the twisting number tw(s1,s2) is cancelled by the twistings obtained from the writhe wr(γ). thus the diagram gives split trivial circles s′1 and s ′ 2. b. efficient unknotting operations if an unknotting operation is applied to one crossing of the dna-link l(s1,s2; γ) with tw(s1,s2) = m, then the resulting link has the number of twistings m−1. consider the following operation. (*) to change consecutive k crossings of a dna-link with 2m crossings, where m = tw(s1,s2). this operation eliminates 2k crossings (see figure 10). when the dna-link is modified and moved so that it is partially rewound, some energy is used. the operation is expressed by a finite number of reidemeister moves (see figure 4). thus it is possible that the necessary energy can be indicated by the number of steps consisting of those local moves. of course, the number of local moves depends on the diagram. as these moves always occur around the center curve of ds-dna, we can assume that the link l is linearly projected onto the plane can be used for describing this operation. under this situation, the reidemeister moves of r2 are used as shown in figure 10. once some r2 are applied along the center curve, count this as one step. then we have the following theorem. biomath 9 (2020), 2001047, http://dx.doi.org/10.11145/j.biomath.2020.01.047 page 5 of 9 http://dx.doi.org/10.11145/j.biomath.2020.01.047 abdul adheem mohamad, tsukasa yashiro, a rewinding model for replicons with dna-links +1 +1 +1 +1 +1 +1 −1 −1 −1 −1 −1 −1 γ′ fig. 9. since lk(s′1,s ′ 2) = 0 and wr(γ ′) = 0, tw(s1,s2) is cancelled by the opposite twistings induced from the writhe. the negative twistings come from the writhe wr(γ). r2 r2 r2 r2 k = 4k = 3 r2 r2 r2 fig. 10. the white arrows indicate the places where unknotting operations are applied; k = 3 (left) and k = 4 (right). the black arrows indicate the places that the reidemeister moves r2 are applied. the both cases need 2 steps. theorem iii.1. for a linear dna-link, applying unknotting operations to 2 consecutive crossings requires the minimal energy and implies the maximal length of rewound segment of the link. proof: let µ(k) denote the number of the steps of the deformation which is a function of k, expressed by µ(k) = ⌈ k 2 ⌉ , (4) where the function d e is the ceiling function. on the other hand, the outcome of the operation is indicated as the length of the rewound segment of the dna. this can be counted as the number of crossings eliminated by the operation. let ν(k) be the number of the eliminated crossings, expressed by ν(k) = 2k (5) an efficient operation should minimize µ and maximize ν. in order to see the efficiency of the operation, take the ratio: ν µ (6) it is easy to see that this ratio is 2 or 4. therefore, it can be justified to take the values µ = 1 and ν = 4 to make the operation efficient. this means k = 2. this is the required result. iv. a rewinding model for replicons it is believed that the replication process starts from a specific domain called a replicon which has a looped shape (see figure 2 (a)). the replicon (loop) is modeled by the 2-component tangle (link) shown in figure 2 (c). we described a topological model for dna replication in [5] but it requires biomath 9 (2020), 2001047, http://dx.doi.org/10.11145/j.biomath.2020.01.047 page 6 of 9 http://dx.doi.org/10.11145/j.biomath.2020.01.047 abdul adheem mohamad, tsukasa yashiro, a rewinding model for replicons with dna-links some over twist absorbing mechanisms which are unknown. if the process described in this section is used in the replication model, then the over twisting absorbing mechanism is not needed. the model in this section can be a solution for the problem unsolved in [5]. a. estimate of the writhe from lemmas iii.1, iii.2 and iii.3, in order to split the dna-link l0(s1,s2; γ0), it is necessary to eliminate the linking number lk(s1,s2). the linking number can be estimated from the length of the dna segment of the replicon. let `γ0 be the length of γ0 in terms of base pairs and let b be the number of base pairs within one full twist of the double strands; b is approximately 10.5. then the linking number is given by: lk(s1,s2) = tw(s1,s2) + wr(γ0) (7) ≈ `γ0 b + wr(γ0) (8) ≈ `γ0 10.5 + wr(γ0). (9) the dna in eukaryotic cell has a form called nucleosome, where the dna is wrapped into two negative supercoils around a histone octamer (see [11] for details). the dna string around the histone octamer forms a double loop with wr = −2 (see figure 11). the total writhe in the replicon 197bp fig. 11. the nucleosome has the length about 197bpand its writhe −2. induced from nucleosomes in it is calculated as follows. each nucleosome has the writhe −2 with 147 bp and a linear piece of dna called a linker which has 50 bp (see [11] for details). therefore, we may assume that a nucleosome appears at least every 197 base pairs. if we compare the number of full twists within the 197 base pairs: 197 10.5 = 18.76 (full twists.) (10) the ratio between the writhe of a nucleosome and the number of full twists of the nucleosome is: |wr| tw ≈ 2 18.76 ≈ 0.107. (11) this means that the ratio of the number of fully twisted supercoils is formed by nucleosomes; that is, the negative writhe wr(γ) induced from the nucleosomes is at most about 10.7%. therefore, by lemma iii.2, the equation lk(s1,s2) = tw(s1,s2) + wr(γ) implies that about 90% of tw(s1,s2) should be reduced by some operations. for instance, it is estimated in [7] that the length of the replicon (loop) is about 100kbp. then the number of full-twists in one replicon (loop) is given by: 100kbp 10.5bp ≈ 9.5k. (12) the length for each nucleosome is about 197bp. suppose that for every 197bp, there is one nucleosome. then the number of nucleosomes is 100kbp/197bp ≈ 0.5k. (13) each nucleosome has the writhe −2. therefore, wr(γ0) is estimated as −1k. this observation implies that about 8.5k of fulltwisting should be reduced by some operations. b. recursive unknotting operations the enzymes topoia and topoii act as unknotting operators to single and double strands respectively. theorem iii.1 suggests the following operation. topoia is activated at the adjacent two crossings as shown in figure 13. this modification releases exactly two and a half twists. this decreases the twisting number 2.5. this relaxation does not require the counter biomath 9 (2020), 2001047, http://dx.doi.org/10.11145/j.biomath.2020.01.047 page 7 of 9 http://dx.doi.org/10.11145/j.biomath.2020.01.047 abdul adheem mohamad, tsukasa yashiro, a rewinding model for replicons with dna-links a b a b ba fig. 12. topoia acts if a and b are single strings. topoii acts if a and b are double strings. topi topi fig. 13. the topological model of the rewinding at the replication origin. topia is activated at the shaded circles in the top diagram. this releases 2.5 full twists. this model keeps the writhe of the single strands. it does not require too many extra rotations. rotation. this is different from the topological model proposed in [5]. the next problem to solve is to know how many times topoia should be applied to change the crossings during the replication in the replicon. from (12), the number of full twists in the replicon is about 9500 denoted by tw0. thus the number of places on the replicon where the topoia is activated is: 9500 2.5 = 3800 (14) the resulting relaxed dna has now about 3800 crossings. for each replicon, about 3800 of activations of topoia will be needed to reduce the linking number. after activating topoia to one place, the linking number is reduced from 2.5 to 0.5. from this observation, if topoia is pairwisely activated at every 2.5 full-twists, the twistings will be reduced to 20% of the twistings tw0 of the replicon. it is denoted by tw1. tw1 = 0.5 × 3800 ≈ 1900 if topoia is applied again at every 2.5 full-twists of the relaxed dna again, the twistings will be biomath 9 (2020), 2001047, http://dx.doi.org/10.11145/j.biomath.2020.01.047 page 8 of 9 http://dx.doi.org/10.11145/j.biomath.2020.01.047 abdul adheem mohamad, tsukasa yashiro, a rewinding model for replicons with dna-links reduced to 4% of tw0 denoted by tw2. tw2 = 1900 2.5 × 0.5 = 380 this means that after applying the unknotting operation twice, only 380 twisings remain. therefore, the unknotting operation gives tw2 which is 4% of tw0. from section iv-a, the writhe induced from the nucleosomes is at most 10% of tw0. the twisting number tw2 is less than half of 10%. therefore, from the formula (3): and lemma iii.3, the link is very close to be split. v. conclusion in this paper, we studied a possible topological model for a replicon equipped with unknotting operations. it is proved that the model is efficient to make the linking number of the replicon close to zero. however, it remains that the existence of this kind of mechanism is unknown. this should be examined through experiments. references [1] m. barbi, j. mozziconacci, h. wong, and j. victor, dna topology in chromosomes: a quantitative survey andits physiological implication, j. math. biol. 68 (2014), 145–179. [2] a. bates and a. maxwell, dna topology, oxford university press, 2005. [3] a. kornberg and t. a. baker, dna replication, second edition, university science book, 2005. [4] s. d. levene, c. donahue, t. c. boles, and n. r. cozzarelli, analysis of the structure of dimeric dna catenanes by electron microscopy, biophysical journal 69 (1995), 1036–1045. [5] a. a. mohamad and t. yashiro, a topological model of dna replication with dna-links, far east j. mathematical sciences 107 (2018), 241–255. [6] k. murasugi, knot theory and its applications, modern birkhäuser classics, birkhäuser, 2008. [7] h. nakamura, t. morita, and c. sato, structural organizations of replicon domains during dna synthetic phase in the mammalian nucleus, exp. cell res. 165 (1986), 291. [8] s. v. razin, a. a. gavrilov, e. s. ioudinkova, and o. v. iarovaia, communication of genome regulatory elements in a folded chromosome, febs letters 587 (2013), 1840–1847. [9] j. c. rivera-mulia, r. hernández-munõz, f. martı́nez, and a. aranda-anzaldo, dna moves sequentially towards the nuclear matrix during dna replication in vivo, bmc cell biology 12:3 (2011), 16 pages. [10] v. v. rybenkov, n. r. cozzarelli, and a. v. vologodskii, propability of dna knotting and the effective diameter of the dna double helix, natl. acad. sci. usa 90 (1993), 5307–5311. [11] r. r. sinden, dna structure and function, academic press, 1994. [12] a. vologodskii, bridged dna circles: a new model system to study dna topology, macromolecules 45 (2012), 4333–4336. [13] j. h. white, self-linking and gauss integral in higher dimensions, amer. j. of math. 91 (1969), 693–728. [14] r. h. c. wilson and d. coverley, relationship between dna replication and the nuclear matrix, gens to cells 18 (2013), 17–31. biomath 9 (2020), 2001047, http://dx.doi.org/10.11145/j.biomath.2020.01.047 page 9 of 9 http://dx.doi.org/10.11145/j.biomath.2020.01.047 introduction knots and links split link linking number unknotting operations dna as a 2-component link dna-link efficient unknotting operations a rewinding model for replicons estimate of the writhe recursive unknotting operations conclusion references original article biomath 1 (2012), 1209041, 1–6 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum constructing one-dimensional continuous models from two-dimensional discrete models of medical implants alicia prieto-langarica∗, hristo v. kojouharov†, liping tang ‡ ∗ department of math. and statistics, youngstown state univ., youngstown, oh, usa email: aprietolangarica@ysu.edu † department of mathematics, the univ. of texas at arlington, arlington, tx, usa email: hristo@uta.edu ‡department of bioengineering, the univ. of texas at arlington, arlington, tx, usa email: ltang@uta.edu received: 17 july 2012, accepted: 04 september 2012, published: 08 october 2012 abstract—medically implanted devices are becoming increasingly important in medical practice. over 4 million people in the united states have long-term biomedical implants. however, many medical implants have to be removed because of infection or because their protein coating causes excessive inflammation and decrease in the immune system response. in this work, a discrete twodimensional model of blood cells and bacteria interactions on the surface of a medical implant is transformed into a discrete one-dimensional model. this one dimensional model is then upscaled into a partial differential equation model. the results from the discrete two-dimensional model and the continuous one-dimensional model are then compared for different protein coating mixtures. two medical treatment alternatives are also explored and the two models are compared again. keywords-upscaling; medical implants; modeling; cellular automata i. introduction a. biological background medically implanted devices are becoming increasingly important in medical practice [18]. since the first applications of biomaterials in medicine, infections represent the most important complications, which still limit the unrestricted use of biomaterials in humans [3]. implant-associated infections account for nearly 50% of the estimated 2 million nosocomial infections in the united states each year, [3]. the most common medical implant infections are due to staphyloccocus epidermidis [17], which is a bacterial colonizer of the skin and mucous membranes of humans and other mammals [10]. s. epidermidis can lead to a wide variety of complications including inflammation, thrombosis, infections and fibrosis [18]. these complications have a direct effect on the stability of the implanted device because they trigger immune responses, including a rapid accumulation of phagocytic cells [18]. if the immune system is not able to eradicate s. epidermidis during the first several hours after it has entered the body then biofilm formation is likely to commence. biofilms represent the most prevalent type of microbial growth in nature and are crucial to the development of clinical infections. they can serve as a nidus for disease and are often associated with high-level antimicrobial resistance of the associated organisms [7]. studies suggest that biofilms are present on the surface of the implant as early as 16 hours after implantation [4]. however young biofilms are more vulnerable to phagocytic cells than mature ones which have been growing for more than 48 hours [4]. in addition, most antibiotics are only effective against the fast growing bacteria which reside in the outer layers of the biofilm, citation: a. prieto-langarica, h. kojouharov, l. tang, constructing one-dimensional continuous models from two-dimensional discrete models of medical implants, biomath 1 (2012), 1209041, http://dx.doi.org/10.11145/j.biomath.2012.09.041 page 1 of 6 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.09.041 a. prieto-langarica et al., constructing one-dimensional continuous models... while the slow growing bacteria deep inside of the biofilm formation tend to be spared and to persist in the body [2]. therefore, it is critical that the immune system destroys the majority of the bacteria before a biofilm begins to form. of all the types of phagocytic cells, the most important to the immune system’s defense against s. epidermidis are the white blood cells called neutrophils. in order to attack the s. epidermidis growing on medical implants, neutrophil cells adhere to the surface of the device and move towards the bacterial formations [18]. previous studies have indicated that the rate of neutrophil locomotion is influenced by adhesive forces between the cells and the substratum [6]. fibrinogen and albumin are two of the most commonly used protein coatings on medically implanted devices. fibrinogen facilitates a strong attachment between neutrophils and the implant since it is readily recognized as a malign substance by the immune system. however, fibrinogen also works as a distraction to the neutrophils because the phagocytes place themselves in one spot attacking the fibrinogen covered implant and move very slowly towards the bacteria [8, 16]. in contrast, albumin is not recognized by the phagocytes as a malign substance and hence the neutrophils cells can move freely around the implant. another important distinction between albumin and fibrinogen is the amount of neutrophils each protein coating attracts. experimental studies suggest that two groups of chemokines macrophage inflammatory protein (mip) and monocyte chemoattactant protein (mcp) appear to play a major role in phagocyte-implant interactions [18]. by releasing chemokines, the neutrophil cells present on the surface of the implant are able to attract more neutrophils to the site. these chemotactic interactions create waves of incoming phagocytic cells, which aid in the fight against the bacterial infection. while fibrinogen covered implants are interpreted as a threat to the body and many phagocytes are attracted to them, the albumin coated implant is not perceived as a threat and thus fewer phagocytes are present to fight the infection. b. mathematical model a discrete two-dimensional cellular automata (ca) model [9] that describes the interactions between neutrophil cells and s. epidermidis cells on the surface of an implant was previously developed [14]. a cellular automaton consists of a regular grid, where each site in the lattice can be in one of a finite number of possible states updated synchronously in discrete time steps according to local, identical rules [9]. a set of rules for the movement of the cells and the growth of the bacteria is given for the two different types of protein coatings. the amounts of albumin and fibrinogen in the mixture are allowed to be varied, since they have different effects on the speed of the neutrophils and their ability to control a bacterial infection. in the ca model neutrophil cells move with greater probability towards larger bacterial concentrations. the model is divided into three parts. the first part simulates the complex s. epidermidis-neutrophils interactions between 4 and 20 hours after the implant is introduced into the body. at this early stage, reproduction of bacteria and early bacterial community formation triggers the immune system response. a series of chemotactic waves of neutrophil cells are then incorporated into the system and are considered in the model. the second part of the model simulates the system dynamics after the s. epidermidis have started forming a biofilm, which takes place between the 20 and the 52 hours. during this part of the simulation, bacteria experience an increase in the reproduction rate while the immune system response gradually decreases effectiveness as the biofilms become stronger. the last part of the model, after the initial 52 hours, the immune system can no longer fight s. epidermidis since they are all gathered in fully formed strong biofilms. an important aspect of the ca model is the different scales being used. two different grids are considered: a 12 × 12 grid for the neutrophil cells and a 144 × 144 grid for the s. epidermidis since neutrophil cells are 12 times larger in size than s. epidermidis cells. however, the two-dimensional ca only models 0.1% of the total experimental implant area. larger areas become almost impossible to simulate since running the ca model is computationally very expensive. in order to model larger areas continuous model needs to be used. by adding the elements on each column and expressing the result on a line grid, the model is then transform into a one-dimensional discrete model. using the upscaling method described in [11, 12, 13] the one-dimensional discrete model is then upscaled into a continuous partial differential equation (pde) model by considering the transition probabilities of each site from one state to another and then taking limits as space and time steps tend to zero. using both models, the discrete two-dimensional model and the continuous one-dimensional model, a variety of mixtures of fibrinogen and albumin implant coatings are examined in order to maximize the effecbiomath 1 (2012), 1209041, http://dx.doi.org/10.11145/j.biomath.2012.09.041 page 2 of 6 http://dx.doi.org/10.11145/j.biomath.2012.09.041 a. prieto-langarica et al., constructing one-dimensional continuous models... tiveness of the immune system response [14]. finding the optimum amounts of each of these two proteins can help the immune system destroy most of the bacteria before they start to form biofilm communities. this can reduce the number of rejections of medically implanted devices and drastically improve the ability of the body’s immune system to combat bacterial infections. the presented simulations can also be used to help determine the appropriate amount of antibiotics to use over the implant area so that an s. epidermidis infection can be successfully controlled, as well as to predict what can happen if biofilm formation is prevented. ii. from two-dimensions to one-dimension let u0(a, b) be the initial number of neutrophil cells, at position (a, b) in the grid and un(a, b) be the expected number of neutrophil cells at position (a, b) after n time steps. also let w0(x, y) be the initial number of s. epidermidis cells, at position (x, y) in the grid and wn(x, y) be the expected number of s. epidermidis cells at position (x, y) after n time steps. as a first step in constructing a one-dimensional discrete model, the initial distributions of the neutrophil cells and the s. epidermidis cells are added in each column of the grid as follows: ûn(a) = ∑12 i=1 u n(a, i), ŵn(x) = ∑144 i=1 w n(x, i). (1) this creates the corresponding one-dimensional initial density profiles (figure 1). the rules for movement, the rates of bacterial reproduction, addition of neutrophils into the system, and neutrophils killing bacteria are all kept the same. a similar approach has been taken before in the literature [15], for transforming a two-dimensional discrete model into a one-dimensional discrete model. next, the corresponding one-dimensional continuous model is constructed. in order to do this, transition probabilities are defined for the state of each point in the grid creating a discrete description of all undergoing processes [11]. taking limits as the time step and the mesh size tend to zero yields the following system of partial differential equations: ∂u ∂t = d ∂2u ∂x2 − ∂ ∂x (v u) + (kw + a)u, ∂w ∂t = dw ∂2w ∂x2 + (r − keu)w, (2) fig. 1. 2-d to 1-d conversion through column addition. where v and d are the advection and diffusion coefficients, respectively, in the neutrophils equation, k is the rate at which neutrophil cells call on other neutrophil cells depending on the presence of s. epdidermidis while a is the rate at which neutrophil cells call on other neutrophil cells independently of the presence of s. epidermidis. dw is the diffusion coefficient in the s. epidermidis equation, r is the growth rate of s. epidermidis, and ke is the rate at which neutrophil cells kill s. epidermidis cells. iii. numerical simulations the one-dimensional continuous model (2) is numerically solved and the results are compared with the results obtained after running 100, 000 simulations of the ca model accounting for the amount of bacteria left in each simulation after t = 20, t = 52 and t = 76 hours. a simulation is considered effective if at least 99% of the implant area is free of bacteria coverage after 76 hours. figure 2 shows the percentage of effective simulations for a variety of mixtures of protein coatings of the implant with albumin percentages between 0% and 100% in 10% increments (% of fibrinogen=100−% of albumin) after t = 20, t = 52 and t = 76 hours for both the discrete 2-d and continuous 1-d models. as figure 2 shows, mixtures with either high albumin or low albumin percentages yield a high percentage of ineffective simulations. however, two treatment strategies can be used to improve result for all protein mixtures: (1) medical devices can be pre-coated with antibiotics before implantation; or (2) biofilm formation can be blocked [5]. accordingly, the mathematical model is modified to include both strategies: • the effect of antibiotics is included in the model by randomly selecting a fixed percentage of bacteria every certain amount of time and eliminating it biomath 1 (2012), 1209041, http://dx.doi.org/10.11145/j.biomath.2012.09.041 page 3 of 6 http://dx.doi.org/10.11145/j.biomath.2012.09.041 a. prieto-langarica et al., constructing one-dimensional continuous models... −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.21.2 30 40 50 60 70 80 90 100 110 fraction of albumin in the mixture p e rc e n ta g e o f e ff e ct iv e s im u la tio n s discrete simulations t=20 t=52 t=76 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11 30 40 50 60 70 80 90 100 110 fraction of albumine in mixture p e rc e n ta g e o f e ff e ct iv e s im u la tio n s continuous model 20 hours 52 hours 76 hours fig. 2. comparison of the discrete cellular automata 2-d model (above) and the continuous 1-d differential equation model (below) for t = 20, t = 52 and t = 76. from the implant. the percentage of bacteria to be eliminated is a parameter in the model and can be modified to represent different antibiotics. figure 3 shows the effects of pre-coating the implant with a sample antibiotic for all different protein coating mixtures. • the second strategy entails disrupting the agr system to prevent bacterial attachment and therefore avoid biofilm formation. no biofilm formation translates into treating both parts in the model, the 20-to-76 hour and the 4-to-20 hour, in a similar way, i.e., all bacteria are treated as free bacteria and neutrophils kill bacteria at the same constant rate throughout the entire simulation. the results of disrupting the agr system are shown in figure 4. as it can be seen in figures 3 and 4, there is a very good agreement between the two-dimensional discrete ca model and the one-dimensional continuous pde model when either of the two treatment strategies is used. a comparison of the cpu time used in matlab r© to compute the corresponding numerical solutions is presented in table i. the pde model was solved using a finite difference method [1] while the ca model was run 100, 000 times. as shown on table i, the two-dimensional discrete ca model simulations takes −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.21.2 30 40 50 60 70 80 90 100 110 fraction of albumin in the mixture p e rc e n ta g e o f e ff e ct iv e s im u la tio n s antibiotic effect 1 dose 2 doses 3 doses 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11 40 50 60 70 80 90 100 110 fraction of albumin in mixture p e rc e n ta g e o f e ff e ct iv e s im u la tio n s antibiotic effect 1 dose 2 doses 3 doses fig. 3. effects of different doses of antibiotics on bacterial growth on the surface of an implant. discrete (above) and continuous (below) models results for t = 76. overwhelmingly longer time to run compared to the numerical solution of the one-dimensional continuous pde model. iv. conclusion in this work, a two-dimensional discrete ca model and a one-dimensional pde model for the interactions between neutrophils and s. epidermidis on the surface of a medical implant under different protein coating were explored. the pde model was solved numerically and the results compared with 100, 000 runs of the ca model. both models where used to determine the protein coating mixture that will allow the immune system to eradicate s. epidermidis within 74 hours after implantation. the models were then modified to include the effect of pre-coating the implants with antibiotics and problem figure pde model ca model cpu (h) cpu (h) implant experiment 2 0.0231 42.57 antibiotic experiment 3 0.0294 49.27 no biofilm experiment 4 0.0227 41.98 table i comparative cost of the continuous and discrete models implemented on an intel r© coretm cpu 2.53ghz. biomath 1 (2012), 1209041, http://dx.doi.org/10.11145/j.biomath.2012.09.041 page 4 of 6 http://dx.doi.org/10.11145/j.biomath.2012.09.041 a. prieto-langarica et al., constructing one-dimensional continuous models... −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.21.2 30 40 50 60 70 80 90 100 110 percentage of albumin in mixture p e rc e n ta g e o f e ff e ct iv e s im u la tio n s no biofilm formation 20 hours 52 hours 76 hours 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 30 40 50 60 70 80 90 100 110 percentage of albumin in mixture p e rc e n ta g e o f e ff e ct iv e s im u la tio n s no biofilm formation 20 hours 52 hours 76 hours fig. 4. effect of no biofilm formation on bacterial growth on the surface of an implant. discrete (above) and continuous (below) models results for t = 20, t = 52, t = 76. disturbing biofilm formation as two different treatment strategies. the two-dimensional ca discrete model represents very accurately the medical implant system of interest. however, as can be seen in table i, running 100, 000 iterations of the discrete model is computationally expensive and it models only 0.1% of the implant area. using the upscaling method described in [11] together with the conversion from two-dimensions to one-dimension, an efficient partial differential equations model can be constructed. the pde model uses the correct parameters from the ca model making it both biologically accurate and very efficient to solve numerically as shown in table i. therefore, constructing a one-dimensional continuous model using what is presented in this paper can be as accurate in representing the biology of the problem and much more computationally 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[17] c. vuong, m. otto, “staphylococcus epidermidis infections”, microbes. infect., vol. 4 no. 4, pp. 481– 489, 2002. http://dx.doi.org/10.1016/s1286-4579(02)01563-0 [18] j. xue, gao, l. tang, “mathematical modeling of phagocyte chemotaxis toward and adherence to biomaterial implants”, bibm 2007, pp. 302–307, 2007. biomath 1 (2012), 1209041, http://dx.doi.org/10.11145/j.biomath.2012.09.041 page 6 of 6 http://dx.doi.org/10.1063/1.3659919 http://dx.doi.org/10.1016/j.physa.2008.10.038 http://dx.doi.org/10.1016/s1286-4579(02)01563-0 http://dx.doi.org/10.11145/j.biomath.2012.09.041 introduction biological background mathematical model from two-dimensions to one-dimension numerical simulations conclusion original article biomath 2 (2013), 1305155, 1–7 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum on some multipoint methods arising from optimal in the sense of kung–traub algorithms nikolay kyurkchiev faculty of mathematics and informatics plovdiv university plovdiv, bulgaria institute of mathematics and informatics bulgarian academy of sciences sofia, bulgaria email: nkyurk@uni-plovdiv.bg anton iliev faculty of mathematics and informatics plovdiv university plovdiv, bulgaria institute of mathematics and informatics bulgarian academy of sciences sofia, bulgaria email: aii@uni-plovdiv.bg received: 20 april 2013, accepted: 15 may 2013, published: 29 june 2013 abstract—in this paper we will examine self-accelerating in terms of convergence speed and the corresponding index of efficiency in the sense of ostrowski–traub of certain standard and most commonly used in practice multipoint iterative methods using several initial approximations for numerical solution of nonlinear equations due to optimal in the sense of the kung–traub algorithm of order 4, 8 and 16. some hypothetical iterative procedures generated by algorithms from order of convergence 32 and 64 are also studied (the receipt and publication of which is a matter of time, having in mind the increased interest in such optimal algorithms). the corresponding model theorems for their convergence speed and efficiency index have been formulated and proved. keywords-solving nonlinear equations; order of convergence; optimal algorithm; efficiency index i. introduction one of the most basic problems in scientific and engineering applications is to find the solution of a nonlinear equation f(x) = 0. (1) in literature, it is known that the computational efficiency of a method is measured by the concept of the efficiency index p 1 n , where p is the order of convergence and n is the whole number of functional evaluations per iteration. subsequently, the maximum efficiency index for newton’s iteration with two functional evaluations is 2 1 2 ≈ 1.414 [30]. according to the conjecture of kung and traub [13], the maximum convergence order of a scheme (without memory) including n evaluations per step is 2n−1. by taking into account the optimality concept, many authors have tried to build iterative procedures of optimal order of convergence p = 4, p = 8, p = 16. the recent results of m. petkovic [20] and m. petkovic and l. petkovic [22], bi, wu and ren [2], geum and kim [7], thurkal and petkovic [29], wang and liu [31], kou, wang and sun [12], chun and neta [3], soleymani and soleymani [24], soleymani [25], bi, ren and wu [1], sargolzaei and soleymani [26], soleymani and mousavi [27], soleymani and sharifi [28], ignatova, kyurkchiev and iliev [10], m. petkovic, neta, l. petkovic and dzunic [21] are presented for optimal multipoint methods for solving nonlinear equations. for other results see dzunic and m. petkovic [6]. m. petkovic [20] gives a useful detailed review about computational efficiency of many methods in the sense of kung–traub hypothesis. for other nontrivial methods for solving nonlinear equations see, kyurkchiev and iliev [14] and iliev and citation: n kyurkchiev, a iliev, on some multipoint methods arising from optimal in the sense of kung–traub algorithms, biomath 2 (2013), 1305155, http://dx.doi.org/10.11145/j.biomath.2013.05.155 page 1 of 7 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2013.05.155 n kyurkchiev at al., on some multipoint methods arising from optimal in the sense of kung–traub algorithms kyurkchiev [11]. in many natural science tasks, from purely physical considerations, the user of numerical algorithms for solving nonlinear equation (1) knows a set of initial approximations x01,x 0 2, . . . ,x 0 k for the root ξ of equation (1). as an example, regula falsi methods and modifications of euler–chebyshev method and halley method with a lower order of convergence use two or three initial approximations for the root ξ. in [16], refined conditions of convergence for the difference analogue of halley method (using three initial approximations) for solving nonlinear equation are given (see, also [32]). an efficient modification of a finite–difference analogue of halley method is proposed in [9]. naturally arises the task of designing and testing multipoint variants of the classical procedures in the light of the achievements over the past five years important theoretical results related to obtaining optimal in the sense of kung–traub algorithms. in this sense the task of detailed refinement of the self-accelerating multipoint methods using several initial approximations become very actual. ii. main results a. optimal algorithm in the sense of kung–traub with order of convergence p = 4 we consider the following nonstationary iterative scheme based on the 4-point iteration function in combination with an optimal algorithm in the sense of kung– traub with order of convergence p = 4: x2n+1 = ϕ1(x2n,x2n−1,x2n−2,x2n−3), x2n+2 = ϕ2(x2n+1). (2) it is known that for the error �i = xi − ξ, i = −3,−2,−1,0,1,2 . . . ; [30] is valid �2n+1 ∼ c1(ξ)�2n�2n−1�2n−2�2n−3, (3) �2n+2 ∼ c2(ξ)�42n+1. (4) let k9 = max{|c1(ξ))|, |c2(ξ)|} , d2n−1 = k 1 3 9 |�2n−1|, d2n = k 1 3 9 |�2n|, and let 0 < d < 1, and x−3, x−2, x−1 and x0 be chosen so that the following inequalities d−3 = k 1 3 9 |x−3 − ξ| ≤ d < 1, d−2 = k 1 3 9 |x−2 − ξ| ≤ d < 1, d−1 = k 1 3 9 |x−1 − ξ| ≤ d < 1, d0 = k 1 3 9 |x0 − ξ| ≤ d < 1 hold true. from (3) and (4), we have d2n+1 = k 1 3 9 |�2n+1| ≤ k 1 3 9 k9|�2n||�2n−1||�2n−2||�2n−3| = k 1 3 9 |�2n−1|k 1 3 9 |�2n|k 1 3 9 |�2n−2|k 1 3 9 |�2n−3| = d2nd2n−1d2n−2d2n−3, d2n+2 = k 1 3 9 |�2n+2| ≤ k 1 3 9 k9� 4 2n+1 = ( k 1 3 9 �2n+1 )4 = d42n+1. (5) evidently, from (5), we find d1 ≤ d4, d2 ≤ d16, d3 ≤ d22, d4 ≤ d88, d5 ≤ d130, d6 ≤ d520, d7 ≤ d760, d8 ≤ d3040, d9 ≤ d4450, d10 ≤ d17800. our results concerning the order of convergence generated by (2) are summarized in the following theorem. theorem a. assume that the initial approximations x0, x−1, x−2, x−3 are chosen so that d−3 ≤ d, d−2 ≤ d, d−1 ≤ d < 1 and d0 ≤ d < 1. then for the error of the sequences {x2n+1}∞n=0 and {x2n+2} ∞ n=0 determined by (2), we have d2n−1 ≤ dτ2n−1, d2n ≤ dτ2n, (6) where τm+4 = 5τm+2 + 5τm, m = 1,2, . . . (7) and the order of convergence of the iteration (2) is τ = 5 + 3 √ 5 2 . proof. it is well known that the recursion: γi+1 = n∑ j=1 ajγi−j+1, i = n−1,n−2, . . . , biomath 2 (2013), 1305155, http://dx.doi.org/10.11145/j.biomath.2013.05.155 page 2 of 7 http://dx.doi.org/10.11145/j.biomath.2013.05.155 n kyurkchiev at al., on some multipoint methods arising from optimal in the sense of kung–traub algorithms (for any initial conditions) corresponds to the characteristic polynomial: ρn = n∑ j=1 ajρ n−j. in our case, for the recursion τm+4 = 5τm+2 + 5τm, the characteristic polynomial is of the type ρ2 −5ρ−5 = 0. (8) equation (8) has the roots: ρ1 = 5 + 3 √ 5 2 , ρ2 = 5−3 √ 5 2 . from the general iterative theory [30], (see, also [8]) it follows that the order of convergence of the iteration procedure, defined by (2) is given by the only real root of equation (8) with magnitude greater than 1. on the other hand, |�2n+1| ≤ k −1 3 9 d2n+1, |�2n+2| ≤ k −1 3 9 d2n+2, and consequently we can conclude that the order of convergence of iteration (2) is τ = 5 + 3 √ 5 2 ≈ 5.8541... thus, the theorem is proven. b. optimal algorithm in the sense of kung–traub with order of convergence p = 8 we consider the following nonstationary iterative scheme based on the 4-point iteration function in combination with an optimal algorithm in the sense of kung– traub with order of convergence p = 8: x2n+1 = ϕ1(x2n,x2n−1,x2n−2,x2n−3), x2n+2 = ϕ3(x2n+1). (9) for the error �i = xi − ξ, i = −3,−2,−1,0,1,2 . . . ; [30], [23] is valid �2n+1 ∼ c1(ξ)�2n�2n−1�2n−2�2n−3, (10) �2n+2 ∼ c3(ξ)�82n+1. (11) let k10 = max{|c1(ξ))|, |c3(ξ)|} , d2n−1 = k 3 17 10 |�2n−1|, d2n = k 7 17 10 |�2n|, and let d > 0, and x−3, x−2, x−1 and x0 be chosen so that the following inequalities d−3 = k 3 17 10 |x−3 − ξ| ≤ d < 1, d−2 = k 7 17 10 |x−2 − ξ| ≤ d < 1, d−1 = k 3 17 10 |x−1 − ξ| ≤ d < 1, d0 = k 7 17 10 |x0 − ξ| ≤ d < 1 hold true. from (10) and (11), we have d2n+1 = k 3 17 10 |�2n+1| ≤ k 3 17 10 k10|�2n||�2n−1||�2n−2||�2n−3| = k 3 17 10 k 3 17 + 7 17 + 7 17 10 |�2n||�2n−1||�2n−2||�2n−3| = d2nd2n−1d2n−2d2n−3, d2n+2 = k 7 17 10 |�2n+2| ≤ k 7 17 10 k10� 8 2n+1 = ( k 3 17 10 �2n+1 )8 = d82n+1. (12) from (12), we find d1 ≤ d4, d2 ≤ d32, d3 ≤ d38, d4 ≤ d304, d5 ≤ d378, d6 ≤ d3024, d7 ≤ d3744, d8 ≤ d29952, d9 ≤ d37098, d10 ≤ d296784. our results concerning the order of convergence generated by (9) are summarized in the following theorem. theorem b. assume that the initial approximations x0, x−1, x−2, x−3 are chosen so that d−3 ≤ d, d−2 ≤ d, d−1 ≤ d < 1 and d0 ≤ d < 1. then for the error of the sequences {x2n+1}∞n=0 and {x2n+2} ∞ n=0 determined by (9), we have d2n−1 ≤ dτ2n−1, d2n ≤ dτ2n, (13) where τm+4 = 9τm+2 + 9τm, m = 1,2, . . . (14) and the order of convergence of the iteration (9) is τ = 3 ( 3 + √ 13 ) 2 . proof. in our case, for the recursion τm+4 = 9τm+2 + 9τm, biomath 2 (2013), 1305155, http://dx.doi.org/10.11145/j.biomath.2013.05.155 page 3 of 7 http://dx.doi.org/10.11145/j.biomath.2013.05.155 n kyurkchiev at al., on some multipoint methods arising from optimal in the sense of kung–traub algorithms the characteristic polynomial is of the type ρ2 −9ρ−9 = 0. (15) equation (15) has the roots: ρ1 = 3 ( 3 + √ 13 ) 2 , 3 ( 3− √ 13 ) 2 . from the general iterative theory it follows that the order of convergence of the iteration procedure, defined by (9) is given by the only real root of equation (15) with magnitude greater than 1. on the other hand, |�2n+1| ≤ k − 3 17 10 d2n+1, |�2n+2| ≤ k − 7 17 10 d2n+2, and consequently we can conclude that the order of convergence of iteration (9) is τ = 3 ( 3 + √ 13 ) 2 ≈ 9.9083... thus, the theorem is proven. c. optimal algorithm in the sense of kung–traub with order of convergence p = 16 we consider the following nonstationary iterative scheme based on the 4-point iteration function in combination with an optimal algorithm in the sense of kung– traub with order of convergence p = 16: x2n+1 = ϕ1(x2n,x2n−1,x2n−2,x2n−3), x2n+2 = ϕ4(x2n+1). (16) it is known that for the error �i = xi − ξ, i = −3,−2,−1,0,1,2 . . . ; [30] is valid �2n+1 ∼ c1(ξ)�2n�2n−1�2n−2�2n−3, (17) �2n+2 ∼ c4(ξ)�162n+1. (18) let k11 = max{|c1(ξ))|, |c4(ξ)|} , d2n−1 = k 1 11 11 |�2n−1|, d2n = k 5 11 11 |�2n|, and let d > 0, and x−3, x−2, x−1 and x0 be chosen so that the following inequalities d−3 = k 1 11 11 |x−3 − ξ| ≤ d < 1, d−2 = k 5 11 11 |x−2 − ξ| ≤ d < 1, d−1 = k 1 11 11 |x−1 − ξ| ≤ d < 1, d0 = k 5 11 11 |x0 − ξ| ≤ d < 1 hold true. from (17) and (18), we have d2n+1 = k 1 11 11 |�2n+1| ≤ k 1 11 11 k11|�2n||�2n−1||�2n−2||�2n−3| = k 1 11 11 k 1 11 + 1 11 + 1 11 11 |�2n|�2n−1||�2n−2|�2n−3| = d2nd2n−1d2n−2d2n−3, d2n+2 = k 5 11 11 |�2n+2| ≤ k 5 11 11 k11� 16 2n+1 = ( k 1 11 11 �2n+1 )16 = d162n+1. (19) from (19), we find d1 ≤ d4, d2 ≤ d64, d3 ≤ d70, d4 ≤ d1120, d5 ≤ d1258, d6 ≤ d20128, d7 ≤ d22576, d8 ≤ d361216, d9 ≤ d405178, d10 ≤ d6482848. our results concerning the order of convergence generated by (19) are summarized in the following theorem. theorem c. assume that the initial approximations x0, x−1, x−2, x−3 are chosen so that d−3 ≤ d, d−2 ≤ d, d−1 ≤ d < 1 and d0 ≤ d < 1. then for the error of the sequences {x2n+1}∞n=0 and {x2n+2} ∞ n=0 determined by (16), we have d2n−1 ≤ dτ2n−1, d2n ≤ dτ2n, (20) where τm+4 = 17τm+2 + 17τm, m = 1,2, . . . (21) and the order of convergence of the iteration (16) is τ = ( 17 + √ 357 ) 2 . proof. in our case, for the recursion τm+4 = 17τm+2 + 17τm, the characteristic polynomial is of the type ρ2 −17ρ−17 = 0. (22) equation (22) has the roots: ρ1 = ( 17 + √ 357 ) 2 , ( 17− √ 357 ) 2 . from the general iterative theory it follows that the order of convergence of the iteration procedure, defined by biomath 2 (2013), 1305155, http://dx.doi.org/10.11145/j.biomath.2013.05.155 page 4 of 7 http://dx.doi.org/10.11145/j.biomath.2013.05.155 n kyurkchiev at al., on some multipoint methods arising from optimal in the sense of kung–traub algorithms (16) is given by the only real root of equation (22) with magnitude greater than 1. on the other hand, |�2n+1| ≤ k − 1 11 11 d2n+1, |�2n+2| ≤ k − 5 11 11 d2n+2, and consequently we can conclude that the order of convergence of iteration (16) is τ = ( 17 + √ 357 ) 2 ≈ 17.9472... thus, the theorem is proven. d. optimal algorithm in the sense of kung–traub with order of convergence p = 32 we consider the following nonstationary iterative scheme based on the 4-point iteration function in combination with an optimal algorithm in the sense of kung– traub with order of convergence p = 32: x2n+1 = ϕ1(x2n,x2n−1,x2n−2,x2n−3), x2n+2 = ϕ5(x2n+1). (23) for the error �i = xi − ξ, i = −3,−2,−1,0,1,2 . . . ; [30] is valid �2n+1 ∼ c1(ξ)�2n�2n−1�2n−2�2n−3, (24) �2n+2 ∼ c5(ξ)�162n+1. (25) let k12 = max{|c1(ξ))|, |c5(ξ)|} , d2n−1 = k 3 65 12 |�2n−1|, d2n = k 31 65 12 |�2n|, and let d > 0, and x−3, x−2, x−1 and x0 be chosen so that the following inequalities d−3 = k 3 65 12 |x−3 − ξ| ≤ d < 1, d−2 = k 31 65 12 |x−2 − ξ| ≤ d < 1, d−1 = k 3 65 12 |x−1 − ξ| ≤ d < 1, d0 = k 31 65 12 |x0 − ξ| ≤ d < 1 hold true. from (24) and (25), we have d2n+1 = k 3 65 12 |�2n+1| ≤ k 1 12 12 k12|�2n||�2n−1||�2n−2||�2n−3| = k 3 65 12 k 3 65 + 31 65 + 31 65 12 |�2n|�2n−1||�2n−2|�2n−3| = d2nd2n−1d2n−2d2n−3, d2n+2 = k 31 65 12 |�2n+2| ≤ k 31 65 12 k12� 32 2n+1 = ( k 3 65 12 �2n+1 )32 = d322n+1. (26) evidently, from (26), we find d1 ≤ d4, d2 ≤ d128, d3 ≤ d134, d4 ≤ d4288, d5 ≤ d4554, d6 ≤ d145728, d7 ≤ d154704, d8 ≤ d4950528, d9 ≤ d5255514, d10 ≤ d168176448. our results concerning the order of convergence generated by (23) are summarized in the following theorem. theorem d. assume that the initial approximations x0, x−1, x−2, x−3 are chosen so that d−3 ≤ d, d−2 ≤ d, d−1 ≤ d < 1 and d0 ≤ d < 1. then for the error of the sequences {x2n+1}∞n=0 and {x2n+2} ∞ n=0 determined by (23), we have d2n−1 ≤ dτ2n−1, d2n ≤ dτ2n, (27) where τm+4 = 33τm+2 + 33τm, m = 1,2, . . . (28) and the order of convergence of the iteration (23) is τ = ( 33 + √ 1221 ) 2 . proof. in our case, for the recursion τm+4 = 33τm+2 + 33τm, characteristic polynomial is of the type ρ2 −33ρ−33 = 0. (29) equation (29) has the roots: ρ1 = ( 33 + √ 1221 ) 2 , ( 33− √ 1221 ) 2 . from the general iterative theory it follows that the order of convergence of the iteration procedure, defined by (23) is given by the only real root of equation (22) with magnitude greater than 1. on the other hand, |�2n+1| ≤ k − 3 65 12 d2n+1, |�2n+2| ≤ k −31 65 12 d2n+2, biomath 2 (2013), 1305155, http://dx.doi.org/10.11145/j.biomath.2013.05.155 page 5 of 7 http://dx.doi.org/10.11145/j.biomath.2013.05.155 n kyurkchiev at al., on some multipoint methods arising from optimal in the sense of kung–traub algorithms and consequently we can conclude that the order of convergence of iteration (23) is τ = ( 33 + √ 1221 ) 2 ≈ 33.9714... thus, the theorem is proven. iii. conclusion if in the literature in terms of optimal kung–traub algorithm of order 64 appeared, we have shown that the acceleration in the light of our considerations is: τ = ( 65 + √ 4485 ) 2 ≈ 65.9851... intensively working scientific groups in this branch of numerical analysis should directed theirs efforts to make new interval numerical algorithms which are based on recently arised schemes which are optimal in the sense of kung–traub. for methodical construction of numerical algorithms with result verification see markov [17], [18] and his coauthors [4], [5], [19], [15]. acknowledgment the work presented here is dedicated to the 70th anniversary of prof. dr. svetoslav markov. this article is partially supported by project ni13 fmi-002 of the department for scientific research, paisii hilendarski university of plovdiv. references [1] w. bi, h. ren and q. wu, three-step iterative methods with eight-order convergence for solving nonlinear equations, j. comput. appl. math. 225 (2009) 105–112. http://dx.doi.org/10.1016/j.cam.2008.07.004 [2] w. bi, q. wu and h. ren, a new family of eight-order iterative methods for solving nonlinear equations, appl. math. comput. 214 (2009) 236–245. http://dx.doi.org/10.1016/j.amc.2009.03.077 [3] c. chun and b. neta, certain improvements of newton’s method with fourth-order convergence, appl. math. comput. 215 (2009) 821–824. http://dx.doi.org/10.1016/j.amc.2009.06.007 [4] n. dimitrova and s. markov, interval methods of newton type for nonlinear equations, pliska (studia math. bulg.) 5 (1983) 105– 117. 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original article spike timing neural network model of conscious visual perception petia koprinkova-hristova, simona nedelcheva institute of information and communication technologies bulgarian academy of sciences sofia, bulgaria petia.koprinkova@iict.bas.bg simona.nedelcheva@iict.bas.bg received: 19 november 2021, accepted: 25 february 2022, published: 8 april 2022 abstract—the aim of the paper is to investigate the influence of thalamo-cortical connectivity on the conscious perception of visual stimuli. we conducted simulation experiments changing the key parameters of our spike timing neural network model of visual perception and decision making that are supposed to be related to conscious perception, namely bottom-up and top-down connections between thalamic relay, including thalamic reticulate nucleus (trn) and lateral geniculate nucleus (lgn), and primary visual cortex (v1). the model output, that is perceptual based decision in the lateral intrapareital (lip) area of the brain for left or right saccade generation, was observed. conclusions about the influence of altered key parameters on the ability of our model to take proper decision were commented in respect to the observed activity in the brain areas responsible for conscious visual perception and decision making. keywords-spike timing neural network; consciousness; thalamo-cortical connections; visual perception; i. introduction since the earliest days of psycho-physiology, there has been a debate about the link between sensation, perception, attention, and consciousness. the main question is: what happens to a sensory signal in the brain when it reaches a conscious stage of processing as opposed to being processed outside of awareness? in search of an answer to this question the concept of “neural correlates of consciousness” is introduced that represents the set of neuronal events and mechanisms generating a specific conscious perception. based on it in [3], [4] consciousness is viewed as a state-dependent property of some complex, adaptive, and highly interconnected biological structures in the brain. a model of consciousness is a theoretical description that relates brain phenomena such as fast irregular electrical activity and widespread brain activation to expressions of consciousness such as qualia [19]. copyright: © 2022 koprinkova et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: petia koprinkova-hristova, simona nedelcheva, spike timing neural network model of conscious visual perception, biomath 11 (2022), 2202258, https://doi.org/10.55630/j.biomath.2022.02.258 page 1 of 10 https://biomath.math.bas.bg/biomath/index.php/biomath mailto:petia.koprinkova@iict.bas.bg mailto:simona.nedelcheva@iict.bas.bg https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ https://doi.org/10.55630/j.biomath.2022.02.258 petia koprinkova-hristova, simona nedelcheva, spike timing neural network model of conscious ... recent studies on neural correlates of consciousness are a continuation of the research initiated at the end of the 19th century [12]. the contemporary trend in this intensively developing nowadays area of research was initiated in the 1990s by the development of an empirical approach focusing on visual awareness because the visual system was already intensively investigated [2], [3], [4], [10]. since then, consciousness research became more diverse but its link to visual perception continued [17], [18]. irrespective of the intense interest and research efforts in studying the consciousness, there is still no consensus about the neural correlates of consciousness, i.e. what are the minimal neural mechanisms that are jointly sufficient for any one conscious perception, thought or memory, under constant background conditions [3]. it is still unclear which brain regions are essential for conscious experience. recently, a dominant trend is to view consciousness as emerging from interactions between distributed networks of neurons and especially from the global activity patterns of cortico-cortical and thalamo-cortical loops [5], [6], [7], [14], [21], [22], [23]. in [15] a hypothesis was proposed that the thalamus is the primary candidate for the location of consciousness since it has been referred to as the gateway of nearly all sensory inputs to the corresponding cortical areas. this theory was supported by numerous works. lately, a view of thalamocortical processing is proposed in [20] where two types of thalamic relays are defined: first-order relays receiving subcortical driver input, e.g. retinal input to the lateral geniculate nucleus, and higherorder relays, receiving driver input from layer 5 of the cortex, that participate in cortico-thalamocortical circuits. recent findings [8] support the important role of the lateral geniculate nucleus in the emergence of consciousness and provide a more complex view of its connections to the other parts of the thalamus and the visual cortex. in [1] it was suggested that the hallmark of conscious processing is the flexible integration of bottomup and top-down thalamo-cortical streams and a novel neurobiology theory of consciousness called dendritic information theory, was proposed. we have already developed a hierarchical spike timing model of visual information perception and decision making including the detailed structure of the thalamic relay composed by laretal geniculate nucleus, thalamic reticulate nucleus and interneurons [11]. the model was implemented in nest simulator [9] on the supercomputer avitohol. here we investigate the influence of the thalamo-cortical connectivity on the conscious perception of visual stimuli by changing the key parameters of our model that are supposed to be related to conscious perception, namely the bottom-up and top-down connections between the thalamic relay and primary visual cortex (v1). the model output perceptual based decision for left or right saccade generation was observed. conclusions about the influence of the altered key parameters on the ability of our model to take proper decision were commented in respect to the observed activity in the areas v1, middle temporal (mt), medial superior temporal (mst) and lip areas. the structure of the rest of the paper is as follows: next section presents briefly our model architecture and its parameters under investigation; the simulation results are presented and discussed in sections iii and iv; the paper finishes with the concluding remarks pointing out the directions of our further work. ii. model structure the details of the structure of our spike timing hierarchical model of visual information processing and perceptual decision making with reinforcement learning were reported in [11]. for the aim of the current simulation investigations we use only its perceptual-based decision part shown on fig. 1. each colored rectangle on fig. 1 represents a group of neurons positioned on a regular twodimensional grid. each group, called further layer, corresponds to a brain structure involved in visual information perception as follows: retinal ganglion cells (rgc); lateral geniculate nucleus (lgn); biomath 11 (2022), 2202258, https://doi.org/10.55630/j.biomath.2022.02.258 page 2 of 10 https://doi.org/10.55630/j.biomath.2022.02.258 petia koprinkova-hristova, simona nedelcheva, spike timing neural network model of conscious ... fig. 1. model structure as in [11]. thalamic reticulate nucleus (trn) and interneurons (in); primary visual cortex (v1); middle temporal (mt) area; medial superior temporal (mst) area; lateral intraparietal cortex (lip). each layer has a special role in the visual information processing as follows: rgc in eyes transform the light to electrical signal fed into the brain via the optic nerve; lgn, trn and in has a role of relay structure forwarding the information to the visual cortex; v1 detects orientation of the visual stimulus; further mt detects the direction of movements while mst has more complicated role to detect the patterns of movement (here expansion/contraction from/to a given focal point); finally lip collects the processed visual information and takes a perceptual based decision (in this case left or right expansion center of the stimulus). the arrows denote the connections between layers, called synapses. the details of connectivity, described in our previous works [11], are based on the literature information. briefly, each neuron has its own receptive field area of neurons from a given layer that it is connected to depending on the function of the layer it belongs to as well as on its position within its layer. the dimensions of all model building blocks are shown in table i table i visual perception and decision making model dimensionality. layer size neurons number rgc 2 × 20 × 20 800 lgn 2 × 20 × 20 800 in 2 × 20 × 20 800 trn 2 × 20 × 20 800 v1 e 2 × 20 × 20 800 v1 i 2 × 10 × 10 200 mt e 2 × 20 × 20 800 mt i 2 × 10 × 10 200 mst 2 × 20 × 20 800 lip 2 × 20 × 20 800 the connections of interest in current investigation are those between the thalamus (including lgn, trn and in layers) and the visual cortex (v1 layer). the v1 neurons have orientation sensitivity due to their elongated receptive fields defined by gabor functions. their orientation and phase parameters were determined so as to achieve the typical for the mammalian brain “pinwheelstructure” (for more information see [16]). they are separated into four groups two excitatory (e1 and e2) and two inhibitory (i1 and i2) connected via lateral connections based on their distance and roles as in [13]. fig. 2 shows the designed in this way strengths of forward connections (from the thalamus to v1). the feedback connections from v1 to the thalamus are proportional to the feedforward once. in the current work we scale the feedforward and feedback connections to investigate their role in the conscious visual information perception. iii. simulation experiments the model input stimulus consisted of moving dots expanding from a left focal point as shown on fig. 3 (for more details see our previous works [11], [16]). the visual stimulation lasted biomath 11 (2022), 2202258, https://doi.org/10.55630/j.biomath.2022.02.258 page 3 of 10 https://doi.org/10.55630/j.biomath.2022.02.258 petia koprinkova-hristova, simona nedelcheva, spike timing neural network model of conscious ... fig. 2. connectivity between thalamus and v1 as in [16]. fig. 3. visual stimulus screen shot. blue arrows denote the imaginary expansion center of the moving white dots on the gray screen. for 1670 milliseconds. first 150 milliseconds were “washed out” since they were needed for rgc spatio-temporal filters to accumulate the visual information at the beginning of the stimulation. we conducted simulation experiments by varying the scaling factors of both feedforward and feedback connections between the thalamus and v1. table ii summarizes the experiments. sign + denotes the cases with successful propagation of the visual information from rgc up to lip area while those marked with − are the cases without transmission of the visual information to v1. the smallest values of the feedforward connections (scaled by 0.1 and below) do not allow propagation of the visual information to the primary visual cortex thus preventing the perceptual based decision while the minimal feedback connectivity does not have such deterioration effect. in order to investigate in details the effect of varying feedforward/feedback connectivity we table ii simulation experiments feedback feedforward scaling scaling 0.1 and below 0.5 1.0 2.0 5.0 0.0 − + + + + 0.01 − + + + + 0.1 − + + + + 0.5 − + + + + 1.0 − + + + + 2.0 − + + + + 5.0 − + + + + monitored the spiking activity in all layers from primary visual cortex v1 up to the decision lip areas. the next section summarizes and discusses the results of the simulation experiments carried out. iv. results and discussion figures 4-7 show the simulation results for the experiments with successful propagation of the visual information (marked by + in table ii) for all considered model layers (v1, mt, mste and lipleft respectively). the top parts of the figures 4-7 show firing rates during stimulation in the observed layers of the model. in order to distinguish clearly differences in spiking activity, bottom parts of the figures 47 show the mean and the variance of the above spiking frequencies. as it was expected, the biggest differences in spiking activity caused by scaling of the feedback/feedforward connectivity were observed in v1 since it is the first structure influenced directly by the thalamus. fig. 4 shows that with the increase of both feedforward and feedback connections the initiation of spiking activity, e.g. first reaction to the visual stimulus, begins earlier. in case of smallest feedforward connectivity without feedback connections (feedback scaling 0.0 and feedforward scaling 0.5) it begins at about 107th millisecond of stimulation, while in the last case (feedback/feedforward scaling 5.0) it begins at about 61st millisecond. the increased feedforward connectivity also led to the increase in spiking activity by the end of stimulation. biomath 11 (2022), 2202258, https://doi.org/10.55630/j.biomath.2022.02.258 page 4 of 10 https://doi.org/10.55630/j.biomath.2022.02.258 petia koprinkova-hristova, simona nedelcheva, spike timing neural network model of conscious ... fig. 4. spiking activity in v1 (top) and its mean (bottom left) and variance (bottom right). biomath 11 (2022), 2202258, https://doi.org/10.55630/j.biomath.2022.02.258 page 5 of 10 https://doi.org/10.55630/j.biomath.2022.02.258 petia koprinkova-hristova, simona nedelcheva, spike timing neural network model of conscious ... fig. 5. spiking activity in mt and its mean (bottom left) and variance (bottom right). biomath 11 (2022), 2202258, https://doi.org/10.55630/j.biomath.2022.02.258 page 6 of 10 https://doi.org/10.55630/j.biomath.2022.02.258 petia koprinkova-hristova, simona nedelcheva, spike timing neural network model of conscious ... fig. 6. spiking activity in mste and its mean (bottom left) and variance (bottom right). biomath 11 (2022), 2202258, https://doi.org/10.55630/j.biomath.2022.02.258 page 7 of 10 https://doi.org/10.55630/j.biomath.2022.02.258 petia koprinkova-hristova, simona nedelcheva, spike timing neural network model of conscious ... fig. 7. spiking activity in lipleft and its mean (bottom left) and variance (bottom right). biomath 11 (2022), 2202258, https://doi.org/10.55630/j.biomath.2022.02.258 page 8 of 10 https://doi.org/10.55630/j.biomath.2022.02.258 petia koprinkova-hristova, simona nedelcheva, spike timing neural network model of conscious ... the spiking activity in the next (mt) layer was influenced in similar manner like that in v1 while the mste and lip areas show almost similar activity for all experimental cases. while the mean of the v1 firing rate increases with both scaling parameters (fig. 4), its variance shows lowest values in cases of middle (2.0) and lowest (0.5) feedforward scaling coefficients. the mean spiking frequency in the mt area shows a bit different tendency: the biggest mean frequency was observed in the case of the biggest feedforward scaling coefficient in combination with lowest feedback scaling coefficient. the variance of spiking activity in mt has also irregular dependence on the thalamus-v1 connectivity showing bigger values not only in the case with the biggest scaling parameters but also in the cases with middle feedforward and low feedback scaling as well as with low feedforward and high feedback scaling coefficients (fig. 5). the observed in the mt area effect spreads also to the mste and lip areas (figures 6 and 7). while the biggest mean frequencies in the mste area were observed for the biggest feedforward and lowest feedback scaling coefficients, its biggest variance was in the case of middle feedforward scaling (2.0) in combination with low feedback scaling (0.01) coefficients. the strongest mean firing activity in the decision area (left lip area in our example) moves further to the lowest feedforward scaling (0.5) in combination with middle (2.0) feedback scaling while the biggest variance in lip spiking activity was observed for relatively low feedback/feedforward connectivity with scaling coefficients 0.5 and 1.0 respectively. in summary, the biggest mean firing rates move through the model areas from the case of biggest feedforward/feedback scaling coefficients in v1 to the case of lowest feedforward and middle feedback scaling coefficients in the lip area. this result shows the significance of feedback connectivity from v1 to the thalamus. in all cases even small feedback connectivity increases the mean spiking activity through all the model areas in comparison with cases of missing feedback thalamo-cortical connectivity. however, after some threshold the higher feedback scaling coefficients have suppressing effect on the spiking activity in the mt, mst and lip areas. v. conclusion the presented in this paper simulation experiments with varying feedforward/feedback thalamo-cortical connectivity demonstrated the significance of the feedback from the visual cortex to the thalamus. our results revealed that the relation between the two way connectivity strengths and the spiking activity in the visual information processing areas mt, mst and lip following the visual cortex v1 is not straightforward and should be investigated in deeper details. we have to account that our model connectivity was designed based on the literature information and needs further experimental clarification. the dimension of the test model was minimal so its scaling up to more realistic number of neurons will make the simulation experiments more realistic too. our planes for future work include exploitation of the experimental data (mri and fmri) revealing the real brain connectivity in order to make our model more realistic. we expect to obtain such a data by cooperation with the partners from cost action “neural architectures of consciousness” in the near future. the implementation of the model on the supercomputer avitohol will allow a significant increase of its dimension that is our second aim for the future work. acknowledgments: the reported work is supported by the bulgarian science fund grant no kp-06-cost/9 from 2021 “supercomputer simulation investigation of a neural model of brain structures involved in conscious visual perception” by a co-funding scheme of cost action ca18106 “neural architectures of consciousness”. sn was supported by the bulgarian national scientific program young scientists and postdocs, module young scientists. biomath 11 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(2016) integrated information theory: from consciousness to its physical substrate. nat. rev. neurosci. 17, 450-461. biomath 11 (2022), 2202258, https://doi.org/10.55630/j.biomath.2022.02.258 page 10 of 10 https://doi.org/10.55630/j.biomath.2022.02.258 introduction model structure simulation experiments results and discussion conclusion references original article biomath 2 (2013), 1312071, 1–6 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum mathematical and numerical analysis of a modified keller-segel model with general diffusive tensors georges chamoun∗,∗∗, mazen saad∗, raafat talhouk∗∗ *ecole centrale de nantes, ed stim, lmjl umr6629 cnrs-un 1, rue de la noé, 44321 nantes, france . emails: georges.chamoun@ec-nantes.fr, mazen.saad@ec-nantes.fr **université libanaise, edst et faculté des sciences i, laboratoire de mathématiques, hadath, liban . email: rtalhouk@ul.edu.lb received: 18 september 2013, accepted: 7 december 2013, published: 23 december 2013 abstract—this paper is devoted to the mathematical analysis of a model arising from biology, consisting of diffusion and chemotaxis with volume filling effect. motivated by numerical and modeling issues, the global existence in time and the uniqueness of weak solutions to this model is investigated. the novelty with respect to other related papers lies in the presence of a two-sidedly nonlinear degenerate diffusion and anisotropic heterogeneous diffusion tensors, where we prove global existence and uniqueness under further assumptions. moreover, we introduce and we study the convergence analysis of the combined scheme applied to this anisotropic keller-segel model with general tensors. finally, a numerical test is given to prove the effectiveness of the combined scheme. keywords-degenerate parabolic equation ; heterogeneous and anisotropic diffusion; global existence of solutions; combined scheme. i. introduction chemotaxis, the directed movement of cells in response to chemical gradients, plays an important role in many biological fields, such as embrogenesis, immunology, cancer growth and wound healing. this behavior enables many living organisms to locate nutrients, avoid predators or find animals of the same species. for example, bacteria often swim towards higher concentration of oxygen to survive (see [7]). in the following, we investigate a system consisting of the parabolic kellersegel equations with general tensors,{ ∂tn −∇· ( s(x) ( a(n)∇n −χ(n)∇c )) = 0, ∂tc −∇· (m(x)∇c) = g(n,c), (1) where ω is a bounded domain in rd; d ∈ n∗ with boundary ∂ω. this system of equations is supplemented by the following boundary conditions on σt= ∂ω × (0,t), s(x)a(n)∇n ·η = 0, m(x)∇c ·η = 0, (2) where ν is the exterior unit normal to ∂ω. the initial conditions on ω, are given by, n(x, 0) = n0(x), c(x, 0) = c0(x). (3) here n and c denote respectively the cell density and the concentration of a chemical. moreover, a(n) denotes the density-dependent diffusion coefficient and χ(n) is usually written in the form χ(n) = nh(n) where h is commonly referred as a chemotactic sensitivity function. s(x) and m(x) are considered as general tensors which may be anisotropic and heterogeneous. in the model (1), the cell density n diffuses and swims upwards chemical gradients. in addition to that, the chemical c also diffuses where g(n,c) is the rate of production and consumption of the chemical. this article is concerned with the global existence in time of weak solutions to the model (1), for any citation: georges chamoun, mazen saad, raafat talhouk, mathematical and numerical analysis of a modified keller-segel model with general diffusive tensors, biomath 2 (2013), 1312071, http://dx.doi.org/10.11145/j.biomath.2013.12.071 page 1 of 6 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2013.12.071 g chamoun, mathematical and numerical analysis of a modified keller-segel model... space dimension, in the presence of anisotropic and heterogeneous tensors, two-sidedly nonlinear degenerate diffusion and modified chemotactic sensitivity χ. the techniques of the proof of global existence are essentially those designed by [4] and [5]. under further assumptions and in the spirit of the duality method used in [8], the uniqueness of these weak solutions is guaranteed. moreover, in order to discretize our model (1), it is wellknown that classical finite volume methods not permit to handle anisotropic tensors in the diffusive terms but it is very useful, especially the technique ”upwind”, to discretize the convective term since it does not allow instabilities in the numerical solution. the intuitive idea is hence to combine two numerical methods (see [2], [3]) by discretizing the diffusive terms with the finite element method enabling the discretization of anisotropic diffusion tensors without any restrictions on the meshes and the other terms with the finite volume method since we avoid numerical instabilities in the convectiondominated regime. finally, a numerical test will be given to illustrate the effectiveness of our combined scheme applied to the anisotropic keller-segel model (1). ii. setting of the problem let us now state the assumptions on the data we will use in the sequel, together with the main results obtained in this paper. we assume that χ(0) = 0 and the chemotactical sensitivity χ(n) vanishes when n ≥ nm. this threshold condition has a clear biological interpretation called the volume-filling effect. in fact, the effect of a threshold cell density or a volume filling effect has been also taken into account in the modeling of chemotaxis phenomenon in [9]. upon normalization, we can assume that the threshold density is nm = 1. the main assumptions are: χ : [0, 1] 7−→ r is continuous and χ(0) = χ(1) = 0 , a : [0, 1] 7−→ r+ is continuous, a(0) = a(1) = 0 and a(s) > 0 for 0 < s < 1 . (4) a standard example for χ is, χ(n) = n(1 −n); n ∈ [0, 1] . (5) the positivity of χ means that the chemical attracts the cells; the repellent case is the one of negative χ. in addition to that, will assume that the rate g is linear, g(n,c) = αn −βc; α, β ≥ 0 . (6) the permeabilities s, m: ω −→ md(r) where md(r) is the set of symmetric matrices d×d, verify: si,j ∈ l∞(ω), mi,j ∈ l∞(ω), ∀i,j ∈{1, ..,d} , (7) and there exist cs ∈ r∗+ and cm ∈ r∗+ such that a.e x ∈ ω, ∀ξ ∈ rd, s(x)ξ · ξ ≥ cs|ξ|2, m(x)ξ · ξ ≥ cm|ξ|2 . (8) definition 2.1: assume that 0 ≤ n0 ≤ 1, c0 ≥ 0 and c0 ∈ l∞(ω). a couple (n,c) is said to be a weak solution of (1) if, 0 ≤ n(x,t) ≤ 1, c(x,t) ≥ 0 a.e. in qt = ω × [0,t], n ∈ cw([0,t]; l2(ω)), ∂tn ∈ l2(0,t; (h1(ω)) ′ ) , a(n) := ∫ n 0 a(r)dr ∈ l2(0,t; h1(ω)), c ∈ l∞(qt ) ∩l2(0,t; h1(ω)) ∩c([0,t]; l2(ω)) , ∂tc ∈ l2(0,t; (h1(ω)) ′ ), and (n,c) satisfy,∫ t 0 < ∂tn,ψ1 >(h1)′,h1 dt+∫∫ qt ( s(x)a(n)∇n−s(x)χ(n)∇c ) ·∇ψ1 dxdt = 0 ,∫ t 0 < ∂tc,ψ2 >(h1)′,h1 dt+ ∫∫ qt m(x)∇c ·∇ψ2 dxdt = ∫∫ qt g(n,c)ψ2 dxdt, for all ψ1, ψ2 ∈ l2(0,t; h1(ω)), where cw(0,t; l 2(ω)) denotes the space of continuous functions with values in (a closed ball of) l2(ω) endowed with the weak topology. our first result is the following existence theorem of weak solutions proved by using a technique of semidiscretization in time (see [4]) for the regularized nondegenerate problem associated to (1). next, when the parameter of regularization tends to zero, a similar strategy as in [5] will be followed to achieve the proof. theorem 2.1: assume that 0 ≤ n0 ≤ 1 and c0 ≥ 0 a.e. in ω, c0 ∈ l∞(ω). then, the system (1) has a global weak solution (n,c) in the sense of definition 2.1. our second result is the following theorem. theorem 2.2: under the assumptions of theorem 2.1, and assume that the functions a and χ are of class c1 and if there exists a constant c > 0 such that, (χ(n1) −χ(n2))2 ≤ c(n1 −n2)(a(n1) −a(n2)), ∀n1,n2 ∈ [0, 1] . (9) biomath 2 (2013), 1312071, http://dx.doi.org/10.11145/j.biomath.2013.12.071 page 2 of 6 http://dx.doi.org/10.11145/j.biomath.2013.12.071 g chamoun, mathematical and numerical analysis of a modified keller-segel model... then the system (1) has a global unique weak solution. outline of the proof: it relies on a classical duality technique. we introduce the subset l20(ω) = {w ∈ l2(ω), ∫ ω wdx = 0} and we denote by nw the unique solution to { −∇· ( s(x)∇nw ) = w s(x)∇nw ·η = 0 . (10) this dual problem (10) and similar guidelines of [8] (subsection 2.2) are followed to apply gronwall lemma and to prove the uniqueness statement of theorem 2.2. in fact, we can give a useful sufficient condition which implies (9). indeed, by a straight application of the mean theorem applied to χ and a which are c1-functions, we can rewrite (9) as follows, (χ(n1)−χ(n2))2 = (χ′(ξ))2 a(ξ1) (n1−n2)(a(n1)−a(n2)) . then it amounts to check the following sufficient condition to prove (9), ∃c > 0; max ξ,ξ1∈]0,1[ (χ′(ξ))2 a(ξ1) ≤ c . (11) this sufficient condition is also mentioned in ( [8], proposition 4 ) for diffusion coefficients with one point degeneracy. example. in the model (1), if a(u) = u(1 −u) and χ(u) = (u(1−u))β then the weak solution of the system (1) is unique provided β ≥ 3 2 . a simple computation is left to the reader to check the sufficient condition (11). iii. numerical scheme this section is devoted to the formulation of a combined scheme for the anisotropic keller-segel model (1). first, we will describe the space and time discretizations, define the approximation spaces and then we will introduce the combined scheme. let ω be an open bounded subset of rd with d = 2 or 3. in order to discretize the problem (1), we consider a family th of meshes of the domain ω, consisting of disjoint closed simplices such that ω̄ = ∪k∈thk̄ and such that if k,l ∈th, k 6= l, then k ∩l is either an empty set or a common face or edge of k and l. we denote by eh the set of all sides, by einth the set of all interior sides, by eexth the set of all exterior sides. the size of the mesh th is defined by h:= size(th)=maxk∈th diam(k), which has the sense of an upper bound for the maximum diameter of the control volumes in th. we also define a geometrical factor, linked with the regularity of the mesh, by making the following shape regularity assumption on the family of triangulations {th}h: there exists a positive constant kt ; min k∈th |k| (diam(k))d ≥ kt , ∀h > 0 . (12) we also use a dual partition dh of disjoint closed simplices called control volumes of ω such that ω̄ = ∪d∈dhd̄. there is one dual element d associated with each side σd = σk,l ∈ eh. we construct it by connecting the barycenters of every k ∈th that contains σd through the vertices of σd. as for the primal mesh, we define fh, finth and f ext h respectively as the set of all dual, interior and exterior mesh sides. for σd ∈fexth , the contour of d is completed by the side σd itself. we refer to the figure 1 for the two-dimensional case. fig. 1. triangles k,l ∈ th and dual volumes d, e ∈ dh associated with edges σd, σe ∈eh. we use the following notations: • |d|= meas(d)= d-dimensional lebesgue measure of d and |σ| is the (d−1)-dimensional measure of σ. • pd is the barycenter of the side σd. • n(d) is the set of neighbors of the volume d. • dinth and d ext h are respectively the set of all interior and boundary dual volumes. the time discretization is the sequence of discrete times tn = n∆t for n ∈ n, where ∆t > 0 is a given time-step. in the sequel, the exponent n denotes the approximation at time tn. next, we define the following finite-dimensional spaces: xh := {ϕh ∈ l2(ω); ϕh|k is linear ∀k ∈th, ϕh is continuous at the points pd, d ∈dinth } , x0h := {ϕh ∈ xh; ϕh(pd) = 0, ∀d ∈d ext h } . the basis of xh is spanned by the shape functions ϕd, d ∈ dh, such that ϕd(pe) = δde, e ∈ dh, δ being the kronecker delta. we recall that the approximations biomath 2 (2013), 1312071, http://dx.doi.org/10.11145/j.biomath.2013.12.071 page 3 of 6 http://dx.doi.org/10.11145/j.biomath.2013.12.071 g chamoun, mathematical and numerical analysis of a modified keller-segel model... in these spaces are nonconforming since xh 6⊂ h1(ω). indeed, only the weak continuity of the solution is provided through the faces of the mesh and therefore the solution may be discontinuous on the faces. we equip xh with the seminorm, ||nh||2xh := ∑ k∈th ∫ k |∇nh|2dx, (13) which becomes a norm on x0h. the approximation of the flux s(x)∇c · ηd,e on the interface σd,e is denoted by δcd,e. now, we have to approximate s(x)χ(n)∇c · ηd,e by means of the values nd,ne and δcd,e that are available in the neighborhood of the interface σd,e. to do this, we use a numerical flux function g(nd,ne,δcd,e). one can find in [1] a way to construct this numerical flux g. finally, a combined finite volume-nonconforming finite element scheme for the discretization of the problem (1) is given by the following set of equations: for all d ∈ dh, n0d = 1 |d| ∫ d n0(x)dx, c 0 d = 1 |d| ∫ d c0(x)dx, (14) and for all d ∈dh, n ∈{0, 1, ...,n}, |d| nn+1d −n n d ∆t − ∑ e∈dh sd,ea(nn+1e ) + ∑ e∈n(d) g(nn+1d ,n n+1 e ; δc n+1 d,e ) = 0 , (15) |d| cn+1d −c n d ∆t − ∑ e∈dh md,ecn+1e = |d|g(n n d,c n+1 d ) . (16) the matrix s (resp. m), of elements sd,e (resp. md,e) for d,e ∈ dinth , is the diffusion matrix which is the stiffness matrix of the nonconforming finite element method. so that: sd,e = − ∑ k∈th (s(x)∇ϕe,∇ϕd)0,k . (17) moreover, δcn+1d,e denotes the approximation of s(x)∇c ·ηd,e on the interface σd,e: δcn+1d,e = sd,e(c n+1 e −c n+1 d ) . (18) definition 3.1: using the values of nn+1d ,∀d ∈ dh and n ∈ [0,n], we will define the approximate finite volume solution ñh,∆t defined as piecewise constant on the dual volumes in space and piecewise constant in time, such that: ñh,∆t(x, 0) = n 0 d for x ∈ d, d ∈dh , ñh,∆t(x,t) = n n+1 d for x ∈ d, d ∈dh, t ∈]tn, tn+1] . now, we state the discrete maximum principle and a convergence result of the combined scheme under the assumption that all transmissibilities coefficients, defined in (17), are positive: sd,e ≥ 0 and md,e ≥ 0, ∀d ∈dh, e ∈n(d) . (19) recall that 0 ≤ n0 ≤ 1 and c0 ≥ 0 a.e. in ω. we have the following classical proposition proved by a simple induction argument as in [3]. proposition 3.1: (discrete maximum principle) under the assumption (19), one has, 0 ≤ nn+1d ≤ 1, c n+1 d ≥ 0, ∀d ∈dh, n ∈{0, 1, ..,n}. theorem 3.1: (convergence of the scheme) assume (4) to (8). consider c0 ∈ l∞(ω), c0 ≥ 0 and 0 ≤ n0 ≤ 1 a.e. on ω. under the assumption (19), 1) there exists a solution (nh,ch) of the discrete system (15)-(16) with initial data (14). 2) any sequence (hm)m decreasing to zero possesses a subsequence such that (nhm,chm ) converges a.e. on qt to a weak solution (n,c) of (1). outline of the proof. adding to proposition 3.1, we establish estimates on the discrete gradient of the function a(ñh,∆t) and ñh,∆t and these discrete properties on the diffusive term allow to construct a priori estimate with respect to the norm xh defined in (13). then, constructing estimates in time and in space imply that the sequence ( a(ñh,∆t) ) h,∆t satisfies the assumptions of the kolmogorov’s compactness criterion, and consequently( a(ñh,∆t) ) h,∆t is relatively compact in l2(qt ). this implies the existence of subsequences of ( a(ñh,∆t) ) h,∆t which converges strongly to some function γ ∈ l2(qt ). using the monotonicity of a, we get γ = a(n). moreover, due to the space translate estimate, ( [6], theorem 3.10 ) gives that a(n) ∈ l2(0,t; h1(ω)). as a−1 is well defined and continuous, applying the l∞ bound on ñh,∆t and the dominated convergence theorem of lebesgue to ñh,∆t = a−1(a(ñh,∆t)), there exists a subsequence of ñh,∆t which converges strongly in l2(qt ) and a.e. in qt to the same function n. then, we conclude by showing that the limit couple (n,c) is a weak solution of the continuous problem (1). biomath 2 (2013), 1312071, http://dx.doi.org/10.11145/j.biomath.2013.12.071 page 4 of 6 http://dx.doi.org/10.11145/j.biomath.2013.12.071 g chamoun, mathematical and numerical analysis of a modified keller-segel model... fig. 2. initial condition for the cell density n0 (left) and for the concentration of the chemo-attractant c0 (right) on the dual mesh dh iv. numerical test through this numerical example, we would like to illustrate the effectiveness of the combined scheme for anisotropic keller-segel model (1). all the computations for this new numerical scheme have been implemented using the software package fortran. the algorithm used to compute numerical solution of the discrete problem is the following: at each time step, we first calculate cn+1 solution of the linear system given by the equation of (16) and next we compute nn+1 as the solution of the nonlinear system defined by the first equation of (15). to this end, a newton algorithm is implemented to approach the solution of nonlinear system and a bigradient conjugate method to solve linear systems arising from the newton algorithm process. we will provide this test on the dual mesh dh of a refined initial admissible mesh th, where the assumption (19) is satisfied. in this test, we consider the following anisotropic diffusion tensors as, s = m = [ 8 −7 −7 20 ] . further, dt = 0.0005, α = 0.01, β = 0.05, a(n) = d(n 2 2 − n 3 3 ) with d = 0.005, χ(n) = cn(1 − n)2 with c = 0.001. finally, the diffusion coefficient of the chemo-attractant is d1 = 10−6. the initial conditions are defined by region. the initial density is defined as n0(x,y) = 1 in the square (x,y) ∈ ( [0.45, 0.55] × [0.45, 0.55] ) and 0 otherwise. the initial chemoattractant is defined as c0(x,y) = 5 in the union of four squares (x,y) ∈ ( [0.2, 0.3] × [0.7, 0.8] ) ∪ ( [0.2, 0.3] × [0.2, 0.3] ) ∪ ( [0.7, 0.8]×[0.2, 0.3] ) ∪ ( [0.7, 0.8]×[0.7, 0.8] ) and 0 otherwise (see figure 2). in the isotropic case (s = m = id), the cells diffuse towards the four regions of the chemoattractant. in this test case and under the influence of the anisotropic diffusion, we observe in fig. 3. test 1the cell density (n), at time t = 1 with 0 ≤ n ≤ 0.1752 (left), at time t = 5 with 0 ≤ n ≤ 0.08036 (right) . fig. 4. test 1the cell density (n), at time t = 10 with 0 ≤ n ≤ 0.1456 (left), at time t = 15 with 0 ≤ n ≤ 0.1666, at time t = 20 with 0 ≤ n ≤ 0.1626 (right) . figures 3 and 4 the movement of the cells only towards two chemoattractant regions during the stage of evolution in time of the cell density. v. conclusion in this article, we have explored the relevance of the keller-segel equations in the modeling of anisotropic chemotactic cell migration. motivated by numerical simulations, we have proceeded to prove results pertaining to the existence and the uniqueness of global weak solutions of the anisotropic keller-segel model with general diffusive tensors and the convergence analysis of a new combined scheme introduced. this numerical scheme is a compromise between the nonconforming finite elements, enabling in particular the use of general meshes and the discretization of anisotropic diffusion tensors, and between the finite volumes enabling to avoid spurious oscillations in the convection-dominated regime. finally, a numerical test was given to illustrate the combined scheme proposed. acknowledgment the authors would like to thank the national council for scientific research (lebanon), the ecole centrale de nantes and the lebanese university for their support to this work. biomath 2 (2013), 1312071, http://dx.doi.org/10.11145/j.biomath.2013.12.071 page 5 of 6 http://dx.doi.org/10.11145/j.biomath.2013.12.071 g chamoun, mathematical and numerical analysis of a modified keller-segel model... references [1] b. andreianov, m. bendahmane and m. saad, finite volume methods for degenerate chemotaxis model. journal of computational and applied mathematics, 235: p. 4015-4031, 2011. http://dx.doi.org/10.1016/j.cam.2011.02.023 [2] p. angot, v. dolejsi, m. feistauer and j. felcman, analysis of a combined barycentric finite volume-nonconforming finite element method for nonlinear convection-diffusion problems. appl.math., 43(4), p. 263-310, 1998. http://dx.doi.org/10.1023/a:1023217905340 [3] r. eymard, d. hilhorst and m. vohralik, a combined finite volume-nonconforming/mixed hybrid finite element scheme for degenerate parabolic problems. numer.math., 105: p. 73-131, 2006. 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[7] i. tuval, l. cisneros, c. dombrowski, c.w. wolgemuth, j.o.kessler and r.e.goldstein, bacterial swimming and oxygen transport near contact lines. proc. natl. acad. sci. usa, 102, pp. 2277-2282, 2005. http://dx.doi.org/10.1073/pnas.0406724102 [8] p. laurencot et d. wrzosek, a chemotaxis model with threshold density and degenerate diffusion. nonlinear differential equations and their applications, vol. 64, birkhauser, boston, p. 273-290, 2005. [9] k. painter and t. hillen, volume filling effect and quorum-sensing in models for chemosensitive movement. canadian appl. math. q. 10, p. 501-543, 2002. biomath 2 (2013), 1312071, http://dx.doi.org/10.11145/j.biomath.2013.12.071 page 6 of 6 http://dx.doi.org/10.1016/j.cam.2011.02.023 http://dx.doi.org/10.1023/a:1023217905340 http://dx.doi.org/10.1016/j.jde.2006.03.023 http://dx.doi.org/10.1073/pnas.0406724102 http://dx.doi.org/10.11145/j.biomath.2013.12.071 introduction setting of the problem numerical scheme numerical test conclusion references original article biomath 3 (2014), 1312281, 1–7 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum numerical analysis of the coupled modified van der pol equations in a model of the heart action beata zduniak faculty of applied informatics and mathematics warsaw university of life sciences (sggw) warsaw, poland email: beata.zduniak@wp.pl received: 11 october 2013, accepted: 28 december 2013, published: 21 may 2014 abstract—in this paper, a modified van der pol equations are considered as a description of the heart action. wide ranges of the model parameters yield interesting qualitative results, e.g. hopf bifurcation, bogdanov-takens bifurcation, transcritical and pitchfork bifurcations but also some stable solutions can be found. the physiological model works in the narrowest range of parameters which allows to obtain a stable behaviour what is important in biological problem. when some kinds of pathologies appear in the heart, it is possible to obtain chaotic behaviour. my aim is to compare the influence of these two types of coupling (unidirectional and bidirectional) on the behaviour of the van der pol system. the coupling takes place in a system with healthy conductivity, between two nodes: sa and av, but in some circumstances, a pathological coupling may occur in the heart. the van der pol oscillator is a type of relaxation oscillator which can be synchronized. in this paper, synchronization properties of such a system are studied as well. for the purpose of a numerical analysis of the system in question, a numerical model was created. keywords-van der pol equation; heart action; coupling; synchronization; i. introduction this paper is related to the research on the electrical conduction system of the human heart. in the heart, in addition to ordinarily working fibres, there are pacemaker centres made of special cells that resemble embryonic cells. these are the cells of the electrical conduction system forming the following concentrations: the sino-atrial node (sa) and the atrioventricular node (av) and his– purkinje system, [1]. the key elements of the conduction system which we consider are the sa node and the av node. each of the two nodes is modelled by the modified van der pol oscillator. this model allows for rendering phenomena observable in clinical experiments, such as holter recordings. the aim of this work is to create a model which is able to render the behaviour typical of the sinoatrial block. the initial pulse in the heart is usually formed in the sa node, and carried through the atria to the av node. in the sa block, the electrical impulse is delayed or blocked on the way to the atria, thus delaying atria depolarization. this is citation: beata zduniak, numerical analysis of the coupled modified van der pol equations in a model of the heart action, biomath 3 (2014), 1312281, http://dx.doi.org/10.11145/j.biomath.2013.12.281 page 1 of 7 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2013.12.281 b zduniak, numerical analysis of the coupled modified van der pol equations... different from av block which occurs in the av node and delays ventricular depolarization. the sa blocks are categorized into three classes, based on the length of the delay. the first degree sa block is characterized by a prolonged conduction time from the sa node to the surrounding atrial tissue. the second degree block has two types: wenckebach block and type ii. the wenckebach block shows an irregular rhythm. the pause of the second degree type i is shorter than twice the minimum length of the period. type ii has a regular rhythm, which may be normal or slow. it is followed by a pause, which is a multiple of the period. conduction across the sa node is normal until the pause, and then it is blocked. the third degree is characterized by lack of atrial activity. heart rhythm is determined by escape rhytm. in biological systems, the phenomenon of synchronization is extremly important. it is responsible for many periodic processes in the body, e.g. the coupling of the pituitary gland is responsible for production of hormones from the thyroid gland that produces them, without synchronization of the two components, the operation of our endocrine system would not be correct . also the main oscillator in the human body, i.e. the heart is subject of synchronization. in this paper, i will discuss the impact of different types of couplings connecting the av node and the sa node, and how they affect the synchronization of the two oscillators. the analysis of synchronization of various modifications of the van der pol model is the aim of many papers. synchronization areas near the main parametric resonance and transition conditions from regular to chaotic motion are presented in paper [1]. the phenomenon of complete synchronization in a network of four coupled oscillators is described in [2]. in paper [3], the authors investigated mechanisms of various bifurcation phenomena observed in the bonhoffer van der pol neurons coupled through the characteristics of synaptic transmissions with a time delay. also synchronization phenomena in van der pol oscillators coupled by a time-varying resistor is researched in paper[4]. however, these articles do not offer any examples of application of this model for recreating pathological behaviour of the electrical-conduction system of the human heart, and therefore the considered ranges of parameters are wider than those applicable for medical applications. in papers [5,6], the authors showed that coupled two van der pol oscillators modelled behaviour of the heart conduction system, and there are described a heart block as pathologies of coupled van der pol oscillators. the van der pol oscillator provides rich dynamical behaviour, which we would like to exploit in the modelling of the heart action [7] also synchronization phenomena. a. mathematical model because each node is a self-exciting pacemaker, it can be described by a relaxation oscillator, i.e. the van der pol oscillator. the model by van der pol and van der mark was created as a model in the electronic circuit theory in 1927: ẍ + 2 f (x)ẋ + x = 0, µ > 0, (1) where f (x) = 12 a(x 2 − 1) is a damping coefficient being a function of the x variable, which is negative for |x| < 1 and positive for |x| > 1. the dynamics of eq. (1) is well-known in literature. the van der pol model needs some changes in order to reproduce the actual features of the action potential. postnov [8] introduced modifications that maintain the required structure of the phase space. to be more precise, he substituted the linear term by a nonlinear cubic force called the duffing term ẍ + a(x2 −µ)ẋ + x(x + d)(x + 2d) d2 = 0, (2) where a, µ, d are positive control parameters. this model can be treated as a sa or av node model. the mutual interaction of the limit cycle present around an unstable focus with a saddle and a stable node is the main property of a modified relaxation oscillator. as a result, the refraction period and the nonlinear phase sensitivity of the action potential of node cells are reproduced correctly. a solution of this equation in terms of biomath 3 (2014), 1312281, http://dx.doi.org/10.11145/j.biomath.2013.12.281 page 2 of 7 http://dx.doi.org/10.11145/j.biomath.2013.12.281 b zduniak, numerical analysis of the coupled modified van der pol equations... time presents the action potential, whereas a solution in terms of velocity enables us to obtain crucial phase portrait. as we can see, the main qualitative difference between eqs. (1) and (2) is the appearance of two additional steady states, i.e. x2 = −d and x3 = −2d. as in the previous case, x1 = 0 is an unstable node or a focus surrounded by a stable unique limit cycle, x2 = −d is a saddle and x3 = −2d forms a focus or a node and can be either stable or unstable, depending on the sign of 4d2 − µ. in the case considered by postnov [8], the first steady state is an unstable focus, while the third is a stable node which attracts all solutions starting on the right hand-side of the stable manifold of the saddle x2. however, in the considered model (2) it is difficult to regulate the location of steady states in the phase space, therefore, a new parameter e is introduced in order to reproduce the heart behaviour. notice that this modification has no influence on the phase portrait, whereas we have the opportunity to modify the location of steady states. in order to simplify frequency regulation and obtain the proper timescale, the ed factor in the denominator is substituted with independent coefficient f , corresponding to harmonic oscillator’s frequency, [9]. below we present the model in its two variable first order form which reads [9] ẋ = y, ẏ = −a(x2 − 1)y − f x(x + d)(x + e). (3) the final system consists of two coupled modified van der pol oscillators. this model can be treated as the sa and av node. the final system that we analyze is given in the following form: ẋ1 = y1 + (k − k1)x1[t − w1] − kx1 + k1 x1[t − w2], ẏ1 = −a1(x12 − 1)y1 − f1 x1(x1 + d1)(x1 + e1)+ +s1(x2 − x1), ẋ2 = y2, ẏ2 = −a2(x22 − 1)y2 − f2 x2(x2 + d2)(x2 + e2)+ +s2(x1 − x2), (4) where k, k1 coupled coefficients, s1, s2 coupled coefficients, w1, w2 delays, a1 = a2 = 5, f1 = f2 = 3, d1 = d2 = 3, e1 = 7, e2 = 4.5 control parameters. parameters values for the modified van der pol model were chosen so that the oscillations frequency correspond to real frequencies of the sa and av nodes. the selection of appropriate parameters was done after the verification of the model by grudziński in [9]. the aim was to recreate the physiological properties of the biological model using mathematical equations. this information is essential for examining stability of our setup because without such limitations the system could have completely different properties and would not recreate physiological properties. modification of the e parameter of the node location influences the distance between consecutive potential needles without changing their shape. this means that the mutual position of the saddle and the node influences the time of spontaneous depolarization, which is one of physiological mechanisms of the regulation of the action potential generation frequency. an increase of the value of the parameter e can be interpreted as an increase of the activity of the nervous system. however, the parameter f is the equivalent of the frequency of the harmonic oscillator. the parameter d adjusts position of a fixed point. the system with delayed feedback (sum of delays with feedback) describes various pathologies observed in the heart action, for example, the sa block, which does not conduct the potential in physiological way. there are situations when the output potential from the sa node influences the input and modifies the action of the system, for example, through injury caused by infarction or instrinsic disease in the sa node. b. types of coupling the origins of the phenomenon of self-exciting reconciliation of vibrations of coupled oscillators back to the seventeenth century. then the christian huygens observed that in the clock with two pendulums after sufficient time has always the situation in which both pendulums oscillated with opposite phases occured, regardless of the initial phase difference. the behaviour of cardiac pacemaker cells resembles that relaxation oscillators. a characteristic property of relaxation oscillators is that they may be synchronized by an external signal, if the latter has a periodicity not differing biomath 3 (2014), 1312281, http://dx.doi.org/10.11145/j.biomath.2013.12.281 page 3 of 7 http://dx.doi.org/10.11145/j.biomath.2013.12.281 b zduniak, numerical analysis of the coupled modified van der pol equations... too much from the spontaneous frequency of the oscillator [7]. synchronization which is defined as an adjustment of rhythms due to weak interaction, is one of the most interesting features displayed by coupled oscillators. we investigate a phenomenological model for the heartbeat consisting of two coupled van der pol oscillators. the coupling between these nodes can be both unidirectional and bidirectional. in addition, feedback may also occur. we know that the sinoatrial node is also reffered to as the pacemaker of the heart. when the impulses generated by the sa node reach the av node, they are delayed. so we have here unidirectional coupling (s1 or s2 is different from zero in eq.4). however, bidirectional coupling (s1 and s2 are different from zero in eq.4) is also possible in the heart, for example, during the wpw syndrome. feedback (a part of eq.4 with k coupling coefficients and with w delays) also appears only in case pathologies, for example, sa and av blocks. the pecorra caroll (pc) theory is considered in this paper. this type of coupling is used, when a state variable from a chaotic system is input into a replica subsystem of the original one,and as a result, both systems can be synchronized identically. ẋ1 = f (x1), ẋ2 = f (x2), (5) where ẋ1 = (u̇1, v̇1), x1 ∈ rn, u1 ∈ rp, v1 ∈ rq. the drive system is presented as: u̇1 = g(u1, v1), v̇1 = h(u1, v1), (6) and response is given as follows:v̇2 = h(u1, v2). the resulting equation for the pc theory gives the following form: ẋ1 = y1 + (k − k1)x1[t − w1] − kx1+ +k1 x1[t − w2], ẏ1 = −a1(x12 − 1)y1 − f1 x1(x1 + d1)(x1 + e1)+ +s1(x2 − x1), ẋ2 = y2, ẏ2 = −a2(x12 − 1)y2 − f2 x1(x1 + d2)(x1 + e2)+ +s2(x1 − x2), (7) fig. 1. time series: red line-without feedback, blue one-with feedback ii. numerical analysis for the purpose of numerical analysis of the discussed system, a numerical model was created using dynamics solver and a program in c++ was developed. a dormand prince 8 integration algorithm was used. this method constitutes a modification of the explicit runge-kutta formula with a variable integration step. in this section, we use some numerical simulations in order to illustrate the pathological behaviour described in the introduction. system with delayed feedback describes various pathologies observed in the heart action, e.g. sa block. when added,the s2 coupling makes the sa node influence the av node rhythm. this type of behaviour is of physiological nature. by adding s1, we arrive at pathological behaviour, and consequently, a reentry wave in our system. it is typical of the wpw syndrome. having included feedback and delay, we obtain a time series which resembles the model presented in figure 1. the red graph presents the initial model without feedback. the blue one presents a modified model with feedback.the parameter values are as follows: k = 1, k1 = 2.85 and w1 = 0.75, w2 = 0.25, and the remaining parameters are the same as in the reference system. in the time series of the modified model with feedback there is a ’delayed impulse’. the period of this oscillator is almost twice as long as in the reference model (the potential period for a single node model of an electrical conduction system with no coupling and feedback is app. 1.4 ), similar like in the second type of the sa block. this is one of the mechanisms causing brachycardia. if we consider the physiological coupling between nodes, then the s2 coupling is introduced to our system. it means that the sa node directs biomath 3 (2014), 1312281, http://dx.doi.org/10.11145/j.biomath.2013.12.281 page 4 of 7 http://dx.doi.org/10.11145/j.biomath.2013.12.281 b zduniak, numerical analysis of the coupled modified van der pol equations... fig. 2. time series: physiological coupling between nodes fig. 3. time series: pathological coupling between nodes av node. with small values of s2, the result is similar to the reference one, but with bigger values, (e.g. 10 or 100) some of the amplitudes are synchronized in-phase with x1 and aperiodic behaviour appears, which is presented in figure 2. the addition of coupling to the y1 term allows us to model the reentry wave, which causes the exceptional situation when av node is the master for sa node. such situation takes place in case of the wpw syndrome. slowing the oscillations down caused by feedback and addition of the s1 coupling, we obtain the aperiodic behaviour. the big arrhythmia occurs. from the medical point of view, it resembles atrial fibrillation. the oscillator begins to work aperiodically, trying to adjust its frequency to the frequency of the av node. it has a tendency to shorten the oscillation period despite the lack of periodic behaviour, as presented in figure 3. the phase portrait is similar to the previous case, but with s1 = 3.5, we observe oscillation death, figure 4. the amplitude death can be understood as a full heart block. no impulse is conducted to the av node. as a result, the av node may take over the function of the pacemaker. figures 5 and 6, which also present various plots of synchronization, indicate that synchronization of the system is greater in case of systems with the s2 coupling than those with the s1 coupling. fig. 4. phase portrait: pathological coupling between nodes fig. 5. synchronization plot for unidirectional case: s1 = 5 2 1 0 1 2 1 0 1 x w fig. 6. synchronization plot for unidirectional case: s2 = 10 biomath 3 (2014), 1312281, http://dx.doi.org/10.11145/j.biomath.2013.12.281 page 5 of 7 http://dx.doi.org/10.11145/j.biomath.2013.12.281 b zduniak, numerical analysis of the coupled modified van der pol equations... fig. 7. time series:bidirectional coupling between nodes fig. 8. synchronization:bidirectional coupling between nodes we applied the pc theory in our system, in spite of the fact that this theory is typical of chaotic behaviour. by applying the theory, we can observe greater synchronization in phase, especially for s2 coupling. the model with the s1 coupling also tries to synchronize in the phase. in the system with two couplings, already in the case of s1 = 1, the oscillator corresponding to the sa node tries to adjust its rhythm to the av node oscillator by shortening its period. with t = 2.3, we obtain biperiodic behaviour, where t = 1.6 and t = 2, figure 7. with s1 = 20 and s2 = 1, we get aperiodic behaviourtypical arrhythmia. with these parameters, there is no synchronization, whereas with values s1 = 2 and s2 = 5, we observe interesting behaviour. periods of both oscillators are shortened. oscillator corresponding to the av node behaves periodically, with its period at the level of 1.6, whereas the one corresponding to the sa node is biperiodic, with periods 1.5 and 1.7. with such system parameters, we observe the antiphase type of synchronization, figure 8. iii. conclusion although the uncoupled van der pol equation has quite trivial dynamics as a stable equilibrium, the system with the coupling can be periodic, but also quasi-periodic, and chaotic. similarly, these couplings of the mathematical system interfere with the work of the heart conduction system (sa block, av block, bradycardia, wpw syndrome). synchronization in the discussed cases is rarely in-phase, but often in anti-phase. the pc theory was also applied to a non chaotic system. this unidirectional coupling affects partial synchronization of our system. bidirectional coupling should be used to describe physiological behaviour of the conduction system, because only this way we can take into account the effect of the av node as a delay element (coupling s1). in our system without couplings, if we have asystole than we can try to give additional s1 coupling. with small values of s1, we observe asystole, while with greater values, the sa rhythm appears but it is not synchronized. the model offered in this study, is a correct reconstruction of heart action pathologies, such as a sa block or a type of the arrhythmia. references [1] j. warmiński, synchronisation effects and chaos in the van der pol-mathieu oscillator, journal of theoretical and aapplied mechanics, 4, 39, 2001. [2] p. perlikowski, a. stefanski, t. kapitaniak, discontinuous synchrony in an array of van der pol oscillators, international journal of non-linear mechanics, 895–901, 45, 2010. [3] k. tsumoto, t. yoshinaga, h. kawakami bifurcations of synchronized responses in synaptically coupled bonhoffervan der pol neurons, physical review e, 65, 2002. [4] y. uwate, y. nishio synchronization phenomena in van der pol oscillators coupled by a time-varying resistor, international journal of bifurcation and chaos, 17(10), 35653569, 2007. [5] j. j. źebrowski, k. grudziński, t. buchner, p. kuklik, j. gac, g. gielerak nonlinear oscillator model reproducing various phenomena in the dynamics of the conduction system of the heart, chaos, 17, 2007. [6] a. m. dos santos, s. r. lopes, r. l. viana rhythm synchronization and chaotic modulation of coupled van der pol oscillators in a model for the heartbeat, physica a, 338, 335 355, 2004. http://dx.doi.org/10.1016/j.physa.2004.02.058 biomath 3 (2014), 1312281, http://dx.doi.org/10.11145/j.biomath.2013.12.281 page 6 of 7 http://dx.doi.org/10.1016/j.physa.2004.02.058 http://dx.doi.org/10.11145/j.biomath.2013.12.281 b zduniak, numerical analysis of the coupled modified van der pol equations... [7] l. henk van der tweel, f. l. meijler, f. j. l. van capelle synchronization of the heart, journal of applied physiology, 34(2), 1973. [8] d. postnov, s.k. han, and h. kook synchronization of diffusively coupled oscillators near the homoclinic bifurcation, physical review e, 60(3), 1999. http://dx.doi.org/10.1103/physreve.60.2799 [9] k. grudziński, modelowanie czynności elektrycznej ukåadu przewodzenia serca, rozprawa doktorska, wydziaå fizyki politechniki warszawskiej, 2007. biomath 3 (2014), 1312281, http://dx.doi.org/10.11145/j.biomath.2013.12.281 page 7 of 7 http://dx.doi.org/10.1103/physreve.60.2799 http://dx.doi.org/10.11145/j.biomath.2013.12.281 introduction mathematical model types of coupling numerical analysis conclusion references www.biomathforum.org/biomath/index.php/biomath original article qualitative analysis of a mathematical model about population of green turtles on the galapagos island candy herrera∗, cosme duque† and hugo leiva‡ ∗ yachay tech university, school of biological science and engineering san miguel de urcuqui-100119. imbabura-ecuador candy.herrera@yachaytech.edu.ec † universidad de los andes, facultad de ciencias, departamento de matemáticas mérida 5101-venezuela cosduq@gmail.com, duquec@ula.ve ‡yachay tech university, school of mathematical and computational sciences san miguel de urcuqui-100119. imbabura, ecuador hleiva@yachaytech.edu.ec received: 8 march 2021, accepted: 29 july 2021, published: 1 october 2021 abstract— according to the iucn, most sea turtles fall into one of the endangered categories. since, sea turtles, like many other reptiles, present an unusual developmental process, marked by the determination of the sex of the offspring by environmental factors, more specifically by temperature. in the temperature sex determination (tsd) system the temperature of an embryo’s environment during incubation period will dictate the embryo’s sex development. this developmental process, together with the complex mating and nesting behavior and the vulnerability of sea turtles to threats of a natural or anthropogenic nature, naturally lead to the study of the population dynamics of the species. for this reason, in this paper, we have developed a continuous model given by a system of three ordinary differential equations to study the dynamics of the green sea turtle population long-term, focusing the mathematical simulations on the data obtained for the nesting species of galapagos islands. through the qualitative analysis of the model, the following is demonstrated: 1) the flow induced by the system is positively invariant on the region of biological interest (ω); and 2) the given condition on f̂ is necessary and sufficient for the unique nontrivial equilibrium point (i∗) to be globally asymptotically stable in that region. when implementing the estimated values for our parameters in the numerical simulations, it was observed that indeed the population of galapagos green sea turtles complies with the condition for which the nontrivial critical point (i∗) is globally asymptotically stable; that is, the asymptotic stabilcopyright: © 2021 herrera et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: candy herrera, cosme duque, hugo leiva, qualitative analysis of a mathematical model about population of green turtles on the galapagos island, biomath 10 (2021), 2107293, http://dx.doi.org/10.11145/j.biomath.2021.07.293 page 1 of 10 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2021.07.293 c herrera, c duque, h leiva, qualitative analysis of a mathematical model about population of ... ity is maintained for any initial value within the set ω. in contrast, when altering the estimated values of the parameters so that the established condition is not met, the trivial critical point (i0) becomes globally stable, and the population falls towards extinction regardless of the values taken within the positively invariant ω set. keywords-population dynamics; sex-structured continuous time model; chelonia mydas; equilibrium point; local stability; global stability. i. introduction sea turtles present a temperature sex determination (tsd) system, this is an evolutionary condition that turtles, like other reptiles, have adopted throughout time. in the tsd system, the temperature of an embryo’s environment during incubation period will dictate the embryo’s sex development. many species of turtles [1], tortoises [2], lizards [3], and crocodiles [4] that exhibit tsd have a thermosensitive period (tsp) during which the embryo sex is developed. for turtles, this period has been observed to take place during the midtrimester of the embryo incubation period [5]. the temperature that defines the 1:1 sex ratio balance, known as pivotal temperature, is ∼ 29.4◦c [6], [7]. when the mean temperature of the nest during tsd is around the pivotal temperature an even distribution of male and female hatchlings occur [6], [7]. below the pivotal temperature, hatchling sex population will be mostly male; and above, it will be mostly female[6], [7], [8]. with a trend towards increasing mean global temperature, species with tsd are particularly affected. the population of sea turtles is facing high egg mortality and feminization of the offspring[9], [10]. in recent years a disproportionate ratio of female to male turtle eggs has been observed in a number of different studies [11]. unfortunately, monitoring sex ratios involves a series of methodological and ethical complications. sex ratios data obtained from the study of adult populations have predicted a complete feminization and a possibly extinction of marine turtles in the future if temperature continues rising [10], [7], [12], [13]. due to the complex nesting behavior of sea turtles and the mechanism by which the sex of the offspring is determined, it is reasonable to look into the factors that could be directly affecting the sex ratios of eggs, such as sand temperatures. in order to understand why there is a bias towards the female population in sea turtles. the most prevalent hypothesis points at climate change as the main factor leading the rising at sand temperatures in nesting sites [11], [10]. the lack of males within the sea turtle population will eventually affect the population dynamics. it is unknown the minimum proportion of males sufficient to support the sea turtle population in such a way to avoid population collapse or even extinction. this problem naturally lends itself to investigation via a sex-structured model to analyse the dynamic of the population. ii. model formulation the green turtle, like other sea turtles, has a remarkable life cycle. individuals inhabit widely separated localities during the course of their lives. these habitats include foraging, migration, breeding, and nesting areas (figure 1). after hatchlings emerge from their nests, they immediately travel to the sea. once in the ocean, the hatchlings are washed away by ocean currents, live a pelagic phase in the open ocean, and are not seen again until they appear as juveniles in foraging areas, probably a decade later [14]. this period of time between the hatchling and juvenile stages is known as the ”lost years” because the migratory route taken by the hatchlings and their behavior remains a mystery [14], [15]. in the foraging areas, they continue to mature until they become subadults. subadults are occasionally seen foraging in the open ocean [14]. once the turtles reach sexual maturity, males and females return to their home beach to mate and nest. this behavior is known as philopatry and has been documented for several species [19]. because during the early stages of the turtle life cycle, hatchlings, juveniles, and subadults are not reproductively active, we can simplify the life cycle of the green turtle and consider only two main stages: adults and eggs. in turn, the adult stage can be divided into two populations: female biomath 10 (2021), 2107293, http://dx.doi.org/10.11145/j.biomath.2021.07.293 page 2 of 10 http://dx.doi.org/10.11145/j.biomath.2021.07.293 c herrera, c duque, h leiva, qualitative analysis of a mathematical model about population of ... fig. 1. schematic diagram of the chelonia mydas life cycle. adults and male adults, which are interesting to analyze by themselves. an schematic diagram of the simplified life cycle of green sea turtle is shown in figure 2. this scheme of the life cycle (figure 2) makes a clear distinction between adult males and females, focusing on their interaction during mating process that eventually will result in the production of eggs. after an incubation period of 7 to 9 weeks [17] the eggs develop into either female or male sea turtles. it also shows the outflows of each stage: the death of adult females, males, and eggs respectively. successful mating of males and females will produce and increase the egg population which in turn will develop into males and females. what determines the proportion of eggs that develop as males or females is the incubation temperature of the eggs. the reproductive biology and nesting behavior of sea turtles comprises a number of aspects that make modeling this interaction truly challenging. the interaction itself between males and females involves a variety of factors that can strongly affect the birth rate of the population and therefore the long-term dynamics. density of adult female (af ) and male populations (am ), the behavioral responses during mating process, and searching efficiency [18], are some of the variables to take into account for a successful mating. this study is particularly concerned on the proportion of eggs allocated to males and females in the population. at time t, we denote the egg population as e(t) and adult population for males and females as am (t), af (t) respectively. the dynamics of the egg population is governed by the following first order ordinary differential equation: de dt = g(af ,am ) − (α + µe)e, (1) where α is the maturity rate of eggs that become adult males or females. µe is the mortality rate of the egg stage. the function g(af ,am ) is the recruitmant rate of eggs and is represented by: g(af ,am ) = f̂(am + af ) ( 1 − am + af k ) . for this particular case, function g is dependent on the size of the population, so it is defined by a logistic type function, where f̂ is the fecundity rate, k is the environmental carrying capacity, the maximum population size that the environment can sustain indefinitely. the saturation constraint shows that when the adult population approaches the carrying capacity, then the egg per-capita approaches zero. biomath 10 (2021), 2107293, http://dx.doi.org/10.11145/j.biomath.2021.07.293 page 3 of 10 http://dx.doi.org/10.11145/j.biomath.2021.07.293 c herrera, c duque, h leiva, qualitative analysis of a mathematical model about population of ... fig. 2. schematic diagram of the mathematical model. for changes in adult male and female population over time linear ordinary differential equations were developed: dam dt = qαem −µam, (2) daf dt = (1 −q)αef −µaf , (3) where the proportion of male and female eggs are given by q and 1−q receptively. µ tell us the death rate for the adult stage. as there is not evidence of different maturity and death rates for males and females, α and µ remains the same for both sexes. from (1) to (3) the next system of differential equations is proposed:   dam dt = qαe −µam ≡ f1(am,af ,e) daf dt = (1 −q)αe −µaf ≡ f2(am,af ,e) de dt = f̂ (af + am ) ( 1 − am + af k ) − (α + µe) e ≡ f3(am,af ,e) (4) iii. qualitative analysis of the model in this section we will give a complete qualitative description of the dynamics of system (4), concretely, we characterize the region where the system is positively invariant and we shall describe completely the global stability of the nontrivial equilibrium point. since the vector field of system (4) is continuously differentiable, our first result, follows from the fundamental existence-uniqueness theorem (see for instance l. perko [24], hale [25]). theorem 3.1: for any initial condition am (0) ≥ 0, af (0) ≥ 0, and e(0) ≥ 0, there exists β > 0 such that system (4) has a unique solution defined on [0,β). the next theorem guarantees that the system (4) is biologically well behaved and that the dynamic of the system is concentrated on a bounded region of r3+. concretely, the following results holds: theorem 3.2: suppose that f̂ < 2µ(α+µe) α , then the region ω defined by ω = { (am,af ,e) ∈ r3 : 0 < af < k, 0 < am < k −af , 0 < e < µk 2α } (5) is positively invariant under the flow induced by (4) (see figure 3). biomath 10 (2021), 2107293, http://dx.doi.org/10.11145/j.biomath.2021.07.293 page 4 of 10 http://dx.doi.org/10.11145/j.biomath.2021.07.293 c herrera, c duque, h leiva, qualitative analysis of a mathematical model about population of ... fig. 3. region where (4) is positively invariant proof: in order to proof this theorem, we will analyze the vector field defined by (4) on ∂ω, this analyze is made in the following table. note that the prism given by ω has five faces and six vertices. i) am ∈ (0,k), af = 0, e = 0 a′m = −µam < 0, a′f = 0, e′ = f̂ ( 1 − am k ) am > 0 ii) am = 0, af ∈ (0,k), e = 0 a′m = 0, a′f = −µaf < 0 e′ = f̂ ( 1 − af k ) af > 0 iii) am = 0, af = 0, e ∈ (0, µk2α ) a′m = qαe > 0 a′f = (1 −q)αe > 0 e′ = −(α + µe)e < 0 iv) am = k, af = 0, e ∈ (0, µk2α ) a′m = −µk + qαe ≤− µk 2 < 0, a′f = (1 −q)αe > 0, e′ = −(α + µe)e < 0, v) am = 0, af = k, e ∈ (0, µk2α ) a′m = −qαe > 0, a′f = −µk + (1 −q)αe ≤− µk 2 < 0 e′ = −(α + µe)e < 0, vi) am = 0, af ∈ (0,k), e = µk2α a′m = qµk 2 > 0, a′f = no matter sign e′ = −(α + µe) µk2α + f̂ ( 1 − af k ) af ≤−(α + µe) µk2α + f̂ k 4 < 0 vii) am ∈ (0,k), af = 0, e = µk2α a′m = no matter sign a′f = (1 −q) µk 2 > 0 e′ = −(α + µe) µk2α + f̂ ( 1 − am k ) am ≤−(α + µe) µk2α + f̂ k 4 < 0 viii) am ∈ (0,k), af = k −am , e = 0 a′m = −µam < 0, af = −µaf < 0, e′ = 0 ix) am ∈ (0,k), af = k −am , e = µk2α note that 〈1, 1, 0〉 is a normal vector of the plane am + af = k. therefore, 〈f1,f2,f3〉 · 〈1, 1, 0〉 = −µk2 < 0, and e′ = −(α + µe) µk2α < 0. so, 〈f1,f2,f3〉 is directed to the interior of ω x) am ∈ (0,k), af = 0, e ∈ (0, µk2α ) a′m = no matter sign a′f = (1 −q)αe > 0 e′ = no matter sign xi) am = 0, af ∈ (0,k), e ∈ (0, µk2α ) a′m = qαe > 0, a′f = no matter sign, e′ = no matter sign. xii) am ∈ (0,k), 0 < af < k −am , e = 0 a′m = no matter sign a′f = no matter sign e′ = f̂ ( 1 − am +af k ) (am + af ) > 0 xiii) am ∈ (0,k), 0 < af < k −am , e = µk2α a′m = no matter sign a′f = no matter sign e′ = −(α + µe) µk2α + f̂ ( 1 − am +af k ) (am + af ) ≤−(α + µe) µk2α + f̂ k 4 < 0 xiv) am ∈ (0,k), af = k −am , e ∈ (0, µk2α ) this case is analogous to (ix) therefore, we have that the vector field given by the right side of system (4) on the boundary of ω is directed to the interior of the set ω, in consequence solutions with initial data in ω remain there for any t ≥ 0. this conclude the proof. � remark 1: from theorem 3.2 we deduced that the solutions of system (4) are defined for all t ≥ 0. hereafter we will assume that f̂ < 2µ(α+µe) α . the equilibria of system (4) are obtained by the solutions of the following algebraic equations   qαe −µam = 0 (1 −q)αe −µaf = 0( 1 − am + af k ) f̂ (af + am ) − (α + µe) e = 0 (6) so, the equilibria of (4) consist of one trivial critical point i0 = (0, 0, 0), that always exists, and a unique nontrivial critical point i∗ = biomath 10 (2021), 2107293, http://dx.doi.org/10.11145/j.biomath.2021.07.293 page 5 of 10 http://dx.doi.org/10.11145/j.biomath.2021.07.293 c herrera, c duque, h leiva, qualitative analysis of a mathematical model about population of ... (a∗m,a ∗ f ,e ∗) which exists if and only if the following condition is true f̂ > µ(α + µe) α (7) in this case, we have a∗m =qk ( 1 − µ (α + µe) f̂α ) , a∗f =(1 −q)k ( 1 − µ (α + µe) f̂α ) , e∗ =µk ( 1 − µ (α + µe) f̂α ) . (8) a. local stability of equilibrium points in this subsection we shall discuss the local stability properties of the equilibria i0 and i∗. theorem 3.3: the equilibrium point i0 is locally asymptotically stable if f̂ < µ(α+µe) α proof: the jacobian matrix of system (4) about the equilibrium point i0 is given by j(i0) =   −µ 0 qα 0 −µ (1 −q) α f̂ f̂ −α−µe   (9) and the characteristic polynomial associated to (9) is p(λ) = a0λ 3 + a1λ 2 + a2λ + a3, (10) where a0 = 1, a1 = α + µe + 2µ, a2 = (−f̂α + µ(α + µe)) + µ(α + µ + µe) and a3 = µ(−αf̂ + µ(α + µe)). since a1, a2 and a3 are positive and a1a2 −a3 = (α + µe + 2µ)(−f̂α + µ(α + µe)) + µ(α + µ + µe)−µ(−αf̂ + µ(α + µe)) > 0, so by the routh-hurwitz criterion (see [16]) we have that the equilibrium point i0 is locally asymptotically stable. � theorem 3.4: the equilibrium point i∗ is local asymptotically stable if and only if f̂ > µ(α+µe) α . proof: the jacobian matrix of system (4) about the equilibrium point i∗ is given by j(i∗) =   −µ 0 qα 0 −µ (1 −q) α γ∗ γ∗ −α−µe   , (11) where γ∗ = 2µ(α + µe) α − f̂. after a simple calculation, one finds the characteristic polynomial is p(l) = a0λ 3 + a1λ 2 + a2λ + a3, (12) where a0 = 1, a1 = µe + α + 2µ, a2 = f̂α + µ2, and a3 = µ(f̂α−µ(α + µe)). since a0, a1 and a3 are positive and a1a2−a3 = (µe + α + 2µ)(f̂α + µ 2) −µ(f̂α−µ(α + µe)) > 0, then we have by the routh-hurwitz criterion (see [16]) that the equilibrium point i∗ is locally asymptotically stable for system (4). therefore i∗ is locally asymptotically stable if and only if it exist, i.e. if and only if f̂ > µ(α+µe) α . � b. global stability analysis now, we are ready to prove that under some conditions the non trivial equilibrium point is globally asymptotically stable, this is done transforming system (4) into a planar system and applying bendixon’s criterion first, and next poincare bendixon’s theorem. after that, we go back to system (4) and prove the global asymptotic stability of the three dimensional equilibrium point. theorem 3.5: if f̂ > µ(α+µe) α , then the equilibrium point i∗ is global asymptotically stable in ω; otherwise i0 is global asymptotically stable. proof: let (am (0),af (0),e(0)) be an arbitrary initial data on ω. if we make the change of biomath 10 (2021), 2107293, http://dx.doi.org/10.11145/j.biomath.2021.07.293 page 6 of 10 http://dx.doi.org/10.11145/j.biomath.2021.07.293 c herrera, c duque, h leiva, qualitative analysis of a mathematical model about population of ... variable a = am + af , then system (4) can be reduced to the following bidimensional system   da dt = αe −µa := g1(a,e) de dt = f̂ ( 1 − a k ) a − (α + µe) e := g2(a,e). (13) since the vector field of system (13) is directed to the interior of the region ω̃ = { (a,e) ∈ r2 : 0<a<k, 0<e< µk 2α } , we have that ω̃ is positively invariant (see figure 4) fig. 4. region where (4) is positively invariant note that (am (0) + af (0),e(0)) ∈ ω̃ and the system (13) has the equilibrium points ĩ0 = (0, 0) and ĩ∗ = (a∗,e∗) where a∗ = αµe ∗ and e∗ is given by (8). since div(g1,g2) = −(µ + α + µe) < 0 we get, by using the bendixon’s criterion (see [24]), that system (13) has not periodic solutions in ω̃. on the other hand, the eigenvalues of the linearized system around ĩ∗ associated to (13) are given by λ= 1 2 ( −(µ+α+µe) ± √ (µ+α+µe)2−4αf̂ ) , which have negative real part (<(λ) < 0). so, ĩ∗ is locally asymptotically stable, and applying poincare-bendixon’s theorem (see [25]), we conclude that the equilibrium point ĩ∗ is globally asymptotically stable on ω̃. that is to say, lim t→∞ a(t) = a∗ and lim t→∞ e(t) = e∗. now, by using the first equations of system (4), we get that am (t) = am (0)e −µt + e−µt ∫ t 0 qαe(s)eµsds. on the other hand, given ε = e∗/2, there exists τ > 0 such that e(t) > e∗/2 for t > τ; hence∫ t 0 qαe(s)eµsds= ∫ τ 0 qαe(s)eµsds+ ∫ t τ qαe(s)eµsds ≥ ∫ τ 0 qαe(s)eµsds+ ∫ t τ qα e∗ 2 eµsds−→∞ as t→∞. therefore, by using the l’hospital rule, we have that lim t→∞ am (t) = lim t→∞ am (0)e −µt + lim t→∞ e−µt ∫ t 0 qαe(s)eµsds qα µ e∗= a∗m. analogously, we obtain that lim t→∞ af (t) = (1 −q)α µ e∗ = a∗f . so, the equilibrium point i∗ = (a∗m,a ∗ f ,e ∗) is globally asymptotically stable in ω. if µ ≥ f̂α α+µe , then i0 is the unique point of equilibrium of system (4). following an analogous reasoning to the one above, we have that i0 is globally asymptotically stable. this conclude the proof. � iv. numerical simulation in this section is devoted to show numerical examples that illustrate our results. in this sense we will use table i, and several initial conditions. if let us pick f̂ = 0.25 and k = 1, then the nontrivial critical point i∗ = (0.24385, 0.24385, 0.025361) is a global attractor (see figure 5) biomath 10 (2021), 2107293, http://dx.doi.org/10.11145/j.biomath.2021.07.293 page 7 of 10 http://dx.doi.org/10.11145/j.biomath.2021.07.293 c herrera, c duque, h leiva, qualitative analysis of a mathematical model about population of ... table i parameters description and values considering the galapagos islands and ecuatorian mainland green sea turtle populations. parameter description estimated values reference α maturity rate of green turtle 0.54 ±0.45 [20] µ per capita death rate for adults 0.052 ± 0.005 [21] µe per capita death rate for eggs 0.79 ± 0.19 [22] q proportion of eggs that become male 0.5 [23] f̂ fertility rate 0.35 ± 0.15 (estimated) fig. 5. solutions of system (4) with f̂ = 0.25 and k = 1 if let us pick f̂ = 0.01 and k = 1, then the trivial critical point i0 is the global attractor (see figure 6). fig. 6. solutions of system (4) with f̂ = 0.01 and k = 1 v. conclusion and final remark the model developed in this paper is characterized by studying the population dynamics through a sex-structured continuous time model. unlike conventional discrete age-structured models, in this paper we seek, through the distinction between sexes, to determine how the different parameters described influence population dynamics and its stability. from the above results of the qualitative analysis of the model, we infer that the condition established determine the persistence and stability of the green turtle population or its extinction. in terms of the biological significance of the proposed condition, we can deduce that the number of offspring a female individual produce over time should be biger than the outflow rates of the population stages. this model can be used as a basis for a more complex models and include a series of parameters that can offer us a better appreciation of population dynamics and how it is affected or distorted by the influence of new parameters, such as temperature. since the proportion of males and females is directly related to the incubation temperature of the eggs, modeling the population dynamics by considering the proportion of male eggs as a function of temperature q(t) and adding details about other stages of the turtle life cycle could reveal new conditions, which help to envision the population situation and take actions for the conservation of the species. the methodology applied here to study the dynamics of the population of green turtles on the galapagos island, can be applied to study the dynamics of other types of biomath 10 (2021), 2107293, http://dx.doi.org/10.11145/j.biomath.2021.07.293 page 8 of 10 http://dx.doi.org/10.11145/j.biomath.2021.07.293 c herrera, c duque, h leiva, qualitative analysis of a mathematical model about population of ... reptiles whose reproduction also depends on the environment and temperature changes. in all these cases, the region of biological interest will be of the polyhedral type, which allows us to study the vector field on each face of the region, and the invariance of this region can be achieved in the same way we did it with the prism in this work. after that, we can study the local stability of the equilibrium points by linearizing the differential equation around them, and looking at the sign of the eigenvalues; then reduce the system to a twodimensional one and apply bendixon’s criterion, and finish it with the poincare-bendixon theory to conclude that the non-trivial equilibrium point is globally asymptotically stable. from the foregoing comments, it is clear that our method can be used to study a broad class of similar problems. acknowledgment we would like to thank the anonymous reviewer who carefully read the paper, his/her comments and suggestions allowed us to improve the final presentation of this manuscript. references [1] s.j. morreale, g.j. ruiz and e.a. standora, temperature-dependent sex determination: current practices threaten conservation of sea turtles. science, 216 (4551), 1245-1247, 1982. 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[6] y. matsumoto, and d. crews, molecular mechanisms of temperature-dependent sex determination in the context of ecological developmental biology, molecular and cellular endocrinology, 354 (1-2), 103-110, 2012. [7] s.y. özdilek, b.e. sönmez, and y. kaska, sex ratio estimations of chelonia mydas hatchlings at samandağ beach, turkey, turkish journal of zoology, 40 (4), 552560, 2016. [8] l.i. wright, k.l. stokes, w.j. fuller, b.j. godley, a. mcgowan, r. snape and a.c. broderick, turtle mating patterns buffer against disruptive effects of climate change, proceedings of the royal society b: biological sciences, 279 (1736), 2122-2127, 2012. [9] n. mrosovsky and c.l. yntema, temperature dependence of sexual differentiation in sea turtles: implications for conservation practices, biological conservation, 18 (4), 271-280, 1980. 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[14] a.b. bolten, and g.h. balazs, biology of the early pelagic stage—the ‘lost year.’. biology and conservation of sea turtles, revised edition, smithsonian institute press, washington, dc, 579, 1995. [15] a. carr, notes on the behavioral ecology of sea turtles. biology and conservation of sea turtles, smithsonian institution press washington, dc, 19-23, 1982. [16] f. gantmacher, the theory of matrices, vol 2, chelsea publishing, new york, 1974. [17] d. green, the east pacific green sea turtle in galapagos, noticias de galápagos, charles darwin research station (28), 9-12, 1978 [18] h.f. hirth and d.a. samson, nesting behavior of green turtles (chelonia mydas) at tortuguero, costa rica. comportamiento de anidamiento de las tortugas verde (chelonia mydas) en tortuguero, costa rica, caribbean journal of science, 23 (3/4), 374-379, 1987. [19] p.l. lee, p. luschi and g.c. hays, detecting female precise natal philopatry in green turtles using assignment methods, molecular ecology, 16(1), 61-74, 2007. [20] d. green, growth rates of wild immature green turtles in the galapagos islands, ecuador, journal of herpetology, 338-341, 1993. [21] j.j. alava, p. jiménez, m. peñafiel, w. aguirre and p. amador. sea turtle strandings and mortality in ecuador: 1994-1999, marine turtle newsletter, 108, 4-7, 2005. biomath 10 (2021), 2107293, http://dx.doi.org/10.11145/j.biomath.2021.07.293 page 9 of 10 http://dx.doi.org/10.11145/j.biomath.2021.07.293 c herrera, c duque, h leiva, qualitative analysis of a mathematical model about population of ... [22] p. zárate, p, k.a. bjorndal, m. parra, p.h. dutton, j.a. seminoff, and a.b. bolten, hatching and emergence success in green turtle chelonia mydas nests in the galápagos islands, aquatic biology, 19 (3), 217-229, 2013. [23] s. zavala montoya, j. belmont, m. hirschfeld and d. alarcón, estimación de la proporción de sexos de latortuga verde (chelonia mydas) en áreas de alimentación en las islas galápagos, in simposio de tortugas marinas de ecuador 2018. [24] l. perko, differential equations and dynamical systems, texts in applied mathematics 7, springer-verlag, 1993. [25] j.k. hale, ordinary differential equations, dover publications, mineola, new york, 2009. biomath 10 (2021), 2107293, http://dx.doi.org/10.11145/j.biomath.2021.07.293 page 10 of 10 http://dx.doi.org/10.11145/j.biomath.2021.07.293 introduction model formulation qualitative analysis of the model local stability of equilibrium points global stability analysis numerical simulation conclusion and final remark references original article biomath 1 (2012), 1209252, 1–7 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum growth in a turing model of cortical folding gregory toole∗, monica k. hurdal∗ ∗ department of mathematics, florida state university, tallahassee, fl 32306-4510, usa emails: gtoole@math.fsu.edu, mhurdal@math.fsu.edu received: 12 july 2012, accepted: 25 september 2012, published: 15 october 2012 abstract—the brain’s cerebral cortex is folded into many gyri (hills) and sulci (valleys). little is known about how the cortex folds or why the folds are located where they are. we have developed a spatio-temporal mathematical model of cortical folding to address this question. our model utilizes a turing reaction-diffusion system on an exponentially growing prolate spheroidal domain. this domain approximates the shape of the lateral ventricle (lv) during cortical development. the intermediate progenitor model (ipm) of cortical folding states that regional patterning of self-amplication of intermediate progenitor cells (ipcs) in the subventricular zone of the lv corresponds with the formation of cortical folding. as selfamplication of ipcs is genetically controlled via chemical gradients, a turing system is a logical choice to create a mathematical representation of the ipm. a growing domain model of cortical folding may be more realistic than previous static domain models of cortical folding since it incorporates the growth that naturally occurs as the brain develops. by comparing patterns generated by our growing prolate spheroid turing system with those generated by a static prolate spheroid turing system, we show that the addition of growth causes a significant change in system behavior; the system produces transient patterns instead of converging to one final pattern. our model illustrates the importance of including growth in a model of cortical folding and can be utilized to explain certain human diseases of cortical folding. keywords-cortical folding; morphology; neurobiology; turing system i. introduction the cerebral cortex of the brain is folded into an intricate pattern of gyri (hills) and sulci (valleys). the pattern of cerebral cortical folds varies from species to species as well as between individuals of the same species. current biological models that attempt to explain the underlying processes of cortical folding conflict with one another; some emphasize the role of physical tension created by axonal connections within the cortex, while others highlight the importance of genetic chemical factors that influence cortical cells and their precursors. furthermore, it is extremely difficult to perform neuroscience experiments to investigate cortical folding in living humans. because of this biological debate and lack of experimental data, we have created a spatio-temporal mathematical model of cortical folding patterns in the brain. our model employs a turing reaction-diffusion system on an exponentially growing prolate spheroidal domain. turing systems have been used to mathematically model pattern formation in many different areas of biological development, such as zebra stripes, giraffe spots, and alligator tooth formation [13], [15]. previous biomathematical models of cortical folding employed turing systems on a static domain [26], failing to capture the growth of the organism that naturally occurs as development progresses. our model employs a growing domain, allowing us to create a more biologically realistic model of cortical folding by incorporating developmental growth. numerical simulations demonstrate that the inclusion of domain growth in a turing system causes a fundamental change in the pattern-generating behavior of the system, causing it to generate transiently evolving patterns rather than one convergent pattern. incorporating domain growth into a turing system model of cortical folding also allows one citation: g. toole, m. hurdal, growth in a turing model of cortical folding, biomath 1 (2012), 1209252, http://dx.doi.org/10.11145/j.biomath.2012.09.252 page 1 of 7 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.09.252 g. toole et al., growth in a turing model of cortical folding to model certain diseases of cortical folding. ii. biological background the ventricular system of the brain consists of four ventricles: the two lateral ventricles (lvs), the third ventricle, and the fourth ventricle [25]. the ventricular zone (vz) lines the lateral wall of the lvs and contains special proliferative cells that play a role in cortical development [17]. continuing outward from the inside of a lv, one passes from the vz to the subventricular zone (svz), another area containing proliferative cells implicated in cortical development [17]. according to the intermediate progenitor model (ipm) of cortical folding [12], intermediate progenitor cells (ipcs) in the svz undergo self-amplifying cell division with the number of rounds of cell division varying regionally throughout the svz. ipcs then divide into neurons which populate the upper layers of the cortex. areas of high ipc selfamplification lead to cortical areas highly populated with neurons, forming gyri, while areas of low ipc selfamplfication lead to cortical areas with fewer neurons, forming sulci. ipcs and hence cortical folding are regulated via a genetic chemical gradient; namely, pax6 and wnt have been shown to affect the number of ipcs and proper cortical development in mice [22]. another important role of the svz in brain development is the production of a structure called the germinal matrix (gm), which is also located along the lateral wall of the lvs [1], [11]. the gm contains precursors of neurons and glial cells and has been observed to grow exponentially from 11 to 23 weeks gestational age (ga) [1], [11]. the period of gm exponential growth overlaps with a period of development during which cortical folding occurs in humans, as primary cortical folds in humans emerge from 10 weeks ga to 30 weeks ga [9]. iii. turing systems turing reaction-diffusion systems are activatorinhibitor systems originally created to model chemical gradient concentrations on the developing embryo [28]. let u (x, t) and v (x, t) represent the concentrations at time t ≥ 0 of an activator morphogen and an inhibitor morphogen which are interacting on a static domain parametrized by position vector x. then the canonical turing system is ∂u ∂t = du∇2u + f (u, v), ∂v ∂t = dv∇2v + g(u, v),   (1) where 0 < du < dv are the respective diffusion coefficients of u and v and f, g are the reaction kinetics. if system (1) possesses a spatially uniform steady state (u0, v0) which is linearly stable in the absence of diffusion but is driven unstable by noise when diffusion is present, then system (1) is capable of generating spatially inhomogeneous patterns. these two properties must be satisfied in order for system (1) to exhibit turing systems’ characteristic pattern-generating behavior [15], [28]. we refer to these properties as turing criteria. since we wish to include developmental growth as a part of our model of cortical folding, a growing domain must be incorporated into system (1). to accomplish this, let st ⊂ r3 be a two-dimensional regular growing surface with position vector x = x (ζ, η, t), where ζ, η parametrize st in space and t ≥ 0. let u (x, t) and v (x, t) be the concentrations of two chemical substances on st with diffusion coefficients du and dv, respectively. if we define d = du/dv, h1 = |xζ| , h2 = |xη|, then system (1) becomes ut = d∆su − ∂t (ln (h1h2)) u + ωf (u, v) , vt = ∆sv − ∂t (ln (h1h2)) v + ωg (u, v) ,   (2) where ω > 0 is the domain scale parameter and ∆sφ = 1 h1h2 [( h2 h1 φζ ) ζ + ( h1 h2 φη ) η ] (3) is the laplace-beltrami operator on st (with φ = u, φ = v) [21]. notice that the incorporation of domain growth into a turing system results in a third term in each reaction-diffusion equation: −∂t (ln (h1h2)) φ (for φ = u, φ = v). this new term represents dilution of the u, v concentrations due to the growth of the domain [21]. system (2) can construct a turing system on any regular growing surface, but since the growing domain of interest for our model is an exponentially prolate spheroid, we shall incorporate prolate spheroidal coordinates into the system. a prolate spheroid is the result of rotating an ellipse about its major axis and can be defined by the prolate spheroidal coordinate system [8], x = f 2 √ (1 − η2) (ξ2 − 1) cos 2πζ, y = f 2 √ (1 − η2) (ξ2 − 1) sin 2πζ, z = f 2 ηξ, where ξ > 1 controls domain shape via eccentricity, η = cos θ ∈ [−1, 1] with polar angle θ, ζ = φ 2π ∈ [0, 1) with azimuthal angle φ, and f is the interfocal distance. we biomath 1 (2012), 1209252, http://dx.doi.org/10.11145/j.biomath.2012.09.252 page 2 of 7 http://dx.doi.org/10.11145/j.biomath.2012.09.252 g. toole et al., growth in a turing model of cortical folding define the position vector x on an exponentially growing prolate spheroid as x(ζ, η, t) = ρ(t)   f0 2 √ (ξ2 − 1)(1 − η2) cos 2πζ f0 2 √ (ξ2 − 1)(1 − η2) sin 2πζ f0 2 ξη   , (4) where f0 is the interfocal distance at t = 0 and ρ (t) = ert is the growth function with growth rate r > 0. using equation (4), the laplace-beltrami operator given in equation (3) becomes ∆s = 1 π2ρ2f 20 (1 − η2) (ξ2 − 1) φζζ + 4 ( 1 − η2 ) ρ2f 20 (ξ 2 − η2) φηη − 4η ( 2ξ2 − η2 − 1 ) ρ2f 20 (ξ 2 − η2)2 φη, and the dilution term reduces to −∂t (ln (h1h2)) φ = −2 ρ̇ ρ φ = −2rφ. overall, system (2) on an exponentially growing prolate spheroidal domain becomes ut = d∆su − 2ru + ωf (u, v), vt = ∆sv − 2rv + ωg(u, v), } (5) where u = u(ζ, η, t) and v = v(ζ, η, t). iv. turing conditions next, we derive mathematical conditions whose satisfaction ensures that system (5) satisfies the two turing criteria and thus can generate patterns. these mathematical conditions are called turing conditions and are derived using linear stability analysis in the method of [10]. a. turing criterion: linear stability to begin, we rewrite system (5) as ut = dρ2 ∆†u − 2ru + ωf (u, v), vt = 1ρ2 ∆†v − 2rv + ωg(u, v), } (6) where ∆† = ρ2∆s. let (u0, v0) be a spatially uniform steady state of system (5) which remains a steady state in both the presence and absence of diffusion; in other words, 0 = −2ru0 + ωf (u0, v0), 0 = −2rv0 + ωg(u0, v0). if w(t) = ( u(t) − u0 v(t) − v0 ) = ( �u �v ) is defined to be a perturbation from (u0, v0), then system (6) can be rewritten as wt = ( ut vt ) = ( −2r(u0 + �u) + ωf (u0 + �u, v0 + �v) −2r(v0 + �v) + ωg(u0 + �u, v0 + �v) ) . performing a taylor expansion of ut, vt around (u0, v0) allows us to write ut ≈ −2r�u + ω [�ufu(u0, v0) + �vfv(u0, v0)] , vt ≈ −2r�v + ω [�ugu(u0, v0) + �vgv(u0, v0)] , from which it follows that wt = −2rw + ωaw, (7) where a = ( fu fv gu gv ) (u0,v0) . next, consider solutions to equation (7) with form w (t) = ceλt. in order to achieve linear stability of (u0, v0) , it must follow that w → 0 as t → ∞. hence, λ must satisfy re (λ) < 0. substituting w (t) = ceλt into equation (7) and simplifying yields the eigenvalue equation λc = ãc, where ã = ωa − 2ri. solving for the characteristic polynomial in λ and using the quadratic formula implies that re (λ) < 0 when trã = ω(fu + gv) − 4r < 0 and det ã = ω2(fugv − fvgu) − 2rω(fu + gv) + 4r2 > 0. these two inequalities constitute the first two turing conditions for system (5). b. turing criterion: diffusion-driven instability linearizing system (6) about (u0, v0) gives wt = dm ∆†w − 2rw + ωaw, (8) where dm = 1 ρ2 ( d 0 0 1 ) . consider solutions to equation (8) with form w(x, t) = ∑ k cke λtyk (x) , (9) biomath 1 (2012), 1209252, http://dx.doi.org/10.11145/j.biomath.2012.09.252 page 3 of 7 http://dx.doi.org/10.11145/j.biomath.2012.09.252 g. toole et al., growth in a turing model of cortical folding where yk are prolate sheroidal harmonics. since yk satisfy ∆†yk = −k2yk, substituting equation (9) into equation (8) and simplifying gives∑ k ck ( λyk + dm k 2yk + 2ryk − ωayk ) = 0. for nontrivial solutions w it must be that ck 6= 0. it then follows that λyk = ( −dm k2 − 2ri + ωa ) yk, which is another eigenvalue equation. nontrivial w occur when det ( ã − dm k2 − λi ) = 0, and evaluating this determinant yields λ2 + λ [ k2 ρ2 (1 + d) − trã ] + h ( k2 ) = 0, (10) where h ( k2 ) = d ρ4 ( k2 )2 + det ã + k2 ρ2 [2r (1 + d) − ω (fu + dgv)] . recalling that w (x, t) = ∑ k cke λtyk (x) , it follows that diffusion-driven instability of (u0, v0) occurs when re (λ) > 0. solving equation (10) for λ tells us that re (λ) > 0 when 2r (1 + d) − ω (fu + dgv) < 0 and r2 [ 4 − (1 + d)2 d ] + ω2 (fugv − fvgu) + rω [ 1 d (1 + d) (fu + dgv) − 2 (fu + gv) ] < ω2 4d (fu + dgv) 2 . these two inequalities constitute the final two turing conditions for system (5). v. the model our mathematical model of cortical folding utilizes system (5), a turing reaction-diffusion system on an exponentially growing prolate spheroidal domain. we select an exponentially growing prolate spheroidal domain to represent the lv and use the surface of the domain to represent the svz. patterns generated by the model’s turing system represent regions of activation and nonactivation for self-amplification of ipcs in the svz; this regional self-amplification of ipcs then leads to cortical folds as described by the ipm. since turing systems were originally created to model chemical morphogen concentration gradient patterns, they are useful for modeling developmental phenomena involving patterns of genetic chemical gradients. as mentioned in section ii, evidence suggests that genes regulate cortical folding via a chemical gradient, making a turing system a reasonable choice for a mathematical model of cortical folding. recall that the gm grows exponentially over a period of development during which cortical folds are formed. since the gm is produced by the svz and the svz is the site of self-amplification of ipcs (a key factor in the ipm), an exponentially growing domain is a reasonable choice for our model. neurogenesis in humans occurs approximately during embryonic days 43 to 120 [23]. early in neurogenesis, the cerebral hemispheres are prolate spheroidal in shape, with the lvs accounting for almost all of the cerebral hemispheres’ volume [26]. thus, the lvs at this time of development are also prolate spheroidal in shape, making a prolate spheroidal domain a reasonable choice for our model. to fully define our model’s turing system given in system (5), we must select reaction kinetics functions f (u, v), g(u, v). we select nondimensional barrio-vareamaini (bvm) kinetics [4], f (u, v) = u + av − cuv − uv2, g(u, v) = bv + hu + cuv + uv2, so that system (5) becomes ut = d∆su − 2ru + ω(u + av − cuv − uv2), vt = ∆sv − 2rv + ω(bv + hu + cuv + uv2). } (11) bvm kinetics are phenomenological and are not modeled after any particular physical or chemical mechanism. since the underlying mechanism of cortical folding is not fully understood, bvm kinetics are an appropriate choice for our model. a. numerical results to observe the patterns produced by system (11), we performed numerical simulations using a forward time, central space finite difference scheme [14]. kinetics parameters were selected from the literature and were as follows: d = 0.516, a = 1.112, b = −1.01, c = 0. initial conditions consisted of random values φ ∈ [−0.5, 0.5] along the domain’s equator; other points on the domain were initialized as zero. the random initial values were seeded due to turing systems’ intrinsic high biomath 1 (2012), 1209252, http://dx.doi.org/10.11145/j.biomath.2012.09.252 page 4 of 7 http://dx.doi.org/10.11145/j.biomath.2012.09.252 g. toole et al., growth in a turing model of cortical folding fig. 1. transient patterns generated by system (11) with r = 0.009, ω = 150. fig. 2. transient patterns generated by system (11) with r = 0.045, ω = 150. sensitivity to initial conditions [29]. initial interfocal distance was chosen to be f0 = 2 and domain shape was fixed by fixing ξ at ξ = 1.3141, giving a domain with initial surface area 4π, identical to that of the unit sphere. simulations were varied only by altering the growth rate parameter r and the domain scale parameter ω. system (11) generated transient patterns that constantly evolve from one pattern to another as elapsed time t progresses (see figures 1–3). as these transient patterns evolve, the number of stripes or spots in the pattern increases with increasing t. the patternfig. 3. transient patterns generated by system (11) with r = 0.045, ω = 60. generating behavior of our growing domain system sharply contrasts with that of a static domain turing system, which eventually converges to one final pattern. it is therefore clear that the addition of growth to a turing system causes a major change in the system’s pattern-generating behavior. increasing the value of r or ω causes system (11) to generate a more complex pattern (more stripes or spots) at a given t (compare figure 1 with figure 2 and compare figure 2 with figure 3). increasing these parameters also increased the frequency of transient pattern change; that is, the system evolved from pattern to pattern more quickly. b. application to diseases of cortical folding 1) polymicrogyria: polymicrogyria (pmg) is a disease of cortical folding in which the cortex is excessively folded into many small folds [2]. common symptoms of pmg include mental retardation, epilepsy, and developmental delay [2]. several different forms of pmg are associated with enlarged lvs, such as megalencephaly pmg with polydactyly and hydrocephalus (mpph) [7], [20], unilateral pmg [19], and bilateral frontoparietal pmg (bfpp) [6], [27]. our model can encapsulate the enlargement of lvs by increasing the growth rate r, leading to a larger prolate spheroid (and hence lv) at any t > 0. increasing the growth rate r leads to smaller and more numerous stripes in the pattern produced by system (11) (see figure 2). by interpreting the stripes as locations and sizes of cortical folds (as described in section v), this change in size and number of stripes biomath 1 (2012), 1209252, http://dx.doi.org/10.11145/j.biomath.2012.09.252 page 5 of 7 http://dx.doi.org/10.11145/j.biomath.2012.09.252 g. toole et al., growth in a turing model of cortical folding represents an increased number of small cortical folds, agreeing with the characteristic brain abnormalities of pmg. 2) norman-roberts syndrome: norman-roberts syndrome (nrs) is a rare, potentially fatal congenital disease affecting brain development [16], [18], [24]. the genetic basis and pathophysiology of nrs are not fully known. symptoms of nrs include reduced head growth rate, microcephaly, type i lissencephaly (too few cortical folds), epilepsy, and mental retardation [3], [5], [16], [24]. since nrs is associated with microcephaly, it is reasonable to assume that the smaller-than-normal nrs brain also has smaller-than-normal lvs. we thus model nrs by decreasing r so that the domain representing the lv is smaller at any t > 0. decreasing r leads to larger and less numerous stripes in the pattern generated by system (11) (see figure 1). by once again interpreting the stripes as cortical folds, this change in size and number of stripes represents a decrease in the number of cortical folds, reproducing the distinguishing brain anomalies of nrs. vi. conclusion we have shown that it is important to consider growth when constructing a biomathematical model of cortical folding, as adding growth to our turing system model of cortical folding not only makes it more biologically realistic but also significantly changes the system’s patterngenerating behavior. furthermore, by appropriately altering the domain growth rate, a growing domain turing system model of cortical folding can model certain diseases of cortical folding. references [1] j.a. anstrom, c.r. thore, d.m. moody, v.r. challa, s.m. block, and w.r. brown,“germinal matrix cells associate with veins and a glial scaffold in the human fetal brain”, dev. brain res., vol. 160, pp. 96–100, 2005. http://dx.doi.org/10.1016/j.devbrainres.2005.07.016 [2] a.j. barkovich, “current concepts of polymicrogyria”, neuroradiology, vol. 52, pp. 479–487, 2010. http://dx.doi.org/10.1007/s00234-009-0644-2 [3] a.j. barkovich, r.i. kuzniecky, g.d. jackson, r. guerrini, and w.b. dobyns, “a developmental and genetic classification for malformations of cortical development”, neurology, vol. 65, pp. 1873–1887, 2005. http://dx.doi.org/10.1212/01.wnl.0000183747.05269.2d [4] r.a. barrio, c. varea, j.l. aragon, and p.k. maini. “a twodimensional numerical study of spatial pattern formation in interacting turing systems”, b math. biol., vol. 61 no. 3, pp. 483–505, 1999. http://dx.doi.org/10.1006/bulm.1998.0093 [5] h. caksen, o. tuncer, e. kirimi, j.p. fryns, a. uner, o. unal, a. cinal, and d. odabas, “report of two turkish infants with norman-roberts syndrome”, genet. counsel, vol. 15 no. 1, pp. 9–17, 2004. 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criterion: diffusion-driven instability the model numerical results application to diseases of cortical folding polymicrogyria norman-roberts syndrome conclusion references www.biomathforum.org/biomath/index.php/biomath original article relationship between blood superoxide dismutase activity and zinc, copper, glutathione and metallothioneines concentrations in calves vladimir safonov∗, vadim ermakov∗, valentina danilova∗, vyacheslav yakimenko† ∗vernadsky institute of geochemistry and analytical chemistry moscow, russian federation safonovladimir8@rambler.ru, vad.ermakov@rambler.ru, valentina danilova142@rambler.ru †voronezh state agricultural university named after emperor peter the great voronezh, russian federation yakimenko0987@rambler.ru received: 29 july 2021, accepted: 24 november 2021, published: 28 december 2021 abstract— redox (reduction-oxidation) processes determine the resistance of the organism to pollutants. the aim of the study was to establish a possible relationship between copper and zinc concentration in the blood of calves and the enzyme activity of superoxide dismutase. the study was conducted in 2019 on 20 calves with a weight of 201250 kg. the samples of venous blood were taken to estimate the level of hemoglobin, glutathione, metallothioneins, as well as zinc, copper, and superoxide dismutase activity. the obtained average values of these substances’ concentrations were compared between each other. a positive correlation between the activity of superoxide dismutase and the concentration of zinc (r = 0.64) and copper (r = 0.87) in the blood of calves has been established. it may be because both metals are obligatory components of superoxide dismutase. there is also a positive relationship between the levels of copper and zinc (r = 0.68). for the other parameters, no reliable relationship was found. the data obtained indicate a positive relationship between superoxide dismutase activity and metal concentrations of copper and zinc in the blood of calves. at the same time, a more significant positive relationship is established for copper. keywordsantioxidant system; calves; metallothioneins; superoxide dismutase i. introduction among all the metabolic processes occurring in animals, one of the key roles is assigned to reduction-oxidation (redox) processes intended copyright: © 2021 safonov et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: vladimir safonov, vadim ermakov, valentina danilova, vyacheslav yakimenko, relationship between blood superoxide dismutase activity and zinc, copper, glutathione and metallothioneines concentrations in calves, biomath 10 (2021), 2111247, http://dx.doi.org/10.11145/j.biomath.2021.11.247 page 1 of 9 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2021.11.247 v safonov, v ermakov, v danilova, v yakimenko, relationship between blood superoxide dismutase ... to meet the body’s energy needs. additionally, redox processes determine oxygen delivery and utilization in the tissues. therefore, oxygen radicals are constantly formed in the body, being quenched/scavenged by endogenous and exogenous antioxidants that maintain homeostasis. increased concentration of oxygen radicals can lead to lipid peroxidation (lpo), leading to negative effects, which manifests in various kinds of damage to cell membranes, organelles (lysosomes), or functional cells (erythrocytes). given the critical functions performed by membranes, organelles, and erythrocytes, the negative impact of excessive lpo should not be underestimated [1]. at the cellular level, the negative effect is expressed in structural and irreversible changes of cell membranes, leading to the death of a single cell, entire tissue, or even the entire organism. another negative characteristic of lpos excessive concentrations is the disruption of such a vital process as cellular respiration. in particular, primary lpo products (diene conjugates) are toxic metabolites, contributing to the damage of such macromolecules as proteins, enzymes, nucleic acids, and lipoproteins. another type of lpo product includes aldehydes and ketone bodies, such as malonic dialdehyde. these compounds are involved in the synthesis of prostaglandins and steroids (progesterone, etc.). the reaction between dialdehydes and free groups of membrane compounds results in the formation of peroxidation end products, such as schiff base. if the formation of such products is prolonged, their concentration increases, and their interaction with free groups of membrane compounds eventually leads to membrane destruction and the death of cells [2]. three groups of antioxidation factors were distinguished: 1) enzymes such as superoxide dismutase, catalase, and glutathione peroxidase); 2) proteins, which include ferritin, ceruloplasmin, transferrin, and albumin; and 3) low-molecularweight compounds such as vitamins (e, c, a) and other compounds (ubiquinone, carotenoids, acetylcysteine, and alipoic acid) [3]. due to the heterogeneity of the antioxidant factors, the mechanisms that regulate oxidative activity are different. in particular, superoxide dismutase has an inactivating effect on the superoxide anion as it contains metal ions at varying valence ratios. such metals include zinc, magnesium, copper, and manganese [4]. another enzyme, catalase, plays a crucial role in preventing the accumulation of hydrogen peroxide excess in cells. the latter is formed by aerobic oxidation of reduced flavoproteins. of extreme importance for studying the physiological processes occurring during the first postnatal development stages in young cattle is understanding the fight against excessive peroxidation product concentration, which may provoke the development of various pathologies [5]. there is another connection between antioxidant enzymes and micronutrients (zinc, copper, manganese, selenium), since the micronutrients mentioned above are a part of the enzymes. in particular, the active center of superoxide dismutase contains copper and manganese. superoxide dismutase, in turn, is one of the main components of the antioxidant system [6]. free radicals are well-known leading players in the pathogenesis of more than one hundred diseases when in increased concentration [7]. most of these diseases are otherwise associated with the adverse effects of environmental factors or aging processes. the increased reactivity of free radicals is associated with the presence of a freeelectron on the outer orbital of an atom. when free radicals inside the body react with macromolecules on the surface of cell membranes, the latter is destroyed. negative effects of free radicals are linked to the processes that occur in cells, such as protein phosphorylation, as well as cell proliferation or dna transcription in cell nuclei. in organs, free radicals are involved in the regulation of vascular tone. in case of increased concentration of free radicals, the synthesis of prostaglandins and metalloproteins, together with growth factors, are activated. usually, only up to 5% of the oxygen used by the body is converted into free radicals. some of them, such as nitric oxide (no), can exhibit antioxidant properties [8]. biomath 10 (2021), 2111247, http://dx.doi.org/10.11145/j.biomath.2021.11.247 page 2 of 9 http://dx.doi.org/10.11145/j.biomath.2021.11.247 v safonov, v ermakov, v danilova, v yakimenko, relationship between blood superoxide dismutase ... it is known that, when blocking highly toxic oxygen radicals formed in the process of oxygen metabolism, the enzyme superoxide dismutase (sod, ec 1.15.1.1) plays a significant role. it belongs to the group of antioxidant enzymes that catalyzes superoxide dismutation to oxygen and hydrogen peroxide and also prevents oxidation of several biologically active substances [9-11]. sod activity determination in the blood of animal species and humans possesses important diagnostic value in developing various pathologies [12,13]. sod is a metalloprotein that catalyzes the conversion of the highly active superoxide radical into less active hydrogen peroxide. three varieties of the enzyme are distinguished by the level of iron, copper, zinc, or magnesium ions [14]. copper and zinc levels play a major role in sod activity. the study of physiological processes, including redox reactions, is equally essential for both medicine and veterinary medicine. furthermore, the results obtained in veterinary medicine can be extrapolated to medicine, following similar experiments. at the same time, there have been no studies in which the role of superoxide dismutase and copper/zinc concentrations in blood have been considered and compared simultaneously. this determined the relevance of the present work. the blood of calves was analyzed to establish the relationship between the concentration of copper and zinc on the one hand, and the activity of superoxide dismutase on the other. the authors suggest that these parameters are more directly related to copper than to zinc. this work aimed to define the existence of a relationship between the concentration of copper and zinc in the blood of calves and the activity of the enzyme superoxide dismutase. considering that an insufficient number of data on the copper and zinc concentrations and sod activity comparison is available, we evaluated this interaction by analyzing the blood of calves. in addition, this study took into account the total glutathione and metallothioneins content in the blood of animal species determined by liquid chromatography and derivatization thereof using n-9 (acridinyl) maleimide. ii. materials and methods a. materials the study was carried in 2019 out at the liskinskiy apc, liskinskiy district, voronezh region in russia. young cattle species were selected weighing 201-250 kg. the study enrolled 20 animals. animal species received a ration consisting of 25 kg of silage, 2 kg of straw and 1.8 kg of combined feed. venous blood samples for laboratory tests were taken from animal species in the morning before feeding. vacuum tubes with edta-na were used for the whole blood samples. all international and national guidelines for the care and use of animals were followed. the study complies with the relevant ethical standards. it was approved at the ethical committee’s meeting of voronezh state agricultural university named after emperor peter the great (minutes of the meeting no. 554). b. analytical methods when determining hemoglobin content, 20 µl of blood were added to 4 ml of 0.04% nh4oh, and the contents were mixed by shaking for 5 minutes. after one hour, solution absorbance was measured at 523 nm in cuvettes with an optical path length of 1 cm using the 320 spectrophotometer (hitachi). hemoglobin content was identified according to the hemoglobin standard calibration chart (sigma 08449 hemoglobin from bovine blood). this method is described in the article [15]. sod activity in whole blood with heparin was evaluated by the adrenaline auto-oxidation method [16,17] after the erythrocytes separation, washing thereof with 0.9% cold sodium chloride solution, and hemolysis at 1 4oc for 15 minutes. absorption alteration was observed at 347 nm in the 320 spectrophotometers temperaturecontrolled cuvette. copper and zinc concentrations in blood were determined by atomic absorption spectroscopy (aas) in the flame version using the cortec biomath 10 (2021), 2111247, http://dx.doi.org/10.11145/j.biomath.2021.11.247 page 3 of 9 http://dx.doi.org/10.11145/j.biomath.2021.11.247 v safonov, v ermakov, v danilova, v yakimenko, relationship between blood superoxide dismutase ... llc certified device after blood mineralization with a mixture of nitric and perchloric acids. glutathione and metallothioneins content was determined by low-pressure liquid chromatography using a device with shimadzu spectrofluorometric detector in the form of fluorescent complexes after the reaction of sh-groups with n-(9-acridinyl) maleimide (nam) in columns with polymeric sorbents [18]. in this case, blood samples were mixed with an equal portion of methanol, and the clear supernatant fluid was used for analysis after homogenization and centrifugation. ss groups restoration to sh groups was carried out with sodium borohydride in the nitrogen atmosphere at 50oc. c. statistical analysis study results were processed by the variation analysis method using ms-excel 2013 software program the arithmetic mean and error of the mean were calculated for each of the studied parameters, i.e., superoxide dismutase activity, copper and zinc concentrations, glutathione, and metallothioneins. also, the variability from minimum to maximum for each of these parameters was indicated. the concentration of the studied parameters was established depending on the hemoglobin level. the correlations between the parameters, i.e., how the concentration of one compound is related to that of another (pearson correlations) was stated as well. at that, correlations greater than 0.50 were considered significant. a p ≤ 0.05 was taken as the baseline level of significantly different data. the data were tested for normality (test for distribution normality). the obtained results allowed stating that the distribution of parameters corresponded to normal, specifying parametric analysis. iii. results data on the calves’ blood components content and sod activity are presented in table 1. hemoglobin content in the blood of the clinically healthy calves varied from 104.8 to 151.7 g/l with an average of 129.2±10.9 g/l, which is considered a physiological norm. ten animal species out of 20 had hemoglobin levels approaching the 120 g/l indicators. hemoprotein content of 107-116 mg/l was found only with three calves. sod activity (in hemoglobin units/mg) varied from 0.80 to 1.51 units, but in most cases was approaching the 1 indicator (0.99±0.11). zinc concentration in the calves’ blood was noticeably varying and ranging from 1.50 to 4.58 mg/l. on average, zinc content in the calves’ blood was 2.60±0.60 mg/l. variation range reached 3 units. however, the maximum zinc concentration in blood was found only with one calf (4.58 and averaging at 1.08±0.22 mg/l. the average copper concentration value in the calves’ blood approached the most common trace element content in the blood of terrestrial mammals. table 1 also presents the results of studies on the content of total glutathione and metallothioneins in the calves’ blood. glutathione content varies from 80.3 to 255.2 mg/l, i.e., in 3 times, as in the case of trace elements. this indicates a significant difference in metabolic processes with rapidly growing animal species. data on mt concentration in the calves’ blood is presented for the first time. mt average content in the calves’ blood is 13.3 ± 2.8 mg/l. this low molecular weight metalloprotein content is significantly lower than that of glutathione and varies from 7.9 to 19.6 mg/l. the relationship between the content of separate blood components with calves is demonstrated in table 2. significant positive correlation between sod activity and zinc (r = 0.64) and especially copper (r = 0.87) concentrations in the blood of the young animal species is noteworthy. in all likelihood, this is because both trace elements are the sod essential components [12, 11]. at the same time, the relationship between the level of copper and zinc is also significant (r = 0.68). correlation between sod activity and glutathione and mt content level was not observed. however, such a relationship exists between glutathione and mt concentrations, which seems to be determined by the difference in metabolic processes of these biologically active components. biomath 10 (2021), 2111247, http://dx.doi.org/10.11145/j.biomath.2021.11.247 page 4 of 9 http://dx.doi.org/10.11145/j.biomath.2021.11.247 v safonov, v ermakov, v danilova, v yakimenko, relationship between blood superoxide dismutase ... table i hemoglobin, zinc, copper, glutathione, and metallothioneins content and sod activity in the calves’ blood hb, g/l sod activity, hb unit/mg zn, mg/l cu, mg/l gl, mg/l mt, mg/l weight of the calves 142.6 0.86 1.99 0.82 255.2 19.2 208 121.0 0.94 2.71 0.88 185.4 12.5 224 151.2 0.94 3.61 1.05 176 15.4 212 149.0 0.88 2.36 1.00 129.9 7.9 239 126.4 0.83 1.88 0.98 140.8 12.1 241 151.7 0.85 1.55 0.92 171.5 12.9 207 126.9 0.94 2.49 0.98 116.5 11.3 222 125.3 0.96 2.15 0.90 131.2 9.6 245 151.7 0.93 3.36 1.16 80.3 9.6 237 118.3 0.80 1.50 0.62 178.1 16.6 201 129.6 0.85 2.31 1.11 153.4 19.6 203 117.2 0.90 2.45 1.02 96.0 10.4 228 123.7 1.06 1.98 0.87 117.3 12.5 216 140.4 0.98 1.99 1.03 156 9.4 238 123.7 0.94 2.29 1.05 132.2 11.3 249 131.8 1.03 2.92 0.78 158.4 15 209 104.8 0.90 3.26 0.96 123.2 13.4 234 115.6 1.51 3.90 2.18 94.4 10.8 243 106.9 1.21 4.58 1.68 99.3 17 239 126.9 1.19 2.70 1.57 184.3 19.1 215 averages (m ± m) 129.2 ± 10.9 0.98± 0.11 2.60± 0.60 1.08± 0.22 144.0± 32.3 13.3± 2.8 legend: hb hemoglobin, gl glutathione, mt metallothioneins, zn zinc, cu copper. the velocity constant in the reaction of mts with hydroxyl radicals is 2 orders of magnitude higher than that of glutathione. the regression model showed that mt concentration increased significantly with calf body weight (fig. 1). a positive trend in the regression was observed between weight and sod activity, with zinc concentration tending to increase (fig. 2). however, a negative trend was observed for mt concentration (fig. 3). the pca analysis (fig. 4) demonstrated that the weight and hemoglobin levels are in different components, indicating their negative relationship. at that, glutathione and mt concentrations are within the same component but at negative levels. it indicates a different direction of the processes involving these substances, biomath 10 (2021), 2111247, http://dx.doi.org/10.11145/j.biomath.2021.11.247 page 5 of 9 http://dx.doi.org/10.11145/j.biomath.2021.11.247 v safonov, v ermakov, v danilova, v yakimenko, relationship between blood superoxide dismutase ... table ii correlation parameters between calves’ blood components hb sod zn cu gl mt 0.348769 -0.27261 -0.26462 0.31708 -0.13691 0.638739 0.874938 0.34148 0.000449 0.684429 0.43311 0.07109 0.40321 0.033942 0.585325 legend: hb hemoglobin, gl glutathione, mt metallothioneins, zn zinc, cu copper. fig. 1. regression model for the considered parameters fig. 2. linear regression between sod activity and weight of the animals, considering zinc and copper concentrations in blood biomath 10 (2021), 2111247, http://dx.doi.org/10.11145/j.biomath.2021.11.247 page 6 of 9 http://dx.doi.org/10.11145/j.biomath.2021.11.247 v safonov, v ermakov, v danilova, v yakimenko, relationship between blood superoxide dismutase ... fig. 3. linear regression between mt and weight of the animals, considering zinc and copper concentrations in blood fig. 4. results of the pca analysis for the considered parameters whereas the activity of sod is related to weight, which was also illustrated in fig. 1. iv. discussion increasing pollution associated with the anthropogenic factor causes the ingress of heavy metals and radioactive substances from the external environment into the body. their combined negative impact on various physiological processes, including redox reactions, is becoming increasingly important. when toxicants enter the body in elevated concentrations, free radicals are produced, further increasing the destruction of individual cells, organs, systems, and, eventually, the entire organism [19]. in this connection, the developments related to the pro-oxidant-antioxidant system of the body are highly relevant since the disturbances in the functioning of these systems will determine the possible pathologies. this work is among similar studies devoted to analyzing the protective mechanisms of the antioxidant system, as well as the role of free radicals and thiol compounds in the maintenance of the antioxidant system [20,21]. the relationship between the concentration of copper ions in the blood of calves and the level of superoxide dismutase activity, the most important enzyme in redox reactions, has been demonstrated. at that, the data obtained on an animal object and applicable to veterinary medicine can also be used as a comparison or model in similar medical studies. thus, this research data can be applied not only in veterinary medicine but also in other disciplines. redox reactions play a principal role in metabolism, providing a dynamic equilibrium between catabolism and anabolism and thus, achieving the constancy of physiological processes in the body. the antioxidant system regulates biochemical processes caused by redox reactions. it includes both high molecular weight components (primarily such enzymes as superoxide dismutase, glutathione peroxidase, catalase, and glutathione reductase) and low molecular weight components, particularly vitamins and bioflavonoids [22]. the regulation of redox processes is performed by endogenous thiols, which include sh-groups, as well as thiols and mercaptans. metallothioneins, glutathione, cysteine, and other compounds are known to be also involved in the regulation process. the interaction of these compounds in the blood has also been considered in this paper, while for metallothioneins, such a study has been performed for the first time. compounds containing sulfhydryl groups are known to be involved in such important processes as cell division, phosphorylation, peroxidation, and muscle contraction [23]. sulfhydryl groups constitute hormones, active enzyme centers, and receptors. thiol compounds’ role consists of their protective effect on the functional groups of macromolecules and cell membranes concerning active oxygen radicals. besides, thiol compounds can form complexes with metal ions, participating in the transport of these ions or the toxicants’ neutralization [24]. the metallothioneins considered in this work have the thiol groups of cysteine amino acid residues biomath 10 (2021), 2111247, http://dx.doi.org/10.11145/j.biomath.2021.11.247 page 7 of 9 http://dx.doi.org/10.11145/j.biomath.2021.11.247 v safonov, v ermakov, v danilova, v yakimenko, relationship between blood superoxide dismutase ... in their composition. they can bind both metals involved in physiological processes (zinc and copper) and toxicant metals (cadmium, mercury, lead, arsenic). pathological processes of such toxicants accumulation can lead to imbalances in the prooxidant-antioxidant system [25]. in this regard, oxidative stress develops since the thiol-disulfide system responds to internal or external influences by changing the indicators of the redox state [26]. it is also worth mentioning that thiol groups serve as targets when exposed to heavy metals because of their high reactivity. hence, sh-groups are being blocked, while mercaptides and disulfide bonds are being formed [27]. if disulfide groups are reduced, thiol groups can be regenerated, thus providing antioxidant homeostasis. v. conclusions the data obtained by the authors indicate a definite relationship between sod activity and copper and zinc trace elements concentration in calves’ blood. it should be noted that the most significant relationship is characteristic for copper. differentiation in the sod trace elements composition in blood of various animal species in the process of their development could be of interest, as well as existence of a dependence of sod activity on status of the indicated trace elements in the environment and feed. as for glutathione and mt, the data obtained indicate a difference in metabolic pathways involving sod and thiol-containing substances. it is of 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a.diouf1269@zig.univ.sn, dngom@univ-zig.sn †université de lorraine cnrs umr 7360, laboratoire interdisciplinaire des environnements continentaux, campus bridoux 8 rue du général delestraint, 57070 metz, france baba-issa.camara@univ-lorraine.fr, vincent.felten@univ-lorraine.fr, jean-francois.masfaraud@univ-lorraine.fr, jean-francois.ferard@univ-lorraine.fr ‡laboratoire de bio-surveillance de l’environnement (lbe), université de carthage, faculté des sciences de bizerte, 7021 zarzouna, bizerte, tunisie toumihela@yahoo.fr received: 8 august 2018, accepted: 17 december 2018, published: 23 december 2018 abstract—the toxicokinetic and toxicodynamic models (tk-td) are very well-known for their ability, at both the individual and the population level, to accurately describe life cycles such as the growth, reproduction and survival of sentinel organisms under the influence of an ecological biomarker. being dynamics, the consistent inference of life history and environmental traits parameters that engender them is sometimes very complex numerically, especially as these parameters vary from one individual to another. in this paper, we estimate the parameters of a survival model tk-td already applied and validated by the implementation of the r package guts (the general unified threshold model of survival) by another coding applied to another very recent implementation of bayesian inference with the r package debinfer in order to evaluate the survival effects of our ecotoxicological biomarker called deltamethrin on our daphnia sample. the study allowed us to evaluate from a population point of view especially the threshold concentration not to be exceeded to observe a survival effect commonly known nec (no effect concentration) and possibly determine the correlations between different variables of life history and the environment traits. keywords-bayesian inference; parameter correlations; daphnia survival; debinfer, deltamethrin; dynamic; nec copyright: c© 2018 diouf et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: abdoulaye diouf, baba issa camara, diène ngom, héla toumi, vincent felten, jean-françois masfaraud, jean-françois férard, bayesian inference of a dynamical model evaluating deltamethrin effect on daphnia survival, biomath 7 (2018), 1812177, http://dx.doi.org/10.11145/j.biomath.2018.12.177 page 1 of 10 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2018.12.177 a. diouf, b.i. camara, d. ngom, h. toumi, v. felten, j. masfaraud, j. férard, bayesian inference of a ... i. introduction statistical methods for the analysis of survival data have continued to flourish over the last two decades [7], [31]. since then, there have been many publications that deal with this hot topic in various fields such as medicine [13], [19], [34], epidemiology [8], [21], criminology [7], [23], business reliability research [11], [29], [35], and the social and behavioral sciences [25], [27], [31], [36]. they are intensively used in biology particulary ecotoxicology as in [4], [6], [12], [15], [16], [24], [32], pharmacology and medical research globally for example in [5], [26], [33]. the simulation of the temporal evolution of processes leading to toxic effects on organisms is the major role of the use of toxicokinetictoxicodynamic models (tk-td models) [17]. there is a diversity of tk-td models for modeling seemingly simple survival according to the underlying assumptions (individual tolerance or stochastic death, speed of toxicodynamic damage recovery, threshold distribution). the general unified threshold model for survival (guts) is the more general survival tk-td model from which a wide range of existing models can be inferred as special cases [17]. it has special cases of very appropriate model that can be adjusted to the survival data. as a result, it is actively contributing to increasing its application in ecotoxicology research as well as in the assessment of environmental risks related to chemicals. however, it is known that in toxicokinetics and pharmacokinetics the evolution of xenobiotics (toxic or therapeutic) in a living organism is qualitative and quantitative. by means of a realistic description (ie anatomical, physiological and biochemical) of the absorption processes (inhalation, skin contact, ingestion or intravenous injection), distribution, metabolism and excretion (adme process), the mechanistic models , which will result, allow the understanding and the simulation of this evolution of the dose of a substance in the various organs and fluids of the body [9]. the action of the organism on the substance defines the toxicokinetics (tk) whereas the opposite effect translates the toxicodynamics (td). the equations that govern them are differential equations. to answer why some individuals survive after exposure of chemicals while others die, ashauer and al., 2015 [2] established the general unified threshold model of survival (guts), a mathematical relationship. in guts, there is two assumptions: the threshold of tolerance is individually distributed and that its overcoming causes sudden death among the individuals of a population and the existence of a certain threshold, above which death occurs stochastically, which all people share. as a result, guts appeared to be a promising development in the analysis of traditional survival curves and dose-response models. recently, roman ashauer and al., 2017 [3] treated the paradigm ”dose is poison”. they illustrated that it is not only the dose that makes the poison but also the sequence of exposure taking into account the toxicokinetic recovery assumptions (the lack of effect that once a chemical is removed from organism) and toxicodynamic recovery (the neglect of the other homeostasis recovery process may be rapid or slow depending on the chemical). to do this, they tested four toxic substances acting on different targets (diazinon, propiconazole, 4,6-dinitro-o-cresol, 4-nitrobenzyl chloride) on the freshwater crustacean gammarus pulex. in this study, special consideration is given to the application of bayesian inference to the evaluation of the effects of deltamethrin (a pesticide) on a toxicokinetic and toxicodynamic (tk-td) survival model. bayesian inference can be a very sophisticated tool for survival data analysis. it is well known for its ability to process data of any sample size, especially small samples as opposed to conventional methods. many statistical methods are currently too complex to be fitted using classical statistical methods, but they can be fitted using bayesian computational methods [14], [23], [28]. however, it may be reassuring that, in many cases, bayesian inference gives answers that numerically closely match those biomath 7 (2018), 1812177, http://dx.doi.org/10.11145/j.biomath.2018.12.177 page 2 of 10 http://dx.doi.org/10.11145/j.biomath.2018.12.177 a. diouf, b.i. camara, d. ngom, h. toumi, v. felten, j. masfaraud, j. férard, bayesian inference of a ... obtained by classical methods. in this article, it is mainly to use, from another angle, a new approach to very recent bayesian implementation [30] allowing the inference of parameters of the model tk-td guts applicable to the adjustment of our survival data collected at the interdisciplinary laboratory of continental environments (liec). it is a very rigorous methodology insofar as it makes it possible to detect the different relations that can exist between the observable quantities of the unobservable quantities, the states and the parameters of the model. simple to implement, it requires a differential equation tk-td or debtox model, experimental data for the calculation of the likelihood on these data and a prior distribution assumption. a markov chain monte carlo procedure (mcmc) describes these inputs to estimate the posterior distributions of the parameters and any derived error variables, including model trajectories. this approach is designed with a mcmc diagnosis of inference, the visualization of posterior distributions of the parameters and trajectories of the model used. this manuscript assesses the long-term survival effects of a toxic substance (a pesticide) called deltamethrin via the use of the highly reputable guts model for assessing the survival of living organisms under stressors such as toxic or pharmaceuticals. the plan adopted for the organization of this article is as follows: in the second section (ii), we explain the experimental protocol established in the laboratory and present the model tk-td guts used to translate our experimental protocol. in the third section (iii), we discuss the results of the bayesian analysis. we end in section (iv) with a conclusion and discussion. ii. materials and methods a. organism test one of the three most widely used biological models for the ecotoxicological risk assessment of toxic substances, daphnia is a major invertebrate of freshwater aquatic ecosystems. the experiments were conducted with clone a of daphnia magna straus 1820 (identified by professor calow, university of sheffield, united kingdom). they are more than 40 years old at liec (university of lorraine, france) [38]. parthenogenetic cultures were carried out in 1l aquaria with lcv medium: a mixture (20/80) of lefevreczarda (lc) medium and french mineral water called volvic (v). this medium is supplemented with i) ca and mg in order to obtain a total hardness of 250 mg.l−1 and a ca/mg molar ratio of 4/1, and ii) a mixture of vitamins (0.1 ml.l−1) containing thiamine hcl (750 mg.l−1), vitamin b12 (10 mg.l−1) and biotin (7.5 mg.l−1). parthenogenetic cultures of daphnids were maintained under a temperature of 20◦c, a photoperiod of 16 − 8 h lightdark and at a density of 40 organism per liter of culture medium. the daphnia medium was renewed at least three times weekly and daphnids were fed with a mixture of three algal species (5×106 pseudokirchneriella subcapitata, 2.5 × 106 desmodesmus subspicatus, and 2.5 × 106 chlorella vulgaris/daphnia/day). these algae were also continuously cultivated in the laboratory using a nutrient lc medium. b. test chemical intensely used in agriculture, deltamethrin is a class ii pyrethroid insecticide that is harmful to freshwater ecosystems, especially the cladoceran daphnia magna (straus 1820) [37], [38]. the deltamethrin (c22h19br2no3) used in the experiments is the technical active substance of the formulation decis ec25 (25 g.l1) commercialized by bayer (germany). stock solutions were prepared by dissolving the toxicant in acetone immediately prior to each experiment. c. data sample the experimental protocol was carried out during 21 days of observation. without the control, five different doses of deltamethrin (9, 20, 40, 80 and 160 ng.l-1, respectively) were administered to daphnia magna, with a replicate of 10 for each dose submitted. the survivor count has allowed us to summarize our data sample in the table i. biomath 7 (2018), 1812177, http://dx.doi.org/10.11145/j.biomath.2018.12.177 page 3 of 10 http://dx.doi.org/10.11145/j.biomath.2018.12.177 a. diouf, b.i. camara, d. ngom, h. toumi, v. felten, j. masfaraud, j. férard, bayesian inference of a ... table i chronic test summary table (21 days) of deltamethrin effects survival. time (day) mean ± standard deviation (sd) of the survivors number during 21 days control 10±0 9 ng.l−1 9.667±0.913 20 ng.l−1 9.619±0.921 40 ng.l−1 9.429±1.121 80 ng.l−1 9.333±1.238 160 ng.l−1 8±1.761 d. model used guts is part of mathematical modeling to quantify the temporal evolution of the survival of an organism population, statistically speaking. it is highly reputed for its ability to assess a population survival effects due to a chemical stressor presence (toxicity in other words) responsible for the individuals mortality in this population. indeed, the toxicokinetic model criterion is explained by the fact that the ingested chemicals will affect a target site within the body before exerting a toxic effect thus causing damage over time. all tk-td models including a damage state use either the assumption of individual tolerance or sd hypothesis (ie the existence of a single threshold not to be exceeded for all individuals). the modeling assumptions are not the same, it is obviously clear that the results and interpretations that will follow will differ thereafter. let us not forget that the term ”hazard” and specific terms of parametrization of the different models (such as killing rate, recovery rate constant or elimination rate constant) will be misinterpreted in both cases [17]. but guts was designed to overcome these confusions because playing a unifying role that merges different concepts of existing models. guts is a synthesis of all these models by mixing the aforementioned hypotheses. more complete documentation of guts formulation hypotheses can be found in [17]. for all these reasons, we take the guts model to adopt it to our survival data study. as in [1], the guts model considered is as follows: (1). ḋ(t) = ke ( c(t)−d(t) ) , (1) where c(t) represent the toxic dose subjected linearly causing the time course of damage d(t). the dominant rate constant denoted ke (in days−1 units) models the slowest process inducing the recovery of the exposed organism. in fact, the more slow the recovery in the individual, the more vulnerable he is to the damage. note that in the body, there are systematically compensation mechanisms and damage repair. the assumption made in this guts model is that damage noted d(t) (′′damage′′) is considered to be the same for all individuals while knowing that once we exceed a certain threshold. the death considered at individual level as a stochastic event will occur and whose probability increases linearly with the damage. at the population level, this threshold is assumed to vary stochastically over the whole population. the hazard rate hz(t) (days−1) for individual with threshold z or nec (no-effect concentration) in equation (2) below represents the ”instantaneous probability to die” at individual level. the nec define the concentration threshold not to be exceeded in the body, an amount that we would like to estimate on average. once it is reached, it affects the health of the living organism. hz(t) = kk max ( 0,d(t)−nec ) + hb, (2) where the proportionality constant kk (in ng.l−1.days−1 units) is well known called killing rate and hb (in days−1 units) is the background mortality rate, that is, the control mortality rate, which is assumed to be constant over time [days−1]. the equation (3) expresses the probability s(t) that an individual of the population considered will survive until time t conditionally at the threshold z or nec. ṡz(t) =−hz(t)sz(t), (3) additional information on guts model modeling assumptions can be found on [1], [2], [3], [17]. e. statistical method in contrast to visual estimation methods, which are often considered biased and not robust, biomath 7 (2018), 1812177, http://dx.doi.org/10.11145/j.biomath.2018.12.177 page 4 of 10 http://dx.doi.org/10.11145/j.biomath.2018.12.177 a. diouf, b.i. camara, d. ngom, h. toumi, v. felten, j. masfaraud, j. férard, bayesian inference of a ... table ii survival model parameters inference. parameter symbol units prior distribution initial value elimination rate ke days−1 g (1; 1) 1 0.001 killing rate kk ng.l −1.days−1 g (1; 14 ) 1 0.015 threshold for effects nec ng.l−1 g (6; 1) 1 1.5 background hazard rate hb days −1 b(0.1; 0.15) 2 0.001 control correction ec1 [−] l n (0; 1) 3 0.005 9 ng.l−1 correction ec2 [−] l n (0; 1) 3 0.005 20 ng.l−1 correction ec3 [−] l n (0; 1) 3 0.005 40 ng.l−1 correction ec4 [−] l n (0; 1) 3 0.005 80 ng.l−1 correction ec5 [−] l n (0; 1) 3 0.005 160 ng.l−1 correction ec6 [−] l n (0; 1) 3 0.005 bayesian statistics using kinetic data have been very successful over the last two decades [9]. for all these reasons, we use in this paper the bayesian approach often considered from a practical point of view as a descriptive statistical analysis technique among the others [22]. in bayesian statistics, any unknown entity is considered as a random variable, in particular parameters of the model used. an assumption of a prior distribution, assigned to each parameter to be estimated, is necessary before the experimental data analysis. via the famous bayes theorem, these prior information will be updated with the experimental data in order to retrieve posterior information. only the bayesian approach allows to integrate the knowledge that one has of a system by taking advantage of the experimental information [22]. it is a conjunction of the information provided by the probabilistic model by a prior distribution and experimental data. the r package used for our model parameters inferring is debinfer [30]. we use the r package desolve [20], [39] as underlined in [30] for the resolution of the implemented tk-td model. to extrapolate likelihood on our experimental data, we use the poisson log-likelihood function as defined in the equation (4). the log-likelihood of the data given the parameters, underlying model, and initial conditions is then a sum over the n observations at 1the gamma distribution 2the beta distribution 3the log-normal distribution each time point in t′: l (y |θ ) = n ∑ t nt log λ −nλ (4) here we use small corrections (eci)i=1,···,6 that are needed because of the differential equations solutions can equal zero, whereas the parameter lambda of the poison likelihood must be strictly positive. we infer them later as suggested in [18], [20]. we set 20,000 iterations for the mcmc procedure, cnt = 500 worth only 1231.06 seconds of execution with an intel (r) core (tm) i3-2350m cpu processor running at 2.30 ghz. the prior distributions assumptions as well as the parameters measures units are presented in the table ii. iii. results and discussion the inference results are presented in tables iii and iv. they were obtained using the major functions ode() of the r package desolve [20] and de_mcmc() of the r package debinfer [30]. tables iii and iv respectively give the empirical mean and standard deviation for each variable, plus standard error of the mean and the quantiles for each variable. the threshold concentration above which there are effects on the survival of our test species (daphnia magna) commonly called nec is estimated cap 6.042 ± 2.418 ng.l−1. it is similar to that estimated in one of our studies on the risk assessment of deltamethrin on growth and reproduction treated separately [10]. this result is biomath 7 (2018), 1812177, http://dx.doi.org/10.11145/j.biomath.2018.12.177 page 5 of 10 http://dx.doi.org/10.11145/j.biomath.2018.12.177 a. diouf, b.i. camara, d. ngom, h. toumi, v. felten, j. masfaraud, j. férard, bayesian inference of a ... table iii empirical mean and standard deviation for each variable, plus standard error of the mean. mean sd naive se time-series se ke 0.428169 0.738795 5.224e-03 0.0827944 kk 0.003064 0.010340 7.311e-05 0.0013127 nec 6.041585 2.418373 1.710e-02 0.1636959 hb 0.002097 0.003396 2.401e-05 0.0001835 ec1 0.723883 0.459832 3.252e-03 0.0303711 ec2 0.593472 0.379989 2.687e-03 0.0211085 ec3 0.662622 0.431101 3.048e-03 0.0245585 ec4 0.699403 0.430309 3.043e-03 0.0244237 ec5 0.938278 0.556387 3.934e-03 0.0313650 ec6 0.808771 0.581670 4.113e-03 0.0385683 table iv quantiles for each variable. 2.5% 25% 50% 75% 97.5% ke 7.848e-04 1.588e-02 9.558e-02 0.542053 2.66691 kk 1.352e-04 2.854e-04 5.183e-04 0.001508 0.02582 nec 2.150e+00 4.184e+00 5.729e+00 7.690363 10.81920 hb 6.774e-18 1.279e-07 9.653e-05 0.003075 0.01194 ec1 1.023e-01 3.698e-01 6.445e-01 0.997039 1.83111 ec2 9.134e-02 3.152e-01 5.145e-01 0.791141 1.52596 ec3 1.150e-01 3.398e-01 5.604e-01 0.883916 1.73556 ec4 1.089e-01 3.752e-01 6.185e-01 0.936333 1.74968 ec5 1.666e-01 5.078e-01 8.321e-01 1.252371 2.27439 ec6 1.106e-01 3.811e-01 6.464e-01 1.111107 2.26996 very consistent in that death stops any evolution process. while the recovery process under the toxic effect is estimated to be around 0.43±0.74. the dominant rate constant is not so negligible as that in contrast to the killing rate and the control mortality rate constants whose respective values are close to 0.003±0.01 and 0.002±0.03. these different estimated values would translate faithfully our experimental realities as shown in the data table i. with 10 replicates for each deltamethrin dose, few deaths were observed in this experimental protocol. the guts model again reflects the reality of the facts in this study. the density plots for the various inferred parameters can be read in figure 1. in this image, some chain trajectories are reasonable and consistent over time in that their posterior distributions are unimodal, sometimes resembling that of a normal distribution. we can cite for example the parameters nec and the small corrections eci=1,···,6. their prior distributions were those of a log-normal distribution. unlike the distributions of the ke, kk and hb unimodal parameters, but suspect because they include a large number of outliers. this aberration would confirm the inference complexity of these types of studies. let’s not forget that these are constant rate. the study results are very consistent overall. the figure 2 perfectly shows a lack of detected correlation between parameters. the highest correlation value is 0.36 between kk and nec, the two most important parameters in guts [17]. for proof purposes, we remove a burnin period of 1500 samples and examine parameter correlations in the figure 2 and overlap between prior and posterior densities. the figure 2 reflects the correlation lack between the different parameters of our dynamics evaluating daphnia survival in the presence of our deltamethrin stressor. from the posterior, we simulate 500 trajectories of our tk-td model while calculating at 95%hdi (highest posterior density intervals) for the debiomath 7 (2018), 1812177, http://dx.doi.org/10.11145/j.biomath.2018.12.177 page 6 of 10 http://dx.doi.org/10.11145/j.biomath.2018.12.177 a. diouf, b.i. camara, d. ngom, h. toumi, v. felten, j. masfaraud, j. férard, bayesian inference of a ... fig. 1. mcmc chains plotting & summarizing fig. 2. parameter correlations plotting biomath 7 (2018), 1812177, http://dx.doi.org/10.11145/j.biomath.2018.12.177 page 7 of 10 http://dx.doi.org/10.11145/j.biomath.2018.12.177 a. diouf, b.i. camara, d. ngom, h. toumi, v. felten, j. masfaraud, j. férard, bayesian inference of a ... fig. 3. median survival fitting with 95% hdi where the black lines represent the solution trajectories of the survival dynamics and the red points reflect the experimental data. terministic part of the model. hdi sets (intervals) contain all values of the parameter θ such that the posterior density fθ|y is larger than some constant cα , where cα ensures that the coverage probability will be 1−α . for each exposure concentration, figure 3 shows in the same graph, the experimental data and the model output describing the dynamics of alive daphnia magna number during the 21 days of experience. it confirms that the inference procedure actually retrieves the model to our data. in addition, the fitted curves are obtained with small estimation errors, see figure 3. post hoc trajectories adjust very well our observational data for different pesticide doses. iv. conclusion this paper is very instructive in that it adapts the guts model to our survival data collected at liec through a more recent implementation of bayesian inference (the r library debinfer). thus we ignored the use of guts (r package guts) implementation. with this new r library, it is easy to encode any toxicico-kinetic and toxicodynamic dynamics (tk-td) then infer the parameters that compose it. once differential encoding is complete, the r package debinfer has a function named de_mcmc() where is integrated that of ode() function of the r package desolve specially designed for system differential solving such that ordinary, partial or delay differential equations. these last facilitate access to a lot of users types whether they are specialists in the field or not. most of the life phenomena are modeled using ordinary differential equations (edo), partial differential equations (pde), or the delaydifferential equations (dde). as a result, this r package debinfer facilitates the transition from determinism to stochastic. as part of our study, it allowed us to consistently address our survival analysis with the guts tk-td model use. it really facilitated the manipulation and inference of the parameters of a mechanistic model to describe the bioaccumulation kinetics and dynamics of survival effects in a contaminated environment of the pesticide deltamethrin. acknowledgments this study was funded by the french cooperation and the african center of excellence in mathematics, computer sciences and tic (ceabiomath 7 (2018), 1812177, http://dx.doi.org/10.11145/j.biomath.2018.12.177 page 8 of 10 http://dx.doi.org/10.11145/j.biomath.2018.12.177 a. diouf, b.i. camara, d. ngom, h. toumi, v. felten, j. masfaraud, j. férard, bayesian inference of a ... mitic). we are very grateful for their financial supports. references [1] albert c, vogel s, ashauer r (2016) computationally efficient implementation of a novel algorithm for the general unified threshold model of survival (guts). plos comput biol 12(6): e1004978. doi:10.1371/journal.pcbi.1004978 [2] ashauer, r., o’connor, i., hintermeister, a., & escher, b. i. 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(2016). devtools: tools to make developing r packages easier. biomath 7 (2018), 1812177, http://dx.doi.org/10.11145/j.biomath.2018.12.177 page 10 of 10 https://doi.org/10.1111/2041-210x.12679 https://doi.org/10.1111/2041-210x.12679 http://dx.doi.org/10.11145/j.biomath.2018.12.177 introduction materials and methods organism test test chemical data sample model used statistical method results and discussion conclusion references www.biomathforum.org/biomath/index.php/biomath original article robust numerical method for a singularly perturbed problem arising in the modelling of enzyme kinetics john j. h. miller, eugene o’riordan 1school of mathematics, trinity college dublin 2, ireland jmiller@tcd.ie 2school of mathematical sciences, dublin city university dublin 9, ireland eugene.oriordan@dcu.ie received: 19 june 2020, accepted: 22 august 2020, published: 12 september 2020 abstract— a system of two coupled nonlinear initial value equations, arising in the mathematical modelling of enzyme kinetics, is examined. the system is singularly perturbed and one of the components will contain steep gradients. a priori parameter explicit bounds on the two components are established. a numerical method incorporating a specially constructed piecewise-uniform mesh is used to generate numerical approximations, which are shown to converge pointwise to the continuous solution irrespective of the size of the singular perturbation parameter. numerical results are presented to illustrate the computational performance of the numerical method. the numerical method is also remarkably simple to implement. keywords-enzyme-substrate dynamics, nonlinear system, shishkin mesh, parameter-uniform convergence. i. introduction the henri-michaelis-menten system of nonlinear differential equations arises in the mathematical modelling of enzyme-substrate dynamics, see, for example, [1], [3], [5], [7]. as analytical solutions are not available, it is necessary to solve this system numerically. this may be difficult when the system is singularly perturbed [1]. asymptotic expansions associated with this singularly perturbed problem are discussed in [9], [7], [8]. here, to establish a parameter-uniform pointwise error bound on the numerical approximations, we construct a shishkin decomposition [6] for the solutions. this can be viewed as an alternative to an asymptotic expansion. based on this decomposition, an efficient finite difference method is constructed, which uses a specially copyright: c©2020 miller et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: john j. h. miller, eugene o’riordan, robust numerical method for a singularly perturbed problem arising in the modelling of enzyme kinetics, biomath 9 (2020), 2008227, http://dx.doi.org/10.11145/j.biomath.2020.08.227 page 1 of 12 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2020.08.227 j. j. h. miller, e. o’riordan, robust numerical method for a singularly perturbed problem arising in ... constructed piecewise uniform mesh and an appropriate standard finite difference operator to deal with the steep gradients that appear initially, see for example [6]. this numerical method is shown to be parameter-uniformly convergent, in the sense that its numerical solutions converge to the exact solution, uniformly with respect to the singular perturbation parameter, with essentially first order. a shishkin mesh was also used in [4] to solve the same system, but it should be noted that the hypotheses [4, (5) and (6)], required by the theory in [4], are not fulfilled by the system considered in the present paper. asymptotic expansions yield useful approximations when the singular perturbation parameter ε is sufficiently small for terms of a certain order o(εn) to be negligible. however, the accuracy in parameter-uniform approximations is valid irrespective of the size of ε and depends only on the number of mesh points n used in the computations. moreover, there can be a debate [7], [8] about how to choose the dimensionless variables when using the quasi-steady-state assumption [8], which becomes irrelevant if one has a parameter-uniform numerical method. the shishkin-mesh method developed below does not require an asymptotic expansion to compute an approximation and, hence, the method is simpler than other approaches which generate individual terms (or approximations) in an asymptotic expansion (e.g., [9]). the structure of the paper is as follows. in the next section the continuous problem is formulated. in §3 the problem is discretised and the error analysis is performed in §4. the main results are stated in theorems 5 and 6. the paper concludes with §5 in which some numerical experiments illustrate the form of the solution, and its initial layer, and also support the theoretical error analysis. ii. continuous problem in the basic model of enzyme reactions (e.g. [7]), a substrate s reacts with an enzyme e to form a complex se which is converted into a product p and the enzyme. the concentrations of these variables vary with time τ and are denoted here by lower case letters s(τ) = [s], e(τ) = [e], c(τ) = [se], p(τ) = [p ]. these reactions can be modelled by the following system of four first order equations for four unknowns with given initial conditions ds dτ = −k1es + k−1c, (2.1a) de dτ = −k1es + (k−1 + k2)c, (2.1b) dc dτ = k1es− (k−1 + k2)c, (2.1c) dp dτ = k2c, (2.1d) s(0) =s0, e(0) =e0, c(0) = 0, p(0) = 0; (2.1e) where the parameters k−1,k1,k2 are reaction rate constants. note that p(τ) = k2 ∫ τ s=0 c(s) ds and e(τ) = −c(τ) + e0. hence, we have only two unknown variables (s and c) to determine. as in [7], we introduce the following scalings and nondimensionless variables u and v: u(t) := s(t) s(0) , v(t) := c(t) e(0) , t := (k1e(0))τ α := k2 k1s(0) , k = k−1 + k2 k1s(0) ,ε := e(0) s(0) . this leads to the following autonomous system of two coupled nonlinear initial value equations ([7, eq. (6.13)]): find (u(t),v(t)) ∈ c∞(0,t), 0 ≤ t ≤ t such that, for all 0 < ε ≤ 1, u′ = −u + (u + k−α)v, t > 0; (2.2a) εv′ = u− (u + k)v, t > 0; (2.2b) u(0) = 1; v(0) = 0;k > α > 0. (2.2c) we observe the following facts u′(0) = −1, εv′(0) = 1, (2.3) εu′′(0) = 1 + k−α + ε, εv′(t) + αv(t) = −u′(t); biomath 9 (2020), 2008227, http://dx.doi.org/10.11145/j.biomath.2020.08.227 page 2 of 12 http://dx.doi.org/10.11145/j.biomath.2020.08.227 j. j. h. miller, e. o’riordan, robust numerical method for a singularly perturbed problem arising in ... and so there will be an initial layer in v and a weak initial layer in u (as εu′′(0) = o(1) and u′(0) = o(1)). the exact solution of problem (2.2) is unknown. lemma 1. for all t > 0, the solutions of (2.2) are bounded as follows: 0 < u(t) < 1 and 0 < v(t) ≤ 1 1 + k . (2.4) proof: given the initial conditions (2.3) and the fact that u′,v′ are continuous, there exists an interval (0, t1), where u′(t) < 0 and v′(t) > 0 for all t ∈ (0, t1). hence, there also must exist some 0 < t0 < t1 such that 0 < u(t) < 1 and 0 < v(t) < u(t) u(t) + k < 1, t ∈ (0, t0). also, for any positive time t > t1, where u(t) + k > 0, v′(t) > 0, if v(t) < u(t) u(t) + k and v′(t) < 0, if v(t) > u(t) u(t) + k . assume that there exists a t∗ > t1 such that 0 < v(t∗) = u(t∗) u(t∗) + k < u(t∗) < 1, where v′(t∗) = 0 and u′(t∗) < 0. if no such t∗ is reached, then we are done. if this time t∗ exists, then v will have a maximum at this point and so v(t) < 1 for all time where u(t) + k > 0 . hence we have that u(t∗) and v(t∗) are both positive. by (2.2a), there does not exist a least time t2 > t∗ where u(t2) = 0,u′(t2) ≤ 0 and v(t2) > 0. moreover, by (2.2b), there does not exist a least time t2 > t∗ where v(t2) = 0,v′(t2) ≤ 0 and u(t2) > 0. hence, if either u or v were to become negative, then u,v would have to simultaneously become zero at t2. hence, by (2.2a) and (2.2b), we would have that u(t2) = u′(t2) = v(t2) = v′(t2) = 0. moreover, by differentiating (2.2a) and (2.2b), we would find that all derivatives of u (and v) were zero at this point t2. for a smooth function u ∈ c∞(0,t], it cannot be that all derivatives of a non-trivial function at a certain time t2 are zero. hence, this point t2 does not exist. it follows that u(t) > 0, v(t) > 0, ∀t ≥ 0. finally, as u(t) + k−α > 0, u′ = −u + (u + k)(1 − α u + k )v ≤ −u + u(1 − α u + k ) = − uα u + k < 0. hence u(t) ≤ 1. note that g′(z) > 0 for g(z) = z z+k. hence, v(t) ≤ v(t∗) ≤ 1 1+k. from the bounds in this lemma, we can deduce that, for i = 1, 2,∥∥∥diu dti ∥∥∥ ≤ c(1 + ε1−i), ∥∥∥div dti ∥∥∥ ≤ cε−i; where ‖g‖ := maxt∈[0,t ] |g(t)|. however, these bounds are not sufficiently sharp for our purposes, as they do not identify the fact that the large derivatives will only occur initially. to generate sharper bounds, we will construct a shishkin decomposition [6] of the solution. from ([7, eq. (6.26)]) and by formally setting ε = 0 in (2.2) and ignoring v(0) = 0, the reduced solution (u0,v0) is given by v0(t) = u0(t) u0(t) + k , u0(t) +k ln u0(t) = 1−αt. (2.5) the reduced solution (u0,v0) or outer solution approximates the solution (u,v) outside a neighbourhood of t = 0. using the stretched variable τ = t/ε, the solution (u,v) is approximated ([7, eq. (6.31)]) initially by the inner solution (ui,vi) ui(t) = 1, vi(t) = 1 1 + k (1 −e−(1+k)t/ε), for t ≤ cε. this motivates the following shishkin decomposition of the solution (u,v). lemma 2. the solutions of problem (2.2) can be decomposed as follows: u(t) = u0(t) + ru(t), (2.6a) where du0 dt = − αu0 u0 + k , u0(0) = 1; biomath 9 (2020), 2008227, http://dx.doi.org/10.11145/j.biomath.2020.08.227 page 3 of 12 http://dx.doi.org/10.11145/j.biomath.2020.08.227 j. j. h. miller, e. o’riordan, robust numerical method for a singularly perturbed problem arising in ... v(t) = u(t) u(t) + k −b(t) + rv(t), (2.6b) b(t) := 1 1 + k e− ∫ t s=0 u(s)+k ε ds; and the remainder terms ru,rv are bounded by ‖ru‖ + ‖rv‖≤ cε, (2.6c)∥∥∥diru dti ∥∥∥ + ∥∥∥dirv dti ∥∥∥ ≤ c(1 + ε1−i), i = 1, 2.(2.6d) proof: the function v(t) can be written in the form v(t) = u(t) u(t) + k − 1 1 + k e− ∫ t s=0 u(s)+k ε ds + rv(t). by inserting this expansion for v into (2.2b), we see that the remainder term rv satisfies the initial value problem ε drv dt +(u+k)rv = ε d dt ( u(t) u(t)+k ) (2.7) = εk (u+k)2 du dt ; rv(0) = 0. hence, rv(t) = ∫ t s=0 g(t)e− ∫ t r=s u(r)+k ε drds, where g(t) = k((u + k−α)v −u) (u + k)2 . note that ‖g‖ ≤ c and by the previous lemma, u(t) + k≥k > 0. then we deduce that∣∣rv(t)∣∣ ≤ c ∫ t s=0 e− k(t−s) ε ds ≤ cε. using the differential equation in (2.7), we conclude that∥∥∥dirv dti ∥∥∥ ≤ c(1 + ε1−i), i = 1, 2. we also have the following decomposition u(t) = u0(t) + ru(t), where du0 dt = − αu0 u0 + k , u0(0) = 1. note that u0(t) is implicitly defined in (2.5). by inserting the expansions for u and v into (2.2a), we see that u′ = −u+(u+k−α)v = −u+u− αu u+k +(u+k−α)(rv−b) = u′0 + αu0 u0 +k − αu u+k +(u+k−α)(rv−b), where r′u = αu0 u0 + k − αu u + k + (u + k−α)(rv −b) = αk(u0 −u) (u0 + k)(u + k) + (u + k−α)(rv −b). hence, the remainder term ru satisfies the initial value problem dru dt + αk (u + k)(u0 + k) ru = (u + k−α)(rv − 1 1 + k e− ∫ t s=0 u(s)+k ε ds); ru(0) = 0. hence, as (u0 +k)(u +k) ≥k2 > 0, by writing out a closed form representation for the function ru, we deduce that ‖ru‖≤ cε and ∥∥∥diru dti ∥∥∥ ≤ c(1 + ε1−i), i = 1, 2. iii. discrete problem consider the following implicit linear finite difference scheme for the continuous problem (2.2): for all tj ∈ ωn , find (u(tj),v (tj)) such that: for all tj > 0 d−u(tj) + (1 −v (tj−1))u(tj) −(k−α)v (tj) = 0, (3.8a) εd−v (tj)−u(tj)+(u(tj−1)+k)v (tj) = 0, (3.8b) and u(0) = 1; v (0) = 0; (3.8c) where d−u(tj) := ( u(tj)−u(tj−1) ) /kj, kj :=tj−tj−1 biomath 9 (2020), 2008227, http://dx.doi.org/10.11145/j.biomath.2020.08.227 page 4 of 12 http://dx.doi.org/10.11145/j.biomath.2020.08.227 j. j. h. miller, e. o’riordan, robust numerical method for a singularly perturbed problem arising in ... and the piecewise uniform mesh ωn is constructed below. note that for any s > 0, since |u0(t)| ≤ 1, u0(s) = 1 + ∫ s 0 du0 dt dt = 1 −α ∫ s 0 u0 u0 + k dt > 1 − α 1 + k s. hence, the initial layer function b(t) is bounded by e− ∫ σ s=0 u(s)+k ε ds ≤ ce− ∫ σ s=0 u0(s)+k ε ds ≤ ce− σ ε (1+k− ασ 2(1+k) ) . based on the decomposition (2.6b) and (2.6a), we will use a piecewise uniform shishkin mesh [6], denoted here by ωn := {tj|0 ≤ tj := tj−1 + kj ≤ t}, where there the fine and coarse mesh sizes k,k are defined by kj = k := 2σ n , j ≤ n 2 ; kj = k := 2(t −σ) n , j > n 2 . the transition point σ is taken to be σ := min{0.5t, ε 0.5 + k ln n}. for simplicity of exposition we assume 1that ε(ln n)2 ≤ c then σ2 ≤ cε and for n sufficiently large, e− ∫ σ s=0 u(s)+k ε ds ≤ ce− ∫ σ s=0 u0(s)+k ε ds ≤ ce ασ2 2ε(1+k) e− σ(1+k) ε ≤ cn−1. remark 3. if ε ≤ cn−1 then for t ≥ σ we have v(t) = u0(t) u0(t) + k + cn−1 and u(t) = u0(t) + cn −1. in other words, outside the initial layer, the solutions (u,v) are computationally close to the 1if ε(ln n)2 > c then a classical argument can be used to deduce the error bound in corollary 7. reduced solutions (u0,v0) when ε is sufficiently small. we next establish that, within the fine mesh, the discrete solutions u,v are bounded by the same bounds as their continuous counterparts u,v. in the next section, we will extend this result to the mesh points outside the fine mesh. these bounds on the discrete solutions ensure that the linear system (3.8) has a unique solution. lemma 4. for the solution of (3.8) and n sufficiently large (independent of ε), we have, for all 0 < tj ≤ σ, 0 < u(tj) < 1 and 0 < v (tj) < 1 1 + k . proof: by explicitly solving the linear system (3.8) at the first internal mesh point, we see that 0 < v (t1) = k/ε 1 + k + k 1+k ε + k 2 ε (1 + α) < k ε 1 1 + k < 1 1 + k 0 < u(t1) = 1 + k(1 + k)/ε 1 + k + k 1+k ε + k 2 ε (1 + α) < 1. define the associated system matrix mj :=i+kj ( 1−v (tj−1) −(k−α) −1 ε u(tj−1)+k ε ) ; (3.9) and write the discrete problem (3.8) in the form mj ( u(tj)) v (tj) ) = ( u(tj−1)) v (tj−1) ) ,( u(t0) v (t0) ) = ( 1 0 ) . note that det(mj) = 1+kj(1−v (tj−1))+kj u(tj−1)+k ε + k2j ε ( u(tj−1)+α−v (tj−1)(u(tj−1)+k) ) = 1+kj(1−v (tj−1))+ kkj ε (1−kj)+ kj ε u(tj−1) + k2j ε ( (1−v (tj−1))(u(tj−1)+k)+α ) biomath 9 (2020), 2008227, http://dx.doi.org/10.11145/j.biomath.2020.08.227 page 5 of 12 http://dx.doi.org/10.11145/j.biomath.2020.08.227 j. j. h. miller, e. o’riordan, robust numerical method for a singularly perturbed problem arising in ... and m−1j = a det(mj) , (3.10) where a:=i+kj ( u(tj−1)+k ε k−α 1 ε 1 −v (tj−1) ) . we now complete the argument by using induction. assume the statement is true for all 1 ≤ i ≤ j − 1. if 0 < v (tj−1) < 1 and u(tj−1) > 0 then det(mj) > 1 and m −1 j > 0. hence, u(tj) > 0,v (tj) > 0. moreover, we can rewrite det(mj) = 1+kj(1−v (tj−1))+ kj ε u(tj−1)(1+k) + k2j ε ( (1 + α−v (tj−1)(u(tj−1) + k) ) + (k−kj)kj ε (1 −u(tj−1)). if (1 + k)v (tj−1) < 1 and u(tj−1) < 1 then v (tj−1)(u(tj−1) + k) < 1 and det(mj) > 1+kj(1−v (tj−1))+ kj ε u(tj−1)(1+k). from this, we deduce that v (tj) = 1 det(mj) ( (1+kj(1−v (tj−1)))v (tj−1) + kj ε u(tj−1) ) < 1 (1 + k)det(mj) ( (1 + kj(1 −v (tj−1))) + kj ε u(tj−1)(1 + k) ) < 1 1 + k . we rewrite det(mj) = 1+kj u(tj−1)+k ε +kj(k−α)v (tj−1) +kj ( 1−(1+k)v (tj−1)+v (tj−1)(α− kj ε (k−α)) + kj ε (u(tj−1) + α)(1 −v (tj−1) )) . if (1 +k)v (tj−1) < 1 and j ≤ n/2 then kj ≤ cεn−1 ln n and, under these assumptions, det(mj)>1+kj (u(tj−1)+k) ε +kj(k−α)v (tj−1). hence u(tj) = 1 det(mj) ( 1 + kj ε (u(tj−1) + k) +kj(k−α)v (tj−1) ) < 1. iv. error analysis consider the error (u − u,v − v)(tj) at each mesh point tj, which satisfies d−(u−u)+(1−v (tj−1))(u−u)−(k−α)(v −v) = u′ −d−u + (v (tj−1) −v(tj))u; εd−(v −v)−(u−u)+(u(tj−1)+k)(v −v) = ε(v′ −d−v) + (u(tj) −u(tj−1))v. we rewrite these equations in matrix form mj ( (u −u)(tj) (v −v)(tj) ) = ( kj(u′ −d−u + (v (tj−1) −v(tj))u) kj ε ( ε(v′ −d−v) + (u(tj) −u(tj−1))v ) ), where the matrix mj is defined in (3.9). the next theorem establishes an error estimate within the fine mesh region [0,σ]. theorem 5. for the solutions of (2.2), (3.8) and n sufficiently large (independent of ε), we have, for all tj ≤ σ, |(u −u)(tj)| + |(v −v)(tj)| ≤ cn−1(ln n)2. proof: for tj ≤ σ, | ( u′ −d−u ) (tj)| ≤ ck‖u′′‖≤ cn−1 ln n, |u(tj) −u(tj−1)| ≤ k‖u′‖≤ cn−1; ε|(v′ −d−v)(tj)| ≤ cεk‖v′′‖≤ cn−1 ln n, |v(tj) −v(tj−1)‖ ≤ k‖v′‖≤ cn−1 ln n. hence, |u′ −d−u + (v (tj−1) −v(tj))u| ≤ cn−1 ln n + c|(v −v)(tj−1)|; |ε(v′ −d−v) + (u(tj) −u(tj−1))v| ≤ cn−1 ln n + c|(u −u)(tj−1)|. biomath 9 (2020), 2008227, http://dx.doi.org/10.11145/j.biomath.2020.08.227 page 6 of 12 http://dx.doi.org/10.11145/j.biomath.2020.08.227 j. j. h. miller, e. o’riordan, robust numerical method for a singularly perturbed problem arising in ... as in the previous lemma, det(mj) > 1 and using the explicit form of the inverse m−1j in (3.10), we deduce that |(u −u)(tj)| ≤ ck ( |(v −v)(tj−1) +cn−1 ln n(1 + |(u −u)(tj−1)|) ) ; |(v −v)(tj)| ≤ c k ε ( |(u −u)(tj−1) +cn−1 ln n(1 + |(v −v)(tj−1)|) ) . hence, using an iterative argument, for all j ≤ n/2 |(u−u)(tj)|+|(v −v)(tj)| ≤ cj( k ε )2 ≤ cn(n−1 ln n)2 ≤ cn−1(ln n)2. it is not as straightforward to establish this error bound in the coarse mesh region. before we examine this error, we present a discrete analogue to the solution decomposition established in lemma 2. we first construct the decomposition in the fine mesh region, as it is relatively straightforward to do so. then, in the next section, we inductively generate this same discrete decomposition and simultaneously establish error bounds at each point tj in the coarse mesh region. we have the following decomposition of the discrete solution v : v (tj) = u(tj) u(tj)+k − 1 1+k bn (tj)+r n v (tj), (4.11a) where εd−bn (tj)+(u(tj−1)+k)bn (tj) = 0, bn (0) = 1. (4.11b) the remainder term rnv satisfies r n v (0) = 0 and the finite difference equation εd−rnv + (u(tj−1) + k)r n v =−εd− u(tj) u(tj)+k +kj u(tj)d −u(tj) u(tj)+k (4.11c) = (−εk(u(tj−1)+k)−1 +kju(tj)) u(tj)+k d−u(tj). for tj ≤ σ and n sufficiently large, u(tj−1) + k = u0(tj−1) + k±cn−1 > 1 − α 1 + k σ + k±cn−1 ≥ 0.5 + k. then in the fine mesh(u(tj−1) + k ε ) k ≥ 2 ln n n and from [6, lemma 5.1] π n/2 j=1 ( 1 + (u(tj−1) + k ε ) k )−1 ≤ cn−1. hence, we have the bound 0 < bn (σ) ≤ cn−1. (4.11d) we next establish a bound on (rnv −rv)(tj) for all tj: εd−(rv−rnv )(tj)+(u(tj−1)+k)(rv−r n v )(tj) = εd−rv(tj) + (u(tj−1) + k)rv(tj)+ εkd−u(tj) (u(tj−1) + k)(u(tj) + k) − kju(tj)) u(tj) + k d−u(tj) =ε(d−rv−r′v)(tj)+(u(tj−1)−u(tj))rv(tj)+g(tj), where g(tj) := εk ( d−u(tj) (u(tj−1)+k)(u(tj)+k) − u′(tj) (u(tj)+k)2 ) − kju(tj) u(tj) + k d−u(tj). also d−u(tj) (u(tj−1) + k)(u(tj) + k) − u′(tj) (u(tj) + k)2 = t(u(tj) + k)2d−u(tj) −tu′(tj)(u(tj−1) + k)(u(tj) + k) = t(u(tj) + k)2(d−u(tj) −u′(tj)) + tu′(tj)((u−u)(tj))((u(tj) + k) + tu′(tj)(u(tj) + k)(u(tj) −u(tj−1), biomath 9 (2020), 2008227, http://dx.doi.org/10.11145/j.biomath.2020.08.227 page 7 of 12 http://dx.doi.org/10.11145/j.biomath.2020.08.227 j. j. h. miller, e. o’riordan, robust numerical method for a singularly perturbed problem arising in ... where t := 1 (u(tj−1) + k)(u(tj) + k)(u(tj) + k)2 . note that u′(tj) = (k−α)v(tj)−(1−v(tj))u(tj) d−u(tj) = (k−α)v (tj)−(1−v (tj−1))u(tj) |u(tj)−u(tj−1)|≤ckj‖u′‖. if 0 < u(tk),v (tk) < c, for tk = tj−1, tj then |g(tj)| ≤ c(ε+ n−1) max k=j−1,j {|(u−u)(tk)|, |(v −v)(tk)|} +cn−1. (4.12) hence, using a discrete comparison principle 2 |(rnv −rv)(tj)| ≤ c(n−1+max k<j |(u−u)(tk)|+max k≤j |g(tk)|). (4.13) since u(tj) > 0, by lemma 3, we have that for all tj ≤ σ |(u −u)(tj)| + |(v −v)(tj)| ≤ cn−1(ln n)2 and hence |(rnv −rv)(tj)| ≤ cn −1(ln n)2, tj≤σ. (4.14) theorem 6. for the solution of (3.8) and n sufficiently large (independent of ε), we have, for all tj ≥ σ, 0 < u(tj) < 1, 0 < v (tj) < 1 1 + k and the following parameter-uniform error bound |(u −u)(tj)| + |(v −v)(tj)| ≤ cn−1(ln n)2. proof: the proof is by induction. by the previous two lemmas the statement is true for tk = σ. 2 if a(tj ) > 0 and w (tj ) is any mesh function such that εd−w (tj ) + a(tj )w (tj ) ≥ 0,∀tj > 0 and w (0) ≥ 0 then w (tj ) ≥ 0,∀tj ≥ 0. assume now that it is true for all σ ≤ tk ≤ tj−1. the argument in lemma 2 to establish the bounds 0 < u(tj), 0 < v (tj) < 1 1 + k is still valid within the coarse mesh. however, the argument to establish the upper bound u(tj) < 1 requires an alternative argument in the coarse mesh region. under the induction assumption, for all σ ≤ tk ≤ tj the decomposition in (4.11a) v (tk) = u(tk) u(tk) + k − 1 1 + k bn (tk) + r n v (tk), where εd−bn (tk) =−(u(tk−1)+k)bn (tk), bn (0) = 1 is still applicable. moreover, as u(tk−1) +k > 0, we see that 0 < bn (tk) < b n (σ) ≤ cn−1 and |(rnv −rv)(tk−1)| ≤ cn −1, for all k ≤ j. returning to the end of the proof of lemma 2, we have that det(mj) = 1 + kj ( u(tj−1) + k ε ) +kj(k−α)v (tj−1)+kj ( 1−(1+k−α)v (tj−1) ) + k2j ε ( u(tj−1) + α− (u(tj−1) + k)v (tj−1) ) . using the decomposition (4.11a), we see that u(tj−1) + α− (u(tj−1) + k)v (tj−1) > α±cn−1 > 0 for n sufficiently large. hence, as in the proof of lemma 2, u(tj) < 1. for each tj, using the decompositions (2.6b) and (4.11a) we can write the error in v in the form (v −v )(tj) = u(tj) u(tj) + k − u(tj) u(tj) + k +cn−1 + (rv −rnv )(tj) = k(u(tj) −u(tj)) (u(tj) + k)(u(tj) + k) +cn−1 + (rv −rnv )(tj). (4.15) biomath 9 (2020), 2008227, http://dx.doi.org/10.11145/j.biomath.2020.08.227 page 8 of 12 http://dx.doi.org/10.11145/j.biomath.2020.08.227 j. j. h. miller, e. o’riordan, robust numerical method for a singularly perturbed problem arising in ... for tj > σ, using (2.6b) we have that ε(v′−d−v)(tj) = 1 kj ∫ kj s=kj−1 ε(v′(tj)−v′(s))ds = 1 kj ∫ kj s=kj−1 (u−(u+k)v)(tj)−(u−(u+k)v(s))ds = (u + k)(tj) kj ∫ kj s=kj−1 v(tj) −v(s) ds + cn−1. hence, using (2.6b) again, we get that ε|(v′ −d−v)| ≤ cn−1. also |u(tj) −u(tj−1)| ≤ cn−1|u′| ≤ cn−1 and for tj > σ |v(tj) −v(tj−1)| ≤∣∣ u(tj) u(tj) + k − u(tj−1) u(tj−1) + k ∣∣ + cn−1 ≤ cn−1. in the coarse mesh | ( u′−d−u ) (tj)| ≤ cn−1 + | ( r′u−d −ru ) (tj)|. the remainder term ru can be further decomposed into the sum ru = su + w, where ‖s′′u‖≤ c,‖w ′‖≤ ce− kt ε . hence, on the coarse mesh, | ( u′ −d−u ) (tj)| ≤ cn−1. hence, the error (u −u)(tj) satisfies d−(u −u) + (1 −v (tj−1))(u −u) − k(k−α) (u(tj) + k)(u(tj) + k) (u −u) = (v −v)(tj−1)u + (k−α)(rnv −rv)(tj) +cn−1 + c kj ε e− ktj−1 ε . by the induction hypothesis and using the expressions in (4.13) and (4.12), we deduce that d−(u−u)+ ( 1−v(tj)− k(k−α) (u(tj)+k)(u(tj)+k) ± c(ε + n−1) ) (u −u)(tj) = (v (tj−1)−v(tj))(u−u)(tj)+cn−1(ln n)2; d−(u −u) + ( k(u(tj) + α) (u(tj) + k)(u(tj) + k) ± c(ε + n−1) ) (u −u)(tj) = cn−1(ln n)2. since the coefficient of the zero order term in this error equation is strictly bounded below by a positive number, we deduce that |(u −u)(tj)| ≤ cn−1(ln n)2. the bound on the error in v (tj) follows from (4.15). the proof is completed by induction. let ū and v̄ denote the piecewise linear interpolants of u and v , respectively. then, as in [6], we can extend the result to a global error bound. corollary 7. for all t ∈ [0,t], we have |(u− ū)(t)| + |(v − v̄ )(t)| ≤ cn−1(ln n)2. remark 8. according to [7, pg. 186], it is of interest to generate an accurate approximation to the rate of reaction u′(t). given the previous results, we have that |(u′−du)(tj)|≤c|v(tj)−v(tj−1)|+cn−1(ln n)2 ≤ckj‖v′‖(tj−1,tj) +cn −1(ln n)2. in the fine mesh kj‖v′‖(tj−1,tj) ≤ cn −1 ln n and in the coarse mesh, where tj > σ, |v(tj) −v(tj−1)| ≤ c ∣∣ u(tj) u(tj) + k − u(tj−1) u(tj−1) + k ∣∣ +cn−1 ≤ cn−1, as ‖u′‖ ≤ c. hence, we have established the nodal error bound |(u′ −du)(tj)| ≤ cn−1(ln n)2. biomath 9 (2020), 2008227, http://dx.doi.org/10.11145/j.biomath.2020.08.227 page 9 of 12 http://dx.doi.org/10.11145/j.biomath.2020.08.227 j. j. h. miller, e. o’riordan, robust numerical method for a singularly perturbed problem arising in ... 0 2 4 6 8 10 12 14 16 18 20 time s 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 u a n d v epsilon = 2.1362e-05, n= 64 u v figure 1. the numerical solution u, v on the interval [0, 20]. moreover, as (2.2a) implies that u′(t) −u′(tj) = (1 −v(tj))(u(tj) −u(t))) +(u(t) + k−α)(v(t) −v(tj)), this nodal error bound can be extended to the global error bound. for all 1 ≤ j ≤ n |u′(t) −du(tj)| ≤ cn−1(ln n)2, t ∈ [tj−1, tj]. v. numerical results in this section we present some numerical results for the numerical method (3.8) applied to the system (2.2) where the parameters in (2.1) have been taken to be k1 = 16847, k−1 = 7, k2 = 12, s(0) = 2.5 × 10−3, e(0) = 5.4 × 10−8. this yields the following parameter values for the non-dimensional system (2.2) k= 0.4511, α= 0.2849 ε= e(0) s(0) = 1.4 × 2−16. these values are used in [1] to fit the henrimichaelis-menten system to experimental data derived from acetylcholine hydrolysis by acetylcholinesterase. we examine the performance of the numerical method over an extensive range of the parameter ε, which is equivalent to varying the initial concentration e(0) of the enzyme. in figure 1 the computed approximations u,v (generated from the finite difference scheme (3.8)) are displayed. the plots confirm that v has an initial layer. larger values of the parameter ε will result in less steep gradients appearing in the plot of v. the global orders of convergence of the finite difference scheme (3.8) are estimated using the double-mesh principle [2, chapter 8, pg. 170]. note that this principle provides estimates of the orders of convergence despite the fact that the exact solution is unknown. we denote by un and u2n the computed solutions on the shishkin meshes ωn and ω2n respectively. these solutions are used to compute the maximum two-mesh global differences d̄nε := ‖ū n − ū2n‖ωn∪ω2n , ‖g‖ωn := max tj∈ωn |g(tj)|, where ūn (ū2n ) denotes the linear interpolant of the discrete solutions un (u2n ) over the mesh ωn (ω2n ), respectively. the uniform global twomesh differences d̄n and their corresponding unibiomath 9 (2020), 2008227, http://dx.doi.org/10.11145/j.biomath.2020.08.227 page 10 of 12 http://dx.doi.org/10.11145/j.biomath.2020.08.227 j. j. h. miller, e. o’riordan, robust numerical method for a singularly perturbed problem arising in ... table i two mesh global differences d̄nε , parameter-uniform two mesh differences d̄ n and orders of parameter-uniform global convergence p̄n for the component u ε n=8 n=16 n=32 n=64 n=128 n=256 n=512 20 0.0388 0.0192 0.0095 0.0069 0.0047 0.0029 0.0018 2−2 0.0784 0.0493 0.0154 0.0073 0.0045 0.0026 0.0015 2−4 0.1155 0.0817 0.0300 0.0068 0.0046 0.0036 0.0021 2−6 0.1278 0.0928 0.0361 0.0100 0.0054 0.0042 0.0025 2−8 0.1312 0.0957 0.0377 0.0109 0.0056 0.0044 0.0026 2−10 0.1320 0.0965 0.0381 0.0111 0.0057 0.0045 0.0027 2−12 0.1322 0.0967 0.0382 0.0112 0.0057 0.0045 0.0027 . . . . . . . . . . . . . . 2−30 0.1323 0.0967 0.0383 0.0112 0.0057 0.0045 0.0027 d̄n 0.1323 0.0967 0.0383 0.0112 0.0057 0.0045 0.0027 p̄n 0.4516 1.3375 1.7734 0.9684 0.3498 0.7491 0.8875 table ii two mesh global differences d̄nε , parameter-uniform two mesh differences d̄ n and orders of parameter-uniform global convergence p̄n for the component v ε n=8 n=16 n=32 n=64 n=128 n=256 n=512 20 0.0428 0.0349 0.0249 0.0163 0.0101 0.0060 0.0035 2−2 0.0573 0.0322 0.0275 0.0207 0.0124 0.0067 0.0034 2−4 0.0484 0.0332 0.0324 0.0260 0.0152 0.0081 0.0042 2−6 0.0454 0.0365 0.0337 0.0272 0.0159 0.0085 0.0045 2−8 0.0461 0.0374 0.0340 0.0275 0.0160 0.0087 0.0045 2−10 0.0463 0.0376 0.0341 0.0276 0.0161 0.0087 0.0045 2−12 0.0463 0.0377 0.0341 0.0276 0.0161 0.0087 0.0046 . . . . . . . . . . . . . . 2−30 0.0463 0.0377 0.0341 0.0276 0.0161 0.0087 0.0046 d̄n 0.0573 0.0377 0.0341 0.0276 0.0161 0.0087 0.0046 p̄n 0.6039 0.1440 0.3050 0.7781 0.8873 0.9358 0.9663 form orders of convergence p̄n are calculated from d̄n := max ε∈s d̄nε , p̄ n := log2 ( d̄n d̄2n ) , where s = {20, 2−1, . . . , 2−30}. the maximum two-mesh global differences d̄nε for each component u,v are, respectively, displayed in tables i and ii. the uniform two-mesh global differences d̄n , and their global orders of convergence p̄n are given in the last two rows of each table. these numerical results are in line with the asymptotic error bound established in theorems 5 and 6. in the final table iii, the uniform two-mesh global differences and their global orders of convergence for the discrete approximations to the rate of reaction u′(t) are displayed, which again show parameter-uniform convergence. in all three tables we observe that, for n sufficiently large, the global orders of convergence p̄n are tending towards the rate associated with the bound n−1(ln n)2. vi. conclusion a numerical method is constructed for a singularly perturbed system of two coupled nonlinear initial value equations. theoretical error bounds are established at all time points, which guarantee that the numerical approximations converge to the continuous solution, irrespective of the size of the singular perturbation parameter. numerical results support these theoretical error bounds. biomath 9 (2020), 2008227, http://dx.doi.org/10.11145/j.biomath.2020.08.227 page 11 of 12 http://dx.doi.org/10.11145/j.biomath.2020.08.227 j. j. h. miller, e. o’riordan, robust numerical method for a singularly perturbed problem arising in ... table iii two mesh global differences parameter-uniform two mesh differences and global orders of parameter-uniform convergence for the discrete derivative d−u ε n=8 n=16 n=32 n=64 n=128 n=256 n=512 20 0.0864 0.0841 0.0695 0.0509 0.0340 0.0212 0.0126 2−2 0.1324 0.0706 0.0484 0.0336 0.0248 0.0138 0.0100 2−4 0.1415 0.0736 0.0546 0.0368 0.0268 0.0149 0.0104 2−6 0.1424 0.0747 0.0566 0.0376 0.0274 0.0152 0.0105 2−8 0.1425 0.0754 0.0572 0.0378 0.0276 0.0153 0.0105 2−10 0.1426 0.0756 0.0573 0.0379 0.0276 0.0153 0.0105 2−12 0.1426 0.0757 0.0573 0.0379 0.0276 0.0153 0.0105 . . . . . . . . . . . . . . 2−30 0.1426 0.0757 0.0573 0.0379 0.0276 0.0153 0.0105 d̄n 0.1426 0.0841 0.0695 0.0509 0.0340 0.0212 0.0126 p̄n 0.7615 0.2742 0.4503 0.5839 0.6812 0.7494 0.7964 references [1] s. dimitrov, s. markov, metabolic rate constants: some computational aspects, math. comput. simulation, 133, 91–110, (2017). [2] p. a. farrell, a. f. hegarty, j. j. h. miller, e. o’riordan, g. i. shishkin, robust computational methods for boundary layers, chapman & hall, new york, 2000. [3] v. henri, lois generales de l’action des diastases, hermann, paris, 1903. [4] r. ishwariya, j. princy merlin, j. j. h. miller, s. valarmathi, a parameter uniform almost first order convergent numerical method for a non-linear system of singularly perturbed differential equations, biomath 5, 1-8, (2016). [5] l. michaelis, m. l. menten, die kinetik der invertinwirkung, biochem. z. 49 333-369, (1913). [6] j.j.h. miller, e. o’riordan and g.i. shishkin, fitted numerical methods for singular perturbation problems, world-scientific, singapore (revised edition), 2012. [7] j. d. murray, mathematical biology. i an introduction, springer, third edition, 2001. [8] l. a. segel and m. slemrod, the quasi-steady-state assumption: a case study in perturbation, siam rev., 31 (3), 446-477, (1989). [9] a. zagaris, h. g. kaper and t. j. kaper, fast and slow dynamics for the computational singular perturbation method, multiscale model. simul., 2 (4), 613–638, (2004). biomath 9 (2020), 2008227, http://dx.doi.org/10.11145/j.biomath.2020.08.227 page 12 of 12 http://dx.doi.org/10.11145/j.biomath.2020.08.227 introduction continuous problem discrete problem error analysis numerical results conclusion references original article biomath 2 (2013), 1311261, 1–6 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum the linear space of hausdorff continuous interval functions dedicated to prof. svetoslav markov on the occasion of his 70th birthday. jan harm van der walt department of mathematics and applied mathematics university of pretoria pretoria, south africa janharm.vanderwalt@up.ac.za received: 11 october 2013, accepted: 26 november 2013, published: 23 december 2013 abstract—in this paper we discuss the algebraic structure of the space h(x) of finite hausdorff continuous interval functions defined on an arbitrary topological space x . in particular, we show that h(x) is a linear space over r containing c (x), the space of continuous real functions on x , as a linear subspace. in addition, we prove that the order on h(x) is compatible with the linear structure introduced here so that h(x) is an archimedean vector lattice. keywords-interval functions; vector space; vector lattice; i. introduction interval functions, and interval analysis in general, is typically associated with numerical analysis and validated computing, see for instance [22], [23]. since the inception of the field in the 1960s, interval functions have been used extensively in mathematics and its applications, including in approximation theory [26], nonsmooth and nonlinear analysis [8], [10], [13], differential inclusions [9], [12], [16], [17], convex analysis [13], [25], nonlinear partial differential equations [5], functional analysis [6] and optimization and optimal control theory [13], [17]. furthermore, interval analysis, and set-valued analysis in general, are essential tools in the design and analysis of mathematical models in the life sciences. as pointed out in [21], biological dynamic systems typically involve uncertain data and / or parameters, numerical and / or inherent sensitivity and structural uncertainties which necessitate model validation. problems related to these issues of uncertainty and sensitivity, including computing enclosures for sets of solutions [24] and estimation of parameter ranges [15], essentially belong to set-valued analysis in general and are often addressed within the setting of interval analysis. we refer the reader to [21] for more details. in [4] it was shown that the set h(x) of hausdorff continuous (h-continuous) interval functions defined on an open subset x of rn is a linear space over r containing c (x) as a linear subspace. the aim of this paper is to extend this result to the most general case. in particular, we will show that h(x) is a linear space over r for every topological space x . as will be seen, this extension of the result in [4] requires an entirely new approach, since the methods used in [4] do not apply in the general setting considered here. while the result as such is not entirely unexpected, the method used in the proof is of intrinsic interest. indeed, our method reveals the essential mathematical mechanism that allows the extension of the linear operations on c (x) to the larger set h(x), namely, a minimality condition satisfied by h-continuous functions which is (almost) preserved by pointwise interval arithmetic [3]. furthermore, our method gives an indication of how to define algebraic operations on certain spaces of setvalued maps that take values in a general metrisable topological vector space. this problem is addressed in [7] in the case when the domain is a baire space, while the method developed here for h-continuous functions citation: jan harm van der walt, the linear space of hausdorff continuous interval functions, biomath 2 (2013), 1311261, http://dx.doi.org/10.11145/j.biomath.2013.11.261 page 1 of 6 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2013.11.261 j h van der walt, the linear space of hausdorff continuous interval functions gives some direction as to how one may proceed in the case of functions defined on a general topological space. ii. preliminaries a. h-continuous functions for the convenience of the reader unfamiliar with interval analysis, we now recall those definitions and results that will be used in subsequent sections. we denote by ir the set of real intervals, that is ir ={a = [a,a] : a,a ∈r, a ≤ a}. for a topological space x , we call a function f : x → ir locally bounded if for every x ∈ x there exists a neighbourhood v of x and a constant m ∈ r so that f (y) ⊆ [−m,m] for every y ∈ v . the set of locally bounded functions f : x →ir is denoted as a(x), while a (x) is the set of real valued locally bounded functions. every f ∈a(x) may be identified with a pair of (real valued) functions f , f ∈a (x) by setting f (x) = inf f (x) and f (x) = sup f (x), x ∈ x . clearly f (x) = [ f (x), f (x)] for each x ∈ x . conversely, by identifying every a ∈ r with the degenerate interval [a,a], we may consider a (x) as a subset of a(x). the relation f ≤ g ⇔ ( ∀ x ∈ x : f (x)≤ g(x), f (x)≤ g(x) ) , (1) with f = [ f , f ], g = [g,g]∈a(x), defines a partial order on the set a(x). note that the order (1) extends the usual pointwise order on a (x). the upper and lower baire operators [1], [11], [26] are mappings s,i : a(x)→ a (x) defined by s( f )(x) = inf{sup{z ∈ f (y) : y ∈v} : v ∈ vx} and i( f )(x) = sup{inf{z ∈ f (y) : y ∈v} : v ∈ vx}, respectively, with vx denoting the set of open neighbourhoods at x ∈ x . the graph completion operator f : a(x)→a(x) is defined by setting f( f ) = [i( f ),s( f )], f ∈a(x). the operators i, s and f are monotone with respect to the order (1) on a(x) and the pointwise order on a (x). that is, f ≤ g ⇒ i( f )≤ i(g), s( f )≤ s(g), f( f )≤ f(g). (2) consequently, f is inclusion isotone so that f (x)⊆ g(x), x ∈ x ⇒ f( f )(x)⊆ f(g)(x), x ∈ x. (3) furthermore, i(i( f )) = i( f ), s(s( f )) = s( f ), f(f( f )) = f( f ) (4) and i( f )≤ f( f )≤ s( f ) (5) for every f ∈ a(x), where we consider the real valued functions i( f ) and s( f ) as interval functions, as explained previously. using the graph completion operator, the following notions of continuity of interval functions are defined, see [1], [26]. definition 2.1: a function f ∈ a(x) is sendov continuous (s-continuous) if f( f ) = f . definition 2.2: a function f ∈ a(x) is hausdorff conitnuous (h-continuous) if f is s-continuous and f = g for every s-continuous function g so that g(x) ⊆ f (x), x ∈ x . the set of all s-continuous functions is denoted by f(x) while h(x) denotes the set of h-continuous functions. we may note that the baire operators characterise lower and upper semi-continuity of real functions. in particular, for every f ∈ a (x) we have f is lower semi−continuous ⇔ i( f ) = f and f is upper semi−continuous ⇔ s( f ) = f . consequently, f = [ f , f ] ∈ a(x) is s-continuous if and only if f is lower semi-continuous and f is upper semicontinuous. therefore if we identify a continuous function f ∈ c (x) with the (degenerate) interval function [ f , f ], it is clear that the resulting interval function is scontinuous and, since its values are degenerate intervals, it is h-continuous. conversely, if f = [ f , f ] ∈ h(x) is such that f (x) is a degenerate interval for every x ∈ x , then f = f is a continuous function. in this way we may identify c (x) in a canonical way with a subset of h(x), namely, the set of h-continuous functions with degenerate values at every x ∈ x . we make no distinction between a continuous function f and the interval function [ f , f ] so that we consider c (x) as a subset of h(x). recall the following results on h-continuous functions [1] which will be used in subsequent sections. theorem 2.3: the following are equivalent for all f = [ f , f ]∈a(x). (i) f is h-continuous. (ii) f( f ) = f( f ) = f . biomath 2 (2013), 1311261, http://dx.doi.org/10.11145/j.biomath.2013.11.261 page 2 of 6 http://dx.doi.org/10.11145/j.biomath.2013.11.261 j h van der walt, the linear space of hausdorff continuous interval functions (iii) s( f ) = f and i( f ) = f . theorem 2.4: if f = [ f , f ] is h-continuous, then the set dε ( f ) ={x ∈ x : f − f < ε} is open and dense in x for every ε > 0. theorem 2.5: the functions f(s(i( f ))) and f(i(s( f )) are h-continuous for every f ∈a(x). we may note that if f = [ f , f ] is s-continuous then, due to (4), we have f(s(i( f ))) = f(s( f )) = [i(s( f )),s( f )]. since f = i( f )≤ i(s( f ))≤ s( f )≤ s( f ) = f it follows that f(s(i( f )))(x)⊆ f (x), x ∈ x. (6) similarly, f(i(s( f )))(x)⊆ f (x), x ∈ x . b. interval arithmetic we now recall the arithmetic operations on ir, see for instance [3]. the sum of two intervals a = [a,a],b = [b,b]∈ ir is given by a + b = [a + b,a + b]. the scalar multiple of a ∈ ir and α ∈r is defined as α a = [min{α a,α a},max{α a,α a}]. with respect to addition and scalar multiplication, defined in this way, ir is not a vector space over r. indeed, the additive identity in ir is clearly the degenerate interval 0 = [0,0]. however, if a = [a,a]∈ ir is a proper interval, that is a 6= a, then a does not have an additive inverse. however, as is shown in [20], ir does have a rich algebraic structure. in fact, ir is a quasilinear space with cancelation law. that is, the following properties are satisfied for all a,b,c ∈ ir and α,β ∈r. (i) a +(b + c) = (a + b)+ c (ii) a + b = b + a (iii) a + b = a + c ⇒ b = c (iv) α(a + b) = α a + α b (v) α(β a) = (α β )a (vi) 1a = a (vii) (α + β )a = α a + β a if α β ≥ 0 iii. the linear space h(x) the linear structure of c (x) is defined, naturally, through the pointwise operations ( f + g)(x) = f (x)+ g(x), x ∈ x and (a f )(x) = a f (x), x ∈ x. the fact that these operations turn c (x) into a linear space over r is due to the fact that addition and multiplication are continuous operations form r×r to r, and that r is a linear space over itself. the first condition implies that c (x) is closed under the operations of pointwise addition and scalar multiplication, while the second ensures that the resulting operations satisfy the axioms of a linear space. for f ,g ∈a(x), let us denote by f ⊕g the pointwise interval sum of f and g. that is, f ⊕g(x) = f (x)+ g(x) = [ f (x)+ g(x), f (x)+ g(x)], x ∈ x. (7) for α ∈r, the pointwise product α � f is defined as α � f (x) = α f (x) = [min{α f (x),α f (x)},max{α f (x),α f (x)}]. (8) we note the following consequence of the fact that ir is a quasi-linear space over r. proposition 3.1: with respect to addition and scalar multiplication given by (7) and (8), a(x) is a quasilinear space with cancellation law. when attempting to extend the pointwise operations on c (x) to h(x) in this way, one encounters a double breakdown of the above situation. firstly, h(x) is not closed under the pointwise operation of interval addition. example 3.2: consider the functions f ,g ∈ h(r) given by f (x) =   0 i f x < 0 [0,1] i f x = 0 1 i f x > 0 and g(x) =   0 i f x < 0 [−1,0] i f x = 0 −1 i f x > 0 biomath 2 (2013), 1311261, http://dx.doi.org/10.11145/j.biomath.2013.11.261 page 3 of 6 http://dx.doi.org/10.11145/j.biomath.2013.11.261 j h van der walt, the linear space of hausdorff continuous interval functions the pointwise sum of f and g is f ⊕g(x) =   0 i f x 6= 0 [−1,1] i f x = 0 which is clearly not h-continuous. it is easily seen that f(x) is closed under the pointwise interval operations. however, here we encounter the second breakdown: the pointwise interval operations on f(x) do not satisfy the axioms of a linear space. this is due to the fact that, as mentioned in section ii (b), ir is not a vector space over r. the main result of this section, namely, that h(x) is a vector space over r for any topological space x , is a consequence of the following. proposition 3.3: if f = [ f , f ] is s-continuous and the set dε ( f ) = { x ∈ x : f (x)− f (x) < ε } contains an open and dense subset of x for every ε > 0, then f contains exactly one h-continuous function. proof: since f is s-continuous, it follows from (2), (3) and (5) that f contains the functions h1 = f(i(s( f ))) and h2 = f(s(i( f ))), and h1 ≤h2. according to theorem 2.5 both h1 and h2 are h-continuous. suppose that g = [g,g] is h-continuous and g(x)⊆ f (x), x ∈ x . then f ≤ g ≤ g ≤ f so that the monotonicity of the operators i and s (2) and theorem 2.3 implies that h1 ≤ g ≤ h2. therefore it is sufficient to shown that h1 = h2. suppose that there exists x0 ∈ x so that h2(x0) > h1(x0). let m = (h2(x0) + h1(x0))/2 so that h2(x0) > m > h1(x0). pick any ε > 0 so that h2(x0) > m + ε/2 > m−ε/2 > h1(x0). since h2 is lower semi-continuous and h1 is upper semicontinuous, there exists v ∈ vx0 so that h2(x) > m + ε/2 > m −ε/2 > h1(x) for every x ∈ v . then h2(x)− h1(x) > ε for every x ∈v . according to (2) and (5), f = s( f )≥ s(i( f )) = h2 ≥ h2 and f = i( f )≤ i(s( f )) = h1 ≤ h1. therefore f (x)− f (x) ≥ h2(x)−h1(x) > ε for every x ∈v . since v is open, it follows that dε ( f ) not dense in x , contrary to the assumption that dε ( f ) contains an open and dense subset x . therefore h2(x) ≤ h1(x) for every x ∈ x . then, since h1 is upper semi-continuous and h2 is lower semi-continuous, (2) and theorem 2.3 imply that h2 = s(h2)≤ s(h1) = h1 and h2 = i(h2)≤ i(h1) = h1 so that h2 ≤ h1. hence h1 = h2. corollary 3.4: if f1,..., fn are h-continuous, then the function f1 ⊕ ...⊕ fn contains exactly one h-continuous function. proof: we give a proof for the case when n = 2. the general case follows in exactly the same way. let f = [ f , f ] and g = [g,g] be h-continuous. therefore f and g are s-continuous, so that f ,g are lower semicontinuous while f ,g are upper semi-continuous. since the sum of lower (upper) semi-continuous functions are lower (upper) semi-continuous, it follows that f ⊕g = [ f ⊕g, f ⊕g] = [ f + g, f + g] is s-continuous. fix ε > 0. it follows from theorem 2.4 that dε ( f ⊕ g) = {x ∈ x : f ⊕g(x)− f ⊕g(x) < ε} contains an open and dense subset of x . in particular, d ε 2 ( f )∩d ε 2 (g) ⊆ dε ( f ⊕g). the result follows from proposition 3.3. in view of corollary 3.4 we define addition in h(x) as follows. definition 3.5: for f ,g ∈ h(x), the sum f + g of f and g is the unique h-continuous function contained in f ⊕g. theorem 3.6: h(x) is a linear space over r with addition defined as in definition 3.5 and scalar multiplication given by (8). furthermore, c (x) is a linear subspace of h(x). proof: consider f ,g,h ∈ h(x) and α,β ∈ r. denote by 0 both the additive identity in r and the hcontinuous function that is identically 0. h(x) is closed under scalar multiplication. if α = 0, then α f = α � f = 0, so α f is h-continuous. suppose that α > 0. clearly α f = [α f ,α f ] is s-continuous. suppose that f1 = [ f 1, f 1] ⊆ α f is s-continuous. then 1 α f1 = [ 1 α f 1 , 1 α f 1] is s-continuous and 1 α f1 ⊆ f so that 1 α f1 = f . hence f1 = α( 1 α ) f1 = α f so that α f is hcontinuous. the case when α < 0 is dealt with in the same way. associativity of addition. since addition in ir is associative, it follows that f ⊕(g⊕h) = ( f ⊕g)⊕h = f ⊕g⊕g. now f +(g + h)⊆ f ⊕(g + h)⊆ f ⊕(g⊕h) = f ⊕g⊕h and, similarly, ( f + g)+ h ⊆ f ⊕g⊕h. it now follows from corollary 3.4 that f +(g+h) = ( f +g)+h. commutativity of addition. since f + g ⊆ f ⊕ g = g⊕ f ⊇ g + f , it follows from corollary 3.4 that f + g = g + f . additive identity. clearly f + 0 = f ⊕0 = f . additive inverse. we have f + (−1 f ) ⊆ f ⊕(−1 f ) = [ f − f , f − f ] so that 0 ⊆ f ⊕ (−1 f ). corollary 3.4 implies that f +(−1 f ) = 0. first distributive law. according to proposition 3.1, α( f + g) ⊆ α( f ⊕g) = (α f )⊕(α g) ⊇ α f + α g. thus biomath 2 (2013), 1311261, http://dx.doi.org/10.11145/j.biomath.2013.11.261 page 4 of 6 http://dx.doi.org/10.11145/j.biomath.2013.11.261 j h van der walt, the linear space of hausdorff continuous interval functions α( f + g) = α f + α g by corollary 3.4. second distributive law. according to definition 3.5, α f +β f ⊆ α f ⊕β f . but by proposition 3.1, α f ⊕β f = (α + β ) f if α β ≥ 0 so that, in this case, (α + β ) f = α f +β f by definition 2.2. now suppose that α > 0 and β < 0. if α + β > 0, then α f + β f ≤ α f + β f = (α + β ) f ≤ (α + β ) f ≤ α f + β f so that (α + β ) f ⊆ α f ⊕β f . corollary 3.4 now implies that (α + β ) f = α f + β f . the case when α + β ≤ 0 follows in the same way. associativity of scalar multiplication. it follows from proposition 3.1 that α(β f ) = (α β ) f . multiplicative identity. it follows from (8) that 1 f = f . if f ,g ∈c (x), then f ⊕g = f +g so that c (x) is closed under addition in h(x), with the sum in h(x) corresponding to the pointwise sum in c (x). furthermore, in view of (8), c (x) is closed under scalar multiplication. therefore c (x), with the usual pointwise operations, is a linear subspace of h(x). finally we show that h(x) is an archimedean vector lattice with respect to the partial order (1), and that c (x) is a sublattice of h(x). theorem 3.7: h(x) is an archimedean vector lattice with respect to the partial order (1). furthermore, c (x) with the usual pointwise order is a sublattice of h(x). proof: first we prove that h(x) is a lattice. consider f = [ f , f ],g = [g,g] ∈ h(x) and let h′(x) = sup{ f (x),g(x)} for every x ∈ x . since h′ is lower semicontinuous, it follows from (2) and theorems 2.3 and 2.5 that h = f(s(h′)) ∈ h(x) and f ,g ≤ h. suppose that f ,g ≤ p for some p ∈ h(x). then f ,g ≤ p so that h′ ≤ p. theorem 2.3 and (2) imply that h = i(s(h′)) ≤ i(s(p)) = p. similarly h ≤ p so that h ≤ p. it therefore follows that h = sup{ f ,g} in h(x). in the same way inf{ f ,g} = f(i(h′′) where h′′(x) = inf{ f (x),g(x)} for every x ∈ x . next we show that h(x) is an archimedean vector lattice. in this regard, consider f ,g,h ∈ h(x) so that f ≤g and a real number α > 0. since α f = [α f ,α f ] and α g = [α g,α g] it is clear that α f ≤ α g. since f ⊕h = [ f + h, f + h] is s-continuous, it follows from (6) that f(s(i( f +h)))(x)⊆ f ⊕h(x), x ∈ x . since, by corollary 3.4, f ⊕h contains exactly one h-continuous function, namely f + h, and f(s(i( f + h))) is h-continuous by theorem 2.5, it follows that f + h = f(s(i( f + h))). in the same way, g + h = f(s(i(g + h))). since f + h ≤ g + h, it follows from (2) that f +h ≤ g+h. therefore h(x) is a vector lattice. since h(x) is dedekind complete [1], it follows from [18, theorem 25.1] that h(x) is archimedean. lastly we show that c (x) is a vector lattice subspace of h(x). since c (x) is a linear subspace of h(x), it is sufficient to show that c (x) is a sublattice of h(x), that is, we must show that the lattice operations in c (x) agree with those in h(x). this is clear from the respective expressions for sup{ f ,g} and inf{ f ,g} in h(x). we may note that theorems 3.6 and 3.7 are known, indirectly, for the class of so-called completely regular weak cb-spaces [19]. the dedkind completion of an archimedean riesz space is an archimedean riesz space [18]. anguelov [1] showed that if x is a completely regular space, then the dedekind completion of (the archimedean riesz space) c (x) is the set hcm(x) of h-continuous functions majorised by a continuous function. dăneţ [14] showed that hcm(x) = h(x) if and only if x is a weak cb-space. therefore, if x is a completely regular weak cb-space, then h(x) is the dedekind completion of the archimedean riesz space c (x), and is therefore an archimedean riesz space, and hence also a vector space over r. here we have given direct proofs of these algebraic results, in the most general case. iv. conclusion it has been shown that h(x) is a linear space, and in fact an archimedean riesz space, over r for an arbitrary topological space, generalising a result of anguelov, markov and sendov [4]. however, this generalisation required a new approach since the method [4] relies on the fact that, for x a baire space, a function f ∈h(x) is pointvalued and continuous on a dense subset of x . in the general case considered here, this property does not hold. our method may also apply to more general situations, namely, to spaces of set-valued maps with values in a metrizable topological vector space. the result presented in this paper also has implications for real analysis and the life sciences. in [2] it is shown that the rational completion of c (x) may be constructed as a space of (nearly finite) h-continuous functions when x is completely regular. our method may be applied to generalise this result to arbitrary topological spaces. furthermore, computations involving uncertainty, which occurs frequently when modeling phenomena in the life sciences, need not be restricted to functions with domain a baire space. biomath 2 (2013), 1311261, http://dx.doi.org/10.11145/j.biomath.2013.11.261 page 5 of 6 http://dx.doi.org/10.11145/j.biomath.2013.11.261 j h van der walt, the linear space of hausdorff continuous interval functions references [1] r. anguelov, dedekind order completion of c(x) by hausdorff continuous functions. quaestiones mathematicae 27, 153 – 170 (2004). http://dx.doi.org/10.2989/16073600409486091 [2] r. anguelov, the rational extension of c (x) via hausdorff continuous functions. thai journal of mathematics 5 no 2, 267 – 272 (2007). 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[26] b. sendov, hausdorff approximations. kluwer, dordrecht, 1990. biomath 2 (2013), 1311261, http://dx.doi.org/10.11145/j.biomath.2013.11.261 page 6 of 6 http://dx.doi.org/10.2989/16073600409486091 http://dx.doi.org/10.1007/s11155-006-9006-5 http://dx.doi.org/10.1016/j.camwa.2006.02.040 http://dx.doi.org/10.2989/16073600509486139 http://dx.doi.org/10.1016/j.camwa.2013.04.008 http://dx.doi.org/10.1016/0021-9045(89)90131-7 http://dx.doi.org/10.1016/0362-546x(91)90176-2 http://dx.doi.org/10.1007/bf01025922 http://dx.doi.org/10.2989/qm.2009.32.3.7.908 http://dx.doi.org/10.11145/j.biomath.2012.10.043 http://dx.doi.org/10.1090/s0002-9939-1965-0177388-1 http://dx.doi.org/10.1023/a:1011418014248 http://dx.doi.org/10.1063/1.3526621 http://dx.doi.org/10.1137/120870359 http://dx.doi.org/10.11145/j.biomath.2013.11.261 introduction preliminaries h-continuous functions interval arithmetic the linear space h(x) conclusion references www.biomathforum.org/biomath/index.php/biomath original article a stochastic model for intracellular active transport raluca purnichescu-purtan∗, irina badralexi† ∗ univeristy politehnica of bucharest raluca.purtan@gmail.com † univeristy politehnica of bucharest irina.badralexi@gmail.com received: 7 august 2018, accepted: 4 december 2018, published: 18 december 2018 abstract—we develop a stochastic model for an intracellular active transport problem. our aims are to calculate the probability that a molecular motor reaches a hidden target, to study what influences this probability and to calculate the time required for the molecular motor to hit the target (mean first passage time). we study different biologically relevant scenarios, which include the possibility of multiple hidden targets (which breed competition) and the presence of obstacles. the purpose of including obstacles is to illustrate actual disruptions of the intracellular transport (which can result, for example, in several neurological disorders [11]). from a mathematical point of view, the intracellular active transport is modelled by two independent continuous-time, discrete space markov chains: one for the dynamics of the molecular motor in the space intervals and one for the domain of target. the process is time homogeneous and independent of the position of the molecular motor. keywords-intermittent search; intracellular active transport; markov process i. introduction the numerous applications of intermittent search problems give way to complex mathematical models. from the behavior of foraging animals ([17], [19], [20]), to search and rescue operations ([12]) to active cell transport ([3], [4], [5], [6], [7], [8], [9], [15]), the stochastic nature of some processes fit perfectly in the framework of intermittent search strategies. this study is related to intracellular transport, which is the directed movement of substances within a cell. microtubules (microscopic hollow tubes) and microfilaments constitute a part of a cell’s cytoskeleton. these represent the roads in intracellular transport. an important element in vesicle transport is motor proteins. these proteins bind to vesicles and organelles and move them along the microtubule or microfilament network. the motor proteins which move vesicles along microtubules are called kinesins and dyneins, and those which copyright: c©2018 purnichescu-purtan et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: raluca purnichescu-purtan, irina badralexi, a stochastic model for intracellular active transport, biomath 7 (2018), 1812047, http://dx.doi.org/10.11145/j.biomath.2018.12.047 page 1 of 9 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2018.12.047 raluca purnichescu-purtan, irina badralexi, a stochastic model for intracellular active transport fig. 1. schematic presentation of the model system with one target domain. move vesicles along microfilaments belong to the myosin family. issues with intracellular transport can have serious consequences. they have been linked to neurological disorders such as alzheimers disease ([11], [21]). in the field of molecular biology, such models were designed to describe active transport of reactive chemicals in cells ([15]), promoter protein searching for a specific target site on dna ([7], [9]), transport of mrna ([8]). depending on the context of the activity which is being modeled, one can consider a one dimensional case, which means that the movement only occurs from left to right or vice-versa ([4]), or a higher dimensional case (two dimensional or three dimensional), where the movement is less restricted ([5]). the intermittent motions that occur at the microscopic level of reaction kinetics within biological cells are modelled in a one-dimensional framework. previous models were based on the assumptions that the search begins from a random point within a specified interval and that the target is reached upon entering the target interval ([3], [6], [15]). also, many of the studies regarding intermittent search problems in microbiology consider bidirectional transport and the condition that the target can be reached from a specified type of movement ballistic (anterograde, with constant velocity) or during a diffusive phase ([8], [16], [2], [13], [10], [18]). unlike the previous studies, in our research, the unidirectional active transport is modelled by two independent continuous-time, discrete space markov chains: one for the dynamics of the molecular motor in the space intervals (outside the target domain) and one for the domain of target. the novelty of our study consists of considering two types of motions inside and outside the target domain: a brownian motion (from which the target can be reached) and a state of active motor-driven transport along microtubules or microfilaments. we also consider an imperfect detection of the target, the scenario of multiple targets (competitivity) and the presence of obstacles. a stochastic algorithm was also developed. for each scenario we calculate the probability that a molecular motor reaches the target and calculate the time required for the molecular motor to hit the target (mean first passage time mfpt). ii. intracellular transport problem − one target domain a. general framework for one target domain consider a single motor-driven particle moving along a one-dimensional path of length l (unidirectional or anterograde movement). a ”hidden target” is located at a fixed (known) location on the path. the term ”hidden” means that the molecular motor may detect the target only when it enters into the target domain which lies fully within the interval i = [0,l], i.e., there is some positive constant � > 0 for which (a− �,a + �) ⊂ i. the mathematical framework for our stochastic model is based on the following general assumptions: a1. for the molecular motor we consider two movement regimes, denoted by: • ”u” uniform one-dimensional movement with velocity v > 0; • ”b” one-dimensional brownian motion (modeled as continuous time random walk). biomath 7 (2018), 1812047, http://dx.doi.org/10.11145/j.biomath.2018.12.047 page 2 of 9 http://dx.doi.org/10.11145/j.biomath.2018.12.047 raluca purnichescu-purtan, irina badralexi, a stochastic model for intracellular active transport a2. the particle motion is subjected to random decision moments which, independent from the time points, could change between the movement regimes; a3. the detection of the target (the searcher is ”absorbed” by the target) can be done only from brownian motion, in the interval (a − �,a+�) ⊂ i, and may take place with a given probability p (we consider an ”imperfect detection” there is always a possibility that the target remains undetected); a4. the first movement regime (at time t0 = 0) is the uniform one; a5. there is a finite time of observation [0,t] (the particle is absorbed by the target, leaves the path though the right point or the observation time elapses). b. choosing and changing the movement regime we consider a homogeneous finite markov chain 1 with the state space s = {u,b,d}, where u and b represent the movement regimes of the molecular motor and d is the absorbing state (the target is reached, the searcher remains in state d with probability one); we shall denote the cardinal of the set s by card(s) ; for the time interval [0,t], we consider a division 0 = t0 < t1 < ... < tn = t (the points t1, ..., tn−1 and the number n of the division points are random, see below the connection with the holding time). at t1, ..., tn−1 the particle probabilistically ”decides” (independent from the time points) if it remains in the previous movement regime or if it changes to the other regime (except from state d). if the state d is not reached by tn = t , we shall consider that the detection of the target failed. the n -step trajectory of the markov chain is s0,s1, ...,sn−1, where sk ∈ s is the state for the time interval (tk, tk+1], k = 1,n−1. the state holding time is denoted by t(k) = tk+1 −tk, for k = 1,n−1 (a random exponential 1for details regarding the theory of markov chains, relations between the probability matrix, rate transition matrix and holding times, see for example [14] variable with parameter λ). the value of the parameter is connected to the values of the diagonal elements of the rate transition matrix q (known): q = (qij)i,j=1,card(s) , with qii = − card(s)∑ j=1 i 6=j qij, with λ ≥ |qii|, for any i ∈ 1,card(s) . the transition probability matrix is: p = (pij)i,j=1,card(s) and the connection between the transition probabilities, the parameter λ and the transition rates is: pij = qij λ , i,j = 1,card(s), i 6= j pii = qii + λ λ , i,j = 1,card(s) (1) c. the movement of the molecular motor the molecular motor will move with sk regime on the interval (tk, tk+1], and we have two possibilities: first, if sk = u, the space position (the net displacement) of the ”searcher” is a continuous function of time and it is obtained by integrating the velocity function, as a function of time: x(tk+1) = x(tk) + tk+1∫ tk v dt (2) alternatively, if sk = b, the time interval (tk, tk+1] is divided into a sequence of times τ0,τ1, ...,τm (with τ0 = tk and τm = tk+1); then, for any fixed τi, i = 1,m, the space position (the net displacement) of the ”searcher” is given by: x(τi) = x(τi−1) + √ τi − τi−1yi, (3) where yi are values of a standard normal (gaussian) random variable, y ∈ n(0,1). biomath 7 (2018), 1812047, http://dx.doi.org/10.11145/j.biomath.2018.12.047 page 3 of 9 http://dx.doi.org/10.11145/j.biomath.2018.12.047 raluca purnichescu-purtan, irina badralexi, a stochastic model for intracellular active transport d. the general stochastic algorithm for one target domain for the stochastic algorithm, we consider the necessary input parameters: l, t , a, �, δτ, p, λ and the rate transition matrix q = (qij)i,j=1,2. in what follows, we briefly outline the steps of the algorithm. step 0: we construct the transition probability matrix p = (pij)i,j=1,2, according to (1), for a fixed value of λ. step 1: the particle enters the path and is moving in the space interval i1 = [0,a−�]. according to the general assumption a4., at t0 = 0 we have s0 = u and x(0) = 0 (the space position of the molecular motor). the holding time in states0 = u is generated using the integral inversion theorem2 ([1]): t(0) = − 1 λ ln(µ), where µ is a value of the random variable u ∈ unif(0,1). during this time, the particle is moving uniformly with the constant speed v along the path. after this time elapses, the new state is chosen (u with the probability p11 or b with probability p12; a monte-carlo discrete sequence is used). the position of the particle after its exits from the state u is calculated according to (2). step 2: while x(tk) ≤ a − �, with k > 0: leftmargin=* • randomly select the new state, using the corresponding row of the probability transition matrix p ; • generate the sojourn time in the current state,t(0) = − 1 λ ln(µ); • if the new state is u, see step 1; • if the new state is b, generate a value y from a standard normal (gaussian) random variable, y ∈ n(0,1) ; generate a sequence of equal step times τ0,τ1, ...,τm (within the sojourn time t(k)), with τ0 = tk and τm = tk+1; 2in [1] is called by the author quantile function theorem and is also known simply as the inversion method or ”smirnov transform”. during each time step δτ = τi+1 − τi, i = 1,m, the particle is moving in one dimension, according to (3); • update the new position of the particle and the time. step 3: the particle is moving in the target domain, interval i2 = (a− �,a + �) ⊂ [0,l]. the first state in this interval is corresponding to the last ”new” state from step 2. in order to include the new possible state d (the absorbing state), we construct the new transition probability matrix: p∗ =   p∗11 p∗12 0p∗21 p∗22 p∗23 0 0 1   . the values of the new transition probabilities can be completely different from the values in step 1 or can be different just in the second row (p21 = p∗21 and p22 = p ∗ 22 +p ∗ 23, for example); the detection of the target may take place with a given probability p = p∗23; as long as a− � < x(tk) < a + �, with k > 0: leftmargin=* • if the state d is not reached, see step 2 (with the matrix p∗ instead of p ); • if the state d is reached, the algorithm stops and the time and position of the particle are recorded; • if the time elapsed, the algorithm stops and the fact that the target was not reached is recorded. step 4: the particle is moving in the space interval i3 = [a+�,l]. if the state d is not reached in step 3 and the time has not elapsed, the motion of the molecular motor is simulated as in step 2. leftmargin=* • if the molecular motor re-enters the target interval from state b, go to step 3; • the algorithm stops when the particle reaches the end of the path, l, or the time has elapsed; the fact that the target was not reached is recorded. remark. the algorithm can be easily adapted for the case of different holding times for the states u and b (in the sequence where the holding time is generated, two different values, λ1 and λ2 should be used). biomath 7 (2018), 1812047, http://dx.doi.org/10.11145/j.biomath.2018.12.047 page 4 of 9 http://dx.doi.org/10.11145/j.biomath.2018.12.047 raluca purnichescu-purtan, irina badralexi, a stochastic model for intracellular active transport fig. 2. schematic presentation of the model system with two targets domain. e. general settings for the simulations depending on the domain of interest, an intermittent search strategy has its own natural mechanism. the use of the algorithm in biology, for example, requires some conditions to be met. thus, the net displacement of the particle in the state u always exceeds the maximum net displacement of the particle during the brownian motion state b: x(tk+1)−x(tk) ≥ max (x(τk+1)−x(τk)) also, the length of the target domain has to be proportional to the maximum net displacement of the particle during the brownian motion state. for our simulations, we consider: 2� ' 10−3 max (x(τk+1)−x(τk)) to ensure the adaptability and flexibility of the algorithm, space and time are measured in arbitrary units; iii. intracellular transport problem − two target domains (competitivity) we shall consider the same general framework as in the ”one target domain” case, but with two ”hidden targets”, located at two fixed but unknown locations on the path, denoted by a and b. the target domains are (a − �1,a + �1) ⊂ i and (b − �2,b + �2) ⊂ i, for some positive constants �1,�2 > 0 and the ”competitivity” is modelled within the overlapping target interval i4. if the molecular motor is in any of the non-overlapping target domains i2 or i3, it can be absorbed with two different given probabilities p1 and p2 (we consider an ”imperfect detection”). the general assumptions a1. − a5. are valid and the state spaces and the transition probability matrices for the markov chains are different for each interval. for intervals i1 and i5, the state space is s = {u,b} and the transition probability matrix is p1 = ( p11 p12 p21 p22 ) . for intervals i2 and i3, the state spaces are s = {u,b,d1} or s = {u,b,d2} and the transition probability matrix is p2 =   p11 p12 0p∗21 p∗22 p∗23 0 0 1   . in this case we left the values for the transition probabilities unchanged from state u (the first row). we consider the same regime of absorption for both targets (the probability of absorption is p1 = p2 = p ∗ 23). if there is some relevant biological reason for considering different absorption rates, p1 6= p2, we should consider two different transition probability matrices of type p2. for interval i4, the state space is s = {u,b,d1,d2} and the transition probability matrix is p3 =   p11 p12 0 0 p∗∗21 p ∗∗ 22 p ∗∗ 23 p ∗∗ 24 0 0 1 0 0 0 0 1   . in this case, the absorption probabilities are p1 = p ∗∗ 23 and p2 = p ∗∗ 24. these probabilities can be considered equal or different, depending on the biological model. biomath 7 (2018), 1812047, http://dx.doi.org/10.11145/j.biomath.2018.12.047 page 5 of 9 http://dx.doi.org/10.11145/j.biomath.2018.12.047 raluca purnichescu-purtan, irina badralexi, a stochastic model for intracellular active transport fig. 3. schematic presentation of the model system with one target domain and one obstacle. iv. intracellular transport problem − one target domain in the presence of one obstacle we shall consider the general framework of the ”one target domain” case, including an ”obstacle” particle, with a fixed but unknown location on the path, denoted c, with an obstacle interval (c − �2,c + �2) ⊂ i, for some positive constant �2 > 0. we shall assume that the obstacle domain is located on the path before the target domain (a − �1,a + �1) ⊂ i (or after the target domain, according to a biologically relevant scenario). we consider that the molecular motor can hit the obstacle only from brownian motion and the result of ”collision” may result in retrograde uniform movement (or other biologically relevant movement could be considered); the general assumptions a2.−a5. are valid. a1. should be modified as follows: a1∗. for the molecular motor we consider three movement regimes, denoted by: • ”u1” uniform one-dimensional movement with velocity v1 > 0 (anterograde); • ”b” one-dimensional brownian motion; • ”u2” uniform one-dimensional movement with velocity −v2 < 0 (retrograde); for intervals i1, i3 and i5, the state space is s = {u1,b} and the transition probability matrix is: p1 = ( p11 p12 p21 p22 ) . for interval i2, the state space is s = {u1,b,u2} and the transition probability matrix is: p2 =   p11 p12 0p∗21 p∗22 p∗23 p31 p32 0   . in this case, we left the values for the transition probabilities from state u1 unchanged and we consider that from the retrograde uniform movement, the molecular motor can reach only the states u1 or b. for interval i4, the state space is s = {u1,b,d} and the transition probability matrix is: p3 =   p11 p12 0p∗∗21 p∗∗22 p∗∗23 0 0 1   . v. numerical simulations numerical simulations were conducted only in the case of one target domain and that of two target domains. simulations for the movement of the molecular motor in the presence of obstacles, complete with interpretations and discussions in relevant biological situations, will be addressed in future work. a. one target domain for the numerical simulations, we considered the input data: v = 1, l = 10, t = 50, λ = 7, δτ = 0.001, p = 0.5714, p1 = ( 0.2857 0.7143 0.1429 0.8571 ) p2 =   0.2857 0.7143 00.1429 0.2857 0.5714 0 0 1   biomath 7 (2018), 1812047, http://dx.doi.org/10.11145/j.biomath.2018.12.047 page 6 of 9 http://dx.doi.org/10.11145/j.biomath.2018.12.047 raluca purnichescu-purtan, irina badralexi, a stochastic model for intracellular active transport target domain(a− �,a + �) ⊂ [0,10] (0.9975,1.0025) (1.4975,1.5025) (1.9975,2.0025) · · · (8.9975,9.0025) hitting probability 0.723 0.697 0.717 · · · 0.708 mfpt 1.4778 2.2873 2.8245 · · · 14.8667 table i the hitting probability and the mfpt depending on the position of the target domain. velocity for the uniform movement v = 1 v = 0.75 v = 0.5 v = 0.25 v = 0.1 v = 0.05 hitting probability 0.185 0.195 0.222 0.239 0.251 0.257 mfpt 14.506 16.245 18.717 22.062 24.981 25.673 table ii the hitting probability and the mfpt depending on the position of the velocity of the uniform movement. we are first interested in the variation of the hitting probability and the mfpt due to the different location of the target domain. we choose the first position of a (center of the target domain) at a = 1 and ”move it” with 0.5 units until the position a = 9. for every situation, � = 0.0025 and n = 10000 runs. we notice that there are no significant variations of the hitting probability due to a different location of the target, as long as the other parameters are fixed. however, the mfpt increases (linearly) with the distance from 0 to the target domain. (see table 1) next, we focus on the variation of the hitting probability and the mfpt due to the different velocities of the uniform movement. because the position of the target domain has no influence on the hitting probability, we choose the target domain (a− �,a + �) = (2.995,3). as expected, both the hitting probability and the mfpt increase as the velocity of the uniform movement decreases (see table 2). b. competitive targets for the numerical simulations, we considered the input data: v = 1; l = 8; λ = 7, δτ = 0.001, �1 = 0.1, �2 = 0.12, n = 10000 runs. we positioned target a at the space point 6.1 and target b at 6.18. for the intervals [0,6) and (6.3,8], the computed transition probability matrix is: p1 = ( 0.2857 0.7143 0.1429 0.8571 ) . for the target domains (6,6.2) and (6.06,6.3), the computed transition probability matrices are: p2 = p3 =   0.2857 0.7143 00.2857 0.1429 0.5714 0 0 1   , therefore, the absorbtion probabilities are equal, p1 = p2 = 0.5714. for the ”competitive” domain (6.06,6.2), the computed transition probability matrix is: p4 =   0.2857 0.7143 0 0 0.2857 0.1429 0.2857 0.2857 0 0 1 0 0 0 0 1   . we focused on comparing the hitting probabilities and the mfpt for the two targets, taking into account our simulation settings. we observe that, for equal absorbtion probabilities, the location of the target is important (target biomath 7 (2018), 1812047, http://dx.doi.org/10.11145/j.biomath.2018.12.047 page 7 of 9 http://dx.doi.org/10.11145/j.biomath.2018.12.047 raluca purnichescu-purtan, irina badralexi, a stochastic model for intracellular active transport target a target b hitting probability 0.405 0.335 mfpt 24.935 25.162 table iii the hitting probability and the mfpt in a two target domain. a which is closer to the starting point has a greater hitting probability than target b which is further from the starting point). likewise, our simulations show that the mfpt is related to the location of the target. (see table 3) remark. in both scenarios, regardless of the other input data, if the ratio between the probability of entering in the state b and the probability of entering in the state u (at a time point) is less or equal to one (in the transition probability matrices), then the probability of hitting the target is close to 0 (the search is inefficient). vi. conclusion and future work we develop and analyze a stochastic model of directed intermittent search for a hidden target on a one-dimensional path within the framework of continuous time, discrete state markov chain. in order to gain flexibility and adaptability to different relevant biological situations, we modelled the stochastic movement of the molecular motor in the target domain with a different markov chain and we have also considered the possibility of different holding times for the markov processes involved. in the markov chain approach, we chose to define the transition probabilities using the transition rates and the holding times. this can be of great interest from a modelling point of view: one can compare the dynamics of the process under two different hypothesis: first, considering that the holding times in the distinct states are different (exponential random variables with different parameters λ) and second, using the same exponential distribution for all holding times (with a single parameter λ). the novelty of our work consists of the framework formulated for three realistic situations for a molecular motor to reach its target. this target lies in a target domain (once the particle reaches this domain, it is ”absorbed” by the target with a known probability). the first of the situations describes the movement of the molecular motor in search of a single target. for the second one, we considered one molecular motor particle which can be absorbed by two different targets (which are seen in competition). the third one introduces the search of one target in the presence of one obstacle (in this case, only the theoretical approach is presented; a more detailed study will be conducted in future work). for our framework, we considered two main types of movement: a uniform movement with constant velocity and a brownian motion. in the case of an obstacle, one more movement is introduced: a retrograde uniform movement with constant velocity. we developed a highly adaptable and flexible algorithm for the study of all three situations of intermittent search introduced. the main output of the algorithm is the hitting probability and the mfpt. the proposed algorithm allows for imperfect detection of the target in its domain, which can be easily converted in a unit probability perfect detection according to different biologically relevant assumptions. in the case of competitive targets, multiple scenarios can be configured through the modification of the ratio between the probabilities of absorption, the imperfect detection or the distinct holding times. this flexibility can lead to determining the parameters which significantly influence the hitting probability and the mfpt. there are many interesting connections to be made between the input parameters of the model and the output, depending on the biological data available and this will be our purpose in a future work. we will also give attention to a combination of multiple obstacles and multiple targets, in a relevant biological context, based on the stochastic algorithm presented in this paper. biomath 7 (2018), 1812047, http://dx.doi.org/10.11145/j.biomath.2018.12.047 page 8 of 9 http://dx.doi.org/10.11145/j.biomath.2018.12.047 raluca purnichescu-purtan, irina badralexi, a stochastic model for intracellular active transport vii. acknowledgement this work has been funded by university politehnica of bucharest, through the excellence research grants program, upb-gex 2017. identifier: upb-gex2017, ctr.no.85/25.09.2017. references [1] j.e. angus, the probability integral transform and related results, siam review 36 (1994), 652–654 [2] h. bannai, t. inoue, t. nakayama, m. hattori and k. mikoshiba, kinesin dependent, rapid bidirectional transport of er sub-compartment in dendrites of hippocampal neurons, j. cell. sci.117, (2004), 163 [3] o. benichou, m. coppey, m. moreau, p.h. suet and r. voituriez, optimal search strategies for hidden targets, phys. rev. lett. 94 (2005), s4275s4286 [4] o. benichou, m. coppey, m. moreau, p.h. suet and r. voituriez, astochas tic model for intermittent search strategies, j. phys.: condens. matter 17 (2005), 234109 [5] o. benichou, c. loverdo, m. moreau and r. voituriez, two-dimensional intermittent search processes: an alternative to lvy flight strategies, physical review e 74 (2006), 020102 [6] o. benichou, c. loverdo, m. moreau and r. voituriez, intermittent search process and teleportation, j. phys.: condens. matter 126 (2007), 234109 [7] o.g. berg, c. blomberg, association kinetics with coupled diffusional flows: special application to the lac repressor-operator system, biophys. chem. 4 (1976), 367–381 [8] p. bressloff, j. newby, directed intermittent search for hidden targets, new journal of physics 11 (2009) 1–22 [9] m. coppey, o. benichou, r. voituriez and m. moreau, kinetics of target site localization of a protein on dna: a stochastic approach, biophys. j. 87 (2004), 1640– 1649 [10] j.l. dynes and o. steward, dynamics of bidirectional transport of arc mrna in neuronal dendrites, j. comp. neurol. 500 (2007), 433 [11] m.a.m. franker, c.c. hoogenraad, microtubule-based transport basic mechanisms, traffic rules and role in neurological pathogenesis, journal of cell science 126, 2319–2329 [12] j.r. frost, l. d. stone, review of search theory: advances and applications to search and rescue decision support, us coast guard research and development center, groton (2001) [13] a. gennerich, d. schild finite-particle tracking reveals submicroscopic-size changes of mitochondria during transport in mitral cell dendrites, phys. biol. 3 (2007), 433 [14] j.w. lamperti, probability: a survey of the mathematical theory, john wiley &sons, usa, 2011. [15] c. loverdo, o. benichou, m. moreau and r. voituriez, enhanced reaction kinetics in biological cells, nat. phys. 4 (2008), 134–137 [16] j.m. newby, p.c. bressloff, directed intermittent search for a hidden target on a dendritic tree, physical review e 80 (2009), 021913 [17] j. revelli, f. rojo, c. e. budde, and h.s. wio, optimal intermittent search strategies: smelling a prey, journal of physics a: mathematical and theoretical 43, 195001 [18] m.s. rook, m. lu and k.s. kosik, camkii 3 untranslated regions-directed mrna translocation in living neurons: visualization by gfp linkage, j. neurosci 20 (2000), 6385 [19] g.m. viswanathan, v. afanasyev, s.v. buldyrev, e.j. murphy, p.a. prince and h.e. stanley, levy flight search patterns of wandering albatrosses, nature 381 (1996), 413–415 [20] g.m. viswanathan, sergey v. buldyrev, shlomo havlin, m.g.e. da luz, e.p. raposo and h. eugene stanley, optimizing the success of random searches, nature 401 (1999), 911–914 [21] k.j. de vos, a.j. grierson, s. ackerley, and c.c.j. miller, role of axonal transport in neurodegenerative diseases, ann. rev. neurosci. 31 (2008), 151–173 biomath 7 (2018), 1812047, http://dx.doi.org/10.11145/j.biomath.2018.12.047 page 9 of 9 http://dx.doi.org/10.11145/j.biomath.2018.12.047 introduction intracellular transport problem one target domain general framework for one target domain choosing and changing the movement regime the movement of the molecular motor the general stochastic algorithm for one target domain general settings for the simulations intracellular transport problem two target domains (competitivity) intracellular transport problem one target domain in the presence of one obstacle numerical simulations one target domain competitive targets conclusion and future work acknowledgement references www.biomathforum.org/biomath/index.php/biomath original article a novel multi-scale immuno-epidemiological model of visceral leishmaniasis in dogs j. shane welker, maia martcheva department of mathematics, university of florida, fl 32611, usa spane@ufl.edu, maia@ufl.edu received: 8 august 2018, accepted: 2 january 2019, published: 23 january 2019 abstract—leishmaniasis is a neglected and emerging disease prevalent in mediterranean and tropical climates. as such, the study and development of new models are of increasing importance. we introduce a new immuno-epidemiological model of visceral leishmaniasis in dogs. the within-host system is based on previously collected and published data, showing the movement and proliferation of the parasite in the skin and the bone-marrow, as well as the igg response. the between-host system structures the infected individuals in time-sinceinfection and is of vector-host type. the within-host system has a parasite-free equilibrium and at least one endemic equilibrium, consistent with the fact that infected dogs do not recover without treatment. we compute the basic reproduction number r0 of the immuno-epidemiological model and provide the existence and stability results of the population-level disease-free equilibrium. additionally, we prove existence of an unique endemic equilibrium when r0 > 1, and evidence of backward bifurcation and existence of multiple endemic equilibria when r0 < 1. keywords-leishmaniasis in dogs, backward bifurcation, immuno-epidemiological model, stability, parameter estimation, immune dynamics ams subject classification: 92d30 i. introduction the leishmaniases are a group of diseases found in over 90 countries around the world, spread by over 30 species of the phlebotomine sand flies and infecting a variety of hosts including humans and dogs. while cutaneous leishmaniasis is more common, visceral leishmaniasis (vl) is lethal if untreated. we focus on zoonotic visceral leishmaniasis (zvl), which has symptoms including enlarged spleen and liver and non-specific symptoms such as fever, weight loss, and anemia [1]. the non-specificity makes diagnosis challenging, particularly in the case of dogs [15]. the leishmaniases are classified as a neglected tropical disease (ntd), with an estimated 0.2-0.4 million new human cases per year [14] and hundreds of millions at risk of new infection [19]. the who has stated that it is one of the most significant tropical diseases in the world [19]. as such, it is imperative to continue the study of vl, including its epidemiology, immunology, control measures, and identification. visceral leishmaniasis is usually caused by the l. donovani and l. infantum protozoa. the l. donovani-induced vl is more common in africa copyright: © 2019 welker et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: j. shane welker, maia martcheva, a novel multi-scale immuno-epidemiological model of visceral leishmaniasis in dogs, biomath 8 (2019), 1901026, http://dx.doi.org/10.11145/j.biomath.2019.01.026 page 1 of 12 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2019.01.026 j. shane welker, maia martcheva, a novel multi-scale immuno-epidemiological model of visceral ... and asia, while the l. infantum-induced vl is more common in the americas and the mediterranean [10]. while dogs and other mammals are infected by the l. donovani-induced vl, it has been shown that dogs are a primary reservoir only for the disease caused by the l. infantum species [2]. in fact, previous work has postulated that dogs are the main contributor to the spread of vl, claiming that 20% of the infected dogs cause 80% of transmission [2]. for these reasons, we focus primarily on the role of dogs in the persistence of vl. to date, there have been few mathematical models for the between-host dynamics of vl [15]. some models have chosen to compartmentalize asymptomatic or latently infected, resistant, and/or recovered classes, but almost all models have been odes. some models, including the first model of vl in dogs by dye includes resistance from birth [3]. another model was developed by ribas et al [4] which studies the population level dynamics between dogs and humans. the model was used to argue that treatment in dogs does not reduce significantly human illness. shimozako et al considered a model of leishmaniasis in dogs and humans [16] and concluded that latent dogs contribute more to the illness than clinically ill dogs. seva et al [12] investigated an outbreak of vl in spain through a model that involved multiple host classes – rabbits, hares, dogs, and humans. all of the vl epidemic models studied to date are single-scale ode models. even fewer models have been made for the within-host dynamics. länger et al [8] developed a model examining the effect of igg1, igg2a, and lymphocytes on the parasite load, concluding that their model could be used in identifying biologically significant parameters [8]. siewe et al [17] modeled macrophages, parasite loads, dendritic cells, t cells, and cytokines, and simulated the effects of various control measures. as a result, they stated that an increase in ifn-γ production should lead to a decrease in parasite load; an implication for potential therapy. we develop, for the first time, an immunoepidemiological model for vl. while still in the early stages of development, we intend for this model to display the effect that the within-host dynamics may have on infection of the vector. additionally, we hope to assess the infectivity of a vector based on parasite reproduction and, eventually, control measure efficacy. similarly, we plan to examine the parasite reproduction inside hosts with respect to the efficacy towards a vector, immune response, and treatment. since these processes occur at a drastically different rate, we adopt the multi-scale approach. our within-host system was designed to fit data from courtenay et al [2]. our between-host system is of vector-host type, with the infected host class structured by time-since-infection. the betweenhost system contains susceptible, infected, and recovered host classes and susceptible, carrier, and infectious vector classes. courtenay et al [2] conclude that, while samples were taken from both the skin tissue and bone marrow to record parasite loads, it is the parasite load in the skin tissue that is the most reliable indicator of vl infection [2]. we use this fact in the linking of the withinand between-host systems. following the introduction of the model in section 2, we present a parasite-free equilibrium of the within-host system and prove it to be unstable in section 3.1. then, in section 3.2, we introduce the basic reproduction number r0, and prove the disease-free equilibrium of the immunoepidemiological model to be locally asymptotically stable when r0 < 1. we show the existence of an endemic equilibrium when r0 > 1, and the existence of multiple endemic equilibria and the presence of backward bifurcation, when r0 < 1. in section 4 we discuss our conclusions. ii. the model a. the within-host system our within-host model is motivated by time series data in courtenay et al [2] pertaining to the parasite loads in the skin tissue and bone marrow of dogs, as well as the immunoglobulin g (igg) concentration. we derive a system coupling the biomath 8 (2019), 1901026, http://dx.doi.org/10.11145/j.biomath.2019.01.026 page 2 of 12 http://dx.doi.org/10.11145/j.biomath.2019.01.026 j. shane welker, maia martcheva, a novel multi-scale immuno-epidemiological model of visceral ... vectors sv (t) cv (t) iv (t) de ath de ath de ath bite τbirth hosts sh(t) ∫ i(t,x)dx rh(t) birth de ath de ath de ath transmission recovery waning immun ity parasite load in skin (ps(x)) parasite load in bone marrow (po(x)) igg level (g(x)) within-host dynamics fig. 1: the flow-chart of the immuno-epidemiological model of vl in dogs dynamics of the parasite load in the skin tissue (ps(x)), the parasite load in the bone marrow (po(x)), and the igg concentration (g(x)), where x is the time-since-infection. the subscript s will indicate skin tissue, while the subscript o will indicate the bone marrow. we first introduce the model below and then we explain and motivate it. a full list of parameter meanings can be found in table i and the variable meanings in table iv: p ′s =rsps ( 1 − ps ks ) + 1 ρ kopo −ksps −εspsg (1) p ′o =ropo ( 1 − po ko ) −kopo + ksρps −εopog (2) g′ =asρps + aopo −dg (3) for the parasite loads, we assume reproduction is limited by a carrying capacities ks and ko. hence the equations for p ′s and p ′ o use logistic terms to model recruitment, with rates rs and ro. we note that to infect a host, a sand fly must deposit parasites into the skin, when it takes a blood meal. similarly, for a sand fly to become infected, it must take a blood meal from an infected host’s skin. as there are parasites in the bone marrow, we can assume mobility of the parasite between the skin tissue and bone marrow. hence our model contains “travel terms,” with rates ko and ks and density conversion coefficient ρ. thus, for every time within-host unit, a fraction of the parasites in the skin move to the bone marrow, and vice versa. while not much is known about the parasite, we assume that the life span is short, and we include the natural death as part of the logistic term. however, we include the clearance of the parasite induced by the immune response as a separate term. hence εs and εo are the igg induced clearance rates of the parasite. lastly, we assume that the igg response is caused by the presence of the foreign parasite, i.e., the basal level of igg present before the introduction of the parasite is not counted towards g. as such, as and ao are the igg production rates caused by the parasite loads. we assume that the clearance rate is the only way igg leave the system, which occurs at rate d. courtenay et al [2] obtained data for the parasite loads (pl) and igg concentration. we present simulations of our within-host system. the behaviors biomath 8 (2019), 1901026, http://dx.doi.org/10.11145/j.biomath.2019.01.026 page 3 of 12 http://dx.doi.org/10.11145/j.biomath.2019.01.026 j. shane welker, maia martcheva, a novel multi-scale immuno-epidemiological model of visceral ... table i: parameters for the within-host system. parameter description unit ao po induced igg production rate igg au / [parasites · time x] as ps induced igg production rate igg au / [parasites · time x] d natural clearance rate of igg 1 / [time x] εo igg induced po clearance 1 / [time x · igg au/ml] εs igg induced ps clearance 1 / [time x · igg au/ml] ko parasite carrying capacity in bone marrow parasites/ml ks parasite carrying capacity in skin tissue parasites/g ko parasite travel rate from bone marrow to skin tissue 1 / [time x] ks parasite travel rate from skin tissue to bone marrow 1 / [time x] ro parasite reproduction rate in bone marrow 1 / [time x] rs parasite reproduction rate in skin tissue 1 / [time x] ρ conversion coefficient [g skin] / [ml bone marrow] of these curves are similar to that of the data provided. the result of our simulation is given by the curves in figures 2(a)-2(c). the parameter values for the simulation can be found in table ii. b. the between-host system few models exist for the between-host dynamics of vl, and even fewer for zoonotic vl in dogs. previous models have largely been ode in structure. some models included an asymptomatic or latently infected class [15]. however, obtaning data on infectious dogs is obstructed by the fact that identifying infectious dogs can be very difficult [13]. since an infectious host is considered only infectious to the vector, the most reliable way to test for vl is through xenodiagnosis, a process in which a susceptible vector population bites a possibly infected host, and is then tested for the presence of the parasite [2]. however, xenodiagnosis is not always feasible or practical [2]. since the symptoms of vl are non-specific, it is difficult to separate the latently infected dogs from the infectious dogs. in our multi-scale model, we structure the infectious hosts by time-since-infection, with the assumption that hosts are less infectious and less likely to display symptoms closer to when they first contract the parasite. the time-since-infection structure provides flexibility; allowing for fitting data given in different time units in the two different scales – the within-host and the between-host. we first introduce the system for the between-host dynamics of vl. definitions of the parameters used can be found in table iii and definitions of the variables used in table iv: s′h = λh − βhashiv n −mhsh + γrh, (4) it + kuix = −(σ(x) + µ(x) + mh)i(t,x), (5) kui(t, 0) = βhashiv n , (6) r′h = ∫ ∞ 0 σ(x)i(t,x)dx− (γ + mh)rh, (7) where x is the time-since-infection and the total number of infected hosts, ih(t), is ih(t) = ∫ ∞ 0 i(t,x)dx. (8) the host system consists of susceptible sh(t), infected i(t,x), and recovered/resistant rh(t) classes. the constant ku in (5) and (6) accounts for the difference in the rates that the time t and timesince-infection x occur, i.e., x = kut. for the sake of initial analysis, we let ku = 1. susceptible hosts are born at rate λh, and move to the infected class with standard incidence βhashiv /n. recovery takes place at rate σ. the integral term in (7) is the total number of recovered individuals per unit time. hosts exit the system either through natural death, mh, or disease-induced mortality, µ. the existence of relapse and reinfection shows the necessity of waning immunity at rate γ (see table iii). the vector system consists of susceptible vectors sv (t), carrier vectors cv (t), who are infected biomath 8 (2019), 1901026, http://dx.doi.org/10.11145/j.biomath.2019.01.026 page 4 of 12 http://dx.doi.org/10.11145/j.biomath.2019.01.026 j. shane welker, maia martcheva, a novel multi-scale immuno-epidemiological model of visceral ... (a) (b) (c) fig. 2: figures (a) and (b) show simulations of the parasites in the skin and bone marrow, respectively, in log10 values. figure (c) shows simulations of log10 igg antibody units. table ii: parameter values used. parameter value unit ao 4.550060 × 10−1 igg au / [parasites · day] as 1.739817 × 10−5 igg au / [parasites · day] d 4.136157 × 10−3 parasites/day εo 8.927579 × 10−7 1 / [day · igg au/ml] εs 6.968101 × 10−7 1 / [day · igg au/ml] ko 1.034155 × 106 parasites/ml ks 1.007303 × 108 parasites/g ko 5.723339 × 10−9 1 / day ks 5.228469 × 10−4 1 / day ro 2.614700 × 10−2 1 / day rs 2.965272 × 10−2 1 / day ρ 1 [g skin] / [ml bone marrow] table iii: parameters for the between-host system. parameter description units a average biting rate bites / [time t · vectors] βh rate of host becoming infected after bite 1 / [bites/hosts] βv rate of vector becoming infected after bite 1 / [bites/vectors] γ rate of becoming susceptible after recovery 1 / [time t] ku time scaling constant [time x] / [time t] λh birth rate of hosts hosts / [time t] λv birth rate of vectors vectors / [time t] mh natural death rate of hosts 1 / [time t] mv natural death rate of vectors 1 / [time t] µ(x) disease induced death rate of hosts 1 / [time t] σ(x) rate of recovery of hosts 1 / [time t] τ rate of moving from carrier to infectious 1 / [time t] biomath 8 (2019), 1901026, http://dx.doi.org/10.11145/j.biomath.2019.01.026 page 5 of 12 http://dx.doi.org/10.11145/j.biomath.2019.01.026 j. shane welker, maia martcheva, a novel multi-scale immuno-epidemiological model of visceral ... table iv: definitions of dependent variables. variable description unit sh(t) susceptible hosts at time t hosts ih(t) infected hosts at time t hosts i(t,x) density of hosts infected x time units ago at time t hosts / [time x] rh(t) recovered hosts at time t hosts sv (t) susceptible vectors at time t vectors cv (t) carrier vectors at time t vectors n(t) host population hosts iv (t) infectious vectors at time t vectors po(x) parasite load of bone marrow at time x parasites/ml ps(x) parasite load of skin tissue at time x parasites/g g(x) igg concentration at time x igg au/ml table v: parameters for linking. parameter description unit ξ rate of exponential decay 1 / [parasites/g] cv maximal transmission coefficient 1 / [bites/vector] δ0 constant 1 / ml κ constant 1 / [igg au/ml · time t] η constant 1 / [igg au/ml · time t] ν constant (unitless) ψo constant 1 / [parasites/ml · time t] ψs constant 1 / [parasites/g · time t] θo constant 1 / parasites θs constant 1 / parasites but not infectious yet, and infectious vectors iv (t). while there are many unknowns about the sand flies and leishmania, it is known that the parasite must make its way through the sand fly after a blood meal before it can be deposited in a host. the time elapsed for the parasite to potentially infect a new host, called extrinsic incubation period, is comparable to the life span of the sand fly. this requires the carrier class for the vector. s′v = λv −mv sv − asv n ∫ ∞ 0 βv (x)i(t,x)dx (9) c′v = asv n ∫ ∞ 0 βv (x)i(t,x)dx−(τ +mv )cv (10) i′v = τcv −mv iv , (11) vectors are born at rate λv , and exit the system only through natural death rate mv . vectors move from the carrier class to the infectious class at rate τ. the integral terms in (9) and (10) represent the force of infection of humans to susceptible vectors. since the rate of infection (roi) is assumed to be dependent on x, the rate of recovery of hosts, σ, the disease-induced death rate of hosts, µ, and the rate of an infected host infecting a susceptible vector at the time of the blood meal, βv , are also dependent on x. c. linking the withinand between-host systems while much analysis is left, we note the different time scales that will be utilized. that of the parasites and vectors will occur much faster than that of the hosts. we introduce the methods in which we initially plan to incorporate the faster time scale into the spread of the virus. since courtenay et al [2] concluded that the parasite load in the dog skin tissue was the best indicator of infectiousness to the vector, we use ps(x) to determine the rate of transmission βv (x): βv (x) = cv e −ξpνs (x), (12) where ξ is the rate of exponential decay, cv is the maximal transmission coefficient, and ν = biomath 8 (2019), 1901026, http://dx.doi.org/10.11145/j.biomath.2019.01.026 page 6 of 12 http://dx.doi.org/10.11145/j.biomath.2019.01.026 j. shane welker, maia martcheva, a novel multi-scale immuno-epidemiological model of visceral ... −2/ ln(10). this relationship was derived by li et al [9], using data for dengue. assuming treatment, the recovery rate also depends on the time-since-infection. we assume recovery occurs when the within-host parasite load becomes zero. thus, we let σ(x) = κg δ0 + θsρps + θopo , (13) where δ0 is a small constant, and θs, θo, and κ are constants [18]. note that when ps and po approach 0, σ becomes large due to δ0. to link the disease-induced mortality, µ, we let µ(x) = ψsρps + ψopo + ηg, (14) where ψs, ψo, and η are constants, [6]. iii. analysis a. analysis of the within-host system the within-host system (1)-(3) has a parasitefree equilibrium e0 = (0, 0, 0). to determine its stability, we consider the jacobian at the parasitefree equilibrium. we have the following result. theorem 1. the parasite-free equilibrium e0 is always unstable. proof: let k̂o = 1 ρ ko, k̂s = ksρ, and âs = asρ. the jacobian of the within-host system evaluated at e0 is j0 :=  rs −ks k̂o 0k̂s ro −ko 0 âs ao −d   , which has the eigenvalue λ1 = −d. the remaining eigenvalues are eigenvalues of the submatrix j1 := ( rs −ks k̂o k̂s ro −ko ) . note that k̂ok̂s = koks. then e0 is stable if and only if det(j1) = (rs−ks)(ro−ko)−ksko > 0 and tr(j1) < 0. suppose that det(j1) > 0. note that det(j1) = rsro −rsko −roks = rs(ro −ko) −roks. so rs(ro−ko)−roks > 0 if and only if rs(ro− ko) > roks. then ro > ko. similarly, we can get rs > ks. hence tr(j1) > 0 when det(j1) > 0. therefore e0 is unstable. this stability result heuristically makes sense as, without the introduction of treatment, infected hosts stay infected. theorem 2. the within-host system (1)-(3) always has at least one parasite equilibrium e∗. this equilibrium is unique if rs < ks. if rs > ks, the equilibrium is unique if k̂o ( εs d âs + rs ks ) > (rs −ks) εs d ao (15) proof: to show existence, we set the withinhost system equal to zero and reduce the system to 0 =rsps ( 1 − ps ks ) + k̂opo −ksps − εs d ps(âsps + aopo) (16) 0 =ropo ( 1 − po ko ) −kopo + k̂sps − εo d po(âsps + aopo). (17) solving (16) for po, we obtain po = psf1(ps), where f1(ps) = 1 φo(ps) [ ks + εs d âsps−rs ( 1− ps ks )] and φo(ps) = k̂o − εs d aops. substituting po in (17), we obtain the following equation for ps: f(ps) := f1(ps)g1(ps) = 0 with g1(ps) =ro ( 1 − psf1(ps) ks ) −ko + k̂s f1(ps) − εo d (âsps + aopsf1(ps)). biomath 8 (2019), 1901026, http://dx.doi.org/10.11145/j.biomath.2019.01.026 page 7 of 12 http://dx.doi.org/10.11145/j.biomath.2019.01.026 j. shane welker, maia martcheva, a novel multi-scale immuno-epidemiological model of visceral ... denote by p̂s the value of ps such that φo(p̂s) = 0. further, denote by p̄s the value of ps such that f1(p̄s) = 0. we have p̄s = rs −ks εsâs d + rs ks , p̂s = k̂o εs d ao . we consider the following cases case 1:] we have rs < ks or ro < ko. then, f1(ps) > 0 and φo(ps) > 0 iff ps ∈ (0, p̂s). we have that f1(ps) is an incrasing function of ps. further g1(ps) is a decreasing function of ps. as f1(ps) > 0 on ps ∈ (0, p̂s) the roots of f(ps) = 0 are the same as the roots of g1(ps) = 0. since g1 is decreasing, if a root exists, it must be unique. since g1(0) = ro−ko + ksko ks −rs = ro + kors ks −rs > 0 if ks > rs, as in this case. on the other hand lim ps→p̂s − g1(ps) = −∞ since g1(ps) is continuous on (0, p̂s), then there is at least one solution of g1(ps) = 0. hence, there exists a unique p∗s ∈ (0, p̂s) such that f(p∗s) = 0 and p ∗ o = p ∗ sf1(p ∗ s) > 0. in this case, it is easy to see that g∗ = âsp ∗ s + aop ∗ o d is positive as well. case 2: we have rs > ks. since f1(0) < 0, the solution, if it exists, lies in a different interval. note f1(ps) > 0 iff ps ∈ (p̄s, p̂s). however, in this case we don’t know whether p̄s < p̂s or vice versa. so we have to consider two subcases. case 2a: assume inequality (15), that is, assume p̄s < p̂s. then f1(ps) is an increasing function of ps with f1(p̄s) = 0, lim ps→p̂s − f1(ps) = ∞. it is easy to see that in this case we also have lim ps→p̄s + g1(ps) =∞, lim ps→p̂s − g1(ps) = −∞. further, g1(ps) is also monotone and continuous as in case 1. hence, there exists a unique p∗s ∈ (p̄s, p̂s) such that f(p ∗ s) = 0 and p∗o = p ∗ sf1(p ∗ s) > 0. in this case it is easy to see that g∗ is positive as well. we also note that f(p̄s) = k̂s > 0. thus, p̄s is not a solution. case 2b: assume p̄s > p̂s. then f1(ps) is a decreasing function of ps. f1(p̄s) = 0, lim ps→p̂s − f1(ps) = ∞ it is easy to see that in this case we also have lim ps→p̄s + g1(ps) =∞, lim ps→p̂s − g1(ps) = −∞. since g1(ps) is also continuous as in case 1, there exists at least one solution in (p̂s, p̄s). however, in this case g1(ps) may not be monotone and the equilibrium may not be unique. hence, there exists at least one p∗s ∈ (p̂s, p̄s) such that f(p∗s) = 0 and p ∗ o = p∗sf1(p ∗ s) > 0. in this case it is easy to see that g∗ is positive as well, and p̄s is not a solution to f(ps) = 0. b. analysis of the full immuno-epidemiological model the immuno-epidemiological model has a disease-free equilibrium e0 = ( λh mh , 0, 0, λv mv , 0, 0 ) . (18) we linearize model (4)-(11) around e0. looking for exponential solutions of the form ~z(t) = zeλt, biomath 8 (2019), 1901026, http://dx.doi.org/10.11145/j.biomath.2019.01.026 page 8 of 12 http://dx.doi.org/10.11145/j.biomath.2019.01.026 j. shane welker, maia martcheva, a novel multi-scale immuno-epidemiological model of visceral ... we obtain the characteristic equation f(λ) = 1, where λ ∈ c and f(λ) = τ mv + λ · am τ + mv + λ βha·∫ ∞ 0 βv (x)e −λxe− ∫ x 0 (σ(ξ)+µ(ξ)+mh)dξdx. (19) then the basic reproduction number is f(0), or r0 = % of vectors that become infectious︷ ︸︸ ︷ τ τ + mv · aβh mv︸ ︷︷ ︸ rv · rh︷ ︸︸ ︷ am ∫ ∞ 0 βv (x)e − ∫ x 0 (σ(ξ)+µ(ξ)+mh)dξdx, (20) where rv is the basic reproduction number of the vectors and rh is the basic reproduction number of the hosts [11]. the parameter m denotes the ratio of the vector to hosts and is defined as m = λv mv mh λh . it should be noted that r0 is dependent on the within-host system. theorem 3. if r0 < 1, then the disease-free equilibrium e0 is locally asymptotically stable. if r0 > 1, then e0 is unstable. proof: suppose r0 < 1. then f(λ) = 1 has a unique negative solution λ∗ ∈ r. for a complex λ, let λ = α1 + iα2, and assume α1 ≥ 0. then |f(λ)| ≤ a2βhmτ |mv + λ| |τ + mv + λ| · ∫ ∞ 0 βv (x) ∣∣∣e−λx∣∣∣π(x)dx ≤ a2βhmτ (mv + α1)(τ + mv + α1) · ∫ ∞ 0 βv (x)e −α1xπ(x)dx =f(α1) ≤ f(0) = r0 < 1. since 1 = |f(λ)| ≤ r0 < 1, a contradiction, we must have α1 < 0. hence every complex solution to f(λ) = 1 must have a negative real part. therefore, e0 is locally asymptotically stable when r0 < 1. now suppose that r0 > 1. then for positive λ ∈ r, f ′(λ) = (mv + λ)(τ + mv + λ)(a 2βhmτ) (mv + λ)2(τ + mv + λ)2 · ∫ ∞ 0 (−x)βv (x)e−λxπ(x)dx − a2βhmτ(τ + 2mv + 2λ) (mv + λ)2(τ + mv + λ)2 · ∫ ∞ 0 βv (x)e −λxπ(x)dx = − a2βhmτ (mv + λ)2(τ + mv + λ)2 · [ (mv + λ)(τ + mv + λ) · ∫ ∞ 0 xβv (x)e −λxπ(x)dx + (τ +2mv +2λ) ∫ ∞ 0 βv (x)e −λxπ(x)dx ] < 0, since the bracketed expression is always positive for non-negative βv (x) 6≡ 0. since r0 > 1, then f(0) > 1. since lim λ→∞ f(λ) = 0 and f is decreasing, f(λ) = 1 has a unique positive solution λ∗ ∈ r. thus e0 is unstable. to study existence of endemic equilibria, we set the time derivatives in the between-host system equal to zero and reduce the system to 0 = λh − βhashiv n −mhsh + γ γ + mh · βhashiv n σ, (21) fsv (iv )sh n2 = q, (22) n2 − λh mh n = − βhashiv mh m, (23) biomath 8 (2019), 1901026, http://dx.doi.org/10.11145/j.biomath.2019.01.026 page 9 of 12 http://dx.doi.org/10.11145/j.biomath.2019.01.026 j. shane welker, maia martcheva, a novel multi-scale immuno-epidemiological model of visceral ... where fsv (iv ) := sv = λv mv − τ + mv τ iv , q = (τ + mv )mv τ · 1 βha2bv , m = ∫ ∞ 0 µ(x)π(x)dx, σ = ∫ ∞ 0 σ(x)π(x)dx, bv = ∫ ∞ 0 βv (x)π(x)dx, π(x) = exp ( − ∫ x 0 (σ(ξ) + µ(ξ) + mh)dξ ) , n = sh + ∫ ∞ 0 i(t,x)dx + rh = λh mh − βhashiv n m. solving (21) for sh gives fsh (iv /n). substituting that into (23) yields n = λh mh − βha [ fsh ( iv n )] iv n mh m =: fn (iv /n). we let x = βhaiv /n and p = 1−γς/(γ+mh). we redefine fsh and fn as functions of x, and obtain fsh (x) = λh mh + px , fn (x) = λh mh [ 1 − mx mh + px ] . we have that qn = sv sh/n = fsv (x)fsh (x)/fn (x). expanding and rearranging, we obtain a0x 2 + b0x + c0 = 0, (24) where a0 = (p−m)2 mh m r0 + τ + mv τβha (p−m), (25) b0 = 2m r0 (p−m) −mp + τ + mv τβha mh, (26) c0 = ( 1 r0 − 1 ) mhm, (27) and m = s0v /n 0. theorem 4. when r0 > 1, there exists a unique positive endemic equilibrium. proof: we have that a0 > 0, since p−m > 0. if r0 > 1, then c0 is necessarily negative. hence, the equation (24) has exactly one positive solution. it is not hard to see that in this case sh = fsh (x) > 0 and n = fn (x) > 0. this also implies that sv > 0. hence, a unique positive equilibrium exists. on the other hand, if r0 < 1, (24) may have two positive soluitons or no positive solutions. two positive solutions are obtained if equation (24) exhibits backward bifurcation. we find a necessary and sufficient condition for the existence of two equilibria: p + abv mh < 2m, noting that m is the disease-induced mortality. thus, backward bifurcation in this model can occur only if m > 0. theorem 5. if r0 < 1 and b0 is negative, then backward bifurcation occurs and two endemic equilibria exist. if r0 < 1 and b0 is positive, there are no endemic equilibria. the existence of the two endemic equilibria is established by the backward bifurcation shown in figure 3 where x is plotted on the y-axis and r0 is plotted on the x axis. the parameter varied in the figure is the mortality rate of the vector mv . it may be important to note the decrease in the level of the upper equilibrium upon increasing mv . this could imply that a stronger control measure on the vector could greatly affect the level of persistence of the disease. iv. discussion and conclusion in this paper, we present a new immunoepidemiological model for zoonotic visceral leishmaniasis (zvl) in dogs, in which the within-host model simulated infectiousness based on parasite load and the between-host model was structured by time-since-infection. biomath 8 (2019), 1901026, http://dx.doi.org/10.11145/j.biomath.2019.01.026 page 10 of 12 http://dx.doi.org/10.11145/j.biomath.2019.01.026 j. shane welker, maia martcheva, a novel multi-scale immuno-epidemiological model of visceral ... fig. 3: backward bifurcation of (24) when r0 < 1 and b0 < 0. various values of mv are shown. the within-host system examined the parasite loads in the skin and bone marrow, as well as the igg concentration. this system agrees well with data provided by courtenay et al [2]. the within-host model was shown to have an unstable parasite-free equilibrium, in which the parasite population dies out within the host. this equilibrium’s stability is in the absence of treatment, consistent with the persistence of zvl without treatment. while the agreement of the model solutions with the data was satisfactory, in future work we will fit the model to the data and examine the biological significance of the values found for the parameters. upon establishing the existence of equilibria and their stability, it is of great importance to examine the effect of control measures on the parasite population, as originally presented by dye [3]. this would include existing medicinal treatments, a hypothetical vaccine, and control measures directly affecting the vector. the basic reproduction number for the immunoepidemiological model r0 was introduced, and the disease-free equilibrium of the between-host system was shown to be locally asymptotically stable when r0 < 1. we then derived a quadratic equation for the equilibria of the full system, based on a reduction of the system. this equation in x := βhaiv /n was used to establish the existence and characterize the endemic equilibria of the immuno-epidemiological model. when r0 > 1, the model was shown to have a unique positive endemic equilibrium. however, when r0 < 1 and the coefficient of the linear term of the quadratic equation was negative, the presence of disease-induced mortality allowed for backward bifurcation to occur, consistent with the results found in ode cases [5]. this provided justification for the existence of two endemic equilibria. the presence of subthreshold equilibria generally obstructs disease eradication. control measures in this case should be directed towards (a) removing the cause of the backward bifurcation which in this case is the disease-induced mortality in dogs or (b) coupling sustain control measures that bring r0 below one with temporary control measures, such as mosquito spraying, to put the disease on elimination path [7]. acknowledgements the authors acknowledge partial support from nsf grant dms-1515661, and the contributions of george pu in the fitting of the within-host model. we also thank the referees for their comments and suggestions. references [1] about leishmaniasis dndi. 2016. url: http : / / www . dndi . org / diseases projects / leishmaniasis/. 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[19] who — leishmaniasis. 2017. url: http : //www.who.int/leishmaniasis/en/. biomath 8 (2019), 1901026, http://dx.doi.org/10.11145/j.biomath.2019.01.026 page 12 of 12 http://dx.doi.org/https://doi.org/10.1006/jtbi.2002.3076 http://dx.doi.org/https://doi.org/10.1006/jtbi.2002.3076 http://www.sciencedirect.com/science/article/pii/s0022519302930766 http://www.sciencedirect.com/science/article/pii/s0022519302930766 http://www.sciencedirect.com/science/article/pii/s0022519302930766 http://dx.doi.org/10.1186/1752-0509-6-1 http://www.biomedcentral.com/1752-0509/6/1 http://www.biomedcentral.com/1752-0509/6/1 http://dx.doi.org/10.1073/pnas.0703678104 http://dx.doi.org/10.1073/pnas.0703678104 http://www.pnas.org/cgi/doi/10.1073/pnas.0703678104 http://www.pnas.org/cgi/doi/10.1073/pnas.0703678104 http://dx.doi.org/10.1007/s11103-011-9767-z.plastid http://dx.doi.org/10.1007/s11103-011-9767-z.plastid http://arxiv.org/abs/nihms150003 https://www.cdc.gov/parasites/leishmaniasis/ https://www.cdc.gov/parasites/leishmaniasis/ http://dx.doi.org/10.1016/j.idm.2017.03.002 http://ac.els-cdn.com/s2468042716300173/1-s2.0-s2468042716300173-main.pdf?{\_}tid=c8ed3798-8db2-11e7-b1dc-00000aacb360{\&}acdnat=1504118647{\_}14d164bb702c07064ef2e80eeff2e974 http://ac.els-cdn.com/s2468042716300173/1-s2.0-s2468042716300173-main.pdf?{\_}tid=c8ed3798-8db2-11e7-b1dc-00000aacb360{\&}acdnat=1504118647{\_}14d164bb702c07064ef2e80eeff2e974 http://ac.els-cdn.com/s2468042716300173/1-s2.0-s2468042716300173-main.pdf?{\_}tid=c8ed3798-8db2-11e7-b1dc-00000aacb360{\&}acdnat=1504118647{\_}14d164bb702c07064ef2e80eeff2e974 http://ac.els-cdn.com/s2468042716300173/1-s2.0-s2468042716300173-main.pdf?{\_}tid=c8ed3798-8db2-11e7-b1dc-00000aacb360{\&}acdnat=1504118647{\_}14d164bb702c07064ef2e80eeff2e974 http://ac.els-cdn.com/s2468042716300173/1-s2.0-s2468042716300173-main.pdf?{\_}tid=c8ed3798-8db2-11e7-b1dc-00000aacb360{\&}acdnat=1504118647{\_}14d164bb702c07064ef2e80eeff2e974 http://ac.els-cdn.com/s2468042716300173/1-s2.0-s2468042716300173-main.pdf?{\_}tid=c8ed3798-8db2-11e7-b1dc-00000aacb360{\&}acdnat=1504118647{\_}14d164bb702c07064ef2e80eeff2e974 http://dx.doi.org/10.1016/j.mbs.2016.02.015 http://dx.doi.org/10.1016/j.mbs.2016.02.015 http://linkinghub.elsevier.com/retrieve/pii/s0025556416000468 http://linkinghub.elsevier.com/retrieve/pii/s0025556416000468 http://linkinghub.elsevier.com/retrieve/pii/s0025556416000468 http://www.who.int/leishmaniasis/en/ http://www.who.int/leishmaniasis/en/ http://dx.doi.org/10.11145/j.biomath.2019.01.026 introduction the model the within-host system the between-host system linking the withinand between-host systems analysis analysis of the within-host system analysis of the full immuno-epidemiological model discussion and conclusion original article biomath 3 (2014), 1407211, 1–8 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum global stability of an epidemic model with two infected stages and mass-action incidence mamadou lamine diouf1,2, abderrahman iggidr1, mamadou sy2 1 inria, université de lorraine, cnrs. institut elie cartan de lorraine, umr 7502. isgmp bat. a, ile du saulcy, 57045 metz cedex 01, france. 2 umi-ird-209 ummisco, and lani université gaston berger, saint-louis, sénégal. e-mail: dioufabu@yahoo.fr, abderrahman.iggidr@inria.fr, mamadou.sy@ugb.edu.sn received: 30 october 2013, accepted: 21 july 2014, published: 31 july 2014 abstract—the goal of this paper is the establishment of the global asymptotic stability of the model si with two classes of infected stages and with varying total population size. the incidence used is the mass-action incidence given by (β1i1 + β2i2) s n . existence and uniqueness of the endemic equilibrium is established when the basic reproduction number is greater than one. a lyapunov function is used to prove the stability of the disease free equilibrium, and the poincarré-bendixson theorem allows to prove the stability of the endemic equilibrium when it exists. keywords-epidemic model, global stability, massaction incidence i. introduction mathematical analysis became a major tool in the study of the evolution of epidemics. indeed, more and more models were developed for the study of some epidemics. in order to model an epidemic disease, the population is divided into various classes. in some cases the population is divided into two senior classes: the class of the susceptible individuals, denoted by s, and the class of the infected individuals, denoted by i. sometimes, the class of the infected can be split into several classes which allow to highlight the state of the disease. in our case, the infected are divided into two categories, denoted i1 and i2, with i1 the first stage of the disease and i2 the worsened case. if β1 and β2 are the per capita transmission rate of the infection in respectively the compartments i1 and i2, there are β1i1 + β2i2 infective contacts. if any contact with a susceptible gives a new infected, then there is (β1i1 + β2i2)p(s) new infected, where p(s) is the probability for an infected to meet a susceptible. the quantity (β1i1 + β2i2)p(s) is known in the literature as citation: mamadou diouf, abderrahman iggidr, mamadou sy, global stability of an epidemic model with two infected stages and mass-action incidence, biomath 3 (2014), 1407211, http://dx.doi.org/10.11145/j.biomath.2014.07.211 page 1 of 8 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2014.07.211 m diouf et al., global stability of an epidemic model with two infected stages... the mass action incidence rate. one can notice that most of the classical models of disease use a bilinear mass action incidence (β1i1 + β2i2)s. for example, some of the most famous: the models of kermack-mckendrick (1927) and that of lotkavoltera (1926). the goal of our study is to analyze the global stability of the si1i2 model. the system considered can represent, for instance, the modeling of the hiv. for this model we suppose that an infected can have s n contact of susceptible, then p(s) is given by s n . also the incidence is given by (β1i1 + β2i2) s n , where n represents the total population size: n = s + i1 + i2. the stability study of systems using this form of incidence is a very interesting subject to which some authors have already devoted some works. the work of c. simon and j. jacquez in [18] can be cited. indeed, these authors addressed the problem for n classes of infected, using some elegant geometrical arguments, but they use a constant recruitment and also they suppose that transition rate from a class of infected to the next class and the rate of disease-induced death are equal. however, in our study, the recruitment is variable and the transition rate (denoted γ) from the first stage of infection i1 to the second stage i2 is different from the rate of disease-induced death (denoted d). this makes that for our system the explicit determination of the endemic equilibrium is very difficult if not impossible. so, the stability around possible endemic equilibrium is also more difficult to check than in the case of a constant recruitment. we can also cite more recent works. particularly, the work of melese and gumel in [17], where for the proof of the endemic equilibrium stability, authors make a very strong assumption, which is very difficult to verify. we cite also and specially the work of m. li, j. graef, l. wang and j. karsai in [15], which deals with a similar system, but the authors used one contact rate. in the works made by c. c. mccluskey (2003) [16] and j. m. hyman and j. li (2005) [8], similar models have been considered, but the authors of [8] did not address the question of the global stability of the endemic equilibrium while in [16] the global stability of the endemic equilibrium was proved under the assumption that γ = d (i.e., the transition rate from i1 i2 is equal to the rate of diseaseinduced death) and β1 = β2. besides, we mention the work of h. guo and m. y. li (2006), where authors established the stability of the disease free equilibrium, but for the endemic equilibrium, they used bilinear incidence. we finish by mentioning the paper [10], where the authors considered similar systems but they used bilinear incidence. the paper is organized as follows. in section ii, we give the differential system governing the time evolution of the number of individuals in different classes is given, we derive the system governing the dynamics of the proportions and we compute the basic reproduction number r0. in section iii, we prove the existence and uniqueness of the endemic equilibrium when r0 is greater than one. the global asymptotic stability of the disease free equilibrium is studied in section iv by using two lyapunov functions. the local stability of the disease free equilibrium is given in section v. we prove in section vi that the system governing the proportions has no periodic orbit and that the endemic equilibrium is globally asymptotically stable. for the stability of the endemic equilibrium, the poincaré-bendixson theory is used. ii. the model the si models are well known in the dynamic of population. in this section, we present the si model used in this paper. the population of size n is divided into subclasses of individuals who are susceptible, infected into the first stage of the disease and infected into the second stage, with sizes denoted by s, i1 and i2. the model we consider is given by the system   ṡ = bn − (β1i1 + β2i2) s n −µs, i̇1 = (β1i1 + β2i2) s n − (µ + γ)i1, i̇2 = γi1 − (µ + d)i2. (1) where n = s + i1 + i2 is the total population size; b and µ represent the per capita birth rate and the per capita natural death rate of the population, respectively. β1 and β2 are respectively the per capita transmission rate of the compartments i1 and i2. γ denotes the per capita rate of transfer of infected individuals from the infected stage 1 to stage 2, and d is the disease induced death rate. the total population size n satisfies the equation: ṅ = (b−µ)n −di2. biomath 3 (2014), 1407211, http://dx.doi.org/10.11145/j.biomath.2014.07.211 page 2 of 8 http://dx.doi.org/10.11145/j.biomath.2014.07.211 m diouf et al., global stability of an epidemic model with two infected stages... the proportions s = s n , i1 = i1 n and i2 = i2 n satisfy the following differential system:  ṡ = b− bs− (β1i1 + β2i2)s + dsi2, i̇1 = (β1i1 + β2i2)s− (b + γ)i1 + di1i2, i̇2 = γi1 − (b + d)i2 + di22. (2) we determine the basic reproduction number, which represents the number of secondary cases produced by one infective host in an entirely susceptible population. we denote by fj(s,i1, i2) the rate of appearance of new infections in compartment j, and by vj(s,i1, i2) the rate of transfer of individuals in and out the compartment j by all other means. the matrices f and v are given by: f =   0(β1i1 + β2i2)s 0   and v =   b− bs− (β1i1 + β2i2)s + dsi2−(b + γ)i1 + di1i2 γi1 − (b + d)i2 + di22   . the jacobian matrices at the disease free equilibrium (1, 0, 0) are: df =   0 0 00 β1 β2 0 0 0   and dv =   −b −β1 −β2 + d0 −(b + γ) 0 0 γ −(b + d)   . let: f = ( β1 β2 0 0 ) and v = ( −(b + γ) 0 γ −(b + d) ) . it is well known [3] that the basic reproduction number is the spectral radius of the next generation matrix for the model, namely −fv −1. the basic reproduction number of system (2) is then r0 = β1 b + γ + β2γ (b + γ)(b + d) . iii. the equilibrium points the disease free equilibrium is given by dfe=(1,0,0). in the following, we show the existence and uniqueness of the endemic equilibrium for the system (2) assuming that b ≥ d. recall that b and d represent the birth and the disease induced rate, respectively. proposition iii.1. if r0 > 1, the endemic equilibrium exists and is unique. proof: at the equilibrium, the third equation of (2) gives: i∗1 = b + d γ i∗2 − d γ i∗22 . (3) replacing i∗1 by its expression in the second equation of (2), we have after simplification by γi∗2: β1(b + d) −β1di∗2 + β2γ)s ∗ − (b + γ)(b + d) +d(b + d)i∗2 + d(b + d)i ∗ 2 −d 2 i∗22 = 0. (4) also, in (4) we replace s∗ by its expression given by: s∗ = 1− i∗1 − i∗2 = 1− b + d γ i∗2 − d γ i∗22 − i∗2, then i∗2 is solution of the polynomial: p(i∗2) = a3(i ∗ 2) 3 + a2 i ∗2 2 + a1i ∗ 2 + a0 = 0, where a3 = −β1 d2 γ , a2 = 2β1d b + d γ + β1d + β2d−d2, a1 = −β1 (b+d)2 γ −β1(b + d) −β1d−β2(b + d) −β2γ + d(b + γ) + d(b + d), and = −r0 (b + d)(b + γ) ( 1 + b + d γ ) 4 −β1d + d(2b + d + γ) a0 = β1(b + d) + β2γ − (b + d)(b + γ) = (b + d)(b + γ) (r0 − 1). using the fact that r0 > 1, it is easy to show that: a3 < 0, a2 > 0, a1 < 0, and a0 > 0. we have p(i∗2) = 0 ⇔ q(i∗2) = r0, where q is the polynomial given by q(i∗2) = − a3 k (i∗2) 3 − a2 k i∗22 − a1 k i∗2 + 1, and k = (b + d)(b + γ). we have: q(0) = 1 biomath 3 (2014), 1407211, http://dx.doi.org/10.11145/j.biomath.2014.07.211 page 3 of 8 http://dx.doi.org/10.11145/j.biomath.2014.07.211 m diouf et al., global stability of an epidemic model with two infected stages... q(1) = β1(b 2 + bγ + dγ) + γ(b(b−d + γ) + β2(b + γ)) k . also q(1) −r0 = b β1b + β2γ + bγ + γ 2 −dγ kγ , which is positive if and only if β1b + β2γ + bγ + γ 2 > dγ. (5) the relation (5) is satisfied thanks to the assumption b ≥ d. thus, 1 = q(0) < r0 < q(1). let us localize exactly the domain of i∗2. we have i∗1 + i ∗ 2 < 1, (6) and since, by relation (3), i∗1 = b + d γ i∗2 − d γ i∗22 , we deduce that i∗2 must verify the following inequality: r(i∗2) = −di ∗2 2 + (b + d + γ)i ∗ 2 −γ < 0. the discriminant of the polynomial r is ∆r = (b + d + γ)2 − 4dγ = b2 + 2b(d + γ) + (d − γ)2 > 0. the roots of r are r1 = (b + d + γ − √ ∆r)/2d and r2 = (b+d+γ+ √ ∆r)/2d. we have: r1 < γ/2d < r2, and with the assumption b ≥ d we have r2 > 1. i∗2 must satisfy o < i∗2 < min{r1, 1} ≤ min{γ/2d, 1}, that is i∗2 must belong to the interval i = (0, min{r1, 1}) ⊂ (0, min{γ/2d, 1}). on the other hand, we have q(r1) −r0 = (1/(2k))[b(b + d + γ + √ ∆r)] > 0. since q(0) = 1 < r0, q(r1) > r0, and q(1) > r0, the graph of q intersects the horizontal line y = r0 at least one time in i. now let us show that there is exactly one intersection in i. the derivative of q is: q′(i∗2) = −(1/k)(3a3 i ∗2 2 + 2a2i ∗ 2 + a1). note that by descartes rules of signs there is no negative root. on the other hand, the discriminant of q′ is ∆ = a22 − 3a3a1, we then have two cases: if ∆ ≤ 0, q′ is positive on r. if ∆ > 0, we have two roots x1 and x2, and x1 + x2 = −(2a2/3a3). however: −2a2 3a3 = 4 3 b + d d + 2 3 { γ d + β2γ β1d − γ β1 } = 2 3 b+d d + 2 3 {b+d d + γ d + β2γ β1d − γ β1 } = 2 3 b + d d + 2 3β1d {β1(b + d) + β2γ +β1γ −dγ} = 2 3 b+d d + 2 3β1d {(b + d)(b + γ)r0 +β1γ −dγ}. thus −2a2/3a3 = 2 3 b + d d + 2 3β1d {b(b + γ)r0 +bdr0 + β1γ + dγ(r0 − 1)}. we know that b(b + γ)r0 = β1b + β2bγ b + d . since b ≥ d, we have −2a2/3a3 > 2, thus there is at least one root of q′ larger than one. all these observations show that the graph of q intersects the line y = r0 only once. i∗1 is deduced by i∗1 = b + d γ i∗2− d γ i∗22 , and s ∗ = 1− b + d + γ γ i∗2+ d γ i∗22 . then, the endemic equilibrium exists and is unique. iv. global stability of the dfe theorem iv.1. if r0 < 1, the dfe is globally asymptotically stable. proof: to prove theorem iv.1, we distinguish two cases, the first case corresponds to β2 ≥ d and the second is β2 < d. in both cases, we use lyapunov functions. case 1: β2 ≥ d.: we consider the following lyapunov function: v = i1 + β2 b + d i2. the derivative of v is: v̇ = (β1i1 + β2i2)s− (b + γ)i1 + di1i2 + β2γ b + d i1 −β2i2 + β2 d b + d i22. since β1i1s ≤ β1i1, we have v̇ ≤ β1i1 + β2i2s− (b + γ)i1 + di1i2 + β2γ b + d i1 −β2i2 + β2 d b + d i22 ≤ (b + γ)[ β1 b + γ + β2γ (b + γ)(b + d) − 1]i1 +β2i2(s− 1) + di1i2 + β2 d b + d i22, ≤ (b + γ)(r0 − 1)i1 + β2i2(s− 1) + di1i2 +β2 d b + d i22. we know that β2i2(s− 1) = −β2i2(i1 + i2), then v̇ ≤ (b + γ)(r0 − 1)i1 + (d−β2)i1i2 +( d b + d − 1)β2i22. biomath 3 (2014), 1407211, http://dx.doi.org/10.11145/j.biomath.2014.07.211 page 4 of 8 http://dx.doi.org/10.11145/j.biomath.2014.07.211 m diouf et al., global stability of an epidemic model with two infected stages... thus v̇ ≤ (b + γ)(r0 − 1)i1 + (d−β2)i1i2 − β2b b + d β2i 2 2 ≤ 0. it follows that v̇ is negative definite when r0 < 1. when r0 = 1, the time derivative of v v̇ is only nonpositive but in this case lasalle invariance principle allows to prove the global asymptotic stability of the dfe. case 2: β2 < d: we consider the following lyapunov function defined on {0 < s ≤ 1, 0 ≤ i1 ≤ 1, 0 ≤ i2 ≤ 1}: v = s− ln s + i1 + ( b + γ γ − β1 γ )i2. we obtain v̇ = ṡ(1 − 1 s ) + i̇1 + ( b + γ γ − β1 γ )i̇2 = (b− bs)(1 − 1 s ) − (β1i1 + β2i2)s + (β1i1 +β2i2) + dsi2 −di2 + (β1i1 + β2i2)s −(b + γ)i1 + di1i2 + (b + γ)i1 − (b + γ)(b + d) γ i2 + d(b + γ) γ i22 −β1i1 +β1 b + d γ i2 −β1 d γ i22, we get : v̇ = − b s (1 −s)2 + β2i2 + di2(s + i1 − 1) − (b + γ)(b + d) γ i2 + bd γ i22 + di 2 2 +β1 b + d γ i2 −β1 d γ i22. we have the followings equalities: b s (1−s)2 = b s (i1 +i2) 2 and di2(s+i1−1) = −di22. then v̇ becomes: v̇ = − b s (i1 + i2) 2 + β2i2 − (b + γ)(b + d) γ i2 + bd γ i22 + β1 b + d γ i2 −β1 d γ i22 = − b s (i1 + i2) 2 + (b + γ)(b + d) γ (r0 − 1)i2 + bd γ i22 −β1 d γ i22 = − b s (i1 + i2) 2 − (b + γ)(b + d) γ (1 −r0)i2 −β1 d γ i22 + bd γ i22. as 1/s ≥ 1 and i2 ≥ i22, we have v̇ ≤ −b(i1 + i2)2 − (b + γ)(b + d) γ (1 −r0)i22 −β1 d γ i22 + bd γ i22 = −bi21 − 2bi1i2 − bi 2 2 − (b + γ)(b + d) γ .(1 −r0)i22 −β1 d γ i22 + bd γ i22 = −bi21 − 2bi1i2 − i22 γ ( bγ + (b + γ)(b + d) .(1 −r0) + β1d− bd ) . denote by d = bγ + (b+γ)(b+d)(1−r0) +β1d−bd, then v̇ ≤−bi21 − 2bi1 i2 −d i22 γ . therefore, v̇ ≤ 0 if d ≥ 0. if bd < bγ + β1d holds then d ≥ 0. if not, we rewrite d in the following form: d = bγ + (b + γ)(b + d) −β1(b + d) −β2γ +β1d− bd = bγ + b2 + bd + bγ + dγ −β1(b + d) −β2γ + β1d− bd = b2 + 2bγ + dγ −β1b−β2γ = b(b−β1) + γ(2b + d−β2). the inequality bd ≥ bγ + β1d gives b > β1, and with the assumption β2 < d we get again d ≥ 0. we conclude that v̇ ≤ 0 if the assumption β2 < d holds. once again lasalle invariance principle allows to conclude. conclusion: in both cases ( β2 ≥ d and β2 < d), we have proved that the disease free equilibrium is globally asymptotically stable. biomath 3 (2014), 1407211, http://dx.doi.org/10.11145/j.biomath.2014.07.211 page 5 of 8 http://dx.doi.org/10.11145/j.biomath.2014.07.211 m diouf et al., global stability of an epidemic model with two infected stages... v. local stability of the endemic equilibrium with the assumption b ≥ d we have the following result: theorem v.1. the endemic equilibrium is asymptotically stable when it exists, i.e., when r0 > 1. proof: since s + i1 + i2 = 1, we can eliminate s in system (2). therefore, we get the following system:   i̇1 = (β1i1 + β2i2)(1 − i1 − i2) − (b + γ)i1 +di1i2, i̇2 = γi1 − (b + d)i2 + di22. (7) the jacobian of system (7) at the endemic equilibrium (ee = (i∗1, i ∗ 2)) is: j(ee) = ( β1 − 2β1i∗1 −β1i∗2 −β2i∗2 − (b + γ) + di∗2 γ β2 − 2β2i∗2 −β2i∗1 −β1i∗1 + di∗1 −b−d + 2di∗2 ) at the endemic equilibrium we have: β1 − 2β1i∗1 −β1i ∗ 2 −β2i ∗ 2 − (b + γ) + di ∗ 2 = −β2i∗2 1 − i∗2 i∗1 −β1i∗1. the determinant of j(ee) is given by: det(j(ee)) = β2(b + d)i ∗ 2 1−i∗2 i∗1 + β1(b + d)i ∗ 1 −2β2di∗22 1 − i∗2 i∗1 − 2β1di∗1i ∗ 2 −β2γ + 2β2γi∗2 + β2γi ∗ 1 +β1γi ∗ 1 −dγi ∗ 1 = β2(b + d)i ∗ 2 1 − i∗2 i∗1 + ( β1(b + d) +β2γ ) i∗1 + 2β2i ∗ 2 ( γ −di∗2 1−i∗2 i∗1 ) −2β1 di∗1 i ∗ 2 −β2γ + β1 γ i ∗ 1 −dγ i∗1. in the first term of the determinant, we replace (b+d)i∗2 by γi∗1 + di ∗2 2 and we get: det(j(ee)) = β2 ( γi∗1 + di ∗2 2 ) 1 − i∗2 i∗1 +(b + d)(b + γ)r0i∗1 +2β2 i∗2 i∗1 ( γi∗1 −di ∗ 2 + di ∗2 2 ) −2β1di∗1i∗2 −β2γ + β1γi∗1 −dγi∗1. we replace again γi∗1 − di∗2 + di∗22 by bi∗2 and by developing the first term of the determinant, we get: det(j(ee)) = β2γ −β2γi∗2 + β2di ∗2 2 1 − i∗2 i∗1 +(b + d)(b + γ)r0i∗1 + 2β2b i∗22 i∗1 −2β1di∗1i ∗ 2 −β2γ + β1γi ∗ 1 −dγi∗1 = β2i ∗ 2 ( −γ + di∗2 1 − i∗2 i∗1 + b i∗2 i∗1 ) +(b + d)(b + γ)r0i∗1 + β2b i∗22 i∗1 −2β1di∗1i ∗ 2 + β1γi ∗ 1 −dγi ∗ 1 = β2 i∗2 i∗1 ( −γi∗1 + (b + d)i ∗ 2 −di ∗2 2 ) +(b + d)(b + γ)r0i∗1 + β2b i∗22 i∗1 −2β1di∗1i ∗ 2 + β1γi ∗ 1 −dγi ∗ 1. thus det(j(ee)) = b(b + d + γ)r0i∗1 + β2b i∗2 i∗1 +dγi∗1(r0 − 1) + β1i∗1(γ − 2di∗2). the determinant is positive because i∗2 ∈ (0, γ 2d ). furthermore the trace is negative, because it is given by: trj(ee) = −β2i∗2 1 − i∗2 i∗1 −β1i∗1 − b−d + 2di ∗ 2. then the endemic equilibrium is asymptotically stable. vi. global stability of the endemic equilibrium since s + i1 + i2 = 1, we can reduce system (2) to a planar system and hence we can use the poincarrébendixson theorem to investigate the global attraction of the endemic equilibrium when r0 > 1. to this end, let us consider the following system:  ṡ = b(1 −s) − (β1i1 + β2(1 − i1 −s))s +ds(1 − i1 −s), i̇1 = (β1i1 + β2(1 − i1 −s))s− (b + γ)i1 +di1(1 − i1 −s), (8) defined on the set ω = {0 ≤ s ≤ 1, 0 ≤ i1 ≤ 1, s + i1 ≤ 1}. we establish by the dulac-bendixson criterium that there is no periodic orbit for (8). biomath 3 (2014), 1407211, http://dx.doi.org/10.11145/j.biomath.2014.07.211 page 6 of 8 http://dx.doi.org/10.11145/j.biomath.2014.07.211 m diouf et al., global stability of an epidemic model with two infected stages... theorem vi.1. system (8) has no periodic orbit. proof: consider the function b(x,y) = 1 xy . we have: b ṡ(s,i1) = b si1 − b i1 −β1 − β2 i1 + β2 + β2s i1 + d i1 −d− ds i1 , thus ∂ ∂s b ṡ(s,i1) = − b s2i1 + β2 i1 − d i1 . and b i̇1(s,i1) = β1 + β2 i1 −β2 − β2s i1 − b + γ s + d s − di1 s −d, so ∂ ∂i1 b i̇1(s,i1) = − β2 i21 + β2s i21 − d s . it leads to ∂b ṡ(s,i1) ∂s + ∂bi̇1(s,i1) ∂i1 = − b s2i1 + β2 i21 (s + i1 − 1) − d i1 − d s ∂b ṡ(s,i1) ∂s + ∂bi̇1(s,i1) ∂i1 < 0 ∀s,i1 ∈ (0, 1]. by dulac-bendixson criterium, we conclude that there is no closed orbit for system (8). thanks to theorem vi.1 and the poincaré-bendixson theorem we have the following result: theorem vi.2. if r0 > 1 the endemic equilibrium exists and is globally asymptotically stable in ω − γ, where γ is the stable manifold of the disease free equilibrium. proof: if r0 > 1, the jacobian matrix of system of (8) at the point (1, 0) has a negative determinant. therefore the dfe is unstable, but the eigenvalues of the jacobian matrix at the dfe are equal to: λ1,2 = β1 − (b + γ) − (b + d) ± √ (β1 − (b + γ) − (b + d)) 2 − 4(b + γ)(b + d)(1 −r0). one of the two eigenvalues is negative, which gives that the disease free equilibrium has one dimensional stable manifold γ. the ω−limit set of the system (8) on ω−γ is reduced to the endemic equilibrium point. because of the local stability of the endemic equilibrium for r0 > 1, the endemic equilibrium is globally asymptotically stable. vii. conclusion the model si is one of the most important epidemiological model. this paper gives a qualitative analysis of the stability of the model with a non-linear incidence. for this incidence, the system is analyzed by considering the differential system satisfied by the proportions, and the theory of poincarré-bendixson is used. it would be interesting to generalize the work to study the system with arbitrary n infected stages. it also would be interesting to find a lyapunov function for proving the global asymptotic stability of the endemic equilibrium. references [1] r. m. anderson and r. m. may. infection diseases of humans. oxford university press, london 1991. [2] l -m. cai, x. -z. li and ghosh. global satbility of staged-structured model with a non linear incidence. applied math. comput., 214:73–82, 2009. [3] p. van den driessche and j. watmough. reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. math. biosci., 180:29–48, 2002. http://dx.doi.org/10.1016/s0025-5564(02)00108-6 [4] h. guo and m. y. li. global dynamics of a staged progression model for infectious diseases. math. biosci. and eng.,3(3):513–525, 2006. http://dx.doi.org/10.3934/mbe.2006.3.513 [5] h. w. hethcote. the mathematics of infectious diseases. siam, 42:599–653, 2000. [6] h. w. hethcote and and h. r. thieme. stability of the endemic equilibrium in epidemic model with subpopulations. math. biosciences, 75:205–227, 1985. http://dx.doi.org/10.1016/0025-5564(85)90038-0 [7] h. w. hethcote and j. a. yorke. gonorrhea transmission dynamics and control. springer verlag, 1984. [8] j. m. hyman, j. li. the reproductive number for an hiv model with differential infectivity and staged progression. linear algebra and its applications, 398:101–116, 2005. [9] j. m. hyman, j. li, e. a. stanley. the differential infectivity and staged progression models for the transmission of hiv12. mathematical biosciences, 155(2):77–109,1999. [10] a. iggidr, j. mbang, g. sallet, and j.-j. tewa. multicompartment models. discrete contin. dyn. syst. supplements, suppl. volume(dynamical systems and differential equations. proceedings of the 6th aims international conference,):506–519, september 2007. biomath 3 (2014), 1407211, http://dx.doi.org/10.11145/j.biomath.2014.07.211 page 7 of 8 http://dx.doi.org/10.1016/s0025-5564(02)00108-6 http://dx.doi.org/10.3934/mbe.2006.3.513 http://dx.doi.org/10.1016/0025-5564(85)90038-0 http://dx.doi.org/10.11145/j.biomath.2014.07.211 m diouf et al., global stability of an epidemic model with two infected stages... [11] a. korobeinikov. lyapunov functions and global properties for seir and seis epidemic models. math.med.biology, 21:75-83, 2004. [12] a. korobeinikov and p. k. maini. non-linear incidence and stability of infectious disease models. math.med.biology, 22:113–128, 2005. http://dx.doi.org/10.1093/imammb/dqi001 [13] j. p. lasalle. the stability of dynamics systems. cbms-nsf regional conf. ser. in appl.math. 25, siam, philadelphia, 1976. [14] s. a. levin. descartes’ rule of sign-how hard can it be? [15] m. y. li, j. r. graef, l wang and j karsai. global dynamics of a seir with varying total population size. mathematical biosciences, 160:191–213, 1999. [16] c. c. mccluskey. a model of hiv/aids with staged progression and amelioration. mathematical biosciences, 181:1–16, 2003. [17] d. y. melesse and a. b. gumel. global asymptotic properties of an seirs model with multiple infectious stages. math. anal. and appl.,366:202–217, 2010. http://dx.doi.org/10.1016/s0025-5564(02)00149-9 [18] c. p. simon and j. a. jacquez. reproduction numbers and the stability of equilibria of si models for heteregeneous population. siam, 52(2):541–576,april 1992. [19] s. m. moghadas and a. b. gumel. global stability of a two stage epidemic model with generalised two stage incidence. mathematics and computers and simulation, 60:107–118, 2002. [20] h. r. thieme. global stability of the endemic equilibrium in infinite dimension: lyapunov functions and positive operator. journal of differentiel equation, 250(9):3772–3801, 2011. biomath 3 (2014), 1407211, http://dx.doi.org/10.11145/j.biomath.2014.07.211 page 8 of 8 http://dx.doi.org/10.1093/imammb/dqi001 http://dx.doi.org/10.1016/s0025-5564(02)00149-9 http://dx.doi.org/10.11145/j.biomath.2014.07.211 introduction the model the equilibrium points global stability of the dfe local stability of the endemic equilibrium global stability of the endemic equilibrium conclusion references editorial biomath 1 (2012), 1210117 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum from the editor-in-chief interdisciplinary research involving mathematical and biological sciences has deep roots at the bulgarian academy of sciences, established well before the contemporary terms of mathematical biology or biomathematics were coined in. the father of molecular biology in bulgaria, rumen tsanev, is also pioneer of bio-mathematical research at the academy. his joint work with blagovest sendov in the nineteen sixties using mathematical modelling and computer simulation in studying the cellular proliferation, differentiation and carcinogenesis productively links new developments in both fields for further advancement of knowledge, [1], [2], [3]. this research is associated with a tradition of scientific biomathematics meetings at the academy. probably the most significant is the international conferences series biomath (www.biomath.bg). up to now there are three biomath conferences, biomath1995 [4], biomath2011 [5], [6], biomath2012, and they are intended to be an annual event in the future. in order to facilitate the dissemination of the results in this fast developing and exciting research field the biomath forum society has decided to publish under the auspices of the bulgarian academy of sciences a fully peer reviewed scientific international journal biomath. the general scope of the journal is research in biosciences based on applications of mathematics as well as mathematics applied to or motivated by biological applications. it includes developing and applying mathematical and computational tools to the study of phenomena in the broad fields of biology, ecology, medicine, biotechnology, bioengineering, environmental science, etc. the journal will follow the tradition of the biomath conferences to present results of interdisciplinary nature in a form which is accessible to researchers from other disciplines. the biomath journal will be published both electronically and in printed form. i am happy to present this first issue of the biomath journal. roumen anguelov references [1] r. tsanev, bl. sendov, a model of the regulatory mechanism of cellular multiplication, j. theoret. biol. 12 (1966), 327–341. http://dx.doi.org/10.1016/0022-5193(66)90146-9 [2] bl. sendov, r. tsanev, computer simulation of the regenerative processes in the liver, j. theoret. biol. 18 (1968), 90–104. http://dx.doi.org/10.1016/0022-5193(68)90172-0 [3] r. tsanev, bl. sendov, a model of cancer studies by a computer, j. theoret. biol. 23 (1969), 124–134. http://dx.doi.org/10.1016/0022-5193(69)90071-x [4] s. markov, c. ullrich (eds), special issue: biomath 1995, computers & mathematics with applications 32(11) (1996). [5] r. anguelov, s. markov (eds), special issue: biomath 2011, computers & mathematics with applications 64(3) (2012). http://dx.doi.org/10.1016/j.camwa.2012.03.110 [6] s. markov, v. beschkov (eds), special issue: biomath 2011, biotechnology & biotechnological equipment 26(5) (2012). http://dx.doi.org/10.11145/j.biomath.2012.10.117 page 1 of 1 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.1016/0022-5193(66)90146-9 http://dx.doi.org/10.1016/0022-5193(68)90172-0 http://dx.doi.org/10.1016/0022-5193(69)90071-x http://www.sciencedirect.com/science/journal/08981221/32/11 http://dx.doi.org/10.1016/j.camwa.2012.03.110 http://www.diagnosisp.com/dp/journals/issue.php?journal_id=1&archive=1&issue_id=41&phpsessid=9crco9j9nvf7okojcefc49blm4 http://dx.doi.org/10.11145/j.biomath.2012.10.117 references communication/review biomath 1 (2012), 1209303, 1–2 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum farewell speech: stefan dodunekov (1945–2012) dear mournful family and all present, we are here to bow and to send to his eternal rest the president of the bulgarian academy of sciences— academician stefan manev dodunekov. we have come to share together the enormous loss every one of us has suffered. the family lost their loving and caring father and husband, the mathematical community lost one of its brightest and most active members, the bulgarian academy of sciences lost one of its hopes for a better and more successful development, bulgaria lost one quiet, but uncompromising patriot. after he was elected president of the bulgarian academy of sciences, stefan invited me in his office as being the director of the institute of mathematics and informatics to discuss about what has happened and about what needs to be done. he told me that for him the most difficult moment in his preparation for the election was to take the decision to run for a president of bas. at that moment, i didn’t pay much attention to this statement, but he was not in the habit to say needless things. now, when i come back to our conversation, i realize that he didn’t simply want to be just the next president of the bas. stefan was aware of the difficult situation in which the bas was and that his mission was to help it by all means. he has taken his decision with difficulty but once he has taken it, he put all his efforts to realize his program.he mobilized all his forces to the extreme. he was very encouraged and optimistic due to the really good attitude towards him shown by the bulgarian president—mr. plevneliev, by the prime minister dr. borisov and by the minister prof. ignatov. this really inspired him. regretfully, the newly elected president of bas didn’t get similar attitude by the general assembly of the bas. academician dodunekov was blamed by the general assembly, when he proposed his deputies team. this was an unexpected and perfidious blow. in the day when he was stricken by the fatal disease, academician dodunekov had appointment with three ministers to discuss the specific details of the co-operation with the bas. he could not attend this meeting. he burned as a sacrificial lamb at the stake. academician dodunekov had to preside over the meeting of academicians in the upcoming election of new academicians and corresponding members of the bas. his desire was to have more young and active members of the academy. after his unexpected death, this becomes a must for us. in this way, the general assembly of the bas will enable to choose a worthy successor. with the death of academician dodunekov we lost one of the hopes for a better future of the bas. we are not able, no matter how much we would like that, to bring back to life stefan. but we can bring back and fulfill this hope! http://dx.doi.org/10.11145/j.biomath.2012.09.303 page 1 of 2 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.09.303 b. sendov, farewell speech: stefan dodunekov (1945–2012) for less than two months, academician dodunekov proved that the main problem of the bas does not come from outside. the main problem of the bas is within the academy itself and it can be solved only by its own resources. according to the present law, the general assembly of the bas is the main governing body of the academy. this democratic achievement, dating from the beginning of the bulgarian democratic transition, with the time deteriorated and in essence it lead to a complete isolation of the members of the academy from taking decisions about its development. it is our national and professional duty, our duty to the memory of academician dodunekov, to build a new image of the bas, which should serve our country bulgaria for its spiritual development, as well as for its prosperity. farewell stefan! we will not forget you! we will not forget you and your hope! rest in peace! academician blagovest sendov bulgarian academy of sciences 8 august 2012 the speech in bulgarian is available at: http://www.bas.bg/fce/001/0079/files/prostalno.pdf biomath 1 (2012), 1209303, http://dx.doi.org/10.11145/j.biomath.2012.09.303 page 2 of 2 http://www.bas.bg/fce/001/0079/files/prostalno.pdf http://dx.doi.org/10.11145/j.biomath.2012.09.303 www.biomathforum.org/biomath/index.php/biomath original article a computational investigation of the ventilation structure and maximum rate of metabolism for a physiologically based pharmacokinetic (pbpk) model of inhaled xylene karen a. yokley∗‡, jaclyn ashcraft∗, and nicholas s. luke† ∗department of mathematics and statistics elon university, elon, nc 27244 †department of mathematics north carolina a&t state university, greensboro, nc 27411 ‡ corresponding author: kyokley@elon.edu received: 29 august 2018, accepted: 6 january 2019, published: 4 february 2019 abstract—physiologically based pharmacokinetic (pbpk) models are systems of ordinary differential equations that estimate internal doses following exposure to toxicants. most pbpk models use standard equations to describe inhalation and concentrations in blood. this study extends previous work investigating the effect of the structure of air and blood concentration equations on pbpk predictions. the current study uses an existing pbpk model of xylene to investigate if different values for the maximum rate of toxicant metabolism, v xylmax, can result in similar compartmental predictions when used with different equations describing inhalation. simulations are performed using v xylmax values based on existing literature. simulated data is also used to determine specific v xylmax values that result in similar predictions from different ventilation structures. differences in ventilation equation structure may affect parameter estimates found through inverse problems, although further investigation is needed with more complicated models. keywords-pbpk modeling, xylene i. introduction physiologically based pharmacokinetic (pbpk) modeling uses ordinary differential equations to describe absorption, distribution, metabolism, and excretion of toxicants following exposure. pbpk models have been developed to estimate internal doses for various toxicants in rodents [1] [9] [11] [14] [17] [18] [25] [34] [37] [40] and in humans [2] [3] [8] [10] [13] [15] [23] [26] [35] [36] [39] [38] [43] [48]. most pbpk model equations use copyright: c©2019 yokley et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: karen a. yokley, jaclyn ashcraft, nicholas s. luke, a computational investigation of the ventilation structure and maximum rate of metabolism for a physiologically based pharmacokinetic (pbpk) model of inhaled xylene, biomath 8 (2019), 1901067, http://dx.doi.org/10.11145/j.biomath.2019.01.067 page 1 of 13 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2019.01.067 k.a. yokley, j. ashcraft, n.s. luke, a computational investigation of the ventilation structure and ... the same basic structure and assumptions, such as the assumption that compartments are well-mixed and that the transfer of some chemicals is at equilibrium. however, the appropriate use of pbpk model estimates in risk assessment is dependent on multiple factors, including the biological basis of the model parameters and structure [6]. the industrial solvent xylene is a component of paints, paint thinners, and related products [42]. workers in specific industries may be at higher risk for xylene exposure and poorly ventilated areas may amplify exposure dangers [16], and xylene may increase the effect of other chemicals when present in mixtures [4]. as in [46], the pbpk model of xylene from [39] is used for the current investigation because of the relative simplicity of the model. the purpose of this project is to expand on the previous pbpk investigation that considered different modeling approaches for ventilation [46] and to also consider variation or errors in the parameter for the maximum rate of metabolism, vmax (mg/h). yokley [46] considered three structures for modeling ventilation, described as “equilibrium,” “non-equilibrium,” and “hybrid” models. the results were very similar for the equilibrium and hybrid models, and therefore only two structures will be used to model ventilation in the current study. the equilibrium model uses the standard quotient for the concentration of toxicant in arterial blood, and the non-equilibrium model allows for a separate lung compartment. the equilibrium and non-equilibrium models are used to predict various xylene concentrations and amounts in the body following inhalation exposure using different values for the maximum rate of metabolism in the liver. results of simulations with different metabolic rates are then used in inverse problems to investigate the success of optimization using different types of data. ii. model investigation a. model background in [46], a pbpk model for xylene exposure in humans from [39] was used with different fig. 1. the structure of the pbpk model of xylene from tardif et al. [39] used in [46]. equations describing ventilation exposure in order to determine the effect of the structure of those equations on model output. the four compartment model described xylene concentration in the slowly perfused tissues, the rapidly perfused tissues, the fat, and the liver. a schematic of the overall model is presented in figure 1. for easy reference, a summary of the notations used throughout the model is presented in the appendix. the amount of xylene within each compartment depicted in figure 1 is modeled by a separate differential equation. additional equations are added to the model to incorporate the ventilation structure. for each model equation, the compartmental concentration is defined as c xyl j = a xyl j vj , (1) where cxylj represents the concentration of the biomath 8 (2019), 1901067, http://dx.doi.org/10.11145/j.biomath.2019.01.067 page 2 of 13 http://dx.doi.org/10.11145/j.biomath.2019.01.067 k.a. yokley, j. ashcraft, n.s. luke, a computational investigation of the ventilation structure and ... chemical, xylene, in compartment j (mg/l); axylj represents the amount of the chemical, xylene, in compartment j (mg); and vj represents the volume of compartment j (l). compartment j is either the slowly perfused tissue (denoted s), the rapidly perfused tissue (denoted r), the adipose tissue or fat (denoted f), or the liver (denoted l). the differential equation for the change in the amount of xylene within each internal compartment j (excluding the liver) takes the form da xyl j dt = qj ( c xyl art − c xyl j p xyl j ) , (2) where qj represents the rate of blood flow to compartment j (l/h); cxylart is the concentration of the chemcial, xylene, in the arterial blood (mg/l); and p xylj is the partition coefficient for xylene in tissue j (unitless). the differential equation representing the change in the amount of xylene within the liver takes a different form because xylene is metabolized in the liver. the structure of the equation is similar to that of the other compartments, with a basic flow in minus flow out, but an additional term is subtracted to represent the metabolism. the form of the equation for the liver is da xyl l dt =ql ( c xyl art − c xyl l p xyl l ) − v xyl max c xyl l p xyl l k xyl m + c xyl l p xyl l , (3) where v xylmax is the maximum rate of metabolism for xylene (mg/h) and kxylm is the concentration of xylene at half saturation (mg/l). the full pbpk model is comprised of three equations of form (2) (one each for the slowly perfused, rapidly perfused, and fat compartments), equation (3), and two or more equations that represent the ventilation structures, which are outlined below. the three ventilation structures used in [46] were classified as “equilibrium,” “nonequilibrium,” and “hybrid” cases. many pbpk models use the “equilibrium” case equations for blood concentrations [14] [23] [39], which are constructed under a few assumptions including that the amount of toxicant leaving the alveolar space is equal to the amount eliminated through the blood. equations similar to (4) and (5) below are typically used in pbpk modeling for concentrations of a particular toxicant, i, in the venous (ven) and arterial (art) blood: ciart = qcc i ven + qpc i inh qc + qp p ibl:air (4) civen = ∑ j qj aij p ij vj qc , (5) where qc represents the cardiac flow rate, defined as the sum of the compartmental flow rates (l/h); qp is the pulmonary flow or alveolar ventilation (l/h); and pbl:air is the blood/air partition coefficient (unitless). the “non-equilibrium” case allows for the amount of toxicant leaving the alveolar space (alv) to not equal the amount entering the alveolar space and thus the alveolar space, arterial concentration, and venous concentration are each represented with their own compartment in the model. this scenario can be modeled using the equations daialv dt = qp ( ciinh − aialv valvp i bl:air ) +qc ( civen − aialv valvp i alv:air ) (6) dciart dt = qc ( aialv valvp i alv − ciart ) (7) dciven dt = 1 vven  ∑ j qj aij p ij vj − qcciven  (8) which are described more fully in [46]. the “hybrid” case involves using the “non-equilibrium” equations (6)-(8) with the alveolar partition coefficient, p ialv, set to be 1. because the results from [46] were very similar between the “equilibrium” and “hybrid” cases, the “hybrid” scenario is omitted from the current investigation. b. fixed parameter values all simulations use a body weight bw of 70 kg, under the assumption that 1 l = 1 kg biomath 8 (2019), 1901067, http://dx.doi.org/10.11145/j.biomath.2019.01.067 page 3 of 13 http://dx.doi.org/10.11145/j.biomath.2019.01.067 k.a. yokley, j. ashcraft, n.s. luke, a computational investigation of the ventilation structure and ... table i the compartmental flows and partition coefficients used in simulations of the xylene pbpk model. compartmental flow value (% qc) source fat 5 [39] slowly perfused 25 [39] richly perfused 44 [39] liver 26 [39] partition coefficient value source blood:air 26.4 [39] fat:blood 77.8 [39] slowly perfused:blood 3.0 [39] richly perfused:blood 4.42 [39] liver:blood 3.02 [39] lung:air 87.4 [42] is a sufficient conversion. physiological parameters used in the pbpk model are presented in tables i and ii. as described in [46], the lung:air partition coefficient (p xylalv:air) was obtained from [42] and was used with the blood:air partition coefficient from [39] in order to find p xylalv (i.e., p xyl alv = p xyl alv:air/p xyl bl:air). metabolic values from the pbpk source model are used as follows, v xylmax = v xyl maxcbw 0.75 kxylm = 0.2, and cardiac output and alveolar ventilation are assumed to follow allometric scaling with scaling factors used from the source model qc = 18.0 · bw 0.70 = qp [39] . c. vmaxc values metabolism of xylene is described using michaelis-menten kinetics in the model [39] which involves a nonlinear term with the parameter v xylmax representing the maximum rate of the enzymatic process (see equation (3)). the value of v xylmax is scaled to body weight as follows: v xylmax = v xyl maxcbw 0.75 . (9) different values of v xylmaxc have been used in pbpk modeling and other research as is shown in table iii. parameter values have been rescaled for table ii the compartmental volumes used in simulations of the xylene pbpk model. all values are the same as used for the “non-aged adult” in [47] except for vr , vbl, valv , vart, and vven. vr was altered to have all volumes sum to 100%. vbl valv , vart, and vven are based on [5]. volume value vl 0.026bw vf 0.214bw vs 0.613bw vr 0.06bw vbl 0.079bw valv 0.008bw vart 0.2vbl vven 0.8vbl consistency to represent v xylmaxc as in (9). note that the pbpk source model for the work in [46] focuses on m-xylene (as do the majority of the references in table iii). the values from [41] are estimated through conversion using different values for liver size. the values in table iii are used as a basis for v xylmaxc values used in the simulations. v xylmaxc values used in simulations or found in optimized fits to simulated output fall within a range of 1-11 mg/(h·kg) which is similar to the range of values presented in table iii. d. investigational methods computational solutions are generated using matlab r2015b [27] with the ode15s solver. with the exception of values for v xylmaxc, all initial conditions, exposure scenarios (50 ppm xylene over 7 hours), and parameters remain unchanged from the previous investigation [46]. solution curves are generated for values of v xyl maxc ranging from v xyl maxc0 = 1 to v xyl maxcfinal = 10, which are chosen to illustrate the trend of output as well as include the value, 8.4, used in the original xylene model [39]. a list of v xylmaxci values is generated with spacing 0.1, and computational solutions are generated for each metabolic constant in this list. the exposure duration is 7 hours, and 20 hours are used for output calculations to allow for the majority of the toxicant to be cleared from the system. maximum output values over the biomath 8 (2019), 1901067, http://dx.doi.org/10.11145/j.biomath.2019.01.067 page 4 of 13 http://dx.doi.org/10.11145/j.biomath.2019.01.067 k.a. yokley, j. ashcraft, n.s. luke, a computational investigation of the ventilation structure and ... table iii table of various values of v xylmaxc values and references citing them. numerical values have been rescaled for consistency. listed sources either used the value given (not necessarily scaled) or used the data from the paper in some way. note that the reference marked (*) cited the given source but used 6.49. primary source scaled v xylmaxc cited by tardif et al. 1993 [40] and 8.4 mg/(h·kg) [12], [19], [21], [22], [24], tardif et al. 1995 [39] [31], [44], [45], [46] tardif et al. 1997 [38] 5.5 mg/(h·kg) [26], [18]* tassaneeyakul et al. 1996 [41] ≈11.3-13.7 mg/(h·kg) [29], [30] price and krishnan 2011 [33] measured: 6.475 mg/(h·kg), [7], [32] predicted: 5.300 mg/(h·kg) haddad et al. 1999 [20] 6.59 mg/(h·kg) [22] 20 hour simulations are found for the amount of xylene in exhaled air, concentration of xylene in the venous blood, and amount of xylene in the liver. output over the 20 hours is generated at each 0.05 step, and a riemann sum is used to estimate area below the curve for the xylene concentration in the liver. output in amount is converted to concentration before the area is estimated. the xylene concentration in the venous blood and amount in the exhaled air are investigated because both could be physically collected, and those data could be used to estimate v xylmaxc values. the predictions for xylene in the liver are investigated because the liver is often a target organ (in general for various toxicants) and for the use of liver predictions in risk assessment. in order to ascertain how different v xylmaxc values could produce similar results with the two different models, the following procedure is employed. simulated data is generated from each model using a beginning v xylmaxc value for one model output that conceivably could be compared to measured data. different levels of noise are added to the data, and then both models are optimized to the simulated data over v xylmaxc, minimizing a least squares cost function j(vmaxc) = min ∑ i ( xi(v xyl maxc) 2 − d2i ) where xi represents the state of the differential equation model (using a particular value of v xylmaxc) at time ti corresponding to data point di. optimization is performed using the fminsearchbnd function [28] from the matlab optimization toolbox. data are simulated for the concentration of xylene in exhaled air and in the venous blood for both the equilibrium and non-equilibrium models and then fit to output of each model. iii. results the amount (or concentration) of a chemical within any given compartment during exposure is characterized by an increase during the interval of exposure followed by a gradual decline or clearing of the chemical after exposure has ceased. between the interval of exposure and the clearance of the chemical, the amount of chemical in any given compartment will reach a maximum level. in order to fit chemical exposure data, it is imperative that the pbpk model under consideration can reach these maximum levels. as a premilinary exercise for this study, the maximum model outputs are generated for both the equilibrium and nonequilibrium models and compared to determine if there is any overlap. figure 2 depicts the relationship between maximum predicted xylene amount in the exhaled air and values of v xylmaxc for both the equlibrium and non-equlibrium models. it is evident from this graph that there is little to no overlap in the maximum predicted xylene amount for both models. biomath 8 (2019), 1901067, http://dx.doi.org/10.11145/j.biomath.2019.01.067 page 5 of 13 http://dx.doi.org/10.11145/j.biomath.2019.01.067 k.a. yokley, j. ashcraft, n.s. luke, a computational investigation of the ventilation structure and ... 1 2 3 4 5 6 7 8 9 10 scaling constant for vmax 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 m a xi m u m p re d ic te d a m o u n t in e xh a le d a ir equilibrium non-equilibrium fig. 2. maximum model output for the amount of xylene in exhaled air for various v xylmaxc values. 1 2 3 4 5 6 7 8 9 10 scaling constant for vmax 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 m a xi m u m p re d ic te d c o n ce n tr a tio n in v e n o u s b lo o d equilibrium non-equilibrium fig. 3. maximum model output for the concentration of xylene in the venous blood for various v xylmaxc values. the maximum amounts generated from the nonequilibrium model are consistently higher than those of the equilibrium model. the significant difference between the model outputs would suggest that the non-equilibrium model would not be capable of fitting data generated by the equilibrium model, and the equilibrium model would not be capable of fitting data that was generated by the non-equilibrium model. figures 3 and 4 illustrate the relationship between values of v xylmaxc with the maximum concentration of xylene in the venous blood and maxi1 2 3 4 5 6 7 8 9 10 scaling constant for vmax 0 1 2 3 4 5 6 7 m a xi m u m p re d ic te d a m o u n t in li ve r equilibrium non-equilibrium fig. 4. maximum model output for the amount of xylene in the liver for various v xylmaxc values. 1 2 3 4 5 6 7 8 9 10 scaling constant for vmax 0 0.2 0.4 0.6 0.8 1 1.2 e st im a te d a re a b e lo w p re d ic te d li ve r co n c. c u rv e equilibrium non-equilibrium fig. 5. estimated area below the predicted liver concentration curve for various v xylmaxc values. mum amount of xylene in the liver, respectively. in contrast to the maximum xylene amount in the exhaled air (figure 2), maximum values of xylene in the venous blood and liver show a considerable amount of overlap between the equilibrium and non-equilibrium models. this indicates that the same maximum level may be predicted using either the equilibrium model or the non-equilibrium model with different values for v xylmaxc. a similar relationship for the area below the curve estimates for concentation of xylene in the liver is presented in figure 5. biomath 8 (2019), 1901067, http://dx.doi.org/10.11145/j.biomath.2019.01.067 page 6 of 13 http://dx.doi.org/10.11145/j.biomath.2019.01.067 k.a. yokley, j. ashcraft, n.s. luke, a computational investigation of the ventilation structure and ... 0 2 4 6 8 10 12 14 16 18 20 time (h) 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 c o n ce n tr a tio n o f xy le n e in e xh a le d a ir ( m g /l ) equilibrium output using simulated equilibrium data (a) equilibrium output 0 2 4 6 8 10 12 14 16 18 20 time (h) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 c o n ce n tr a tio n o f xy le n e in e xh a le d a ir ( m g /l ) non-equilibrium output using simulated equilibrium data (b) non-equilibrium output fig. 6. model output for the concentration of xylene in exhaled air after optimization over data simulated from the equilibrium model with noise level 0.001 and v xylmaxc = 6. figure (a) contains equilibrium model output and figure (b) contains nonequilibrium model output. 0 2 4 6 8 10 12 14 16 18 20 time (h) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 c o n ce n tr a tio n o f xy le n e in e xh a le d a ir ( m g /l ) equilibrium output using simulated non-equilibrium data (a) equilibrium output 0 2 4 6 8 10 12 14 16 18 20 time (h) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 c o n ce n tr a tio n o f xy le n e in e xh a le d a ir ( m g /l ) non-equilibrium output using simulated non-equilibrium data (b) non-equilibrium output fig. 7. model output for the concentration of xylene in exhaled air after optimization over data simulated from the nonequilibrium model with noise level 0.001 and v xylmaxc = 8. figure (a) contains equilibrium model output and figure (b) contains non-equilibrium model output. based on these comparisons of the maximum value of xylene for each variation of the model, it seems that the equilibrium and non-equilibrium models would fit the opposite data set more efficiently for the concentration of xylene in the venous blood and the amount of xylene in the liver than they would for the amount of xylene in the exhaled air. while the liver is a target organ for risk assessment, experimental data for the amount of a chemical within the liver is not easily collected; thus it is often unavailable. for these reasons, more focus is placed on predictions of blood concentrations for this study. examples of graphical output for xylene in exhaled air using simulated data from the equilibrium model are contained in figure 6, and examples using simulated data from the non-equilibrium model are contained in figure 7. these figures provide examples of best fitting curves to the simulated data. a summary of simulation results for exhaled biomath 8 (2019), 1901067, http://dx.doi.org/10.11145/j.biomath.2019.01.067 page 7 of 13 http://dx.doi.org/10.11145/j.biomath.2019.01.067 k.a. yokley, j. ashcraft, n.s. luke, a computational investigation of the ventilation structure and ... table iv optimal cost using simulated data for the concentration of xylene in exhaled air. the v xylmaxc value used for simulation is listed in column 1 of the table, and initial guesses for v xylmaxc for the optimization routine were 3, 6, and 10. the value for j∗ that was lowest for the three is listed below. in cases marked with (†) the same cost was found for various v xylmaxc values, all of which were close to zero. equilibrium simulated data v xylmaxc noise best fit to “equilibrium” best fit to “non-equilibrium” 2 0.001 v xylmaxc = 1.9125, j ∗ = 1.6885e−6 v xylmaxc = 14.9393, j ∗ = 0.0224 4 0.001 v xylmaxc = 3.7144, j ∗ = 1.8447e−6 v xylmaxc = 14.9386, j ∗ = 0.0281 6 0.001 v xylmaxc = 5.1232, j ∗ = 2.1612e−6 v xylmaxc = 14.9383, j ∗ = 0.0299 non-equilibrium simulated data v xylmaxc noise fit to “equilibrium” fit to “non-equilibrium” 4 0.001 v xylmaxc = 0.0014 †, j∗ = 0.0151 v xylmaxc = 3.5138, j ∗ = 4.9275e−5 6 0.001 v xylmaxc = 0.0013 †, j∗ = 0.0123 v xylmaxc = 5.7957, j ∗ = 4.3432e−6 8 0.001 v xylmaxc = 0.0051 †, j∗ = 0.0114 v xylmaxc = 7.0188, j ∗ = 1.5291e−6 0 2 4 6 8 10 12 14 16 18 20 time (h) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 c o n ce n tr a tio n o f xy le n e in v e n o u s b lo o d ( m g /l ) equilibrium output using simulated equilibrium data (a) equilibrium output 0 2 4 6 8 10 12 14 16 18 20 time (h) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 c o n ce n tr a tio n o f xy le n e in v e n o u s b lo o d ( m g /l ) non-equilibrium output using simulated equilibrium data (b) non-equilibrium output fig. 8. model output for the concentration of xylene in venous blood after optimization over data simulated from the equilibrium model with noise level 0.05 and v xylmaxc = 2. figure (a) contains equilibrium model output and figure (b) contains non-equilibrium model output. air concentration is presented in table iv. in figure 6, it can be observed that the equilibrium model (depicted in figure 6(a)) provides a much better fit to the equilibrium data than the non-equilibrium model (depicted in figure 6(b)) does. when fit to the equilibrium data, the equilibrium model produces a least squares cost of 2.1612e−6 and the non-equilibrium model produces a cost of 0.0299. similary, figure 7 shows that the non-equilibrium model provides a much better fit to the non-equilibrium data than that of the equilibrium model. the non-equilibrium model yields a least squares cost of 1.5291e−6 with the non-equilbrium data, compared to the equilibrium model’s cost of 0.0114. data are also simulated for the concentration of xylene in venous blood for both the equilibrium and non-equilibrium models. the fits to these models show some success for v xylmaxc in the range of 3-7. examples of graphical results are contained biomath 8 (2019), 1901067, http://dx.doi.org/10.11145/j.biomath.2019.01.067 page 8 of 13 http://dx.doi.org/10.11145/j.biomath.2019.01.067 k.a. yokley, j. ashcraft, n.s. luke, a computational investigation of the ventilation structure and ... 0 2 4 6 8 10 12 14 16 18 20 time (h) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 c o n ce n tr a tio n o f xy le n e in v e n o u s b lo o d ( m g /l ) equilibrium output using simulated non-equilibrium data (a) equilibrium output, noise=0.02 0 2 4 6 8 10 12 14 16 18 20 time (h) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 c o n ce n tr a tio n o f xy le n e in t h e v e n o u s b lo o d ( m g /l ) equilibrium output using simulated equilibrium data (b) equilibrium output, noise=0.05 0 2 4 6 8 10 12 14 16 18 20 time (h) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 c o n ce n tr a tio n o f xy le n e in v e n o u s b lo o d ( m g /l ) non-equilibrium output using simulated non-equilibrium data (c) non-equilibrium output, noise=0.02 0 2 4 6 8 10 12 14 16 18 20 time (h) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 c o n ce n tr a tio n o f xy le n e in v e n o u s b lo o d ( m g /l ) non-equilibrium output using simulated non-equilibrium data (d) non-equilibrium output, noise=0.05 fig. 9. model output for the concentration of xylene in venous blood after optimization over data simulated from the non-equilibrium model with given noise level and v xylmaxc = 6. figures (a)-(b) contain equilibrium model output and figures (c)-(d) contain non-equilibrium model output. in figure 8 and figure 9, and a summary of overall results for the concentration of xylene in venous blood is presented in table v. figure 8 illustrates the best fits of the equilibrium and non-equilibrium models to venous blood concentration data that was generated by the equilibrium model with a v xylmaxc value of 2. unlike the previously presented results for the model fits to exhaled air, both models seem to provide an adequate fit to the venous blood data. the results from the equilibrium model (in figure 8(a)) seem to capture the maximum more efficiently. the best fit for the equilibrium model is produced using a v xyl maxc value of 1.8299, with a cost function value of 0.0058. the best fit for the non-equilibrium model is produced with a v xylmaxc value of 1.8514, yielding a cost of 0.0203. the fits of the equilibrium and non-equilibrium models to non-equilibrium data are presented in figure 9. figures 9(a) and 9(b) display the fit of the equilibrium model to non-equilibrium data with noise levels of 2% and 5%, respectively. results for the non-equilibrium model fitted to the nonequilibrium data with noise levels of 2% and 5% are depicted in figures 9(c) and 9(d). a visual inspection of the graphs would suggest that both the equilibrium model and the non-equilibrium model can provide an adequate fit to the simulated biomath 8 (2019), 1901067, http://dx.doi.org/10.11145/j.biomath.2019.01.067 page 9 of 13 http://dx.doi.org/10.11145/j.biomath.2019.01.067 k.a. yokley, j. ashcraft, n.s. luke, a computational investigation of the ventilation structure and ... table v optimal cost using simulated data for the concentration of xylene in venous blood. the v xylmaxc value used for simulation is listed in column 1 of the table, and initial guesses for v xylmaxc for the optimization routine were 3, 6, and 10. the value for j∗ that was lowest for the three is listed below. equilibrium simulated data v xylmaxc noise best fit to “equilibrium” best fit to “non-equilibrium” 2 0.001 v xylmaxc = 1.9687, j ∗ = 2.2307e−4 v xylmaxc = 2.0752, j ∗ = 0.0247 2 0.01 v xylmaxc = 1.9596, j ∗ = 4.7682e−4 v xylmaxc = 2.0700, j ∗ = 0.0234 2 0.02 v xylmaxc = 1.9173, j ∗ = 7.6803e−4 v xylmaxc = 2.0190, j ∗ = 0.0237 2 0.05 v xylmaxc = 1.8299, j ∗ = 0.0058 v xylmaxc = 1.8514, j ∗ = 0.0203 3 0.001 v xylmaxc = 3.0000, j ∗ = 7.1896e−6 v xylmaxc = 4.8376, j ∗ = 0.0139 3 0.01 v xylmaxc = 2.9006, j ∗ = 2.3456e−4 v xylmaxc = 4.5075, j ∗ = 0.0132 3.5 0.05 v xylmaxc = 3.1610, j ∗ = 0.0058 v xylmaxc = 5.4462, j ∗ = 0.0127 non-equilibrium simulated data v xylmaxc noise fit to “equilibrium” fit to “non-equilibrium” 6 0.001 v xylmaxc = 3.3016, j ∗ = 0.0128 v xylmaxc = 6.0000, j ∗ = 8.5450e−6 6 0.01 v xylmaxc = 3.3067, j ∗ = 0.0154 v xylmaxc = 6.2866, j ∗ = 7.8847e−4 6 0.02 v xylmaxc = 3.3163, j ∗ = 0.0201 v xylmaxc = 6.6882, j ∗ = 0.0050 6 0.05 v xylmaxc = 3.3067, j ∗ = 0.0489 v xylmaxc = 6.2866, j ∗ = 0.0466 8 0.001 v xylmaxc = 3.6121, j ∗ = 0.0111 v xylmaxc = 10.6876, j ∗ = 8.9750e−4 8 0.01 v xylmaxc = 3.5750, j ∗ = 0.0111 v xylmaxc = 10.2790, j ∗ = 0.0017 non-equilibrium data. the non-equilibrium data in these graphs were generated with a v xylmaxc value of 6. the best fit of the equilibrium model to the data with 2% noise is found with v xylmaxc equal to 3.3163 and has a least squares cost of 0.0201. a v xylmaxc value of 3.3067 leads to the optimal fit of the equilibrium model to the non-equilibrium data with 5% noise, resulting in a cost of 0.0489. the best fits for the non-equilibrium model to the non-equilibrium data with 2% and 5% noise are produced with v xylmaxc values of 6.6882 and 6.2866 and result in cost values of 0.0050 and 0.0466, respectively. as previously stated, the amount of xylene present in the liver following an inhalation exposure is not easily measured and therefore not reported. for this reason, a comparison of the equilibrium and non-equilibrium model results for the amount of xylene in the liver is not conducted. based on the maximum model output for the amount of xylene in the liver (depicted in figure 4), it is hypothesized that adequate fits to data can be found using both the equilibrium and nonequilibrium models (similar to the results reported for the venous blood concentrations above). iv. discussion and conclusions simulations suggest that different v xylmaxc values could be used to make similar predictions for the concentration of xylene in venous blood with the different ventilation structures, but different v xylmaxc values are not found to produce similar model output for the amount in exhaled air. specifically, using v xylmaxc around 3 in the “equilibrium” model produces similar blood concentration results as using v xylmaxc around 6 in the “non-equilibrium” model. as shown in figure 3 and in figure 9, the maximum xylene concentration is predicted to be about 0.4 mg/l by both “equilibrium” and “non-equilibrium” models when v xylmaxc is around 3 or 6, respectively. additionally, optimization to simulated data results in best fits with similar numbers (see results in table v for simulated v xyl maxc=3, 3.6, and 6). hence, blood data may not be as informative as exhaled air data for identifying the maximum rate of metabolism when biomath 8 (2019), 1901067, http://dx.doi.org/10.11145/j.biomath.2019.01.067 page 10 of 13 http://dx.doi.org/10.11145/j.biomath.2019.01.067 k.a. yokley, j. ashcraft, n.s. luke, a computational investigation of the ventilation structure and ... the exposure method is inhalation. results may differ if a toxicant is administered in the blood directly or ingested or exposed dermally as in [36]. in cases involving exposure through methods other than inhalation, the ventilation equation structure would be expected to be less critical and so are not investigated in the current study. however, in those cases, the equations describing the entrance of toxicant into the body could be critical as well. figures 4 and 5 contain predictions related to xylene in the liver. when these predictions are calculated using the “equilibrium” model with v xylmaxc around 3 and the “non-equilibrium” model with v xyl maxc around 6, however, the liver predictions are higher for the “equilibrium” model. although the “equilibrium” model for the xylene pbpk model used here may then provide higher predictions for internal liver doses, the “equilibrium” ventilation structure may not provide higher internal dose estimates when used with more complicated pbpk models. for example, the results in figures 4 and 5 contain predictions for the parent chemical, and concentrations of metabolites may show different dynamics. although the pbpk models of xylene used in the current study do not focus on the excretion of toxicants, some models do make predictions of the parent chemical or metabolites in the urine. data of excreted toxicants could also be problematic for use in optimizing metabolic parameters. additionally, the model used in the current study is a more simplistic pbpk model, and more investigation is needed to be able to make conclusions about toxicants with harmful metabolites or that require models with more compartments. pbpk models have been used to describe exposure to a mixture of chemicals (such as in [12] [18] [24] [33][39] [43]) which would also involve a more complicated investigation than in the current study. v. acknowledgements the authors would like to thank elon university funding for research and development for support for this project. appendix abbreviations: xyl xylene s slowly perfused tissues f adipose tissue or fat r rapidly perfused tissues l liver ven venous blood art arterial blood bl blood (mixed) alv alveolar space or respiratory compartment model notations: qj blood flow in/to compartment j (l/h) qc cardiac flow, ∑ j qj (l/h) qp pulmonary flow/alveolar ventilation (l/h) vj volume of compartment j (l) p ij the partition coefficient for chemical i in tissue j (dimensionless) p xyl bl:air the blood/air partition coefficient for chemical xyl (dimensionless) a xyl j the amount of chemical xyl in tissue j (mg) c xyl j the concentration of chemical xyl in compartment j (mg/l) c xyl inh the concentration of chemical xyl inhaled (mg/l) v xylmax the maximum rate of metabolism for chemical xyl (mg/h) kxylm the concentration of xyl at half saturation (mg/l) references [1] abbas, r, fisher, jw (1997) a physiologically based pharmacokinetic model for trichloroethylene and its metabolites, chloral hydrate, trichloroacetate, dichloroacetate, trichloroethanol, and trichloroethanol biomath 8 (2019), 1901067, http://dx.doi.org/10.11145/j.biomath.2019.01.067 page 11 of 13 http://dx.doi.org/10.11145/j.biomath.2019.01.067 k.a. 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[48] yokley, ka, evans, mv (2007) an example of model structure differences using sensitivity analyses in physiologically based pharmacokinetic models of trichloroethylene in humans, b math biol 69(8):25912625. biomath 8 (2019), 1901067, http://dx.doi.org/10.11145/j.biomath.2019.01.067 page 13 of 13 http://www.mathworks.com/ http://www.mathworks.com/matlabcentral/fileexchange/8277-fminsearchbnd--fminsearchcon/ http://www.mathworks.com/matlabcentral/fileexchange/8277-fminsearchbnd--fminsearchcon/ http://dx.doi.org/10.11145/j.biomath.2019.01.067 introduction model investigation model background fixed parameter values vmaxc values investigational methods results discussion and conclusions acknowledgements appendix references biomath https://biomath.math.bas.bg/biomath/index.php/biomath b f biomath forum original article topological process of splitting dna-links abdul adheem mohamad1, tsukasa yashiro2 1college of arts and sciences, university of nizwa, oman mohamad@unizwa.edu.om 2independent mathematical institute, miyota, kitasaku, nagano, japan t-yashiro@dokusuken.com received: 30 november 2021, accepted: 28 march 2022, published: 16 may 2022 abstract—a dna replicon is modeled by a special type of 2-component link, called a dna-link, in which two circles form a double helix around a trivial center core curve. the dna replication process is semi-conservative, which is interpreted as a splitting process of the dna-link. to split this non-trivial link, the linking number must become zero, and thus an unknotting operation is necessary. some families of enzymes act as the unknotting operation. the present paper considers two topological problems; one is to know how the linking number is reduced and the other, how the enzymes are allocated at appropriate places. for the first problem, we suggest a reduction system of the linking number of a dnalink. from this system, the number of repetitions of the procedure is obtained and this could be reduced when the dna is previously relaxed by type i topoisomerases. for the second problem, we propose a possible conformation of the dna-link in which the unknotting operation does not change the knot type of the core curve but decreases the writhe. this conformation could allocate type ii topoisomerases to appropriate places. these models suggest that the combination of type i and type ii topoisomerases efficiently reduces the linking number and it is possible to allocate enzymes by the conformation of dna strands. keywords-dna; replication; link; topological model; replicon msc2010-92b99 i. introduction a dna molecule has a double helical structure [24] along a curve that causes several topological problems when it is unwound [4], [1], [23]. during the transcription or replication process, tangled (catenated) strands occur. as the replication forks advance, the axial rotation introduces positive supercoils ahead of each of the replication forks, while the negatively supercoiled daughter dnas are introduced behind the forks [7], [16], [18], [23]. it is known that topoisomerases are responsible for reducing stress and supercoils [1], [2], [16], [17], [22], [23]. rybenkov et. al. [16] revealed the ability of the enzyme topoisomerase type ii to simplify dna topology. it has been pointed out that there are several topological problems caused by the double helical structure of dna itself (see [23]). in this paper, we focus on the unwinding process of the double-strand dna introducing supercoils and their reduction during copyright: © 2022 abdul adheem mohamad, tsukasa yashiro. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: abdul adheem mohamad, tsukasa yashiro, topological process of splitting dna-links, biomath 11 (2022), 2203288, https://doi.org/10.55630/j.biomath.2022.03.288 page 1 of 12 https://biomath.math.bas.bg/biomath/index.php/biomath mailto:mohamad@unizwa.edu.om mailto:t-yashiro@dokusuken.com https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ https://doi.org/10.55630/j.biomath.2022.03.288 abdul adheem mohamad, tsukasa yashiro, topological process of splitting dna-links dna replication. in knot theory, a link is a set of embedded disjoint circles in 3-dimensional space [12], and its projected image into a plane is called a link diagram (section iv-a) [12]. it is known that for any link diagram, there is a finite sequence of crossing changes (unknotting operations), which deforms the link into separated trivial circles called a trivial link. however, this does not mean that we can specify the set of crossings that should be changed to obtain a trivial link (example iv.1). mathematically, if we are given a piece of global information such as a link diagram in a plane, it is possible to find such a set, but it is not known how to find such a set from only local information. since each replicon is fixed at substructure of the nucleus, topologically, it is modeled by a special 2-component link called a dna-link (see section iv) and also, the semi-conservative replication is interpreted in terms of the dna-link; that is, the dna-link is deformed into a disjoint union of trivial circles (lemma iv.2). an oriented link is characterized as an algebraic invariant called the linking number (see section iv-c for the definition) which is obtained as the sum of the total number of full-twists and the writhe of the core curve (lemma iv.3). to split the dna-link is equivalent to having the linking number zero (lemma iv.4). therefore, the following problem arises. p1: how is the linking number of the dna-link reduced? to solve problem p1, we propose a procedure to reduce the linking number of dna-link. this procedure considers the situation that only type ii topoisomerase is used for the reduction. using this procedure, we will obtain a formula for the number of repetitions of the procedure (proposition v.2), by this formula, it is at least 11.5. this observation suggests that a relaxation of the double strand by type i topoisomerase should accelerate the reduction of the linking number. the procedure above focuses only on the numeral calculations, not the location of unknotting operations. it should be noted that a random choice of crossings to apply the unknotting operations does not always reduce the linking number (see example 5). thus the next problem arises. p2: how are the enzymes allocated to the right positions of crossings that need to be changed to resolve a tangled dna? to answer p2, we propose a topological conformation of dna in which the chromatin fibre forms a zig-zag shaped (juxtaposed) formation (see figure 8). this model suggests that a special local conformation enforces the enzyme to locate the right crossing. this paper is organized as such: section ii is a brief description of dna replication. section iii briefly describes topoisomerases type i and ii. section iv introduces dna-links and unknotting operations. section v introduces a possible procedure to reduce the linking number and estimate the number of repetitions of the procedure. section vi introduces a topological model for unknotting operations. section vii discusses the previously obtained results. section viii states the conclusion. ii. dna replication the dna replication is done along the loopshaped dna called a replicon [1], [4], [20], the ends of which are anchored at a substructure of the nucleus called a nuclear matrix (nm) [14], [15]. a group of replicons forms a replicon domain or simply domain [20]. each replicon has a specific site called a replication origin or simply origin to unwind (relax) and to separate the double strand into a pair of single strands. we sometimes write the double strand dna as a ds-dna, and the single strand dna as an ss-dna. the separation occurs at two branching points called replication forks or simply forks. it is known that there are several models of forks whether the forks move or not [4], [15]. in this paper, ‘ahead of a fork’ means one side of the fork in which the ds-dna has not yet separated into ss-dnas. as synthesizing proceeds, it is either continuous along a single strand, called a leading strand or discontinuous along with the other, called a biomath 11 (2022), 2203288, https://doi.org/10.55630/j.biomath.2022.03.288 page 2 of 12 https://doi.org/10.55630/j.biomath.2022.03.288 abdul adheem mohamad, tsukasa yashiro, topological process of splitting dna-links lagging strand. along the lagging strand, short synthesized segments (100 bp to 200 bp) called okazaki fragments [13] are constructed. the separated single strands of their parent dna are preserved in its daughter dnas. this is said to be semi-conservative [8]. a nucleosome is an essential element of the chromatin and it is formed with the double strand dna (ds-dna) wrapping around the histone octamer [1], [20]. the dna is stored in the nucleus through a compactification involving nucleosomes [3], [5], [6]. in the interphase of the eukaryotic cell cycle, the dna exists in the format with nucleosomes called chromatin fibre [3], [5], [11], [6]. although a variety of models of organizations for chromatin fibre has been suggested [3], [5], [6], [11], [21], the exact conformation of the chromatin fibre has not yet been settled. iii. topoisomerases there are two types of topoisomerases, type i and type ii, and they have several subfamilies such as types ia and ib [1], [2]. type i is to release the rotational stress of single strands while type ii releases the stress of the double strands [1], [7], [16], [17], [23]. in this section, we briefly describe type ia, type ib, and type ii. the topoisomerase type ia acts on a ds-dna to cut one single strand of it and lets the other single strand pass through the gap and reseal the gap. this reduces one fulltwist of the ds-dna. the topoisomerase type ib acts on a ds-dna to cut one single strand so that it creates free ends and let one of the ends rotate around the other complete strand multiple times and reseal the gap at the end. this reduces multiple full-twists of the ds-dna. the topoisomerase ii acts on a pair of ds-dnas to capture the pair of ds-dna segments and cut one of them to make a gap, then lets the other ds-dna pass through the gap and reseal it. finally, those ds-dnas are released (see figure 1). it is observed in [17] that topoisomerase type ii is more efficient than topoisomerase type i to untangle dna strands. note that when the type ii topoisomerase acts on the pair of ds-dna segments, these two segments must be close enough so that the enzyme can capture both segments (see figure 1-(a)). iv. dna-links a. links a link is a disjoint union of circles embedded in r3. each of the circles is called a component. if the number of components is n, then the link is called an n-component link. a 1-component link is called a knot. if a link l can be deformed into a link l′ without any cut or intersecting in r3, then l and l′ are in the same link type and l and l′ are said to be equivalent and they have the same type (see [12] for details). if a knot bounds a disc in 3-space, then it is said to be trivial. if a link l is equivalent to a disjoint union of trivial knots, then the link is said to be trivial. the image of a link l under the orthogonal projection from r3 to r2 by omitting the last coordinate, is a diagram with finite number of crossings every which is formed by two short segments; one is higher and the other is lower with respect to the last coordinate. this is called crossing information. the image of l under the orthogonal projection with the crossing information is called a link diagram denoted by dl. if every component of a link l has an orientation, (oriented), l is called an oriented link. on a link diagram, there are three elementary moves ω1, ω2 and ω3, called reidemeister moves (see figure 2). lemma iv.1 ([12]). two diagrams of equivalent links are deformed into each other by a finite sequence of reidemeister moves. proof: a proof can be found in [12]. b. topological semi-conservative scheme a ds-dna can be viewed as the boundary components {s1,s2} of a long thin twisted strip with the trivial centre circle γ. we assume here, the centre curve γ is oriented and the components are parallelly oriented along γ. we write this as l = l(s1,s2; γ), where s1 and s2 represent the single strands of the dna. to define a topological biomath 11 (2022), 2203288, https://doi.org/10.55630/j.biomath.2022.03.288 page 3 of 12 https://doi.org/10.55630/j.biomath.2022.03.288 abdul adheem mohamad, tsukasa yashiro, topological process of splitting dna-links (a) (b) (c) (d) fig. 1. (a) topoisomerase ii captures the pair of ds-dna strands. (b) it cuts one of those strands to make a gap. (c) the other strand passes through the gap. (d) reseals the gap and releases them. ω1 ω2 ω3 fig. 2. two diagrams of equivalent links are moved to each other by a finite sequence of three moves ω1, ω2 and ω3. model of a replicon, we assume that the model satisfies the following conditions. a1. the core curve γ is unknotted (trivial) and oriented. a2. the link l(s1,s2; γ) is an oriented 2component link in which s1 and s2 are parallely oriented along the orientation of γ and they form a positive double helical structure around γ. we call l(s1,s2; γ) satisfying all the assumptions above, a dna-link. let l0 = l0(s1,s2; γ0) be a dna-link. after the dna is replicated and distributed into two daughter cells, there are two identical dna-links representing daughter dnas, l1 = l1(s ′ 1, s̄1,γ1), (1) l2 = l2(s ′ 2, s̄2,γ2), (2) where s′1 and s ′ 2 are single strands (templates) inherited from l0. s̄1 and s̄2 represent counterparts of s1 and s2 respectively and γ1 and γ2 are centre curves of the strips for l1 and l2 respectively. the semi-conservative scheme is interpreted in terms of dna-link l(s1,s2; γ): lemma iv.2 ([10]). the semi-conservative scheme is interpreted as such: the dna-link l0(s1,s2; γ) is deformed into the split 2component link {s′1,s ′ 2}, where s ′ i is obtained from si (i = 1, 2) by applying unknotting operations to l0. c. linking number let dl be a link diagram of an oriented link l. at a crossing point of dl, there are two types of crossings formed by short subarcs of dl; positive and negative crossings (see figure 4). let l(s1,s2) be an oriented link with link components s1 and s2. let c(dl) be the set of crossings of the diagram dl. the linking number is defined by lk(s1,s2) = 1 2 ∑ c∈c(dl) ε(c)d(c) where c is a crossing of the link diagram, ε(c) is the sign ±1 according to the diagrams in figure 4, biomath 11 (2022), 2203288, https://doi.org/10.55630/j.biomath.2022.03.288 page 4 of 12 https://doi.org/10.55630/j.biomath.2022.03.288 abdul adheem mohamad, tsukasa yashiro, topological process of splitting dna-links l(s1, s2; γ) s′ 1 s ′ 2 fig. 3. the semi-conservative scheme is topologically interpreted as splitting the dna-link. +1 −1 k2 k1 k2k1 fig. 4. crossings with signs and also d(c) =   1 if the crossing c consists of distinct components, 0 otherwise. let k be a knot and let dk be a knot diagram. the total sum of signs w(dk): w(dk) = ∑ c∈c(dk) ε(c) is called a writhe of k. note iv.1. the linking number does not depend on the choice of the diagram of l (see [12]). if a 2-component link is split, then the linking number between the components is of course zero but the converse is not always true (see [12]). the writhe depends on the choice of diagram (see [12]). white proved in [25] the following formula of the linking number lk(s1,s2): lemma iv.3 (white [25]). for a dna modeled by the dna-link l(s1,s2; γ), the following formula of the linking number holds: lk(s1,s2) = tw(s1,s2) + wr(γ), (3) where tw(s1,s2) is the number of full-twists of the curves {s1,s2} along the centre curve γ and wr(γ) is the writhe of γ. proof: a proof can be found in [25]. the following is easily verified (see [9]). corollary iv.1. suppose that a dna-link l(s1,s2; γ) has a trivial γ. then lk(s1,s2) = 0 if and only if l(s1,s2; γ) is split. therefore, the splitting process of the dna-link is equivalent to that making the linking number zero. lemma iv.4. the semi-conservative scheme is interpreted to make the linking number of the dna-link zero. proof: combining lemma iv.2 and corollary iv.1, the result follows. we understand that the contribution to the writhe from the conformation of dna of a degree higher than the nucleosomes should be considered. however, as this paper focuses on an individual replicon, and for sake of simplicity, we ignore the contribution from the conformation of a degree higher than the nucleosomes. d. unknotting operations there is an operation to exchange the over arc and the under arc, called an unknotting operation (see figure 5-(a)). biomath 11 (2022), 2203288, https://doi.org/10.55630/j.biomath.2022.03.288 page 5 of 12 https://doi.org/10.55630/j.biomath.2022.03.288 abdul adheem mohamad, tsukasa yashiro, topological process of splitting dna-links lemma iv.5 (proposition 4.4.1 [12]). for any knot diagram of a non-trivial knot, it is deformed into a knot diagram of a trivial knot by applying a finite number of unknotting operations. proof: a proof of this lemma can be found in [12]. note that this lemma gives a guarantee to modify every knot into a trivial knot by applying a finite number of unknotting operations. however, it should be emphasized that lemma iv.5 does not say which crossings should be changed to obtain the trivial knot. it should be noted that a random choice of a sequence of the deformation does not work. for instance, as we can see in the diagram in figure 5-(b), it is a trivial knot but even one crossing change creates a non-trivial knot. conversely, if a non-trivial knot is given, if we choose a wrong sequence of crossings to exchange, then we cannot reach a trivial knot. this observation suggests that the conformation of a supercoil must have a certain format and size to allocate the enzymes to the right place. example iv.1. the diagram (b) in figure 5 is a diagram of a trivial knot γ with loops and some crossings. if we change the crossing at the top, then we obtain a non-trivial knot (the figureeight knot). on the other hand, if we change two crossings at the bottom (figure 5(c)), then it keeps the triviality and decreases the writhe by 4. as we can see in the proof of proposition 4.4.1 in [12], if a link diagram is given, then there is a way to specify the set of crossings to be changed to obtain a diagram of a trivial link. however, we do not know how to detect such a set of crossings from only local information such as a set of crossings. v. a reduction process we make the following assumption. a3 the number of unwound full twists is equal to the number of crossings in the positive supercoil ahead of the fork. the semi-conservative scheme implies that the dna-link must be deformed into a trivial link. by corollary iv.1, to split a dna-link, the linking number must be zero. we define the following procedure to reduce the linking number consisting of the following steps. s1 unwind n full twists at a specified point of the dna-link to create a pair of forks. s2 create positive n crossings (supercoil) in front of each fork. s3 apply the unknotting operations on the n crossings of the supercoil to obtain −n crossings. s4 if the linking number is not zero, then go back to s1. to determine the number n, it is natural that n is proportional to the initial twisting number tw0. so, n = ctw0, 0 < c < 1 (4) we start the process with the initial linking number lk0: lk0 = tw0 + wr0, (5) where wr0 is the initial writhe of γ. following the steps from s1 to s4, we obtain the following sequence of values twk and wrk, k = 0, 1, . . .. tw1 = tw0 − ctw0 = tw0(1 − c) (6) wr1 = wr0 − ctw0 (7) lk1 = tw1 + wr1 (8) = tw0(1 − c) + wr0 − ctw0 (9) = lk0 − 2ctw0 (10) if the procedure ends at this stage; that is, lk1 = 0, then the number n is half of lk0. this is almost half of tw0. however, this is not possible without changing tw0 because, to form a crossing in which two segments of double strand dna become very close, a dna segment with a certain length must necessarily be bent. this implies that the number of unwound twists becomes much larger than the possible number of crossings introduced ahead of the fork. biomath 11 (2022), 2203288, https://doi.org/10.55630/j.biomath.2022.03.288 page 6 of 12 https://doi.org/10.55630/j.biomath.2022.03.288 abdul adheem mohamad, tsukasa yashiro, topological process of splitting dna-links wr = 2wr = 2 wr = 0 wr = −2 unknotting operation(a) (c)(b) fig. 5. we cannot change the randomly selected crossings to obtain a required knot type. (a) the unknotting operation. (b) the unknotting operation at the top crossing leads to a non-trivial knot. (c) the uknotting operations at the bottom crossings keep decreasing the writhe and keep the triviality of γ. a. the number of repetitions next, we consider how many times we have to repeat the procedure. we assume that the number of crossings unwound in each cycle is the same as the number of nucleosomes existing ahead of the fork. this assumption will be justified in section vi. then we have the identity: n = ctw0 = tw0 l , (11) where l is the number of full twists within the dna around a nucleosome and its linker dna. note that if the nucleosomes are distributed uniformly along a replicon, then tw0 l is the number of nucleosomes in the replicon. we denote this by τ0. although, a recent study in [19] shows that the writhe contributed to each nucleosome is −1.26, for sake of generality, here we use α > 0 as the contribution of writhe for each nucleosome. thus the initial writhe for the replicon is wr0 = −ατ0 = −αctw0 (12) proposition v.1. let l be a dna-link with the initial twists tw0 and the initial writhe wr0. if the reduction process is applied to l k times, then the linking number lkk is given by the following. lkk = tw0 [ 2 (1 − c)k − (1 + αc) ] , (13) where α is the writhe contribution to each nucleosome, and c is the rate of unwond full twists to tw0. proof: after applying the unknotting operations to the chromatin fibre at the first stage, the number of full-twists tw1 is given by tw1 = tw0 − τ0 = tw0 − ctw0 = tw0 (1 − c) the number of nucleosomes τ1 at the second stage is given by τ1 = ctw1 = ctw0 (1 − c) after applying the unknotting operations to the chromatin fibre at the second stage, tw2 = tw1 − τ1 = tw0 (1 − c) − ctw0 (1 − c) = tw0 (1 − c)2 at the kth stage, twk = tw0 (1 − c) k , (14) whrere k is the number of repetition of the deformation cycles. on the other hand, the initial writhe wr0 is given by the following. wr0 = −ατ0 = −αctw0 biomath 11 (2022), 2203288, https://doi.org/10.55630/j.biomath.2022.03.288 page 7 of 12 https://doi.org/10.55630/j.biomath.2022.03.288 abdul adheem mohamad, tsukasa yashiro, topological process of splitting dna-links for further stages, wr1 = wr0 − τ0 = −αctw0 − ctw0 = ctw0(−α− 1) (15) wr2 = wr1 − τ1 = ctw0(−α− 1) − τ1 = ctw0 (−α− 1 − (1 − c)) (16) wr3 = ctw0 ( −α− 1 − (1 − c) − (1 − c)2 ) (17) ... wrk = ctw0 (−α− 1 − (1 − c) −··· −(1 − c)k−1 ) (18) applying the formula of a geometric series, we obtain the following. wrk = −ctw0 ( α + 1 − (1 − c)k 1 − (1 − c) ) = −tw0 (1 + αc) + tw0 (1 − c)k (19) therefore, the sum of (14) and (19) is the linking number after applying the procedure k times. lkk = twk + wrk = 2tw0 (1 − c)k − tw0 (1 + αc) = tw0 [ 2 (1 − c)k − (1 + αc) ] (20) proposition v.2. suppose that the reduction system is applied to a dna-link multiple times to obtain the linking number zero. the number of the repetitions k is given by k = ln ( 1+αc 2 ) ln (1 − c) (21) proof: suppose lkk = 0, we obtain tw0 [ 2 (1 − c)k − (1 + αc) ] = 0 2 (1 − c)k = (1 + αc) k ln (1 − c) = ln ( 1 + αc 2 ) k = ln ( 1+αc 2 ) ln (1 − c) (22) since the number α is a constant, k is determined by the parameter c, which is given by tw0. we consider the nucleosome in which the dna wraps around the histone core as a unit of the bending. the diameter of the histone core is about 6.4 nm, and the diameter of dna is 2 nm [1], [20]. the dna wraps around a histone core about 1.8 times [1], [20] and each nucleosome is associated with a linker dna. the total length is 197 bp. therefore, we obtain: l = 197 10.5 ≈ 18.8 (23) the number of unwound full twists depends on the parameter l; that represents the relaxation of the ds-dna. a recent study in [19] shows that the writhe contributed to each nucleosome is −1.26. substituting (23) and α = 1.26 to the formula (21) of k, we obtain k ≈ 11.5. this implies that if we apply the reduction process on the core curve of the dna-link, then we need to repeat the process 11.5 times. the linking number lkk in (20) is a function of c = 1/l which is determined by the relaxation of the double strand dna done by type i topoisomerase. in fact, from the formula (21) with α = 1.26, if a dnalink is relaxed so that l = 3.26, then k = 1. vi. topological model a. ε-crossings let γ be an oriented knot. let x,y ∈ γ be two distinct points, and let ε > 0 be some number. let b(x; r) denote an open 3-ball in r3, centered at the point x and with radius r. suppose that there is a point z ∈ r3 \ γ such that x,y ∈ b(z; ε/2). if γ ∩ b(z; ε/2) is a pair of line segments e1 and e2 such that x ∈ e1 and y ∈ e2, then we say e1 and e2 form an ε-crossing. the ε-crossing has a sign + or − according to the orientation (see left two diagrams in figure 6). we do not admit the exact parallel cases (the middle two diagrams in figure 6) as the ε-crossings. a loop is a simple sub-arc of a knot, from an εcrossing to itself (see figure 6-(c)). biomath 11 (2022), 2203288, https://doi.org/10.55630/j.biomath.2022.03.288 page 8 of 12 https://doi.org/10.55630/j.biomath.2022.03.288 abdul adheem mohamad, tsukasa yashiro, topological process of splitting dna-links − b(z; ε) z α + (a) ε-crossings (b) + − (c) loops (d) e fig. 6. singed ε-crossings and loops. suppose that the boundary of a disc e, ∂e is the union of portion of α \ b(z; ε) and a simple arc on ∂b(z; ε) (see figure 6-(d)), then we say the loop α based at an ε-crossing bounds a disc e. note that we have the following fact. lemma vi.1. let γ be an oriented knot. let α be a loop of γ based at an ε-crossing. if α bounds an embedded disc e in r3, and the interior of e does not meet γ, then applying an unknotting operation at the ε-crossing does not change the knot type of γ. proof: without loss of generality, we can assume the ε-crossing of the oriented loop is positive. since the loop α with the ε-crossing bounds a disc e in r3, the loop can be deformed into the diagram shown in the left diagram of figure 6(c). then we can apply the reversed reidemeister move i to remove the ε-crossing and apply the reidemeister move i to create a loop based at the negative ε-crossing (see the right diagram of figure 6-(c)). the resulting diagram is equivalent to that obtained by a crossing change at the εcrossing. since the move does not change the knot type (see [12] for details), the unknotting operation at the ε-crossing does not change the knot type. b. a modeling policy as example iv.1 demonstrates, it is not easy to deform a non-trivial knot into a trivial knot. therefore, it is natural to assume that the center curve γ remains trivial during the replication. we assume the following. a4 the imaginary core curve γ keeps its triviality during the replication process. note that although the core curve itself is trivial, it may have a certain complexity. c. elementary twists in order to solve the problem p2, a special type of conformation of dna needs to be introduced so that the conformation allocates the enzymes to the suitable positions. suppose a short segment of dna is u-shaped (see figure 7 (a)) and one end is fixed while the other side is rotated around the axial curve. this rotational stress will introduce a loop, based at ε-crossing (figure 7 (b)) so that the writhe is increased by 1. then apply the unknotting operation at the crossing (figure 7). here the writhe is decreased by −1. then the segment returns to the initial position. during the move, the segment is fully twisted twice around the axis. this deformation will be called an elementary twist. the chromatin fibre has a sequence of nucleosomes [20]. we suppose that the chromatine fibre has a zig-zag shape (juxtapositioned) shown as the diagram in figure 8 (see also [17]). then near a nucleosome, it has a u-shape region. we can apply the elementary twist around each nucleosome so that the obtained ε-crossings near nucleosomes is changed (see figure 8 (c) and (d)). in this conformation, as each ε-crossing is close enough to the histone core, we can assume the following. supposition 1. the loop based at the ε-crossing near the histone core bounds an embedded disc e in r3 and the interior of e does not meet the dna strand. this means that the histone core plays the biomath 11 (2022), 2203288, https://doi.org/10.55630/j.biomath.2022.03.288 page 9 of 12 https://doi.org/10.55630/j.biomath.2022.03.288 abdul adheem mohamad, tsukasa yashiro, topological process of splitting dna-links (a) (b) (c) operation unknotting fig. 7. (a) the u-shaped string is twisted around the axial curve. (b) a positive loop is created. (c) applying unknotting operations to the crossing, the negative crossing is obtained. (a) (c) (b) topo ii (d) fig. 8. schematic diagrams showing the unknotting operation done by topo ii at crossings near the nucleosomes. e fig. 9. the loop around the histone octamer with a ε-crossing close enough to the histone octamer is supposed to bound an embedded disc e. biomath 11 (2022), 2203288, https://doi.org/10.55630/j.biomath.2022.03.288 page 10 of 12 https://doi.org/10.55630/j.biomath.2022.03.288 abdul adheem mohamad, tsukasa yashiro, topological process of splitting dna-links role of a disc e in lemma vi.1. therefore, by lemma vi.1, the unknotting operation at the εcrossing does not change the knot type of γ. we call this process an unknotting process of a dnalink (replicon). the process follows the steps below. s1 unwinding n full-twists at the replication fork will introduce a positive supercoil with n crossings ahead of the fork (see figure 8c)). s2 when the number of the crossings reaches the maximum, activate the type ii topoisomerases to the crossings. s3 continue the move to return the conformation to the original shape. this unknotting process keeps γ trivial. as discussed in section v, the number of repetitions of the deformation above depends on the numbers l and α. using this process in the reduction process explained in section v, the assumption that the number of crossings unwound in each cycle is the same as the number of nucleosomes existing ahead of the fork, is justified. vii. discussion to solve problem p1, a schematic procedure of reducing the linking number of dna is considered. the procedure gives the formula (13) in proposition v.1 to obtain the reduced linking number. on the other hand, proposition v.2 gives the formula (21), and it implies that if only the unknotting operations, namely, the type ii topoisomerase, is used, the number of repetitions is about 11.5. although it is difficult to say whether this number is large or small, the formula (21) depends on two parameters α and c = 1/l, which may vary under the relaxation of the double strand dna [1]. this suggests that if the ds-dna is relaxed in prior, then the number of repetitions could be much smaller. for instance, the authors proposed in [10] the reduction of tw with type i topoisomerase, in which tw is reduced to 20 % of the initial twists by the end of the first stage. this gives l′ = l/5; that is, c′ = 5c. assuming the same contribution of writhe from the nucleosome, α = 1.26, substituting c′ and α = 1.26, to (21), the number of repetitions is about 1.3. combining type i and ii topoisomerases to simplify dna has been pointed out in researches [2], [7], [22], [26] from different viewpoints. next, in order to solve p2, we proposed a model of a mechanism to allocate type ii topoisomerases to suitable crossings. as we have seen in example iv.1, we cannot randomly choose the set of crossings to apply unknotting operations to deform a non-trivial knot into a trivial knot. therefore, it is natural to assume that the model does not change the triviality of the core curve of a dna-link. the model has zig-zag shaped (juxtaposed) nucleosomes in which the axial rotation introduces a trivial loop with a crossing near each of the nucleosomes. for this crossing, the unknotting operation reduces the writhe, but it does not change the triviality of the core curve. also, this guarantees that the core curve is always trivial during the replication process. viii. conclusion from the observation of the proposed procedure, we obtained 11.5 as the necessary number of repetitions to make the dna-link split. this number is parametrized by two parameters, α and c = 1/l, and c will be changed by relaxation of double strand dna by type i topoisomerase. therefore, a combination of two types of topoisomerases efficiently reduce the linking number. as we have seen in example iv.1, specifying the location of topoisomerase ii is an essential issue to make the linking number zero. our model provides the mechanism that allocates enzymes to the right position and the action of type ii topoisomerase does not change the knot type of the core curve. from the arguments about the procedure and the model, we can conclude that the linking number is efficiently reduced when two types of topoisomerases are combined, and it is possible to allocate type ii topoisomerase to the appropriate places by the conformation of dna. in this research, we have not considered the reduction process of the negative supercoils behind the forks. this should be done in further study. biomath 11 (2022), 2203288, https://doi.org/10.55630/j.biomath.2022.03.288 page 11 of 12 https://doi.org/10.55630/j.biomath.2022.03.288 abdul adheem mohamad, tsukasa yashiro, topological process of splitting dna-links acknowledgement the authors would like to thank dr jagir hussan for giving valuable suggestions for the earlier version of the paper. references [1] a.d. bates, a. maxwell, et al. dna topology. oxford university press, usa, 2005. 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[24] j.d. watson and f.h.c. crick. “genetical implications of the structure of deoxyribonucleic acid”. in: nature 171.4361 (1953), pp. 964-967. [25] j.h. white. “self-linking and the gauss integral in higher dimensions”. in: american journal of mathematics 91.3 (1969), pp. 693-728. [26] g. witz, g. dietler, and a. stasiak. “dna knots and dna supercoiling”. in: cell cycle 10.9 (2011). pmid: 21393995, pp. 1339-1340. doi: 10.4161/cc.10.9.15293. biomath 11 (2022), 2203288, https://doi.org/10.55630/j.biomath.2022.03.288 page 12 of 12 https://doi.org/10.3390/molecules26113375 https://doi.org/10.1016/j.jmb.2021.166847 https://doi.org/10.1073/pnas.1106834108 https://doi.org/10.4161/cc.10.9.15293 https://doi.org/10.55630/j.biomath.2022.03.288 introduction dna replication topoisomerases dna-links links topological semi-conservative scheme linking number unknotting operations a reduction process the number of repetitions topological model e-crossings a modeling policy elementary twists discussion conclusion references www.biomathforum.org/biomath/index.php/biomath original article non-monotonicity of fano factor in a stochastic model for protein expression with sequesterisation at decoy binding sites michal hojcka, pavol bokes department of applied mathematics and statistics comenius university bratislava, slovakia michal.hojcka@fmph.uniba.sk, pavol.bokes@fmph.uniba.sk received: 11 july 2017, accepted: 21 october 2017, published: 2 november 2017 abstract—we present a stochastic model motivated by gene expression which incorporates unspecific interactions between proteins and binding sites. we focus on characterizing the distribution of free (i.e. unbound) protein molecules in a cell. although it cannot be expressed in a closed form, we present three different approaches to obtain it: stochastic simulation algorithms, system of odes and quasisteady-state solution. additionally we use a largesystem-size scaling to derive statistical measures of approximate distribution of the amount of free protein, such as the fano factor. intriguingly, we report that while in the absence of or in the excess of decoy binding sites the fano factor is equal to one (suggestive of poissonian fluctuations), in the intermediate regimes of moderate levels of binding sites the fano factor is greater than one (suggestive of super-poissonian fluctuations). we support and illustrate all of our results with numerical simulations. keywords-gene expression; master equation; small noise approximation; stochastic simulation i. introduction the number of proteins and other species present in the biological processes inside the cells such as gene expression is usually small [6], [28]. therefore, deterministic modeling of such reactions can be quite inaccurate and we often turn to stochastic methods [18]. they account for discrete number of molecules and can easily be simulated through stochastic simulation algorithms, in particular the gillespie algorithm [10], [11]. being a very timely topic, gene expression spurred a revival of interest in markovian models of chemical kinetics, e.g. [25]. we assume that the protein is produced with a constant rate and that the rate of its decay is proportional to the number of proteins. we study the protein dynamics in presence of so-called decoy binding sites [17] on the dna. our model takes into account protein binding/unbinding reactions with these binding sites. similar models have already been studied previcopyright: c© 2017 hojcka et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: michal hojcka, pavol bokes, non-monotonicity of fano factor in a stochastic model for protein expression with sequesterisation at decoy binding sites, biomath 6 (2017), 1710217, http://dx.doi.org/10.11145/j.biomath.2017.10.217 page 1 of 17 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2017.10.217 michal hojcka, pavol bokes, non-monotonicity of fano factor in a stochastic model for protein ... ously; in particular [9] investigated the model with protected complexes, i.e. the case when bound proteins were immune to degradation, showing that the steady-state distribution is poissonian. our model allows bounded proteins to degrade, which introduces additional noise into the model [2], [3]. for simplicity, we ignore effects of burst-like protein synthesis or transcriptional auto-regulation [2], [4], [26]. in section ii we formulate the master equation for the probability distribution of free protein; unfortunately, we are unable to find its solution in a closed form. however, biochemical reactions often operate on different timescales as was already thoroughly investigated in works such as [14], [12], [13]. specifically, in the current context the interactions between the protein and its binding sites occur on a substantially faster timescale than the turnover of protein (by transcription and decay) does [1]. this allows us to successfully use singular perturbation methods [5], [22], [23] to obtain the quasi-steady-state solution to our problem. obtaining the quasi-steady-state approximations in our model involves finding an equilibrium of binding/unbinding reaction, which is a specific case of a reversible bimolecular reaction studied by laurenzi in [16]. in section iii we expand the master equation using the linear noise expansion as proposed in [29]. ii. stochastic approach the number of proteins expressed from a single gene can often be quite small [31]; it would therefore be inaccurate to use deterministic approach, which treats reactants as continuous variables. instead we take a stochastic approach, in which we model each species as a discrete random variable and each reaction as a random event with a given probability to occur. some protein species are present at even less than 10 copies per e. coli cell [28], therefore we also work with mean number of proteins below 100 in this paper. as we use discrete variables it is quite clear that we need to simulate these reactions through a simulation algorithm. such algorithms gained a wider recognition after the work of gillespie [10]. a. model description in this section we present a minimalistic model for gene expression. we neglect the effect of mrna translation and production of proteins in bursts; instead we focus on the interaction between the proteins and decoy binding sites. unlike in [26], we assume that bounded proteins are subject to degradation processes. let us use the following notation for our variables: x protein (free or bound), xf free protein, y binding site, yf free binding site, c complex (protein bound to the binding site). for the sake of simplicity we omit ’decoy’ from the binding site notation as we do not take into account any other binding sites. we assume that three reversible reactions can take place: 1) protein production/decay. ∅ k−⇀↽− γ xf 2) protein binding/unbinding reaction. xf + yf k+−⇀↽− k− c 3) decay of the complex (whereby a binding site is vacated). c γ −→ yf we use upper-case letters in italics to represent a number of corresponding species throughout this paper. we reserve the corresponding lower-case letter as a notation for a concentration of a given species. in order to avoid confusion with x , we use n instead of xf as the number of free protein. although we mentioned five different variables, the problem is just two-dimensional, with the following straightforward conservation laws held between the variables: biomath 6 (2017), 1710217, http://dx.doi.org/10.11145/j.biomath.2017.10.217 page 2 of 17 http://dx.doi.org/10.11145/j.biomath.2017.10.217 michal hojcka, pavol bokes, non-monotonicity of fano factor in a stochastic model for protein ... fig. 1: simulation using the gillespie algorithm. y is a known constant (the number of binding sites) c = x −n (the number of complexes is the same as the number of bound protein) yf = y −c = y −x + n (the number of free binding sites is the same as the number of all binding sites without the complexes) therefore we can express the master equation of the system in terms of x and n (with a constant total number of binding sites y ): ṗx ,n = kpx−1,n−1 −kpx,n +γ(n +1)px+1,n+1−γnpx,n +k+(n +1)(y −x +n +1)px,n+1 −k+n (y −x +n )px,n +k−(x−n +1)px,n−1−k−(x−n )px,n +γ(x−n +1)px+1,n −γ(x−n )px,n , (1) where px ,n is the abbreviation of p(x(t) = x , xf (t) = n ). we can use the gillespie algorithm to simulate this process (parameters of the reactions are set to y = 10, k = 3, γ = 0.1, k+ = 1, k− = 10 with no proteins at the biomath 6 (2017), 1710217, http://dx.doi.org/10.11145/j.biomath.2017.10.217 page 3 of 17 http://dx.doi.org/10.11145/j.biomath.2017.10.217 michal hojcka, pavol bokes, non-monotonicity of fano factor in a stochastic model for protein ... beginning: px ,n (0) = δx ,0δn ,0. we illustrate the individual components of the process in figure 1. b. total protein distribution our first goal is to obtain the marginal distribution of the total protein number. it can be obtained as the sum of px ,n through all possible n ’s: px = ∞∑ n =−∞ px ,n . let us use an abbreviation ∑ n for the sum through all integers. in order to derive an equation px from (1) let us sum both sides of the equation through all n ’s. as the sum goes through all integers (if n < 0 is the probability is naturally equal to zero), we can use the fact that∑ n f(n ) = ∑ n f(n ± 1), obtaining ṗx =k ∑ n (px−1,n −px,n ) +γ ∑ n n (px+1,n −px,n) +γ ∑ n ((x−n +1)px+1,n −(x−n )px,n) , which simplifies to ṗx =k (px−1−px )+γ ((x +1)px+1−xpx ) . (2) a system of differential equations in this form can be solved using the method of generating functions [15]. first we multiply both sides by sx and sum them through all integer x ’s. ∂ ∂t ∑ x sx p(x, t) =k (∑ x sx px−1− ∑ x sx px ) +γ ( (x +1) ∑ x sx px +1−x ∑ x sx px ) . (3) now we can use the definition of the generating function g(s,t) = ∑ x sx px and its derivative ∂g ∂s = ∑ x xsx−1px in order to transform (3) into a partial differential equation: ∂g ∂t = k(s− 1)g + γ(1 −s) ∂g ∂s = (s− 1) ( kg−γ ∂g ∂s ) . (4) in the steady state we can omit the left-hand side of the equation (as ∂g ∂t = 0) and easily solve it using the separation of variables. we come to the solution g(s,∞) = e k γ (s−1), which can be recognized as the probability generating function of the poisson distribution parametrized by λ = k γ . using the notation 〈x〉 = λ for the distribution’s mean we can write px (∞) = 〈x〉x e−〈x〉 x ! . away from the steady state, (4) is an example of a nonhomogeneous first-order linear partial differential equation, which can be solved for suitable initial conditions. using a common extension of the method of characteristics for (quasi-)linear pde’s (see e.g. [8]) we introduce an auxiliary function u = u(s,t,g), which satisfies ∂u ∂t + γ(s− 1) ∂u ∂s + k(s− 1)g ∂u ∂g = 0. the characteristic system of this equation has the form: ṫ = 1, ṡ = γ(s− 1), ġ = k(s− 1)g, t(τ) = τ + c1 s(τ) = c2e γτ + 1 dg g = k(c2e γτ )dτ (5) by finding the functions which are constant on the characteristics, we obtain solutions in the form g(s,t) = ψ ( t− ln(s−1) γ ) ·e ks γ . we can specify the results for any particular initial conditions. let us investigate the case when there is no protein at the beginning, i.e. p(x, 0) = δx,0, which gives us an initial condition g(s, 0) = 1 also for the generating function. using the initial condition we come to the solution g(s,t) = e− k γ (e−γt(s−1)+1)+ ks γ = e −k γ (s−1)(e−γt−1) , which is again the poisson distribution, just with different, time-dependent, parameter meaning that 〈x (t)〉 = k γ · ( 1 −e−γt ) . taking t → ∞ we can easily see that the timedependent solution converges to the steady-state one. biomath 6 (2017), 1710217, http://dx.doi.org/10.11145/j.biomath.2017.10.217 page 4 of 17 http://dx.doi.org/10.11145/j.biomath.2017.10.217 michal hojcka, pavol bokes, non-monotonicity of fano factor in a stochastic model for protein ... c. free protein distribution searching for an exact formula for the free protein distribution yields no apparent results: the master equation (1) does not have solution in the closed form (unless y = 0). therefore we have to look for alternative ways to obtain it. 1) stochastic simulation algorithms: the first approach consists of using the techniques of stochastic simulation algorithms, in particular the gillespie algorithms (see [7] for a practical guide). with their help we can simulate the time evolution of all species involved in a system of reactions. the simulation technique is based upon the following scheme: we generate two uniformly distributed random numbers, the first of which determines the time of the next reaction and the second one determines which reaction will occur. when we use a sufficient number of sample trajectories, we obtain a robust estimate for the distribution of any variable. the main problem of this approach is its limited computational capacity. in order to obtain a robust distribution of variables we have to run a substantial number of simulations; this can be very time-consuming and even unrealistic when calculating results for many different parameter sets. 2) system of odes: the second option is to transform the master equation, which is an infinite system of equations, into a finite system of ordinary differential equations. the most straightforward way to obtain this is to set threshold values of the discrete variables, replacing the unknown probability by zero whenever these thresholds are exceeded (see [20], in our case we set all px ,n to zero, whenever x > 100). in this way, the overall sum of probability distribution is no longer conserved at one, but in return we get a finite system of differential equations. we use the matlab ode15s solver for stiff problems to calculate the solution. but this system is also very timeconsuming, as the number of equations and the computational difficulty grows rapidly with raising the thresholds for the non-zero probabilities of px ,n . 3) quasi-steady-state solution: in order to obtain the solution in a closed form, which is as close to exact solution as possible, we proceed to obtain the so-called quasi-steady-state solution [24], [27]. we utilize the fact from biological background that binding/unbinding reactions are fast compared to protein production/degradation (k− � γ). in order to use this fact we perform the singular perturbation reduction [9] using the ratio γ/k− as a small parameter. let us introduce new dimensionless time and the parameters: ε = γ k− , kb = k− k+ , t = τ γ . the master equation then reads ε d dτ px ,n = ε k γ (px−1,n−1 −px ,n ) + ε ((n + 1)px +1,n +1 −γnpx ,n ) + 1 kb ( (n + 1)(y −x + n + 1)px ,n +1 −n (y −x + n )px ,n ) + (x −n + 1)px ,n−1 − (x −n )px ,n + ε ((x −n + 1)px +1,n − (x −n )px ,n ) . (6) as ε tends to zero, we obtain an equation for the leading-order approximation of px ,n . the number of total protein x is constant after this approximation, so that we can abbreviate px ,n by pn , writing (n + 1)(y −x + n + 1)pn +1 = (n (y −x + n ) + kb(x −n )) pn− −kb(x −n + 1)pn−1. (7) solving (7) subject to boundary conditions (see e.g. [16]) pn (n <0) = 0 and pn (n >x ∨n <x−y ) = 0 yields pn =px,n = knb c(x ) n !(x −n )!(y −x +n )! , (8) where c(x ) is a constant with respect to n , dependent only on the value of x . the multiplicative constant c(x ) can be deterbiomath 6 (2017), 1710217, http://dx.doi.org/10.11145/j.biomath.2017.10.217 page 5 of 17 http://dx.doi.org/10.11145/j.biomath.2017.10.217 michal hojcka, pavol bokes, non-monotonicity of fano factor in a stochastic model for protein ... fig. 2: quality of approximation by the quasi-steady-state solution as function of k−/γ mined from the relation px = x∑ n =max{x−y ,0} px ,n , since px = x∑ i=max{0,x−y} px ,i = c(x ) x∑ i=max{0,x−y} kib (x−i)!(y −x +i)!i! , and thus c(x ) =px   x∑ i=max{0,x−y} kib (x−i)!(y−x+i)!i!  −1. since we already know about the poisson character of steady-state solution for px , we are ready to calculate the number of free protein in quasisteady state: pn = n +y∑ x =n px · knb n !(x−n )!(y−x +n )! x∑ i=max{0,x−y} kib (x−i)!(y−x +i)!i! (9) this represents a major improvement in terms of computational requirements necessary to get the distribution of free protein amount. in order to illustrate the quality of this approximate solution, let us investigate the differences between the simulated distribution (by gillespie algorithm) and the approximate quasi steady state biomath 6 (2017), 1710217, http://dx.doi.org/10.11145/j.biomath.2017.10.217 page 6 of 17 http://dx.doi.org/10.11145/j.biomath.2017.10.217 michal hojcka, pavol bokes, non-monotonicity of fano factor in a stochastic model for protein ... fig. 3: mean of free protein probability distribution solution (9) for selected values of ε−1 = k− γ , which is assumed to be large as justified above. we assumed the following values of the parameters: y = 10; k = 3,γ = 0.1 ⇒ 〈x〉 = 30; k+ = a,k− = 10a ⇒ kb = 10. we vary the value of a in order to change the value of ( ε−1 ) while keeping kb fixed; thus the distribution obtained from (9) remains the same. the results are shown in figure 2, where the green histogram is estimated from repeated simulations (105 trajectories) of the gillespie algorithm on a sufficiently long timescale. the blue line is the probability distribution of free protein in quasi-steady state (9). it is easy to see that with the lowering of k− γ , we diverge from exact distribution results estimated by the gillespie algorithm histogram. to evaluate the goodness of the fit, we calculate the `2 distance between these two realizations of free protein distribution for all n ’s from 0 to 100 (see table i). k−/γ diff 1000 0.0031 100 0.0036 10 0.0071 1 0.048 0.1 0.1483 table i: the `2 distance between the quasisteady state approximation and the exact distribution estimated by stochastic simulation. d. moments of free protein distribution in this section we focus on the analysis of the free protein probability distribution in the quasisteady state. we focus on four basic statistical characteristics: mean ((figure 3)), variance, the fano factor (f = σ 2 µ ) (figure 4) and the squared coefficient of variation (cv 2 = σ 2 µ2 ). we use the number of binding sites as a free parameter, which we vary on the horizontal axis. we sometimes biomath 6 (2017), 1710217, http://dx.doi.org/10.11145/j.biomath.2017.10.217 page 7 of 17 http://dx.doi.org/10.11145/j.biomath.2017.10.217 michal hojcka, pavol bokes, non-monotonicity of fano factor in a stochastic model for protein ... use an abbreviation bs instead of binding site. we study the change in these characteristics for selected choices of the total protein production 〈x〉 and the dissociation constant kb. the values γ = 0.1 and k− = 10 are fixed. different choices of k and k+ are used to modify 〈x〉 and kb. without any binding sites (y = 0) the poisson distribution follows, so that µ = σ2 and thus f = 1. very large dissociation constants imply that proteins bind weakly to the binding sites and thus the system exhibits characteristics that are consistent with the poisson distribution. it is apparent that for large values of y , the mean and variance tend to 0, fano factor tends to 1 and cv 2 diverges to infinity. the convergence of the fano factor to one suggests an approximate poisson distribution in the excess of decoy binding sites. below, we show with asymptotics that this is indeed the case. provided that y � x , we can use the approximation for the factorial (y − x + n )! ≈ y !y n−x . substituting this into (9) and using binomial theorem and basic algebraic manipulation, we obtain a simplified equation for the probability distribution of the free protein: pn = n+y∑ x=n px · ( x n )( kb kb+y )n ( y kb+y )x−n . this formula can also be rewritten as ∑ px ·pn |x , where px has a poisson distribution with parameter 〈x〉; we see that pn |x has a binomial distribution with probability of a bernoulli trial given by kb kb+y and the number of trials given by x . therefore, we can express the free protein distribution to be that of a random n = x∑ i=0 ξi, where p(ξi = 1) = kb kb+y . after some algebraic manipulation with probability generating functions (such as the rule for calculating probability generating function of the sum of independent random variables) we arrive at: gn (s) = gx (gξ(s)) . (10) it is clear now that probability generating function for free protein is given as a composition of generating functions for gx (poisson distribution) and gξ (bernoulli distribution). substituting the well-known formulae for generating functions of these distributions into (10), we obtain gn (s) = exp (〈x〉(gξ(s) − 1)) = exp (( kb〈x〉 kb + y ) (s− 1) ) , (11) which again indicates the poisson distribution, but with a different parameter 〈n〉 = 〈(n −〈n〉)2〉 = kb〈x〉 kb + y . (12) computing the probability distribution in quasisteady state for large values of binding sites is infeasible as the formula works with the terms of the order y !; therefore, we use the gillespie algorithm (with 105 repetitions) to estimate the mean and the variance of the distribution. the results are provided in table ii. other parameters used are 〈x〉 = 10 and kb = 10). as the distribution is poissonian, the fano factor tends to one and cv 2 = 1〈n〉, which tends to infinity. y approximate 〈n〉 〈(n −〈n〉)2〉 100 0.9091 1.0027 1.0124 500 0.1961 0.1999 0.2009 1000 0.099 0.0999 0.0998 5000 0.02 0.02 0.02 10000 0.01 0.0101 0.01 table ii: the approximate value (12) against simulated estimates of the mean and variance in the large y regime. e. comparison of the methods in this section we compare the probability distributions generated by the three methods mentioned earlier: stochastic simulation by gillespie algorithm (referred to as gill in the table iii), transforming master equation into the finite system of odes (ode in table iii) and calculating the solution in quasi-steady state (quasi in table iii). obviously we use the same set of parameters in all three approaches: y = 10, k = 3, k+ = biomath 6 (2017), 1710217, http://dx.doi.org/10.11145/j.biomath.2017.10.217 page 8 of 17 http://dx.doi.org/10.11145/j.biomath.2017.10.217 michal hojcka, pavol bokes, non-monotonicity of fano factor in a stochastic model for protein ... fig. 4: fano factor of free protein probability distribution 1, k− = 10 and γ = 0.1. initially, the system is set to contain no proteins: px ,n = δ(0,0). the output of each method is the probability surface px ,n in all given time points. in table iii we record the differences in the probability surfaces obtained by the three methods. we define the distance between the two methods as the `2-matrix norm, i.e.: ‖a‖2 = √√√√ n∑ i=1 m∑ j=1 |aij|2, which we apply to the differences between the two probability surfaces. iii. small noise approximation in the second part of this paper we focus on the limit case when the size ω of the system is large enough and try to obtain the statistical moments of free protein distribution in the closed form for this case. the noise in free protein distribution has two sources; the first one is the reversible time gill-ode gill-quasi ode-quasi 0 0 0 0 1 0.02 0.0111 0.0219 2 0.0058 0.0055 0.0071 3 0.0027 0.0035 0.0031 4 0.0022 0.0034 0.0027 5 0.0066 0.0024 0.0064 10 0.0044 0.0018 0.0043 20 0.0016 0.0019 0.00061 50 0.0014 0.0016 0.00061 100 0.0018 0.0019 0.00055 table iii: distances between various numerical implementations of the probability surface. association of protein and binding site and the second one is new protein production/degradation. let us first concentrate on the reversible association xf + yf k+−⇀↽− k− c. in the course of this reaction, we can treat the values of x (total protein concentration) and y (total binding sites concentration) biomath 6 (2017), 1710217, http://dx.doi.org/10.11145/j.biomath.2017.10.217 page 9 of 17 http://dx.doi.org/10.11145/j.biomath.2017.10.217 michal hojcka, pavol bokes, non-monotonicity of fano factor in a stochastic model for protein ... as constants. we would like to underscore the fact that lowercase letters denote the concentration of a given reactant denoted by corresponding uppercase letter. a. deterministic case if we assume large protein numbers, we can obtain the mean of the distribution by finding the stationary state of deterministic reaction kinetics. this approach is used since the inception of chemical kinetics [19]. we know that in the stationary state (see e.g. [21]), we have n · yf = c, where the concentrations xf of free protein, yf of free binding site, and c of their complex are measured in units of the dissociation constant. if we combine this equation with conservation laws, n + c = x for the total protein and yf + c = y for the total binding-site concentration, we will obtain a system of three equations with three unknown variables. this system has a single non-negative solution, which we refer to as n̄, ȳf, c̄. we can obtain it by solving the quadratic equation c̄2 + c̄(−x−y − 1) + xy = 0, which gives us the solution in the form c̄(x,y) = x+y+1− √ x2 +y2 +1+2x+2y−2xy 2 , n̄(x,y) = x− c̄, ȳf (x,y) = y − c̄. (13) for the sake of parsimony, we omit the explicit notation of dependence of c̄, n̄, ȳf on the total concentration of x and y in all non-ambiguous cases. b. stochastic component as we mentioned before, we were not able to obtain the distribution of free protein in a closed form; therefore we have to choose an alternative approach. the number of free protein is affected by binding and unbinding reactions as well as by the production/degradation of new proteins. as the timescales of these two processes are diametrically different, we can treat them separately. let us divide the stochastic component into the part corresponding to binding/unbinding and the part corresponding to total protein production/degradation. 1) constant x (total protein count): first we assume that the number of total protein remains constant, focusing only on the binding/unbinding reactions: xf + yf k+−⇀↽− k− c. using the shift-operator notation from [29] (eif(x) = f(x+i)), we can write down the master equation in the steady state for this reaction as 0 = (e−1)k−cpc +(e−1−1)k+n yf pc , (14) where pc denotes probability mass function of c , xf = x − c and yf = y − c , and the shifting is executed with respect to c . we further assume that size ω of the system is large, whereby we identify the system size with the dissociation constant ω = k− k+ . this will naturally lead to the macroscopic dissociation constant being equal to unity after the system-size reduction is performed. now we can write down the relation between the total species count (uppercase letters in italics) and their corresponding concentrations (lowercase letters): n = ω ·n, yf = ω ·yf and c = ω · c. inserting this scaling into (14) yields 0 = (e− 1)cpc + (e−1 − 1)nyfpc, (15) using the taylor expansion together with the fact that c is large (i.e. ω−1 is small), the shift operators can formally be expanded as follows: e=e∂c =eω −1∂c = 1+ω−1∂c+ ω−2 2 ∂2c +. . . e−1=e−∂c =e−ω −1∂c = 1−ω−1∂c+ ω−2 2 ∂2c−. . . (16) inserting (16) into (15) and neglecting terms of order of ω−1 we obtain ∂c [ (c−nyf )pc + ω−1 2 ∂c ((c + nyf )pc) ] = 0. (17) biomath 6 (2017), 1710217, http://dx.doi.org/10.11145/j.biomath.2017.10.217 page 10 of 17 http://dx.doi.org/10.11145/j.biomath.2017.10.217 michal hojcka, pavol bokes, non-monotonicity of fano factor in a stochastic model for protein ... if we now integrate this equation with respect to c (with zero-flux conditions) we come to (c−nyf )pc + ω−1 2 ∂c ((c + nyf )pc) = 0. we denote by terms a = (c−nyf ), b = c + nyf the terms which appear in the above equation. since ω−1 � 1, we can use a small-noise approximation (variance of c is of order ω−1), which is based on taylor-expanding a and b around the deterministic mean value: a(c) = c− (x− c)(y − c) a′(c) = 1 + x + y − 2c = 1 + n + yf a(c) ' a(c̄) + a′(c̄)(c− c̄) = a′(c̄)(c− c̄) = (1 + n̄ + ȳf )(c− c̄) b(c) ' b(c̄) = c̄ + n̄ȳf = 2n̄ȳf. (18) substituting approximations (18) into (17) yields (1 + n̄ + ȳf )(c− c̄)pc + ω−1n̄ȳf∂cpc = 0, i.e. ∂cpc = − 1 + n̄ + ȳf ω−1n̄ȳf · (c− c̄)pc (19) this equation can be easily solved by separation of variables, which yields pc ∝ exp ( 1 2 · (1 + n̄ + ȳf ) ω−1n̄ȳf · (c− c̄)2 ) . we have found an approximate distribution of c; to put it in other words, we can say that c ∼n ( c̄, ω−1n̄ȳf 1 + n̄ + ȳf ) (20) as ω →∞. as n and yf can be computed directly from c as x− c, resp. y − c, they have the same variance as c. 2) fluctuating x (total protein count): in the second part, instead of taking x to be a constant, we assume that total protein count fluctuates due to creation of new protein molecules and decay of old ones, which is denoted by a reversible pair of chemical reactions ∅ γ −⇀↽− k x. we already know that these processes operate on a much slower timescale that the association/dissociation processes. furthermore, we have already found the steady-state distribution of x (from ii-b); the result is that it is poissonian with 〈x〉 = var(x ) = k γ . we use the systemsize scaling k γ = 〈x〉 · ω. as ω → ∞, we can approximate the poissonian distribution by a small-noise gaussian distribution x = x ω ∼n ( 〈x〉, ω−1 · 〈x〉 ) . (21) in order to calculate the statistics of n (free protein concentration) we have to combine the results (20) and (21) given that n is expressed in terms of slowly fluctuating x (total protein concentration). the free protein concentration also naturally depends on y (total concentration of binding sites), but as it is a constant through the whole process, we can neglect it from our notation for the sake of simplicity. we can use the variance decomposition theorem in order to solve this problem (see [30]). it expresses the total (unconditional) variance as the sum of the expectation of conditional variances and the variances of conditional expectations, in our case, we can write var(n) = e (var(n|x)) + var (e(n|x)) . (22) we already know (from (20)) the solution for e(n|x) and var(n|x); we utilize the fact that x fluctuates much more slowly that n and thus we can obtain results for n subject to constant x. using the large ω approximation, we can write e(n|x) = n̄ = n̄(x) var(n|x) = ω−1n̄(x)ȳf (x) 1 + n̄(x) + ȳf (x) . (23) since the variance of x is of order ω−1 and we neglect all terms of order higher than ω−1, we can use the approximations n̄(x) ' n̄(〈x〉) and ȳf (x) ' ȳf (〈x〉) in the formula for the conditional biomath 6 (2017), 1710217, http://dx.doi.org/10.11145/j.biomath.2017.10.217 page 11 of 17 http://dx.doi.org/10.11145/j.biomath.2017.10.217 michal hojcka, pavol bokes, non-monotonicity of fano factor in a stochastic model for protein ... variance, which yields e(var(n|x)) ' var(n|〈x〉) = ω−1 · n̄ȳf 1 + n̄ + ȳf , (24) evaluated at 〈x〉 (mean of total protein concentration) and y (constant concentration of binding sites). the final term left to calculate is var(n̄(x)). if we used the approximation n̄(x) = n̄(〈x〉), we would end with zero, which would incorrectly neglect all the variance. therefore we also include terms of next order by using the linear variance approximation, i.e. by writing n̄(x) ' n̄(〈x〉) + dn̄ dx · (x−〈x〉). from this expression we can find that var(n̄(x)) ' ( dn̄ dx )2 |x=〈x〉 ·v ar(x) = ( dn̄ dx )2 |x=〈x〉 · ω−1〈x〉 (25) hence, the last item to calculate is the derivation of n̄ with respect to x. in order to find it, let us differentiate the equations n̄ȳf = c̄, n̄ + c̄ = x, ȳf + c̄ = y that define the dependence of n̄, among others, on x, with respect to variable x; this yields dn̄ dx · ȳf + dȳf dx · n̄ = dc̄ dx , (26) dn̄ dx + dc̄ dx = 1, (27) dȳf dx + dc̄ dx = 0. (28) substituting (28) into (26), we eliminate dȳf dx and obtain dn̄ dx · ȳf = dc̄ dx · (1 + n̄). now using equation (27) we get dn̄ dx = 1 + n̄ 1 + n̄ + ȳf . (29) thus we are now in a position to find the moments of n. we get the mean by substituting 〈x〉 into the deterministic results (13) and we get the variance by substituting the partial results (24), (25) and (29) into the variance decomposition theorem (22). for mean we obtained the formula e(n) =n̄(〈x〉) = 〈x〉−y−1+ √ 〈x〉2 +y2 +1+2〈x〉+2y−2〈x〉y 2 . (30) variance can be expressed in the form var(n) = ω−1 ( n̄ȳf 1+n̄+ȳf + ( 1+n̄ 1+n̄+ȳf )2 〈x〉 ) . (31) combining these two results gives us the possibility to obtain the fano factor as f = var(n ) e(n ) ' var(ω ·n) e(ω ·n) = ω · var(n) e(n) . (32) after some simplifying steps we come to the expression f = 1 + n̄ȳf (1 + n̄) (1 + n̄ + ȳf ) 2 , (33) where 1 can be interpreted as the poissonian noise and the residual fraction as an additional nonpoissonian noise present due to the interaction with decoy binding sites. c. numerical simulations in the current section we perform numerical simulations in order to visualize of our results and to confirm the validity of the approximation scheme presented above. 1) fano factor based on system-size approach: in the first simulation we investigate the relation between 〈x〉 and the fano factor. in the first figure (figure 5) we plot the dependance of fano factor on the number of binding sites. we calculate the fano factor for different values of 〈x〉. all values are meant as the concentrations; therefore to obtain the corresponding number of molecules we have to multiply them by ω. we clearly see that for larger values of 〈x〉 we are able to reach larger values of f . an interesting point to observe in biomath 6 (2017), 1710217, http://dx.doi.org/10.11145/j.biomath.2017.10.217 page 12 of 17 http://dx.doi.org/10.11145/j.biomath.2017.10.217 michal hojcka, pavol bokes, non-monotonicity of fano factor in a stochastic model for protein ... fig. 5: fano factor for large system size. the graphs is the slope near y = 0. we see that for small values of 〈x〉 the slope increases with increasing 〈x〉 , but for larger values of 〈x〉 the slope starts to decrease. in order to investigate this phenomenon into further depth, let us calculate taylor expansion near f(x, 0): f(x, ∆y) = 1 + ∆y · x (1 + x)2 . we can find the maximum of this value by differentiating ∂f(x, ∆y) ∂x = ∆y · 1 −x (1 + x)3 , which confirms that the maximal slope is obtained for 〈x〉 = 1. in figure 6 we take the ratio y/〈x〉 (number of bs divided by mean number of total protein) as the independent variable and plot fano factor again for several different choices of 〈x〉. we can see that for larger values of 〈x〉 the maximum of fano factor is achieved near y/〈x〉 = 1. in order to investigate this phenomenon further, let us express the fano factor in terms of new variables a = y〈x〉 and b = 1 x . this substitution yields f(a,b) = 1 + 1 2 · a(1 −a− b + √ ?) ? , ? = a2 + b2 + 1 + 2b + 2a− 2ab. (34) the problem of finding the maximum of fano factor with respect to a is equivalent to finding the solution of ∂f(a,b) ∂a = 0, with f(a,b) from (34). this yields a very complex implicit function. as we are interested in cases with large values of 〈x〉, we want to investigate its solution for small values of b. as we are unable to express the value of b in terms of a from the implicit function in reasonable way, we have to settle for numerical and graphical solution for this equation, which is provided in figure 7. this verifies our hypothesis that for large 〈x〉 the maximal fano factor is achieved near the point where y = 〈x〉. 2) quality of system size approach: in this section we investigate whether the linear noise approximation is consistent with the stochastic biomath 6 (2017), 1710217, http://dx.doi.org/10.11145/j.biomath.2017.10.217 page 13 of 17 http://dx.doi.org/10.11145/j.biomath.2017.10.217 michal hojcka, pavol bokes, non-monotonicity of fano factor in a stochastic model for protein ... fig. 6: fano factor for large system size. fig. 7: maximum of fano factor. biomath 6 (2017), 1710217, http://dx.doi.org/10.11145/j.biomath.2017.10.217 page 14 of 17 http://dx.doi.org/10.11145/j.biomath.2017.10.217 michal hojcka, pavol bokes, non-monotonicity of fano factor in a stochastic model for protein ... fig. 8: fano factor for different system sizes. results for quasi-steady state: pn = n+y∑ x =n px · knb n !(x−n )!(y −x +n )! ·   x∑ i=max{0,x−y} kib (x−i)!(y −x + i)!i!  −1 (35) using this formula we can obtain probability distribution of free protein and therefore calculate its fano factor. but it is not that straightforward. in order to get the results for big values of n , we have to rewrite the sum in a way that we get around the problems of calculating big factorials and multiplying number close to 0 with an extremely large number. for this purpose, (9) can be rewritten as: pn = n +y∑ x =n px · · (∑ i ki−nb · n !(x−n )!(y −x +n )! i!(x−i)!(y −x +i)! )−1 . (36) now we can proceed with the calculations: we use 1, 5, 10 and 100 as the value for ω and compare the numerically calculated exact fano factor with the lna expression for fano factor (33) for values of y between 0 and 10. in the case where ω = 1 we can only use integer values of y, for ω = 5 we can use multiples of 0.2 for y, in other cases we use multiples of 0.1 for y in order to plot the graphs. we use the same setup for the different values of 〈x〉, in particular 0.1, 0.5, 1 and 2. results for scenario with 〈x〉 = 2 is shown in the figure 8; results corresponding to other values of 〈x〉 are analogous. furthermore, in table iv we present the sum of squared residuals in the points y = 1, . . . , 10. (if y = 0, then f = 1 always, so we can omit this point.) we see that the differences between approximations increase with 〈x〉, which is no surprise as fano factor achieves higher values in such cases. iv. conclusion we introduced three methods to obtain free protein distribution of our gene expression model. in addition to stochastic simulation and numerical simulation of the master equation, we also employed singular-perturbation reduction techbiomath 6 (2017), 1710217, http://dx.doi.org/10.11145/j.biomath.2017.10.217 page 15 of 17 http://dx.doi.org/10.11145/j.biomath.2017.10.217 michal hojcka, pavol bokes, non-monotonicity of fano factor in a stochastic model for protein ... ω 〈x〉 = 0.1 〈x〉 = 0.5 〈x〉 = 1 〈x〉 = 2 1 1.8 · 10−5 2.2 · 10−4 8.0 · 10−4 0.0032 5 5.6 · 10−7 8.9 · 10−6 3.3 · 10−5 1.4 · 10−4 10 1.4 · 10−7 2.2 · 10−6 8.4 · 10−6 3.6 · 10−5 100 1.3 · 10−9 2.2 · 10−8 8.5 · 10−8 3.6 · 10−7 table iv: sum of squares from lna. niques to obtain a quasi-steady-state approximation, which was helpful in finding a relatively simple explicit formula for the free protein distribution. using this formula we were able to observe its statistical moments for many different input parameters. very interesting results were obtained for the fano factor, which substantially differed from poissonian case. in the second part of the paper we considered the case of large system size, using the dissociation constant as the measure of size. this approach yielded a tractable expression for the fano factor of free protein distribution. through numerical simulations we showed that this solution is consistent with results from the first part of the paper. while we have employed our methodologies to explore the properties of a relatively simple model, we expect that the same approaches can yield valuable insights in stochastic gene-expression models in particular as well as problems of mathematical biology more widely. acknowledgment pb and mh are supported by the slovak research and development agency grant apvv14-0378. pb acknowledges the support from the vega grant agency (grant no. 1/0319/15). pb would like to thank a. singh for helpful conversation about questions related to the research presented herein. references [1] alon, u.: an introduction to systems biology: design principles of biological circuits, chapman & hall/crc, 2007. dx.doi.org/10.1063/1.2754606 [2] bokes, p., singh, a.: protein copy number distributions for a self-regulating gene in the presence of decoy binding sites, plos one vol. 10, no. 3: 1-19, 2015. dx.doi.org/10.1371/journal.pone.0120555 [3] burger, a., walczak, a.m., wolynes, p.g.: abduction and asylum in the lives of transcription factors, p. natl. acad. sci. usa vol. 107: 4016-4021, 2010. dx.doi.org/10.1073/pnas.0915138107 [4] burger, a., walczak, a.m., wolynes, p.g.: influence of decoys on the noise and dynamics of gene expression, physical review e vol. 86(4): 041920, 2012. dx.doi.org/10.1103/physreve.86.041920 [5] cao, y., gillespie, d.t., petzold, l.r.: the slow-scale stochastic simulation algorithm, journal of chemical physics, 122(1): 014116, 2005. dx.doi.org/10.1063/1.1824902 [6] elowitz, m.b., levine, a.j., siggia, e.d., swain, p.s.: stochastic gene expression in a single cell, science vol. 297: 1183-1186, 2002. dx.doi.org/10.1126/science.1070919 [7] erban, r., chapman, s.j., maini, p.k.: a practical guide to stochastic simulations of reaction-diffusion processes, arxiv:0704.1908, 2007. 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system of odes quasi-steady-state solution moments of free protein distribution comparison of the methods small noise approximation deterministic case stochastic component constant x (total protein count) fluctuating x (total protein count) numerical simulations fano factor based on system-size approach quality of system size approach conclusion references original article biomath 2 (2013), 1309089, 1–11 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum quantitative structure-activity relationships: linear regression modelling and validation strategies by example sorana d. bolboacă department of medical informatics and biostatistics ”iuliu haţieganu” umf cluj-napoca cluj-napoca, romania email: sbolboaca@gmail.com lorentz jäntschi department of physics and chemistry technical university of cluj-napoca cluj-napoca, romania email: lorentz.jantschi@gmail.com received: 4 may 2013, accepted: 8 september 2013, published: 18 september 2013 abstract—quantitative structure-activity relationships are mathematical models constructed based on the hypothesis that structure of chemical compounds is related to their biological activity. a linear regression model is often used to estimate and/or to predict the nature of the relationship between a measured activity and some measure or calculated descriptors. linear regression helps to answer main three questions: does the biological activity depend on structure information; if so, the nature of the relationship is linear; and if yes, how good is the model in prediction of the biological activity of new compound(s). this manuscript presents the steps on linear regression analysis moving from theoretical knowledge to an example conducted on sets of endocrine disrupting chemicals. keywords-robust regression; validation; diagnostic; predictive power; quantitative structure-activity relationships (qsars); i. linear regression on qsar analysis quantitative structure-activity relationships (qsars) are mathematical models linking chemical structure and pharmacological activity/property in a quantitative manner for a series of compounds [1]. the approaches are based on the assumption that the structure of chemical compounds (such as geometric, topologic, steric, electronic properties, etc.) contains features responsible for its physical, chemical and/or biological properties [2]. this assumption could be summarized as ”similar compounds have similar properties” [3]. the two main fields where linear regression analysis found its applicability are drug discovery [4], [5] and toxicology prediction [6], [7]. in both of these fields, the linear regression is used mainly to predict not to estimate (the model is used to quickly determine the activity/property of new/un-investigated compounds) [8]. the linear regression is used in qsar analysis to linearly link the activity/property of chemical compounds (measured or observed value outcome variable abbreviated as y) and some values translated from the structure of the compounds and generally called descriptors (assumed error non-affected independent variables abbreviated as x(s)). the multiple linear regression (mlr) expression is presented in eq(1): ŷ = b0 + k∑ i=1 bixi (1) where ŷ = estimated activity/property; b0 = intercept; bi = coefficient of the ith independent variable / descriptor variable (1 ≤ i ≤ k, 5 × k ≤ n [9]), k = number of descriptors (independent/descriptor variables) in the model, n = number of observations in the sample. the regression coefficients bi could be interpreted as the change in y when xi increased or decreased by 1 unit citation: sorana d. bolboacă, lorentz jäntschi, quantitative structure-activity relationships: linear regression modelling and validation strategies by example, biomath 2 (2013), 1309089, http://dx.doi.org/10.11145/j.biomath.2013.09.089 page 1 of 11 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2013.09.089 s d bolboacă, l jäntschi, quantitative structure-activity relationships: linear regression modelling and ... when all other independent variables are held constant (b0 and b1 estimate the population parameters β0 and βi, [10]). the identified values of b0 and bi are calculated to minimize the squared error for all n observations. a. linear regression assumptions the main assumptions of linear regression (table i) could be summarized as: 1) linearity. the relation between y and each of descriptors xi are linear. 2) independence of the errors. both the experimental values (y ) and experimental/calculated descriptors (xi) are measured without errors. 3) homoscedasticity. the variance of the errors is constant. 4) normality. the dependent variable (y ) is normal distributed. 5) absence of multicolinearity. the independent variables (xi) are linearly independent of each other. please note that this constrain did not exclude a certain degree of collinearity. since it has been recognized that ”normal law ... is not valid in a great many cases which are both common and important” [11] a series of transformation could be used to reach normal distribution [29] (see table ii). 1) model selection and diagnostic: selection of the regression model is an important task that researchers must to accomplish. the main criteria useful in this step are: • determination coefficient (r2) and its adjustment form (r2adj r 2 adjusted with the number of coefficients in the model → the value will not necessary increase with the addition of x’s). generally, the r2 increase with the number of parameters in the model but r2adj penalizes according to the number of parameters (the model with higher number of descriptors does not necessary has the higher value of r2adj). • standard error of the estimate: the average error predicting the activity/property of interest by the identified model. • statistics of overall model performances (f -value and associated p-value): assess the overall ability of a model to explain as much as possible from the observed variability in y . • models performances in cross-validation by the leave-one-out analysis. it is say that a model with q2 (determination coefficient in cross-validation by the leave-one-out analysis) >0.6 and |r2 −q2| < 0.1 is a desired model in qsar analysis [30]. however, the value of f -statistics and its associated probability are as important as q2 in assessment of internal validation of a qsar model. • mallows cp-statistic (cp = ssres/msres −n + 2 · (k + 1), k = number of descriptor variables in the model) [31], [32], [33]: measures the overall bias or mean square error in the estimated model parameters. this is a useful parameter when models with different x(s) are compared on the same sample of compounds. a low cp value indicates good model prediction or a model with a small positive/negative discrepancy between cp and (k+1) could be used in evaluating candidate regression models. • akaikes information criterion and derivative formulas: assess the degree of fit by involving the goodness-of-fit of the model (r2): akaike information criterion (aic = n · ln(rss/n) + 2 · (k + 1) for the model with intercept and aic = n · ln(rss/n) + 2 ·k for the model without intercept, where n = sample size, rss = residual sum of squares; k = number of xi) [34]; aic based on the determination coefficient (aicr2 = ln[(1 − r2)/n] + 2·(k + 1)); mcquarrie and tsai corrected aic (aicu = ln[rss/(n − k + 1)] + (n + k + 1)/(n − k − 1)) [35]; bayesian information criterion (bic = n · ln[rss/(n − k + 1)] + (k + 1)·ln(n)) [36]; amemiya prediction criterion (apc = rss/n · (n−k + 1)/(n + k + 1)) [37]; hannan-quinn criterion (hqc = n·ln(rss/n)+ 2 · (k + 1) · ln[ln(n)] [38]. the smallest the aic, bic, apc and hqc values are the better the model is considered. in addition to aic values, the akaike weights are also used in models assessment: wi = [exp(−0.5 · ∆i)/[σjj=1exp(−0.5 · ∆j)]] [39] where ∆i = aicimin(aic), ∆i = difference between the aic of the best fitting model and that of the model ith, min(aic) = minimum aic value out of all models, j = the number of the models. • kubinyi function (fit ) [40], [41]: fit = [r2 · (n−k)]/[(n+ (k + 1)2)·(1−r2)]. the highest the fit value the better the model is considered. the diagnosis of a regression model when the dependent variable is continuous could be conducted by analyzing of residuals or rescaled residuals: • look to the largest and/or smallest experimental values ← detect if the values are in the plausible range. also look to descriptive statistics value: mean, standard deviation, histogram. biomath 2 (2013), 1309089, http://dx.doi.org/10.11145/j.biomath.2013.09.089 page 2 of 11 http://dx.doi.org/10.11145/j.biomath.2013.09.089 s d bolboacă, l jäntschi, quantitative structure-activity relationships: linear regression modelling and ... table i assumptions of linear regression: effect identification methods • plot the independent variable(s) vs dependent variable. • plot the values associated to studentized residuals (si), leverage (hi), cook’s (di) vs individual xi values. the hat values (0 ≤ hi ≤ 1) are used to evaluate the leverage of observations in the dimensional space of independent variables (covariates). if the hi value of a compound exceeds the threshold value (2·(k+1)/n for a regression model with intercept and 2·k/n for a model without intercept, where k = number of xi [42]) it is considered influential whenever if by its removal determine a significant improvement of the model. cook’s distance consider in its formula both residuals and hat matrix to identify influential compound(s) (threshold di > 4/n, where di = 1/(k+ 1)·s2i ·[hi/(1−hi)] for the model with intercept and di = 1/k·s2i ·[hi/(1−hi)] for the model without intercept, si = studentized residuals [43]). several parameters that can found their usefulness in diagnosis of a mlr are presented in table iii. several parameters presented in table iii are also used by some authors as measures of model predictivity power (see for example mae [44]). b. model predictive power the ability to predict the activity/property of new compounds is of major importance in qsar/qspr analysis. several parameters were proposed and are used to assess model predictivity power and are presented in table iv. the diagnosis of a linear regression model could be conducted using a series of statistical parameters calculated on contingency table [58] after transformabiomath 2 (2013), 1309089, http://dx.doi.org/10.11145/j.biomath.2013.09.089 page 3 of 11 http://dx.doi.org/10.11145/j.biomath.2013.09.089 s d bolboacă, l jäntschi, quantitative structure-activity relationships: linear regression modelling and ... table ii methods for data transformation transformation applied to: appropriate when: 'log' y' = logy stabilize the variance of y normalized the dependent variable ← positive skewed distribution of the residuals for y linearize the regression model y have positive values 'square root' y' = √y stabilize the variance (the variance is proportional with the mean of y) y has the poisson distribution 'reciprocal' y' = 1/y stabilize the variance the variance is proportional to the fourth power of the mean of y 'square' y' = y2 stabilize the variance (the variance decrease with the mean of y) normalized the dependent variable ← negative skewed distribution of the residuals for y linearize the regression model ← the original relation with some independent variable is curvilinear downward (such as decrease of slope with the increase of independent variable) 'arcsine' y' = asin√y stabilize the variance y is a proportion or a percentage tion of the observed and estimated/predicted logrba as dichotomial variables using criteria for classification of compounds as active or inactive. the total fraction of compounds correctly classified (parameter called concordance / accuracy / non-error rate) is one parameter that could bring useful information in choosing which model to be applied. ii. practical considerations three data sets of endocrine disrupting chemicals with experimental values of relative binding affinity expressed in logarithmic scale (logrba) [59] were used for exemplification. the investigated compounds could be classified according to their logrba values as weak binders (logrba < −2.0), moderate binders (−2.0 = logrba = 0) and strong binders (logrba > 0) [60]. the following descriptors were previously calculated on the investigated structures [59] and were used here to illustrate how linear regression analysis works: tie = estate topological parameter; tic1 = total information content index (neighbourhood symmetry of 1-order); ats4m = broto-moreau autocorrelation of a topological structure lag 4 / weighted by atomic masses; eeig02d = eigenvalue 02 from edge adj. matrix weighted by dipole moments; e1s = 1st component accessibility directional whim index / weighted by atomic electrotopological states; and dv = total accessibility index / weighted by atomic van der waals volumes. the first set was used to identify the model and comprised 132 compounds (training set; 1 withdrawn, 60 weak binders, 41 moderate binders and 30 strong binders). the second dataset was used to test the performances of the model (test set) and comprised 23 compounds (3 weak binders, 16 moderate binders and 4 strong binders). the third dataset was used as external validation set and consists of 9 compounds (4 weak binders and 5 moderate binders). a. mlr in training sets the first step in the linear regression analysis was to investigate the distribution of logrba in training set. one out of three tests rejected the null hypothesis of normality (chi-square statistics = 14.862, p-value = 0.03781). no outlier had been identified when the grubbs test was applied but there was one compound with studentized residuals higher than 3 standard deviations. the experimental data in training test proved not normal distributed according just with the chi-square test (see table v), the normality test that is known to be affected by the presence of outlier(s) [12], even if in this example no outlier has been identified. the normality was not achieved even by withdrawing that compounds but the correlation coefficient increased from 0.810 to 0.837. the studentized residuals, hat matrix and cook’s distance values were plotted against logrba to identify how data were distributed (figure 1). three models obtained on the same datasets were investigated: biomath 2 (2013), 1309089, http://dx.doi.org/10.11145/j.biomath.2013.09.089 page 4 of 11 http://dx.doi.org/10.11145/j.biomath.2013.09.089 s d bolboacă, l jäntschi, quantitative structure-activity relationships: linear regression modelling and ... table iii statistical parameters for diagnosis of mlr parameter (abbreviation) formula [ref] remarks residual mean square (rms) error variance kn yy rms n i ii − − = ∑=1 2)ˆ( rms: the smaller the better 0 < rms < ∞ average prediction variance (apv) )( kn n rms apv +⋅= [45] the smaller the better total squared error (tse) nk yy tse n i ii −⋅+ − = ∑= 2 ˆ )ˆ( 2 1 2 σ [46] 2)2( +⋅−−= kn mse sse tse [33] the smaller the better tse > (k+1) → bias due to incompletely specified model tse< (k+1) → the model is over specified (contains too many variables) average prediction mean squared error (apmse) 1−− = kn rms apmse [47] the smaller the better mean absolute error (mae) measures the average magnitude of the errors; could be also used to compare two models n yy mae n i ii∑= −= 1 | ˆ| mae = 0 → perfect accuracy 0 < mae < ∞ root mean square error (rmse): measures the average magnitude of the error ( ) n yy rmse n i ii∑= −= 1 2ˆ rmse > mae → variation in the errors exists 0 < rmse < ∞ mean absolute percentage error (mape) measure of accuracy expressed as percentage n yyy mape n i iii∑= −= 1 |/) ˆ(| [48], [49] mape ~ 0 → perfect fit standard error of prediction (sep) ( ) 1 ˆ 1 2 − − = ∑= n yy sep n i ii the smaller the better relative error of prediction (rep%) ( ) n yy y rep n i ii∑ = −= 1 2ˆ100 (%) the smaller the better n = sample size; k = number of independent variables in the model; y = the mean of estimated/predicted activity/property; iŷ = predicted value of the ith compound in the sample; yi = observed/measured activity/property of i th compound; sse = sum of squared errors; mse = mean of squared errors full-model (the model comprised all compounds assigned to training test), di-model (the model comprised just the compounds that did not exceeded the imposed cooks distance threshold), and hi-model (the model comprised just the compounds that did not exceeded the imposed hat matrix threshold). the cook’s distance and hat matrix approaches were applied to withdrawn compounds of the training sample until two criteria were accomplished: logrba proved normal distributed and withdrawing the compound(s) did not led to an improvement in determination coefficient. both models proved smaller rmse and rmsep values. the characteristics of all investigated models are presented in table v. the analysis of the models (table v) revealed that none model proved collinearity (the highest correlation coefficient did not exceeded 0.8 and vif values are less than 10). the di-model is twice better in terms of internal validity when the |r2 − q2| difference is evaluated compared to hi-model and three times better compared to the full-model. the mallows cp-statistic did not found its applicability in our example because the same descriptors are used in all models. the smallest values of information criteria parameter were systematbiomath 2 (2013), 1309089, http://dx.doi.org/10.11145/j.biomath.2013.09.089 page 5 of 11 http://dx.doi.org/10.11145/j.biomath.2013.09.089 s d bolboacă, l jäntschi, quantitative structure-activity relationships: linear regression modelling and ... table iv statistics for assessment the predictive power of mlr parameter (abbr.) formula [ref] remarks predictive squared correlation coefficient in training set (qf1 2) ∑ ∑ = = − − −= ts ts n i tri n i ii f yy yy q 1 2 1 2 2 )( )ˆ( 1 1 [50] predictive squared correlation coefficient in test set (qf2 2) ∑ ∑ = = − − −= ts ts n i tsi n i ii f yy yy q 1 2 1 2 2 )( )ˆ( 1 2 [52] external predictive ability (qf3 2) tr n i tri ts n i ii f nyy nyy q ts ts /)( /)ˆ( 1 1 2 1 2 2 3 ∑ ∑ = = − − −= [53] prediction is considered accurate if the predictive power of the model is > 0.6 [51] rm 2 metrics [ ]20222 1 rrrrm −−⋅= [44], [54] || 2'22 mmm rrr −=δ values higher than 0.5 indicate an acceptable model [44] , [54] 2 mrδ indicate an acceptable model concordance correlation coefficient (ccc) 2 1 2 2 1 1 )ˆ()ˆˆ()( )ˆˆ()(2 yynyyyy yyyy ccc n i i n i i i n i i −⋅+−+− −⋅−⋅ = ∑∑ ∑ == = [55] strength of agreement between observed and predicted values [56]: > 0.99 almost perfect; [0.95; 0.99) substantial; [0.90; 0.95) moderate; < 0.90 poor predictive power (pp): fisher's approach tsts ts nresstdev res t /)( 0− = [57] p = tdist(abs(t),nts-1,1) evaluate if the mean of residual is statistically different by the expected value (0) n = sample size; v = number of independent variables in the model; y = the mean of observed/measured activity/property; ŷ = the mean of estimated/predicted activity/property; iŷ = predicted value of the i th compound in the sample; yi = observed/measured activity/property of ith compound; res = mean of residuals; stdev = standard deviation; tr = training set; ts = test set; r 2 m = a metric calculated using observed (y-axis) and estimated/predicted (x-axis)values; r′2m = a metric calculated using observed (x-axis) and estimated/predicted (y-axis)values; r20 = determination coefficient calculating by forcing the origin of axis; δr 2 m = absolute difference between r2m and r′ 2 m; ext = external set; abs = absolute value ically obtained by di-model which was follow by himodel while the full-model systematically obtained the highest values (see table v). the concordance correlation coefficient for training sets had values closed to the correlation coefficients and for all models were higher than 0.80 (see table 5). looking to the weights of akaike’s information criteria, which can be interpreted as probability that a certain model is the best model, it could not be identify any model with robust inference (none of the model had the values of weights higher than 0.9 [61]). the dimodel had the weights around 0.37 that is far away from 0.90 but are a little higher than those obtained by the full model where the weights are around 0.30 or by those obtained by the hi-model which are around 0.32. recall that the di-model could be considered the preferred model and from the inspection of the akaike weights in table v, this model is 1.2 (wi −aicr2) to 1.4 (wi−aicc) times more likely in terms of kullbackleible discrepancy, a measure of distance between the probability generated by the model and reality [62], compared with hi-model. significant differences between models could also been observed if the bic and hqc parameters are analyzed; the smallest value of bic was obtained by dimodel while the smallest value of hqc was obtained by hi-model. the plots of residuals versus predicted values for the investigated models are presented in figure 2. the analyses of residuals allow to identify if the assumptions of the regression appear to have been met or not (specifically linearity and homoscedascity) the residual plot look like a horizontal band. thus, according biomath 2 (2013), 1309089, http://dx.doi.org/10.11145/j.biomath.2013.09.089 page 6 of 11 http://dx.doi.org/10.11145/j.biomath.2013.09.089 s d bolboacă, l jäntschi, quantitative structure-activity relationships: linear regression modelling and ... -5 -4 -3 -2 -1 0 1 2 3 4 -5 -4 -3 -2 -1 0 1 2 3 logrba st ud en ti ze d re si du al s s i>3 → 1compound 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 -5 -4 -3 -2 -1 0 1 2 3 logrba c oo k' s di st an ce di>4/n→ 9 compounds 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 -5 -4 -3 -2 -1 0 1 2 3 logrba h at m at ri x va lu e hi>2(k+1)/n → 6 compounds a) b) c) fig. 1. studentized residuals (a), cook’s distance (b) and hat matrix values (c) versus logrba in model with all compounds in training set (n=132) table v mlr in training sets: models characteristics statistical parameter full-model (n=132) di-model (n=115) a hi-model (n=123) b normality tests: ks-ad-cs 0.116* 2.409* 14.862** 0.124* 2.432* 12.613* 0.120* 2.428* 12.083* durbin-watson 1.275 1.292 1.263 collinearity: highest r higher vif & lower t 0.7700 tie: 3.367& 0.297 0.7889 ats4m: 4.082&0.245 0.7752 ats4m: 4.516&0.221 r2 0.6559 0.7797 0.6928 r2adj 0.6394 0.7675 0.6769 rmse 1.0701 0.8293 0.9977 f-value (p-value) 39.711 (9.89·10-27) 63.721 (3.12·10-33) 43.59 (1.62·10-27) q2 0.5832 0.7543 0.6497 rmsep 1.1827 0.8764 1.0668 floo-value (p-value) 28.74 (9.49·10 -22) 55.17 (1.85·10-31) (1.62·10-27) |r2-q2| 0.0727 0.0254 0.0431 concordance correlation coefficient (ccc) 0.8108 [0.7476 to 0.8595] 0.8762 [0.8278 to 0.9117] 0.8185 [0.7545 to 0.8671] r2m (δr 2 m) 0.6071 (0.1324) 0.7797 (0.1278) 0.6921 (0.1586) cp-statistic 7.00 7.00 7.00 aic (wi-aic) 18.9639 (0.2856) 18.3078 (0.3965) 18.7490 (0.3180) aicr2 (wiaicr2) 8.0504 (0.3137) 7.7421 (0.3659) 8.0077 (0.3204) aicc (wiaicc) 1.2657 (0.2990) 0.7766 (0.3819) 1.1358 (0.3191) bic 52.0750 9.8317† 33.1255 hqc 26.2887 34.7113† 7.8043 fit 1.3058 2.3097 1.5076 * p ≥0.05; ** p = 0.0378; † = absolute values; ks = kolmogorow-smirnov; ad = anderson darling; cs = chi-square; r = correlation coefficient; vif = variance inflation factor; t = tolerance; r2 = determination coefficient; r2adj = adjusted determination coefficient; rmse = root mean square error; f-value = fisher's statistics; q2 = determination coefficient in cross-validation by the leave-one-out analysis; rmsep = root mean square error in prediction; ccc = concordance correlation coefficient [95% confidence interval]; cp-statistic = mallows’ statistic; aic = akaike’s information criterion; aicr2 = aic based on the determination coefficient; aicc = aic corrected by mcquarrie and tsai; bic = bayesian information criterion; hqc = hannan-quinn criterion; fit = kubinyi's function; a 56 weak binders, 35 moderate binders, and 24 strong binders; withdrawn (16 compounds): 4 weak binders, 6 moderate binders and 6 strong binders; b 57 weak binders, 38 moderate binders, and 28 strong binders; withdrawn (8 compounds): 3 weak binders, 3 moderate binders and 2 strong binders; to the pattern of the residuals [63], the most appropriate model is the di-model since the distribution indicates a homoscedastic model. furthermore, both full-model and hi-model showed evidence of heteroscedascity, the error in estimating logrba increasing as the value of logrba increase. however, both these models could be accepted because none of them showed the presence of systematic errors or inadequacy [63]. if assumption of linearity biomath 2 (2013), 1309089, http://dx.doi.org/10.11145/j.biomath.2013.09.089 page 7 of 11 http://dx.doi.org/10.11145/j.biomath.2013.09.089 s d bolboacă, l jäntschi, quantitative structure-activity relationships: linear regression modelling and ... and/or of homoscedascity is violated, the residual plots show an increasing and narrow pattern if systematic error exists or depict a gaussian trend when the model is inadequate [64]. other proposed plot methods, such as linear residual plots, show to be useful in identification of non-linearity while squared residual plots proved utility in detection of non-constant variances [65]. the normal probability plots (right graphical representations in figure 2) can be used to verify normality assumption of the residuals. figure 2 showed that the hi-model fit better a straight line compared to both fullmodel and di-model. the results obtained on our data associated to the statistical parameters useful in model diagnosis introduced in table iii are presented in table vi. the total square error is the single parameter that has the same value for all models and in all cases is equal to 7 (obtained by adding 1 to the number of descriptors in the model 6 in our example), indicating that none of the models were not over-specified or did not contain bias due to incompletely specified model. the classification of our models based on parameters presented in table vi led to the classification obtained according to the parameters presented in table v: di-model, hi-model, and full model. several parameters were used to assess the predictive power of the models and their results are presented in table vii. the analysis of results presented in table vii revealed the followings: • external predictive ability parameter (q2f3) [53] systematically took negative values for both external and withdrawn sets. at least for the external set, this result could be explained by the distribution of logrba values (min=-3.3, max=-0.6) compared to training (min=-4.5, max=2.6) and test (min=2.51, max=1.41) sets. it could be also of interest to analyze how different are the compounds containing in external and withdrawn data sets compared to the compounds from training set (in terms of similarity of their structure for example). • di-model achieve the criterion of exceeding 0.6 [52] in just one of 6 possible case while the himodel reach this criterion in four out of 6 cases. the hi-model accomplished more frequently the criteria of having values higher than 0.6 while the full-model did not accomplished at all this criterion. thus, it seems that the compounds in test and external sets are uniformly distributed over the range of training set at least in hi-model, in view of the fact that otherwise the q2f1 and the q 2 f2 suffer from drawbacks [66]. • the concordance correlation coefficients obtained values higher than 0.70 in test sets. the abilities of prediction the external sets proved smaller than 0.5 for all investigated models but had values higher than 0.50 (di-model and hi-model) when the withdrawn set is investigated. • the residual of the models proved significantly different by zero in test set for full-model and dimodel and in external set for all models. both diand hi-models proved to have residual not significantly different by zero in samples that contain the withdrawn compounds. according to this criterion, just hi-model proved prediction power. the r2m metric and associated ∆r 2 m obtained in test sets were as follows: 0.3726 (0.1743) for full model, 0.3134 (0.1796) for di-model, and 0.5248 (0.1494) for hi-model. these metrics showed that the hi-model is acceptable model. the r2m is a parameter computed by forcing the regression through origin [54] with certain applicability and limitations (fails to detect the differences between experimental and predicted values when the slopes of the regression line are not near to 1) [67]. the values of these metrics were smaller than the determination coefficient in all investigated models and the highest value was observed in di-model when training (see table v) set was investigated but acceptable values were obtained just by the hi-model when the test set was investigated (r2m > 0.5 and ∆r 2 m < 0.2). the classification of the models according to results presented table vii is as follows: hi-model, di-model, and full-model. one remark about the parameters used to assess the predictive power, namely q2f1, q 2 f2 and q 2 f3, can be made. even the symbols contain ”square”, these parameters could take both positive and negative values according to their formula (see table iv). a simulation study of these parameters needs to be done to identify their possible values as well as their proper interpretation. the best way to see the abilities of a mlr model is to plot the measured values against the estimated / predicted values to visualize how well each model works (see figure 3). with one exception, represented by hi-model in external set (p-value = 0.0632), all other correlation coefficients proved statistically significant (p < 0.04). the analysis of models presented in figure 3 revealed the followings: • the distribution of compounds in training set is narrower in di-model compared to both full-model and hi-model. biomath 2 (2013), 1309089, http://dx.doi.org/10.11145/j.biomath.2013.09.089 page 8 of 11 http://dx.doi.org/10.11145/j.biomath.2013.09.089 s d bolboacă, l jäntschi, quantitative structure-activity relationships: linear regression modelling and ... • di-model obtained higher determination coefficients in training and external sets while the himodel obtained the higher determination coefficients in training and withdrawn sets. • the hi-model in more stable compared to di-model if the difference in determination between training and test set is concerned. • both di-model and hi-model performed better in training and test sets compared to full-model. whenever applicable, the accuracy of a model will show its ability in correct classification of compounds. the overall accuracy as well as the accuracy on each class (weak binder, moderate binder and strong binder) were computed and the obtained results are presented in figure 4. the analysis of figure 4 revealed the followings: • the accuracy of all three models was identical for strong binders in test set (75%) and weak binders in external set (25%). overall, out of 16 possibilities, all models (full-model, di-model, and hi-model) proved highest accuracy in almost 38% of cases. • full-model proved highest overall accuracy in both test and external sets, and highest accuracy for moderate binders in test and external sets. • di-model proved highest overall accuracy in training set, highest accuracy for strong binders in training set, highest accuracy for weak binders in training set, and highest accuracy of moderate binders in training set. • hi-model proved highest overall accuracy, as well as higher accuracy for weak binders, moderate binders and strong binders for withdrawn compounds. • no model proved abilities in correct classification of weak binders in test set or of strong binders in external set. regarding the accuracy of investigated models it is impossible to classify them since their performances are generally the same (38%). it could be observed that models had abilities to accurately identify the compounds on average of two sets out of three or four. the absence of accurate classification of weak binders in test set and strong binders in externals set could be explained by differences in the chemical structure or measured logrba of compounds included in these sets. iii. summary and furher work choosing a proper linear model is crucial in qsar analysis because a model able to predict accurately the activity of interest of new chemical compounds is desired under the hypothesis that changes in molecular structure directly reflect in the compound activity/property. input data and data preparation for regression analysis are of great importance but these subjects were beyond the aim of the present manuscript. linear regression analyses identify in qsar analysis the linearity between compound’s activity and calculated descriptors based on chemical structure. regression analysis answer to the following questions: does the biological activity depend on structural information? if so, the nature of the relationship is linear? if yes, how good is the model in prediction of the biological activity of new compounds? in this manuscript, some rules had been presented: 1© test the assumption of linear regression (normality, linearity, independence, homoscedascity, and/or collinearity); 2© construct the model(s) if assumptions are accomplished analyze the data (choose the best performing model); 3© assess and diagnose the alternative models analyze the mlr; 4© decide which model fit best to your objectives. following these steps in linear regression analysis certainly led to a performing estimation model but the prediction power of the model will always depend on the structure of compounds and their biological activity on which the model is used to predict; in other words, will be dependent by similarity in terms of structure and activity. researches on linear regression analysis are of general interest since mlr found its applicability in many research fields. the classical approach implemented in available dedicated software deal with maximization of correlation coefficient. maximization of the observed probability under assumption of random error affecting all variables in the model is an ongoing research and will be reported somewhere else. it is known that the classical method is exposed to type i errors (to accept a regression model obtained by maximization of determination correlation even if it does not exist) while this new approach does not because it maximize just the observation chance having as hypothesis that the errors between observed value and value obtained by the model is random and depend just by the observed/measured value (therefore being symmetric relative to its arithmetic mean). acknowledgment the authors are grateful to the organizers of the biomath 2013 for the opportunity to present our results. biomath 2 (2013), 1309089, http://dx.doi.org/10.11145/j.biomath.2013.09.089 page 9 of 11 http://dx.doi.org/10.11145/j.biomath.2013.09.089 s d bolboacă, l jäntschi, quantitative structure-activity relationships: linear regression modelling and ... references [1] l. p. hammett, ”the effect of structure upon the reactions of organic compounds. benzene derivatives,” j. am. chem. soc., vol. 59, no. 1, 1937, pp. 96-103. http://dx.doi.org/10.1063/1.1749914 [2] p. gramatica, ”a short history of qsar evolution,” [online] [accessed january 26, 2012]. available from: http://qsarworld. com/temp fileupload/shorthistoryofqsar.pdf. 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n’gaoundéré, p.o.box 455 n’gaoundéré, cameroon 2 department of basic and technical engineering sciences egcim – university of n’gaoundéré, p.o.box 454 n’gaoundéré, cameroon kouakep@aims-senegal.org 3 department of mathematics and computer sciences, faculty of sciences university of n’gaoundéré, p.o.box 454 n’gaoundéré, cameroon received: 3 december 2020, accepted: 22 july 2021, published: 12 september 2021 abstract— we develop a new model of integro-differential equations coupled with a partial differential equation that focuses on the study of the naturally acquiring immunity to malaria induced by exposure to infection. we analyze a continuous acquisition of immunity after infected individuals are treated. it exhibits complex and realistic mechanisms precised mathematically in both disease free or endemic context and in several numerical simulations showing the interplay between infection through the bite of mosquitoes. the model confirms the (partial) premunition of the human population in the regions where malaria is endemic. as common in literature, we indicate an equivalence of the basic reproduction rate as the spectral radius of a “next generation” operator. keywords-malaria, premunition, modeling, endemic msc2010: 35k20, 92b05 i. introduction one of the major health challenges in africa is the management of malaria endemicity [15], [9], [13], [21], [11], [3], [15], [18]. as underlined by langhorne [11] “preventative and treatment strategies are continuously hampered by the issues of the ever-emerging parasite resistance to newly introduced drugs, considerable costs and logistical problems”. understanding the mechanisms of naturally acquiring immunity to malaria [11] even with modeling, is important to analyze its hidden mechacopyright: © 2021 tchoumi et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: s y tchoumi, y t kouakep, d j m fotsa, f g t kamba, j c kamgang, d d e houpa, mathematical model for acquiring immunity to malaria: a pde approach, biomath 10 (2021), 2107227, http://dx.doi.org/10.11145/j.biomath.2021.07.227 page 1 of 14 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2021.07.227 s y tchoumi et al., mathematical model for acquiring immunity to malaria: a pde ... nisms, and justifies our study [19], [9], [13], [21], [11], [3], [15], [18]. the “premunition” could be defined as the natural acquired immunity capacity to live with a relatively low concentration of malaria parasite, and resist to falling sick. the immunity is not a sterilizing type in that the infection persists longer than the symptoms and individuals can exhibit relapses or recrudescences or become reinfected. moreover, chronic infection persists, although the maximum parasite load reached is low. even if it adds a little in terms of reduction of parasite load as compared to innate resistance, this additional immunity is substantial in terms of morbidity as it keeps the parasite load low, below the threshold of pathogens. super infection can occur, but it remains at a low grade” [19]. obi et al. [19] precised that “premunition is independent of transmission levels provided it occurs at least once a year”. it is rapidly lost: exactly one year without re-challenge is enough to lose this protective state. it is strain independent and clearly immunoglobulin g (igg) dependent. the delay of acquisition is remarkably long, compared to the rate of transmission”. moreover, [19] states that epidemiological studies in africa and papua new guinea have helped to define three clinical periods: a short period of 0 5 years where mortality can occur; a long period of 0 to 15-20 years where morbidity is ”frequent” (though decreaing in frequency with age); thereafter a longer period of premunition where the diseases in any form is absent. this paper addresses the impact of a continuous level of premunition on a scale of 0 (non immune) to 1 (high degree of immune responsiveness to infection in terms of premunition) in recovered individuals. it will be interesting to see which mechanisms support the evolution of the number of (partially) immune humans to the natural acquisition of premunition. it is common to see that an individual from an endemic region (central africa, south america..) who goes out to a malaria epidemic region (west europe for e.g), has a great chance to fall sick if he turns back to another endemic (malaria) region [22]. [14] claims that ”after a couple of more infections, anti-disease immunity develops and causes suppression of clinical symptoms even in the presence of heavy parasitemia and also reduces the risk of severe disease. frequent and multiple infections slowly lead to the development of anti-parasite immunity that results in very low or undetectable parasitemia. (...). premunition suggests an immunity mediated directly by the presence of the parasites themselves and not as much as the result of the previous infections” [13], [9]. malaria affects more than 40 % of the world population in over 90 countries [14]. we adopt the partial differential approach to model the premunition acquisition. more specifically, we use an integro-differential modeling coupled with a partial differential equation because they track very well the continuity of the temporary immunity level. one could rather consider several discrete states of the immunity level and obtain a huge number of differential equations that are more complex to use. after some mathematical analysis of the model, we made several numerical simulations. our mathematical analysis and simulations support the fact that the premunition is a continuous process because the relative resistance to a low level of parasite seems to increase with time in an endemic situation. but it is clear that the level of premunition is somehow a probability with a possibility to suffer from the severe malaria disease if other negative factors are dramatically considered (physiological lacks, physical or psychological injuries . . . ) even if the level of premunition is high. moreover, the non frequently mosquitoes-human biting contact like in the use (properly or not) of the bed biomath 10 (2021), 2107227, http://dx.doi.org/10.11145/j.biomath.2021.07.227 page 2 of 14 http://dx.doi.org/10.11145/j.biomath.2021.07.227 s y tchoumi et al., mathematical model for acquiring immunity to malaria: a pde ... nets could lead to a partial immunity with a possibility to suffer from the severe malaria disease. as a biological and modeling assumption, we consider that usage of bed net should be rigorous to avoid instability in the premunition acquisition process. following [14], [13], [9], more recovered people in endemic area seems to have a high degree of immune responsiveness after being continuously exposed to mosquitoes biting. then, in endemic areas, the adult population (of more than 5 years) could live with the parasite in an asymptomatic carriers status. it explains why public health strategies in high endemicity region of malaria, concerns children from 0 to 5 years old and pregnant women. our main result supports the fact that more recovered people in endemic area seem to have a high degree of immune responsiveness after being continuously exposed to mosquitoes biting. finally, it’s likely to see that in malaria endemic areas, the population (in adulthood likely more that five years old) could live mainly as ”recovered” even within the presence of heavy parasitemia as biologically pointed out by [14], [13], [9]. our work is subdivided as follow: the second section describes our model and some extensions, the third section presents the main results. the fourth section shows the simulations and analyses them, mainly in the cases where the transmission functions a, m, c, c̃(., θ) are periodic (even constant). the fifth section discuss the important results and in the last section, we conclude our work. ii. model description and extension similarly as in [8], [18], the model subdivides the total human population at time t, denoted by nh(t), into the following sub-populations of susceptible sh(t), symptomatic infectious with sickness ih(t) and recovered individuals rh(t, θ) with temporal immunity level θ ∈ [0, 1] at a time t ≥ 0. biologically, it is logical to assume that the level of acquired immunity cannot be 0 in endemic areas like africa. so that nh(t) = sh(t) + ih(t) + ∫ 1 0 rh(t, θ)dθ. the total vector (mosquito) population at time t, denoted by nv(t), is sub-divided into susceptible sv(t) and infectious mosquitoes iv(t). thus, nv(t) = sv(t) + iv(t). susceptible individuals are recruited at a constant rate γh(t). we define the force of infection from mosquitoes to humans by β(t) as the product of the transmission rate per contact with infectious mosquitoes m(t) and the successful biting rate after a contact a(t) (seen in [8] as the product of the mosquito contact rate α with the mosquito biting rates θm h(t)) and the probability that the mosquito is infectious iv /nh [18]. the natural death rate of human is µh. recovered individual loses immunity at a rate γ(t, θ). susceptible mosquitoes are generated at a per capita rate γv(t) at time t and acquire malaria through contacts with infectious humans with the force of infection ϕ(t). mosquitoes are assumed to suffer death due to natural causes at a rate µv(t) at time t. g(t, θ)rh(t, θ) represents the total number of recovered leaving the level θ to the greater level θ′′ ∈ ]θ, 1] and g(t, θ) < γ(t, θ). we emphasize on the term rh(t, θ) which is the number of recovered individuals with an acquired temporal immunity level θ ∈ [0, 1] at a time t ≥ 0. it’s dynamics is described by ∂rh(t, θ) ∂t + ∂j(θ)rh(t, θ) ∂θ = β(t) ∫ θ 0 rh(t, θ ′)dθ′− γ(t, θ)rh(t, θ) that includes through β(t), the impact of the ”new bites of mosquitoes” on all the rh(t, s)’recovered individuals who moves to a next stage of new individuals with a greater level of immunity θ > s. biomath 10 (2021), 2107227, http://dx.doi.org/10.11145/j.biomath.2021.07.227 page 3 of 14 http://dx.doi.org/10.11145/j.biomath.2021.07.227 s y tchoumi et al., mathematical model for acquiring immunity to malaria: a pde ... a. model equation the formulation and construction of the model’s equations (for t, θ > 0) are given by:  dsh(t) dt = γh(t)+ ∫ 1 0(γ(t,θ ′)−g(t,θ))rh(t,θ′)dθ′ −β(t)sh(t)− µh(t)sh(t) dih(t) dt = β(t)sh(t)−δ(t))ih(t)−µ̃h(t)ih(t) ∂rh(t, θ) ∂t + ∂j(θ)rh(t, θ) ∂θ = β(t) ∫ θ 0 rh(t, θ ′)dθ′− γ(t, θ)rh(t, θ) dsv(t) dt = γv(t)−(ϕ(t)+µv(t))sv(t) div(t) dt = ϕ(t)sv(t)−µv(t)iv, (1) where β(t) = a(t)m(t)iv(t) nh(t) , j(θ) = αθ and ϕ(t) = a(t) ( c(t)ih(t) + ∫ 1 0 c̃(t, θ ′)rh(t, θ ′)dθ′ ) nh(t) with µ̃ = µh + d and (1) initial conditions (p1) or (p2) exclusively, that is (p1)   sh(0) = s 0 h ih(0) = i 0 h rh(t, 0) = δ(t)ih(t) rh(0, θ) = r 0 h(θ),∀θ ∈ [0, 1] iv(0) = i0v sv(0) = s0v or (p2)   sh(0) = s 0 h ih(0) = i 0 h ∂rh(t,0) ∂t = δ(t)ih(t) rh(0, θ) = r 0 h(θ),∀θ ∈ [0, 1] iv(0) = i0v sv(0) = s0v. remark 2.1: the function j(θ) is a factor characterizing the rate of entering the rh(t, θ) compartment if other influxes are neglected. a generalization of the function j could include the state of a withinhost model of blood cells with the malaria pathogen [16]. the function g(t, θ) could be β(t)z(θ), but one could also consider that g(t, θ) = (1 − k)γ(t, θ) with k ∈ [0, 1] as in the section iv of simulations. the first and third equation re-stated become:  dsh(t) dt = γh(t) + ∫ 1 0 kγ(t, θ ′)rh(t, θ ′)dθ′ −β(t)sh(t)− µh(t)sh(t) ∂rh(t, θ) ∂t + ∂j(θ)rh(t, θ) ∂θ = β(t) ∫ θ 0 rh(t, θ ′)dθ′− γ(t, θ)rh(t, θ), (2) where k represents the fraction of recovered individuals r(t, θ) who move to other levels θ′ ∈ ]θ, 1]. we will explore some of this particular cases: case 1. values of γ, g and c̃ motivated by the biological references therein: • the level of infection θ satisfies dθdt = αθ and ∫ 1 0 ρ(θ) = 1, with ρ a probability density [26], [24] ρk,l(θ) = l (−l.ln(θ))k−1 θl−1 γ(k) e−lθ . • consider g(t, θ) = (1 − k).γ(t, θ), (3) where γ(t, θ) = γ(t) = he−ht 1 − e−ht , h is the annual rate of infections of individuals [12], 0 < k ≤ 0.4466 and 0.4466 = 76.6×58.3 is the probability to be likely protected from the common perfect possession and use of the bed nets in cameroon [2], [5] over three years of use of the long-lasting insecticide-treated bed nets (llins). we have the average biomath 10 (2021), 2107227, http://dx.doi.org/10.11145/j.biomath.2021.07.227 page 4 of 14 http://dx.doi.org/10.11145/j.biomath.2021.07.227 s y tchoumi et al., mathematical model for acquiring immunity to malaria: a pde ... table i: table of parameters parameters interpretation average value reference e immigration rate of humans 12365.25 [7] f relative birth rate of humans 40365250 [7] j relative rate of acquiring premunition [16] q immigration rate of mosquitoes 1000021 [5] x birth rate of new adult female mosquitoes 1301000 [7], [4] µh humans per capita death rate 1 59×365.25 modified [7], [8] µv mosquitoes per capita death rate 121 [7], [18] γv(t) recruitment of mosquitoes q + x.nv(t) [7] γh(t) recruitment of humans e + f .nh(t) [7] a(t) man biting rate 0.5 × 19 [7], [1] m(t) prob. of dis. transm. from inf. mosq. to human 0.022 [7] c(t) prob. of dis. transm. from inf. humans to mosq. 0.48 [7] c̃(t) prob. of dis. transm. from recovered humans to mosq. 0.048 [7] d(t)) human disease induce mort.rate 9 × 10−5 [1] γ(t, θ) average per cap. rate of lost of immu. [0.027-0.0146] [1] h rate of infectious per year for sh to become ih h = ln ( 1 1−e180.〈γ〉 ) 180 [12] δ recovery rate of humans [0.035-0.03704] [12] value 〈γ〉 = 1180 ∫ 180 0 γ(t)dt (180 days is estimated by raoult [22]) of γ that leads to h = ln ( 1 1−e180.〈γ〉 ) 180 . see values of 〈γ〉 in table i. it is also possible to consider the average value as γ (seen as constant); • c̃(t, θ) could be either a constant or c(t)ρ(θ). case 2. γh(t) = e + f (t)sh(t) and γv(t) = q + x(t)sv(t). here, these forms combines the proportional and constant influxes (birth, migration, ...) in the humans and mosquitoes compartments. case 3. γh(t) = γh > 0 and γv(t) = γv > 0 seen as constants. case 4. in the references below, we collect these values of the parameters: this kind of models which track the global dynamics according to the temporal level of immunity (inducing premunition) has not been developed before. langhorne, pinkevych and mandal’s reviews [11], [12], [21] described the lost of immunity in a local aspect (some discrete values of θ) by focusing only on the dynamics of infected individuals. we go beyond that by studying continuously the impact of the reverse effect of acquiring immunity not discretely (for a fixed θ ∈ [0, 1] ) but globally with all the interactions between the different individuals (susceptible or infected at all the stages of premunition) in order to see the overall dynamics. b. extension of model (1) to different malaria strains with same initial conditions we shall consider a single strain of malaria (such as plasmodium falciparum) in this study since it is the major cause of mortality and morbidity in the tropical and sub-tropical areas of the globe [23]. several strains exists (falciparum, gambiae, coustani, balabacensis, funestus, nili, ...) [7, p.1290, table a.3] but we focus on the afore mentioned. if we name ”i” the strain in a decreasing level of infectiousness depending on the global prevalence (with ”1” for p. falciparum, ”2” for the next..., (i), ... until the strain ”n” with the lowest infectiousness and prevalence. i could also be the patch index where the dominant strain is also called i), biomath 10 (2021), 2107227, http://dx.doi.org/10.11145/j.biomath.2021.07.227 page 5 of 14 http://dx.doi.org/10.11145/j.biomath.2021.07.227 s y tchoumi et al., mathematical model for acquiring immunity to malaria: a pde ... one could get the following model:   dsh(t) dt = γh(t)! +σ n 1 ∫ 1 0 kγi(t,θ ′)r(i)h (t,θ ′)dθ′ −σn1 βi(t)sh(t)− µh(t)sh(t) dih(t) dt = σn1 βi(t)sh(t)−δ(t))ih(t)−µ̃h(t)ih(t) ∂r(i)h (t, θ) ∂t + ∂αθ r(i)h (t, θ) ∂θ = βi(t) ∫ θ 0 r (i) h (t,θ ′)dθ′−[γi(t,θ)+ βi(t)g(θ)]r (i) h (t,θ) dsv(t) dt = γv(t)− (σn1 ϕi(t) + µv(t))sv(t) div(t) dt = σn1 ϕi(t)sv(t)− µv(t)iv, (4) where βi(t) = ai(t)mi(t)iv(t) nh(t) and ϕi(t) = ai(t) ( ci(t)ih(t) + ∫ 1 0 c̃i(t, θ ′)r(i)h (t, θ ′)dθ′ ) nh(t) with initial condition (p1) or (p2). in the sequel, we will study some mathematical and biological properties of our model (1) simplified as (2) and present the analysis derived. iii. main results under initial condition p1 a. well-posedness of the model (1) with initial condition p1 and g(t, θ) = (1 − k)γ(t, θ) assumption 3.1: assume that the functions a, m, δ, c, k, γ(., θ), µh, µ̃h, µv are positive and bounded (as x, f of the case 2), for example in l∞+(0, +∞). moreover, g(t, θ) = (1 − k)γ(t, θ). assumption 3.2: assume that the functions δ, µh, µ̃h, µv are positive, g = 0, e, q = 0 and constant (as x, f of the case 2), and the functions a, m, c̃, γ, c are positive and bounded (as x, f of the case 2), for example in l∞+(0, +∞) for a, m, c and l ∞ +(0, +∞)×(0, 1) for c̃, γ. moreover, min{x, f , µ̃h}− [ kγ + esssu p[0,+∞){β} ] > 0. proposition 3.1: under assumptions 3.1, we have n(t) = sh(t) + ih(t) + ∫ 1 0 rh(t, θ)dθ + sv(t) + iv(t) and a suitable differentiation under the integral: (e+q)− ( max { x, f , µ̃h, sup [0,+∞) {δ} }) n≤ dn dt ≤ (e+q)− ( min{x, f ,µh}− [ kγ+esssup [0,+∞) {β} ]) n. moreover, one could re-write model (1) with initial conditions as the following abstract non autonomous cauchy problem u′(t)= a(t,u(t))+v(t)u(t)+ h(t,u(t)), t > 0, u(0) =   s0h i0h r0h 0 s0v i0v  ∈ x0+, with [v(t).] bounded and [v(t).] + h(t, .) locally lipschitzian in t. proof: the proof is obtained by straightforward computations with this strategy. in order to deal with (1)-(initial condition), let us introduce the banach spaces x = r × r × l1(0, 1)× r × r × r and x0 = r × r × l1(0, 1)×{0}× r × r endowed with the usual product norm, as well as its positive cone x+ defined by x+ = [0, +∞)× [0, +∞)× l1(0, 1) ×[0, +∞)× [0, +∞)× [0, +∞) and x0+ = x+ ∩ x0. consider the linear operator a : d(a) ⊂ x → x defined by d(a) = r2 × l1(0, 1)× r3 biomath 10 (2021), 2107227, http://dx.doi.org/10.11145/j.biomath.2021.07.227 page 6 of 14 http://dx.doi.org/10.11145/j.biomath.2021.07.227 s y tchoumi et al., mathematical model for acquiring immunity to malaria: a pde ... and a  t,   αsh αi z 0 αsv αiv     = la(t) where rh(t, 1) = 0, nh = sh + ih + ∫ 1 0 r(θ)dθ, nv = sv + iv and: case 2: (γh(t) = e + f (t)nh(t) and γv(t) = q + x(t)nv(t)), la(t)=   −µh(t)αsh + ∫ 1 0 kγ(t, y)z(y)dy + f (t)nh −(δ(t) + µ̃h(t)) αih −α. × z′− (γ(t, .) + α) z −z(0) −µv(t)αsv + x(t)nv −µv(t)αiv .   . consider also the non-linear map f  t,   αsh αi z 0 αsv αiv    =   e − β∗(t)αsh β∗(t)αsh β∗(t) [∫ θ 0 z(y)dy−g.z ] δ(t)αih q + ϕ∗(t)αsv , ϕ∗(t)αsv   with β∗(t) = a(t)m(t)αiv (t) αsh (t) + αih (t) + ∫ 1 0 z(y)dy and ϕ∗(t) = a(t) ( c(t)αih (t) + ∫ 1 0 c̃(t, θ ′)z(y)dy ) αsh (t) + αih (t) + ∫ 1 0 z(y)dy . case 3: (γh(t) = γh(:= e) > 0 and γv(t) = γv(:= q) > 0 seen as constants.) la(t) =   −µh(t)αsh + ∫ 1 0 kγ(t, y)z(y)dy −(δ(t) + µ̃h(t)) αih −α. × z′− (γ(t, .) + α) z −z(0) −µv(t)αsv −µv(t)αiv   consider also the non-linear map f  t,   αsh αi z 0 αsv αiv     =   γh − β∗(t)αsh β∗(t)αsh β∗(t) [∫ θ 0 z(y)dy−g.z ] δ(t)αih γv + ϕ∗(t)αsv , ϕ∗(t)αsv   with β∗(t) = a(t)m(t)αiv (t) αsh (t) + αih (t) + ∫ 1 0 z(y)dy and ϕ∗(t) = a(t) ( c(t)αih (t) + ∫ 1 0 c̃(t, θ ′)z(y)dy ) αsh (t) + αih (t) + ∫ 1 0 z(y)dy . in fact, the non linear term can further be broken in two terms: v(t) the linear part describing the initial transmission θ = 0 and h = f − v the very non linear part of f corresponding essentially to the horizontal transmission: v  t,   αsh αi z 0 αsv αiv     =   0 0 0 δ(t)αih 0 0  . now identifying   αsh αi z 0 αsv αiv   with u(t)=   αsh αi z αsv αiv  , re-writes (1)+initial condition as the following abstract non autonomous cauchy problem u′(t)= a(t,u(t))+v(t)u(t)+ h(t,u(t)), t > 0, u(0) =   s0h i0h r0h 0 s0v i0v  ∈ x0+ remarks 3.1: if we consider β(t) with nv(t) at the denominator like didjou [8] biomath 10 (2021), 2107227, http://dx.doi.org/10.11145/j.biomath.2021.07.227 page 7 of 14 http://dx.doi.org/10.11145/j.biomath.2021.07.227 s y tchoumi et al., mathematical model for acquiring immunity to malaria: a pde ... instead of nh, then property 3.1 provides the boundedness due to the gronwall inequality. the choice of the denominator of β(t) is an interplay between modeling considerations and the biological explications of the models in literature. proposition 3.2: under assumptions in case 2, we have the following: i) a is an infinitesimal generator of a strongly continuous semigroup (ta(t))t>0 that is exponentially stable with domain d(a). ii) n is bounded. iii) under more restrictive conditions, one could obtain that ||v(t)|| < ||δ||∞ and (a + v(t))t>0 generates an evolution family. moreover the non linear map f is locally lipschitz-continuous and for all initial condition in x, there exists an interval of time [0, tmax) in which the problem (1)-(initial condition) has a unique mild solution [26], [20] and is wellposed. moreover, the mild solution is a global solution on [0, +∞). proof: 1. the existence and uniqueness of the strongly continuous and infinitesimal semigroup of a in the item i) is proved from theorem a.7 of [17], retrieved by [26] in page 673. 2. item ii) and the exponential stability come from (pa) in proposition 3.1 and gromwall inequalities. 3. item iii) this local solution is bounded on bounded time intervals by the boundedness of the mild solution. then theorem a.8 [20], [26] implies that the maximum time interval on which the solution exists is infinite. in other words the mild solution is a global solution on [0, +∞). remarks 3.2: to study the problem (1)(initial condition p2), one can consider the methods in the work of chekroun [6]. b. steady states of model (1) for g(t, θ) = (1 − k)γ(t, θ). notation 3.1: we set the following: 1y0+={x∈c ([0,1] ,r)∩c ((0,1) ,r) , x≥0}; 2f = 0, x = 0, c̃ ≡ c is constant, γ̃ = γ + α and a, m, µh, µv, γ are positive constants; 3when it exists, the steady states x∗ = (s∗h , i ∗ h , r ∗ h , s ∗ v , i ∗ v ) is either the disease free equilibrium (dfe) xf := (sfh , i f h , r f h , s f v , i f v ) or an endemic equilibrium (ee) xe := (seh , i e h , r e h , s e v , i e v ); 4δ̃ := δ + µ̃h, φ∗ = ac(i∗h + ∫ 1 0 r ∗ h(δ)dθ) s∗h+ ∫ 1 0 r ∗ h(δ)dθ)+ih and β∗ = ami∗v s∗h+ ∫ 1 0 r ∗ h(δ)dθ)+ih ; 5the ”next generation operator” k : y0+ → x0+ such that ∀r∗h ∈ x0+: k(r∗h)[θ] = e − (α+γ)α ln(θ)r∗h(0) +vp ∫ θ 0 β? αθ′ e− (α+γ) α [ln(θ)−ln(θ ′)] ∫ θ′ 0 r∗h(σ)dσdθ ′. (”vp” of ∫ θ 0 ...dθ ′ is the ”pricipal value” seen literally as ”lime−→0 of ∫ θ e ...dθ′”.) we could see k as k(r∗h)[θ] = θ − (α+γ)α [ r∗h(0) + β? α ∫ θ 0 θ′ γ α ∫ θ′ 0 r∗h(σ)dσdθ ′ ] . 6precisely with β̃ = e+kγ ∫ 1 0 r ∗ h(θ ′)dθ′ seh − β∗, k(r∗h)[θ] = θ − (α+γ)α [ r∗h(0) + 1 α ( e+kγ ∫ 1 0 r ∗ h(θ ′)dθ′ seh −β̃ )∫ θ 0 θ′ γ α ∫ θ′ 0 r∗h(σ)dσdθ ′ ] 7r(k) is the spectral radius of the next generation operator [10]. 8λ1 = i∗h + ∫ 1 0 r ∗ h(θ)dθ and λ2 = i e v . in the literature, it is common to consider r(k) as an equivalence of the basic reproduction rate. under the condition on compactness and the supporting properties biomath 10 (2021), 2107227, http://dx.doi.org/10.11145/j.biomath.2021.07.227 page 8 of 14 http://dx.doi.org/10.11145/j.biomath.2021.07.227 s y tchoumi et al., mathematical model for acquiring immunity to malaria: a pde ... of k [10] [proposition 6.6, page 321], it is possible to prove the existence of the endemic steady states and study the stability of the steady states depending on the sign of r(k)− 1. . some biological remarks on the possible steady states and the basic reproduction threshold is summarized in the following proposition with straightforward computation. proposition 3.3: following the notations above in notation iii-b, we have the following results: 1possible steady states seen as{ s∗h , i ∗ h , r ∗ h , s ∗ h , i ∗ v } , are: xf ={ e β̃ , 0, 0, qµ̃v , 0 } for the disease free equilibrium, and xe = { e β̃ , i eh , r e h , q µ̃v , i ev } for the endemic equilibrium; 2endemic steady state xe is such that: a) sev = q−µv iv µ̃v with sev < s f v ; b) e + kγ ∫ 1 0 r e h (δ)dδ = µh s e h + δ̃i e h . remarks 3.3: the existence problem of the endemic steady state(s) is a solution to the fixed-point problem f(reh ) = r e h where f is an operator, in a subspace of y0+. iv. numerical simulations in this section, our initial conditions are: sh(0) = 100000 individuals (humans), ih(0) = 1000 individuals (humans), rh(0, θ) = δ(0) ∗ ih(0) individuals (humans), sv(0) = 10000 mosquitoes, iv(0) = 10000 mosquitoes. we run the simulation over t = 1500days. we set also: g(t, θ) = (1 − k).γ(t, θ) following (3) and considering individuals living with mosquitoes without clinical malaria by reinforcing premunition, we run the simulations. a. simulation for (1)-(initial condition p1) with g(t, θ) = (1 − k)γ(t, θ) with more infectious humans for a long period of time in this case, immigration rate in humans is e = 100059∗365 , [5, table 1, page 4]. for other parameter values, see table i with α = 0.85 (assumed). after several simulations, all the compartments go to extinction (zero) and there is just nothing we can say further. this type of results is not interesting as seen in the literature ([19] and references therein). the problem (1)-(initial condition p2) the is more interesting as we see below. b. simulations of model (1) with initial condition p2, g(t, θ) = (1 − 0.2466) γ(t, θ) and less infectious humans for a long period of time the immigration rate in humans is given to be e = 159∗365 , [5, table 1, page 4]. for other values of parameters, see table i with α = 0.85 (assumed). numerical simulations give the following results: in figures 1a, 1b, 1c, 1d, fig 2 and other figures not shown here, we observe that the parameters e and k play a great role in the dynamics of the model. we also see that the number of infectious humans could go to zero with a large number of recovered individuals installed for a long time: the disease subsisting in mosquitoes and human and they do not fall sick although they bear the parasite. we see graphically that the proportion p1 of recovered individuals with a premunition level θ ∈ [ 1 2 , 1 ] is greater than the proportion p2 of recovered individuals with a premunition level θ ∈ [ 0, 12 ] . it suggests that people in areas where malaria is endemic, are more likely to get premunition if they are continuously bitten by mosquitoes. c. simulation of model (1) with initial condition p2 and g(t, θ) = ( 1 − 110 ) γ(t, θ) in this case, the immigration rate in humans is e = 100059∗365 (see table 1, page 4, [5]). for other parameter values, see table i with α = 0.85 (assumed). in figures 3a, 3b, 3c, 3d and fig 4, one could again say that the proportion p1 of recovered individuals with a premunition biomath 10 (2021), 2107227, http://dx.doi.org/10.11145/j.biomath.2021.07.227 page 9 of 14 http://dx.doi.org/10.11145/j.biomath.2021.07.227 s y tchoumi et al., mathematical model for acquiring immunity to malaria: a pde ... (a) suceptibles humans, (e = 159∗365 , g(t, θ) = (1 − 0.2466)γ(t, θ)) (b) infectious humans (e = 159∗365 , g(t, θ) = (1 − 0.2466)γ(t, θ)) (c) susceptible mosquitoes (e = 159∗365 , g(t, θ) = (1 − 0.2466)γ(t, θ)) (d) infected mosquitoes (e = 159∗365 , g(t, θ) = (1 − 0.2466)γ(t, θ)) fig. 1: dynamics of the model (1) using p2 with e = 159∗365 , and g(t, θ) = (1 − 0.2466)γ(t, θ) level θ ∈ [ 1 2 ; 1 ] is greater than the proportion p2 of recovered individuals with a premunition level θ ∈ [ 0, 12 ] , and support the fact that the factor e and k are very important. v. discussion a) : a model for naturally acquiring immunity to malaria diseases was studied in the present work. the ability or capacity of a person who acquires natural immunity to live with a relatively low concentration of malaria and resist to falling sick was considered to be the premunition. susceptibility and death are high during childhood but as children grow, they gradually begin to have intermittent absence of parasitemia, followed by lower density parasitemia, splenomegally and finally premunition. on the other hand, pregnant women especially primigravids (first pregnancy) are highly susceptible to malaria infections and serious diseases since the natural defense biomath 10 (2021), 2107227, http://dx.doi.org/10.11145/j.biomath.2021.07.227 page 10 of 14 http://dx.doi.org/10.11145/j.biomath.2021.07.227 s y tchoumi et al., mathematical model for acquiring immunity to malaria: a pde ... fig. 2: the number of recovered individuals (e = 159∗365 , g(t, θ) = (1 − 0.2466)γ(t, θ)) mechanisms are reduced during pregnancy. adolescents and adults sometimes have parasitemia and occasionally clinical symptoms but their premunition depends on the individual’s antibodies. antibodies as a protective measure can also boost the immunity of an individual to malaria. the pathogen replication cycle assumes a typical viral pathogen that replicates using a machinery of host cells called target cells [16]. the ability of the target cells to fight the disease is what gives premunition to an individual’s organism. b) : in section i we gave a review of previous work concerning the loss of immunity and pointed that this work on acquiring immunity for malaria is a pioneer approach. in section ii, we derive an integro-differential equation for acquiring immunity for malaria in humans taking into account all possible strains of the pathogens indexed by i. in section iii, we obtained three main results from the analysis of the model namely: the wellposedness of the model was established with initial conditions p1, the generator term a was bounded and found to be exponentially stable in the domain d(a) defined in proposition 3.1, n was shown to be bounded in proposition p2. numerical simulations were done in section iv, with initial conditions p1 and p2. the results with p1 was quiet. simulations with initial conditions p2 gave more interesting results. in subsections iv-b and iv-c we suggested that the premunition is a continuous process because the relative resistance to a low level of parasite seems to increase with time in an endemic situation. it is clear that the level of premunition is somehow a probability with a possibility to suffer from the severe malaria disease if other negative factors are dramatically considered (physiological lacks, physical or psychological injuries . . . ) even if the level of premunition is high. c) : the use of bed nets(represented by k) is recommended without interruption to avoid a severe disease due to a possible loss of immunity. at least, a partial immunity could be acquired if the bed net is not properly used and rigorous. we observed from subsections iv-b and iv-c, that more recovered people in endemic areas seem to have a high degree of immune responsiveness after being continuously exposed to mosquitoes bites. this is why obi et al. [19] said that “humans repeated infection by plasmodium falciparum induce a progressive modulation of the immune response, eventually leading to an anti-parasite immunity characteristic of premunition”. beside that, we observed in figures 2 and 4 a relatively increase in the number of recovered individuals with temporary immunity, for a long period of time (about 1000 units). vi. conclusion a) : in this paper, we have analyzed the consideration of continuous acquisition of immunity (from level 0 to level 1). our paper confirms that living in malaria endemic areas implies a premunition in the body of people exposed to infected mosquitoes bites. if mosquitoes bites stops for a long time, biomath 10 (2021), 2107227, http://dx.doi.org/10.11145/j.biomath.2021.07.227 page 11 of 14 http://dx.doi.org/10.11145/j.biomath.2021.07.227 s y tchoumi et al., mathematical model for acquiring immunity to malaria: a pde ... (a) suceptibles humans, (e = 100059∗365 , g(t, θ) = ( 1 − 110 ) γ(t, θ)) (b) infectious humans (e = 100059∗365 , g(t, θ) = ( 1 − 110 ) γ(t, θ)) (c) susceptible mosquitoes (e = 100059∗365 , g(t, θ) = ( 1 − 110 ) γ(t, θ)) (d) infected mosquitoes (e = 100059∗365 , g(t, θ) = ( 1 − 110 ) γ(t, θ)) fig. 3: dynamics of the model (1)-p2 with (e = 100059∗365 , g(t, θ) = ( 1 − 110 ) γ(t, θ)) susceptible can loose their relative immunity to malaria disease. people should be conscious and rigorous in the using bed nets especially in malaria endemic areas to avoid super infection which and failure of premunition to malaria disease. a perspective of this work could be to include a space variable in the model to target places with high risk of infection (e.g water pools, ...). in fact, the choice of g depends on the context one has. to find the right function g, one needs to solve a kind of inverse problem from the data. that is a huge perspective where the least square method could be used. finally, it’s common to see in malaria endemic areas that the population (in adulthood likely more than five years old) could live mainly as ”recovered” even within the presence of heavy parasitemia and this is the reason why public health strategies in biomath 10 (2021), 2107227, http://dx.doi.org/10.11145/j.biomath.2021.07.227 page 12 of 14 http://dx.doi.org/10.11145/j.biomath.2021.07.227 s y tchoumi et al., mathematical model for acquiring immunity to malaria: a pde ... fig. 4: the number of recovered individuals (e = 100059∗365 , g(t, θ) = ( 1 − 110 ) γ(t, θ)) high endemicity region of malaria, concerns children from 0 to 5 years old and pregnant women: a new model with age of infection is then more relevant. references [1] r. anguelov, y. dumont, j. lubuma and e. mureithi. stability analysis and dynamics preserving nonstandard finite difference schemes for a malaria model, mathematical population studies: an international journal of mathematical demography, 20:2, (2013), 101-122. 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[19] r. k. obi, c. c. okangba, f. c. nwanebu, u. u. ndubuisi and n. m. orji, premunition in plasmodium falciparum malaria, african journal of biotechnology vol. 9(10), pp. 1397-1401, 8 march, 2010 [20] a. pazy. semigroups of linear operators and applications to partial differential equations, applied mathematical sciences vol. 44, springerverlag, new york, 1983. biomath 10 (2021), 2107227, http://dx.doi.org/10.11145/j.biomath.2021.07.227 page 13 of 14 http://dx.doi.org/10.11145/j.biomath.2021.07.227 s y tchoumi et al., mathematical model for acquiring immunity to malaria: a pde ... [21] m. pinkevych, j. petravic, k. chelimo, j. w. kazura, a. m. moormann and m. p. davenport. the dynamics of naturally acquired immunity to plasmodium falciparum infection, plos comput biolv.8 (10); (2012). [22] d. raoult, epidémies: vrais dangers et fausses alertes, éditions michel lafon, 2020. 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[26] n. ziyadi, s. boulite and m. lhassan hbid. mathematical of a pde epidemiological model applied to scrapie transmission, comm. on pure and appl. an., volume 7, number 3, pp. 659–675, may 2008. biomath 10 (2021), 2107227, http://dx.doi.org/10.11145/j.biomath.2021.07.227 page 14 of 14 http://dx.doi.org/10.11145/j.biomath.2021.07.227 introduction model description and extension model equation extension of model (1) to different malaria strains with same initial conditions main results under initial condition p1 well-posedness of the model (1) with initial condition p1 and g(t,)=(1-k)(t,) steady states of model (1) for g(t,)=(1-k) (t,). numerical simulations simulation for (1)-(initial condition p1) with g(t,)=(1-k)(t,) with more infectious humans for a long period of time simulations of model (1) with initial condition p2, g(t,)=( 10.2466) (t,) and less infectious humans for a long period of time simulation of model (1) with initial condition p2 and g(t,)=(1110) (t,) discussion conclusion references www.biomathforum.org/biomath/index.php/biomath original article modeling outbreak data: analysis of a 2012 ebola virus disease epidemic in drc boseung choi∗, sydney busch†, dieudonné kazadi††‡‡, benoit ilunga‡‡, emile okitolonda††, yi dai‡, robert lumpkin§, omar saucedo¶, wasiur r. khudabukhsh‡¶, joseph tien§, marcel yotebieng‖, eben kenah‡, grzegorz a. rempala‡§¶∗∗ ∗department of national statistics, korea university sejoung campus sejoung, republic of korea †department of mathematics, augsburg college minneapolis, mn, usa ††ministry of health, democratic republic of the congo ‡‡school of public health, university of kinshasa kinshasa, democratic republic of the congo ‡division of biostatistics, college of public health §department of mathematics ¶mathematical biosciences institute ‖division of epidemiology, college of public health the ohio state university, columbus, oh, usa ∗∗corresponding author: rempala.3@osu.edu received: 8 september 2019, accepted: 3 october 2019, published: 15 october 2019 abstract—we describe two approaches to modeling data from a small to moderate-sized epidemic outbreak. the first approach is based on a branching process approximation and direct analysis of the transmission network, whereas the second one is based on a survival model derived from the classical sir equations with no explicit transmission information. we compare these approaches using data from a 2012 outbreak of ebola virus disease caused by bundibugyo ebolavirus in city of isiro, democratic republic of the congo. the branching process model allows for a direct comparison of disease transmission across different environments, such as the general community or the ebola treatment unit. however, the survival model appears to yield parameter estimates with more accuracy and better precision in some circumstances. keywords-parameter estimation; branching process; markov chain monte-carlo methods; survival dynamical system; i. introduction on august 1, 2018, the ministry of health of the democratic republic of the congo (drc) copyright: c©2019 choi et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: boseung choi, sydney busch, dieudonné kazadi, benoit ilunga, emile okitolonda, yi dai, robert lumpkin, omar saucedo, wasiur r. khudabukhsh, joseph tien, marcel yotebieng, eben kenah, grzegorz a. rempala, modeling outbreak data: analysis of a 2012 ebola virus disease epidemic in drc, biomath 8 (2019), 1910037, http://dx.doi.org/10.11145/j.biomath.2019.10.037 page 1 of 12 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2019.10.037 b. choi et al., modeling outbreak data: analysis of a 2012 ebola virus disease epidemic in drc reported an outbreak of ebola virus disease (evd) in north kivu province. at the time of writing about one year later, confirmed and probable cases have been reported in nine health zones of north kivu and ituri provinces (including the provincial capital city of goma), threatening further spread of the epidemic into neighboring provinces and the countries of uganda and rwanda. the current outbreak area is roughly 780 miles away from equateur province, where an earlier ebola outbreak was reported in may 2018. this persistent reoccurrence of ebola in the drc as well as elsewhere in africa is a reminder that another large pandemic like the 2013-2016 west african ebola epidemic remains possible. to control future outbreaks and to better understand patterns of transmission in households and at health care facilities, it is essential to carefully analyze welldocumented historic data from past outbreaks. the current paper is concerned with modeling data from a small 2012 ebola virus disease (evd) outbreak caused by bundibugyo ebolavirus (bdbv) in the isiro municipality in drc. the interesting feature of this dataset is that it includes partial contact information on ebola cases treated either in the community or in healthcare facilities, which allows for network-based inference. despite the fact that such inference has been an extremely active area of research in the past 20 years [1]– [5], there have been relatively few well documented historical datasets from real epidemics. for our purpose of analyzing a relatively small network, we here apply the edge-based approach of miller and volz [15]–[17] and compare it with the recently proposed simple non-network survival dynamical system model [18]. the paper is organized as follows. in the remainder of this section we give some basic background information on the ebola virus and describe the isiro outbreak dataset to be analyzed. in sections 2 and 3, we outline the proposed statistical models, use them to analyze the isiro outbreak data, and describe the results. in section 4, we offer some brief concluding remarks. a. ebola virus disease evd in humans is a severe hemorrhagic fever caused by four of the five viruses of the genus ebolavirus in the viral family filoviridae [19]. the virus was initially characterized in 1976 during an outbreak in drc (known then as zaire) and has subsequently caused at least 26 additional outbreaks in human populations [20], [21]. since 2000, there have been 8 recorded outbreaks in the drc. the ongoing outbreak in the north kivu province is the most serious to date, and maganga et al. [22] describe an earlier serious outbreak that occurred in the boende region of équateur province in 2014. data is available in the literature for at least 4 out of these 8 outbreaks, although these data are not considered fully reliable because the majority of the outbreaks occurred in remote areas. although evd outbreaks in drc have been limited to fewer than 500 cases to date, the potential for dangerous spread across the african continent was demonstrated by the 2013-2016 west african ebola epidemic, which resulted in more than 28,600 cases and 11,313 deaths across ten countries [23] . the introduction of the virus into human communities is likely the result of sporadic zoonotic events. several species of fruit bats native to areas endemic to ebola have been implicated as the natural reservoir for the disease [19]. upon infection, the virus will typically incubate for a period of two to 21 days [23]. after this period, the typical clinical presentation is a mix of severe symptoms that may include fever, nausea, diarrhea, vomiting, chest pain, dyspnea, cough, ocular edema, hypotension, conjunctivitis, headaches, coma, and hemorrhaging [24]. human-to-human ebola transmission is believed to occur via close contact with infected bodily fluids and, therefore, individuals such as family members of cases and healthcare workers have significantly increased risk of infection [25]–[28]. in addition, viral load (which changes throughout the course of illness and is at the highest level immediately after death) is known to impact the probability of transmission [29], [30]. transmisbiomath 8 (2019), 1910037, http://dx.doi.org/10.11145/j.biomath.2019.10.037 page 2 of 12 http://dx.doi.org/10.11145/j.biomath.2019.10.037 b. choi et al., modeling outbreak data: analysis of a 2012 ebola virus disease epidemic in drc sion dynamics are further complicated by the fact that an infectious individual’s contacts may vary throughout disease progression. accounting for these dynamics through contact tracing and network analysis is vital to understanding the disease, as studies of previous outbreaks have noted disproportionately high rates of infection among women, healthcare workers, and family members [23], [25], [27], [31]. practical strategies have been implemented to reduce the risk of transmission, including barrier nursing methods, safe burial practices, and the creation of isolation units within treatment centers [31], [32]. nevertheless, the scope of the 2013-2016 west african ebola epidemic suggests that a deeper quantitative understanding of these dynamics might be needed to control transmission more effectively. while an effective vaccine is now available [33], the question of how and when to most effectively use available resources during an ongoing epidemic remains in need of further quantitative study. prior to the west african ebola epidemic, only a handful of mathematical models for ebola had been studied [34]–[37]. chowell et al. [34] and lekone and finkenstadt [35] used an seir framework to determine the effect of interventions on the 1995 drc and 2000 uganda outbreaks. legrand et al. [36] formulated a stochastic compartmental model that accounted for transmission in several epidemiological settings by introducing compartments for hospitalized individuals and dead ebola cases, who can transmit the disease during funerals. the west african epidemic highlighted the critical need for a better understanding of the ebola transmission dynamics and potential control measures. consequently, there has been an outpouring of ebola models including deterministic compartmental models [38]–[40], stochastic models [41]–[44] and multi-type branching process models [45]. many of these adapted the seihrf framework, which includes hospital and funeral compartments [26], [46]–[48]. recent work has considered spatial aspects of ebola virus transmission and the effect of clustering in the population. the spatiotemporal spread of ebola has been studied with a county-level multipatch model employing mobile data [41], with spatial individual-based models for international spread [43], and transmission between households [44]. scarpino et al. [26] used a phylodynamic model to reconstruct chains of transmission for cases occurring in sierra leone in june 2014. their fit of an seir model with the rand-style [1] pair approximation gave evidence for the presence of clustering within the population. the same pair approximation was employed by wells et al. [46] in their investigation of the effectiveness of case isolation and ring vaccination. their model had 10 compartments and required 65 equations after closure at the level of pairs. b. isiro evd outbreak data isiro is a municipality in the north-east part of drc that is the capital of haut-uele district. it is situated between equatorial forest and savannah. the isiro dataset contains information about the 2012 evd outbreak caused by bundibugyo ebolavirus (bdbv). in total, there were 62 cases of infection listed as either probable or confirmed with 52 of them having proper clinical information allowing for a more detailed study. as shown in figure 1(a), these were divided into community cases and ebola treatment center (etc) cases based on the source of the infecting contact. among community cases, there was overrepresentation of females (85.3%) and of individuals aged 15-54 years (82.4%). the mean duration of evd was 18 days, and the mean incubation period was 11.3 days [49]. we were able to obtain additional information concerning contacts for most individual cases from the drc ministry of health. however, unlike traditional contact tracing, these additional records contained a list of potentially infecting individuals for each case. using this information, we were able to track the likely number of people each case infected as well as the overall contact network (for most of the evd cases) in both the community and the etc. to refine the transmission network, biomath 8 (2019), 1910037, http://dx.doi.org/10.11145/j.biomath.2019.10.037 page 3 of 12 http://dx.doi.org/10.11145/j.biomath.2019.10.037 b. choi et al., modeling outbreak data: analysis of a 2012 ebola virus disease epidemic in drc (a) (b) fig. 1: 2012 isiro evd data and model. panel (a): summary of available isiro cases used in current analysis of the transmission dynamics in the community and etc. this data is a subset of 52 cases described in [49]. panel (b): example of transmission data reconstructed from the isiro outbreak files. dark figures represent primary cases and secondary cases who infected others. all others represent infected who did not transmit. all cases of transmission ambiguity (multiple in-arrows) were resolved uniformly at random. we used occupation and other socioeconomic information as available. of the 52 documented infections in the isiro epidemic, clinical documentation of contacts among 48 cases (17 probable, 31 confirmed) could be retrieved. out of these, there were 37 community cases1 (13 suspected or probable and 24 confirmed) who never reached the etc. the 11 cases that reached the etc were all confirmed as evd [49]. ii. statistical models in this paper, we consider two different statistical models for the isiro evd outbreak. the first one is based on a branching process approximation of the virus transmission network, which appears especially appropriate for directly comparing outbreak parameters in different environments such as the village community and the etc. the second model is based on a survival analysis approach that assumes homogenous contact patterns among susceptibles and infectives. this assumption is more likely to be appropriate among community cases than etc cases. 1we note here a slight discrepancy with [49] where only 34 cases were classified as community. a. branching model 1) primary cases: for a primary (or index) case i represented by a node of degree di, the distribution of the number of secondary infections created by i (say, xi) conditionally on the degree and the infection period (say, ti) is given by p(xi = xi|ti,di) = ( di xi ) pxiti (1 −pti) di−xi. (1) we see therefore that the distribution function of xi is binomial with di trials and the probability of success pti. under the assumption that infectious contacts follow a poisson process with rate β, the probability of a successful infection of a given neighbor by case i before time ti is pti = 1 −e −βti. the infectious period (i.e., time from symptom onset to removal) for each case is assumed to follow an exponential distribution with rate γ, so its density is f(ti) = γe −γti. (2) for the i-th index case the joint conditional probability distribution of (xi, ti) given di is the product biomath 8 (2019), 1910037, http://dx.doi.org/10.11145/j.biomath.2019.10.037 page 4 of 12 http://dx.doi.org/10.11145/j.biomath.2019.10.037 b. choi et al., modeling outbreak data: analysis of a 2012 ebola virus disease epidemic in drc of (1) and (2): πdi(xi, ti) = p(xi = xi|ti,di)f(ti) = ( di xi ) γ(1 −e−βti)xie−(β(di−xi)+γ)ti. without any prior information on the degree di of the i-th index case, the unconditional joint probability distribution of the number of infections and the infection period is then given by f(xi, ti) = ∑ di≥xi qdiπdi(xi, ti) (3) where qd is the probability of degree d. here, we consider the poisson distribution with mean λ,which appears to fit well the transmission network for the isiro epidemic. with this degree distribution, the final form of (3) admits the following closed-form expression parametrized by the triple θ = (β,γ,λ) ∈ r3>0: fθ(xi, ti) = ∑ di≥xi λdie−λ di! ( di xi ) γ(1−e−βti)xie−(β(di−xi)+γ)ti = [λ(1 −e−βti)]xi xi! e−λ(1−e −βti)γe−γti. (4) 2) secondary cases: consider now the joint distribution of the number of infections and the infection period for a secondary (i.e., non-index) case. for tractability, we assume the random network follows the configuration model (cm), so edges are formed uniformly at random (excluding multiple edges and self-loops) given the degrees of all nodes. for such networks, we define a secondary case as an individual (node) to whom the infection was successfully transmitted (see also [50]). note that since by definition the secondary case has one infecting neighbor, its degree available for further infection decreases by one. this decreased degree is often referred to as the excess degree in the literature. for secondary cases, we can modify the primary case model by replacing the degree with the excess degree. let q′dj denote the excess degree probability for a secondary case with degree dj > 0, and let µ ∈ (0,∞) be the mean of the degree distribution. then it is known ( [51]) that for the cm network q′dj = djqdj µ . in the case of the poisson degree distribution, λ = µ and it is easy to check that q′dj = qdj−1. (5) let x′j and t ′ j be the number of infections and the infection period for the secondary case j. equations (3) and (5) give us f(x′j, t ′ j) = ∑ dj>x ′ j q′djπdj−1(x ′ j, t ′ j) = ∑ dj>x ′ j qdj−1πdj−1(x ′ j, t ′ j) = fθ(x ′ j, t ′ j), (6) where fθ is given by (4). for known θ = (β,γ,λ), we can calculate the key characteristic of an epidemic known as the basic reproduction number (r0). it may be interpreted as the average number of secondary infections caused by a primary case. for an arbitrary degree distribution the definition of r0 for cm graph is given, for instance, in [52]. for the poisson degree distribution with exponential infectious periods, we have simply r0 = βλ β + γ . (7) 3) likelihood and estimation: suppose the observed numbers of infections and infectious periods for m primary cases are x = {x1,x2, . . . ,xm} and t = {t1, t2, . . . , tm}, and those for n secondary cases are x′ = {x′1,x ′ 2, . . . ,x ′ n} and t′ = {t′1, t ′ 2, . . . , t ′ n}. recall that θ = (β,γ,λ) is the vector of parameters that need to be estimated. here β is the transmission rate of infection, and γ−1 is the mean infectious period, and λ is the mean degree. for the observed data, the likelihood function for θ can be constructed using (4) and (6): l(θ|x,t,x′,t′) ∝ m∏ i=1 fθ(xi, ti) n∏ j=1 fθ(x ′ j, t ′ j). biomath 8 (2019), 1910037, http://dx.doi.org/10.11145/j.biomath.2019.10.037 page 5 of 12 http://dx.doi.org/10.11145/j.biomath.2019.10.037 b. choi et al., modeling outbreak data: analysis of a 2012 ebola virus disease epidemic in drc we could calculate the maximum likelihood estimates (mles) of θ by maximizing the expression above using numerical optimization. however, for better stability and reproducibility of results, we will use a bayesian markov chain monte carlo (mcmc) procedure. we assign independent gamma prior distributions for the components of θ. specifically, let π(β) ∼ γ(aβ,bβ) π(γ) ∼ γ(aγ,bγ) π(λ) ∼ γ(aλ,bλ). (8) given the data x, t, x′, and t′, the bayesian estimate of θ is the mean of the joint posterior distribution lp(θ|x,t,x′,t′) ∝l(θ|x,t,x′,t′)π(β)π(γ)π(λ). (9) this mean may be approximated with an empirical average from the following converged mcmc sampler: algorithm 1 mcmc posterior sampler for the branching process model 1: initialize θ with θ0 = (β0,γ0,λ0). 2: sample β via a metropolis-hastings step [53] with truncated normal proposal for the target conditional distribution of β given x,t,x′,t′,λ. 3: sample γ directly from its conditional distribution γ|x,t,x′,t′ ∝ γm+n+aγ−1e−γ( ∑ ti+ ∑ t′j+bγ). 4: sample λ via another metropolis-hastings step (similar to step 2) for the target conditional distribution of λ given x,t,x′,t′,β. 5: return to step 2 and repeat until convergence. b. survival model under the survival model, it is assumed that the population at risk (which is possibly much smaller than the set of all initially susceptible individuals) interacts homogeneously according to the kermack-mckendrick model (see below). hence it is problematic to apply this model in the case of an etc where the homogenous mixing is likely not satisfied. consequently, we apply this model only to the community outbreak data. in what follows it is convenient to write the usual kermackmckendrick model in the following integral form: st = exp ( −β ∫ t 0 ιudu ) = exp (−rort) ιt = ρe −γt − ∫ t 0 sue −γ(t−u)du rt = γ ∫ t 0 ιudu where ρ = ι0, and r0 = β/γ. by interpreting the strictly decreasing function st as an improper survival function, it follows [18] that the conditional density of infection time is given by fτ (t) = −ṡt/τ (10) where τ = limt→∞rt < 1 is the final size of the epidemic [18], which is the unique solution of 1 − τ = e−r0(τ+ρ). (11) thus, for a collection of n individuals initially at risk, out of which k are infected at respective times t1 < ... < tk < t (where t < ∞ is the maximum follow-up time), we have the following log-likelihood function for infection times li (t1, . . . tk|θ,n) = (n−k) log st + k∑ i=1 log fτ (ti) . note that this likelihood is conditional on the number of individuals at risk, which is often unknown. however, given the number of infections (k) by time t and under the assumption of independence of infection times (see [18] for discussion), we may consider n as the realization of the negative binomial distribution: n ∼ negbinom(k,τ). denote by wi the ith individual’s removal time, defined as minimum of the times of individual’s recovery, hospitalization, or death. assuming r recoveries given k infections, the log-likelihood is lr (w1, . . . ,wr|θ,k) = (k −r) log hγ(t) + r∑ i=1 log hγ(wi), biomath 8 (2019), 1910037, http://dx.doi.org/10.11145/j.biomath.2019.10.037 page 6 of 12 http://dx.doi.org/10.11145/j.biomath.2019.10.037 b. choi et al., modeling outbreak data: analysis of a 2012 ebola virus disease epidemic in drc where hγ(·) is the survival function and hγ(·) is the pdf of an exponential distribution with rate γ. the complete log-likelihood is then the sum of li and lr [18]: l0 (t1, . . . , tk,w1, . . . ,wr|θ,n) = ll (t1, . . . , tk|θ,n) + lr (w1, . . . ,wr|θ,k) . under this survival model, we may estimate the model parameters and the size of the population at risk n using another bayesian mcmc. the model parameters are now θ′ = (β,γ,ρ), because τ is fully determined by θ′ via (11). the prior distributions for β and γ are of the form (8) and the prior distribution for ρ is taken as uniform(0, 1). all parameters are assumed independent a priori. the mcmc algorithm used to obtain the posterior sample of the parameters is similar to that in previous section. algorithm 2 mcmc posterior sampler for the survival model 1: initialize θ′ = (β,γ,ρ) from the prior distribution and set n = k. 2: perform a metropolis-hastings step (using the truncated normal proposal) for the target conditional distribution of θ′ given n using the complete log-likelihood `0 = `i + `r. 3: calculate τ based on the current value of θ′ as the solution to final size equation (11). 4: sample the conditional distribution of n given θ′ by drawing n ∼ negbinom(k,τ). 5: return to step 2 and repeat until convergence. iii. data analysis the analysis was conducted separately for the two models (branching process and survival) based on the 48 available cases of evd described in section 1. parameter estimates were obtained using the mcmc algorithms for each model described in section 2. for the branching process model, we separately analyzed the community and etc outbreaks. for the survival model, we only analyzed the community outbreak. a. branching process model to perform the separate analyses of the community and etc outbreaks, the contact and infection data were partitioned into two subsets depending on the location of the infective contact (community or etc). although the same uninfected individuals were allowed to be in both outbreak networks, all the evd cases were assigned either to the community or to the etc. when reconstructing the transmission network, all ambiguous contact tracing was resolved uniformly at random as shown in figure 1(b). for some individuals, the complete infection period was unknown and needed to be imputed. all such imputations were based on the density (2) and performed between steps 3 and 4 in mcmc algorithm. for estimating β and γ, we assigned noninformative gamma prior distributions with location and rate parameters of 0.001. however, due to limited contact information, we assigned a relatively informative prior to λ with location parameter of 6 and a rate parameter of 1. this assignment was based on the empirical mean of the primary and secondary cases in the dataset. finally, for the metropolis-hastings steps in the mcmc sampler algorithm we used the truncated normal distribution as proposal distribution and tuned its standard deviation to achieve an acceptance ratio of 44% as recommended in [54]. the final results of the mcmc were based on 55,000 iterations of the sampler with first 5,000 iterations removed as “burn-in”. the posterior samples were thinned by keeping only the results from every 10th iteration, resulting in a final set of 5,000 posterior samples that were used to estimate the parameters and calculate approximate posterior credible intervals. the convergence of the mcmc algorithm was diagnosed based on the r statistic and trace and autocorrelation (acf) plots. to conserve space, these plots are not shown here. table i summarizes the results of branching process analyses of community and etc epidemics. for each parameter, the posterior mean, standard deviation, and 95% credible interval (ci) are provided. we note that the comparisons bebiomath 8 (2019), 1910037, http://dx.doi.org/10.11145/j.biomath.2019.10.037 page 7 of 12 http://dx.doi.org/10.11145/j.biomath.2019.10.037 b. choi et al., modeling outbreak data: analysis of a 2012 ebola virus disease epidemic in drc table i: results under the evd branching process model for the community and etc outbreaks community infections etc infections mean std dev 95% ci mean std dev 95% ci β 0.0741 0.0389 (0.0331, 0.1806) 0.0387 0.0182 (0.0152, 0.0851) γ 0.1936 0.0397 (0.1254, 0.2811) 0.2205 0.0634 (0.1170, 0.3605) λ 5.4460 1.4460 (2.9690, 8.6310) 5.9030 1.4200 (3.4200, 8.9820) r0 1.3730 0.2951 (0.8510, 2.0230) 0.8592 0.3200 (0.3700, 1.6040) table ii: results under the evd survival model for the community outbreak only mean std dev 95% ci β 0.1964 0.0324 (0.1403, 0.2626) γ 0.1774 0.0296 (0.1258, 0.2381) ρ 0.0039 0.0017 (0.0017, 0.0079) n 163.20 35.35 (113.00, 252.00) r0 1.1080 0.0316 (1.0570,1.1650) tween parameters in the two epidemics may be conducted informally by comparing their respective cis bounds. if a particular parameter’s ci bounds for the community are contained within the respective ci bounds for the etc, or viceversa, one would consider the corresponding posterior distributions as statistically (i.e., for given data) equal. for β, which represents the transmission rate of ebola virus, the posterior mean for household infection was approximately 0.0741— about twice as large as etc infection rate—so the two posterior distributions may be considered statistically different. this is in contrast with the parameter γ, which represents the reciprocal of the mean infectious period, for which the estimated respective posterior means of 0.1936 and 0.2205 for the community and etc were not found to be statistically different based on their respective 95% credible bounds. the parameter λ represents the average degree of the degree distribution and its posterior mean in the etc is slightly (but not significantly) larger that its posterior mean in the community. this may reflect additional contacts of the individuals at the etc with patients, visitors, and etc staff. finally, the posterior means of the basic reproduction numbers r0 for the community and etc outbreaks were found to be significantly different at 1.373 and 0.8592, respectively. as expected, the posterior mean of r0 for the community outbreak is higher than one for the etc. in both settings, r0 was calculated according to equation (7). however, we also note that the 95% ci bounds for both posterior distributions are quite wide, indicating a lack of precision in the branching process model. b. survival model under the survival model outlined in section ii-b, the analysis simplifies in that we are no longer concerned with estimating the network average degree λ. hence our model parameters are θ′ = (β,γ,ρ) as well as the size of the population at risk (or effective population size) denoted by n. in this model, r0 = β/γ. the results of the analysis for the 37 community cases are presented in table ii. we note that the estimates of the parameters (β,γ,r0) under the survival model can be compared with the estimates of (βλ,γ,r0) under branching process model. despite considerable differences in their respective posterior mean values, the posterior distributions of all three parameters are statistically equivalent based on the respective 95% ci bounds. this underscores the lack of precision of the estimates based on the branching process model. biomath 8 (2019), 1910037, http://dx.doi.org/10.11145/j.biomath.2019.10.037 page 8 of 12 http://dx.doi.org/10.11145/j.biomath.2019.10.037 b. choi et al., modeling outbreak data: analysis of a 2012 ebola virus disease epidemic in drc c. model validation to perform our model validation and fit assessment, we compared the distributions of 5,000 samples from the posterior distributions of the final outbreak size obtained from our branching process analysis to the values of 37 and 11 observed in the community and etc outbreaks respectively. figure 2 presents the histograms of the posterior size distributions for the respective final sizes. the vertical lines are plotted for reference at the observed outbreak values of 37 and 11. the histogram plots show that the observed values are close to the modes of the respective posterior distributions for both models, indicating adequate model fit [50]. the last panel of figure 2 presents a comparison of observed depletion of susceptibles over the course of the community evd outbreak with that predicted by the survival model. both the observed and predicted curves are initiated on day one at the estimated mean of total population at risk. the two curves are very close to each other, indicating good fit of the survival model. iv. summary and conclusions we presented two statistical models for analyzing patterns of evd transmission in a small community. the branching process model was based on partial contact network tracing data, whereas survival model used an aggregate (network-free) approach. the two models allowed for a more detailed analysis of the 2012 isiro evd epidemic data that was summarized in [49]. the branching process model was based on a configuration model random graph with a poisson degree distribution, and it explicitly described the direct and indirect contacts of the evd cases in the community and at the etc. in particular, the model made it possible to derive and directly compare the characteristics of the evd isiro outbreaks at these two different locations. although the comparison provided some evidence of the usefulness of the etc in controlling evd outbreaks, the analysis also suggests that the type of basic contact tracing performed in isiro may not be sufficient to provide precise estimates of the epidemic force via the basic reproduction number (r0). the survival model was derived from the standard sir equations. since this model did not require contact tracing or estimation of mean degree, it did not suffer from the same problem of low precision as the branching process model. in fact, the estimate of r0 provided by the survival model was more precise and likely also more accurate (based on a resampling analysis not shown here) than the values obtained from the branching process model. however, the drawback of the survival model was that it could not be used for comparison of transmission in the community and the etc because the etc was unlikely to satisfy the homogeneous mixing assumption. it appears that an effective approach to modeling the type of outbreak dynamics described by the isiro data might be to combine the two models presented here, so as to retain the precision of the survival one but also incorporate the transmission network information. this will be likely the focus of our future research. acknowledgment this research was partially funded by the us national science foundation and the us national institutes of health under grants dms1853587, u54 gm111274 and r01 ai116770. the funds were also received from the mershon center for international security studies at osu, the korea university and the ohio five-osu summer undergraduate research experience (sure) fund. the authors would like to especially thank the mathematical biosciences institute (mbi) at osu for providing additional resources and logistical help. mbi receives funds from the national science foundation under grants dms1440386 and dms1757423. references [1] h. andersson and t. britton, stochastic epidemic models and their statistical analysis. springer science & business media, 2012, vol. 151. 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[54] a. gelman, j. b. carlin, h. s. stern, d. b. dunson, a. vehtari, and d. b. rubin, bayesian data analysis. crc press boca raton, fl, 2014, vol. 2. biomath 8 (2019), 1910037, http://dx.doi.org/10.11145/j.biomath.2019.10.037 page 12 of 12 http://rsif.royalsocietypublishing.org/content/9/70/890 http://dx.doi.org/10.11145/j.biomath.2019.10.037 introduction ebola virus disease isiro evd outbreak data statistical models branching model primary cases secondary cases likelihood and estimation survival model data analysis branching process model survival model model validation summary and conclusions references www.biomathforum.org/biomath/index.php/biomath original article analysis of a virus-resistant hiv-1 model with behavior change in non-progressors rabiu musa, robert willie, nabendra parumasur school of mathematics, statistics and computer science university of kwazulu-natal, durban, south africa. rabiumusa003@gmail.com, willier@ukzn.ac.za, parumasurn1@ukzn.ac.za received: 26 february 2020, accepted: 14 june 2020, published: 8 august 2020 abstract—we develop a virus-resistant hiv-1 mathematical model with behavior change in hiv1 resistant non-progressors which was analyzed for both partial and total abstinence cases. the model has both disease-free and endemic equilibrium points that are locally asymptotically stable depending on the value of the associated threshold quantities rt and r ′ t . in both cases, a nonlinear goh–volterra lyapunov function was used to prove that the endemic equilibrium point is globally asymptotically stable for special case while the method of castillo-chavez was used to prove the global asymptotic stability of the disease-free equilibrium point. in both the analytic and numerical results, this study shows that in the context of resistance to hiv/aids, total abstinence can also play an important role in protection against this notorious infectious disease. keywords-resistance; behavior change; partial & total abstinence; goh–volterra lyapunov function. ams subject classification: 92bxx, 92b05. i. introduction as it was reported in the 1980s, the human immunodeficiency virus (hiv), and the later stage of infection through cell depletion known as aids has continue to play a leading role in the series of the greatest ever infectious disease. united nations program on hiv/aids (unaids) and the world health organization (who) have already provided the estimates of the number of cases since the 1980s. more than 30 million people are currently hiv positive. according to the current trends, at least 7300 people are infected with hiv and a minimum of 5000 die from aids-related causes including at-least 690 children on a daily basis (unaids, 2009). this means that for every five hiv positive individuals, at least four of them including adults and children die from the infection daily [10], [32]. the two main types of hiv are hiv-1 and hiv2. the most dangerous that has spread worldwide is hiv-1 while the latter is less pathogenic and less spread since it’s confined to west african countries. the test carried out on one can not copyright: c©2020 musa et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: rabiu musa, robert willie, nabendra parumasur, analysis of a virus-resistant hiv-1 model with behavior change in non-progressors, biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 page 1 of 17 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2020.06.143 r. musa, r. willie, n. parumasur, analysis of a virus-resistant hiv-1 model with behavior change in ... sufficiently detect the other due to large genetic differences between them. immediately after hiv infection, the lymphocytes, or white blood cells, known as cd4+ t cells are the major target. therefore, the antihiv antibody and cytotoxic t cell production by the immune system is consequently initiated. an hiv positive individual is not classified as having aids until cd4+t cell count which is approximately around 1000mm−3 depletes to 150mm−3 or thereabout. since cd4+t plays a very important role in the body immune mechanism, deterioration and depletion result in acquired immunodeficiency syndrome called aids. the average number of times it takes hiv to develop to aids is dependent on the strength of the immune system of the victim [23]. it is therefore pertinent to study methods of hiv prevention. different control strategies such as behavior change due to hiv awareness campaign, reduction in sexual partners, anti retroviral treatment art etc. have collectively played important roles in combating the menace. they are still very much relevant due to unavailability of vaccine. the use of condom has also played an important role and can possibly prevent hiv transmission almost perfectly. other intervention methods that can concurrently prevent both sexes are still very much needed. recently, an experimental product containing a drug that can prevent rectal and vaginal transmission of hiv and other sexually transmitted diseases was detected but unfortunately did not see the light of the day due to the fact that the gel is ineffective with high hiv infection risk [24]. other efforts such as the hiv vaccine and diaphragm technique fail to manifest to any meaningful impact [6], [21]. from biological point of view, hiv resistance is known as the genetic mutation in the dna that delays aids progression or aids production of permanent immunity (i.e. no progression) to aids. this kind of mutation which is known as ccr5-delta 32 plays an important role in the development of the two kinds of hiv resistance known. this ccr5-delta 32 breaks and distorts the hiv’s ability to deplete and destroy the immunity of the cd4+t cells. the mutation makes the ccr5 co-receptor on the outside of cells to develop at a smaller rate than usual and no longer sit outside of the cell. this co-receptor is similar to a door that allows hiv passage into the cell where within a second locks “the door” which consequently prevents hiv entrance into the cd4+t cells [14]. this genetic mutation has been reported to be inborn. there are very few paper on resistant mathematical model, some of them are [25], [13] and [15] but the resistance was modeled on influenza and sars which is quite different from hiv/aids. this still remains a biological research question needed to be answered. research has shown that some people develop resistance to the killer hiv-1 virus [22], [28]. in fact, a report in [18] shows that though this resistance is rare but actually exists. virus resistance can be understood in two scenarios. first, there are cases of individuals that are exposed to hiv but after a long period of times, diagnosis shows that they are uninfected. this case of exposed uninfected have been detected from among infants of infected mothers, health workers during treatment of infected individuals, commercial sex workers, individuals having unprotected sex with seropositive partners etc. the second category is hiv infected individuals with low or no progression to aids as expected under normal circumstances. they live with the virus for many years with an absolutely low level of hiv-1 rna or no loss of cd4+ cells that has been identified among various individuals such as children and homosexual men and women mutation [18], [8]. in 2014, the report in [12] confirms that some people show partial or absolutely complete inborn resistance to the hiv virus . the major or main contributor to this strange development is a mutation of the gene encoding ccr5 which acts as a co-receptor for hiv. ccr5 may even be defective in some individuals which will enhance protection against disease. these individuals live a normal life since the hiv-1 virus cannot bind itself to it and its perhaps here that the key to biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 page 2 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 r. musa, r. willie, n. parumasur, analysis of a virus-resistant hiv-1 model with behavior change in ... overcome the disease lies hidden. estimation later shows that the proportion of individuals under this category is less than 1%. similar occurrences make leading oxford university researcher sarah rowland-jones to believe continual exposure is a requirement for maintaining immunity after which 15 proteins were identified to be unique to those virus-free sex workers [3], [2]. a genetic mutation that blocks hiv which may hold the key to future treatment was also studied in 2016 by [9]. in 2010, [4] identified factors such as apobec3g, toll-like receptors, acute-phase amyloid a protein, interleukin-22, apobec3g and natural killer cells as the main reason why some people do not even seroconvert let alone progressing to aids despite multiple hiv exposure. more interesting reasons behind this strange occurrence has been examined by the university of minnesota in 2014 [33] and by [17] in 2013. another interesting factor that influence the spread of hiv/aids is change in sexual behavior towards sex. this is caused by the infectiousness nature, high death rate and stigmatization encountered by victims of hiv/aids. this has subsequently affect the transmission of the disease in recent years. behavior change intervention will help individuals change their drug-using behaviors and sexual behavior that put them at a high risk of contracting hiv. it also creates skills and knowledge that can influence their motivation and ability to kick start behavior change. couples, peer groups, individuals, communities or institutions can be targeted on a multiple level. this behavior change can also be motivated through skills-building, motivational or educational approach. interventions can target different kind of behaviors such as condom usage, number of sexual partners, correct use of best prevention approach etc. though many researchers have developed different models to examine the dynamics of the virus, hiv-1 mathematical model where infected individuals gain resistance to acquisition of hiv and resistance to deterioration of hiv incorporating behavior change in form of partial and total abstinence is still a biological question needed to be answered. researchers like [31], [19] have done commendable work in tackling the menace of the deadly virus, in this research, we present a new virusresistant hiv-1 model with behavior change. this behavior change to avoid infection happens as a result of the wide spread of the agony and death caused by hiv/aids. this change happens either partially or totally. those who show partial abstinence are those that only reduced their sexual partners but still involve in hiv-risk activities or live in endemic environment while those who totally abstain are those who maintain only one sexual partner and do away from all hiv-risk activities or exposed and endemic environment. mathematical modeling has become an effective tool in studying infectious disease by many researchers. it shall be used again here to study the dynamics of resistance in hiv-1 transmission and how it produce significant reduction rate in the community. we hope it helps policy-makers and public health workers in the epidemic control. several researchers like [20], [19], [1] and references therein have published commendable research output about transmission dynamics of hiv/aids. they have also studied control and prevention strategies of this notorious epidemic. in order to further extend, compliment and contribute to the work of the aforementioned researchers, a new comprehensive model has been designed. the model extends the work of the aforementioned researchers by, for instance, 1) considering the influence of virus-resistance i.e. resistance to acquisition and resistance to deterioration. 2) incorporating the change of behavior class whose rate of progression is either through partial abstinence or total abstinence. 3) including a compartment (i1) for slow progressors. these are the category of people with partial resistance to the virus. 4) including a compartment (i2) for non progressors. these are the category of people with complete resistance to the virus and do not move to aids compartment (a). 5) including a compartment (i3) for fast probiomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 page 3 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 r. musa, r. willie, n. parumasur, analysis of a virus-resistant hiv-1 model with behavior change in ... gressors. these are the category of people with no resistance to the virus. all these instances have not been considered before. the paper is organized as follows. section 2 entails model formulation and assumptions while section 3 contains basic properties of the model. this is followed by the analysis of the sub-model (model with total abstinence) and that of the full model (model with partial abstinence) in section 4. section 5 presents the numerical simulation and discussion of results while the last section contains the conclusion, acknowledgment and disclosure statement. ii. model formulation and model assumptions we formulate an hiv-1 resistant and behavior change model by splitting the total human population at time t, denoted by n(t), into six mutuallyexclusive compartments of susceptible individuals s(t), slow progressor hiv-1 infected class i1(t), non progressor hiv-1 infected class i2(t), fast progressor hiv-1 infected class i3(t), behavior change class i4 and aids class a such that n(t) = s(t) +i1(t) +i2(t) +i3(t) +i4(t) +a(t). it is worth noting that the aids class consists of weak and unhealthy infected individuals that are assumed to be sexually inactive. sexually active individuals are recruited into the susceptible population at a constant rate b. the susceptible individuals acquire the virus through effective contact with an hiv-1 positive and infectious individuals at the rate λ given by λ = β(i3 + σ1i1 + σ2i2 + σ3i4) n , (1) where β in (1) denotes the effective contact rate that is capable of leading to infection, 0 ≤ σ1 ≤ 1 denotes the modification parameter that account for the assumed reduction in the transmission of virus by the slow progressor hiv-1 infected class i1 in comparison to the fast progressor hiv-1 infected individuals in i3, 0 ≤ σ2,σ3 ≤ 1 are the modification parameters accounting for the assumed reduction of infectiousness by i2 and i4 classes in comparison to the slow-progressor and fast progressor classes i1 and i3 respectively. so that σ3 < σ2 < σ1 < 1, σ3 ≥ 0. (2) the acquisition of infection by the slow progressor hiv-1 infected individuals i1 occur at the rate α1λ, that of i2 occur at the rate α2λ and that of i3 at the rate α3λ. natural death occur constantly to anybody at the rate µ and rate of progression from i1 to aids class a at the rate ρ1. therefore, the rate of change of the total population of the susceptible and and slow progressor classes is respectively given by ṡ(t) =b − (α1 + α2 + α3)λs −µs, i̇1(t) =α1λs −ρ1i1 −µi1, where · represents derivative with respect to time. the non-progressor hiv-1 infected class is generated by the break-through of infection of susceptible class at the rate α2λ, total abstinence due to behavior change at the rate γ1, partial abstinence from i4 due to behavior change at the rate γ2 and natural death at the rate µ so that we have i̇2(t) = α2λs −γ1i2 + γ2i4 −µi2. similarly, we compose the fast progressor class by the break-through of infection of the susceptible class at the rate α3λ, aids acquisition at the rate ρ2 so that the class is given by i̇3(t) = α3λs −ρ2i3 −µi3. the behavior change class is formulated through the total abstinence of non progressors at the rate γ1 and partial abstinence at the rate γ2 given by i̇4(t) = γ1i2 −γ2i4 −µi4. while incorporating the behavior change in the model, we deliberately focused on the behavior change of the non-progressors hiv-1 infected individuals even though, it is imperative that all individuals can change their behavior at any given time. this is because this class of individuals are the most dangerous class just that they won’t show biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 page 4 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 r. musa, r. willie, n. parumasur, analysis of a virus-resistant hiv-1 model with behavior change in ... fig. 1. flow chart of the model. any sign of aids. and finally, the aids class is given by ȧ(t) = ρ1i1 + ρ2i3 − (µ + τ)a, where τ is the aids-induced death rate. since progression are not the same, we have α3 > α1 > α2, α1 + α2 + α3 = 1 (3) where 0 < α1, α2, α3 < 1. the resultant mathematical model for the transmission dynamics of hiv-1 incorporating virus resistance and behavior change through partial and total abstinence using a set of non-linear autonomous set of differential equations is given by: ds dt =b − (α1 + α2 + α3)λs −µs, (4) di1 dt =α1λs −k1i1, (5) di2 dt =α2λs + γ2i4 −k2i2, (6) di3 dt =α3λs −k3i3, (7) di4 dt =γ1i2 −k4i4, (8) da dt =ρ1i1 + ρ2i3 −k5a, (9) where k1 = ρ1 + µ,k2 = γ1 + µ,k3 = ρ2 + µ, k4 = γ2 + µ,k5 = µ + τ, with initial condition s(0) > 0, i1(0) > 0, i2(0) > 0, i3(0) > 0, i4(0) > 0, a(0) > 0. (10) the flow chart of this model is given in figure 1. iii. basic properties of the model since the model is a dynamical system, it it is therefore imperative to ensure that it is biologically meaningful through the establishment of its positivity solution and boundedness at all time t ≥ 0. a. positivity and boundedness of the model. lemma iii.1. the closed set γ= { (s,i1,i2,i3,i4,a)∈r6+|s+i1+...+i4+a≤ b µ } is attracting and positively invariant with respect to the model equation (4)-(9). proof: from (4), we define an integrating factor as ξ(t) = exp {∫ t o [µ + (α1 + α2 + α3)λ(η)]dη } , biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 page 5 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 r. musa, r. willie, n. parumasur, analysis of a virus-resistant hiv-1 model with behavior change in ... where λ(η) = λ(i1,i2,i3,i4). so that the solution of (4) is given by s(t)ξ(t) = b ∫ t o ξ(t)dt, which can be re-written as s(t) exp {∫ t o [µ+(α1 +α2 +α3)λ(η)]dη } = s(0) +b ∫ t o [ exp {∫ s o [µ+(α1 +α2 +α3)λ(η)]dη }] ds, which implies s(t) exp { µt+ ∫ t o (α1 +α2 +α3)λ(η)dη } = s(0) +b ∫ t o [ exp { µs+ ∫ s o (α1 +α2 +α3)λ(η)dη }] ds so that s(t) =b ∫ t o [ exp { µs+ ∫ s o (α1 +α2 +α3)λ(η)dη }] ds × exp { −µt− ∫ t o (α1 +α2 +α3)λ(η)dη } + s(0) exp { −µt− ∫ t o (α1 +α2 +α3)λ(η)dη } , where s(0) is an initial condition for s(t) and hence it is a constant. this expression guarantees the positivity of the state variable s(t) under the condition that s(0) > 0 which consequently ensures the positivity of i1(t), i2(t), i3(t), i4(t) and a(t) provided that (10) is satisfied for all time t ≥ 0. furthermore, addition of (4)-(9) gives dn(t) dt = b −µn(t) − τaw� (11) dn(t) dt ≤ b −µn(t), whose solution is n(t) ≤ b µ + [ n(0) − b µ ] exp (−µt), (12) lim t→∞ n(t) ≤ b µ + lim t→∞ [ n(0)− b µ ] exp (−µt) = b µ . this shows the boundedness of the solution above by b µ in the domain defined by the provision of lemma iii.1. therefore, the model is epidemically well-posed and mathematically meaningful since all the state variables are non-negative for all t ≥ 0. hence, it is sufficient to study and analyze the model in γ [26], [27]. this completes the proof. iv. analysis of the model a. analysis of the model with total abstinence of non-progressors here, we analyze the model for non-progressors that change their behavior through total abstinence from all means of contracting hiv-1 and from all hiv-1 endemic environments i.e. γ2 = 0,σ3 = 0 so that equation (4)-(9) becomes ds dt =b − (α1 + α2 + α3)λ1s −µs, (13) di1 dt =α1λ1s −k1i1, (14) di2 dt =α2λ1s −k2i2, (15) di3 dt =α3λ1s −k3i3, (16) di4 dt =γ1i2 −µi4, (17) da dt =ρ1i1 + ρ2i3 −k5a, (18) where λ1 = β(i3 + σ1i1 + σ2i2) n . (19) all model parameters are positive. b. local stability of disease-free equilibrium (dfe) the disease-free equilibrium of (13)-(18) is given by ψ∗1 = (s ∗,i∗1,i ∗ 2,i ∗ 3,i ∗ 4,a ∗) = ( b µ , 0, 0, 0, 0, 0 ) . (20) this shows that n∗ = s∗ = b µ and s∗ n∗ = 1 biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 page 6 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 r. musa, r. willie, n. parumasur, analysis of a virus-resistant hiv-1 model with behavior change in ... at disease-free equilibrium point ψ∗1. by employing the next generation method [7], [34], f1 (the new infection terms) and v1 (transfer terms) are expressed as f1 =   α1σ1β βσ2α1 βα1 0 0 α2σ1β βσ2α2 βα2 0 0 α3σ1β βσ2α3 βα3 0 0 0 0 0 0 0 0 0 0 0 0   , v1 =   k1 0 0 0 0 0 k2 0 0 0 0 0 k3 0 0 0 −γ1 0 µ 0 −ρ1 0 −ρ2 0 k5   . taking ρ as the spectral radius (magnitude of the dominate eigenvalue) of the next generation matrix f1v−11 , the reproduction number is given by r ′ t = β(α1k2k3σ1 + α2k1k3σ2 + α3k1k2) k1k2k3 . (21) the quantity r ′ t represents the measure of average number of new virus infection of hiv-1 developed by a single hiv-1 infected individual in a population where there are people who practice total abstinence and are completely susceptible. hence, we present the following lemma. lemma iv.1. the dfe of the reduced model (13)(18) with total abstinence is locally asymptotically stable (las) if r ′ t < 1, and unstable if r ′ t > 1. the proof is standard and can be established using theorem 2 of [34]. c. existence of endemic equilibrium the reduced model with total abstinence has a unique positive endemic equilibrium point (eep). this is the point where at least one of the virus infected compartments is non-zero. let ψ∗∗1 = (s ∗∗,i∗∗1 ,i ∗∗ 2 ,i ∗∗ 3 ,i ∗∗ 4 ,a ∗∗) (22) be the endemic equilibrium point. we further define the force of infection as λ∗∗1 = β(i∗∗3 + σ1i ∗∗ 1 + σ2i ∗∗ 2 ) n∗∗ . (23) solving equation (13)-(18) in terms of the force of infection λ∗∗1 at steady-state gives: s∗∗= b µ + (α1 + α2 + α3)λ ∗∗ 1 , i∗∗1 = α1bλ ∗∗ 1 k1[µ + (α1 + α2 + α3)λ ∗∗ 1 ] , i∗∗3 = bλ∗∗1 k3[µ + (α1 + α2 + α3)λ ∗∗ 1 ] , i∗∗4 = γ1bα2λ ∗∗ 1 k2µ[µ + (α1 + α2 + α3)λ ∗∗ 1 ] , (24) a∗∗= bλ∗∗1 (ρ1α1k3 + k1ρ2) k1k3k5[µ + (α1 + α2 + α3)λ ∗∗ 1 ] , i∗∗2 = α2bλ ∗∗ 1 k2[µ + (α1 + α2 + α3)λ ∗∗ 1 ] , n∗∗= bk1k3k5f1−τbλ∗∗1 (ρ1α1k3 +k1ρ2) µk1k3k5f1 , where f1 = µ + (α1 + α2 + α3)λ∗∗1 . substituting all the equations in (24) into (23), it can be shown that the non-zero equilibria of the model satisfy the following linear equation in terms of λ∗∗1 : aoλ ∗∗ 1 + a1 = 0, (25) where ao = α1µk2k3(µ + τ + ρ1) + k1k2[µα3(ρ2 + µ + τ) + k3k5α2], (26) a1 = µk1k2k3k5(1 −r ′ t ). (27) clearly, ao > 0, a1 ≥ 0 if and only if r ′ t ≤ 1 so that λ∗∗1 = − a1 ao ≤ 0. this shows that no existence of positive endemic equilibrium whenever r ′ t ≤ 1. hence, the endemic equilibrium point ψ∗∗1 exists and unique whenever r ′ t > 1. we claim the following result. lemma iv.2. the endemic equilibrium point (eep) of the reduced model (13)-(18) with total abstinence is locally asymptotically stable (las) if r ′ t > 1. d. global stability of dfe to establish the global stability of dfe points, we adopt the approach of [5] to re express (13)(18) in the following vector form ẋ = l(x,y ), (28) biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 page 7 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 r. musa, r. willie, n. parumasur, analysis of a virus-resistant hiv-1 model with behavior change in ... ẏ = m(x,y ),m(x, 0) = 0, (29) where the vector x = (s) denotes the hiv-1 uninfected compartment of the system and y = (i1,i2,i3,i4,a) ∈ r5+ represents the hiv1 infected compartments. using the dfe point to establish the stability analysis, the following two conditions must be satisfied: n1 : for ẋ(t) = l(xo, 0), xo is globally asymptotically stable. n2 : m(x,y ) = jy −m̂(x,y ), m̂(x,y ) ≥ 0 for x,y ∈ ωm where j = ∂m∂y (x o, 0). for this analysis, the expressions for j1,y1,m̂1 and m1 are for the reduced model with the same definition as above while expressions for j,y,m̂ and m are for the full model with the same definition. from our model equation, we obtain the jacobian matrix of only the infected compartment at dfe as follows: j1 =  α1σ1βs ∗ n∗ −k1 βσ2α1s ∗ n∗ βα1s ∗ n∗ 0 0 α2σ1βs ∗ n∗ α2σ2βs ∗ n∗ −k2 βα2s ∗ n∗ 0 0 α3σ1βs ∗ n∗ α3σ2βs ∗ n∗ βα3s ∗ n∗ −k3 0 0 0 γ1 0 −µ 0 ρ1 0 ρ2 0 −k5   j1y1 =j1   i1 i2 i3 i4 a  =   βα1(σ1i1+σ2i2+i3)s ∗ n∗ −k1i1 βα2(σ1i1+σ2i2+i3)s ∗ n∗ −k2i2 βα3(σ1i1+σ2i2+i3)s ∗ n∗ −k3i3 γ1i2 −µi4 ρ1i1 + ρ2i3 −k5a   m̂1(x,y ) = j1y1 −m1(x,y ) ≥ 0 where m1(x,y ) =   βα1(σ1i1+σ2i2+i3)s n −k1i1 βα2(σ1i1+σ2i2+i3)s n −k2i2 βα3(σ1i1+σ2i2+i3)s n −k3i3 γ1i2 −µi4 ρ1i1 + ρ2i3 −k5a   and m̂1(x,y ) =   βα1(σ1i1 +σ2i2 +i3) ( 1− s n ) βα2(σ1i1 +σ2i2 +i3) ( 1− s n ) βα3(σ1i1 +σ2i2 +i3) ( 1− s n ) 0 0   , since s ≤ n, this shows that m̂1(x,y ) ≥ 0. it can be seen that limt→∞x(t) = xo and j is an m-matrix, thus xo is globally asymptotically stable, hence, n1 is satisfied. also, m̂1(x,y ) ≥ 0 for (x,y ) ∈ ωm. hence, n2 is satisfied and eo is globally asymptotically stable whenever r ′ t < 1. e. global stability of endemic equilibrium point following the provision of lemma iv.2, we establish the following theorem. theorem iv.3. the endemic equilibrium point of the reduced model (13)-(18) is globally asymptotically stable (gas) whenever r ′ t > 1. proof: using the idea of [1], we construct the lyapunov function: b = 6∑ k=1 akbk, ak > 0, (30) where ak is a constant and bk is given by bk = ∫ f f∗∗k ( 1 − f∗∗k x ) dx, (31) for f∗∗k ∈ w = {s,i1,i2,i3,i4,a} , where k = 1, 2, 3, 4, 5, 6. this vividly shows that bk is positive definite, continuous and differentiable in γ. hence, bk ∈ c ′ [γ,r+]. differentiating b partially with respect to each fk we have ∂b ∂fk = ak ( 1 − f∗∗k fk ) , (32) so that ∂b ∂fk = 0 =⇒ ak ( 1 − f∗∗k fk ) = 0. biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 page 8 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 r. musa, r. willie, n. parumasur, analysis of a virus-resistant hiv-1 model with behavior change in ... differentiating (32) again partially with respect to each fk gives ∂2b ∂f2k = akf ∗∗ k f2k , k = 1, ..., 6. (33) from (32), if fk = f∗∗k , then s = s ∗∗,i1 = i∗∗1 ,i2 = i ∗∗ 2 ,i3 = i ∗∗ 3 ,i4 = i ∗∗ 4 ,a = a ∗∗. this clearly shows that the endemic equilibrium point is the only stationary point of b. since (30) is always positive, it means that the endemic equilibrium is a global minimum point of the function b for all fk ∈ γ ⊆ r6+. next is to establish that the function b is a lyapunov function which can be done by proving that b is negative definite. the time derivative of b is given by db dt = 6∑ k=1 ak ( 1 − f∗∗k fk ) ḟk, (34) which is negative definite for all time t > 0. it is worth noting here that for all f∗∗k ∈ γ, ḟk ≤ ṅ which makes equation (34) to be db dt ≤ 6∑ k=1 ak ( 1 − f∗∗k fk ) ṅ. (35) from equation (12), we obtain the derivative dn dt = µ ( b µ −n(0) ) exp(−µt). (36) substituting (36) in (35), we have db dt ≤ 6∑ k=1 ak ( 1− f∗∗k fk ) µ ( b µ −n(0) ) exp(−µt). (37) when t → ∞, db dt ≤ 0 which means that the total initial population n(0) is within the basin γ i.e. n(0) ≤ b µ . also when the initial population is outside the basin of attraction i.e. n(0) ≥ b µ as t → ∞, then db dt ≤ 0 and hence, the righthand side of (37) is negative definite. this proves that irrespective of the size of the initial population n(0), the left hand side is always less or equal to zero as t > 0. this consequently clarifies that the constructed function b is a lyapunov type and can be used to establish the global stability of the system. moreover, db dt = 0 if and only if s = s∗∗,i1 = i ∗∗ 1 ,i1 = i ∗∗ 1 ,i2 = i ∗∗ 2 , i3 = i ∗∗ 3 ,i4 = i ∗∗ 4 ,a = a ∗∗, and the largest positive invariant subset of γ that satisfies db dt = 0 is the singleton ψ∗∗1 . hence, ψ ∗∗ 1 is a unique endemic equilibrium point of the system (13)-(18) which is gas in γ. f. analysis of the full model g. local stability of dfe in this section, we shall analyze the full model just as we did for the sub-model in the previous section. it is worth noting that the full model has the same dfe as the sub-model given by equation (20) which exists in the same region γ. we employ the same next generation matrix to establish the reproduction number as follows: f =   α1σ1β βσ2α1 βα1 βα1σ3 0 α2σ1β βσ2α2 βα2 βα2σ3 0 α3σ1β βσ2α3 βα3 βα3σ3 0 0 0 0 0 0 0 0 0 0 0   , v =   k1 0 0 0 0 0 k2 0 −γ2 0 0 0 k3 0 0 0 −γ1 0 k4 0 −ρ1 0 −ρ2 0 k5   . taking ρ as the spectral radius (magnitude of the dominate eigenvalue) of the next generation matrix fv−1, the reproduction number is given by rt = [ p + q k1k3(k2k4−γ1γ2) ] , (38) where p = (k2k4−γ1γ2)(α1k3σ1 +α3k1), q = α2k1k3(γ1σ3 +k4σ2). lemma iv.4. the disease-free equilibrium point (dfe) of the full model (4)-(9) with partial abstinence is locally asymptotically stable (las) if rt < 1 and unstable otherwise. biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 page 9 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 r. musa, r. willie, n. parumasur, analysis of a virus-resistant hiv-1 model with behavior change in ... h. existence of endemic equilibrium the full model with partial abstinence has a unique positive endemic equilibrium point (eep). this is the point where at least one of the virus infected compartments is non-zero. let ψ∗∗ = (s∗∗,i∗∗1 ,i ∗∗ 2 ,i ∗∗ 3 ,i ∗∗ 4 ,a ∗∗) (39) be the endemic equilibrium point. we further define the force of infection as λ∗∗ = β(i∗∗3 + σ1i ∗∗ 1 + σ2i ∗∗ 2 + σ3i ∗∗ 4 ) n∗∗ . (40) solving equation (4)-(9) in terms of the force of infection λ∗∗ at steady-state we obtain: s∗∗ = b µ + (α1 + α2 + α3)λ∗∗ , i∗∗1 = α1bλ ∗∗ k1[µ + (α1 + α2 + α3)λ∗∗] , i∗∗3 = bλ∗∗ k3[µ + (α1 + α2 + α3)λ∗∗] , (41) a∗∗ = bλ∗∗(ρ1α1k3 + k1ρ2) k1k3k5[µ + (α1 + α2 + α3)λ∗∗] , i∗∗2 = α2bλ ∗∗k4 f1(k2k4 −γ1γ2) , n∗∗= bk1k3k5f1−τbλ∗∗(ρ1α1k3 +k1ρ2) µk1k3k5f1 , i∗∗4 = γ1bα2λ ∗∗k4 k4[µ+(α1 +α2 +α3)λ∗∗](k2k4−γ1γ2) , where f1 = µ + (α1 + α2 + α3)λ∗∗. substituting all the equations in (41) into (40), it can be shown that the non-zero equilibria of the model satisfy the following linear equation in terms of λ∗∗: a2λ ∗∗ + a3 = 0, (42) where a2 =α3µk1(ρ2 +µ+τ)+α1k3µ(µ+ρ1 +τ) + k1k3k5α2 > 0 (43) a3 =µk1k3k5(1 −rt ). (44) clearly, a2 > 0, a3 ≥ 0 if and only if rt ≤ 1 so that λ∗∗ = −a3 a2 ≤ 0 which shows no existence of positive endemic equilibrium whenever rt ≤ 1. hence, the endemic equilibrium point ψ∗∗ exists and unique whenever rt > 1. we claim the following result. lemma iv.5. the endemic equilibrium point (eep) of the full model (4)-(9) with partial abstinence is locally asymptotically stable (las) if rt > 1. i. global stability of dfe of the full model we will establish the proof using the same approach as in section iv.c as follows: m̂(x,y ) = jy −m(x,y ) =   βα1(σ1i1 +σ2i2 +i3 +σ3i4)s ∗ n∗ −k1i1 βα2(σ1i1+σ2i2+i3+σ3i4)s ∗ n∗ −k2i2 +γ2i4 β(σ1i1 +σ2i2 +i3 +σ3i4)s ∗ n∗ −k3i3 γ1i2 −k4i4 ρ1i1 + ρ2i3 −k5a   −   βα1(σ1i1 +σ2i2 +i3 +σ3i4)s ∗ n∗ −k1i1 βα2(σ1i1+σ2i2+i3 +σ3i4)s ∗ n∗ −k2i2 +γ2i4 β(σ1i1 +σ2i2 +i3 +σ3i4)s ∗ n∗ −k3i3 γ1i2 −k4i4 ρ1i1 + ρ2i3 −k5a   =   βα1(σ1i1 +σ2i2 +i3 +σ3i4) ( 1− s n ) βα2(σ1i1 +σ2i2 +i3 +σ3i4) ( 1− s n ) βα3(σ1i1 +σ2i2 +i3 +σ3i4) ( 1− s n ) 0 0   ≥ 0, where s ∗ n∗ ≤ 1 at dfe and since s ≤ n, this shows that m̂(x,y ) ≥ 0. it can be seen that limt→∞x(t) = x o and j is an m-matrix, thus xo is globally asymptotically stable, hence, n1 is satisfied. also, m̂(x,y ) ≥ 0 for (x,y ) ∈ ωm. hence, n2 is satisfied and eo is globally asymptotically stable whenever rt < 1. biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 page 10 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 r. musa, r. willie, n. parumasur, analysis of a virus-resistant hiv-1 model with behavior change in ... j. global stability of the endemic equilibrium we consider the special case where the virusinduced death rate τ is negligible. this is very much realistic since hiv-1 positive individuals under treatment can live many years hail and healthy without dying of the virus. substituting τ = 0 into (11), as t →∞ gives n → b µ . putting this in equation (1), we have λ = β1(i3 + σ1i1 + σ2i2 + σ3i4) (45) where β1 = βµ b . theorem iv.6. the endemic equilibrium point of the full model (4)-(9) is globally asymptotically stable (gas) whenever rt > 1. proof: since lemma iv.5 has already been established, we construct the following non-linear lyapunov function for the system (4)-(9) as follows: l = β1 f2µ ∫ s s∗ ( 1− s∗ x ) dx+ 1 α1α2 ∫ i1 i∗1 ( 1− i∗1 x ) dx + γ2 α2 ∫ i2 i∗2 ( 1− i∗2 x ) dx+ α1 β1α3 ∫ i3 i∗3 ( 1− i∗3 x ) dx + β21 k2k3γ1 ∫ i4 i∗4 ( 1− i∗4 x ) dx+ 1 ρ1ρ2 ∫ a a∗ ( 1− a∗ x ) dx. the derivative of l along the solution of the system (4)-(9) is given by l̇ = β1 f2µ ( 1 − s∗ s ) ṡ + 1 α1α2 ( 1 − i∗1 i1 ) i̇1 + γ2 α2 ( 1 − i∗2 i2 ) i̇2 + α1 β1α3 ( 1 − i∗3 i3 ) i̇3 + β21 k2k3γ1 ( 1 − i∗4 i4 ) i̇4 + 1 ρ1ρ2 ( 1 − a∗ a ) ȧ. using (4)-(9), we have l̇= β1 f2µ [ b−(f2λ+µ)s− s∗∗ s {b−(f2λ+µ)s} ] + 1 α1α2 [ α1λs−k1i1− i∗∗1 i1 {α1λs−k1i1} ] + γ2 α2 [ α2λs+γ2i4−k2i2− i∗∗2 i2 {α2λs+γ2i4−k2i2} ] + α1 β1α3 [ α3λs −k3i3 − i∗∗3 i3 {α3λs −k3i3} ] + β21 k2k3γ1 [ γ1i2−k4i4− i∗∗4 i4 {γ1i2−k4i4} ] + 1 ρ1ρ2 [ ρ1i1 + ρ2i3 −k5a − a∗∗ a {ρ1i1 + ρ2i3 −k5a} ] , (46) where f2 = α1 +α2 +α3. at endemic equilibrium point of (4)-(9), we have the following expressions. b = µs∗∗+f2(i ∗∗ 3 +σ1i ∗∗ 1 +σ2i ∗∗ 2 +σ3i ∗∗ 4 )s ∗∗, k1 = α1β1(i ∗∗ 3 + σ1i ∗∗ 1 + σ2i ∗∗ 2 + σ3i ∗∗ 4 )s ∗∗ i∗∗1 , k2 = γ2i ∗∗ 4 +α2β1(i ∗∗ 3 +σ1i ∗∗ 1 +σ2i ∗∗ 2 +σ3i ∗∗ 4 )s ∗∗ i∗∗2 , k3 = α3β1(i ∗∗ 3 + σ1i ∗∗ 1 + σ2i ∗∗ 2 + σ3i ∗∗ 4 )s ∗∗ i∗∗3 , k4 = γ1i ∗∗ 2 i∗∗4 , k5 = ρ1i ∗∗ 1 + ρ2i ∗∗ 3 a∗∗ (47) substituting expressions in (47) into (46), after some simplifications and factorization, we have l̇ = β1s ∗∗ f2 ( 2 − s s∗∗ − s∗∗ s ) + β21 µ (i∗∗3 +σ1i ∗∗ 1 +σ2i ∗∗ 2 +σ3i ∗∗ 4 )s ∗∗ ( 2− s s∗∗ − s∗∗ s ) + β1 α2 [ (i∗∗3 +σ1i ∗∗ 1 + σ2i ∗∗ 2 +σ3i ∗∗ 4 )s ∗∗ ( 2− i1 i∗∗1 − i∗∗1 i1 ) + (i3+σ1i1+σ2i2+σ3i4)s ( 2− i1 i∗∗1 − i∗∗1 i1 )] + ( 1 ρ2 + 1 ρ1 )( 2 − a a∗∗ − a∗∗ a ) biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 page 11 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 r. musa, r. willie, n. parumasur, analysis of a virus-resistant hiv-1 model with behavior change in ... +γ2 [ β1(i3+σ1i1+σ2i2+σ3i4)s ( 2− i2 i∗∗2 − i∗∗2 i2 ) + β1(i ∗∗ 3 +σ1i ∗∗ 1 +σ2i ∗∗ 2 +σ3i ∗∗ 4 )s ∗∗ ( 2− i2 i∗∗2 − i∗∗2 i2 )] + ( γ22 α2 + α1β 2 1 k2k3 ) i22i ∗∗ 3 i∗∗4 ( 3− i∗∗2 i2 − i2i ∗∗ 3 i∗∗2 i3 − i∗∗2 i3i4 i2i ∗∗ 3 i ∗∗ 4 ) . consequently, since the arithmetic mean exceeds the geometric mean, then we have 2 − s s∗∗ − s∗∗ s ≤ 0, 2 − a a∗∗ − a∗∗ a ≤ 0, 2 − i1 i∗∗1 − i∗∗1 i1 ≤ 0, 2 − i2 i∗∗2 − i∗∗2 i2 ≤ 0, 3 − i∗∗2 i2 − i2i ∗∗ 3 i∗∗2 i3 − i∗∗2 i3i4 i2i ∗∗ 3 i ∗∗ 4 ≤ 0. since s ≥ 0,i1 ≥ 0,i2 ≥ 0,i3 ≥ 0,i4 ≥ 0,a ≥ 0 and lemma iv.5 is satisfied, it follows that l̇ ≤ 0 since all other model parameters are non-negative for rt > 1. furthermore, l̇ = 0 if and only if s = s∗∗,i1 = i∗∗1 ,i2 = i ∗∗ 2 ,i3 = i∗∗3 ,i4 = i ∗∗ 4 ,a = a ∗∗. thus, l is a lyapunov function of the subsystem (4)-(9) on γ. it therefore follows by lasalle’s invariance principle [16] that the subsystem (4)-(9) has a globally asymptotically stable endemic equilibrium point ψ∗∗. the result presented here shows that for a special case (τ = 0), the virus will consistently persist in the community whenever the associated reproduction number rt > 1. v. numerical simulation and discussion of results in this section, we shall carry out the numerical simulation of the model to corroborate the analytic results. we shall solve the model equation (4)(9) numerically and present the results graphically using maple 18 and python mathematical software. a 3d surface plot shall also be presented to examine the relationship between the reproduction number, the partial abstinence rate γ2 and σ3 which is the modification parameter which account for the assumed reduction of infectiousness by the behavior change class i4. table 1: hypothetical value of parameters parameter value (per year) source b 5600 estimated α1 0.25 estimated α2 0.10 estimated β 0.015 estimated µ 0.016 estimated α3 0.65 estimated ρ1 0.12 estimated γ1 1.00 [30], [20] γ2 0.95 estimated ρ2 0.75 estimated τ 0.0909 [11] σ1 0.85 estimated σ2 0.55 estimated σ3 0.008 [29] table 2: initial conditions s(0) i1(0) i2(0) i3(0) i4(0) a(0) 450 10 8 5 10 15 400 40 25 20 10 5 300 70 50 40 30 10 200 95 65 53 47 40 100 120 88 67 65 60 to start with, we will show numerically that the disease-free equilibrium ψ∗ is locally asymptotically stable. the parameter values presented in table 1 and the initial conditions shown in table 2 shall be used. considering the case when the reproduction number is less than unity i.e. rt = 0.025 < 1, the graphical solution of model equation (4)-(9) is given in fig.2 fig.7. it can be seen that only the susceptible population s = 500 survive while the infected population in the slow progression class i1, non progression class i2, fast progression class i3, behavior change class i4 and aids class a goes into extinction. this confirms that the dfe of (4)-(9) as presented in lemma (iv.4) is locally asymptotically stable whenever rt < 1. biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 page 12 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 r. musa, r. willie, n. parumasur, analysis of a virus-resistant hiv-1 model with behavior change in ... fig. 2 and fig. 3 showing the behavior of both susceptible and slow progression populations when rt is less than unity. fig. 4 and fig. 5 showing the behavior of both non progression and fast progression populations when rt is less than unity. fig. 6 and fig. 7 showing the behavior of both fast progression and aids populations when rt is less than unity. biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 page 13 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 r. musa, r. willie, n. parumasur, analysis of a virus-resistant hiv-1 model with behavior change in ... fig. 8 and fig. 9 showing the behavior of both susceptible and slow progression populations when rt is greater than unity. fig. 10 and fig. 11 showing the behavior of both non progression and fast progression populations when rt is greater than unity. fig. 12 and fig. 13 showing the behavior of both fast progression and aids populations when rt is greater than unity. biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 page 14 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 r. musa, r. willie, n. parumasur, analysis of a virus-resistant hiv-1 model with behavior change in ... considering the case when the reproduction number is greater than unity i.e. rt = 5.903 > 1, the graphical solution of model equation (4)-(9) is given in fig.8 fig.13. it can be observed that when the reproduction number rt > 1, all the populations survive that’s [s∗∗,i∗∗1 ,i ∗∗ 2 ,i ∗∗ 3 ,i ∗∗ 4 ,a ∗∗] = [0.0155, 0.6575, 0.9452, 0.8105, 0.5421, 0.7813], which clearly indicates that the population converges or tends to the endemic equilibrium points ψ∗∗ whenever rt > 1. it can also be seen that the susceptible population reduces drastically because of the reproduction number being greater than unity while all the remaining infected populations increases with time. this confirms that the endemic equilibrium points ψ∗∗ is locally asymptotically stable and thus, confirms the analytic results presented in lemma iv.5. a. effect of partial and total abstinence in hiv/aids transmission here, we will observe the effect of partial and total abstinence in the transmission of hiv/aids. we simulate the reproduction number in equation (21) and (38). for the case of partial abstinence (i.e. when σ3 = γ2 6= 0), with the parameter values in table 1, when γ2 = 0.95 and σ3 = 0.008, rt = 0.025. for the case of total abstinence (i.e. when σ3 = γ2 = 0), with the parameter values in table 1, r ′ t = 0.020 meaning that r ′ t for the total abstinence is less than rt for the partial abstinence. since our aim in epidemiology is to find all possible means to reduce the reproduction number of infectious disease, it means that those that changed their sexual attitude through total abstinence from hiv/aids and all factors that can cause its transmission are at lower or no risk of contacting hiv/aids. hence, total abstinence is one of the key factors to be safe from hiv/aids. figure 14 below shows a 3d surface plot to understand more about the relationship between the reproduction number and γ2 and σ3. we can easily observe that the higher the value of both σ3 and γ2, the higher the reproduction number and the lower their values the lower the reproduction number. the lowest reproduction number 0.018 is gotten when σ3 = γ2 = 0 i.e. (total abstinence). hence, total abstinence is essential in the protection against hiv/aids transmission. vi. conclusion, acknowledgment and disclosure statement in this study, a new virus resistant hiv-1 model with behavior change was proposed and systematically analyzed for both partial and total abstinence from hiv/aids. basic analysis of the model such as positivity solution, reproduction number, invariant region, establishment of both disease-free and endemic equilibrium points for both scenarios were carried out. the local asymptotic stability of the dfe and ee for both models whenever the associated reproduction number is less than unity and greater than unity respectively were proved. a non-linear goh–volterra lyapunov function is used to prove that the endemic equilibrium point is globally asymptotically stable for the case when the virus-induced death rate τ = 0 while the method of castillo-chavez is used to prove the global asymptotic stability of the disease-free equilibrium point whenever the reproduction number is less than unity. in the numerical simulation, it was established that people with total abstinence are more protected against hiv/aids than those with partial abstinence and also established that the reproduction number is minimal under this same condition. since those with resistance to hiv/aids do not proceed to the aids compartment, this also highlight the importance of hivresistance which plays an important role in the protection against hiv/aids. acknowledgment the authors really acknowledge and appreciate the efforts of the unknown reviewers. the first author, rabiu musa, acknowledges funding from the nrf and dst of south africa through grant number 48518. disclosure statement the research work forms part of the first authors biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 page 15 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 r. musa, r. willie, n. parumasur, analysis of a virus-resistant hiv-1 model with behavior change in ... fig. 14 showing the effect of σ3 and γ2 on the reproduction number using the parameter values on table 1 when σ3 = γ2 = [0, 1]. phd work and the co-authors are his supervisors. data availability statement the numerical data and hypothetical value of parameters used to support the findings of this research are included within the article. they are either properly referenced, assumed or estimated in table 1. conflict of interest no conflict of interest as far as this research is concerned. references [1] afassinou, k., chirove, f., & govinder, k. s. 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(2013). modeling the spread of hiv-aids with infective immigrants and time delay. international journal of nonlinear science , 16, 313-322. [32] unaids. aids epidemic update. technical report, 2009. www.unaids.org/en/knowledgecentre/hiv\ data/epiupdate/epiupdarchive/2009/default.asp. [33] university of minnesota (2014). ”why some people may be immune to hiv-1: clues.” sciencedaily. sciencedaily, 20 november 2014. www.sciencedaily.com/ releases/2014/11/141120141750.htm. [34] van den driessche, p., & watmough, j. (2002). reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. mathematical biosciences, 180(1-2), 29-48. biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 page 17 of 17 https://www.nature.com/scitable/blog/viruses101/hiv_resistant_mutation/?isforcedmobile=y https://www.nature.com/scitable/blog/viruses101/hiv_resistant_mutation/?isforcedmobile=y https://doi.org/10.1016/j.mcm.2008.09.013 https://doi.org/10.1016/j.mcm.2008.09.013 www.unaids.org/en/knowledgecentre/hiv\data/epiupdate/epiupdarchive/2009/default.asp www.unaids.org/en/knowledgecentre/hiv\data/epiupdate/epiupdarchive/2009/default.asp www.sciencedaily.com/releases/2014/11/141120141750.htm www.sciencedaily.com/releases/2014/11/141120141750.htm http://dx.doi.org/10.11145/j.biomath.2020.06.143 introduction model formulation and model assumptions basic properties of the model positivity and boundedness of the model. analysis of the model analysis of the model with total abstinence of non-progressors local stability of disease-free equilibrium (dfe) existence of endemic equilibrium global stability of dfe global stability of endemic equilibrium point analysis of the full model local stability of dfe existence of endemic equilibrium global stability of dfe of the full model global stability of the endemic equilibrium numerical simulation and discussion of results effect of partial and total abstinence in hiv/aids transmission conclusion, acknowledgment and disclosure statement references original article biomath 2 (2013), 1307247, 1–14 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum on the distribution of transcription times marc r. roussel alberta rna research and training institute department of chemistry and biochemistry, university of lethbridge lethbridge, alberta, canada email: roussel@uleth.ca received: 22 april 2013, accepted: 24 july 2013, published: 15 september 2013 abstract—a previously studied model of prokaryotic transcription [roussel and zhu, bull. math. biol. 68 (2006) 1681–1713] is revisited. the first four moments of the distribution of transcription times are obtained analytically and analyzed. a gaussian is found to be a poor approximation to this distribution for short transcription units at typical values of the rate constants, but a good approximation for long transcription units. an approximate form of the distribution is obtained in which the slow steps are treated exactly and the fast steps are lumped together into a single lag term. this approximate form might be particularly useful as a function to be fit to experimental transcription time distributions. multipolymerase effects are also studied by simulation. we find that the analytic model generally predicts the behavior of the multi-polymerase simulations, often quantitatively, provided termination is not rate-limiting. keywords-gene transcription; stochastic model i. introduction for many years, proteins were regarded as the “hardware” of the cell, with dna as the “software” [1, p. 111]. the central dogma of molecular biology relegated rna to a secondary role, that of a messenger between the software and hardware layers. ribosomal and transfer rna (rrna and trna) have of course been known for a long time but, being involved in the translation machinery, they were still considered, implicitly at least, to be a means to an end, the end being proteins. in fact, “gene expression” was often explicitly defined to mean the expression of proteins as a result of transcription and translation [2, p. 327]. the roles of messenger rna (mrna), rrna and trna in translation of course continue to be studied, and with just cause, but our appreciation for the versatility of rna has expanded tremendously in recent years with the discovery of the diverse cellular roles of rnas, including catalysis [3], [4], sensing [5], and regulation [6], [7], [8], [9], all roles traditionally believed to be the exclusive province of proteins. with this greater appreciation of rna has come an increased interest in rna synthesis and processing processes. gene transcription has always been of interest because of its role in the expression of proteincoding genes, but now we are equally interested in the transcription of non-coding rnas (ncrnas) [10], [11]. transcription thus claims for itself a larger portion of the stage, with gene expression now including processes in which an rna, rather than a protein, is the “gene product”. in the last few years, several models of the transcription process have appeared [12], [13], [14], [15], [16], [17], [18]. the unifying theme of these models has been a stochastic formulation designed to facilitate a study of the statistical properties of transcription. why should we think of transcription as a stochastic process? there are a number of reasons. first and foremost is that stochasticity is an inescapable property of transcription. if we consider a typical gene in a diploid cell, there are either zero, one or two copies of the gene active at any given time. small molecular populations inevitably lead to large fluctuations, which in this case manifest themselves as transcriptional bursts [19], [20]. the inherent stochasticity of transcription leads us to a second reason to consider stochastic models of this process, namely that citation: marc r. roussel, on the distribution of transcription times, biomath 2 (2013), 1307247, http://dx.doi.org/10.11145/j.biomath.2013.07.247 page 1 of 14 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2013.07.247 m r roussel, on the distribution of transcription times the stochasticity at this level is an important determinant of fluctuations in protein expression levels [21], [22], [23], and these fluctuations can have functional significance [24]. if we want to understand fluctuations in rna or protein levels, or if we want to model a genetic control system in which these fluctuations are likely to be important, then we need a strong understanding of the statistical properties of transcription. if we can, for example, obtain an easily evaluated formula for the transcription time, we can apply delay-stochastic modeling methods [25], [26], [27], [28], [29] to simulate a gene expression system [30], [31]. one of the benefits of a mathematical model, and particularly of one with analytic solutions, is that it is easy to vary parameters. among other benefits, this sometimes allows us to rapidly discover unexpected behaviors, or to provide specific criteria for the observation of some particular phenomenon. in this spirit, we have pursued models that both provide a reasonable cartoon of a biological system, and that are, in some useful limit, analytically solvable [13], [18]. while the analytic solutions are correct only under particular conditions, they often provide vital clues for interesting regimes to be studied by simulation and, ultimately, in the experimental laboratory. in the next section, our model, which was originally presented elsewhere [13], is described, and its first four moments are obtained analytically. in section iii, the moments are analyzed using the analytic expressions obtained in section ii, with particular emphasis on shorter transcripts in which some interesting statistical effects are observed. an approximate form of the distribution of transcription times is also obtained, which involves an analytic expression for the distribution of the slow steps and an empirical lag phase. in section iv, the analytic predictions of this model are compared to stochastic simulations. the concluding section reviews some of the findings and offers some perspectives on this area of research. ii. a prokaryotic transcription model a. model description in our models, the nucleotides of the dna template strand are numbered 1 to n, where n is the length of the transcription unit. conceptually, we track the position of the active site of the rna polymerase. in multipolymerase simulations, a rigid polymerase is assumed (consistent with the lack of “inchworm” movements in transcription [32]), and the leading and trailing edges of the polymerase are located relative to the active 6 u 7 u 8 u 9 u ... 1 u 2 u 3 u 4 o 5 u fig. 1. schematic diagram illustrating the relationship between the polymerase active site and the template strand states. the polymerase is represented by the green rounded rectangle, and its active site by the hooked arrow. here, the active site occupies site 5, while all other sites are “unoccupied” in the sense discussed in the text. this illustration is not intended to realistically portray the geometry of the transcription complex. in particular, a much longer stretch of dna passes through the polymerase than is shown here. site [18]. polymerases can be prevented from overlapping by imposing a minimum distance constraint between the active sites of adjacent polymerases [18], or by adopting a more sophisticated labeling system than the one used here [16]. each site on the dna template strand can be labeled u (unoccupied), o (occupied by the active site of the polymerase), or a (activated for translocation). figure 1 illustrates the relationship between the polymerase and the u and o nucleotide states. the a state can be thought of as a variant of the o state, in the sense that it represents a state in which the polymerase active site occupies a template site and the polymerase has been activated for translocation. we use subscripts to indicate the site to which a given state applies. thus, ui indicates an unoccupied site i, oi indicates an occupied site i, and so on. in the independent polymerase case, we only need to track the location of the active site. any site not occupied by the active site of a polymerase is labeled ui, even though the polymerase covers many sites on the template. as noted above, we can maintain this notation and deal with the multi-polymerase case by adding distance constraints between active sites to our model, as we do in the stochastic simulations in section iv. in the first step of our model, the polymerase locates the transcription start site (tss): rnap + u1 k0−→ o1. (1) this of course is not a single-step process. minimally, it would involve binding of appropriate initiation factors to the promoter, loading of the polymerase, and positioning of the polymerase at the tss, which would typically also involve conformational changes of the dna (e.g. unwinding) [33], [34], [35]. as an initial model however, we assume that this multi-step process has a single ratebiomath 2 (2013), 1307247, http://dx.doi.org/10.11145/j.biomath.2013.07.247 page 2 of 14 http://dx.doi.org/10.11145/j.biomath.2013.07.247 m r roussel, on the distribution of transcription times limiting step, represented as shown above. note also that in a delay stochastic model of gene expression [26], [27], [28], we might separate out the binding of initiation factors and the initial binding of the polymerase to the promoter as steps to be explicitly modeled, particularly since these steps are often regulated [34]. in the present case, we further assume that the local rna polymerase concentration is constant, so that this process reduces to a pseudo-first-order process. these assumptions can easily be relaxed, as discussed later. once the polymerase is positioned, the initiation complex is activated by binding the two nucleotide triphosphates complementary to the first two nucleotides of the expressed sequence. assuming a constant pool of free nucleotides, this process can be represented as the pseudo-first-order reaction o1 k1−→ a1. (2) note that the first two steps are being assigned a slightly different interpretation than in our original paper [13]. this is a conscious choice which will enable us to more easily use the results of our analysis in other contexts, particularly in delay stochastic simulations. consequently, we expect k0 to be smaller than k1, since the former includes the rate-limiting transition from a closed to an open promoter complex [36]. once the polymerase has been activated, formation of the phosphodiester bond between the two bound nucleotides provides the free energy required to drive the polymerase forward, i.e. to translocate: a1 + u2 k′2−→ o2 + u1. (3) although shown as involving two “reactants”, the reaction is represented this way only to maintain the logic of the labeling of sites. it is in fact a first-order process if polymerases are sufficiently widely spaced along the template strand. otherwise, it is still a first-order process, but one whose rate constant is contingent on the position of a downstream polymerase (if any). in the multipolymerase case, we number the polymerases in the order in which they loaded onto the dna, so polymerase 1 is the one furthest downstream, and the polymerase with the highest index is the last one to have loaded. let xj be the current position of the active site of polymerase j. then, k′2 = { k2 if xj−1 −xj > ∆, 0 otherwise. (4) here, k2 is the value of the rate constant for a polymerase that is free to move, and ∆ is the minimum distance between polymerase active sites. after the first translocation, nucleotides are added one at a time in a process that alternates between activation by a nucleotide and translocation driven by phosphodiester bond formation: oi k3−→ ai, i = 2, 3, . . . ,n; (5) ai + ui+1 k′2−→ oi+1 + ui, i = 2, 3, . . . ,n− 1. (6) these two steps together model the elongation phase of transcription. note that we assume here that translocation, whether from the first site (reaction 3) or from subsequent sites (reaction 6), occurs with a common rate constant, an assumption which can easily be relaxed as we discuss below. finally, we model termination as a single-step process although, as with initiation, we could contemplate much more complex models. in our simplified model, we assume that a polymerase becomes activated for termination in much the same way and with a similar rate as in activation for translocation. however, once the polymerase has been activated and the last phosphodiester bond formed, the polymerase, rna and template strand dissociate from each other: an k4−→ rnap + rna + un. (7) many criticisms could be made of this model. however, as a minimal model, it allows us to explore the effects of various steps on the overall statistical properties of transcription. b. model analysis methodology the single-polymerase model is solvable, in the sense that the moments of the distribution of transcription times can be obtained analytically. this case arises when transcription initiation is sufficiently infrequent that polymerases only rarely interact with each other [13]. as pointed out by greive et al. [17], our current model belongs to the class of brownian ratchets, undergoing a set of irreversible transitions from one state to the next. we can obtain the distribution of “jump” times from one site to the next, and then using some straightforward mathematical tricks, obtain the moments of the distribution of total transcription time. we have refined this procedure somewhat since our original publication [13], so we work through the details here. let ρi(τi) be the distribution of jump times (τi) from site i to site i + 1. we adopt the convention that ρ0(τ0) biomath 2 (2013), 1307247, http://dx.doi.org/10.11145/j.biomath.2013.07.247 page 3 of 14 http://dx.doi.org/10.11145/j.biomath.2013.07.247 m r roussel, on the distribution of transcription times is the distribution of initiation times, and ρn(τn) is the distribution of termination times. the jump from site i to site i + 1 is independent of all the other jumps, so the joint probability distribution for the jump times is just the product ρ(τ0,τ1, . . . ,τn) = n∏ i=0 ρi(τi). (8) the total transcription time is defined by τ = n∑ i=0 τi. (9) by a standard theorem of statistics, the distribution of the total transcription time is given by the convolution ρ(τ) = ∫ · · · ∫ ∑ τi=τ ρ(τ0,τ1, . . . ,τn) dτ1 . . .dτn = ∫ · · · ∫ ∑ τi=τ n∏ i=0 ρi(τi) dτ1 . . .dτn. (10) for the class of models considered here, this convolution is not straightforwardly computable except in some special cases. however, the convolution theorem for laplace transforms [37] allows us to convert this problem into a tractable form. let f̃(s) ≡ ls[f(t)], the laplace transform of f(t). then, ρ̃(s) = n∏ i=0 ρ̃i(s). (11) we therefore only need the laplace transforms of the distributions of jump times from one site to the next in order to obtain the laplace transform of the overall distribution of transcription times. the jump distribution laplace transforms are easy to obtain from the laplace transforms of the master equations for the survival of the occupation of a site, as illustrated in the next section. from here, we have two options: 1) the laplace transform can sometimes be numerically (or semi-numerically) inverted to obtain the distribution of transcription times. 2) the moments of the distribution can always be obtained by differentiation of ρ̃(s) [13]. from the definition of the laplace transform, we have ρ̃(s) = ∫ ∞ 0 e−sτρ(τ) dτ. (12) by definition, the moments of the distribution are given by 〈τm〉 = ∫ ∞ 0 τmρ(τ) dτ. (13) taking successive derivatives of the laplace transform, we get 〈τm〉 = (−1)m dmρ̃ dsm ∣∣∣∣ s=0 . (14) differentiation is a simple mechanical operation which, in the worst case, can be carried out reliably by a symbolic algebra system. arbitrarily high moments can be obtained by this method. both approaches are illustrated below. we note here that our models are modular in precisely the same sense used by greive et al. [17] and that, moreover, the analysis has a corresponding modular structure: we can replace any part of the model with one describing different chemistry, and the only effect is to replace the corresponding jump distribution laplace transform in equation (11). thus, studying model variations requires adjustments only to those parts of the analysis directly concerned with the parts of the model that have been changed. for example, rather than assuming that all translocation steps [reactions (3) and (6) in the current model] have a common rate constant, we could assume, for example, that there were different rate constants at each of the first several sites, until the polymerasedna-rna complex had stabilized, after which these rate constants could reach a constant value for the remainder of the transcription process. the cost of such an assumption would be a number of added parameters, and some additional mathematical derivations for the ρ̃i in the region of variable translocation rate constant. c. jump distributions in this section, we work out the laplace transforms of the jump distributions for our model. formally, each jump distribution is the solution of a survival problem [38] for the occupation of a given site by the polymerase active site. in the following work, we denote by pi,σ the probability that site i is in state σ ∈{u, o, a}. because we treat the case of a single polymerase, we do not need to consider joint probabilities (e.g. the probability that site i is in state a while the site to which the leading edge of the polymerase will move is unoccupied). the construction of the master equations for the joint probabilities was discussed in our previous work [13], although extending the methods used here to that case is still very much an open problem. we start with ρ̃0, which is associated with initiation or, in our model, reaction (1). here, the relevant survival problem is the survival time for an unoccupied site 1. biomath 2 (2013), 1307247, http://dx.doi.org/10.11145/j.biomath.2013.07.247 page 4 of 14 http://dx.doi.org/10.11145/j.biomath.2013.07.247 m r roussel, on the distribution of transcription times thus we assume that at t = 0, site 1 is unoccupied and monitor this site until it becomes occupied. under the pseudo-first-order (constant polymerase pool) assumption, the corresponding master equation is dp1,u dt = −k0p1,u, (15a) dp1,o dt = k0p1,u, (15b) with initial conditions p1,u(0) = 1, p1,o(0) = 0. while equation (15b) is apparently redundant, it plays a very important role in the survival problem: since o1 is a sink state in this simplified master equation, p1,o(t) is the cumulative probability distribution for initiation. thus, p1,o(t) = ∫ t 0 ρ0(τ0) dτ0, (16) or, using the fundamental theorem of calculus, ρ0(t) = dp1,o dt . (17) in the space of laplace transforms, and using the identity [37] ls [df/dt] = sf̃(s) −f(0), (18) equation (17) becomes ρ̃0(s) = sp̃1,o(s). (19) the laplace transform of equations (15) subject to the appropriate initial conditions is sp̃1,u − 1 = −k0p̃1,u, (20a) ρ̃0(s) = sp̃1,o = k0p̃1,u. (20b) solving these equations, we get ρ̃0(s) = k0 s + k0 . (21) this is, not surprisingly, the laplace transform of an exponential probability distribution [37]. once the polymerase has reached site 1, the sequence of steps (2) and (3) is required to reach site 2. the relevant master equation is dp1,o dt = −k1p1,o, (22a) dp1,a dt = k1p1,o −k2p1,a, (22b) dp2,o dt = k2p1,a. (22c) here, p2,o is the cumulative probability distribution for the transcription time. the initial conditions used to determine the jump time distribution are p1,o(0) = 1, p1,a(0) = p2,o(0) = 0. taking the laplace transform of these equations, we get sp̃1,o − 1 = −k1p̃1,o, (23a) sp̃1,a = k1p̃1,o −k2p̃1,a, (23b) ρ̃1(s) = sp̃2,o = k2p̃1,a. (23c) the last of these equations follows from the interpretation of p2,o as a cumulative probability distribution for the jump times. solving these equations, we get ρ̃1(s) = k1k2 (s + k1)(s + k2) . (24) the elongation phase, which in this model extends from nucleotides 2 to n − 1, consists of reactions (5) and (6). the algebra required to derive the equation for ρ̃i(s) in this region is of course essentially identical to that required to obtain ρ̃1(s), with the result ρ̃i(s) = k2k3 (s + k2)(s + k3) , i = 2, 3, . . . ,n− 1. (25) termination consists of steps (5) and (7). again, the algebra to be carried out is not too different from the foregoing. we get ρ̃n(s) = k3k4 (s + k3)(s + k4) . (26) if we assemble the laplace transforms of the jump time distributions according to equation (11), we get ρ̃(s) = k0k1k n−1 2 k n−1 3 k4(s + k0) −1(s + k1) −1 × (s + k2)−(n−1)(s + k3)−(n−1)(s + k4)−1. (27) this equation differs from one presented in [13] by the term (21) due to the different interpretation of the two initial reactions mentioned above. d. moments of the distribution using equation (14), for the laplace transform (27), the first two moments about zero work out to 〈τ〉 = 1 k0 + 1 k1 + n− 1 k2 + n− 1 k3 + 1 k4 ; (28) 〈τ2〉 = 〈τ〉2 + 1 k20 + 1 k21 + n− 1 k22 + n− 1 k23 + 1 k24 . (29) higher moments can easily be computed, although their algebraic forms are much more complicated. biomath 2 (2013), 1307247, http://dx.doi.org/10.11145/j.biomath.2013.07.247 page 5 of 14 http://dx.doi.org/10.11145/j.biomath.2013.07.247 m r roussel, on the distribution of transcription times the central moments actually have somewhat simpler forms than the moments about zero: σ2 = 〈τ2〉−〈τ〉2 = 1 k20 + 1 k21 + n− 1 k22 + n− 1 k23 + 1 k24 ; (30) µ3 = 〈τ3〉− 3σ2〈τ〉−〈τ〉3 = 2 ( 1 k30 + 1 k31 + n− 1 k32 + n− 1 k33 + 1 k34 ) ; (31) µ4 = 〈τ4〉− 4µ3〈τ〉− 6σ2〈τ〉2 −〈τ〉4 = 3σ4 + 6 ( 1 k40 + 1 k41 + n− 1 k42 + n− 1 k43 + 1 k44 ) . (32) iii. analytic results a. noise minimization analytic expressions are of course amenable to deeper analysis than simulation results. accordingly, despite the approximations made to obtain these expressions, analysis plays a prominent role in the work of my laboratory on these problems. we can then carry out computational experiments to verify whether the properties discovered by analysis are robust, in particular to interactions between polymerases which are neglected in the foregoing work. one interesting observation we have made repeatedly in our work is that there typically exist combinations of parameters in these models that minimize the variability in the transcription time [18], in the following sense: the coefficient of variation (cv) is defined by cv = σ/〈τ〉. (33) this is a measure of the relative variability in the transcription time, and is therefore one of many measures of transcriptional noise. from equations (28) and (30), we have cv = √ 1 k20 + 1 k21 + n−1 k22 + n−1 k23 + 1 k24 1 k0 + 1 k1 + n−1 k2 + n−1 k3 + 1 k4 . (34) we have observed that the cv frequently displays a minimum when plotted against one or the other of the rate constants [18]. in fact, there is a global minimum value for the cv in the single-polymerase case, as we shall now see. to minimize the cv, we differentiate with respect to each of the ki and set the derivatives equal to zero. in general, from (33), we have ∂(cv) ∂ki = 1 〈τ〉2 ( ∂σ ∂ki 〈τ〉−σ ∂〈τ〉 ∂ki ) , (35) which is obviously zero when ∂σ ∂ki 〈τ〉 = σ ∂〈τ〉 ∂ki . (36) if we evaluate equation (36) for each ki using equations (28) and (30), we get, in each case and after some simplification, σ2 〈τ〉 = 1 ki (37) which, intriguingly, is a condition on the fano factor, another commonly used measure of noise [21], [39]. since each of the ki is equal to 〈τ〉/σ2 at the critical point, then they must be equal to each other. at this critical point (which is in fact a degenerate line in the space of rate constants, parameterized by the value of the common rate constant, corresponding to the choice of time scale), the cv becomes cvmin = (2n + 1) −1/2. (38) an easy if somewhat tedious calculation shows this critical point to be a minimum, hence the label in equation (38). thus we reach the interesting conclusion that the cv is minimized when the rate constants for all the steps of transcription are identical. note that we need not simultaneously optimize with respect to all of the rate constants. for example, if we suppose that the rate constants k0 and k4 are most directly subject to evolutionary pressures, we could minimize with respect to just those two rate constants. equation (37) still holds for i = 0 and 4, so we must have k0 = k4, i.e. the cv reaches a local minimum when the initiation and termination rates are matched. in this case, we calculate k0 = k4 = 1 k1 + n−1 k2 + n−1 k3 1 k21 + n−1 k22 + n−1 k23 . (39) in figure 2, the cv is plotted vs k0 and k4. from equation (39), we calculate that the minimum cv should occur at k0 = k4 = 5.24 s−1 for the parameters used to generate the figure, and this is indeed what we observe. the value of the cv at this minimum is 0.067, which is somewhat larger than the global minimum cv of 0.050 given by equation (38) for the value of n used here. note that the cv surface shown in the figure increases only slowly away from the minimum. thus, cvs approaching the theoretical constrained minimum can be obtained for a wide range of values of the initiation and termination rate constants, provided neither of these is too small. the value of k0 at which the minimum occurs is perhaps a little large [40], but noting the scale of the figure, we see biomath 2 (2013), 1307247, http://dx.doi.org/10.11145/j.biomath.2013.07.247 page 6 of 14 http://dx.doi.org/10.11145/j.biomath.2013.07.247 m r roussel, on the distribution of transcription times 2 4 6 8 10 2 4 6 8 10 0.067 0.068 0.069 0.07 0.071 0.072 cv k 0 /s -1 k 4 /s -1 cv fig. 2. cv plotted vs k0 and k4 with k1 = 0.5 s−1, k2 = 10 s−1, k3 = 10 s −1 and n = 200 nt. table i biologically reasonable ranges for the rate constants k0 10 −2–100 s−1 [36], [40] k1 1 20 k3– 110k3 [36] k2 5–800 s−1 [13] k3 5–800 s−1 [13] k4 ≥ k0 [13] that cvs of the same order of magnitude as the minimum are available within the biologically plausible range (k0 up to about 1 s−1 [40]). if we numerically minimize the cv (using the optimization[minimize] routine in maple 15 [41]) while constraining the rate constants to lie within biologically plausible ranges (table i), we find the minimum value to be cv=0.055 for our small 200-nucleotide transcription unit when k0 = 1.00 s−1, k1 = 0.65 s−1, k2 = 5.00 s −1, k3 = 6.51 s−1 and k4 = 4.55 s−1, which is very close to the global minimum. selective pressures favoring reasonably constant intervals between rna syntheses would therefore tend to favor rapid initiation (i.e. efficient promoters) but slow elongation. the question then occurs of whether such pressures exist for any transcription units. in the foregoing and in much of the rest of this paper, we focus on small values of n. the main reason for doing so is that many of the statistical properties studied here assume interesting extreme values in this regime. small sequences are of biological interest since regulatory rnas and other ncrnas are often transcribed from small transcription units [8]. however, we will examine the effect of varying n as appropriate. for example, we can consider the effect of n on the cv of the transcription time. at large n, equation (34) tends to cv → √ k22 + k 2 3 k2 + k3 1 √ n . (40) thus, we see that the cv must eventually decrease with n. moreover, the coefficient of n−1/2 is strictly bounded between 2−1/2 (when both rate constants are equal) and 1 (when one of the coefficients is much smaller than the other, a situation that can be closely approached by taking the extreme values from table i). this means that larger transcription units have smaller cvs, although the cv does not decay quickly with transcription unit length. b. shape of the distribution when faced with the problem of choosing a distribution with which to model a random process, a theoretician will generally, absent specific reasons to the contrary, choose a gaussian. the central limit theorem certainly suggests that this is a sensible default choice [38]. however, when we have a detailed, solvable statistical model, we can explore the shape of the distribution in detail and give explicit conditions under which a gaussian is not expected to be a good model. to study these questions, it is useful to introduce two additional statistical parameters. the first of these is the skewness, defined by γ1 = µ3/σ 3, (41) where µ3 is the third central moment [equation (31)]. the name of this quantity suggests its interpretation: it is a measure of asymmetry of the distribution. a symmetric distribution such as a gaussian has a skewness of zero, while the exponential distribution has a skewness of two [42]. because there is a sharp cutoff in the distribution of transcription times at zero, but there is no such cut-off for the long-time tail, the skewness of this distribution will always be positive, as is evident from equation (31). however, the skewness can be large or small, and if reasonably small, then a gaussian approximation may be sufficiently accurate for generating transcription times in a delay stochastic simulation. the second statistical parameter of interest is the excess kurtosis defined by γ2 = µ4 σ4 − 3, (42) where µ4 is the fourth central moment [equation (32)]. the excess kurtosis measures the heaviness of the tails of a distribution [43]. the excess kurtosis of a gaussian biomath 2 (2013), 1307247, http://dx.doi.org/10.11145/j.biomath.2013.07.247 page 7 of 14 http://dx.doi.org/10.11145/j.biomath.2013.07.247 m r roussel, on the distribution of transcription times is zero, a laplace (double exponential) distribution has an excess kurtosis of three, but the simple exponential distribution has an excess kurtosis of six, showing that the kurtosis is a subtle quantity whose precise value depends both on the tail and on the overall shape of the distribution. combining equations (30) and (32), we find that the distribution of transcription times has an excess kurtosis of γ2 = 6 σ4 ( 1 k40 + 1 k41 + n− 1 k42 + n− 1 k43 + 1 k44 ) > 0. (43) a positive γ2 implies that the distribution of transcription times always has heavier tails than a gaussian. but how much heavier? in other words, under what conditions will γ2 be small, so that a gaussian approximation is appropriate, and when do we expect it to be large, so that a more careful choice of statistical models will be necessary? from table i, we see that k0 will typically be the smallest rate constant of the model. suppose that k0 � min ( k1,k2(n− 1)−1/4,k3(n− 1)−1/4,k4 ) . (the conditions on k2 and k3 are more restrictive than they need be, but will be needed to obtain results for the kurtosis below. the point of this demonstration is that there is a common set of conditions that makes the distribution distinctly non-gaussian.) then σ ≈ k−10 and µ3 ≈ 2k −3 0 , from which we obtain γ1 ≈ 2, a value representing a highly skewed distribution. under the same conditions as above, γ2 ≈ 6, a value consistent with an exponential distribution. we therefore conclude that when k0 is sufficiently small and n is not too large, the distribution is significantly nongaussian, with large skewness and excess kurtosis. given the ranges in table i, these conditions will typically be realized for short transcription units provided k4 is not too similar to k0. accordingly, it will typically be the case that the distribution of transcription times of short transcription units is poorly modeled by a gaussian. so what do these highly skewed, large-kurtosis distributions look like? figure 3 shows an example of one of these distributions, for values of the parameters that give γ1 = 1.90 and γ2 = 5.56, with a cv of 0.75. this distribution was computed by the semi-numerical inversion of the laplace transform using maple 15 [41]. (for a laplace transform of the form (27), maple is able to obtain the inverse laplace transform exactly. the only difficulty is that the coefficients of the final expression involve complicated combinations of the original parameters, which must be evaluated carefully to get accurate results. it is therefore necessary to use extra precision, fig. 3. distribution of transcription times obtained by seminumerical laplace inversion of equation (27) for k0 = 0.04 s−1, k1 = 8.0 s −1, k2 = k3 = 100.0 s−1, k4 = 0.2 s−1 and n = 200 nt. the inverse laplace transform was computed in maple [41], and stability of the result with respect to the number of floating-point digits used was verified. the inset shows a semi-log plot of the tail (the upper quartile of the distribution). and to verify that the coefficients are stable to variation in the number of digits used in the calculation.) it is clear that no gaussian could correctly capture the behavior of this distribution, both because of the strong asymmetry and because of the exponential decay of the tail. the excess kurtosis calculated for these parameters is almost as large as that of an exponential distribution, which is related to the slow, exponential decay of the tail. indeed, in the limit of large τ, d ln ρ/dτ → −0.04 s−1 for the distribution shown in figure 3 (evaluated numerically), which is −k0. this is not a coincidence: the tail in this sequential model of transcription is dominated by the slowest step, which in the case studied here is the initial location of the tss by the polymerase. we can also ask ourselves what happens to the skewness and excess kurtosis in the limit of large n. it is easy to see from equations (30), (31) and (41) that γ1 → 2(k32 + k 3 3)( k22 + k 2 3 )3/2 1√n, (44) and from equations (30) and (43) that γ2 → 6(k42 + k 4 3) (k22 + k 2 3) 2 1 n . (45) biomath 2 (2013), 1307247, http://dx.doi.org/10.11145/j.biomath.2013.07.247 page 8 of 14 http://dx.doi.org/10.11145/j.biomath.2013.07.247 m r roussel, on the distribution of transcription times taking these two results together, we see that in the limit of large n, both the skewness and excess kurtosis tend to zero, so that a gaussian approximation becomes increasingly accurate. we also see that the skewness goes to zero much more slowly than the excess kurtosis. there will therefore be a regime where it might be important to take the skewness of the distribution into account, but where the slowly decaying tail might not be so significant. c. a limiting case while we cannot, in general, analytically invert the laplace transform (27), we can obtain approximate distributions valid in some special cases. suppose for example that k0 and k4 are substantially smaller than all of the other rate constants (the case considered in figure 3). then, for s � min(k1,k2,k3), ρ̃ is approximately ρ̃ ≈ k0k4 (s + k0)(s + k4) . (46) this approximation would cease to hold at larger values of s. from the definition of the laplace transform (12), we see that, because of the exponential term, the laplace transform at large s is only significantly dependent on the value of the function at small τ. flipping this observation on its head, the breakdown of the approximation given above at large s will only affect the distribution at small transcription times. if we invert the laplace transform (46), we get the two-exponential distribution ρ(τ) ≈ k0k4 k4 −k0 ( e−k0τ −e−k4τ ) ≡ ρa(τ) (47) if k0 6= k4. (when k0 = k4, we get a gamma distribution.) the exact and approximate distributions are compared in figure 4. as expected, the two distributions differ at small τ: while the shapes of the distributions are very similar, the exact distribution has a region of negligible probability density up to about τ = 4 s. this discrepancy is due to elongation, which we neglect completely in the approximate model. although any single elongation step is fast, the large number of elongation steps results in a lag, i.e. an essentially nil probability that transcription will terminate before a certain time has elapsed, even if, by luck, the initiation and termination steps happen quickly. a much better fit to the exact distribution can be obtained by modifying the naive approximation (47) as follows: ρ(τ) ≈ h(τ − τmin)ρa(τ − τmin), (48) 0 0.005 0.01 0.015 0.02 0.025 0.03 0 20 40 60 80 100 120 140 ρ( τ) /s -1 τ/s fig. 4. exact distribution of transcription times (solid curve replotted from figure 3), two-exponential approximate distribution (47) (dashed curve), and lag-corrected approximate distribution (48) (dotted, almost coincident with the exact distribution). the two-exponential distribution uses the exact values of k0 and k4, i.e. k0 = 0.04 s−1 and k4 = 0.2 s−1. for the lag-corrected distribution, k0, k4 and τmin are treated as fitting parameters, with least-squares estimates k0 = 0.039 987 ± 0.000 021 s−1, k4 = 0.2001 ± 0.0003 s−1, and τmin = 4.111 ± 0.004 s−1. where τmin is an additional fitting parameter and h(·) is the heaviside function (0 for negative arguments, 1 for positive arguments). we call this the lag-corrected distribution. equation (48) was fit to the exact distribution using the marquardt-levenberg algorithm as implemented in gnuplot 4.6 [44]. both of the rate constants as well as the lag time were used as fitting parameters, resulting in the following estimates: k0 = 0.039 987±0.000 021 s−1, k4 = 0.2001 ± 0.0003 s−1, and τmin = 4.111 ± 0.004 s. note the excellent agreement between the least-squares values of k0 and k4 and the values used to generate the exact distribution. moreover, from figure 4, we see that the lag-corrected distribution closely reproduces the exact distribution. our simple ansatz involving the two small rate constants and a lag therefore gives an excellent account of the overall shape of the distribution. by fitting, we can recover k0 and k4, as well as the empirical parameter τmin. what is the physical meaning of τmin? several consecutive rapid steps have a narrow distribution, converging to a dirac delta distribution as n → ∞ [45], [46], [47]. the mean time for the fast steps is therefore the value of τmin, at least to the extent that the lag-corrected distribution represents the exact distribution. here, the consecutive rapid steps are the binding of the initial pair of nucleotide triphosphates to the polymerase (reaction 2) and the elongation steps (reactions 3/6 and 5). biomath 2 (2013), 1307247, http://dx.doi.org/10.11145/j.biomath.2013.07.247 page 9 of 14 http://dx.doi.org/10.11145/j.biomath.2013.07.247 m r roussel, on the distribution of transcription times thus, τmin = 1 k1 + n− 1 k2 + n− 1 k3 . (49) for the parameters of figures 3 and 4, this analytic estimate of τmin is 4.105 s, which is in excellent agreement with the value obtained by fitting equation (48) to the exact distribution. iv. stochastic simulations a number of criticisms could be leveled at our model and at its analysis thus far. perhaps the most serious criticism might be that our analysis assumes a single polymerase not interacting with other molecular machines. the case where many polymerases may transcribe the same transcription unit at the same time can easily be dealt with by stochastic (gillespie) simulation [48]. we can then compare the predictions of our analytic theory with those of the stochastic simulations. the stochastic model requires an additional parameter, namely the minimum distance between polymerase active sites, ∆. the bacterial rna polymerase protects approximately 35 bp (base pairs) of the dna duplex from cleavage by nucleases [49]. this is the length of dna that is inaccessible to other macromolecular machines while rnap is transcribing a gene. moreover, the dna takes a 90◦ bend on its way through the polymerase [50] (figure 5). the minimum distance between polymerase active sites is clearly 35 bp. however, there may be additional steric factors limiting the distance of closest approach, such as dna conformational requirements, or the need to accommodate the rna exiting the leading polymerase. for the sake of argument, we suppose that polymerases must be spaced by at least ∆ = 40 bp. the computed distributions of transcription times are not greatly sensitive to this choice (figure 6). in the simulations, the transcription time was taken to be the time from clearance of the tss by the previous polymerase to completion of the transcript. this is a direct analog of the transcription time considered in the single-polymerase analytic theory. figure 6 also compares the distributions from gillespie simulations to the analytic distribution (replotted from figure 3). despite the inclusion of interactions between polymerases in the simulations, which are absent from the analytic theory, the distributions obtained are not greatly different. the statistics of the distributions are compared in table ii. the statistics confirm our visual assessment that the distribution of transcription times is not greatly affected by ∆, nor is the distribution obtained fig. 5. schematic diagram of a pair of polymerases simultaneously transcribing a gene. the green boxes represent the polymerases, the blue curve is the template strand, the cyan curve is the coding strand, and the magenta curve is the nascent rna. the arrow indicates the direction in which the dna is pulled through the polymerases. approximately 35 bp are protected from cleavage, i.e. this length of rna is sufficiently inside the polymerase to be inaccessible to other macromolecular machines. 0 0.005 0.01 0.015 0.02 0.025 0.03 0 50 100 150 200 ( )/s -1 /s analytic distribution = 20 nt = 40 nt = 60 nt fig. 6. analytic distribution of transcription times (replotted from figure 3), and simulated distributions for three different values of ∆. each simulated distribution was obtained from stochastic simulations continued until 106 rnas had been synthesized. table ii statistics for the distributions at various values of ∆ compared to the statistics of the analytic distribution for the parameters of figure 6. ∆/nt 〈τ〉 σ γ1 γ2 20 35.37 24.82 1.98 5.97 40 35.36 24.84 1.98 6.03 60 35.34 24.84 1.98 6.02 analytic 34.11 25.50 1.90 5.56 biomath 2 (2013), 1307247, http://dx.doi.org/10.11145/j.biomath.2013.07.247 page 10 of 14 http://dx.doi.org/10.11145/j.biomath.2013.07.247 m r roussel, on the distribution of transcription times 0.4 0.5 0.6 0.7 0.8 0.9 1 10-4 10-3 10-2 10-1 100 101 102 103 104 c v k0/s -1 fig. 7. cv [equation (33)] vs k0 for k1 = 5 s−1, k2 = 10 s−1, k3 = 100 s −1, k4 = 0.01 s−1, n = 200 nt and ∆ = 40 nt. the solid curve is the cv computed from the analytic theory, i.e. equation (34). the points are from stochastic simulations. for each value of k0, the model was simulated until 100 000 rnas had been synthesized. from the analytic theory dramatically different from the distributions computed by stochastic simulation. as in the analytic theory, in some stochastic simulations of related models, a minimum cv has been observed as one of the rate constants is varied [18]. figure 7 shows the cv plotted vs k0 both for the analytic model and from stochastic simulations. you will note that the curve computed from the simulations does not have the pronounced minimum of the analytic result. at low initiation frequencies, the density of polymerases on the template strand is small and the single-polymerase analytic theory predicts many properties of the model, including the cv, reasonably well. however, as the initiation frequency increases, interactions between polymerases become more frequent, and the single-polymerase equations become less and less accurate. as k0 increases beyond k4, the initiation frequency exceeds the termination frequency, and a traffic jam ensues in the simulations. interestingly, the traffic jam condition results in a lower minimum cv than is observed at these parameters in the single-polymerase case. this is perhaps not surprising. once a traffic jam has formed, most of the time in the queue is spent waiting, with motion limited to short bursts when the leading motor (in this case, a polymerase) exits the jam (terminates transcription). note that some pairs of rate constants, notably k0 and k4, appear symmetrically in the single-polymerase equation (34). thus, if we set k0 to the value of k4 we used in figure 7, and vary k4, the analytic theory 0.4 0.5 0.6 0.7 0.8 0.9 1 10-4 10-3 10-2 10-1 100 101 102 103 104 c v k4/s -1 fig. 8. cv vs k4. all parameters and simulation conditions are as in figure 7, except k0 = 0.01 s−1. the solid curve is the cv computed from the analytic theory, i.e. equation (34). the points are from stochastic simulations. predicts an identical curve to that obtained by varying k0. (compare the solid curves in figures 7 and 8.) when there are many polymerases however, varying k0 or k4 is not the same because it matters whether k0 < k4, in which case termination is faster than initiation and no traffic jam occurs, or k0 > k4, which leads to a traffic jam. when varying k4, the single-polymerase theory is therefore accurate for large values of k4. as outlined above, the reason that the singlepolymerase theory fails when either k0 is large or k4 is small is that termination becomes rate limiting, which causes the polymerases to pile up along the strand. on the other hand, if we vary one of the other rate constants under conditions in which one of the two initiation steps [reactions (1) and (2)] is rate limiting, we get good agreement between the single-polymerase theory and the stochastic simulations. in particular, the predicted minimum in the cv as we vary the parameters becomes a robust feature of the model, as seen in figure 9. v. discussion and conclusions if we look at figures 7, 8 and 9, we see that the single-polymerase analytic theory gives excellent results provided initiation [reactions (1) and (2)] is rate limiting. miller’s classic electron micrograph snapshots of bacterial transcription in action always show the polymerases well spaced [51]. thus, for typical transcription units, we do not observe the slow termination processes that would cause the traffic jams shown here to lead to significant deviations from the single-polymerase theory. of course, transcriptional pauses, particularly if they occur late in a biomath 2 (2013), 1307247, http://dx.doi.org/10.11145/j.biomath.2013.07.247 page 11 of 14 http://dx.doi.org/10.11145/j.biomath.2013.07.247 m r roussel, on the distribution of transcription times 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 10-4 10-3 10-2 10-1 100 101 102 103 104 c v k1/s -1 fig. 9. cv vs k1 computed from the analytic theory (equation (34), solid line) and from stochastic simulations (dots). the parameters were as follows: k0 = 0.01 s−1, k2 = 10 s−1, k3 = 100 s−1, k4 = 0.1 s−1, n = 200 and ∆ = 40. in the stochastic simulations, statistics were collected until 100 000 rnas had been synthesized. transcription unit, could have a similar effect. pauses are known to occur during transcription in prokaryotes [52], [53], and there are specific sites on the template that are pause-prone [54]. the effect of pausing on singlepolymerase transcription statistics has been studied by voliotis and coworkers [15], where pausing was found to cause a heavy-tailed distribution of transcription times. ribeiro’s group studied pausing in a gene expression model and found, among other things, that a pause in the middle of a gene can cause a trimodal distribution of intervals between transcript completions, with the middle mode corresponding to “normal” spacing of the polymerase, one mode corresponding to microbursts (two or three transcriptions completing in an unusually short interval), a less extreme version of the traffic jams observed here, and another corresponding to the long intervals occasioned when one polymerase runs through the pause site without pausing, while the next one does pause [55]. clearly, it would be interesting to look at the similarities and differences between the effects of pausing and of simple traffic jams caused by slow termination. traffic is a particular problem for the ribosomal rna genes, which are transcribed at very high rates [56]. klumpp and hwa have studied traffic in a model of rrna transcription, where they found that short pauses, which are common events during transcription, would cause traffic jams were it not for the action of the antiterminator (at) complex which, among other things, inhibits pausing [14]. to maintain high rates, it is not enough to inhibit pausing in most elongation complexes; it is also necessary to remove complexes that have not assembled with at, lest they cause traffic jams. here, klumpp and hwa found that the termination factor rho can have precisely the desired effect by removing slow, pause-prone polymerases from the template, thus allowing elongation complexes properly complexed with at to work at the optimal rate. for rapidly transcribed genes, of which the rrna genes are an extreme, traffic may thus prove to be a rate-limiting process. in the large n regime, the cv, skewness and excess kurtosis all go to zero [equations (40), (44) and (45)]. at large n then, the distribution is at once relatively narrow, minimally skewed, and roughly as heavy-tailed as a gaussian. long transcription units therefore pose few modeling difficulties. their transcription time distributions are reasonably gaussian, so two parameters, the mean and variance, are sufficient to describe these distributions. it might even be tolerable, because of the small cv, to use a fixed delay in these cases, just as can be done for ordinary differential equation subsystems consisting of a linear decay chain [47]. the transcription time distributions of short transcription units on the other hand may be highly skewed and have large excess kurtoses. in such distributions, the mode, median and mean are quite different, so that there is no uniquely defined “typical” behavior. accordingly, modeling the expression of short transcription units requires a careful approach. we may in some cases be able to obtain the exact distribution by inversion of the laplace transform (27) (or the equivalent expression for a model involving additional biochemical steps) by a semi-numerical method, as was done for figure 3 using maple [41], or by a fully numerical method [57]. alternatively, we could use the lag-corrected ansatz (48) as an approximation to the exact distribution. any of the above representations of the distribution can then be used to generate random deviates in a delay-stochastic code like sgn sim [29]. of course, all approaches based on the computation of a single-polymerase transcription time distribution assume that perturbations due to interactions between polymerases, or between polymerases and ribosomes [58], are insignificant. if this is not the case, then eventually we will want to develop a many-body theory that yields approximate analytic distributions or moments thereof. the approach of section iii-c can be generalized to any small number of slow steps since it is easy to work out the inverse laplace transform in these cases with the assistance of a computer algebra system like maple [41]. biomath 2 (2013), 1307247, http://dx.doi.org/10.11145/j.biomath.2013.07.247 page 12 of 14 http://dx.doi.org/10.11145/j.biomath.2013.07.247 m r roussel, on the distribution of transcription times if we have a mechanistic model in which the slow steps are identified, fitting a lag-corrected distribution to an experimental distribution will give estimates of the slow rate constants and of the lag time, which combines all the fast processes. these lag-corrected distributions are much simpler to handle than their exact counterparts, and summarize all of the information that can reliably be extracted from a sequential set of reaction processes with a few slow steps and many fast steps. note that experimental distributions of transcription times are beginning to appear in the literature [59], [60]. as mentioned earlier, our model is modular. more complicated modules can be substituted for some of the modules in the current, highly simplified model, leading to more complex survival time problems. our eukaryotic transcription model [18] contains some modules not present in our current prokaryotic model, such as abortive initiation and pausing modules. because of the product form of the laplace transform (11), replacing or adding modules to the model is easy. in the model studied here, the various moments and derived quantities of interest (cv, skewness and kurtosis) adopt particularly simple forms. however, if we insert more complex modules into the model, especially ones that represent alternative pathways rather than simply additions to the sequential chain of events, then the expressions obtained for the moments become less straightforward. many interesting modules can still be treated in the framework presented here. future publications will describe this work, with the current publication, as well as our original paper [13], serving as an important baseline against which new models can be compared. acknowledgments this work was supported by the natural sciences and engineering research council of canada. references [1] b. alberts, d. bray, j. lewis, m. raff, k. roberts, j. d. watson, molecular biology of the cell, garland, 1983. 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http://dx.doi.org/10.4161/rna.6.4.8952 http://dx.doi.org/10.1016/j.camwa.2004.08.003 http://dx.doi.org/10.1126/science.1184939 http://dx.doi.org/10.1088/1478-3975/9/2/026004 http://dx.doi.org/10.1016/j.febslet.2012.09.014 http://dx.doi.org/10.11145/j.biomath.2013.07.247 introduction a prokaryotic transcription model model description model analysis methodology jump distributions moments of the distribution analytic results noise minimization shape of the distribution a limiting case stochastic simulations discussion and conclusions references original article biomath 3 (2014), 1404211, 1–6 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum on a bivariate poisson negative binomial risk process krasimira kostadinova1, leda minkova2 1faculty of mathematics and informatics, shumen university ”k. preslavsky” shumen, bulgaria, email: kostadinova@shu-bg.net 2faculty of mathematics and informatics, sofia university ”st. kl.ohridski” sofia, bulgaria, email: leda@fmi.uni-sofia.bg received: 19 february 2014, accepted: 21 april 2014, published: 30 may 2014 abstract—in this paper we define a bivariate counting process as a compound poisson process with bivariate negative binomial compounding distribution. we investigate some of its basic properties, recursion formulas and probability mass function. then we consider a risk model in which the claim counting process is the defined bivariate poisson negative binomial process. for the defined risk model we derive the distribution of the time to ruin in two cases and the corresponding laplace transforms. we discuss in detail the particular case of exponentially distributed claims. keywords-bivariate negative binomial distribution; compound birth process; ruin probability i. introduction we consider the stochastic process n(t), t > 0 defined on a fixed probability space (ω,f , p) and given by n(t) = x1 + x2 + . . . + xn1 (t), (1) where xi, i = 1, 2, . . . are independent, identically distributed (iid) as x random variables, independent of n1(t). we suppose that the counting process n1(t) is a poisson process with intensity λ > 0 (n1(t) ∼ po(λt)). in this case n(t) is a compound poisson process. the probability mass function (pmf) and probability generating function (pgf) of n1(t) are given by p(n1(t) = i) = (λt)ie−λt i! , i = 0, 1, . . . (2) and ψn1 (t)(s) = e −λt(1−s). (3) the compound poisson distribution is analyzed by many authors; see johnson et al. [3], grandell, [2], minkova [8]. the corresponding compound poisson process is commonly used as a counting process in risk models; see for example klugman et al. [5], minkova [9]. in this paper we suppose that the compounding random variable x has a bivariate negative binomial distribution, given in the next section ii. in section iii we define a counting process with the bivariate poisson negative binomial distribution (bpnb). we derive the moments and the joint pmf. then, in section iv, two types of ruin probability are considered for the risk model with bpnb distributed counting process. we derive the laplace transforms and analyze the case of exponentially distributed claims. citation: krasimira kostadinova, leda minkova, on a bivariate poisson negative binomial risk process, biomath 3 (2014), 1404211, http://dx.doi.org/10.11145/j.biomath.2014.04.211 page 1 of 6 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2014.04.211 k kostadinova et al., on a bivariate poisson negative binomial... ii. bivariate negative binomial distribution let us consider the bivariate negative binomial distribution, defined by the following pgf, given in kocherlakota and kocherlakota [6] ψ1(s1, s2) = ( γ 1 −αs1 −βs2 )r , (4) where γ = 1 − α − β and r ≥ 1 is a given integer number. we use the notation (x, y ) ∼ bn b(r,α,β). the pmf of (x, y ) is given by p(x = k, y = l) = ( k + l l )( r + k + l − 1 k + l ) αkβlγr (5) for k, l = 0, 1, . . . , (k, l) , (0, 0), and p(x = 0, y = 0) = γr. the marginal distributions are again negative binomial with pgfs ψ1(s1) = ψ1(s1, 1) = ( γ 1 −β−αs1 )r (6) and ψ1(s2) = ψ1(1, s2) = ( γ 1 −α−βs2 )r . (7) denote by ρ1 = α 1−β and ρ2 = β 1−α the corresponding parameters of x and y. in terms of ρ1 and ρ2, the pgfs (6) and (7) have the form ψ1(s1) = ( 1 −ρ1 1 −ρ1 s1 )r and ψ1(s2) = ( 1 −ρ2 1 −ρ2 s2 )r . the pgf of the sum x + y is given by ψ1(s, s) = ( γ 1 − (α + β)s )r = ( γ 1 − (1 −γ)s )r . (8) from (8) it follows that x + y has a negative binomial distribution with parameters r and γ, say x + y ∼ n b(r,γ), and for j = 0, 1, . . . , pmf p(x + y = j) = ( r + j − 1 j ) γr(1 −γ) j. denote the pgf in (8) by ψ1(s) = ψ1(s, s). iii. the bivariate counting process in this section we consider a compound poisson process with bivariate negative binomial compounding distribution. the resulting process is a bivariate counting process (n1(t), n2(t)), defined by the pgf ψ(s1, s2) = e −λt(1−ψ1 (s1,s2 )), (9) where ψ1(s1, s2) is the pgf of the compounding distribution, given in (4). we say that the counting process defined by (9) has a bivariate poisson negative binomial distribution with parameters λt,α and β, and use the notation (n1(t), n2(t)) ∼ bpn b(λt,α,β). the marginal distributions are defined by the following pgfs ψ(s1) = e −λt(1−ψ1 (s1 )) and ψ(s2) = e −λt(1−ψ1 (s2 )), where ψ1(s1) and ψ1(s2) are given by (6) and (7). the means are given by e(n1(t)) = rαλt γ and e(n2(t)) = rβλt γ , while the variances are v ar(n1(t)) = αr γ2 [1 + rα − β]λt and v ar(n2(t)) = βr γ2 [1 −α + rβ]λt. from (9) we obtain ∂2ψ(s1, s2) ∂s1∂s2 = ψ(s1, s2)rαβλt × [ rγ2rλt (1 −αs1 −βs2)2r+2 + (r + 1)γr (1 −αs1 −βs2)r+2 ] . upon setting s1 = s2 = 1 we obtain the product moment of n1(t) and n2(t) to be e(n1(t)n2(t)) = rαβ γ2 (rλt + r + 1)λt, which yields the covariance between n1(t) and n2(t) as cov(n1(t), n2(t)) = r(r + 1)αβ γ2 λt. for the correlation coefficient we have corr(n1(t), n2(t)) = (r + 1) √ αβ (1+rα−β)(1+rβ−α). biomath 3 (2014), 1404211, http://dx.doi.org/10.11145/j.biomath.2014.04.211 page 2 of 6 http://dx.doi.org/10.11145/j.biomath.2014.04.211 k kostadinova et al., on a bivariate poisson negative binomial... in terms of ρ1 and ρ2, the covariance and the correlation coefficient have the forms cov(n1(t), n2(t)) = r(r + 1)ρ1ρ2 (1 −ρ1)(1 −ρ2) λt and corr(n1(t), n2(t)) = (r + 1) √ ρ1ρ2 (1 + rρ1)(1 + rρ2) . a. joint probability mass function the probability function of the joint distribution of (n1(t), n2(t)) is given by expanding the pgf ψ(s1, s2) in powers of s1 and s2. denote by f (i, j) = p(n1(t) = i, n2(t) = j), i, j = 0, 1, 2, . . . , the joint probability mass function of (n1(t), n2(t)). we rewrite the pgf of (9) in the form ψ(s1, s2) = e−λt ∞∑ m=0 (λt)m m! ψm1 (s1, s2) = e−λt ∞∑ m=0 (λtγr)m m! 1 (1 −αs1 −βs2)rm . (10) denote by ψ(i, j)(s1, s2) = ∂i+ jψ(s1, s2) ∂si1∂s j 2 , for i, j = 0, 1, . . . , and (i, j) , (0, 0), the derivatives of ψ(s1, s2). from (10) we get the following: ψ(i, j)(s1, s2) = e−λtαiβ j ∑ ∞ m=1 (λtγr )m m! × rm(rm+1)...(rm+i−1)(rm+i)...(rm+i+ j−1) (1−αs1−βs2 )rm+i+ j . (11) from johnson et al. [4], it is known that for i, j = 0, 1, . . . , (i, j) , (0, 0), f (i, j) = ψ(i, j)(s1, s2) i! j! ∣∣∣∣∣∣ s1=s2=0. (12) the result is given in the next theorem. theorem 1. the probability mass function of (n1(t), n2(t)) is given by f (i, j) = ( i + j j ) αiβ j × ∞∑ m=1 ( rm + i + j − 1 i + j ) (λtγr)m m! e−λt, i, j = 0, 1, . . . , (i, j) , (0, 0), (13) and f (0, 0) = e−λt(1−γ r ). proof. the initial value f (0, 0) = e−λt(1−γ r ) follows simply from the pgf ψ(0, 0) = f (0, 0). then (13) follows from (11) and (12). � iv. bivariate riskmodel consider the following bivariate surplus process u1(t) = u1 + c1t − ∑n1 (t) j=1 z 1 j u2(t) = u2 + c2t − ∑n2 (t) j=1 z 2 j for two lines of business. here u1 and u2 are the initial capitals, c1, c2 represent the premium incomes per unit time and z1, z11, z 1 2, . . . , and z2, z21, z 2 2, . . . are two independent sequences of independent random variables, independent of the counting processes n1(t) and n2(t), representing the corresponding claim sizes. the univariate case of this model was analyzed in kostadinova [7]. let µ1 = e(z1) and µ2 = e(z2) be the means of the claims. denote by s 1(t) = ∑n1 (t) j=1 z 1 j and s 2(t) = ∑n2 (t) j=1 z 2 j the corresponding accumulated claim processes. the model, analyzed in chan et al., [1] is the case when n1(t) = n2(t) = n(t). here we consider two possible times to ruin τmax = inf{t|max(u1(t), u2(t)) < 0} and τsum = inf{t|u1(t) + u2(t) < 0}, and the corresponding ruin probabilities ψmax(u1, u2) = p(τmax < ∞) and ψsum(u1, u2) = p(τsum < ∞). for the event of τmax we have the following: biomath 3 (2014), 1404211, http://dx.doi.org/10.11145/j.biomath.2014.04.211 page 3 of 6 http://dx.doi.org/10.11145/j.biomath.2014.04.211 k kostadinova et al., on a bivariate poisson negative binomial... {max(u1(t), u2(t)) < 0} = {u1(t) < 0, u2(t) < 0} = {u1 + c1t − s 1(t) < 0, u2 + c2t − s 2(t) < 0} = {s 1(t) > u1 + c1t, s 2(t) > u2 + c2t}. it follows that the ruin probability ψmax(u1, u2) is the joint survival function of (s 1(t), s 2(t)). in a similar way, we obtain the event for the τsum : {u1(t) + u2(t) < 0} = {s 1(t) + s 2(t) > u1 + u2 + (c1 + c2)t}, i.e., the ruin probability ψsum(u1, u2) is the survival function of the sum s 1(t) + s 2(t). according to the definition of ψmax(u1, u2), for the no initial capitals, we have the following ψ1(0)ψ2(0) ≤ ψmax(0, 0) ≤ min{ψ1(0), ψ2(0)}, where ψ1(0) and ψ2(0) are the ruin probabilities of the models u1(t) and u2(t) with no initial capitals. the univariate poisson negative binomial risk model is analyzed in [7], where it is given that ψ1(0) = ρ1 c1(1 −ρ1) λµ1r and ψ2(0) = ρ2 c2(1 −ρ2) λµ2r. it follows that the upper bound of the ruin probability is given by ψmax(0, 0) ≤ min{ ρ1 c1(1 −ρ1) λµ1r, ρ2 c2(1 −ρ2) λµ2r}. the lower bound has the form ψmax(0, 0) ≥ ρ1ρ2 c1c2(1 −ρ1)(1 −ρ2) λ2µ1µ2r 2. a. laplace transforms denote by ltz1 (s1) and ltz2 (s2) the laplace transforms of the random variables z1 and z2. then, the laplace transform of (s 1(t), s 2(t)) is given by lt(s 1 (t),s 2 (t))(s1, s2) = e[e −s1 s 1 (t)−s2 s 2 (t)] = ∑ ∞ i=0 ∑ ∞ j=0 e[e −s1 (z11 +...+z 1 i )−s2 (z 2 1 +...+z 2 j )] ×p(n1(t) = i, n2(t) = j). according to the construction of the counting process, for the laplace transform of (s 1(t), s 2(t)) we have lt(s 1 (t),s 2 (t))(s1, s2) = ψ(n1 (t),n2 (t))(ltz1 (s1), ltz2 (s2)) = e −λt [ 1− ( γ 1−αlt z1 (s1 )−βltz2 (s2 ) )r] . using the parameters γ 1 −β = 1 −ρ1 and γ 1 −α = 1 −ρ2 of the marginal compounding distributions, we obtain the laplace transforms of the marginal compound distributions to be lts 1 (t)(s1) = lt(s 1 (t),s 2 (t))(s1, 0) = e −λt [ 1− ( 1−ρ1 1−ρ1 ltz1 (s1 ) )r] and lts 2 (t)(s2) = lt(s 1 (t),s 2 (t))(0, s2) = e −λt [ 1− ( 1−ρ2 1−ρ2 ltz2 (s2 ) )r] . we need the following result about laplace transforms, given in omey and minkova ([10]) lemma 1. for the joint survival function p(s 1(t) > x, s 2(t) > y) we have∫ ∞ 0 ∫ ∞ 0 e−s1 x−s2 y p(s 1(t) > x, s 2(t) > y)d xdy = 1 − lts 1 (t)(s1) − lts 2 (t)(s2) + lt(s 1 (t),s 2 (t))(s1, s2) s1 s2 . (14) biomath 3 (2014), 1404211, http://dx.doi.org/10.11145/j.biomath.2014.04.211 page 4 of 6 http://dx.doi.org/10.11145/j.biomath.2014.04.211 k kostadinova et al., on a bivariate poisson negative binomial... in our case we have ∫ ∞ 0 ∫ ∞ 0 e−s1 x−s2 y p(s 1(t) > x, s 2(t) > y)d xdy = 1s1 s2 [ 1 − e −λt [ 1− ( 1−ρ1 1−ρ1 ltz1 (s1 ) )r] −e −λt [ 1− ( 1−ρ2 1−ρ2 ltz2 (s2 ) )r] + e −λt [ 1− ( γ 1−αlt z1 (s1 )−βltz2 (s2 ) )r]] . lemma 2. for the survival function p(s 1(t) + s 2(t) > x) we have∫ ∞ 0 e−sx p(s 1(t) + s 2(t) > x)d x = 1 s [1 − lts 1 (t)+s 2 (t)(s)]. (15) then, for the ruin probability ψsum we have: lts 1 (t)+s 2 (t)(s) = lts 1 (t),s 2 (t)(s, s) = e −λt [ 1− ( γ 1−αlt z1 (s)−βlt z2 (s) )r] . and hence∫ ∞ 0 e−sx p(s 1(t) + s 2(t) > x)d x = 1 s [ 1 − e −λt [ 1− ( γ 1−αlt z1 (s)−βlt z2 (s) )r]] . b. exponentially distributed claims let us consider the case of exponentially distributed claim sizes, i.e. fz1 (x) = 1 − e − x µ1 , x ≥ 0 and gz2 (y) = 1 − e − y µ2 , y ≥ 0, and µ1,µ2 > 0. denote by e(n, x) = n∑ k=0 xk k! = exγ(n + 1, x) γ(n + 1) , where γ(n) is a gamma function and γ(a, x) =∫ ∞ x ta−1e−tdt is the incomplete gamma function, the truncated exponential sum function. for the ruin probability ψmax we have p(s 1(t) > x, s 2(t) > y) = ∑ ∞ i, j=0 f ∗i (x)g ∗ j (y)p(n1(t) = i, n2(t) = j), where f ∗i (x) = e− x µ1 e(i − 1, x µ1 ), i = 1, 2, . . . is the tail distribution of z11 + . . . + z 1 i and g ∗ j (y) = e− y µ2 e( j − 1, y µ2 ), j = 1, 2, . . . is the tail distribution of z21 + . . . + z 2 j . in this case we have p(s 1(t) > x, s 2(t) > y) = e−λt(1−γ r ) + ∑ ∞ i=1 α ie(i − 1, x µ1 ) × ∑ ∞ m=1 ( rm+i−1 i ) (λtγr )m m! e − x µ1 e−λt + ∑ ∞ j=1 β je( j − 1, y µ2 ) × ∑ ∞ m=1 ( rm+ j−1 j ) (λtγr )m m! e − y µ2 e−λt + ∑ ∞ i=1 ∑ ∞ j=1 α iβ je(i − 1, x µ1 )e( j − 1, y µ2 ) ( i+ j j ) × ∑ ∞ m=1 ( rm+i+ j−1 i+ j ) (λtγr )m m! e − x µ1 − y µ2 e−λt. substituting x = u1 + c1t and y = u2 + c2t in the last expression, we obtain the ruin probability ψmax(u1, u2), as it was shown in the previous section. for the ruin probability ψsum(u1, u2) in the case z1 = z2 = z, and hence µ1 = µ2 = µ, we obtain for the laplace transform of the sum s 1(t)+s 2(t) : lts 1 (t)+s 2 (t)(s) = e −λt [ 1− ( γ 1−(α+β)ltz (s) )r] . this means that s 1(t) + s 2(t) = z1 + . . . + zn(t), where n(t) = x1 +. . .+ xn1 (t), n1(t) ∼ po(λt), and xi ∼ n b(r,γ). the survival function of the sum s 1(t) + s 2(t) is the survival function of the sum of claims, i.e. p(s 1(t) + s 2(t) > x) = p(z1 + . . . + zn(t) > x) = ∑ ∞ i=0 f ∗i (x)p(n(t) = i), where f ∗i (x) = e− x µ e(i − 1, x µ ), i = 1, 2, . . . is the biomath 3 (2014), 1404211, http://dx.doi.org/10.11145/j.biomath.2014.04.211 page 5 of 6 http://dx.doi.org/10.11145/j.biomath.2014.04.211 k kostadinova et al., on a bivariate poisson negative binomial... tail distribution of z1 + . . . + zi. hence we have p(s 1(t) + s 2(t) > x) = e−λt(1−γ r ) + ∑ ∞ i=1(1 −γ) ie(i − 1, x µ ) × ∑ ∞ m=1 ( rm+i−1 i ) (λtγr )m m! e − x µ e−λt = e−λt(1−γ r ) + ∑ ∞ i=1(α + β) ie(i − 1, x µ ) × ∑ ∞ m=1 ( rm+i−1 i ) (λtγr )m m! e − x µ e−λt and for x = u1 + u2 + (c1 + c2)t, we obtained the ruin probabilities ψsum. v. conclusion in this study we introduce a compound poisson process with bivariate negative binomial compounding distribution. also, we find the moments and the joint probability mass function. then we define the bivariate risk model with bivariate poisson negative binomial counting process. we find the laplace transform of the ruin probability and investigate a special case of exponentially distributed claims. acknowledgment the authors are thankful to the anonymous reviewers and the editor for making some useful comments and suggestions. this work was supported by the european social fund through the human resource development operational programme under contract bg051po001-3.3.060052 (2012/2014) and by grant rd-08-255/2013 of shumen university, bulgaria. references [1] chan w–s., yang h. and zhang l. 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(2002). a generalization of the classical discrete distributions, commun.statist. theory and methods, 31, 871–888. [9] minkova l.d. (2004). the pólya-aeppli process and ruin problems, j. appl. math. stoch. anal., 3, 221–234. [10] omey e. and minkova l.d. (2013). bivariate geometric distributions, (submitted). biomath 3 (2014), 1404211, http://dx.doi.org/10.11145/j.biomath.2014.04.211 page 6 of 6 http://dx.doi.org/10.11145/j.biomath.2014.04.211 introduction bivariate negative binomial distribution the bivariate counting process joint probability mass function bivariate risk model laplace transforms exponentially distributed claims conclusion references original article biomath 3 (2014), 1403241, 1–11 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum numerical analysis of a size-structured population model with a dynamical resource oscar angulo∗, j.c. lópez-marcos∗ and m.a. lópez-marcos∗ ∗departamento de matemática aplicada universidad de valladolid, valladolid, spain emails: oscar@mat.uva.es, lopezmar@mac.uva.es, malm@mac.uva.es received: 19 october 2013, accepted: 24 march 2014, published: 28 may 2014 abstract—in this paper, we analyze the convergence of a second-order numerical method for the approximation of a size-structured population model whose dependency on the environment is managed by the evolution of a vital resource. optimal convergence rate is derived. numerical experiments are also reported to demonstrate the predicted accuracy of the scheme. also, it is applied to solve a problem that describes the dynamics of a daphnia magna population, paying attention to the unstable case. keywords-structured population models; numerical methods; convergence; daphnia magna i. introduction physiologically structured population models are based on the use of one or more attributes that structure the individuals in the population. size is one of the most natural and important attributes structuring the population for many species: typical examples being fishes and trees. in such species the ability of an individual to obtain the necessary resources to survive and reproduce depends strongly on its size. structured population models reflect the effect of physiological state of individuals on the population dynamics. in addition, the use of nonlinear structured population models allows us to take into account the effect of competition for natural resources in the structured-specific growth, mortality and fertility rates. we can find an extensive study of physiologically structured population models, with analytical studies of aspects such as existence and uniqueness, smoothness and the asymptotic behaviour of solutions in [1], [2], [3], [4], [5]. in this paper, we consider a size-structured population model nonlinearly coupled with an integroordinary differential equation accounting for substrate consumption and/or product formation. it was introduced first in [6] for modeling a daphnia magna population. the model involves a nonlinear hyperbolic partial differential equation ut + (g(x, s (t), t) u)x = −µ(x, s (t), t) u, (1) 0 < x < xm (t), t > 0, a nonlinear and nonlocal boundary condition which reflects the reproduction process g(0, s (t), t) u(0, t) = ∫ xm (t) 0 α(x, s (t), t) u(x, t) d x, (2) t > 0, and an initial size-distribution for the population u(x, 0) = u0(x), 0 ≤ x ≤ xm (0). (3) citation: oscar angulo, j.c. lópez-marcos, m.a. lópez-marcos, numerical analysis of a size-structured population model with a dynamical resource, biomath 3 (2014), 1403241, http://dx.doi.org/10.11145/j.biomath.2014.03.241 page 1 of 11 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2014.03.241 o angulo et al., numerical analysis of a size-structured population model... the influence of the environment on the life history of the individuals is given by a function s (t) which represents a physiological resource. its dynamics is managed by the next initial value problem, s ′(t) = f (s (t), i(t), t), t > 0, (4) s (0) = s 0, which is coupled with (1)-(3). the evolution of the resource also depends on the population, which is performed by means of the nonlocal term i(t) defined by i(t) = ∫ xm (t) 0 γ(x, s (t), t) u(x, t) d x, t ≥ 0. (5) also, we consider that the maximum size xm (t) that an individual could have at time t, changes with time and its dynamics is described by d dt xm (t) = g(xm (t), s (t), t), t > 0, (6) xm (0) = xm. the independent variables x and t represent size and time, respectively. the dependent variable u(x, t) is the size-specific density of individuals with size x at time t. in general, the size of any individual varies according to the following ordinary differential equation d x dt = g(x, s (t), t). (7) functions g, α and µ represent the growth, fertility and mortality rates, respectively. these are usually called the vital functions and define the life history of an individual. functions α and µ are nonnegative. note that all the vital functions (g, µ and α) depend on size x (the structuring internal variable), on time t and on the value of the resource at time t, which can reflect the influence of the environmental changes on the vital functions. function f on the right-hand side of (4) depends on the value of the resource at time t, on the total amount of individuals in the population by means of the weighted functional i(t) (which represents the way of weighting the size distribution density in order to model the different influence of individuals of different sizes on such dynamics) and on time t. the paper is structured as follows. in the next section we introduce some background on theoretical and numerical results. in section iii, we describe the numerical method, which is completely analyzed in section iv. finally, numerical results that confirm the expected order of convergence and show the biological example dynamics are included in section v. ii. preliminary results theoretical analysis (1)-(5) which includes existence, uniqueness and long-time behaviour, is highly difficult. a theoretical study of this model appeared first time in a h.thieme’s [7] presentation and authors in [5] also pointed that, for an analysis of such models, we had to perform as it was made in [8]. however, only a simpler model, in which a positive growth was employed, was analyzed in [9]. the numerical solution to the model, due to its obvious mathematical complexity, entails a serious challenge. in [10], a simple modification of the scheme succesfully employed in [11], for the solution of general size-structured population models, was considered. the daphnia magna test in section v was also introduced. furthermore, the theoretical steady state of the model for this test was provided. this scheme was shown not suitable for a long-time integration with this biological test. in this work, the numerical method described in section iii was proposed. some simulations was performed to show values of the parameters which led us to an asymptotically stable equilibrium and to an asymptotically stable periodic situation, in both cases solutions were bounded. the purpose of the current work is to validate such numerical method by means of an analysis of its convergence. on the other hand, a modification of this numerical procedure was introduced in [12] in order to approximate singular asymptotic states for these kinds of models. in this work, we showed stable and unstable steady state solutions. the study of these singular states is not the aim of present work. biomath 3 (2014), 1403241, http://dx.doi.org/10.11145/j.biomath.2014.03.241 page 2 of 11 http://dx.doi.org/10.11145/j.biomath.2014.03.241 o angulo et al., numerical analysis of a size-structured population model... in our convergence analysis, we shall use the general discretization framework introduced by lópez-marcos et al. [13]. we present it rather tersely, the reader is referred to this paper and the references therein for a more critical and detailed treatment. thus, once we introduced a fixed given problem concerning a differential equation, we denote u a theoretical solution to such a problem. we denote ũh a numerical approximation to u. the subscript h reflects the dependency on a parameter h (mesh size) which takes values in a set h of positive numbers with inf h = 0. the approximation is reached by solving the discretized problem φh(ũh) = 0 (8) (this family of discrete problems with h is referred as discretization), where, for each h ∈ h, the mapping φh is fixed with domain dh ⊂ ah and taking values in bh. here, ah and bh are vector spaces with the same finite dimension. we further assume that, for each h ∈ h, we have chosen a norm in both spaces and an element ũh ∈ dh which is a suitable discrete representation of u in ah. furthermore, we introduce the global and local discretization error, ẽh and lh, respectively, and the consistency, stability and convergence properties of a discretization. the following result is crucial and uses a deep topological lemma due to stetter [14], theorem 1. assume that (8) is consistent and stable with thresholds rh. if φh is continuous in b(ũh, rh) and ‖lh‖bh = o(rh) as h → 0, then: i) for h sufficiently small, the discrete equations (8) possess a unique solution in b(ũh, rh). ii) as h → 0, the solutions converge and ‖ẽh‖ah = o(‖lh‖bh ). iii. the numericalmethod the numerical method we employ to approximate the solution to (1)-(5) is based on the discretization of a representation of the solution along the characteristic curves [10]. first of all we rewrite the partial differential equation (1) in a more suitable form for its numerical treatment. so we define µ∗(x, z, t) = µ(x, z, t) + gx(x, z, t). thus equation (1) has the form ut + g(x, s (t), t) ux = −µ ∗(x, s (t), t) u, (9) 0 < x < xm (t), t > 0. we denote by x(t; t∗, x∗) the characteristic curve of equation (9) that takes the value x∗ at time t∗. such a characteristic curve is the solution to the initial value problem d dt x(t; t∗, x∗) = g(x(t; t∗, x∗), s (t), t), t ≥ t∗, (10) x(t∗; t∗, x∗) = x∗. now we consider the function that represents the solution to (9) along the characteristic curves u(x(t; t∗, x∗), t), t ≥ t∗, which satisfies the initial value problem d dt u(x(t; t∗, x∗), t)= −µ∗ (x (t; t∗, x∗) , s (t), t) u(x(t; t∗, x∗), t), t ≥ t∗, u(x(t∗; t∗, x∗), t∗) = u(x∗, t∗), and, therefore, it can be represented in the following integral form u(x(t; t∗, x∗), t) = (11) u(x∗, t∗)exp { − ∫ t t∗ µ∗(x (τ; t∗, x∗) , s (τ),τ) dτ } , t ≥ t∗. given a constant step k > 0, we introduce the discrete time levels tn = n k, n = 0, 1, 2, . . .. we also take j a positive integer, as a parameter related to the size variable which describes the number of points in the uniform initial grid. the diameter of such a mesh grid is h = xm/j, and the initial grid nodes are x0j = j h, 0 ≤ j ≤ j. in order to start the integration, we consider as an approximation to the density at initial time (t0), the grid restriction of the initial condition in (3), u 0j = u0(x 0 j ), 0 ≤ j ≤ j. also, we use s 0 in (4) as the initial value of the resource. then, the numerical method provides, at each discrete time level, a mesh grid on the size interval in xn, the approximation to the density on such a biomath 3 (2014), 1403241, http://dx.doi.org/10.11145/j.biomath.2014.03.241 page 3 of 11 http://dx.doi.org/10.11145/j.biomath.2014.03.241 o angulo et al., numerical analysis of a size-structured population model... mesh grid in un and the approximation of the value of the resource s n, from the approximations we computed at the previous time level, by using discretizations of equations (10), (11), (2), (4), (5). thus, for n = 0, 1, 2, . . ., the numerical solution at time tn+1 = tn + k, is obtained from the known values of the numerical solution at time tn as follows, xn+10 = 0, (12) xn+1j+1 = x n j + k g(x n+1/2 j+1 , s n+1/2, tn+1/2), (13) 0 ≤ j ≤ j, (14) s n+1 = s n + (15) k f (s n+1/2,q(xn+1/2,γn+1/2.un+1/2), tn+1/2), (16) u n+1j+1 = u n j exp { −k µ∗(xn+1/2j+1 , s n+1/2, tn+1/2) } , (17) 0 ≤ j ≤ j, (18) u n+10 = q(xn+1,αn+1.un+1) g(xn+10 , s n+1, tn+1) , (19) where we have to compute approximations at time level tn+1/2 = tn + k/2, xn+1/20 = 0, xn+1/2j+1 = x n j + k 2 g(xnj , s n, tn), 0 ≤ j ≤ j, s n+1/2 = s n + k 2 f (s n,q(xn,γn.un), tn), u n+1/2j+1 = u n j exp { − k 2 µ∗(xnj , s n, tn) } , 0 ≤ j ≤ j, u n+1/20 = q(xn+1/2,αn+1/2.un+1/2) g(xn+1/20 , s n+1/2, tn+1/2) . note that the general step of the method increases the number of grid points and also the dimension of the vector with the numerical densities: at time tn, we have j + 1 grid nodes in xn and the (j + 1)dimensional vector un, and at time tn+1 we obtain j +2 grid nodes in xn+1 and the (j +2)-dimensional vector un+1. in order to maintain the number of grid points suitable to perform the next step, we eliminate at time tn+1 the first grid node xn+1l which satisfies |xn+1l+1 − x n+1 l−1 | = min1≤ j≤j+1 |xn+1j+1 − x n+1 j−1 |. (20) we reproduce the same reduction in the corresponding vector un+1. in the description of the method, we use the following notation; vectors αp and γp contain the evaluations of the functions α and γ in (2) and (5), respectively, at the grid points in xp, at the resource value s p and at time t p. products γp.up and αp.up must be considered componentwise. in order to approximate integrals over the interval [0, xm (t p)], we use the composite trapezoidal quadrature rule based on the grid points xp = [x p0 , x p 1 , . . . , x p j ], that is q(xp, vp) = j∑ j=1 x pj − x p j−1 2 ( v pj−1 + v p j ) . (21) note that the method is implicit: all the expressions provide explicit equations for the numerical values at the highest time level, except those which involve the numerical density u p0 at the first grid point, but it is easy to implement the method in an explicit form. iv. convergence analysis below, we will analyze numerical methods based on the integration along characteristics that use a general quadrature rule with suitable properties to approximate the integral terms. the proofs of every result are heavily laborious and they would be included in a more technical work. we assume the following regularity conditions on the data functions and the solution to the problem (1)-(5): (h1) u ∈c2([0, xm (t)]× [0, t ]), u(x, t) ≥ 0, x ∈ [0, xm (t)], t ≥ 0. (h2) s ∈c2([0, t ]), s (t) ≥ 0, t ≥ 0. (h3) γ ∈c2([0, xm (t)]×d×[0, t ]), where d is a compact neighbourhood of {s (t) , 0 ≤ t ≤ t}. (h4) µ ∈c2([0, xm (t)]× d×[0, t ]), is nonnegative and d is a compact neighbourhood of {s (t) , 0 ≤ t ≤ t}. (h5) α ∈c2([0, xm (t)]×d×[0, t ]), is nonnegative and d is a compact neighbourhood of {s (t) , 0 ≤ t ≤ t}. (h6) f ∈ c2(d × di × [0, t ]), is nonnegative, d is a compact neighbourhood of biomath 3 (2014), 1403241, http://dx.doi.org/10.11145/j.biomath.2014.03.241 page 4 of 11 http://dx.doi.org/10.11145/j.biomath.2014.03.241 o angulo et al., numerical analysis of a size-structured population model... {s (t) , 0 ≤ t ≤ t}, and di is a compact neighbourhood of{∫ xm (t) 0 γ(x, s (t)) u(x, t) d x, 0 ≤ t ≤ t } . (h7) g ∈c3([0, xm (t)]×d×[0, t ]), where d is a compact neighbourhood of {s (t) , 0 ≤ t ≤ t} and g(0, s, t) ≥ c > 0, t ≥ 0, s ∈ r. in addition, the characteristic curves x(t; t∗, x∗) defined in (7) are continuous and differentiable with respect to the initial values (t∗, x∗) ∈ [0, t ] × [0, xm (t)]. the above hypotheses may be based on three possible reasons. first, biological assumptions such as the nonnegativity of some of the vital functions or, in (h7), to reflect that any individual in the studied population could shrink [2]. second, the mathematical requirements to obtain the existence and uniqueness of solutions for the problem (1)(5) [2]. finally, the regularity properties needed in the numerical analysis to derive optimal rates of convergence [11]. we also assume that the spatial discretization parameter, h, takes values in the set h = {h > 0 : h = xm/j, j ∈n}. now, we suppose that the time step, k, satisfies k = r h, where r is an arbitrary and positive constant, fixed throughout the analysis. in addition, we set n = [t/k]. for each h ∈ h, we define the spaces ah = n∏ n=0 ( rj+n ×rj+n+1 ) ×rn+1, bh = ( rj×rj+1×r ) ×rn × n∏ n=1 ( rj+n ×rj+n ) ×rn. both spaces have the same dimension. in order to measure the size of the errors, we define ‖η‖∞ = max1≤ j≤p |η j|, η ∈ rp, ‖vn‖1 =∑j+n j=0 h |v n j |, v n ∈ rj+n+1. thus, we endow the spaces ah and bh with the following norms. if( y0, v0, . . . , yn, vn, a ) ∈ah, then ‖ ( y0, v0, . . . , yn, vn, a ) ‖ah = max ( max 0≤n≤n ‖yn‖∞, max 0≤n≤n ‖vn‖∞,‖a‖∞ ) . on the other hand, if( y0, z0, a0, z0, y1, z1, . . . , yn, zn, a ) ∈bh, ‖ ( y0, z0, a0, z0, y1, z1, . . . , yn, zn, a ) ‖bh =‖y0‖∞ + ‖z0‖∞ + |a0| + ‖z0‖∞ + n∑ n=1 k ‖zn‖∞ + n∑ n=1 k ‖yn‖∞ + n∑ n=1 k |an|. now, for each h ∈ h, we define xh = (x0, x1, x2, . . . , xn ), xn = (xn1, . . . , x n j+n) ∈ r j+n, x0j = j h, 1 ≤ j ≤ j, xnj = x(t n; tn−1, xn−1j−1 ), 1 ≤ j ≤ j + n, 1 ≤ n ≤ n. (22) also, xn+ 1 2 = (x n+ 12 1 , . . . , x n+ 12 j+n+1) ∈r j+n+1, x n+ 12 j = x(t n+ 12 ; tn, xnj−1), 1 ≤ j ≤ j + n + 1, (23) 0 ≤ n ≤ n − 1. we denote xn0 = x n+ 12 0 = 0, n ≥ 0. in addition, if u represents the theoretical solution to (1)-(5) we define uh = (u0, u1, u2, . . . , un ), un = (un0, u n 1, . . . , u n j+n) ∈r j+n+1, unj = u(x n j, t n), 0 ≤ j ≤ j + n, 0 ≤ n ≤ n, (24) and un+ 1 2 = (u n+ 12 0 , u n+ 12 1 , . . . , u n+ 12 j+n+1) ∈r j+n+2, u n+ 12 j = u(x n+ 12 j , t n), 0 ≤ j ≤ j + n + 1, 0 ≤ n ≤ n−1. (25) finally, if s is the theoretical solution to (4) then we define sh = (s0, s1, s2, . . . , sn ), sn = s (tn), 0 ≤ n ≤ n, (26) and sn+ 1 2 = s (tn+ 1 2 ), 0 ≤ n ≤ n − 1. (27) therefore ũh = (x0, u0, x1, u1, . . . , xn, un, sh) ∈ ah. next, we introduce the discretization operator. let r be a positive constant and we denote by bah (ũh, r h p) ⊂ ah the open ball with center ũh and radius r hp, 1 < p < 2, φh : bah (ũh, r h p) →bh, biomath 3 (2014), 1403241, http://dx.doi.org/10.11145/j.biomath.2014.03.241 page 5 of 11 http://dx.doi.org/10.11145/j.biomath.2014.03.241 o angulo et al., numerical analysis of a size-structured population model... φh ( y0, v0, . . . , yn, vn, a ) = ( y0, p0, a0, p0, y1, p1, . . . , yn, pn, a ) , (28) defined by the following equations: y0 = y0 − x0 ∈rj, (29) p0 = v0 − u0 ∈rj+1, (30) a0 = a0 − s 0 ∈r. (31) vectors x0, u0 and value s 0 represent approximations at t = 0, respectively, to the initial grid nodes, to the theoretical solution at these points and to the initial resource. also, pn+10 = v n+1 0 − q ( yn+1,αn+1 · vn+1 ) g ( 0, an+1, tn+1 ) , (32) y n+1j+1 = 1 k { yn+1j+1 −ynj − k g(y n+ 12 ,∗ j+1 , a n+ 12 ,∗, tn+ 1 2 ) } , (33) pn+1j+1 = 1 k { v n+1j+1 (34) − v nj exp ( −k µ∗ ( y n+ 12 ,∗ j+1 , a n+ 12 ,∗, tn+ 1 2 ))} , (35) 0 ≤ j ≤ j + n − 1, an+1 = 1 k { an+1 − an −k f ( an+ 1 2 ,∗,q(yn+ 1 2 ,∗,γn+ 1 2 ,∗ · vn+ 1 2 ,∗), tn+ 1 2 )} , (36) 0 ≤ n ≤ n−1. where, with the notation introduced in section iii, y n+ 12 ,∗ j+1 = y n j + k 2 g(ynj, a n, tn), (37) v n+ 12 ,∗ j+1 = v n j exp ( − k 2 µ∗ ( ynj, a n, tn )) , (38) 0 ≤ j ≤ j + n − 1, v n+ 12 ,∗ 0 = q(yn+ 1 2 ,∗,αn+ 1 2 ,∗ · vn+ 1 2 ,∗) g(0, an+ 1 2 ,∗, tn+ 1 2 ) , (39) an+ 1 2 ,∗ = an + k 2 f (an,q(yn,γn · vn), tn) , (40) 0 ≤ n ≤ n − 1. we denote by q(x, v) = m∑ l=0 ql(x) vl the general quadrature rule employed in (32)-(40). note that, φh takes into account all the possible nodes and their corresponding solution values at each time level, and it employs quadrature rules possibly based on a subgrid. if ũh = (x0, u0, x1, u1, . . . , xn, un, s) ∈ bah (ũh, r h p), satisfies φh(ũh) = 0 ∈bh, (41) the nodes and the corresponding values of the solution at such nodes of ũh are a numerical solution to the scheme defined by (13)-(19) when the composite trapezoidal quadrature rule is given. on the other hand, the numerical solution of the scheme defined by (13)-(19) satisfies (41). henceforth, c will denote a positive constant, independent of h, k (k = r h), j (0 ≤ j ≤ j + n) and n (0 ≤ n ≤ n); c may have different values in different places. now, we assume that the quadrature rules satisfy the following properties: (p1) |i(tn) −q (xn,γn · un)| ≤ c h2, when h → 0, 0 ≤ n ≤ n. (p2) ∣∣∣∣i(tn−12 ) −q(xn−12 ,γn−12 · un−12 )∣∣∣∣ ≤ c h2, when h → 0, 1 ≤ n ≤ n. (p3) ∣∣∣∣∣∣ ∫ xm (tn ) 0 α(x, s (tn), tn) u(x, tn) d x −q (xn,αn · un) ∣∣∣∣∣∣ ≤ c h2, when h → 0, 0 ≤ n ≤ n. (p4) ∣∣∣∣∣∣∣∣ ∫ xm (tn− 12 ) 0 α(x, s (tn− 1 2 ), tn− 1 2 ) u(x, tn− 1 2 ) d x −q ( xn− 1 2 ,αn− 1 2 · un− 1 2 )∣∣∣∣ ≤ c h2, when h → 0, 1 ≤ n ≤ n. (p5) |q j(xn)| ≤ q h, where q is a positive constant independent of h, k, j (0 ≤ j ≤ j + n) and n (0 ≤ n ≤ n), for 0 ≤ j ≤ j + n, 0 ≤ n ≤ n. (p6) |q j(xn− 1 2 )| ≤ q h, where q is a positive constant independent of h, k, j (0 ≤ j ≤ j + n) and n (0 ≤ n ≤ n), for 0 ≤ j ≤ j + n, 1 ≤ n ≤ n. (p7) let r and p be positive constants with 1 < p < 2. the quadrature weights q j are lipschitz continuous functions on b∞(xn, r hp), biomath 3 (2014), 1403241, http://dx.doi.org/10.11145/j.biomath.2014.03.241 page 6 of 11 http://dx.doi.org/10.11145/j.biomath.2014.03.241 o angulo et al., numerical analysis of a size-structured population model... 0 ≤ j ≤ j + n, 1 ≤ n ≤ n and on b∞(xn− 1 2 , r hp), 0 ≤ j ≤ j + n, 1 ≤ n ≤ n. (p8) let r and p be positive constants with 1 < p < 2. if yn, zn ∈ b∞(xn, r hp), vn ∈ b∞(un, r hp) and an ∈ b∞(sn, r hp), then∣∣∣∣∣∣∣ j+n∑ i=0 (qi(yn) − qi(zn)) γ(zni , a n, tn) v ni ∣∣∣∣∣∣∣ ≤ c‖yn − zn‖∞, when h → 0, 0 ≤ n ≤ n. (p9) let r and p be positive constants with 1 < p < 2. if yn, zn ∈ b∞(xn, r hp), vn ∈ b∞(un, r hp) and an ∈ b∞(sn, r hp), then∣∣∣∣∣∣∣ j+n∑ i=0 (qi(yn) − qi(zn)) α ( zni , a n, tn ) v ni ∣∣∣∣∣∣∣ ≤ c‖yn − zn‖∞, when h → 0, 0 ≤ n ≤ n. (p10) let r and p be positive constants with 1 < p < 2. if yn− 1 2 , zn− 1 2 ∈ b∞(xn− 1 2 , r hp), vn− 1 2 ∈ b∞(un− 1 2 , r hp), and an− 1 2 ∈ b∞(sn− 1 2 , r hp), then∣∣∣∣∣∣∣ j+n+1∑ i=0 ( qi(yn− 1 2 ) − qi(zn− 1 2 ) ) γ ( z n−12 i , a n−12 , tn− 1 2 ) v n−12 i ∣∣∣∣∣∣∣ ≤ c ‖yn− 1 2 − zn− 1 2‖∞, when h → 0, 1 ≤ n ≤ n. (p11) let r and p be positive constants with 1 < p < 2. if yn− 1 2 , zn− 1 2 ∈ b∞(xn− 1 2 , r hp), vn− 1 2 ∈ b∞(un− 1 2 , r hp), and an− 1 2 ∈ b∞(sn− 1 2 , r hp), then∣∣∣∣∣∣∣ j+n+1∑ i=0 ( qi(yn− 1 2 )− qi(zn− 1 2 ) ) α ( z n−12 i , a n−12 , tn− 1 2 ) v n−12 i ∣∣∣∣∣∣∣ ≤ c‖yn− 1 2 − zn− 1 2‖∞, when h → 0, 1 ≤ n ≤ n. these are the enough properties the quadrature rules have to satisfy to carry out our convergence analysis. the following result establishes that the composite trapezoidal rule used in our experiments satisfies them. theorem 2. assume that the hypotheses (h1)(h7) hold. if the quadrature rules are the composite trapezoidal quadrature on subgrids { xnjnl }m(n) l=0 , 0 ≤ n ≤ n with the property (sr) there exists a positive constant c such that, for h sufficiently small, xnjnl+1 − xnjnl ≤ c h, 0 ≤ l ≤ m(n) − 1, xnjn0 = 0, xnjnm(n) = xj+n, with{ xnjnl }m(n)−1 l=1 contained in xn, 0 ≤ n ≤ n. then, properties (p1)-(p11) hold. now we introduce the following result over the numerical values at the half-level time. proposition 1. assume that the hypotheses (h1)(h7) hold and that the considered quadrature rules satisfy properties (p1)-(p11). let be yn ∈ b∞(xn, r hp), vn ∈ b∞(un, r hp) and an ∈ b∞(sn, r hp). then, as h → 0, yn+ 1 2 ,∗ ∈ b∞(xn+ 1 2 , r′ hp), an+ 1 2 ,∗ ∈ b∞(sn+ 1 2 , r′ hp), and vn+ 1 2 ,∗ ∈ b∞(un+ 1 2 , r′ hp) where xn+ 1 2 , sn+ 1 2 , and un+ 1 2 are defined by (23), (27) and (25), respectively. now, we define the local discretization error as lh = φh(ũh) ∈bh, and we say that the discretization (28) is consistent if, as h → 0, lim‖φh(ũh)‖bh = lim‖lh‖bh = 0. the following theorem establishes the consistency of the numerical scheme defined by equations (29)-(36). theorem 3. assume that hypotheses (h1)-(h7) hold and that the considered quadrature rules satisfy properties (p1)-(p11). then, as h → 0, the local discretization error satisfies, ‖φh(ũh)‖bh = ‖u 0 − u0‖∞ + ‖x0 − x0‖∞ + |s0 − s 0| + o(h2 + k2). (42) another notion that plays an important role in the analysis of the numerical method is the stability with h-dependent thresholds. for h ∈ h, let rh be a real number ( the stability threshold) with biomath 3 (2014), 1403241, http://dx.doi.org/10.11145/j.biomath.2014.03.241 page 7 of 11 http://dx.doi.org/10.11145/j.biomath.2014.03.241 o angulo et al., numerical analysis of a size-structured population model... 0 < rh < ∞: we say that the discretization (28) is stable for ũh restricted to the thresholds rh, if there exist two positive constants h0 and s ( the stability constant) such that, for any h ∈ h with h ≤ h0, the open ball bah (ũh, rh) is contained in the domain of φh, and, for all ṽh, w̃h in that ball, ‖ṽh − w̃h‖ ≤ s ‖φh(ṽh) −φh(w̃h)‖. below, we introduce the theorem that establishes the stability of the discretization defined by equations (29)-(36). theorem 4. assume that hypotheses (h1)-(h7) hold and that the considered quadrature rules satisfy properties (p1)-(p11). then, the discretization is stable for ũh with rh = r hp, 1 < p < 2. finally, we define the global discretization error as ẽh = ũh − ũh ∈ah, we say that the discretization (28) is convergent if there exists h0 > 0 such that, for each h ∈ h with h ≤ h0, (41) has a solution ũh for which, as h → 0, lim‖ũh − ũh‖ah = lim‖ẽh‖ah = 0. we propose the following theorem which establishes the convergence of the numerical method defined by equations (29)-(36). theorem 5. assume that hypotheses (h1)-(h7) hold and that the considered quadrature rules satisfy properties (p1)-(p11). then, for h sufficiently small, the numerical method defined by equations (29)-(36) has a unique solution uh ∈ b(uh, rh) and ‖uh − uh‖ah ≤ c ( ‖x0 − x0‖∞ + ‖u0 − u0‖∞ +|s0 − s 0| + o(h2 + k2) ) . (43) the proof of theorem 5 is immediately derived by means of the consistency (theorem 3), the stability (theorem 4) and theorem 1. next, we can establish an error bound for the the numerical and the theoretical solution at the numerical values of the grid nodes. theorem 6. assume that hypotheses (h1)-(h7) hold and that the considered quadrature rules satisfy properties (p1)-(p11). for h sufficiently small, let be u∗h = (u 0 ∗, u 1 ∗, u 2 ∗, . . . , u n ∗ ) ∈ n∏ n=0 rj+n+1, defined by un∗ = ( u(xn0, t n), u(xn1, t n), . . . , u(xnj+n, t n) ) ∈rj+n+1, 0 ≤ n ≤ n, where xnj , 0 ≤ j ≤ j + n, 0 ≤ n ≤ n, are the grid nodes given by the scheme (29)-(36). then, ‖un − un∗‖∞ ≤ c ( ‖x0 − x0‖∞ + ‖u0 − u0‖∞ +|s0 − s 0| + o(h2 + k2) ) . (44) this theorem follows immediately from theorem 5. specifically, if x0 = x0, u0 = u0 and s 0 = s0, the proposed numerical scheme is secondorder accurate. at this moment, we have obtained convergence of the numerical method (29)-(36) which does not employ selection at each time level. also, we have proven the convergence of numerical methods which employ a selection criterion, whenever the positions, which are determined by the criterion we have chosen, lead us to subgrids which satisfy property (sr). for the criterion presented in this paper, this property may be shown in two stages. first, as proved in [11], it leads us to subgrids with such a property when we applied it over nodes which are in a neighbourhood of the theoretical ones with radius r hp. in a second stage, it is proven that the nodes, which in fact the numerical method computes, are in such neighbourhoods. in order to do this, it is enough to realize that such nodes could be seen, up to each time level, as the solutions obtained by a discrete operator which has the form of that defined in (28). v. numerical results we have carried out different numerical experiments with the scheme defined in section iii. we have considered a theoretical test problem with meaningful nonlinearities (both from a mathematical and biological point of view). the numerical integration for the numerical experiment was biomath 3 (2014), 1403241, http://dx.doi.org/10.11145/j.biomath.2014.03.241 page 8 of 11 http://dx.doi.org/10.11145/j.biomath.2014.03.241 o angulo et al., numerical analysis of a size-structured population model... carried out on the time interval [0, 10]. the size interval was taken as [0, 1]. the size-specific growth, fertility and mortality moduli are chosen as g(x, z, t) = λ2 1+z z (( z 1+z )2 − x2 ) + xr1+z ( 29 30 − z k ) , α(x, z, t) = 32λ 1+ ( z c( 2930 − zk ) )−29λ 30r 1+2 ( z c( 2930 − zk ) )−29λ 30r , µ(x, z, t) = λ 1+zz ( z 1+z + 2x ) − 3r 1+z ( 29 30 − z k ) . the weight function is taken as γ(x, z, t) = x2 and, finally, f (z, i, t) = rz ( 1 − zk ) − rzi (1+z) 5 z5 (1+4e−λt ) . with this choice of data functions, the problem (1)-(5) has the following solution u(x, t) = ( s (t) 1 + s (t) − x )2 − e−λt  ( s (t) 1 + s (t) )2 − x2 , s (t) = 29 30 ce29rt/30 1 + ce29rt/30/k , r = 0.1, c = 24, k = 5, λ = 0.3. since we know the exact solution to the problem, we can show numerically that our method is second-order accurate by means of an error table. in table i, each entry in columns two to five represents, at the upper value, the global error eh,k = max { max 0≤ j≤j |u(x0j , t 0) − u 0j |, |s (t 0) − s 0|, max 1≤n≤n { max 0≤ j≤j |u(xnj , t n) − u nj | } , |s (tn) − s n| } and, at the lower number, the experimental order s of the method as computed from s = log (e2h,2k/eh,k )log 2 . each column and each row of the table correspond to different values of the spatial and time discretization parameter, respectively. the results in the table clearly confirm the expected secondorder convergence. on the other hand, the numerical integration of the model with an efficient method allows us to consider a more realistic test problem. the property of convergence in finite time interval is important to carry out experiments in which the long-time behaviour of the population is investigated. therefore, the numerical method has been employed to describe the dynamics of a population k\h 1.25e-2 6.25e-2 3.13e-2 1.56e-2 1.25e-2 1.67e-4 1.27e-4 1.22e-4 1.21e-4 6.25e-2 2.08e-4 4.18e-5 3.15e-5 3.03e-5 2.00 2.01 2.01 3.13e-2 2.02e-4 5.26e-5 1.05e-5 7.85e-6 1.98 2.00 2.00 1.56e-2 2.08e-4 5.05e-5 1.32e-5 2.62e-6 2.00 1.99 2.00 table i error and experimental order of convergence. of ectothermic invertebrates. this is the case of the water flea, daphnia magna. in this particular case, the functions data are given by g(x, s, t) = g( s1+s − x), µ(x, s, t) = µ, α(x, s, t) = α s 1+s x 2, f (s, i, t) = rs ( 1 − sk ) − i s1+s , γ(x, s, t) = x 2, and the values of the parameters are given by g = 1, µ = 0.1, α = 0.75, r = 3, k = 8.3 and xm = 1 [6]. this set led us to unbounded solutions [12]. nevertheless, for the parameter value g = 0.0075, the solution is bounded. as it was pointed in [10], as the value of the parameter k increases, the equilibrium state becomes unstable. we have performed a numerical experiment with the discretization parameters k = 0.0625, j = 4000 and the interesting value k = 9.64. in this case, we observe the unstability of the equilibrium and the solution evolving towards a cycled situation (figure 1). taking into account that the numerical solution is attracted to a limit cycle, considering a sufficiently large time, the numerical solution obtained after this long time integration lies practically on such a cycle. in this way, the numerical method provides an approximation to the limit cycle. in figure 2, the representation of such a cycle in the tridimensional space defined by the total population, the maximum individual size and the dynamical resource, is drawn. from the numerical results obtained for the total population, the maximum size and the dynamical resource, we can obtain an in depth analysis of these quantities throughout a period of the limit cycle. for example, we have estimated the period of the solution by interpolation and it is about 64.6824. biomath 3 (2014), 1403241, http://dx.doi.org/10.11145/j.biomath.2014.03.241 page 9 of 11 http://dx.doi.org/10.11145/j.biomath.2014.03.241 o angulo et al., numerical analysis of a size-structured population model... fig. 1. evolution of the numerical solution. case of a bounded unstable steady state. 46 48 50 52 54 0.76 0.77 0.78 0.79 0.8 0.81 0.82 0.83 0.84 2 3 4 5 6 maximum size total population re s o u rc e fig. 2. limit cycle appearing in the case of a bounded unstable steady state. vi. conclusions we have analyzed a second-order numerical method for a problem that describes a population with a possible shrinking size and with a dependency on the environment managed by the evolution of a vital resource. the second-order convergence has been theoretically proven by means of an argument of consistency and stability of the scheme. we have reported numerical experiments which demonstrate the predicted accuracy of the scheme. this knowledge leads us to employ this secondorder numerical method that, experimentally, shows good stability properties for the study of the behaviour of a real population. we have consider the long time numerical approximation of a particular model which describes the dynamics of a daphnia magna population. moreover, when the steady state is unstable we have used it to analyzed the dynamics, appearing a limit cycle, and a good approximation to the bifurcation process has been obtained. acknowledgments this work was supported in part by the ministerio de ciencia e innovación (spain), project mtm2011-25238, and by the junta de castilla y león (spain), project va191u13. references [1] g. f. webb, theory of nonlinear age-dependent population dynamics, marcel dekker, eds, new york, 1985. [2] j. a. j. metz and e. o. dieckmann, editors, the dynamics of physiologically structured populations, springer lecture notes in biomathematics, 68, springer, heildelberg, 1986. [3] m. iannelli, mathematical theory of age-structured population dynamics, applied mathematics monographs. c.n.r., giardini editori e stampatori, pisa, 1995. [4] j. m. cushing. an introduction to structured populations dynamics, cmb-nsf regional conference series in applied mathematics. siam, 1998. [5] b. perthame, transport equations in biology, birkhäuser verlag, basel, 2007. [6] a. m. de roos, numerical methods for structured population models: the escalator boxcar train, numer. methods partial differential equations, 4 (1988), 173–195. [7] h. thieme, http://math.la.asu.edu/∼dieter/workshop/schedule.html, 2003. [8] a. calsina and j. saldaña. a model of physiologically structured population dynamics with a nonlinear individual growth rate, j. math. biol., 33(4) (1995) 335–364. [9] o. diekmann, m. gyllenberg, j.a.j. metz, s. nakaoka, a.m. de roos, daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example, j. math. biol. 61 (2010), 277-318. [10] l. abia, o. angulo, j. c. lópez-marcos and m. a. lópez-marcos, long-time simulation of a sizestructured population model with a dynamical resource, math. model. nat. phenom., 5 (2010), 1–21. [11] o. angulo and j.c. lópez-marcos. numerical integration of fully nonlinear size-structured population models, app. num. math., 50 (2004) 291-327. biomath 3 (2014), 1403241, http://dx.doi.org/10.11145/j.biomath.2014.03.241 page 10 of 11 http://math.la.asu.edu/~dieter/workshop/schedule.html http://dx.doi.org/10.11145/j.biomath.2014.03.241 o angulo et al., numerical analysis of a size-structured population model... [12] o. angulo, j. c. lópez-marcos and m. a. lópezmarcos, numerical approximation of singular asymptotic states for a size-structured population model with a dynamical resource, math. comput. modelling, 54 (2011), 1693–1698. [13] j. c. lópez-marcos and j. m. sanz-serna, stability and convergence in numerical analysis iii: linear investigation of nonlinear stability, ima j. numer. anal., 8 (1988), 71–84. [14] h. stetter, analysis of discretization methods for ordinary differential equations, springer, berlin, 1973. http://dx.doi.org/10.1007/978-3-642-65471-8 [15] l.m. abia, o. angulo, j. c. lópez-marcos and m. a. lópez-marcos, numerical integration of a hierarchically size-structured population model with contest competition, j. comp. appl. math., 258 (2014), 116–134. biomath 3 (2014), 1403241, http://dx.doi.org/10.11145/j.biomath.2014.03.241 page 11 of 11 http://dx.doi.org/10.1007/978-3-642-65471-8 http://dx.doi.org/10.11145/j.biomath.2014.03.241 introduction preliminary results the numerical method convergence analysis numerical results conclusions references www.biomathforum.org/biomath/index.php/biomath review article optimal control strategies for a class of vector borne diseases, exemplified by a toy model for malaria sebastian aniţa∗, edoardo beretta†, vincenzo capasso‡ ∗faculty of mathematics, “alexandru ioan cuza” university of iaşi and “octav mayer” institute of mathematics of the romanian academy iaşi, romania sanita@uaic.ro †cimab, interuniversity centre for mathematics applied to biology, medicine, and environmental sciences, italy edoardo.beretta@uniurb.it ‡adamss, centre for advanced applied mathematical and statistical sciences università degli studi di milano la statale milano, italy vincenzo.capasso@unimi.it received: 16 february 2019, accepted: 15 september 2019, published: 13 october 2019 abstract—this paper contains a unified review of a set of previous papers by the same authors concerning the mathematical modelling and control of malaria epidemics. the presentation moves from a conceptual mathematical model of malaria transmission in an homogeneous population. among the key epidemiological features of this model, two-ageclasses (child and adult) and asymptomatic carriers have been included. as possible control measures, the extra mortality of mosquitoes due to the use of long-lasting treated mosquito nets (llins) and indoor residual spraying (irs) have been included. by taking advantage of the natural double time scale of the parasite and the human populations, it has been possible to provide interesting threshold results. in particular, key parameters have been identified such that below a threshold level, built on these parameters, the epidemic tends to extinction, while above another threshold level it tends to a nontrivial endemic state. the above model has motivated further analysis when a spatial structure of the relevant populations is added. inspired by the above, additional model reductions have been introduced, which make the resulting reactiondiffusion system mathematically affordable. only the dynamics of the infected mosquitoes and of the infected humans has been included, so that a two-component reaction-diffusion system is finally copyright: c©2019 aniţa et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: sebastian aniţa, edoardo beretta, vincenzo capasso, optimal control strategies for a class of vector borne diseases, exemplified by a toy model for malaria, biomath 8 (2019), 1909157, http://dx.doi.org/10.11145/j.biomath.2019.09.157 page 1 of 19 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2019.09.157 s. aniţa, e. beretta, v. capasso, optimal control strategies for a class of vector borne diseases ... taken. the spread of the disease is controlled by three actions (controls) implemented in a subdomain of the habitat: killing mosquitoes, treating the infected humans and reducing the contact rate mosquitoes-humans. to start with, the problem of the eradicability of the disease is considered, while the cost of the controls is ignored. we prove that it is possible to decrease exponentially both the human and the vector infective population everywhere in the relevant habitat by acting only in a suitable subdomain. later the regional control problem of reducing the total cost of the damages produced by the disease, of the controls and of the intervention in a certain subdomain is treated for the finite time horizon case. in order to take the logistic structure of the habitat into account the level set method is used as a key ingredient for describing the subregion of intervention. here this subregion has been better characterized by both area and perimeter. the authors wish to stress that the target of this paper mainly is to attract the attention of the public health authorities towards an effective and affordable practice of implementation of possible control strategies. keywords-malaria; nonlinear ode models; qualitative analysis; reaction-diffusion system; zerostabilization; regional control; optimal control; epidemic systems; i. introduction ”mathematical models are lies that make us reach the truth”. (paraphrased from picasso) human malaria is caused by one or a combination of four species of plasmodia: plasmodium falciparum, p. vivax, p. malariae, and p. ovale. the parasites are transmitted through the bite of infected female mosquitoes of the genus anopheles. mosquitoes can become infected by feeding on the blood of infected people, and the parasites then undergo another phase of reproduction in the infected mosquitoes. according to [https://data.unicef.org/topic/ child-health/malaria/], “prevention has been mainly carried out by the use of bed nets. sleeping under insecticide-treated mosquito nets (itns) on a regular basis is one of the most effective ways to prevent malaria transmission and reduce malaria related deaths. since 2000, production, procurement and delivery of itns, particularly long lasting insecticide treated nets (llins) have accelerated, resulting in increased household ownership and use. ... household ownership of itns/llins is uneven across countries in the region, with an average coverage of 66 per cent in sub-saharan africa ranging from less than 30 per cent to approximately 90 per cent but most countries have made considerable progress in the past decade. the proportion of sub-saharan african households with at least one itn increased to 67 per cent in 2016, thus, a third of households where itns are the main method of vector control did not have access to a net. additionally, only 42 per cent of households had sufficient itns for all household members which is drastically short of the universal access of 100 per cent to this preventive measure.” this shows the relevance of studies as the ones proposed in this paper, concerning the identification of regions where universal access might be implemented, with respect to areas where this is not affordable because of various restrictions due to financial and logistic affordability. the earliest attempt to provide a quantitative understanding of the dynamics of malaria transmission was that of ross [41]. ross’ models consisted of a few differential equations to describe changes in densities of susceptible and infected people, and susceptible and infected mosquitoes. macdonald [34], [35] extended ross’ basic model, analyzed several factors contributing to malaria transmission, and concluded that “the least influence is the size of the mosquito population, upon which the traditional attack has always been made”. the work of macdonald had a very beneficial impact on the collection, analysis, and interpretation of epidemic data on malaria infection [36] and guided the enormous global malaria-eradication campaign of his era. the classical ross-macdonald model has inspired many contributions considering a variety of additional epidemiological features of the contagion [12], [26] (see also [13], [20], [22], [23], [27], biomath 8 (2019), 1909157, http://dx.doi.org/10.11145/j.biomath.2019.09.157 page 2 of 19 https://data.unicef.org/topic/child-health/malaria/ https://data.unicef.org/topic/child-health/malaria/ http://dx.doi.org/10.11145/j.biomath.2019.09.157 s. aniţa, e. beretta, v. capasso, optimal control strategies for a class of vector borne diseases ... [31], [42], [44]). a realistic mathematical model for an epidemic malaria system would require, at least for the human population, splitting the population with respect to stages of the disease, e. g. susceptibles, infectives, immunes, and alike; age structure, spatial structure, and alike. even though we may suppose that indeed such a model is realistic, it might exhibit a complexity that make it hard for a sound mathematical analysis, and its validation as far as fitting real data is concerned. expected uncertainty in the parameters would add additional complexity, so that the design of sound control strategies appears to be unaffordable. reduction of complexity has then become a key issue, though one must take into account that reduction has not to lead to meaningless models with respect to real epidemic systems (a great interpretation of this concept has been offered by picasso http://www.dailyartmagazine. com/pablo-picassos-bulls-road-simplicity/). this paper is meant to present a unified review of a set of previous papers by the same authors, the crucial issue being the implementation of optimal control strategies solely on an optimally chosen subregion of the habitat. in order to include specific issues of malaria epidemic, we have started by considering a spatially homogeneous population with two age groups (children and adults) for the human population, and two classes of human individuals with respect to symptoms (asymptomatic and symptomatic) [39], [15]. in [39] the role of uncertainty in data was analyzed with respect to optimal control strategies. but the mathematical analysis of the full model was far from satisfactory, due to its intrinsic nontrivial structure. in [15], by taking advantage of the natural double time scale of the parasite and the human populations, the authors have been able to obtain a significant model reduction and then to provide interesting threshold results. in particular, there it is shown that key parameters can be identified such that below a threshold level the epidemic tends to extinction, while above another threshold level it tends to a nontrivial endemic state. in [8], [10] a spatial structure has been taken into account, but no age structure. though the above mentioned simplifications lead de facto to a toy model, we have been using it as the basis for implementing sound realistic control strategies, inspired by the outcomes of the results in [39], [15]. the implementation of control strategies on an optimally chosen subregion of the habitat is a crucial addition of our proposal. although we have tried to make clear the assumptions underlying our model, it has not been validated yet, by comparison with experimental data. therefore it has to be cautioned that it is far from being the last word on a real malaria epidemic. however it is hoped that our model, with additional features that make it more realistic, when combined with efficient numerical methods, might provide the foundations for designing optimal control strategies by public health authorities. we wish to clarify that the reference to malaria does not exclude possible applications to other vector borne diseases, with relevant modifications that may take into account their specific epidemiological issues. for optimal control problems related to population dynamics we refer to the monographs [2], [3], [33], and references therein. a control problem of malaria for a space independent model may be found in [1]. for basic results and methods in shape optimization we refer to [25], [30], [40], [46]. it can be just mentioned that the models presented here might be extended to other vector borne diseases, though this would require taking into account specific features of other diseases. this paper is organized as follows. in section ii a spatially homogeneous population with two age groups (children and adults) for the human population, and two classes of human individuals with respect to symptoms (asymptomatic and symptomatic). by taking advantage of the double scale structure of the model, a reduced epidemic model is obtained, for which a qualitative analysis is carried out so to obtain a key threshold theorem. biomath 8 (2019), 1909157, http://dx.doi.org/10.11145/j.biomath.2019.09.157 page 3 of 19 http://www.dailyartmagazine.com/pablo-picassos-bulls-road-simplicity/ http://www.dailyartmagazine.com/pablo-picassos-bulls-road-simplicity/ http://dx.doi.org/10.11145/j.biomath.2019.09.157 s. aniţa, e. beretta, v. capasso, optimal control strategies for a class of vector borne diseases ... attention is paid to the behaviour of the total human population. in section iii the modelling, analysis and possible control strategies of the epidemic system with spatial structure has been reported. in particular a theorem is reported concerning the eventual eradicability of the disease, guaranteed by the proposed control strategies. in section iv optimal regional control problems are discussed, including possible control strategies, and the optimal choice of the region of intervention. different from previous papers, here the region of intervention has been characterized by both its area and perimeter. ii. a compartmental model with two age groups the model developed in this paper combines two sub-models, the vectorial dynamics and the human host dynamics. similar to ross-macdonald model [34], [32], once it is assumed that the vector population is comprised of only female anopheles mosquitoes, they are categorized into sv : uninfected (susceptible) mosquitoes, and iv : infected mosquitoes. therefore the total mosquito population is nv = sv + iv. the total human population nh has been divided into six classes: x : susceptible children , y : susceptible adults, x1 : asymptomatic infected children, x2 :symptomatic infected children, y1 : asymptomatic infected adults, y2 : symptomatic infected adults (y2). hence, nh = x + x1 + x2 + y + y1 + y2. the parameters of the model are: ηh: per capita birth rate of humans. ηv: per capita birth rate of mosquitoes. λh: transmission rate from infected mosquitoes to susceptible humans. λv: transmission rate from infected humans to susceptible mosquitoes. a: mosquito biting rate. b: probability of transmission of infection from an infectious mosquito to a susceptible human. c: probability of transmission of infection from an infectious human to a susceptible mosquitoes. fx: proportion of symptomatic infection in children. fy: proportion of symptomatic infection in adults. νx: child natural recovery rate from asymptomatic infection. νy: adult natural recovery rate from asymptomatic infection. rx: child natural recovery rate from symptomatic infection. ry: adult natural recovery rate from symptomatic infection. φ: modification parameter for recovery rate from symptomatic infection. g: human maturity rate. µv: mosquito natural mortality rate. µh: human natural mortality rate. biomath 8 (2019), 1909157, http://dx.doi.org/10.11145/j.biomath.2019.09.157 page 4 of 19 http://dx.doi.org/10.11145/j.biomath.2019.09.157 s. aniţa, e. beretta, v. capasso, optimal control strategies for a class of vector borne diseases ... δy: disease induced mortality rate in adults. δx: disease induced mortality rate in children. in the mathematical model we have denoted by λh the infection rate, due to the infective humans, acting on each susceptible mosquito, while by λv we have denoted the infection rate, due to the infective mosquitoes, acting on each susceptible human individual. we will adopt here the infection rates as in the classical ross-mcdonald model [34] so that λh = probability per unit time that a susceptible human is infected due a bite by an infective mosquito = relative abundance of mosquitoes with respect to the human population × probability per unit time a human is bitten by a mosquito × prob{the biting mosquito is infective| a mosquito has bitten a human} × prob{a susceptible human is infected| he has been bitten by an infective textnormalmosquito} = nv nh a iv nv b. similarly, λv = probability per unit time that a susceptible mosquito is infected due to a bite to an infective human = probability per unit time a mosquito bites a human × prob{the bitten human is infective| a mosquito has bitten a human} × prob{a susceptible mosquito is infected| he has bitten an infective human} = a ih nh c. where ih = x1 + x2 + y1 + y2. it is assumed that the individuals in the adult subpopulation have acquired an immunity and hence they are able to clear fast the infections (ry > rx), may have prolonged period of asymptomatic p. falciparum infection (1/νy > 1/νx) and suffers less disease induced mortality (δx > δy) compared to children. an extra death rate αv of the total mosquitoes population has been included, which may take into account the additional mortality due to the use of llins and the indoor residual spraying (irs) of mosquitoes. as in the force of infection of humans by mosquitoes, we have made the choice that this extra death rate of mosquitoes depends itself upon the relative abundance of mosquitoes with respect to the total human population. this modification takes into account the epidemiological issues of the ross-mcdonald model. in this framework it is more convenient to deal with fractional quantities by scaling each subpopulation with the total population at any time t as x(t) = x(t) nh(t) , y(t) = y (t) nh(t) , · · · , iv(t) = iv(t) nv(t) . in this way, if we set m(t) := nv(t) nh(t) , the infection rates become λh(t) = bam(t)iv(t), λv(t) = acih(t). the compartmental mathematical model for malaria transmission is then represented by the system (1) of non-linear ordinary differential equations, where x = 1 − (x1 + x2 + y + y1 + y2). biomath 8 (2019), 1909157, http://dx.doi.org/10.11145/j.biomath.2019.09.157 page 5 of 19 http://dx.doi.org/10.11145/j.biomath.2019.09.157 s. aniţa, e. beretta, v. capasso, optimal control strategies for a class of vector borne diseases ... dx1 dt = (1 −fx)bamivx + (1 −φ)rxx2 − (g + νx)x1 −ηhx1 + x1(δxx2 + δyy2), dx2 dt = fxbamivx− (g + rx + δx)x2 −ηhx2 + δxx22 + δyx2y2, dy dt = gx + νyy1 + φryy2 − bamivy −ηhy + y(δxx2 + δyy2), dy1 dt = (1 −fy)bamivy + (1 −φ)ryy2 + gx1 −νyy1 −ηhy1 + y1(δxx2 + δyy2), (1) dy2 dt = fybamivy + gx2 − (ry + δy)y2 −ηhy2 + δxx2y2 + δyy22, div dt = λv − (λv + ηv)iv, sv = 1 − iv system (1) is complemented by the two equations for the total human and mosquito populations dnh dt = (ηh −µh −δxx2 −δyy2)nh, (2) dnv dt = (ηv −µv −αvm)nv. (3) it is not difficult to show that system (1) is well posed, i.e., given suitable initial conditions, it admits a unique solution. moreover if the initial conditions are such that x(0),x1(0),x2(0),y(0),y1,y2(0) ≥ 0, with x(0) + x1(0) + x2(0) + y(0) + y1(0) + y2(0) = 1, and sv(0), iv(0) ≥ 0, with sv(0) +iv(0) = 1, then the same holds at any later time t > 0. we may then continue our analysis by referring only to the lower dimensional vector xh := (x1,x2,y,y1,y2) t ∈ π := {xh ∈ r5+|x1 + x2 + y + y1 + y2 ∈ [0, 1]}, for the human population, and iv ∈ [0, 1] for the mosquito population. the total fraction of human susceptibles is sh = x + y. (4) if we introduce the vector of all fractions of infective humans ih := (x1,x2,y1,y2) t , (5) the total fraction of human infectives is given by ih = x1 + x2 + y1 + y2, (6) so that sh + ih = 1. (7) for the mosquito population we have just sv + iv = 1. (8) our analysis may be concentrated on the time behaviour of the functions ih(t), and iv(t) describing the fractions of total infective human population, and the fraction of infective mosquito population, respectively. a. the time scales according to the parameter values reported in [39], we can see that the natural birth rate ηh for the human population is of the order 10−5days−1, much lower to the natural birth rate ηv for the mosquito population which is of the order 10−2days−1, so that their ratio ε := ηh ηv � 1, say ε ∼ 10−3. it is then meaningful the introduction of two adimensional time scales as follows τh := ηht, τv := ηvt. the relationship between these two time scales is τh = ετv, so that τv, the natural time scale of the mosquito population results to be the fast time scale, while biomath 8 (2019), 1909157, http://dx.doi.org/10.11145/j.biomath.2019.09.157 page 6 of 19 http://dx.doi.org/10.11145/j.biomath.2019.09.157 s. aniţa, e. beretta, v. capasso, optimal control strategies for a class of vector borne diseases ... τh, the natural time scale of the human population, is the slow time scale. by taking into account all the above, we can obtain the time evolution of the functions ih(t), and iv(t) with respect to these time scales. let ωh := {ih ∈ r 4 +| x1 + x2 + y1 + y2 ∈ [0, 1]} (9) and ωv := [0, 1]. (10) introduce the function f : ωh × ωv → r, (11) such that, for (ih, iv) ∈ ωh × ωv, f(ih, iv) := 1 ηh [f(ih, iv) −r(ih)], (12) with f(ih, iv) := bamiv(1 − ih) −ηhih, (13) and r(ih) := (1 − ih)(δxx2 + δyy2) +(νxx1 + νyy1) + φ(rxx2 + ryy2). (14) because of (1), the time evolution of ih is then given by dih dτh = f(ih, iv), (15) coupled with div dτv = g(ih, iv), (16) where g(ih, iv) := 1 ηv [acih − (ηv + acih)iv]. (17) b. qualitative analysis of the mathematical model at first we may provide some information about the time behaviour of m = nv nh . it can be shown that, at the fast time scale lim τv→+∞ nv(τv) nh(τv) = m∗, (18) for m∗ := ηv −µv αv = σv αv , (19) if we denote σv := ηv −µv > 0. (20) as a consequence, we may then assume, at the “slow” time scale τh, m(τh) := nv(τh) nh(τh) ≡ m∗. (21) 1) the slow time scale: with respect to the same slow time scale τh, equations (15) and (16) can be rewritten as follows dih dτh = f(ih, iv), ih(0) ∈ ωh; (22) ε div dτh = g(ih, iv), iv(0) ∈ ωv. (23) 2) the fast time scale: viceversa, at the fast time scale the above equations become dih dτv = εf(ih, iv) (24) div dτv = g(ih, iv), (25) under the same initial conditions. due to the fact that ε ' 0, at this scale ih can be considered constant, so that equation (25) admits the equilibrium i∗v ≡ ϕ(ih) := acih ηv + acih , for any ih ∈ [0, 1]. (26) notice that, for any ih ∈ [0, 1], ϕ(ih) is the only solution of g(ih, iv) = 0 biomath 8 (2019), 1909157, http://dx.doi.org/10.11145/j.biomath.2019.09.157 page 7 of 19 http://dx.doi.org/10.11145/j.biomath.2019.09.157 s. aniţa, e. beretta, v. capasso, optimal control strategies for a class of vector borne diseases ... in [0, 1]. by the usual lyapunov methods, the following proposition has been proven in [15]. proposition ii.1. for any choice of ih ∈ [0, 1], the point i∗v = ϕ(ih) is a globally asymptotically stable equilibrium solution for equation (25). it is though interesting to directly evaluate the rate of convergence to the equilibrium i∗v = ϕ(ih) of equation (25). if we take zv(τv) := iv(τv) − i∗v, (27) its evolution equation is dzv dτv = −k(ih)zv, (28) subject to zv(0) = iv(0) − i∗v. (29) here k(ih) := ηv + acih ηv ≥ 1. (30) as a consequence, the solution of (28)-(29) is such that zv(τv) = zv(0) exp(−k(ih)τv) ≤ zv(0) exp(−τv), (31) i.e. |iv(τv) − i∗v| ≤ |iv(0) − i ∗ v|exp(−τv). (32) equation (32) shows that an upper estimate of the convergence rate can be made, independent of ih. furthermore, since |iv(0)−i∗v| ≤ 1, at the slow time scale τh this will be seen as |iv(τh) − i∗v| ≤ exp(−τh/ε). (33) which tends to zero extremely fast; indeed for ε ' 10−3, and τh = 10−1, exp(−τh/ε) ' exp(−100) ' 0. c. qualitative analysis at the slow time scale proposition ii.1, and the fast convergence of iv(τh) to i∗v = ϕ(ih) at the slow time scale justify, by tikhonov theorem (see e.g.[14] ), the direct substitution of iv by i∗v in equation (22). in addition, the above analysis allows us to assume that the ratio m(τh) := nv(τh) nh(τh) has reached its asymptotic value m∗ = σv αv , so that dih dτh = f̃(ih), ih(0) ∈ ωh, (34) where f̃(ih) := 1 ηh [f̃(ih)) −r(ih)] (35) with f̃(ih) = −i2hα + ihβ ηv + acih . (36) here α = ba2cm∗ + acηh, (37) and β = ba2cm∗ −ηhηv. (38) r(ih) is taken from (14). due to the structure of (14), equation (34) has always to be considered coupled with system (1) with (2) and (3), which makes its qualitative analysis rather complicated. a way to reduce such complexity is given below. we may observe that the function f̃ can be minorized by the function f`(ih) := 1 ηh [f̃(ih)) −kih], (39) since r(ih) ≤ r(ih) := kih, ih = ‖ih‖1 ∈ [0, 1], for k := max{δx + φry,νx} > 0. (40) on the other hand the function f̃ can be majorized by the function fu(ih) := 1 ηh [f̃(ih)) −kih], (41) since r(ih) ≥ r(ih) := kih, ih = ‖ih‖1 ∈ [0, 1], biomath 8 (2019), 1909157, http://dx.doi.org/10.11145/j.biomath.2019.09.157 page 8 of 19 http://dx.doi.org/10.11145/j.biomath.2019.09.157 s. aniţa, e. beretta, v. capasso, optimal control strategies for a class of vector borne diseases ... for k := min{νy,φrx} > 0. (42) we have denoted by ‖ih‖1 = x1 + x2 + y1 + y2. for the qualitative analysis it is then more convenient to study the minorant equation di`h dτh = f`(i`h), (43) together with the majorant equation diuh dτh = fu(iuh). (44) they are such that f`(ih) ≤ f(ih) ≤ fu(ih). (45) these equations are now decoupled from system (1). if we denote by ih(τh; ih(0)) the solution of equation (34) ( coupled with system (1)-(3)), subject to the inital condition ih(0); by i ` h(τh; ih(0)) the solution of (43), subject to the initial condition ih(0) = ‖ih(0)‖1; and by iuh(τh; ih(0)) the solution of (44), subject to the initial condition ih(0) = ‖ih(0)‖1, then, by classical comparison theorems we may claim that for any τh ≥ 0, i`h(τh; ih(0)) ≤ ih(τh; ih(0)) ≤ i u h(τh; ih(0)). then we have proved that, for any ih(0) ∈ (0, 1] : lim τh→+∞ [ i`h(τh; ih(0)), i u h(τh; ih(0)) ] = [̂ i`h, î u h ] where [̂ i`h, î u h ] is the ”attractor interval” for the total fraction ih(τh; ih(0)) of human infectives. d. the main threshold theorem by using the majorant equation, in [15] the authors have been able to show the following extinction theorem. theorem ii.2. if αv a2 ≥ bc ηv −µv ηv(ηh + k) (46) then the attractor interval [̂ ilh, î u h ] collapses to {0} and for all i.c. ih(0) ∈ (0, 1] we have: lim τh→+∞ ih(τh; ih(0)) = 0. (47) since all parameters, but αv and a, are specific of the relevant populations, as expected we might control the malaria eradication acting on αv, the killing rate of mosquitoes by the use of llin’s and irs, and on a, the mosquito biting rate, by the use of llin’s, and any other device/policy of contact reduction between mosquitoes and humans. we have to point out that theorem ii.2 offers only a sufficient condition for the eventual eradication of the epidemic. by the minorant equation it has been also possible to find a sufficient condition for the existence of a globally attractive endemic interval for the human infectives ih. e. about the total human population in the previous paragraphs we have reported a threshold theorem for the possible extinction of the malaria epidemic. an important issue is the persistence of the total human population, otherwise fractions may loose any epidemiological meaning. as a first step, let us rewrite equation (2) at the slow time scale dnh dτh (τh) = σh − (δx x2(τh) + δy y2(τh)) ηh nh(τh), (48) for τh ≥ 0, subject to an initial condition nh(0) = nh0 > 0. as above, we have denoted by σh := ηh−µh, that we assume to be strictly positive. in the sequel of this section we will write τ instead of τh. a minorant equation for (48) can be obtained as follows. if we denote by δ := max{δx,δy}, (49) and by ih(τ; ih0) the total fraction of human infectives, solution of equation (34) (with (1)-(3)), biomath 8 (2019), 1909157, http://dx.doi.org/10.11145/j.biomath.2019.09.157 page 9 of 19 http://dx.doi.org/10.11145/j.biomath.2019.09.157 s. aniţa, e. beretta, v. capasso, optimal control strategies for a class of vector borne diseases ... subject to an initial condition ih(0) ∈ ωh, it is not difficult to realize that the following equation for a function n`(τ), τ ≥ 0, dn` dτ (τ) = δ ηh [ σh δ − ih(τ; ih0) ] n`(τ), (50) subject to the initial condition n`(0) = nh0 > 0, is a minorant equation of (48) hence, for any τ ≥ 0, the following inequality holds nh(τ) ≥ n`(τ), (51) where n`(τ) :=nh0 exp { δ ηh [ σh δ τ− ∫ τ 0 ih(u; ih0)du ]} . a sufficient condition for the “persistence in time” of the total human population nh(τ), i.e. a sufficient condition which let us exclude that nh(τ) ↓ 0+, as τ → +∞ (52) is (see [15]) σh > δ̂i u h, (53) where îuh is the upper extreme of the ”attractor interval”. since in theorem ii.3 it occurs that î`h = î u h = 0 then we may claim that theorem ii.3. under the assumptions of theorem ii.2 a sufficient condition for the persistence in time of nh is σh > 0. (54) unfortunately we have not been able to show that nh is upper bounded, so that it might eventually explode to +∞, though not in a finite time. theorem ii.2 suggests relevant control strategies for more detailed and realistic mathematical models. next sections are devoted to the case of spatially structured models; eradication theorems will be shown, and an optimal control problem will be analyzed. iii. a spatially structured model we now report about the modelling, analysis and possible control strategies of a malaria epidemic system with spatial structure, as from [8], [10]. about the modelling let us anticipate that, in order to keep the model affordable for the mathematical analysis further reductions have been introduced, still (we hope) in the spirit of picasso. indeed we have analyzed realistic control strategies, inspired by the above outcomes taken from [39], [15]. by considering a spatially structured system, we will refer to an habitat ω ⊂ r2 (a nonempty bounded domain with a sufficiently smooth boundary ∂ω). the two populations of humans and mosquitoes will be described in terms of their spatial densities. specifically, we will consider a population of infected mosquitoes with spatial density u1(x,t) at a spatial location x ∈ ω and a time t ≥ 0; while u2(x,t) will denote the spatial distribution of the human infective population. for our model reduction we have assumed that the spatial density c(x) of the total human population has been taken essentially constant in time, independent of removals by possible acquired immunity, isolation, or death, so that c(x)−u2(x,t) provides the spatial distribution of susceptible humans, at a spatial location x ∈ ω and a time t ≥ 0. further it has been assumed that the total susceptible mosquito population is so large that it can be considered time and space independent. as far as the “local incidence” for humans, at point x ∈ ω, and time t ≥ 0, it is taken of the form (i.r.)h(x,t) = g(x,u1(x,t),u2(x,t)) = (c(x) −u2(x,t))h ( u1(x,t) c(x) ) , depending upon the local densities of both populations via a suitable functional response h that will be better described later. here we have taken into account that, in accordance with the ross-mcdonald model [34] the function h, describing the force of infection of humans by mosquitoes, depends upon the relative biomath 8 (2019), 1909157, http://dx.doi.org/10.11145/j.biomath.2019.09.157 page 10 of 19 http://dx.doi.org/10.11145/j.biomath.2019.09.157 s. aniţa, e. beretta, v. capasso, optimal control strategies for a class of vector borne diseases ... concentration of the total mosquito population with respect to the total human population, because of the specific biting habits of humans by mosquitoes. on the other hand, we have here extended the (linear) response of the ross-mcdonald model, by using a possibly nonlinear functional response h( u1(x,t) c(x) ). this choice may allow possible saturation effects, σ−type responses, etc. (see [19], [17] ). for example behavioral changes can be taken into account; for a very large density of the infective population the force of infection represented by h(u1(x,t) c(x) ) may tend to reduce itself because of reduction of open exposure of the human population. this aspect had been already proposed in [19], which has motivated a large recent literature (see [28], and references therein). as pointed out in [36], seasonality of the aggressivity to humans by the mosquito population might also be considered in the functional response h; this has been a topic of the authors research programme in [18], [17], and [6]. as far as the “local incidence” for mosquitoes, at point x ∈ ω, and time t ≥ 0, as in previous models [16], we assume that it is due to contagious bites to human infectives at any point x′ ∈ ω of the habitat, within a spatial neighborhood of x represented by a suitable probability kernel k(x,x′), depending on the specific structure of the local ecosystem (see also [43]); as a trivial simplification one may assume k(x, ·) as a gaussian density centered at x; hence the “local incidence” for mosquitoes, at point x ∈ ω, and time t ≥ 0, is taken as (i.r.)m (x,t) = ∫ ω k(x,x′)u2(x ′, t)dx′ . as proposed in [13], we have included spatial diffusion of the infective mosquito population (with constant diffusion coefficient to avoid purely technical complications), but we assume that the human population does not diffuse. more refined models may include some kind of mobility of the human population, due to migration mechanisms; but this would require further nontrivial complicacies, that we leave to further investigations. finally, we have denoted by η(x)u1(x,t) the (possibly space dependent) natural decay of the infected mosquito population, while a22u2(x,t) denotes the removal of the human infective population (by acquired immunity, isolation or death). all the above leads to the following over simplified model for the spatial spread of malaria epidemics, in which we have ignored various additional features, such as the possible differentiation of the mosquito population, etc. (see e.g. [31])   ∂tu1(x,t) = d∆u1(x,t) −η(x)u1(x,t) + ∫ ω k(x,x′)u2(x ′, t)dx′ ∂tu2(x,t) =−a22u2(x,t)+g(x,u1(x,t),u2(x,t)) in ω × (0, +∞), where η(x) is the natural decay rate of the infected mosquitoes at x ∈ ω (it is negative), a22 > 0, d > 0 are constants. the reader may recognize that the above system, apart from its spatial structure, corresponds to system (22)-(23), under additional simplifications. a. regional control strategies the public health concern consists of providing methods for the eradication of the disease in the relevant population, as fast as possible. on the other hand, very often the entire domain ω, of interest for the epidemic, is unknown, or difficult to reach for the implementation of suitable environmental sanitation programmes. this has led the authors to suggest that implementation of such programmes might be done only in a given subregion ω ⊂ ω, conveniently chosen so to lead to an effective eradication of the epidemic in the whole habitat ω (“think globally, act locally”, as in [4], [5], [6], [7]). this practice may have an enormous importance in real cases with respect to both financial and practical affordability. more recently, in [24], a patch control approach has been suggested, via the identification of zones with different disease burden. we assume here homogeneous neumann boundary condition, to mean complete isolation of the biomath 8 (2019), 1909157, http://dx.doi.org/10.11145/j.biomath.2019.09.157 page 11 of 19 http://dx.doi.org/10.11145/j.biomath.2019.09.157 s. aniţa, e. beretta, v. capasso, optimal control strategies for a class of vector borne diseases ... habitat: ∂νu1(x,t) = 0 on ∂ω × (0, +∞), where ∂ν denotes the normal derivative. we also impose initial conditions{ u1(x, 0) = u 0 1(x) u2(x, 0) = u 0 2(x) in ω, where u01 and u 0 2 are the initial densities of the population of infective mosquitoes and of the human infected population, respectively. let ω ⊂ ω be a nonempty open subset; we denote by χω the characteristic function of ω and use the convention χω(x)w(x) = 0, x ∈ r2 \ω, even if function w is not defined on the whole set r2 \ω. our goal is to study the controlled system  ∂tu1(x,t) −d∆u1(x,t) = η(x)u1(x,t) + ∫ ω k(x,x′)u2(x ′, t)dx′ −γ1(x,t)χω(x)u1(x,t), ∂tu2(x,t) =−(a22 +γ2(x,t)χω(x))u2(x,t) +(1−γ3(x,t)χω(x))g(x,u1(x,t),u2(x,t)), (55) for (x,t) ∈ q, together with ∂νu1(x,t) = 0, (x,t) ∈ σ (56) and u1(x, 0) = u 0 1(x), u2(x, 0) = u 0 2(x),x ∈ ω, (57) where q = ω × (0, +∞), σ = ∂ω × (0, +∞). the idea underlying the controls is the following: γ1(x,t)u1(x,t)χω(x) represents the additional killing of mosquitoes by the combined action of a general insecticide spraying, and the use of treated nets in the subregion ω; γ2(x,t)u2(x,t)χω(x) describes the treatment of the infected population in the subregion ω; γ3(x,t)χω(x) is the reduction of the contact rate mosquitoeshumans by means of treated nets. as far as the epidemiological significance of the various control terms, let us consider at first the quantity γ1(x,t). this may be interpreted as a harvesting rate of infective mosquitoes, by the use of chemical-physical devices such as insecticides, and γ1(x,t)u1(x,t) as the corresponding distroyed mosquitoes population at location x ∈ ω, and time t > 0. the control γ2(x,t) represents the recovery rate due to the medical treatment at location x ∈ ω, and time t > 0; the corresponding cured population is γ2(x,t)u2(x,t). the control γ3(x,t) gives the additional segregation rate for the population of infective mosquitoes and for the human susceptible population at location x ∈ ω, and time t > 0, due to the use of the treated bed nets (see e.g. [45], [47], and references therein). 1) wellposedness: we have been working under the following technical assumptions: (h1) η ∈ l∞(ω), k ∈ l∞(ω × ω), k(x,x′) ≥ 0 a.e. in ω × ω,∫ ω k(x,x′)dx > 0 a.e. x′ ∈ ω; (h2) c ∈ l∞(ω), c(x) ≥ τ > 0 a.e. in ω, where τ is a positive constant; (h3) h : r → [0, +∞), and h|[0,+∞) is continuously differentiable and increasing, h(x) = 0, for x ∈ (−∞, 0], h(x) ≤ a21x, for any x ∈ [0, +∞), where a21 is a positive constant; (h4) u01, u 0 2 ∈ l ∞(ω), 0 ≤ u01(x), 0 ≤ u02(x) ≤ c(x) a.e. x ∈ ω. based on epidemiological considerations ([10]), the control functions γ = (γ1,γ2,γ3), introduced above, have been assumed to belong to g = {γ = (γ1,γ2,γ3)∈l∞(q)×l∞loc(q)×l ∞(q) : 0 ≤ γ1(x,t) < γ1, 0 ≤ γ2(x,t), 0 ≤ γ3(x,t) < γ3, a.e. (x,t) ∈ q}, for suitable constants γ1, γ3 > 0. biomath 8 (2019), 1909157, http://dx.doi.org/10.11145/j.biomath.2019.09.157 page 12 of 19 http://dx.doi.org/10.11145/j.biomath.2019.09.157 s. aniţa, e. beretta, v. capasso, optimal control strategies for a class of vector borne diseases ... theorem iii.1. [existence and uniqueness] for any γ ∈g, problem (55) has a unique (weak) solution (uγ1,u γ 2 ). the solution satisfies 0≤uγ1 (x,t), 0≤u γ 2 (x,t)≤c(x), a.e.(x,t)∈q. 2) eradicability: definition iii.2. we say that the disease is eradicable if for any u10 and u 2 0 satisfying assumption (h4) there exists a γ ∈ g such that the solution (uγ1,u γ 2 ) to the controlled system satisfies lim t→+∞ u γ 1 (·, t) = lim t→+∞ u γ 2 (·, t) = 0 in l∞(ω). in [10] it has been shown that theorem iii.3. for any u10,u 2 0 satisfying hypothesis (h4) and such that u10(x),u 2 0(x) > 0 a.e. x ∈ ω \ω, and for any γ = (γ1,γ2,γ3) ∈ g, the solution (uγ1,u γ 2 ) satisfies u γ 1 (·, t) → 0 and u γ 2 (·, t) → 0 in l∞(ω) (as t → +∞) slower than or as slow as exp (−a22t). the above theorem provides information on the lower bound for the convergence to zero of the epidemic. additional analysis based on the magnitude of the principal eigenvalue of an appropriate operator, provides information about an upper bound on the convergence. for a given nonempty open subregion ω of ω, such that ω\ω 6= ∅, a given constant control γ = (γ1,γ2,γ3) ∈ g, and ε ∈ (0,a22), consider the following eigenvalue problem   −d∆ψ1(x) − (a22 −ε)ψ1(x) −η(x)ψ1(x) − ∫ ω k(x,x′)ψ2(x ′)dx′ + γ1χω(x)ψ1(x) = λψ1(x),x ∈ ω ∂νψ1(x) = 0,x ∈ ∂ω (ε + γ2χω(x))ψ2(x) −(1 −γ3χω(x))a21ψ1(x) = 0,x ∈ ω, by the krein-rutman theorem, we may state that proposition iii.4. the principal eigenvalue λω1ε of the above problem is real, simple and admits at least an eigenvector (ψ1,ψ2) such that ψ1(x) ≥ ψ01, ψ2(x) ≥ ψ 0 2, a.e. x ∈ ω, for suitable positive constants ψ01,ψ 0 2. by usual comparison theorems, the following monotonicity properties can be obtained. proposition iii.5. for a given set of control parameters γ ∈g, (i) the mapping ε ∈ (0,a22) 7→ λω1ε is continuous and strictly increasing. (ii) the mapping ω ⊂ ω 7→ λω1ε is increasing, with respect to the ordering by inclusion. (iii) if {ωn}n∈n∗ is an increasing sequence of open subsets of ω such that ∪∞n=1ωn = ω, then lim n→+∞ λωn1ε = λ ω 1ε. so that we may finally state our main eradicability result, i.e. an exponential eradicability. theorem iii.6. if ε ∈ (0,a22), γ ∈ g, and ω ⊂ ω are such that the principal eigenvalue of the above problem is λω1ε > 0, then lim t→+∞ u γ 1 (·, t) = lim t→+∞ u γ 2 (·, t) = 0, in l∞(ω) faster than or as fast as exp(−(a22 − ε)t). the following remark guarantees that this theorem is indeed applicable. remark iii.7. if γ1 > a22 + ‖η‖l∞(ω), then for ε ∈ (0,a22) sufficiently small, γ1 sufficiently close to γ1, γ2 sufficiently large and ω sufficiently large, we have that λω1ε ≥ 0, and consequently the constant control γ = (γ1,γ2,γ3) (γ3 is arbitrary but fixed in [0, γ3)) can make the epidemic exponentially eradicable. biomath 8 (2019), 1909157, http://dx.doi.org/10.11145/j.biomath.2019.09.157 page 13 of 19 http://dx.doi.org/10.11145/j.biomath.2019.09.157 s. aniţa, e. beretta, v. capasso, optimal control strategies for a class of vector borne diseases ... iv. optimal regional control of the epidemic for an optimal control problem during a finite time interval of intervention [0,t], we need to introduce a cost functional which takes into account all relevant costs concerning on one side the costs of intervention associated with the functions (γ1,γ2,γ3) ∈ g; on the other side they must be taken also into account the costs deriving from loss of work hours, hospitalization, and alike associated with the infected human population (`(u2)), and the general costs associated with the specific choice of the subregion of intervention ω ⊂ ω ( by assigning a suitable function α(x), x ∈ ω, the specific costs related to the logistic structure of the habitat can be taken into account) . it should be then of the form j(γ,ω) = ∫ t 0 ∫ ω ζ1(γ1(x,t))c(x)χω(x)dxdt + ∫ t 0 ∫ ω ζ2(γ2(x,t)u2(x,t))χω(x)dxdt + ∫ t 0 ∫ ω ζ3(γ3(x,t))c(x)χω(x)dxdt + ∫ t 0 ∫ ω `(u2(x,t))dx dt + ∫ ω α(x)χω(x)dx. (58) based on suitable epidemiological considerations (see [10]), the following choices can be made for the specific cost functionals ζ1,ζ2,ζ3: (h5) ζ1 : [0, γ1) → [0, +∞) is a continuously differentiable function, bijective, strictly increasing; ζ3 : [0, γ3) → [0, +∞) is a continuously differentiable function, bijective, strictly increasing; (h6) ζ2 : [0, +∞) → (γ2, γ̃2] is a continuous differentiable function, bijective, and strictly decreasing. a. optimal control for a fixed region of intervention at first let the subregion ω ⊂ ω be fixed, so that the cost functional j(γ,ω) is just a function of γ, say j(γ). we shall denote by gt = {γ = (γ1,γ2,γ3)∈l∞(qt )3 : 0 ≤ γ1(x,t) < γ1, 0 ≤ γ2(x,t), 0 ≤ γ3(x,t) < γ3 a.e. (x,t) ∈ qt}, where qt = ω × (0,t). under the above assumptions the following theorem holds. theorem iv.1. the optimal control problem minimizeγ∈gt j(γ) admits at least one solution γ∗ = (γ∗1,γ ∗ 2,γ ∗ 3 ). if we denote by dj(γ) the gradient of j(γ), an optimal choice γ∗ is such that dj(γ∗)(w) ≥ 0, for any w as in next proposition. proposition iv.2. for any γ ∈ gt and w ∈ l∞(qt )×l∞(qt )×l∞(qt ) such that γ+θw ∈ gt for any θ > 0 sufficiently small, we have that dj(γ)(w) = ∫ t 0 ∫ ω w1(x,t)u γ 1(x,t)q1(x,t)χω(x)dxdt + ∫ t 0 ∫ ω w1(x,t)c(x)ζ ′ 1(γ1(x,t))χω(x)dxdt + ∫ t 0 ∫ ω w2(x,t)ζ ′ 2(γ2(x,t)u γ 2(x,t))u γ 2(x,t)χω(x)dxdt + ∫ t 0 ∫ ω w2(x,t)u γ 2 (x,t)q2(x,t)χω(x)dxdt + ∫ t 0 ∫ ω w3(x,t)c(x)ζ ′ 3(γ3(x,t))χω(x)dxdt + ∫ t 0 ∫ ω (w3(x,t)(c(x) −u γ 2 (x,t)) ×h( u γ 1 (x,t) c(x) )q2(x,t)χω(x))dxdt, where (q1,q2) is the solution to (59) biomath 8 (2019), 1909157, http://dx.doi.org/10.11145/j.biomath.2019.09.157 page 14 of 19 http://dx.doi.org/10.11145/j.biomath.2019.09.157 s. aniţa, e. beretta, v. capasso, optimal control strategies for a class of vector borne diseases ...   ∂tq1(x,t) + d∆q1(x,t) = −η(x)q1(x,t) + γ1(x,t)χω(x)q1(x,t) −(1 −γ3(x,t)χω(x)) c(x) −uγ2 (x,t) c(x) h′ ( u γ 1 (x,t) c(x) ) q2(x,t), ∂tq2(x,t) = − ∫ ω k(x′,x)q1(x ′, t)dx′ + (a22 + γ2(x,t)χω(x))q2(x,t) +(1−γ3(x,t)χω(x))h ( u γ 1 (x,t) c(x) ) q2(x,t)+ζ̃ ′ 2(γ2(x,t)u γ 2 (x,t))γ2(x,t)χω(x)+` ′(u γ 2 (x,t)), for (x,t) ∈ qt ; ∂νq1(x,t) = 0, (x,t) ∈ σt ; q1(x,t) = q2(x,t) = 0, x ∈ ω. (59) numerical results have been obtained by the gradient method, for suitable choices of all parameters (see [10]). b. optimal choice of the region of intervention the optimization with respect to both the control functions and the subregion of intervention by the gradient method, requires the evaluation of the directional derivative of the cost functional with respect to both the parameters γ1,γ2,γ3, and the region ω. a convenient way to handle the shape and position of ω is to use the level set method [40]. according to the implicit representation of subsets of ω, we consider those subsets ω for which there exists a smooth function ϕ : ω → r such that ω = {x ∈ ω; ϕ(x) > 0} and ∂ω = {x ∈ ω; ϕ(x) = 0}. hence, instead of investigating the total cost function j defined in the first section, we shall deal with (h denotes the usual heaviside function) j̃(γ,ϕ) = ∫ t 0 ∫ ω ζ1(γ1(x,t))c(x)h(ϕ(x))dxdt + ∫ t 0 ∫ ω ζ2(γ2(x,t)u2(x,t))h(ϕ(x))dxdt + ∫ t 0 ∫ ω ζ3(γ3(x,t))c(x)h(ϕ(x))dxdt + ∫ t 0 ∫ ω `(u2(x,t))dxdt + ∫ ω α(x)h(ϕ(x))(x)dx. (60) where now γ ∈ gt , ϕ : ω → r is continuously differentiable, and (u1,u2) is the solution to   ∂tu1(x,t) −d∆u1(x,t) = −η(x)u1(x,t) + ∫ ω k(x,x′)u2(x ′, t)dx′ −γ1(x,t)h(ϕ(x))u1(x,t), ∂tu2(x,t) =−(a22 +γ2(x,t)h(ϕ(x)))u2(x,t) +(1−γ3(x,t)h(ϕ(x)))g(x,u1(x,t),u2(x,t)), (61) for (x,t) ∈ qt , subject to the boundary and initial conditions ∂νu1(x,t) = 0, (x,t) ∈ σt (62) u1(x, 0) =u 0 1(x), u2(x, 0) =u 0 2(x), x∈ω, (63) where qt = ω × (0,t), σt = ∂ω × (0,t). actually h is usually replaced by its mollified version hε(s) = 12 (1 + 2 π arctan s ε ) (ε > 0 is a sufficiently small number). it is possible to prove that [10] biomath 8 (2019), 1909157, http://dx.doi.org/10.11145/j.biomath.2019.09.157 page 15 of 19 http://dx.doi.org/10.11145/j.biomath.2019.09.157 s. aniţa, e. beretta, v. capasso, optimal control strategies for a class of vector borne diseases ... theorem iv.3. for any γ ∈gt and w ∈ l∞(qt ) ×l∞(qt ) ×l∞(qt ) such that γ + θw ∈gt for any θ > 0 sufficiently small, and for any smooth functions ϕ,ψ : ω → r we have that dj̃ε(γ,ϕ)(w,ψ) = ∫ t 0 ∫ ω w1(x,t)hε(ϕ(x))u γ,ϕ 1 (x,t)q1(x,t)dxdt + ∫ t 0 ∫ ω w1(x,t)c(x)ζ ′ 1(γ1(x,t))hε(ϕ(x))dxdt + ∫ t 0 ∫ ω w2(x,t)ζ ′ 2(γ2(x,t)u γ,ϕ 2 (x,t))u γ,ϕ 2 (x,t)hε(ϕ(x))dxdt + ∫ t 0 ∫ ω w2(x,t)u γ,ϕ 2 (x,t)q2(x,t)hε(ϕ(x))dxdt + ∫ t 0 ∫ ω w3(x,t)c(x)ζ ′ 3(γ3(x,t))hε(ϕ(x))dxdt + ∫ t 0 ∫ ω w3(x,t)(c(x) −u γ,ϕ 2 (x,t))hε(ϕ(x))h ( u γ,ϕ 1 (x,t) c(x) ) q2(x,t)]dxdt + ∫ ω ψ(x)[δε(ϕ(x))ψ(x)]dx, (64) where δε(x) = h ′ ε(x) = 1 π ε x2 + ε2 , and ψ(x) is given by ψ(x) = ∫ t 0 [ c(x)ζ1(γ1(x,t))+ζ2(γ2(x,t)u γ,ϕ 2 (x,t))+c(x)ζ3(γ3(x,t))+α(x)+γ1(x,t)u γ,ϕ 1 (x,t)q1(x,t) +γ2(x,t)u γ,ϕ 2 (x,t)q2(x,t) + γ3(x,t)(c(x) −u γ,ϕ 2 (x,t))h ( u γ,ϕ 1 (x,t) c(x) ) q2(x,t) ] dt. here (uγ,ϕ1 ,u γ,ϕ 2 ) is the solution to (61), and (q1,q2) is the solution to  ∂tq1(x,t) + d∆q1(x,t) = −η(x)q1(x,t) + γ1(x,t)hε(ϕ(x))q1(x,t) −(1 −γ3(x,t)hε(ϕ(x)) c(x) −uγ,ϕ2 (x,t) c(x) h′( u γ,ϕ 1 (x,t) c(x) )q2(x,t), ∂tq2(x,t) = − ∫ ω k(x′,x)q1(x ′, t)dx′ + (a22 + γ2(x,t)hε(ϕ(x))q2(x,t) +(1−γ3(x,t)hε(ϕ(x))h ( u γ,ϕ 1 (x,t) c(x) ) q2(x,t)+ζ̃ ′ 2(γ2(x,t)u γ,ϕ 2 (x,t))γ2(x,t)hε(ϕ(x)) + ` ′(u γ,ϕ 2 (x,t)), for (x,t) ∈ qt ; ∂νq1(x,t) = 0, (x,t) ∈ σt ; q1(x,t) = q2(x,t) = 0, x ∈ ω. (65) remark 1: we may notice that for a given γ = (γ1,γ2,γ3) ∈gt the quantity ∂sϕ(x,s) = −δε(ϕ(x,s))ψ(x) (66) for x ∈ ω, s > 0, where (u1,u2) is the solution of the state equation and (q1,q2) is the solution to the adjoint equation as above, in terms of the fictitious ”time” variable s, gives the descent of the optimal search with respect to ϕ. biomath 8 (2019), 1909157, http://dx.doi.org/10.11145/j.biomath.2019.09.157 page 16 of 19 http://dx.doi.org/10.11145/j.biomath.2019.09.157 s. aniţa, e. beretta, v. capasso, optimal control strategies for a class of vector borne diseases ... we may introduce additional costs due to shape of the subregion ω, by considering the perimeter of that region. as in [29] and [9], this would add the following term in the cost functional j̃(γ,ϕ) (see (60)) ∫ ω β(x)δ(ϕ(x)) |∇ϕ(x)|dx. (67) as a consequence, in the gradient dj̃ε(γ,ϕ)(w,ψ) (see (64)) last term would then become∫ ω ψ(x)[δε(ϕ(x))ψ̂(x)]dx + ∫ ∂ω ψ(x) δε(ϕ(x)) |∇ϕ(x)| ∂νϕ(x)d` (68) where now ψ̂(x) = ∫ t 0 [c(x)ζ1(γ1(x,t)) + ζ2(γ2(x,t)u γ,ϕ 2 (x,t)) +c(x)ζ3(γ3(x,t)) + γ1(x,t)u γ,ϕ 1 (x,t)q1(x,t) +γ2(x,t)u γ,ϕ 2 (x,t)q2(x,t) +γ3(x,t)(c(x)−u γ,ϕ 2 (x,t))h ( u γ,ϕ 1 (x,t) c(x) ) q2(x,t) +α(x) −β(x)div ( ∇ϕ(x) |∇ϕ(x)| ) ]dt. under these circumstances equation (66) becomes ∂sϕ(x,s) = −δε(ϕ(x,s))ψ̂(x), x ∈ ω,s > 0, (69) subject to the boundary condition β(x) δε(ϕ(x,s)) |∇ϕ(x)| ∂νϕ(x) = 0, x ∈ ∂ω,s > 0 (70) (see [29] and [9]). we may recognize that this is strictly related to the literature on image segmentation (see [38], [37], [21]). v. conclusions • based on the main ideas of relevant literature concerning epidemiological issues of malaria, we have proposed a mathematical model describing the dynamics of infected mosquitoes and humans in a spatially structured habitat. • in order to reduce the number of infected mosquitoes and humans, we have taken into account three possible control measures to be implemented only in a suitable subdomain ω of the relevant global habitat. • to start with, we have shown that if such a subdomain of intervention is sufficiently large and if the magnitude of the control efforts is sufficiently large, then eventual eradication of both infected populations is possible at an exponential rate. • we have then analyzed the optimal control problem for the reduction of the infected human population (in a finite time interval) at a minimum cost. • further analysis for the identification of an optimal subdomain ω, together with optimal control efforts, has been left to future investigations. — acknowledgment this work has been carried out in the framework of the european cost project ca16227: “investigation and mathematical analysis of avantgarde disease control via mosquito nano-techrepellents”. the authors would like to thank the anonymous referees for their valuable comments. references [1] agusto fb, marcus n, okosun ko. application of optimal control to the epidemiology of malaria. electron. j. diff. eqs. 2012;81:1–22. 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[48] who malaria report; world health organization; 2017. biomath 8 (2019), 1909157, http://dx.doi.org/10.11145/j.biomath.2019.09.157 page 19 of 19 http://dx.doi.org/10.11145/j.biomath.2019.09.157 introduction a compartmental model with two age groups the time scales qualitative analysis of the mathematical model the slow time scale the fast time scale qualitative analysis at the slow time scale the main threshold theorem about the total human population a spatially structured model regional control strategies wellposedness eradicability optimal regional control of the epidemic optimal control for a fixed region of intervention optimal choice of the region of intervention conclusions references www.biomathforum.org/biomath/index.php/biomath original article operator splitting and discontinuous galerkin methods for advection-reaction-diffusion problem. application to plant root growth emilie peynaud cirad, umr amap, yaoundé, cameroun amap, university of montpellier, cirad, cnrs, inra, ird, montpellier, france university of yaoundé 1, national advanced school of engineering, yaoundé, cameroon emilie.peynaud@cirad.fr received: 19 september 2017, accepted: 3 december 2018, published: 18 december 2018 abstract—motivated by the need of developing numerical tools for the simulation of plant root growth, this article deals with the numerical resolution of the c-root model. this model describes the dynamics of plant root apices in the soil and it consists in a time dependent advection-reactiondiffusion equation whose unique unknown is the density of apices. the work is focused on the implementation and validation of a suitable numerical method for the resolution of the c-root model on unstructured meshes. the model is solved using discontinuous galerkin (dg) finite elements combined with an operator splitting technique. after a brief presentation of the numerical method, the implementation of the algorithm is validated in a simple test case, for which an analytic expression of the solution is known. then, the issue of the positivity preservation is discussed. finally, the dgsplitting algorithm is applied to a more realistic root system and the results are discussed. keywords-time dependent advection-reactiondiffusion; operator splitting; discontinuous galerkin method; plant root growth simulation; i. introduction the article is devoted to the numerical modeling of plant root growth. this work has been originally motivated by the need of developing numerical tools for the simulation of plant growth dynamics. due to the difficulty of doing nondestructive observations of the underground part of plants (that allow to do long term studies of the dynamics of tree roots for example), mathematical models are achieving an essential role. several theoretical and numerical challenges arise in the field of the simulation of the dynamics of plant roots [48], [47], [38], [2], [39]. the mathematical description of plant root is not trivial, due to the presence of many interactions arising in the rhizosphere and also due to the diversity of plant root types. mathematical models based on the use of partial differential equations are useful tools to simulate the evolution of root densities in copyright: c© 2018 peynaud. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: emilie peynaud, operator splitting and discontinuous galerkin methods for advection-reaction-diffusion problem. application to plant root growth, biomath 7 (2018), 1812037, http://dx.doi.org/10.11145/j.biomath.2018.12.037 page 1 of 19 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2018.12.037 emilie peynaud, operator splitting and discontinuous galerkin methods for advection-reaction... space and time [43], [44], [44], [45], [46], [41], [40], [1]. this formalism facilitates the coupling with physical models such as water and nutrient transports [42], [43], [44], [41], [49]. and the computational time for the simulation of such models is not dependent on the number of roots which is useful for applications at large scale. the c-root model [1] is a generic model of the dynamics of root density growth. this model takes only one unknown which is related to root densities such as the density of apices, root length density or biomass density. it has only three parameters. the model is said to be generic in the sense that it can apply to a wide variety of root system types. the model consists in a single time-dependent advection-reaction-diffusion equation, and one of the challenge is to numerically solve the equation. in [1] and [2] the authors solved the problem with the finite difference method on cartesian mesh grids combined with an operator splitting technique. unfortunately, cartesian mesh grids do not allow easily to mesh complex soil geometries. from the theoretical and computational point of view, cartesian grids also lead to difficulties for a rigorous study and validation of the model. that is why this article focus on the development and implementation of a suitable numerical method for the resolution of the c-root model on triangular mesh grids, that allow to mesh complex geometries. however, one of the main difficulties in the c-root model is that the advection and diffusion terms are not always of the same order of magnitude. it depends on the phase of the root system development [2]. as a result, the properties of the equation may vary along the simulation: it can be either close to a hyperbolic problem or close to a parabolic problem. in a previous work [3], the use of the discontinuous galerkin method has been implemented and validated. indeed, the usual choice of the classical lagrange finite element method suffers from a lack of stability when the advection term is dominant [4]. for this reason, we implemented a discontinuous galerkin (dg) method for both the advection and diffusion terms. all the three operators where solved simultaneously using the same time approximation scheme (θ-scheme). however, as explained in [6], for multibiophysic problems it is not efficient to use the same numerical scheme for the different operators of the system. for example, we may want to use the euler explicit scheme for the advection term and an euler implicit scheme for the diffusion. the operator splitting technique [7], [8] is a well known alternative for the resolution of equations having a multi-biophysic behaviour that allows the use of different time schemes for each operator of the equation. the idea of the splitting technique is to split the problem into smaller and simpler parts of the problem so that each part can be solved by an efficient and suitable time scheme. this methods has been used for a wide range of applications dealing with the advection-reaction-diffusion equation [9]. operator splitting techniques have been extensively used in combination with finite difference methods [10], [2], finite volume methods [11], [12] but also with continuous galerkin methods [13], [14], [15], [16], [17]. to the best of my knowledge, only very few articles deal with the use of the operator splitting technique in combination with the discontinuous galerkin approximation [18], [19], [20], [21]. in this paper, we present a new application of the operator splitting technique combined with discontinuous finite elements. the paper is structured as follows. in section ii, the c-root growth model [1], [2], [3] is briefly described. an analysis is also provided, where i showed the existence and uniqueness of a positive real solution. in section iii, the splitting operator technique is introduced and applied to the c-root model, combined with the use of discontinuous galerkin approximations. in section iv, the algoritm is implemented and validated using a simple test case for which an analytic expression of the solution is known. as an application, i provide simulations of the development of eucalyptus roots in section v. finally, the paper ends with a conclusion and further improvements. biomath 7 (2018), 1812037, http://dx.doi.org/10.11145/j.biomath.2018.12.037 page 2 of 19 http://dx.doi.org/10.11145/j.biomath.2018.12.037 emilie peynaud, operator splitting and discontinuous galerkin methods for advection-reaction... ii. the model a. modelling root growth with pde: the c-root model the c-root model [1] was developed to simulate the growth of dense root networks, usually composed of fine roots, with negligible secondary thickening. as presented in [1], the unknown variable u is the number of apices per unit volume, but it can also stand for the density of fine root biomass. the soil is considered as a subdomain of rd (with d = 1, 2 or 3). it is assumed that ω has smooth boundaries (lipschitz boundaries) denoted ∂ω. the c-root model combines advection, diffusion and reaction, which aggregate the main biological processes involved in root growth, such as primary growth, ramification and root death. the reaction operator gives the quantity of apices (or root biomass) produced in time, whereas advection and diffusion operators spatially distribute the whole apices (or biomass) in the domain. the reaction operator describes the evolution in time of the root biomass in a given domain. in the c-root model it is a linear term characterized by the scalar parameter ρ which is the growth rate of the root system. the diffusion corresponds to the spread of the root biomass over space. it is described by the parameter σ which is a d × d matrix that characterizes the growth of the root biomass in any direction exploiting free space in the soil. the advection corresponds to the displacement of the root biomass in a direction and velocity given by v which is a vector in rd. on the boundaries of ω, what happens for the quantity being transported is different depending if the growth makes the roots to come inside ω or to go outside of ω. if v is going inside ω (at the inlet boundary) the root biomass u will enter the domain and increase. on edges where v is going out of the domain (outlet boundary) the root biomass u is going to be pushed out of ω. since this phenomena is oriented (causality) and the behaviour of the solution is different on inlet and outlet boundaries, we need to specify in the model these parts of the boundaries. mathematically, it is required to define the inlet boundary with respect to v as ∂ω− = {x ∈ ∂ω : (v ·n)(x) < 0} . (1) the outlet boundary ω+ is given by ∂ω+ = ∂ω\∂ω−. the dynamics of the root system is studied between the time t0 and t1 with 0 ≤ t0 < t1. the problem reads as follow: find u such that  ∂tu + v ·∇u−∇· (σ∇u) + ρu = 0 in ]t0, t1[×ω u(t0) = u0 at {t0}× ω n ·σ∇u = g on ]t0, t1[×∂ω (n ·v)u = gin on ]t0, t1[×∂ω− (2) where g ∈ l2(∂ω) and gin ∈ l2(∂ω−) are given. and u0 is the given initial solution. problem (2) is known as the time dependent advectionreaction-diffusion problem and belongs to the class of parabolic partial differential equations. this equation is a model problem that often occurs in fluid mechanics but also in many other applications in life sciences (see for instance [22], [23], [24]). depending on the boundary conditions, the problem has different meanings. to simplify the presentation we only consider the neumann boundary condition combined with an inlet boundary condition at the inlet of the domain. the neumann condition specifies the value of the normal derivative of the solution at the boundary of the domain. the inlet boundary condition specifies the quantity of u convected by v that enters in the domain. b. the weak problem since the goal is to solve the problem on unstructured meshes, the spatial operators are approximated using finite element methods. within this framework, it is classical to write the problem in a variational form. let us first introduce some functional spaces [50]. • the space h1(ω) defined such that h1(ω) = {v ∈ l2(ω) : ∇v ∈ l2(ω)} is a hilbert space when equipped with the norm ‖ · ‖1,ω. we recall that ∀v ∈ h1(ω), ‖v‖1,ω = (v,v)1,ω biomath 7 (2018), 1812037, http://dx.doi.org/10.11145/j.biomath.2018.12.037 page 3 of 19 http://dx.doi.org/10.11145/j.biomath.2018.12.037 emilie peynaud, operator splitting and discontinuous galerkin methods for advection-reaction... and the scalar product (·, ·)1,ω is defined by ∀v ∈ h1(ω), (u,v)1,ω = ∫ ω uv dx + ∫ ω ∇u ·∇v dx. • we denote l2(]t0, t1[,h) the space of hvalued functions whose norm in h is in l2(]t0, t1[). this space is a hilbert space for the norm ‖u‖l2(]t0,t1[,h) = (∫ t1 t0 ‖u(t)‖2h )1/2 . • let b0 ⊂ b1 be two reflexive hilbert spaces with continuous embedding, we denote w(b0,b1) the space of functions v : ]t0, t1[−→ b0 such that v ∈ l2(]t0, t1[,b0) and dtv ∈ l2(]t0, t1[,b1). equipped with the norm ‖u‖w(b0,b1) = ‖u‖l2(]t0,t1[,b0) +‖dtu‖l2(]t0,t1[,b1), the space w(b0,b1) is a hilbert space [25]. using the previous functional spaces, i now define the problem in the following weak form: find u in w such that ∀v ∈ h 〈dtu(t),v(t)〉h′,h +a(t,u,v) =`(t,v) a.e. t∈]t0,t1[ u(t0) = u0, (3) where w = w(h1(ω), (h1(ω))′) and h = h1(ω) and `(t,v) = ∫ ∂ω g(t)vdγ (4) a(t,u,v) =aa(t,u,v)+ad(t,u,v)+ar(t,u,v) (5) with aa(t,u,v) = ∫ ω v(v(t,x) ·∇u) dx, (6) ad(t,u,v) = ∫ ω ∇v ·σ(t,x) ·∇udx, (7) ar(t,u,v) = ∫ ω ρ(t)uv dx. (8) one can prove that problems (3) and (2) are equivalent almost everywhere in ]t0, t1[×ω. let us assume that there is a constant σ0 > 0 such that ∀ξ ∈ rd, d∑ i,j=1 σijξiξj ≥ σ0‖ξ‖2d a.e. in ω. (9) in addition, i assume that inf x∈ω ( σ − 1 2 (∇·v) ) > 0 and inf x∈∂ω (v ·n) ≥ 0. (10) under assumption (9) and (10), one can prove that the problem is well-posed for sufficiently smooth v, σ and ρ (see for instance [25]). c. the positivity preserving property of the solution in the framework of our applications to the simulation of root biomass densities one of the crucial property of the problem is the preservation of the positity of the solution along time. for a positive initial solution u0, the solution of (3) stays positive. proposition ii-c.1. let u0 ∈ l2(ω) and f ∈ l2(]t0, t1[,l 2(ω)). we consider u the solution of (3) in w . we assume that u0(x) ≥ 0 a.e. in ω and g(t,x) ≥ 0 a.e. in ]t0, t1[×∂ω. then u(t,x) ≥ 0 a.e in ]t0, t1[×ω. proof: i follow [25]. see also [26], [27]. we consider the function u− defined by u− = 1 2 (|u|−u). let us note that u−= { 0 a.e in ]t0,t1[×ω, if u≥0 a.e in ]t0,t1[×ω, −u a.e in ]t0,t1[×ω, if u<0 a.e in ]t0,t1[×ω. that is we have u− ≥ 0 a.e. in ]t0, t1[×ω. (11) we verify that u− is an admissible test function in w . using the following obvious equations (∇|u|)2 = (∇u)2 ∇|u| ·∇|u| = ∇u ·∇u u∇|u| = |u|∇u u∇u = |u|∇|u| biomath 7 (2018), 1812037, http://dx.doi.org/10.11145/j.biomath.2018.12.037 page 4 of 19 http://dx.doi.org/10.11145/j.biomath.2018.12.037 emilie peynaud, operator splitting and discontinuous galerkin methods for advection-reaction... that are valid a.e in ]t0, t1[×ω we can verify that a(t,u−,u−) = −a(t,u,u−). by adding the same quantity on both sides of the equation we get 〈dtu−,u−〉+a(t,u−,u−) =〈dtu−,u−〉−a(t,u,u−). since u satisfy (3) we have 〈dtu−,u−〉+a(t,u−,u−) =〈dtu−,u−〉+〈dtu,u−〉 − `(t,u−). one can notice that 〈dtu−,u−〉 + 〈dtu,u−〉 = 0. then we have 1 2 dt‖u−‖20,ω + a(t,u −,u−) = −`(t,u−) ≤ 0, with g(t,x) ≥ 0 a.e. in ]t0, t1[×∂ω. now from the coercivity of the bilinear form a we obtain 1 2 dt‖u−‖20,ω + c‖u −‖20,ω ≤ 1 2 dt‖u−‖20,ω + a(t; u −,u−) ≤ 0, where c is a strictly positive constant. the estimate is then 1 2 dt‖u−‖20,ω ≤−c‖u −‖20,ω. by the gronwall lemma we have that ∀t ∈ [t0, t1] × ω,‖u−(t)‖20,ω ≤ e −2ct‖u−(0)‖20,ω. since c > 0 and t ≥ t0 ≥ 0 , we have that e−2ct ≤ 1 , so we obtain ∀t ∈ [t0, t1] × ω,‖u−(t)‖20,ω ≤‖u −(0)‖20,ω. since u(0) = u0 ≥ 0 a.e in ω we have u−(0) = 0 a.e in ω. so we deduce that ∀t ∈ [t0, t1] × ω,‖u−(t)‖20,ω ≤ 0. but from the definition of u− we have u− ≥ 0 a.e in ]t0, t1[×ω. so we deduce that ‖u−(t)‖20,ω = 0 and thus u−(t) = 0 a.e in ]t0, t1[×ω. it means that u ≥ 0 a.e in ]t0, t1[×ω by definition of u−. iii. approximation of the model a. the operator splitting technique here we focus on the implementation of the operator splitting technique. the time interval [t0, t1] is divided in n subspaces of size δt such that [t0, t1] = ∪n=1,n ]tn, tn+1[ with ∩n=1,n ]tn, tn+1[= ∅. at each iteration step we solve the following problems • find ua ∈ h such that ∀v ∈ h, for a.e t ∈ ]tn, tn+1[, 〈dtua(t),v(t)〉h′,h + aa(t,ua,v) = 0 ua(tn) = u(tn). • find ud ∈ h such that ∀v ∈ h, for a.e t ∈]tn, tn+1[, 〈dtud(t),v(t)〉h′,h + ad(t,ud,v) = `(t,v) ud(tn) = ua(tn+1). • find ur ∈ h such that ∀v ∈ h, for a.e t ∈ ]tn, tn+1[, 〈dtur(t),v(t)〉h′,h + ar(t,ur,v) = 0 ur(tn) = ud(tn+1). • set u(tn+1) = ur(tn+1). the bilinear forms aa(t,u,v), ad(t,u,v) and ar(t,u,v) are respectively given by (6), (7) and (8). and `(t,v) is the linear form (4). if the operators are commutative, then the splitting error vanishes. otherwise, if the operators are not commutative, then the splitting error does not vanish and a second order splitting would be required (see [6]). in the following, i present the different schemes related to each operator. b. the advection step: dg upwind scheme the advection step consists in solving the following transport problem : find u such that ∀v ∈ h, for a.e t ∈]tn, tn+1[, 〈dtu(t),v(t)〉h′,h + aa(t,u,v) = 0 (12) ua(tn) = ur(tn) (13) where aa(t,u,v) is the bilinear form (6). for the space approximation of this problem, we implemented the dg upwind method presented below. biomath 7 (2018), 1812037, http://dx.doi.org/10.11145/j.biomath.2018.12.037 page 5 of 19 http://dx.doi.org/10.11145/j.biomath.2018.12.037 emilie peynaud, operator splitting and discontinuous galerkin methods for advection-reaction... let th be a regular family of decomposition in triangles of the domain ω such that ω = n⋃ i=1 k̄i and ki ∩kj = ∅,∀i 6= j. the h subscript in th denotes the size of the mesh cells and it is defined by h = max k∈th hk where hk is the diameter of the element k. let eh be the set of edges of the elements of th. among the elements of eh we denote by ebh the set of edges belonging to ∂ω. the sets eb,−h and e b,+ h are the sets of edges belonging to ∂ω− and ∂ω+ respectively. and eih is the set of interior edges. let us consider an element of eih. we denote by t + and t− the two mesh elements sharing the edge e so that e = ∂t + ∩∂t− where the minus and plus superscripts depend on the direction of the advection vector. by convention we suppose that v goes from t− to t + that is v ·n+e < 0 and v · n−e > 0 where n+e (resp. n−e ) is the outward normal vector of e in t + (resp. t−). when it is not necessary to distinguish the orientation of the normal vectors n+e and n − e we denote by n the unitary normal of e. let us consider the advection problem on each element ki of the domain : for all ki, i = 1,n we look for u the solution of the equation (12) defined on ki. similarly to the problem defined on all the domain ω, we look for a solution u that is in l2(ki) and such that ∇u is in l2(ki) for all ki in th. let us introduce the following broken sobolev space: h1(th) = { v ∈ l2(ω) : ∇v ∈ l2(ki) and v ∈ h1/2+ε(ki),∀ki ∈th } with ε a positive real number. the trace of the functions of h1(th) are meaningful on e ⊂ ki, ∀ki ∈ th. the functions v of h1(th) have two traces along the edges e. we denote v+e the trace of v along e on the side of triangle t + and v−e the trace of v along e on the side of t−. on edges that are subsets of ∂ω the trace is unique and we can note v+e = v if e ∈e b,− h and v − e = v if e ∈e b,+ h , and by convention, we set v−e = 0 if e ∈e b,− h and v + e = 0 if e ∈e b,+ h . the jump of functions of h1(th) across the internal edge e is defined by: jvk = v+e −v − e ,∀e ∈e i h. for edges belonging to the boundary of ω we take jvk = ve,∀e ∈e b,− h and jvk = −ve,∀e ∈e b,+ h , with ve the trace of v along e. the mean value of u on e is defined by {{v}} = 1 2 (v+e + v − e ),∀e ∈e i h. besides for edges on the boundaries we take {{v}} = ve,∀e ∈ebh. let us denote by x the functional space defined such that x = {v : ]t0, t1[−→ h1(th) : v ∈ l2(]t0, t1[,h1(th)); and dtv ∈ l2(]t0, t1[,h1(th)′)}. this space is a hilbert space equipped with the norm ‖v‖x = ‖v‖l2(]t0,t1[,h1(th)) +‖v‖l2(]t0,t1[,h1(th)′). the dg variational formulation of the advection step written on the broken sobolev space takes the following form: find u in x such that for a.e t ∈]t0, t1[, ∀v ∈ h1(th) 〈dtu(t),v(t)〉h1(th)′,h1(th) +a up h (t;u,v) =` up h (t;v), u(t0) = u0, where the form auph (t; u,v) is the approximation of the advection term. it consists in the upwind formulation of the dg method [28]. it reads: a up h (t; u,v) = ∑ k∈th ∫ k u(ρv−v ·∇v)dx − ∑ e∈eb,±h ∪e b,+ h ∫ e |v ·n+e |u − e jvkds. (14) biomath 7 (2018), 1812037, http://dx.doi.org/10.11145/j.biomath.2018.12.037 page 6 of 19 http://dx.doi.org/10.11145/j.biomath.2018.12.037 emilie peynaud, operator splitting and discontinuous galerkin methods for advection-reaction... the approximated linear form of the the right hand side reads ` up h (t; v) = − ∑ e∈eb,−h ∫ e (v ·n+e )ginv + e ds. the dg-formulation (14) is consistent and stable, see for example [32]. the discontinuous galerkin method consists in searching the solution in the approximation space xh defined such that xh = { v :]t0, t1[−→ wkh ; v ∈ l 2(]t0, t1[,w k h ); and dtv ∈ l2(]t0, t1[, (wkh ) ′) } , where wkh is given by wkh = { vh ∈ l2(ω);∀k ∈th,vh|k ∈ pk } . let us note that the functions of wkh can be discontinuous from one element of the mesh to the other. let us note that wkh is embedded in h 1(th) so that xh ⊂ x . this problem can be written in a matrix form. let us denote (λi)i=1,n the basis of the finite dimensional subspace wkh where n = dim(wkh ). in this basis the approximated solution takes the form: uh(t,x,y) = n∑ i=1 ξi(t)λi(x,y), where the ξi(t) are the degrees of freedom. let us define x the vector of degrees of freedom: x(t) = (ξ1(t), . . . ,ξn(t)) t . the approximated problem then reduces to find x(t) ∈ [c2(0,t)]n such that m dx(t) dt + aup(t)x(t) = l up h (t) mx(0) = mx0 where m and aup(t) are two matrices defined such that m = (mi,j)i,j and mi,j = ∑ t∈th ∫ k λiλjdx, (15) aup = ( a up i,j ) i,j and aupi,j = a up h (t; ,u,v), (16) and luph (t) is the vector of size n defined such that( l up h (t) ) i = ` up h (t; λi) for i = 1,n. the problem reduces to a linear system of ordinary differential equations. the time approximation is based on a finite difference scheme. at each iteration step we solve the following problem: find xn+1 ∈ rn such that 1 δt m ( xn+1 −xn ) + (1 −θ)aupxn + θaupxn+1 (17) = (1 −θ)lup,nh + θl up,n+1 h and mx0 = mx0, where θ is a real parameter taken in [0, 1]. for θ = 0, we have the explicit euler schema. for θ = 1, it is the implicit euler schema. for θ = 1/2, it is the crank-nicolson schema. c. the diffusion step the diffusion step consists in solving the following problem : find u such that ∀v ∈ h, for a.e t ∈]tn, tn+1[, 〈dtu(t),v(t)〉h′,h + ad(t; u,v) = `(t; v) u(tn) = ua(tn) where ad(t; u,v) is the bilinear form (7) and `(t; v) is the linear form (4). in the setting introduced before, the dg variational formulation of the diffusion step written in the broken sobolev space takes the following form: find u in x such that ∀v ∈ h1(th), for a.e. t ∈]t0, t1[ 〈dtu(t),v〉h1(th)′,h1(th) + a ip h (t; u,v) = ` ip h (t; v) u(t0) = u0. the form aiph (t; u,v) is the approximation of the diffusion term. it consists in the interior penalty biomath 7 (2018), 1812037, http://dx.doi.org/10.11145/j.biomath.2018.12.037 page 7 of 19 http://dx.doi.org/10.11145/j.biomath.2018.12.037 emilie peynaud, operator splitting and discontinuous galerkin methods for advection-reaction... formulation (ip) that reads a ip h (t; u,v) = ∑ k∈th ∫ k σ∇u ·∇v dx − ∑ e∈eih ∫ e {{σ∇u}} ·n+e jvk ds + ∑ e∈eih ∫ e {{σ∇v}} ·n+e juk ds + ∑ e∈eih η he ∫ e jukjvk ds, where η is a positive penalization factor. the linear form `iph (t; v) is given by ` ip h (t; v) = ∑ e∈ebh ∫ e gv ds. this formulation was introduced in [31] and is known as the non-symmetric interior penalty (nsip) formulation, see [30], [32]. in matrix form the problem reduces to find x(t) ∈ [c2(0,t)]n such that m dx(t) dt + aip(t)x(t) = l ip h (t) mx(0) = mx0 where m is defined by (15) and aip is defined such that aip = ( a ip i,j ) i,j and aupi,j = a ip h (t; ,u,v). the vector liph (t) is such that ( l ip h (t) ) i = ` ip h (t; λi) for i = 1,n. similarly to the advection step, the time approximation of the problem is based on a finite difference scheme of the form (17). d. the reaction step the reaction step consists in solving the following problem : find u such that ∀v ∈ h, for a.e. t ∈]tn, tn+1[ 〈dtu(t),v(t)〉h′,h + ar(t; u,v) = 0 u(tn) = ud(tn) where ar(t; u,v) is the bilinear form (8). this problem takes the following matrix form find x(t) ∈ [c2(0,t)]n such that dx(t) dt + ρx(t) = 0 x(0) = x0 where we recall that ρ is a constant real parameter. this problem can be solved by an exact scheme (a kind of schemes that provide exact solutions, i.e. a solution equal to the analytical solution). at each iteration we find xn+1 such that 1 φ(δt) ( xn+1 −xn ) = −ρxn with φ(δt) = 1 ρ (1 − exp(−ρδt)). this scheme is unconditionally stable, meaning that we can choose the time step independently from the space step. it is also positively stable, meaning that if xn ≥ 0 so is xn+1. iv. validation of the splitting algorithm with a simple test case problem (3) has been already solved using discontinuous galerkin elements (dg) [3]. advection and diffusion operators were solved simultaneously using the crank-nicolson scheme providing stable results. however, even for simple test cases some simulations did not always provide positive numerical solutions. one reason is that the same time approximation scheme is not necessarily suitable for both the advection and for the diffusion. that is why a new operator splitting algorithm has been implemented with a different time scheme for each operator. the goal of this section is to validate the implementation of the code. to this end i compare the convergence of the approximation with and without the splitting technique. i briefly explore the question of the positivity of the approximated solution. a. description of the simple test-case first let me introduce a simplified test-case for the validation of the splitting algorithm. set l > 0, and ω =] − l; l[2. let v = (v1,v2) ∈ r2 and biomath 7 (2018), 1812037, http://dx.doi.org/10.11145/j.biomath.2018.12.037 page 8 of 19 http://dx.doi.org/10.11145/j.biomath.2018.12.037 emilie peynaud, operator splitting and discontinuous galerkin methods for advection-reaction... d ∈ r be a constant and 0 ≤ t0 ≤ t1. find u such that ∂u ∂t + v ·∇u + ρu = d∆u in ]t0, t1[×ω, u(x,y, 0) = u0(x,y) on {t0}× ω, (18) n ·∇u = g on ]t0, t1[×∂ω. n ·vu = gin on ]t0, t1[×∂ω−. the initial condition and the boundary condition are chosen such that the solution of problem (18) is explicitly given by ∀(x,y,t) ∈ ω×]t0, t1[ u(x,y,t) = c0 ( a2 a2 + td ) κ(x,y,t)e−ρt. with κ(x,y,t) = c0 exp ( − (x−x0 − tv1)2 + (y −y0 − tv2)2 4(a2 + td) ) where c0 > 0, a > 0, x0 and y0 are real parameters and v1 and v2 are the two components of v. notice that u(x,y,t) > 0 for all (x,y,t) in ω×]t0, t1[. b. numerical validation and convergence to validate the implementation of the splitting technique, i ran the previous test case with different mesh sizes and time steps and i computed the global l2-errors such that eh = ( δt n∑ k=1 ‖u(tk) −uh(tk)‖20,ω )1/2 where tk = t0 + kδt, with k ∈ n+∗ and tn = t1. the flexibility of the splitting technique allows to choose different time schemes for each operator. i consider a θ-scheme with θ = 0 (explicit euler), θ = 1 (implicit euler), and θ = 1 2 (cranknicolson) for both the advection step and the diffusion step, and i consider an exact scheme for the reaction step. for the simulations i took the parameters such that v = (0.1, 0)t , σ = 0.01 and ρ = −1. the triangular meshes used for the simulations are identified by h which is the size of the biggest triangle of the mesh. table i, page 9, gives the number of triangles and the number of nodes of each mesh used for the simulations. choosing h (≈) number of triangles number of nodes 2.63 × 10−1 68 45 1.31 × 10−1 272 157 6.57 × 10−2 1 088 585 3.29 × 10−2 4 352 2 257 1.64 × 10−2 17 408 8 865 8.22 × 10−3 69 632 35 137 4.11 × 10−3 278 528 139 905 table i triangular meshes used for the simulations. fig. 1. solution of the validation test case at t = t0 (left) and t = t1 (right) computed using the dg method with p1finite elements and the euler implicit scheme (θ = 1) and the operator splitting technique with h ≈ 8.2 × 10−3 and δt = 10−2. l = 1/2, the simulations are performed between t0 = 0 and t1 = 1 for different values of the time step δt. fig. 1, page 9, shows the solution at t = t0 and t = t1. the code is implemented in fortran 90 and it is run under a 64-bit linux operating system on a 8-core processor intel r©coretmi77820hq at a frequency of 2.9ghz and with 32 gb of ram. the sparse matrices resulting from the finite element approximation are inverted using a solver provided by the library mumps [51], [52]. according to fig. 2, page 10, all the three temporal schemes provide results with approximately the same level of accuracy with a spatial convergence rate of 2 computed with the global l2biomath 7 (2018), 1812037, http://dx.doi.org/10.11145/j.biomath.2018.12.037 page 9 of 19 http://dx.doi.org/10.11145/j.biomath.2018.12.037 emilie peynaud, operator splitting and discontinuous galerkin methods for advection-reaction... fig. 2. convergence of the solution with respect to the mesh size: plot of the total l2-error computed between t = 0 and t = 1 with and without the splitting technique for the explicit euler scheme (θ = 0), the implicit euler scheme (θ = 1) and the crank-nicolson scheme (θ = 1/2) for δt = 5 × 10−5. fig. 3. convergence of the solution with respect to the time step: plot of the total l2-error computed between t = 0 and t = 1 with and without the splitting technique for the implicit euler scheme (θ = 1) and the crank-nicolson scheme (θ = 1/2) for h = 4.1 × 10−3. norm. the same order of convergence is obtained when the problem is solved without the splitting technique. figure 3 on page 10 shows that the cranknicolson scheme (θ = 1/2) converges in δt2 while the euler implicit scheme (θ = 1) converges in δt, with and without the splitting technique. the convergence rate in time has to be computed with a really refined mesh grid (here h ≈ 4.1 × 10−3). fig. 4. validation of the test case: plot of the cpu time against the mesh size (h) for the computations performed with a processor intel r©coretmi7-7820hq at 2.9 ghz and ram 32 gb, between t = 0 and t = 1 with and without the splitting technique for the explicit euler scheme (θ = 0), the implicit euler scheme (θ = 1) and the crank-nicolson scheme (θ = 1/2) for δt = 5 × 10−5. fig. 5. validation of the test case: plot of the cpu time against the time step (δt) for the computations performed with a processor intel r©coretmi7-7820hq at 2.9 ghz and ram 32 gb, between t = 0 and t = 1 with and without the splitting technique for the explicit euler scheme (θ = 0), the implicit euler scheme (θ = 1) and the crank-nicolson scheme (θ = 1/2) for h ≈ 4.1 × 10−3 and δt ranging from 5 × 10−1 to 5 × 10−4. note that the computations performed here with θ = 0 gave unstable results. biomath 7 (2018), 1812037, http://dx.doi.org/10.11145/j.biomath.2018.12.037 page 10 of 19 http://dx.doi.org/10.11145/j.biomath.2018.12.037 emilie peynaud, operator splitting and discontinuous galerkin methods for advection-reaction... δt global l2-errors min dof t+ cpu time 1 · 10−3 unstable unstable 95 s 2 · 10−4 1.24 · 10−4 −1 · 10−4 0.221 475 s 1 · 10−4 1.22 · 10−4 −1 · 10−4 0.218 838 s 5 · 10−5 1.21 · 10−4 −1 · 10−4 0.216 1672 s 2.5 · 10−5 1.20 · 10−4 −9 · 10−5 0.215 3376 s 1 · 10−5 1.20 · 10−4 −9 · 10−5 0.215 10183 s table ii computations performed with the splitting technique and the explicit euler scheme (θ = 0) with h ≈ 1.6 · 10−2 (intel r©coretm i7-7820hq at 2.9 ghz, ram 32 gb). it results an additional cost in term of cpu time, since it behaves like 1/h2, as shown on figure 4 page 10. for bigger values of h the plot of the errors gave convergence order in time less than 1 and 2 for the implicit euler scheme and the crank-nicolson scheme respectively. as expected, the explicit euler scheme is conditionally stable, such that, when the cfl condition is fulfilled, the computational time becomes prohibitive. indeed, it behaves like 1/δt, as shown on figure 5. for instance, the computation with h ≈ 4.1 × 10−3 and δt = 10−5 takes more than 30 hours with the device specified above. that is why, in the rest of the paper, we will only focus on implicit euler and crank-nicolson schemes. however i present here additional computations performed with a bigger mesh size (h ≈ 1.6×10−2) and smaller time steps chosen such that the cfl condition is fulfilled. the global l2-errors and the cpu time are shown on table ii. clearly, the mesh is not fine enough to recover the convergence order in δt, indeed decreasing the time step results only in an increase of the computational time but not in a significant decrease of the errors. c. some comments on the positivity 1) positivity of the full problem: table iii on page 11 and table v on page 12 give the minimum values of the degrees of freedom (dof) obtained during the simulations performed respectively with and without the splitting technique. the minimum value of the dof is defined such that mintk (mini=1,n x k i ) where x k i is the i th dof at time tk. this quantity gives an idea about the δt h≈1.6·10−2 h≈8.2·10−3 h≈4.1·10−3 5 · 10−1 −7 · 10−1 −7 · 10−1 −7 · 10−1 2 · 10−1 −2 · 10−1 −2 · 10−1 −2 · 10−1 1 · 10−1 −8 · 10−4 −1 · 10−3 −2 · 10−3 4 · 10−2 −2 · 10−4 −6 · 10−4 −1 · 10−3 2 · 10−2 −3 · 10−13 −2 · 10−4 −6 · 10−4 1 · 10−2 −6 · 10−5 −2 · 10−20 −1 · 10−4 4 · 10−3 −3 · 10−4 4 · 10−65 5 · 10−66 2 · 10−3 −3 · 10−4 −9 · 10−11 5 · 10−88 1 · 10−3 −2 · 10−4 −8 · 10−10 8 · 10−114 1 · 10−4 −9 · 10−5 −1 · 10−11 −4 · 10−35 crank-nicolson scheme (θ = 1/2) δt h≈1.6·10−2 h≈8.2·10−3 h≈4.1·10−3 5 · 10−1 −5 · 10−8 −9 · 10−9 −2 · 10−9 2 · 10−1 4 · 10−9 1 · 10−9 4 · 10−10 1 · 10−1 3 · 10−11 3 · 10−11 1 · 10−11 4 · 10−2 6 · 10−17 5 · 10−17 5 · 10−17 2 · 10−2 3 · 10−23 2 · 10−23 2 · 10−23 1 · 10−2 1 · 10−31 4 · 10−32 3 · 10−32 4 · 10−3 −3 · 10−5 1 · 10−48 6 · 10−49 2 · 10−3 −5 · 10−5 2 · 10−65 2 · 10−66 1 · 10−3 −7 · 10−5 −1 · 10−13 2 · 10−88 1 · 10−4 −9 · 10−5 −7 · 10−12 −3 · 10−43 implicit euler scheme (θ = 1) table iii minimum value of the dof (mini,k x k i ) computed with the splitting algorithm with the crank-nicolson scheme (top) and the implicit euler scheme (bottom). stability and the positivity preserving behaviour of the schemes. tables iii and v clearly show that the schemes are not always positivity preserving. in case where the approximated solution is not positive for all t > t0 i also check if it becomes non-negative for larger time ie. if there is t+ > t0 biomath 7 (2018), 1812037, http://dx.doi.org/10.11145/j.biomath.2018.12.037 page 11 of 19 http://dx.doi.org/10.11145/j.biomath.2018.12.037 emilie peynaud, operator splitting and discontinuous galerkin methods for advection-reaction... δt h≈1.6·10−2 h≈8.2·10−3 h ≈4.1·10−3 5 · 10−1 2 · 10−1 1 · 10−1 4 · 10−2 2 · 10−2 0.24 1 · 10−2 0.07 0.10 4 · 10−3 0.172 t0 t0 2 · 10−3 0.202 0.044 t0 1 · 10−3 0.210 0.079 t0 1 · 10−4 0.2140 0.0939 0.0297 crank-nicolson scheme (θ = 1/2) δt h≈1.6·10−2 h≈8.2·10−3 h≈4.1·10−3 5 · 10−1 1 1 1 2 · 10−1 t0 t0 t0 1 · 10−1 t0 t0 t0 4 · 10−2 t0 t0 t0 2 · 10−2 t0 t0 t0 1 · 10−2 t0 t0 t0 4 · 10−3 0.072 t0 t0 2 · 10−3 0.132 t0 t0 1 · 10−3 0.173 0.029 t0 1 · 10−4 0.2102 0.0874 0.0217 implicit euler scheme (θ = 1) table iv positivity threshold (t+) computed with the splitting algorithm and the crank-nicolson scheme (top) and the implicit euler scheme (bottom). such that xki ≥ 0,∀i = 1,n for all tk > t+ > t0. the smallest such t+, if it exists, is referred as the positivity threshold, as defined in [36]. table iv on page 12 and table vi on page 13 give the positivity thresholds computed with and without the splitting technique respectively. for the crank-nicolson scheme (θ = 1/2) and the implicit euler scheme (θ = 1) the positivity is obtained under a specific condition on the time step and the mesh size. for a given mesh size, the time step δt must be bounded from above, but also from below to guarantee that the solution stays positive all along the simulation. in the case of the splitting technique those bounds are more restrictive than in the case of the resolution of the full problem without splitting. those bounds are also more restrictive in the case of the cranknicolson (θ = 1/2) scheme than in the case of the δt h≈1.6·10−2 h≈8.2·10−3 h≈4.1·10−3 5 · 10−1 −4 · 10−1 −4 · 10−1 −4 · 10−1 2 · 10−1 −1 · 10−1 −1 · 10−1 −1 · 10−1 1 · 10−1 1 · 10−13 1 · 10−13 1 · 10−13 4 · 10−2 1 · 10−21 1 · 10−21 9 · 10−22 2 · 10−2 3 · 10−30 1 · 10−30 1 · 10−30 1 · 10−2 −6 · 10−5 1 · 10−42 9 · 10−43 4 · 10−3 −3 · 10−4 3 · 10−64 5 · 10−65 2 · 10−3 −3 · 10−4 −8 · 10−11 3 · 10−87 1 · 10−3 −2 · 10−4 −7 · 10−10 3 · 10−113 1 · 10−4 −9 · 10−5 −1 · 10−11 −8 · 10−35 crank-nicolson scheme (θ = 1/2) δt h≈1.6·10−2 h≈8.2·10−3 h≈4.1·10−3 5 · 10−1 2 · 10−5 2 · 10−5 2 · 10−5 2 · 10−1 1 · 10−7 1 · 10−7 1 · 10−7 1 · 10−1 6 · 10−10 5 · 10−10 5 · 10−10 4 · 10−2 1 · 10−15 1 · 10−15 1 · 10−15 2 · 10−2 6 · 10−22 5 · 10−22 5 · 10−22 1 · 10−2 1 · 10−30 7 · 10−31 6 · 10−31 4 · 10−3 −2 · 10−5 2 · 10−47 8 · 10−48 2 · 10−3 −4 · 10−5 2 · 10−64 2 · 10−65 1 · 10−3 −7 · 10−5 −3 · 10−13 2 · 10−87 1 · 10−4 −9 · 10−5 −7 · 10−12 −1 · 10−42 implicit euler scheme (θ = 1) table v minimum value of the dof (mini,k x k i ) computed without the splitting algorithm and the crank-nicolson scheme (top) and implicit euler scheme (bottom). implicit euler scheme (θ = 1). refining the mesh results in less restrictions on the time step but also lead to additional computational time. with the crank-nicolson scheme (θ = 1/2), for a given mesh size, if δt is too big, there is no threshold of positivity in tk ∈]t0, t1] and the computed solution is not non-negative all along the simulation. for θ = 1/2 and θ = 1, still with a given mesh size, if δt is too small, the simulations showed that there is a threshold of positivity t+ such that the approximated solution becomes non-negative for tk ≥ t+. the thresholds of positivity slightly depend on the time step and tend to increase when the time step δt is decreased. the computations clearly showed that the positivity thresholds diminish with the mesh size h (see for example [36]). altogether, the positivity of the approximated biomath 7 (2018), 1812037, http://dx.doi.org/10.11145/j.biomath.2018.12.037 page 12 of 19 http://dx.doi.org/10.11145/j.biomath.2018.12.037 emilie peynaud, operator splitting and discontinuous galerkin methods for advection-reaction... δt h≈1.6·10−2 h≈8.2·10−3 h≈4.1·10−3 5 · 10−1 1 1 2 · 10−1 0.8 0.8 0.8 1 · 10−1 t0 t0 t0 4 · 10−2 t0 t0 t0 2 · 10−2 t0 t0 t0 1 · 10−2 0.08 t0 t0 4 · 10−3 0.188 t0 t0 2 · 10−3 0.208 0.050 t0 1 · 10−3 0.213 0.083 t0 1 · 10−4 0.2143 0.0942 0.0300 crank-nicolson scheme (θ = 1/2) δt h≈1.6·10−2 h≈8.2·10−3 h≈4.1·10−3 5 · 10−1 t0 t0 t0 2 · 10−1 t0 t0 t0 1 · 10−1 t0 t0 t0 4 · 10−2 t0 t0 t0 2 · 10−2 t0 t0 t0 1 · 10−2 t0 t0 t0 4 · 10−3 0.0720 t0 t0 2 · 10−3 0.1380 t0 t0 1 · 10−3 0.1760 0.0290 t0 1 · 10−4 0.2104 0.0877 0.0221 implicit euler scheme (θ = 1) table vi positivity threshold (t+) computed without the splitting algorithm and the crank-nicolson scheme (top) and the implicit euler scheme (bottom). solution is obtained at the expense of the computational cost, but for a given mesh size h computations performed with too small time step can also lead to a loss of positivity for small tk. in [36] (and references therein), thomée showed that threshold values of tk > 0 may exist such that x(t) > 0 when t > tk. at this stage, one may wonder how each term of the splitting behaves in terms of positivity preservation. the reaction term is approximated using an exact scheme, so obviously the positivity of the solution is preserved. what about the diffusion and the advection term ? 2) positivity of the pure diffusion problem: here i set v = (0, 0) and ρ = 0, while keeping all others parameters to the same values as previously. table vii clearly shows that the crank-nicolson scheme (θ = 1/2) is positivity preserving under a δt h≈1.6·10−2 h≈8.2·10−3 h ≈4.1·10−3 5 · 10−1 −5 · 10−1 −5 · 10−1 −5 · 10−1 2 · 10−1 −2 · 10−1 −2 · 10−1 −2 · 10−1 1 · 10−1 4 · 10−15 4 · 10−15 4 · 10−15 4 · 10−2 6 · 10−23 5 · 10−23 4 · 10−23 2 · 10−2 2 · 10−31 8 · 10−32 6 · 1032 1 · 10−2 −4 · 10−5 1 · 10−43 6 · 1043 4 · 10−3 −3 · 10−4 4 · 10−65 5 · 10−66 2 · 10−3 −3 · 10−4 −5 · 10−11 5 · 10−88 1 · 10−3 −2 · 10−4 −6 · 10−10 8 · 10−114 1 · 10−4 −9 · 10−5 −1 · 10−11 −1 · 10−34 crank-nicolson scheme (θ = 1/2) δt h≈1.6·10−2 h≈8.2·10−3 h≈4.1·10−3 5 · 10−1 4 · 10−6 4 · 10−6 4 · 10−5 2 · 10−1 2 · 10−8 2 · 10−8 2 · 10−8 1 · 10−1 2 · 10−11 2 · 10−11 2 · 10−11 4 · 10−2 5 · 10−17 5 · 10−17 5 · 10−17 2 · 10−2 3 · 10−23 2 · 10−23 2 · 10−23 1 · 10−2 1 · 10−31 4 · 10−32 3 · 10−32 4 · 10−3 −2 · 10−5 1 · 10−48 6 · 10−49 2 · 10−3 −4 · 10−5 2 · 10−65 2 · 10−66 1 · 10−3 −7 · 10−5 −3 · 10−13 2 · 10−88 1 · 10−4 −9 · 10−5 −7 · 10−12 −2 · 10−42 implicit euler scheme (θ = 1) table vii minimum value of the dof (mini,k x k i ) computed for the pure diffusion problem with the crank-nicolson scheme (top) and the implicit euler scheme (bottom). cfl-like condition with upper and lower bounds, like in the previous test. the implicit euler scheme (θ = 1) seems to be more favorable, since it preserves the positivity even for big values of the time step. for both the crank-nicolson (θ = 1/2) and implicit euler (θ = 1) schemes, the approximated solution suffers from a loss of positivity for small values of tk when the time step is too small. according to table viii, there are positivity thresholds, like in [36] which indeed deals with the heat equation. 3) positivity of the pure advection problem: here i set σ = 0 and ρ = 0, while keeping all others parameters to the same values as in the first test. table ix shows that none of the computations performed gave a non negative solutions, even though the minimum value of the dof can be really close to zero for small mesh sizes. besides, i did biomath 7 (2018), 1812037, http://dx.doi.org/10.11145/j.biomath.2018.12.037 page 13 of 19 http://dx.doi.org/10.11145/j.biomath.2018.12.037 emilie peynaud, operator splitting and discontinuous galerkin methods for advection-reaction... δt h≈1.6·10−2 h≈8.2·10−3 h≈4.1·10−3 5 · 10−1 2 · 10−1 0.8 0.8 0.8 1 · 10−1 t0 t0 t0 4 · 10−2 t0 t0 t0 2 · 10−2 t0 t0 t0 1 · 10−2 0.1 t0 t0 4 · 10−3 0.2 t0 t0 2 · 10−3 0.216 0.054 t0 1 · 10−3 0.219 0.085 t0 1 · 10−4 0.2203 0.0956 0.0303 crank-nicolson scheme (θ = 1/2) δt h≈1.6· 10−2 h≈8.2·10−3 h≈4.1·10−3 5 · 10−1 t0 t0 t0 2 · 10−1 t0 t0 t0 1 · 10−1 t0 t0 t0 4 · 10−2 t0 t0 t0 2 · 10−2 t0 t0 t0 1 · 10−2 t0 t0 t0 4 · 10−3 0.084 t0 t0 2 · 10−3 0.148 t0 t0 1 · 10−3 0.184 0.032 t0 1 · 10−4 0.2165 0.0893 0.0224 implicit euler scheme (θ = 1) table viii positivity threshold (t+) computed for the pure diffusion problem with the crank-nicolson scheme (top) and the implicit euler scheme (bottom). not observe any positivity threshold. the approximated solution stays non positive all along the simulation. however i run additional simulations with even smaller mesh size (h ≈ 2.0 × 10−3 and δt = 10−4). this time the computed solution was positive at the beginning of the simulation (before t− = 1.9 × 10−3), pointing the existence of a threshold of negativity, to finally reaching a negative minimum values of dof (around −10−44). unfortunately, this threshold of negativity is really small compared to the ending time of the computation (t1 = 1), while the computational time was reaching more than 14 hours (intel r©coretmi77820hq at 2.9 ghz, ram 32 gb) for both the crank-nicolson and the implicit euler schemes. in fact it is well known that for the advection term the solution can be polluted by overshoot and undershoot oscillations near a discontinuity or a δt h≈1.6·10−2 h≈8.2·10−3 h≈4.1·10−3 5 · 10−1 −2 · 10−1 −2 · 10−1 −2 · 10−1 2 · 10−1 −4 · 10−2 −4 · 10−2 −4 · 10−2 1 · 10−1 −4 · 10−3 −2 · 10−3 −1 · 10−3 4 · 10−2 −1 · 10−3 −4 · 10−7 −5 · 10−7 2 · 10−2 −1 · 10−3 −6 · 10−8 −2 · 10−13 1 · 10−2 −1 · 10−3 −4 · 10−8 −3 · 10−28 4 · 10−3 −1 · 10−3 −3 · 10−8 −1 · 10−28 2 · 10−3 −1 · 10−3 −3 · 10−8 −1 · 10−28 1 · 10−3 −1 · 10−3 −3 · 10−8 −1 · 10−28 1 · 10−4 −1 · 10−3 −3 · 10−8 −1 · 10−28 crank-nicolson scheme (θ = 1/2) δt h≈1.6·10−2 h≈8.2·10−3 h≈4.1·10−3 5 · 10−1 −1 · 10−4 −1 · 10−10 −8 · 10−34 2 · 10−1 −2 · 10−4 −6 · 10−10 −3 · 10−32 1 · 10−1 −4 · 10−4 −1 · 10−9 −2 · 10−31 4 · 10−2 −6 · 10−4 −4 · 10−9 −9 · 10−31 2 · 10−2 −8 · 10−4 −7 · 10−9 −2 · 10−30 1 · 10−2 −9 · 10−4 −1 · 10−8 −5 · 10−30 4 · 10−3 −1 · 10−3 −1 · 10−8 −8 · 10−30 2 · 10−3 −1 · 10−3 −2 · 10−8 −2 · 10−29 1 · 10−3 −1 · 10−3 −2 · 10−8 −4 · 10−29 1 · 10−4 −1 · 10−3 −3 · 10−8 −9 · 10−29 implicit euler scheme (θ = 1) table ix minimum value of the dof (mini,k x k i ) computed for the pure advection problem with the crank-nicolson scheme (top) and the implicit euler scheme (bottom). sharp layer, see [34], [33], [35], [30]. for low order accurate spacial approximations one can prove the positivity preserving property of the scheme [33]. but for high order schemes slopes limiters are often required to guarantee the positivity of the approximated solution. when slope limiters are used, explicit time schemes seem to be suitable for the advection [6]. however, in the next section we will only privilege a numerical scheme that is unconditionally stable, i.e. the crank-nicolson scheme, that is a two-order scheme. v. application to the simulation of root system growth in this section, i apply the previous dg-splitting approach to solve numerically the c-root model. first, i detail the parameters used for the simulations, then, i present and validate the results of the biomath 7 (2018), 1812037, http://dx.doi.org/10.11145/j.biomath.2018.12.037 page 14 of 19 http://dx.doi.org/10.11145/j.biomath.2018.12.037 emilie peynaud, operator splitting and discontinuous galerkin methods for advection-reaction... simulations. a. the c-root parameters for eucalyptus root growth the parameters and operators’ coefficients are chosen based on the previous calibration done in [2]. the diffusion coefficient, σ, is build using the following gaussian function fα,µ(x,y) = α √ 2π exp ( − (r(x,y) −µ)2 2 ) where r(x,y) = √ (x−x0)2 + (y −y0)2 and (x0,y0) ∈ ω =] − l,l[. the function fα,µ(x,y) depends on two real and positive parameters: α, related to the maximum amplitude of fα,µ, and µ, the distance from (x0,y0) to the point where the function fα,µ reaches its maximum. the diffusion tensor is taken such that σ(x,y) = fαd,µd (x,y) ( 1 0 0 1 ) , for all (x,y) ∈ ω, and αd, µd ∈ r+ are given parameters. the advection vector is taken such that v(x,y) = (0,−v0)t , for all (x,y) ∈ ω, with v0 a positive constant. the reaction term is constant in space and splited into two contributions: βr and µr, the branching and mortality rates, respectively. that is ρ = βr −µr ∈ r. the branching rate, βr, is estimated from biological knowledge: it is equal to zero before 9 months and equal to 1/3 after, since no roots die before 9 month. however, for the following simulations we will not distinguish the contribution of βr and µr, so that the reaction term will only be described by the parameter ρ. fig. 6. density of apices computed at t = 6, t = 12, t = 18 and t = 24 months (from the left to the right and from the top to the bottom). b. some simulations for the simulation the initial solution is chosen equal to the following function: u0(x,y) = a [ exp(b(1 −x)) (exp(−b(1 −x)) + exp(b(1 −x))) − exp(b(−1 −x)) (exp(−b(−1 −x)) + exp(b(−1 −x))) ] × [ exp(b(1 −y)) (exp(−b(1 −y)) + exp(b(1 −y))) − exp(b(−1 −y)) (exp(−b(−1 −y)) + exp(b(−1 −y))) ] with a = 2 · 10−4 and b = 1. the parameters’ values µr, αd , µd are estimated using the code biomath 7 (2018), 1812037, http://dx.doi.org/10.11145/j.biomath.2018.12.037 page 15 of 19 http://dx.doi.org/10.11145/j.biomath.2018.12.037 emilie peynaud, operator splitting and discontinuous galerkin methods for advection-reaction... fig. 7. l2-error with respect to the solution obtained with the mesh of size h ≈ 9.87 × 10−2 and δt = 10−3 computed at t = 6, t = 12, t = 18 and t = 24 months and plotted against the mesh size. described in [2]. i run the simulations from t0 = 1 to t1 = 24 months, with l = 13. the simulations are performed for different values of the mesh size. fig. 6, page 15, shows the solution computed at four different stages of the root system development. one can notice the diffusion of the apices in the soil and also the transport of the apices from the top to the bottom of the soil layer. since there is no analytic solution, the convergence of the computation is evaluated by measuring the l2errors with respect to the approximated solution computed with the finest mesh (h ≈ 8.97×10−2) and with δt = 10−3. the curves of the errors against the mesh size are plotted on figure 7 and clearly show that the dg-splitting algorithm converges with a convergence rate of almost two. however, one can note that the mesh sizes and the time steps chosen for the simulations presented here might not be small enough. the positivity of the solution is not preserved at all times and the full convergence might not be acheived. unfortunatly, refining the mesh sizes and the time steps can lead to prohibitive computational time as shown on table x. on top of that simulation of root system growth can last for a long period of time, particularly for trees. finally, this application shows promising results for future simulations of h (≈) δt = 10−1 δt = 10−2 δt = 10−3 1.44 1 s. 7 s. 66 s. 7.18 × 10−1 16 s. 46 s. 6 min. 3.59 × 10−1 70 s. 3.5 min. 27 min. 1.79 × 10−1 7 min. 17 min. 2 h. 8.97 × 10−2 50 min. 2h30 9 h. table x computational times for the simulations of a root system growth performed (with the processor intel r©coretm i7-7820hq at 2.9 ghz, ram 32 gb) between t = 1 and t = 24 months with the dg-splitting algorithm and the crank-nicolson scheme (θ = 1/2). the root system growth, provided that the computational cost is not limiting. further simulations requiring much more computational power has to be done to check if the convergence is acheived. this application also point out the difficulties related to the rigorous simulation validation in realistic test-cases of root system growth. vi. conclusion in this work, a discontinuous galerkin approximation method based on unstructured mesh combined with operator splitting has been described, implemented and tested, to solve an advectiondiffusion-reaction equation used to model the growth of root systems. the code has been validated in a simple test case for which an analytic expression of the solution is known. the computations showed that the method convergences with a convergence rate of two in space with p 1-finite elements. a convergence rate of one and two in time were obtained for respectively the implicit euler scheme and the crank-nicolson scheme both with and without the splitting technique. the computations of those convergence rates required the use of fine mesh grids. for the explicit euler scheme, such fine mesh computations were not performed since they require really small time steps to fulfill the cfl condition, resulting in huge additional computational cost. indeed the computational time of the dg-splitting algorithm behaves like 1/δt and 1/h2 where δt and h are respectively the time step and the mesh size. biomath 7 (2018), 1812037, http://dx.doi.org/10.11145/j.biomath.2018.12.037 page 16 of 19 http://dx.doi.org/10.11145/j.biomath.2018.12.037 emilie peynaud, operator splitting and discontinuous galerkin methods for advection-reaction... similarly, the positivity of the approximated solution is obtained at the expense of the computational time since it requires meshes of small size and small time steps. in fact, there is a cfllike condition for positivity that has to be fulfilled to guarantee the positivity of the approximated solution. but for a given mesh size computations performed with too small time step can also lead to a loss of positivity at the beginning of the computation [36]. in that cases, the computations showed that there is a positivity threshold in time after which the solution becomes positive. this positivity threshold clearly appeared to diminish with the mesh size. this behavior is specific to the diffusion term. for the advection term, the computations also showed that the positivity of the solution can be preserved, but only at the beginning of the simulation and it required a really small mesh size and time step leading to huge computational time. further studies in terms of numerical analysis has to be done in that direction. i also performed a more realistic simulation of root system growth. the computations showed that the algorithm converged but additional simulations with smaller time steps and mesh sizes might be performed to recover the full convergence order and positivity. validation of the computation, but above all the computational time appeared to be the major limitations of the root growth simulation based on the c-root model, particularly when it comes to deal with trees for which the life span is rather a long period of time. 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https://doi.org/10.1007/978-0-387-70914-7 https://doi.org/10.1007/978-0-387-70914-7 https://doi.org/10.1137/s0895479899358194 https://doi.org/10.1137/s0895479899358194 https://doi.org/10.1016/j.parco.2005.07.004 http://dx.doi.org/10.11145/j.biomath.2018.12.037 introduction the model modelling root growth with pde: the c-root model the weak problem the positivity preserving property of the solution approximation of the model the operator splitting technique the advection step: dg upwind scheme the diffusion step the reaction step validation of the splitting algorithm with a simple test case description of the simple test-case numerical validation and convergence some comments on the positivity positivity of the full problem positivity of the pure diffusion problem positivity of the pure advection problem application to the simulation of root system growth the c-root parameters for eucalyptus root growth some simulations conclusion references original article biomath 2 (2013), 1312301, 1–9 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum cell growth models using reaction schemes: batch cultivation svetoslav markov institute of mathematics and informatics bulgarian academy of sciences sofia, bulgaria smarkov@bio.bas.bg received: 27 october 2013, accepted: 30 december 2013, published: 18 january 2014 abstract—simple structured mathematical models of bacterial cell growth are proposed. the models involve fractions of bacterial cells related to their physiological phases. reaction schemes involving the biomass of the cell fractions, the substrate and the product are proposed in analogy to reaction schemes in enzyme kinetics. applying the mass action law these reaction schemes lead to dynamical models represented by systems of ode’s. all parameters of the models are rate constants with clear biological or biochemical meaning. the proposed models generalize classical bacterial growth models and offer more flexible tools for modelling and control of biotechnological processes. in this paper the study is focused on batch cultivation models. we formulate a hypothesis that cell growth models can be entirely based on reaction schemes. i. introduction bacterial cell growth involves reaction steps such as cell phase transition, reproduction and mortality. we propose reaction schemes combining such reaction steps. we then apply the mass action law to formulate simple dynamical models of bacterial growth which account for the physiological states of bacterial cells. the cells are classified into fractions (compartments) related to their physiological states. classical bacterial cell growth models make use of the assumption that all cells are in the same physiological state at a given moment t. therefore these models involve a single variable representing the biomass (population density) of the organisms. it has been recognized since long that most classical cell growth models describe adequately real processes under favorable environmental conditions, whenever organisms are steadily at log (exponential) phase when the cells actively divide and grow at a maximal rate. however, the environment in bio-reactors may be perturbed by various factors and the cells change their physiological state. under perturbed conditions classical cell growth models may fail to reflect adequately the dynamics of the bio-processes. modifications of these models in various directions have been proposed in the literature. a wellstudied direction is to allow some of the parameters in the model, such as the “specific growth rate”, to depend on the nutrient substrate and on other quantities such as temperature, ph etc. [3]. another modelling direction— not so well explored—is based on the assumption that the bacterial cells are not simultaneously in the same physiological state (as assumed in classical models). such models are named “structured” [9]. a. structured cell growth models an important feature of bacteria is their physiological state/phase (lag, log, stationary, dead). the physiological state depends on the environmental conditions for the cell population. we assume that not all cells are simultaneously in the same state. for example, the log (exponential) phase is the time period when the cells reproduce themselves. it is realistic to assume that not all cells are in log state at a given moment t. assume that the environmental conditions have been unfavorable and then have rapidly improved so that the cells start citation: svetoslav markov, cell growth models using reaction schemes: batch cultivation, biomath 2 (2013), 1312301, http://dx.doi.org/10.11145/j.biomath.2013.12.301 page 1 of 9 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2013.12.301 s markov, cell growth models using reaction schemes: batch cultivation changing their physiological state from lag to log phase. some cells that are in better physiological condition than others and will start to reproduce earlier. there could exist cells that are not able to reproduce at all (genetically damaged, defect cells), but still viable (in lag phase). we assume that certain fractions of cells may react differently under similar conditions changing dynamically their physiological states (say, from lag to log phase and conversely). thus organisms can be classified into two (or more) fractions corresponding to their physiological states and degree of adaptation ability. mathematically this means to assign different phase variables for the biomass (population density) of the respective fractions of organisms and to study the dynamics of these fractions as separate cell populations. structured models inspired by models of phytoplankton [2] have been proposed and used in [4], [8], [9]. we accept the idea of structured models in so far that cell population is subdivided into fractions. in addition our modelling approach is tightly related to enzyme kinetics. this contributes to more clarity in the biological interpretation of the partaking terms in the models. our approach aims to assign clear biological meaning to all parameters involved. in particular, no parameters depend on other quantities, that is, only numeric parameters are used. we propose reaction schemes for the transition dynamics of the cell fractions and the substrate/products involved. we believe that such reaction schemes will be helpful in understanding the biological mechanisms of the cell growth dynamics and, in particular, in understanding the biological nature of the physiological states of the fractions involved—for instance, to what extend the phases are related to the cell cycle or to cell communications. cell fractions are assumed to be strongly related to the individual cell cycle in the works [4], [8], [9]. our biological interpretations are closer to those in [13]; there an interesting structured threephase mathematical model stressing on the enzymatic competence of the cells is proposed and discussed. b. relation to enzyme kinetics once postulating that cell growth is tightly related to enzyme activity of the cells, then it is natural to make use of basic enzyme kinetics in the modelling of cell growth. from the perspective of cell growth dynamics bacteria can be viewed as complexes of enzymes processing nutrient substrates. thus cells are similar to enzymes in that both process specific substrates. the only big difference between cells and enzyme complexes is the reproduction property that does not exist in the world of enzymes (viewed as chemical substances). we recall some familiar knowledge from enzyme kinetics. the michaelis-menten differential equation describing the uptake of a substrate by a specific enzyme ds/dt = −const s/(km + s) will be briefly referred as the mm-model. the mmmodel is used to represent monod specific growth rate function µ(s) = const s/(km + s) which is an integral part of various classical cell growth models [11]. for the construction of our models we make use of the familiar reaction scheme between an enzyme e with a single active site and a substrate s, forming an enzymesubstrate complex c, which then yields product p : s + e k1−→←− k−1 c k2−→ p + e. we shall call this scheme henri-michaelis-menten (hmm) reaction scheme in tribute to the prominent scholars victor henri, leonor michaelis and maud menten [5], [6], [10], cf. also [14],[16]. the hmmscheme leads to the familiar enzyme kinetics model involving four phase variables: the concentrations of the substrate s, the product p and the concentrations of the two forms of the enzyme—free e and bound c. denoting the concentrations: s = [s], e = [e], c = [c], p = [p ], and applying the mass action law we obtain a system of four ode’s: ds/dt = −k1es + k−1c, de/dt = −k1es + (k−1 + k2)c, dc/dt = k1es− (k−1 + k2)c, dp/dt = k2c, to be further called the hmm-model. in contrast to the mm-model the hmm-model describes not only the dynamics of the concentration of the substrate s, but also the dynamics of the concentrations of the two fractions of the enzyme—free e and bound c plus the product p . knowing that the solution of the mm-model is an approximation of the solution for the substrate in the hmm-model [12], it is natural to come to the idea to use the exact hmm-model—instead of the approximate mm-model—when constructing a cell growth model. the idea of using the hmm-mechanism for modelling cell growth requires to introduce phase variables corresponding to certain fractions of the cell population which we relate to cell phases. our modelling approach to bacterial growth has been explained in some detail biomath 2 (2013), 1312301, http://dx.doi.org/10.11145/j.biomath.2013.12.301 page 2 of 9 http://dx.doi.org/10.11145/j.biomath.2013.12.301 s markov, cell growth models using reaction schemes: batch cultivation in [1], where some initial variants of structured twophase models are proposed. here we revisit some of these models by basing them more tightly on the hmm reaction scheme, resp. hmm-model. let us briefly sketch the idea of our modelling approach. recall that in a familiar classical cell growth model the utilization of the substrate s by the cell biomass z is described by: ds/dt = − δµz, where µ is monod specific growth rate function µ(s) = const s/(k + s) [11]. if the cell biomass z is nearly constant—as is usually the desired case under a continuous cultivation process—then this model coincides with the mm-model for the substrate uptake by corresponding enzymes. several remarks can be made. first, the connection between the two models is obvious as substrate nutrients are utilized by the cell via corresponding enzyme complexes in the cell cytoplasm; this explains why classical monod type models are so successful and popular. second, recall that the mm-model is an approximation of the accurate hmm-model, hence classical models are likely to be approximations of certain more accurate models involving additional phase variable(s). third, we know that the above mentioned approximation is a good one only if the ratio enzyme/substrate is small, which is rarely the case when realistic cell growth is considered [15]. fourth, passing from the mm-model to the more general hmm-model we get free of rational functions in the right hand side of the model such as s/(k +s) (and have only simple polynomial expressions), however, for the expense of introducing additional phase variable(s). having in mind these remarks one can ask what would be the natural way to upgrade classical cell growth models in order to make them more adequate. to answer this question we propose suitable reaction schemes for the cell growth dynamics based on the hmm-model. this paper focuses on cell growth models related to batch cultivation. continuous cultivation models are proposed and studied in a forthcoming work. c. model assumptions there may be various biological interpretations of the phases depending on the particular situations (bacterial species, nutrient substrate, environmental conditions etc). the following interpretations seem plausible in certain conditions. we start with the following abstractions. enzyme complexes are considered as the most important components of the cells (from the perspective of their production abilities). apart of enzymes bacterial cells contain other substances such as specific proteins, nucleotides, water, minerals etc. for the purposes of this study we shall consider these substances as playing auxiliary role in the metabolic processes. we assume that the total biomass of the bacteria is proportional to the total mass of enzymes in the bacteria cells. hence, in cell growth dynamics bacteria can be considered as organized complexes of enzymes that perform specific metabolic functions generalizing the functions of particular enzymes. as already mentioned the functions of specific enzymesubstrate pairs are modelled by enzyme kinetics systems of ode’s, e. g. the hmm-model. considered as sets of enzymes, cells can also be modelled by the enzyme kinetics systems of ode’s. the total microbial biomass increases for the expense of the utilized nutrient substances and the product substances produced by the bacteria. from an abstract perspective, proteins reproduce themselves—via the ribosome mechanisms. part of the newly built products “come into life” as newborn cell components. for simplicity we may assume that one of the newly formed daughter cells coincides with the mother cell (which corresponds to reproduction by cellular budding). classical bacterial growth models typically make use of a single variable for the biomass concentration. the use of a single variable for the biomass does not permit to distinguish cells of different physiological states at a given moment t. in this work we distinguish between two main phases of bacterial cells: — cells in lag phase denoted as x-cells; — cells in log phase denoted as y -cells. let us recall some characteristics of these two phases. lag phase x-cells do not reproduce and their metabolic activity is limited. x-cells appear as dormant being stressed by environmental perturbations, nutrient limitation etc. the moment when x-cells start to be metabolically active by consuming nutrient substrates and forming products p , then we consider these cells as log phase y -cells, that is x-cells transform into y -cells. y -cells are metabolically active; they consume nutrient substrate s from the environment to form product p . the latter can be viewed as the set of all biochemical substances (like proteins, nucleotides, etc) needed for the cell functioning, including reproduction. we assume that y -cells are in reproduction state at the current moment. reproduction can be viewed as part of the production process; since the newly formed substances p are component part of the newborn cells we assume that this part of p transforms into new cells. biomath 2 (2013), 1312301, http://dx.doi.org/10.11145/j.biomath.2013.12.301 page 3 of 9 http://dx.doi.org/10.11145/j.biomath.2013.12.301 s markov, cell growth models using reaction schemes: batch cultivation after reproduction (by binary fission) the newborn daughter cells are considered as y-cells [7]. thus newly built metabolic products become live y -cells. the ultimate goal of the cell activities is cell reproduction. log phase y -cells are those that have reached this goal, in contrast to lag phase x-cells that are in an apparently dormant phase. both fractions of xand y -cells are assumed to be homogeneously distributed (stirred) in the volume, so that we can work with their biomass concentrations (population densities) x,y, respectively. the same refers to the substances s and p which are also considered well stirred. in agreement with the above assumptions we assign the following biological/biochemical meanings to the variables used in the proposed model: s = s(t) is the concentration of all nutrient substrates s in the bioreactor (fermenter) at moment t that are used for the bacterial growth; x = x(t) is the biomass (concentration or population) of x-cells being in lag phase at time t; these bacteria are not metabolically active. x-cells do not consume s and do not produce p , neither they reproduce at time t. (nevertheless, x is not constant, due to transition of the phases.) y = y(t) is the biomass concentration of y -cells in log phase at time t; y -cells utilize product s and produce substances p . y -cells are in the state of reproduction at time t. the substances metabolized by the y -cells are part of the cell products denoted p . z = z(t) = x + y is the total biomass concentration that is the sum of the two fractions x = x(t) and y = y(t). p = p(t) is the concentration of all product substances p metabolized by the y -cells including those excreted and those build up for growth and reproduction. products p are partly included in the newborn daughter cells. the above model assumptions, abstractions and interpretations suggest direct relations to enzyme kinetics. thus x-cells can be related to free enzymes, as they are not involved in production or reproduction at a given time moment t; y -cells can be related to bound enzymes, as their enzyme complexes are actively engaged with production and reproduction at time t. our biological assumptions are close to the ones discussed in [13], [4], [8], [9]. our proposed models are also based on biochemical arguments related to enzyme kinetics. however, in contrast to the cited works, we make use of the henri-michaelis-menten mechanism involving the concentrations of substrate, enzymes (free and bound) and product (and not just of the single michaelis-menten differential equation for the substrate). d. reaction steps of the cell growth process in the course of model construction we formulate certain possible reaction steps of the cell growth process, e. g.: cell growth reduces or decreases owing to limitation of nutrient substrate; cells stop product formation and stop reproducing themselves; cell growth increases due to abundance and utilization of nutrient substrate; cell population rapidly increases due to reproduction (by binary fission); transition of substrate into product via cell metabolic enzymes; transition of metabolized product into living cells; excretion of waste products of cell metabolism; death and disintegration of living cells; cell growth inhibition due to excess of substrate. some of the above reaction steps are related to the transition of cells from one phase to another. especially important are the transitions from lag to log phase and vice versa. the last two steps concern the stationary and death phases. these two phases are also significant, but in this work we try to ignore them for the sake of simplicity, mainly focusing on the reaction steps involving lag-log transitions. ii. batch cultivation: reaction schemes a. the lag–log cell phase transition we shall borrow the basic lag–log cell phase transition reaction scheme from henri-michaelis-menten enzyme kinetic [12], [14], identifying free enzymes with xcells (bacteria in lag phase) and bound enzymes with y -cells (log phase bacteria). following the familiar reaction scheme for the substrate-enzyme kinetics when the enzymes possess a single active site we have: s + x k1−→←− k−1 y k2−→ p + x, (rsp) wherein k1,k−1,k2 are rate constants. passing from a model describing a specific substrateenzyme reaction to a model describing a substrate-cell activity is a jump from molecular level to cell population level. a possible justification of such a jump is that: biomath 2 (2013), 1312301, http://dx.doi.org/10.11145/j.biomath.2013.12.301 page 4 of 9 http://dx.doi.org/10.11145/j.biomath.2013.12.301 s markov, cell growth models using reaction schemes: batch cultivation i) cells can be viewed as (organized) complexes of enzymes, ignoring thus other components of the cells; ii) nutrient substrates can be restricted to a few (sometimes a single) limiting substrate(s); iii) the cell enzymes process the nutrient substrate s to produce new complex substances (like proteins, via the ribosome mechanism) partly included in product p in the right hand-side of reaction scheme (rsp); iv) x-cells that become engaged in metabolic/reproductive processes transform (change their status)from lag into log phase y -cells. the product p in the scheme (rsp) can be viewed as (part of) the highly organized substances needed to complete the enzymatic competence of the cell to be able to reproduce. we can think of the y -cells as having reached the highest inner protein-nucleotide organization needed to reproduce. we shall thus assume that the y cells are those engaged in production and reproduction at the current moment t, in contrast to x-cells that do not (re)produce at moment t. in the scheme (rsp) the y -cells correspond to bound enzymes c = e-s producing p . a formal analogy between the s-e pairs and the s-x pairs is that an s-e pair is an enzyme in a special temporal state (bound enzyme, engaged in a product formation process) and similarly a s-x pair is a cell in a temporal state (occupied in a (re)production process). the (rsp) reaction scheme represents three reaction steps in the cell growth process, involving the two x −→←− y transitions plus the s −→ p transformation. the scheme shows that y -cells are entirely dependant on the available nutrient substrate s. indeed, the presence of s stimulates (initiates) the transition of lag phase xcells into log phase y -cells. on the other side, substrate limitation leads to a decrease (up to disappearance) of y -cells, resp. of product formation (including reproduction). the reverse transition x ←− y can be interpreted as the case when x-cells absorb a certain amount of nutrient substrate but then do not process it (and excrete back the substrate). this reaction step seems to have minor impact on the total cell growth process. applying the mass action law to reaction scheme (rsp) leads to the following system of ordinary differential equations: ds/dt = −k1xs + k−1y dx/dt = −k1xs + k−1 y + k2y dy/dt = k1xs−k−1 y −k2y dp/dt = k2y (1) familiar from enzyme kinetic textbooks [12]. appropriate initial conditions corresponding to a batch cultivation process are: s(0) = s0, x(0) = x0, y(0) = y0, p(0) = 0. the terms with k−1 in the right hand-side of (1) can be suppressed as k−1 is expected to be small, then we obtain a simpler model: ds/dt = −k1xs dx/dt = −k1xs + k2y dy/dt = k1xs−k2y dp/dt = k2y (2) corresponding to the reaction scheme: s + x k1−→ y k2−→ p + x, (rsrs) systems (1), (2) modelize a batch mode bioreactor under the assumption that cells do not reproduce nor die. the basic models (1), (2) take into account substrate limitation in so far that only the log phase biomass declines due to substrate depletion; however the total biomass concentration z = x + y = const remains constant in spite of substrate depletion. as in enzyme kinetics here we see again the conservation law for the substrate ds/dt + dy/dt + dp/dt = 0 as well. a realistic cell growth model should include the above mentioned additional reaction steps, such as reproduction and mortality in order to encompass the complete cell growth process. b. reproduction reproduction can be viewed as the most important part of the total “production” process—during reproduction a mother cell produces two daughter cells, thus one more cell appears as result of this process. reproduction: a simple reaction scheme. y -cells are engaged in formation (biosynthesis) of products p , which are important components of newborn cells. thus we may view at reproduction as part of the production process. we assume that: a) cell growth is mainly due to reproduction—hence to y -cells; b) y -cells utilize product p to reproduce; c) newborn cells are y -cells, that is newborn cells do reproduce (by binary fission) [7]. following these assumptions a simple reaction scheme for the growth-reproduction process is: p + y k3−→ 2y. (rsg) from a “mechanistic” point of view reaction scheme (rsg) can be interpreted as follows: the mother y -cell transforms into one of the daughter cells, while the other biomath 2 (2013), 1312301, http://dx.doi.org/10.11145/j.biomath.2013.12.301 page 5 of 9 http://dx.doi.org/10.11145/j.biomath.2013.12.301 s markov, cell growth models using reaction schemes: batch cultivation daughter cell is built by means of product p components. in other words, a certain part of product p “comes into life” as a newborn cell. putting reaction schemes (rsp) and (rsg) together: s + x k1−→←− k−1 y k2−→ p + x, p + y k3−→ 2y, (rsp–rsg) and applying the mass action law, we obtain the model: ds/dt = −k1xs + k−1y dx/dt = −k1xs + k−1 y + k2y dy/dt = k1xs−k−1 y −k2y + k3py dp/dt = k2y −k3py (3) with initial conditions (corresponding to a batch cultivation process): s(0) = s0, x(0) = x0, y(0) = y0, p(0) = 0. respectively, assuming k−1 = 0 we obtain the simpler model: ds/dt = −k1xs dx/dt = −k1xs + k2y dy/dt = k1xs−k2y + k3py dp/dt = k2y −k3py (4) induced by the reaction scheme: s + x k1−→ y k2−→ p + x, p + y k3−→ 2y. (rsps)–(rsg) c. comparison to classical models we next compare model (3) with the classical model ds/dt = −δµ(s)z dz/dt = δµ(s)z µ(s) = s/(km + s). (5) proposition. classical model (5) is a special case of model (3) under the assumption that the biomass x and the product p are (nearly) constant during the process. proof. we sum up the second and the third equations in (3) to obtain a relation for the total biomass z = x+y: dz/dt = dx/dt + dy/dt = k3py. integrating in the interval [0, t] we have z = x + y = x0 + y0 + k3i, i(t) = ∫ t 0 p(τ)y(τ)dτ. let us assume that the biomass x is at a steady state: dx/dt = −k1xs + k−1 y + k2y = 0. from (3) this implies ds/dt = −k2y. substituting x by z −y we have −k1xs+k−1 y+k2y = −k1(z−y)s+k−1 y+k2y = 0, which gives y = sz/(km + s), km = (k−1 + k2)/k1. substituting the above expression for y in ds/dt = −k2y we have ds/dt = −k2sz/(km + s), which gives the first equation of (5): ds/dt = −δµ(s)z with δ = k2. to obtain the second equation of (5) we substitute y = sz/(km + s) in the last equation of (3): dp/dt = k2y −k3py to obtain (using dz/dt = k3py): dp/dt = k2zs/(km + s)−dz/dt or dz/dt = k2zs/(km + s)−dp/dt. assuming a constant production of p, i. e. approximately dp/dt = 0, we arrive at the classical model (5) for the substrate consumption and the biomass dynamics in a batch reactor. � remark. note that if the biomass is nearly constant, i. e. z = z0, then the ode for the consumption of s in (5) turns into the michaelis-menten ode of the substrate uptake in a chemostat: ds/dt = −k2z0s/(km + s). proposition 1 shows that the basic monod model (5) is an approximation of model (3) under the assumption of (nearly) constant biomass and product. in other words, model (3) generalizes model (5). d. mortality to model the process of cell mortality, we first should decide which cell fractions are most susceptible of dying. one possibility is to introduce a separate cell fraction, as done in [13]. in this work we shall try to keep our model mathematically simple and decide against introducing a new cell fraction, resp. a new phase variable. from the two fractions x and y we have to choose which one is more likely to die. in this work we accept that the x-cell fraction is the one that is more likely to be affected (but we leave this decision open for future examination). we thus accept here that only x-cells die and disintegrate. in addition, we assume that disintegrated cells transform partially to substrate s and partially to product p : x ks−→ s, x kp−→ p. (rsd) putting all three reaction schemes (rsp), (rsg) and (rsd) together we have: biomath 2 (2013), 1312301, http://dx.doi.org/10.11145/j.biomath.2013.12.301 page 6 of 9 http://dx.doi.org/10.11145/j.biomath.2013.12.301 s markov, cell growth models using reaction schemes: batch cultivation s + x k1−→←− k−1 y k2−→ p + x, p + y k3−→ 2y, x ks−→ s, x kp−→ p. wherein k1,k−1,k2,k3 and ks,kp are rate constants. applying the mass action law the above global scheme leads to the dynamical model ds/dt = −k1xs + k−1y + ksx dx/dt = −k1xs + k−1 y + k2y − (ks + kp)x dy/dt = k1xs−k−1 y −k2y + k3py dp/dt = k2y −k3py + kpx (6) wherein the parameters k1,k−1,k2,k3 and ks,kp are rate constants. biological meaning of the terms in the models. the reaction schemes make the biological interpretation of the terms in model (6) almost obvious. here they are: k1xs represents the consumption of s by bacteria x and the transition of bacteria x into bacteria y ; k−1y describes an amount of substrate concentration temporal;y stored by the cells but not further processed; k2y describes the amount of product concentration formed by y -cells and the transition of y -cells into xcells; k3py describes the increase of the cell population due to reproduction and the decrease of p due to its transition to newly formed y -cells (reaction scheme (rsg)); ksx describes the decay of bacteria x and the disintegrated part of the dead cells transforming into substrate s—reaction scheme (rsd); kpx describes the decay of bacteria x and the disintegrated part of the dead cells transforming into product p —reaction scheme (rsd). the parameters have the meaning of specific rate constants as follows: k1 is the substrate utilization rate, k−1 is the substrate non-utilization rate, k2 is the production rate, k3 is the reproduction rate, ks and kp are death rates. whenever appropriate we may also use specific reaction steps describing waste product excretion, e. g. p + y δ−→ q, (rsw) implying corresponding terms δpy in the dynamical system. we finish this section with the following hypothesis: hypothesis. cell growth models for batch cultivation can be based on reaction schemes, involving reaction steps such as (rsg), (rsp), (rsd), (rsw). iii. computer experiments we present the results of two computer experiments based on the above reaction schemes. for the computer experiments real experimental data for the biomass and product concentrations have been used. the graphics of the solutions demonstrate a good fit to the experimental data. in particular the lag phase data for the biomass are fitted better than when classical models such as (5) are used. computer experiment 1. the model is: ds/dt = −k1xs + k−1y dx/dt = −k1xs + k−1y + k2y dy/dt = k1xs−k−1y −k2y + βpy dp/dt = k2y −γpy (7) fig. 1. numerical solution of cell growth model (7) using matlab the values of the parameters used in model (7) are as follows: k1 = 3.48691,k−1 = 0,k2 = 7,β = 41.5095,γ = 86.4424;s0 = 1.4,x0 = 0.00450748. the graphs of the solutions are represented in fig. 1. model (7) is close to model (6) but is without mortality terms and there we have β 6= γ. to explain why in this experiment β < γ we need to introduce a waste product excretion step, as done in the next experiment. computer experiment 2. the model is as follows: ds/dt = −k1xs + k−1y dx/dt = −k1xs + k−1y + k2y −kdx dy/dt = k1xs−k−1y −k2y + (α−δ)py dp/dt = k2y − (α + δ)py + kdx (8) biomath 2 (2013), 1312301, http://dx.doi.org/10.11145/j.biomath.2013.12.301 page 7 of 9 http://dx.doi.org/10.11145/j.biomath.2013.12.301 s markov, cell growth models using reaction schemes: batch cultivation the reaction scheme leading to model (8) involves a decay step (rsd) and a waste product excretion step (rsw) as follows: s + x k1−→←− k−1 y k2−→ p + x, p + y α−→ 2y, p + y δ−→ q, x kd−→ p, fig. 2. numerical solution of cell growth model (8) using matlab wherein q represents waste product. the values of the parameters used in model (8) are as follows: k1 = 4.18191,k−1 = 0,k2 = 4.50413,α = 60.6611,δ = 21.3493,kd = 0.136806;s0 = 1.55196,x0 = 0.00490581. the graphs of the solutions fitting real experimental data are visualized in fig. 2. a very good fit can be observed. iv. conclusion in this work we formulate certain reaction steps in the cell growth process and try to base our models entirely on proposed reaction schemes. our reaction schemes for the phase transition of the cells make use of appropriate reaction schemes from enzyme kinetics. elements of the michaelis-menten enzyme kinetics can be observed in classical cell growth models using the monod specific growth rate function µ. in this paper instead of the (approximate) michaelis-menten enzyme kinetics for the substrate uptake we use the (exact) henri-michaelismenten enzyme kinetics (hmm-reaction mechanism). it has been noted in the literature that classical monod type models often fail to describe adequately bio-reactors under perturbed conditions. it has been also recognized that monod models lack (do not fit well) the lag phase of the cells. our numerical experiments make us believe that structured models provide more flexibility and can be better fitted to real data. the use of reaction schemes makes the construction of particular cell growth models simple and instructive; it also contributes to understanding the underlying biological mechanism. our hypothesis is that cell growth models can be entirely based on reaction schemes. v. acknowledgements . the author is indebted to prof. v. beschkov from the institute of chemical engineering, bas, and prof. m. kamburova from the institute of microbiology, bas, for providing real experimental data on microbial growth and for useful discussions on the behavior of the microorganisms and the proposed reaction schemes. references [1] alt, r., s. markov, theoretical and computational studies of some bioreactor models, computers and mathematics with applications 64 (2012), 350–360. http://dx.doi.org/10.1016/j.camwa.2012.02.046 [2] droop, m., vitamin b12 and marine ecology. iv. the kinetics of uptake, growth and inhibition in monochrysis lutheri. journal of marine biology 48, 689–733. [3] gerber, m., r. span, an analysis of available mathematical models for anaerobic digestion of organic substances for production of biogas, proc. igrc, paris 2008. [4] gouzé, j.-l., v. lemesle, two simple growth models in the chemostat. arima 9, 145–155 (2008). [5] henri v., recherches sur la loi de laction de la sucrase. c. r. hebd. acad. sci., 133, 891–899 (1901). [6] henri v., ueber das gesetz der wirkung des invertins. z. phys. chem., 39 (1901), 194–216. [7] jocelyn e. krebs, elliott s. goldstein, stephen t. kilpatrick, lewin’s genes xi, jones & bartlett publ., 2014. [8] lemesle, v., j.-l. gouzé, a simple unforced oscillatory growth model in the chemostat. bulletin of mathematical biology 70, 344–357 (2008). http://dx.doi.org/10.1007/s11538-007-9254-5 [9] lemesle, v., j-l. gouzé, a biochemically based structured model for phytoplankton growth in the chemostat. ecological complexity 2, 21–33 (2005). http://dx.doi.org/10.1016/j.ecocom.2004.08.001 [10] michaelis, l., m. l. menten. die kinetik der invertinwirkung. biochem. z. 49, (1913), 333–369. [11] monod, j., the growth of bacterial cultures, annual reviews of microbiology 3, 371–394 (1949). http://dx.doi.org/10.1146/annurev.mi.03.100149.002103 [12] murray j. d., mathematical biology: i. an introduction, third edition, springer, 2002. biomath 2 (2013), 1312301, http://dx.doi.org/10.11145/j.biomath.2013.12.301 page 8 of 9 http://dx.doi.org/10.1016/j.camwa.2012.02.046 http://dx.doi.org/10.1007/s11538-007-9254-5 http://dx.doi.org/10.1016/j.ecocom.2004.08.001 http://dx.doi.org/10.1146/annurev.mi.03.100149.002103 http://dx.doi.org/10.11145/j.biomath.2013.12.301 s markov, cell growth models using reaction schemes: batch cultivation [13] sissons, c. j., m. cross, s. robertson, a new approach to the mathematical modelling of biodegradation processes, appl. math. modelling 10 (1986), 33–40. http://dx.doi.org/10.1016/0307-904x(86)90006-5 [14] schnell, s., chappell, m. j., evans, n. d., m. r. roussel, the mechanism distinguishability problem in biochemical kinetics: the single-enzyme single-substrate reaction as a case study. c. r. biologies 329, 51–61 (2006). http://dx.doi.org/10.1016/j.crvi.2005.09.005 [15] schnell, s., p. k. maini, enzyme kinetics at high enzyme concentration, bulletin of mathematical biology 62, 483–499 (2000). http://dx.doi.org/10.1006/bulm.1999.0163 [16] schnell, s., p. k. maini, a century of enzyme kinetics: reliability of the km and vmax estimates, comments on theoretical biology 8, 169–187 (2003). doi: 10.1080/08948550390206768 biomath 2 (2013), 1312301, http://dx.doi.org/10.11145/j.biomath.2013.12.301 page 9 of 9 http://dx.doi.org/10.1016/0307-904x(86)90006-5 http://dx.doi.org/10.1016/j.crvi.2005.09.005 http://dx.doi.org/10.1006/bulm.1999.0163 http://dx.doi.org/10.11145/j.biomath.2013.12.301 introduction structured cell growth models relation to enzyme kinetics model assumptions reaction steps of the cell growth process batch cultivation: reaction schemes the lag–log cell phase transition reproduction comparison to classical models mortality computer experiments conclusion acknowledgements references original article biomath 2 (2013), 1210071, 1–6 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum dynamical analysis of the microrna-mediated protein translation process ivan jordanov∗, elena nikolova∗, nikolay k. vitanov∗ ∗ institute of mechanics bulgarian academy of sciences, sofia, bulgaria emails: i jordanov@email.bg, elena@imbm.bas.bg, vitanov@imbm.bas.bg received: 15 july 2012, accepted: 7 october 2012, published: 5 january 2013 abstract—mathematical modeling of kinetic processes with different time scales allows a reduction of the governing equations using quasi-steady-state approximations (qssa). a qssa theorem is applied to a modified mathematical model of the microrna-mediated protein translation process. by an appropriate normalized procedure the system of seven nonlinear ordinary differential equations is rewritten in a form suitable for model reduction. in accordance with the terminology of the qssa theorem, it is established that two of the protein concentrations are “fast varying”, such that the corresponding kinetic equations form an attached system. the other four concentrations are “slow varying”, and form a degenerate system. another variable appears to be a constant. analytical relationships between the steady-state values of the fast varying concentrations and the slow varying ones, are derived and interpreted as restrictions on the regulatory role of micrornas on the protein translation process. keywords-nonlinear dynamics; microrna; protein translation; qssa theorem i. introduction micrornas are short (an average of 20–22) ribonucleic acid (rna) molecules that regulate the function of eukaryotic messenger rnas (mrnas) and thereby play an important role in development, cancer, stress responses, and viral infections. recently, remarkable progress was made in the understanding of microrna biogenesis, its functions and mechanisms of action. micrornas affect gene expression by specific inhibition of target mrnas. the exact mechanism of this inhibition is still a matter of debate. in the past few years, several mechanisms have been reported, some of which are contradictory [1], [2], [3]. they include in particular inhibition of translation initiation (acting at the level of cap-40s or 40s-aug-60s association steps), inhibition of translation elongation or premature termination of translation. in order to verify some of these hypotheses, two simple mathematical models of protein translation are proposed as systems of ordinary differential equations in [4]. by their analysis the authors in [4] demonstrated that it is impossible to distinguish alternative biological hypotheses using the steady state data on the rate of protein synthesis. in [5], however, it is shown that dynamical data allow to discriminate some of the mechanisms of microrna action. the authors in [5] demonstrated this fact using the same models as those in [4] for the sake of comparison but they applied different methods. as a result of their investigation, they formulated a hypothesis that the effect of microrna action is measurable and observable only if it affects the dominant system (generalization of the limiting step notion for complex networks) of the protein translation machinery. following the last investigation here we consider one of the models proposed in [4] (so called eif4f/subunit joining model) as our aim is to show that considerations of time hierarchy in genetic interactions allows us to reduce the number of differential equations of the above-mentioned model and thereby to determine the driving reactions and control parameters in the microrna repression mechanism. for this purpose we will use the method of the quasi-steady-state approximation (qssa) theorem proved in the basic paper [6]. citation: i. jordanov, e. nikolova, n. k. vitanov, dynamical analysis of the microrna-mediated protein translation process, biomath 2 (2013), 1210071, http://dx.doi.org/10.11145/j.biomath.2012.10.071 page 1 of 6 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.10.071 i. jordanov et al., dynamical analysis of the microrna-mediated protein translation process ii. qssa theorem in general the mathematical modeling of biochemical reactions with different time scales leads to hierarchical system of the form: � d~x dt = ~f (~x, ~y) (1) d~y dt = ~g(~x, ~y) (2) where ~x ∈ rn , ~y ∈ rm and 0 < � << 1. the terminology of the qssa theorem related to the system (1–2) is as follows. the subsystem of equations having � in the numerator is called an attached system, and its variables are fast variables with respect to the other part of equations, which form a degenerate system, and its variables are slow variables. the set of both subsystems forms the complete system. in accordance with this terminology, the qssa theorem [7] claims that the solution of the complete system (1–2) tends to the solution of the degenerate system (2) at � → 0, if the following conditions are satisfied: a) there is an isolated equilibrium (steady state) solution of the attached system (1)(i.e., there is not another solution in its neighborhood). b) the existing equilibrium solution of the attached system is a stable one for each value of the slow variables ~cs. c) the initial conditions (states) are contained in a region of influence (a basin) of the equilibrium solution of the attached system. d) the solution of the complete system is singlevalued and its right-hand side is continuous. the essence of the qssa theorem is that the character of the solution of (1–2) does not change when the small parameter � converges to zero. thus, we can assume � = 0 in (1)and instead of differential equations obtain algebraic ones for the steady-state value of fast variables, i.e., ~f (~x, ~y) = 0, ~x = ~ϕ(~y) d~y dt = ~g[~ϕ(~y), ~y] (3) in this way, the complete system (1–2) is reduced to the degenerate system (3). for every fixed ~y , the equation ~f (~x, ~y) has a unique solution that depends continuously (or smoothly, if needed) on ~y. thus the variables ~y play the role of a driver of the subordinated variables ~x. according to the qssa theorem, when the stationary solution of the attached system is isolated and stable, then the solution of the reduced (degenerate) system depends only on the post-initial values of the slow variables. the term “post-initial” is introduced in sense of the considerations of initial and later intervals of validity of the complete and degenerate systems, respectively. certainly, the complete system (1–2) holds any time, and the degenerate system (3) will be valid from some later period of time. but it can reveal new properties of the investigated processes near their stationary states as we will show in the following paragraphs. iii. reduction of dimensionality of a modified mathematical model of microrna-mediated protein translation process a. applying the qssa theorem to the model we apply the qssa theorem to a modified ordinary differential equation model of microrna-mediated protein translation process, presented in [4]. it explicitly takes into account recycling of initiation factors and ribosomal subunits. in accordance with fig. 1 there are four reactions in the model, all considered to be irreversible: 1) 40s + eif4f −→ mrna 40s, assembly of the initiation complex, cap-dependent initiation steps (rate k1). 2) mrna 40s −→ aug, some later and capindependent initiation steps, such as scanning the 5’utr by the start codon aug recognition (rate k2). 3) aug −→ 80s, assembly of ribosomes and protein translation (rate k3). 4) 80s −→ 60s+40s, recycling of ribosomal subunits (rate k4). the model is described by the following system of nonlinear ordinary differential equations: dm1 dt = k4m4 − k1m1m6 dm2 dt = k1m1m6 − k2m2 dm3 dt = k2m2 − k3m3m5 dm4 dt = k3m3m5 − k4m4 (4) dm5 dt = k4m4 − k3m3m5 dm6 dt = k2m2 − k1m1m6 dm7 dt = k3m3m5 − k5m7 biomath 2 (2013), 1210071, http://dx.doi.org/10.11145/j.biomath.2012.10.071 page 2 of 6 http://dx.doi.org/10.11145/j.biomath.2012.10.071 i. jordanov et al., dynamical analysis of the microrna-mediated protein translation process fig. 1. biochemical diagram of protein translation process repressed by micrornas where m1 is the concentration of free 40s ribosomal subunits, m2 is the concentration of 40s subunit bound to the initiation site of mrna, m3 is the concentration of aug, the initiation complex bound to the start codon of mrna, m4 is the concentration of 80s, the ribosomes translating protein, m5 is the concentration of 60s subunit joining factors, m6 is the concentration of eif4f, the free translation initiation factors and m7 is the protein concentration. in fact, our modification consists of adding a term for protein decay in the last equation of (4). otherwise, the protein will constantly increase, which is biologically impossible. the notations ki (i = 1, 2, ..., 5) present the rate constants of bimolecular reactions involved in the process and have the following numerical values k1 = 2; k2 = 2; k3 = 5; k4 = 1; k5 = 1 (5) the numerical values of ki (i = 1, ..., 4) are taken from [4], and the rate constant for the protein decay k5 is taken from [8]. the methodology for separation of the complete system to fast and slow subsystems [6] lies on pure mathematical basis. in principle it involves separate scaling (normalization) of reaction rates as well as chemical concentrations on the basis of well known data of their numerical values. we propose a normalized (scaling) procedure, which is a similar to the dimensionless principle. it requires each term in the right-hand side of the system equations to have an order of one. unlike most models in systems biology, including cell signaling pathways, immunological processes, etc., where the rate constants differ by at least one numeric order, the rate constants of the protein translation process have one and same order. therefore we do not normalize them. we will normalize (scale) only the system variables. for the purpose we simulate dynamics of the considered process for a period of 10 seconds taking into account the numerical values of the parameters (4) and the initial values [100, 0, 0, 25,0, 6, 0] of the variables, taken from [4]. next, we select the values near the settled (steady state) ones in order to use them as characteristic values of state variables in accordance with the qss assumption they are: m01 = 82.04; m 0 2 = 5, 93; m 0 3 = 0, 18; m04 = 11, 85; m 0 5 = 13, 14; (6) m06 = 0, 07; m 0 7 = 11.85; the parameters and concentration values shown above are given here without units in view of the fact that we do not intend to compare them. what is of interest for us is not to compare parameters or concentrations, but the terms in (4). in accordance with the normalized (scaling) procedure, which we apply here each term in the righthand side of the system equations must have an order of one. towards this end, we introduce normalized (scaling) substitutions only for the model variables. they have the form m1 = x1 � ; m2 = x2; m3 = x3; m4 = x4 � ; (7) m5 = x5 � ; m6 = �x6; m7 = x7 � ; where � = 0.01. after replacing (7) in (4) we obtain the following system in a normalized form dx1 dt = k4x2 − �k1x1x6 (8) dx2 dt = k1x1x6 − k2x2 (9) � dx3 dt = �k2x2 − k3x3x5 (10) dx4 dt = k3x3x5 − k4x4 (11) dx5 dt = k4x4 − k3x3x5 (12) � dx6 dt = k2x2 − k1x1x6 (13) dx7 dt = k3x3x5 − k5x7 (14) here xi (i = 1, 2, ..., 7) are scaling state variables of order of unity. this means that, in accordance with the terminology of qssa theorem, we can say that the equations (10) and (13) form an attached system, and the biomath 2 (2013), 1210071, http://dx.doi.org/10.11145/j.biomath.2012.10.071 page 3 of 6 http://dx.doi.org/10.11145/j.biomath.2012.10.071 i. jordanov et al., dynamical analysis of the microrna-mediated protein translation process equations (8), (9), (11), (12) and (14) form a degenerated one. the set of both systems is called a complete system. in this way aug, the initiation complex bound to the start codon of mrna and the free translation initiation factors eif4f are fast participants in the micrornamediated protein translation process and the free 40s ribosomal subunits, the 40s mrna complex, the ribosomes translating protein 80s, the subunit joining factors 60s and the protein are its slow components. next, following the qssa theorem for the equilibrium values x03 and x 0 6 the following expressions are valid: x03 = �k2x2 k3x5 > 0 x06 = k2x2 k1x1 > 0 (15) as it is seen in the right-hand sides of (15) the slow varying (not equilibrium) concentrations xi (i = 1, 2, 5) are involved in accordance with the qssa theorem. the equilibrium solution (x03, x 0 6 ) is unique and stable one for the attached system of equations in view of the fact that the variational equations dξ dt = − a3 � x05ξ; dη dt = − a1 � x01η (16) tend asymptotically to zero, where ξ and η are variations around the stationary values x03 and x 0 5, respectively. thus the assertions (a) and (b) of the theorem are satisfied. that allows us to substitute the expressions (15) in the five equations of the degenerated system. as a result the following qssa of the original model (4) is obtained by using reverse substitutions (7) : dm1 dt = k4x4 − k2mi2 dm2 dt = k2x2 − k2m2 = 0 dm4 dt = k2m i 2 − k4m4 (17) dm5 dt = k4m4 − k2mi2 dm7 dt = k2m i 2 − k5m7 it is seen the last system consists already four linear ordinary differential equations. after our mathematical transformations the variable m2, representing dynamics of the [40s-mrna] complex appears to be a constant. by this reason we express it by its initial (post-initial) value in the other equations of (17). so far, our theoretical result related to the persistent behaviour of the [40s-mrna] complex has not reported in similar investigations of the microrna-mediated protein translation dynamics. in this way it could fig. 2. coincidence of the graphs of complete (solid lines) and reduced degenerate (dotted lines) system solutions be experimentally verified. however, as it can be seen from the complete (4) and the degenerate (17) systems, it determines the behaviour of the fast varying components, as well as slow varying concentrations. coincidence between the complete and degenerate systems observes after the 1/10 second from the beginning of the protein translation process as it can be seen from fig. 2. coincidence between the graphs of m4 and m7 exists too (see fig. 2). b. a consequence of the quasi-stationary system on the dynamical behaviour of the microrna-mediated protein translation process in this section we will demonstrate the advantages from the made quasi-steady state approximation for understanding of the main reaction mechanism of microrna -mediated protein translation in details. in the simplest case, the reduced (degenerate) system is a subsystem of the complete system, as it can be seen from equations (1–2). however, from a biological point of view, it also includes new driving reactions with kinetic rates expressed through the parameters of the complete model, and rates of some reactions are renormalized. moreover, the quasi-stationary system is conducted by limited number variables (slow variables), playing the role of “drivers” of the subordinated fast variables of the original system according the terminology of the qssa theorem. in the concrete case the driving reactions in the quasi-stationary protein translation process are the scanning the 5’utr by the start codon aug recognition (reaction 2 from the subsection a, biomath 2 (2013), 1210071, http://dx.doi.org/10.11145/j.biomath.2012.10.071 page 4 of 6 http://dx.doi.org/10.11145/j.biomath.2012.10.071 i. jordanov et al., dynamical analysis of the microrna-mediated protein translation process promoting by the rate k2 ) and the recycling of the ribosomal subunits (reaction 4 from the same subsection, promoting by the rate k4 ). therefore we can conclude that micrornas act on one (or both) of these later stages of the translation initiation. analysis of the reduced system can also answer an important question: which are control parameters in the microrna mediated protein translation process? many of the parameters of the original model (complete system) are no longer presented and we can ignore them in some later time period. parameters presented in the reduced system (control parameters) can provoke changes in the dynamical behavior of the quasi-stationary system as well as of the original model. in this case the control parameters reduce to two: k2 and k4. it is of interest to see their influence mainly on the protein behaviour. for the purpose numerical simulations of the protein dynamics for different values of k2 and k4 are presented in fig. 3 and fig. 4. as we mentioned above, although k2 and k4 are not presented in the corresponding equation for the protein production of the complete model they influence on the protein dynamics. fig. 3 and fig. 4 show that the protein production decreases at essentially lower values of the rate constants. moreover, unlike the results shown in fig. 3, changes in protein production, depending on k4 differ by at least one order (the changes in the protein dynamics start to observe just at k4 = 0.3). in addition, the reduced system is guided only by the variables m2 (its post-initial value) and m4, in view of the fact that they involve in the right-hand sides of the equations (17) excepting the equation of protein production, where m4 is not presented. c. analytical derivation of the degenerate system solution here we will support the arguments made in the previous paragraph in a pure mathematical aspect. for the purpose the degenerate system (17) is analytically solved. further we consider its solution in infinity and derive analytical relationships between steady-state (denoted by ’0’ upper indexes) and initial (denoted by ’i’ upper indexes) values of the slow varying genetic concentrations. they are: m01 = m i 1 + (m i 4 − k2m i 2 k4 ; m02 = m i 2; m 0 4 = k2m i 2 k4 ; m05 = m i 5 + (m i 4 − k2m i 2 k4 ); m07 = k2m i 2 k5 (18) fig. 3. graphs of protein production for k2=0.1; 0.5; 1; 2 fig. 4. graphs of protein production for k4=0.01; 0.3;0.1; 1 the last values, however, take a place in the formulas for the steady-state values of the fast varying concentrations m3 and m6. the corresponding formulas are easy to obtain from (15) by substituting there the corresponding relations (7) and taking into account the relations (18). they are: m03 = k2m i 2 k3(mi5 + m i 4 − k2m i 2/k4) m06 = k2m i 2 k3(mi1 + m i 4 − k2m i 2/k4) (19) it is seen that the control parameters k2 and k4 as well as the “driver” concentrations m2 and m4 are involved in (18–19). this means, by changing their values we can essentially control the quasi-stationary genetic process biomath 2 (2013), 1210071, http://dx.doi.org/10.11145/j.biomath.2012.10.071 page 5 of 6 http://dx.doi.org/10.11145/j.biomath.2012.10.071 i. jordanov et al., dynamical analysis of the microrna-mediated protein translation process in terms of input (initial values) and output (stationary values) relationships. iv. conclusion it is shown that the considerations of time hierarchy in biomolecular reactions allows us to find the simplest model of the microrna-mediated protein translation process, which can substitute a multiscale genetic network such that the dynamics of the complete network can be approximated by the simpler one. this is achieved by applying a well-known qssa theorem as a basic approach for system reduction. analysis of the degenerate model help us to derive the following conclusions: 1) micrornas act on the later stages of the translation initiation, such as aug recognition (cap-independent initiation steps) or recycling of the ribosomal subunits; 2) the rate constants k2 and k4 can be considered as control parameters of the protein translation system; 3) the post-initial concentrations of 40s subunit bound to the initiation site of mrna and ribosomes translating protein [80s] become an important factors when the genetic process approaches its quasi-stationary state; 4) the obtained relationships between the steady-state and initial values of the biomolecular concentrations can be considered as restrictions on participants in the protein translation process, repressed by microrna. they can be experimentally verified and can be used for direct computation of steady states of the genetic components, especially when kinetic information is incomplete. references [1] a. m. chekulaev and w. filipowicz, “mechanisms of mirnamediated post-transcriptional regulation in animal cells”, curr. opin. cell biol., vol. 21, pp. 452–460, 2009. http://dx.doi.org/10.1016/j.ceb.2009.04.009 [2] a. eulalio, e. huntzinger and e. izaurralde, “getting to the root of mirna-mediated gene silencing”, cell , vol. 132, pp. 9–142, 2008. http://dx.doi.org/10.1016/j.cell.2007.12.024 [3] w. filipowicz, s. n. bhattacharyya and n. sonenberg, “mechanisms of post-transcriptional regulation by micrornas: are the answers in sight?”, nat. rev. genet., vol. 9, pp. 102–114, 2008. http://dx.doi.org/10.1038/nrg2290 [4] t. nissan and r. parker. “computational analysis of mirnamediated repression of translation: implications for models of translation initiation inhibition”, rna, vol. 14, pp. 1480–1491, 2008. http://dx.doi.org/10.1261/rna.1072808 [5] a. zinovyev, n. morozova, n. nonne, e. barillot, a. harelbellan, a. n. gorban, “dynamical modeling of microrna action on the protein translation process”, bmc syst biol., vol. 4, 24 p., 2010. [6] v. petrov, e. nikolova and o. wolkenhauer, “reduction of nonlinear dynamic system with an application to signal transduction pathways”, iet systems biology, vol. 1, pp. 2–9, 2007. http://dx.doi.org/10.1049/iet-syb:20050030 [7] a. n. tichonov, “systemy differentsialnyh uravneniy, soderjashchie malye parametry pri proizvodnyh”, matematicheskiy sbornik, vol. 31, pp. 575–586, 1952 (in russian). [8] r. khanin and d. j. higham, “a minimal mathematical model of post-transcriptional gene regulation by micrornas. report to glasgow university, 2007 (in internet). biomath 2 (2013), 1210071, http://dx.doi.org/10.11145/j.biomath.2012.10.071 page 6 of 6 http://dx.doi.org/10.1016/j.ceb.2009.04.009 http://dx.doi.org/10.1016/j.cell.2007.12.024 http://dx.doi.org/10.1038/nrg2290 http://dx.doi.org/10.1261/rna.1072808 http://dx.doi.org/10.1049/iet-syb:20050030 http://dx.doi.org/10.11145/j.biomath.2012.10.071 introduction qssa theorem reduction of dimensionality of a modified mathematical model of microrna-mediated protein translation process applying the qssa theorem to the model a consequence of the quasi-stationary system on the dynamical behaviour of the microrna-mediated protein translation process analytical derivation of the degenerate system solution conclusion references www.biomathforum.org/biomath/index.php/biomath review article some mathematical tools for modelling malaria: a subjective survey jacek banasiak1,2, rachid ouifki1, woldegebriel assefa woldegerima1 1 department of mathematics and applied mathematics university of pretoria, south africa 2institute of mathematics, łódź university of technology, poland correspondence: wa.woldergerima@up.ac.za, assefa@aims-cameroon.org received: 9 june 2021, accepted: 2 october 2021, published: 13 october 2021 abstract— in this paper, we provide a brief survey of mathematical modelling of malaria and how it is used to understand the transmission and progression of the disease and design strategies for its control to support public health interventions and decisionmaking. we discuss some of the past and present contributions of mathematical modelling of malaria, including the recent development of modelling the transmission-blocking drugs. we also comment on the complexity of the malaria dynamics and, in particular, on its multiscale character with its challenges and opportunities. we illustrate the discussion by presenting a curve fitting using a 95% confidence interval for the south african data for malaria from the years 2001−2018 and provide projections for the number of malaria cases and deaths up to the year 2025. keywords: mathematical modelling, malaria, south africa, data fitting i. malaria and itsmathematicalmodelling malaria is an indirectly transmitted disease of humans that requires the interaction of three distinct living organisms (the components) to sustain transmission. these are: 1) parasites of genus plasmodium that causes malaria disease in humans. four species of plasmodium have long been recognized to infect humans and cause illness, namely, p. falciparum, p. malariae, p. vivax and p. ovale. the fifth one, p. knowlesi, that naturally infects macaques have recently been recognized to cause zoonotic malaria also in humans, but no cases of the humanmosquito-human transmission have been reported. of these species, p. falciparum is responsible for most malaria deaths globally and is the most prevalent species in subsaharan africa, [77]. p. vivax is the second most significant species and is prevalent in southeast asia and latin america. p. vivax and p. ovale have the added complication of the dormant liver stage which, after some time, can be reactivated in the absence of a copyright: ©2021 banasiak et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: jacek banasiak, rachid ouifki, woldegebriel assefa woldegerima, some mathematical tools for modelling malaria: a subjective survey, biomath 10 (2021), 2110029, http://dx.doi.org/10.11145/j.biomath.2021.10.029 page 1 of 19 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2021.10.029 j banasiak, r ouifki, w a woldegerima, some mathematical tools for modelling malaria: a subjective ... mosquito bite, leading to clinical infection, [29], [77]. 2) humans (host), where parasites grow and multiply first in the liver cells and then in the red cells of the blood, producing merozoites which cause symptoms and illness, and gametocytes which is the form of the parasites that can be transferred to the mosquito during a blood meal. 3) female anopheles mosquitoes (vector), which are the agents responsible for transmitting the disease from one human to another. they become infected when they feed and ingest human blood that contains mature gametocytes. after a mosquito picks up the gametocytes, they start another cycle of growth and multiplication in the mosquito, eventually producing sporozoites. when the mosquito takes a blood meal on another human, it injects the sporozoites with its saliva and infects humans. thus, the mosquito carries the parasite from one human to another, acting as a vector. despite malaria being preventable and treatable, it remains one of the most prevalent and deadliest human infection in developing countries, especially in the sub-saharan africa, where young children and pregnant women are most affected, [78]. according to the latest who malaria report, released on the 30 november 2020, there were 229 million cases of malaria in 2019 compared to 228 million cases in 2018. the estimated number of malaria deaths stood at 409 000 in 2019, while there were 405 000 deaths in 2018. the who african region carries a disproportionately high share of the global malaria burden with as much as 94% of malaria cases and malaria deaths recorded there in 2019, [77]. it is feared that the number of malaria deaths may increase in the years 2020/2021 due to the covid-19 pandemic, especially in the sub-saharan africa, where the number of covid-19 cases is rising, [69]. first, malaria and covid-19 share several common symptoms such as fever, breathing difficulties, tiredness or acute headache, which may lead to misdiagnosis, particularly when the clinicians rely mainly on the symptoms, [43]. further, covid-19 can disturb and affect the control and treatment of malaria in different ways including delays in the distribution of the insecticide-treated bed nets (itns), difficulties in testing and treatment in hospitals over-crowded with covid-19 patients and due to restricted availability of the health workers. fighting malaria requires coordinated research across many disciplines, with mathematics and mathematical modelling playing an important role. here we use mathematical tools to represent and analyze real-world processes to make predictions or otherwise provide insights about their dynamics. in mathematical epidemiology, we create simplified representations, called models, of diseases such as malaria in a population, to understand how the infection may progress in the future. mathematical models are the cheapest and often the only way to test different scenarios for the development of a disease and various interventions such as vaccination programs, [1], [16], [52]. thus, they are crucial to study the malaria transmission mechanism and the dynamics of its progression, predict and estimate the prevalence and incidence and evaluate strategies for control. hence, they help to inform the public health interventions policy-decision making. for instance, results obtained from mathematical models may predict which populations are most vulnerable to the disease, allowing for focusing antimalarial drugs and preventative treatments on high-risk groups. however, the effectiveness of such models and their robustness largely depend on the choice of model and the researcher’s adherence to the assumptions governing the chosen model’s use. over the past century, many mathematical models of various complexity levels, some discussed in more detail in section 2, have been developed. they have been used to answer the following questions related to its development, growth, progression and transmission in endemic and high transmission areas. biomath 10 (2021), 2110029, http://dx.doi.org/10.11145/j.biomath.2021.10.029 page 2 of 19 http://dx.doi.org/10.11145/j.biomath.2021.10.029 j banasiak, r ouifki, w a woldegerima, some mathematical tools for modelling malaria: a subjective ... 1) what are the predicted peak number of cases and the time of its occurrence? what is the expected cumulative number of cases over the epidemic (i.e. final attack rate)? 2) what is the basic reproduction number (r0) and the current value, or the time course, of the effective reproduction number? 3) what might be/is the potential impact of antimalarial drugs? 4) how should antimalarial drugs be prioritized for the distribution among the population subgroups to minimize infectiousness for those infected, prevent the development of resistance, and shorten the duration of the illness? 5) when should the vaccination be introduced, even though there is no currently approved vaccine for malaria? what would be the impact of vaccine timing on the benefits of vaccination? can the vaccine/drug effectiveness be predicted? mathematical analysis can be applied at various levels, such as the disease transmission dynamics between humans and mosquitoes, the in-host immuno-pathogenesis dynamics of the malaria parasites, or pharmacokinetics (pk) and pharmacodynamics (pd) properties of the drugs and vaccines. we emphasize here that though there is no currently approved vaccine for malaria, the search for malaria vaccine continues and mathematical models can help in the theoretical design, clinical trials, and the vaccine’s deployment. withinhost modelling of infectious diseases has drawn significant attention in the last half-century due to its significance in improving our understanding of how the microscopic processes develop and affect the host health. besides, it is crucial to understand how the within-host dynamical processes (immunological processes) of the parasite impact on the population-level dynamics of the disease spread (epidemiological processes), [53]. in this paper, however, we will focus on the populationlevel compartmental malaria transmission models. ii. a brief reviewofmathematicalmodels of malaria a. history and background of mathematical models of malaria mathematical epidemiology (modelling of infectious diseases) can be traced back to the work of daniel bernoulli. he formulated and solved a model for smallpox in 1760 and used it to evaluate the effectiveness of the inoculation of healthy people against the smallpox virus, [1]. much later, william h. hamer [36] formulated and analyzed a model to understand the recurrence of measles epidemics. he was the first to propose that the spread of infection should depend on the numbers of susceptible and infective individuals and suggested the mass action law for the rate of new infections. his ideas have served as a foundation of compartmental models since then, [15]. in 1927 william o. kermack and anderson g. mckendrick, [44], developed a general epidemic model and described the relationship between susceptible, infected and immune individuals in a population. the kermack–mckendrick epidemic model has successfully predicted the behaviour of outbreaks in many epidemics, [15], [45]. mathematical modelling of malaria dates back to the work of sir ronald ross. he was the first to understand malaria’s human-mosquito transmission mechanism, winning for this achievement the nobel prize in medicine in 1902. subsequently, in 1911 he developed a compartmental differential equation model of malaria as a host-vector disease, [67]. ross divided the human and mosquito populations into the susceptible (s) and infectious (i) classes and used the so-called s h ihs h model for humans and s v iv for mosquitoes, where s h, ih, s h and s v, iv represent susceptibles and infectious humans and mosquitoes, respectively. using his model, he showed that reducing the mosquito population below a critical level would be sufficient for malaria elimination. in other words, contrary to the common belief of that time, there is no need to kill all mosquitoes to control the disease. this result has been since known as the mosquito theorem. biomath 10 (2021), 2110029, http://dx.doi.org/10.11145/j.biomath.2021.10.029 page 3 of 19 http://dx.doi.org/10.11145/j.biomath.2021.10.029 j banasiak, r ouifki, w a woldegerima, some mathematical tools for modelling malaria: a subjective ... ross’ model was extended in 1957 by george macdonald, who included the exposed (infected but not infectious) mosquito class, [50]. the rossmacdonald model of malaria transmission has had a significant influence on malaria control. one of its main conclusions is that malaria’s endemicity is most sensitive to the changes in the mosquito survival rate. the ross-macdonald model allows for several other conclusions. first, it shows that malaria can persist in a population only if the number of mosquitoes is greater than a given threshold. second, the prevalence of infections in humans and mosquitos depend directly on the basic reproductive number calculated in the paper. the basic reproduction number, usually denoted by r0, is defined as the average number of infections produced by one infectious individual introduced into a fully susceptible population over the duration of the infectious period, [26], [73]. it depends on the duration of the infectious period, the probability of infecting a susceptible individual upon contact, and the number of new susceptibles contacted in a unit of time, i.e., the number of newly infected individuals per unit time, [39], [40]. in the malaria context, r0 should take into account that an infectious human must first infect a mosquito, which then generates secondary infections among humans, thus determining it is not straightforward. r0 helps determine whether an infectious disease can spread through a population. in particular, if r0 < 1, then each infectious individual produces on average less than one new infectious individual, and thus the disease should die out in the long run. on the other hand, if r0 > 1, then each infectious individual infects on average more than one individual, and thus the infection should be able to spread, [26], [73]. it is worth noting that the researchers such as r.a. ross, w.h. hamer, a.g. mckendrick or w.o. kermack who, between 1900 and 1935, laid the foundations of the mathematical approach to epidemiology, were not mathematicians but public health physicians, [15]. this observation emphasizes the fact that though biological and epidemiological research has greatly improved our knowledge of the life cycle of the malaria parasite within humans and mosquitoes, it cannot achieve a complete understanding of the complexities of malaria on its own. hence, resorting to mathematical models that can integrate various multiscale and intertwined aspects of the disease and, by using mathematical tools such as local and global stability analysis, bifurcation analysis, sensitivity analysis and data fitting, yield shortand longterm predictions about its progress, proves to be a necessity. the models, however, should reflect the current epidemiological knowledge and thus, they evolve to incorporate the latest empirical findings. to wit, the ross-macdonald model assumed that infected humans cannot develop immunity against malaria and that the human and mosquito populations are homogeneous. with the evidence showing that this was an oversimplification, the model has been extended in different directions by incorporating acquired immunity, variability in the mosquito and human populations, demography and age-structure, the environment or other relevant factors such as more realistic transmission mechanisms. we briefly mention some of these extensions. in [7], the authors considered only the human population but also incorporated the class of immunized humans and included an age structure. the work in [4] included the exposed classes and introduced a s h eh ihs h and s v ev iv compartment model for humans and mosquitoes, respectively. (ngwa & shu, 2000) [57] also study an s h eh ihs h and s v ev iv compartment model but considering varying population sizes. in [20], the authors extended the latter by including constant human immigration. an important addition to the malaria transmission models was the inclusion of the immunity function to the human population in [27]. socioeconomic factors in the human population and environmental factors in the mosquito population have been considered in [83] and others. in [58], the authors formulated a mathematical model that incorporated one aquatic stage of the mosquitos and availability of the adult vector and human treatment. they introduced, in particular, the so-called basic offspring number whose biomath 10 (2021), 2110029, http://dx.doi.org/10.11145/j.biomath.2021.10.029 page 4 of 19 http://dx.doi.org/10.11145/j.biomath.2021.10.029 j banasiak, r ouifki, w a woldegerima, some mathematical tools for modelling malaria: a subjective ... magnitude determines the existence of a thriving mosquito population in the sense that when this number is below unity, then the mosquito population becomes extinct. in recent years, the climate change has attracted attention in the field, and this resulted in a series of papers considering this aspect in modelling, see, e.g. [31], [62], [61], [56], [83]. other malaria transmission models incorporate treatment in their models, see, e.g. [24], [60]. it is reassuring that in [32], the authors used current modelling techniques to show that there is no bistable equilibrium in the malaria transmission and therefore, that elimination is feasible. mathematical modelling of malaria can also include information about how different parts of the population, such as adults and children, interact, and how these interactions influence the infection’s spread, resulting in age-structured mathematical models, where the hosts are grouped into compartments composed of individuals in the same age-group and infection status, see e.g., [12], [28], [34], [33], [37]. age-structured models are important to address intervention strategies involving the treatment with novel antimalarial drugs and vaccines. in [34] the authors fitted real data to an age-structured model to investigate the effectiveness of intervention strategies in reducing malaria parasites’ count. in the subsequent paper, [33], an age-structured model was used to estimate the changing age-burden of p. falciparum malaria using real data in sub-saharan africa. it is important to note that the process of complete eradication of an infectious disease such as malaria is divided into five main phases, defined in [55] and summarized in [76, p.2]. these are the transmission control, the disease elimination, the infection elimination, the eradication and the extinction. mathematical modelling of malaria can be applied at each of these phases. several models, including the ones discussed above, were developed for the phases ”the transmission control” or ”the disease elimination ”. some recently published mathematical models considered the other phases. for example, (white, et al., 2009), the authors used a mathematical model that required the input data in the form of a single estimate of parasite prevalence to consider ”the infection elimination” phase. their model included critical interventions targeting malaria transmission, which are currently available or in the final development stages. their results showed that a simple model has a similar short-term dynamic behaviour to complex models. they also demonstrated that the population level protective effect of multiple controls was crucial to overcoming failed elimination attempts. however, it is important to realize that even though a simple mathematical model could be suitable for situations where the data are sparse, more complex models, populated with new data, would provide more information, especially in the longterm, [75]. a recent development in new anti-malaria drug design opens new avenues for mathematical modelling. for instance, in [17] the authors estimated the transmission reduction that can be achieved by using drugs of varying chemo-prophylactic or transmission-blocking activity and they concluded that transmission reductions and eradication of malaria depend strongly on the deployment strategy, treatment coverage and endemicity level. in [80], the authors formulated a mathematical model that incorporates the effects of transmissionblocking drugs (tbds) on reducing the number of malaria infections. our mathematical results predict, in particular, that treating with tbds both clinical and subclinical malaria infections is required malaria eradication. b. different types of mathematical models in malaria generally, several types of mathematical models have been used to describe the transmission dynamics of infectious diseases, such as malaria. these are the deterministic models, stochastic models, statistical models, computer-based models, etc. several modelling approaches consist in using the available data to derive the process’s statistical parameters and a fitted curve describing its dynamics. we refer to them as statistical modelling; they can be used mainly for predictions, information extraction, and description biomath 10 (2021), 2110029, http://dx.doi.org/10.11145/j.biomath.2021.10.029 page 5 of 19 http://dx.doi.org/10.11145/j.biomath.2021.10.029 j banasiak, r ouifki, w a woldegerima, some mathematical tools for modelling malaria: a subjective ... of stochastic structures, see [46], [22]. however, statistic modelling mostly reveals correlations and not causations in the studied process. for that we need mechanistic models which attempt to reflect the causal relations in the process and thus, by using appropriate conservation laws and constitutive relations, give a description of the process’s dynamics, based on the understanding of its driving mechanisms. in deterministic models of this kind, the model’s output is fully determined by the parameter values and the initial conditions, without any room for random variation or uncertainty. deterministic models with continuous time can be written using ordinary, partial, or delay differential equations, [1], [16], while if we use discrete time intervals, the model takes the form of a difference equation, see [48] for an example of a discrete malaria model. on the other hand, while recognizing and using the causal relations in the described process, stochastic models allow for randomness in one or more inputs, making the evolution not precisely predictable. this implies that the same set of parameter values and initial conditions lead to an ensemble of different outputs, [1], [16] and thus the outcome can be predicted only with some probability. some authors have used a stochastic process to model malaria, e.g. [47], [54], [64], [11], [35], just to mention a few. since mathematical models are approximations of real-life phenomena, each type of mathematical model has its advantages and disadvantages. we have listed some of the limitations of mathematical models in the conclusion section, section 6. however, we emphasize that the models based on our understanding of the mechanisms driving the described processes tend to have better potential for making robust and reliable predictions. though we recognize that for successful control and eradication of malaria it may be necessary to use a combination of different mathematical modelling approaches [2], [51], in this paper, we shall focus on deterministic compartmental models discussed above and discuss some of them in more detail. such models, consisting of appropriately constructed systems of ordinary differential equations, are the most used mathematical tools in modelling malaria transmission dynamics. iii. genericmathematicalmodel ofmalaria in this section, we present a generic malaria model and explain how to use it to represent the disease’s complex transmission dynamics. mathematical modelling of malaria begins with collecting and understanding basic biological facts relevant to the disease. these facts form the model’s assumptions and rephrased in mathematical terms, become its constitutive relations. the model then consists of an appropriate system of conservation laws with the coupling described by these constitutive relations. for instance, in a population of humans of size nhin a region affected by malaria, we can distinguish a group of individuals who are not infected by the malaria parasites but are at risk of getting it; they are called susceptibles and denoted by s h. the remaining individuals, who are infected and can pass the infection to others are called infectious and denoted by ih. individuals in this compartment may fail to recover and die or can recover (either due to natural causes or medication), and then move to the recovered, rh, group. however, it is known that recovering from malaria induces only temporary immunity, so, after a certain time, the recovered can get reinfected upon being bitten by an infectious mosquito. such individuals will move from the rh class back to the s h class. we can refine the description of this process by introducing the exposed class eh of infected but not infectious humans to account for the malaria incubation time. models describing such a progression of malaria in humans are called s h ihrhs h or s h eh ihrhs h models, respectively. analogous models can be constructed for the mosquito population. the progression between the compartments is a conservative process in the sense that the inbiomath 10 (2021), 2110029, http://dx.doi.org/10.11145/j.biomath.2021.10.029 page 6 of 19 http://dx.doi.org/10.11145/j.biomath.2021.10.029 j banasiak, r ouifki, w a woldegerima, some mathematical tools for modelling malaria: a subjective ... (a) simple sir-si flow diagram for transmission dynamics of malaria. (b) flow diagram for s h eh ihrhs h − s v ev iv transmission dynamics of malaria. figure 1: simple sir-si flow diagram (a) and a flow diagram for s h eh ihrhs h−s v ev iv (b) transmission dynamics of malaria. dividuals in, say, the class s h that become infected, vanish from this class but must reappear in the infectious class. the mechanism of the progression is modelled then by appropriate constitutive relations. we must rephrase these notions using mathematical functions, expressions and equations. usually, the first step in constructing a model is representing the relevant processes in a flow diagram. the diagram in figure 1a shows the basic structure of such s h ihrhs h −s v iv model for human-mosquito transmission dynamics of malaria, while figure 1b shows a flow diagram for an s h eh ihrhs h − s v eh iv malaria model with demographic phenomena such as natural births and deaths, see, e.g. a typical example of a mathematical s h eh ihrhs h − s v ev iv model for transmission dynamics of malaria with demography, see, e.g. biomath 10 (2021), 2110029, http://dx.doi.org/10.11145/j.biomath.2021.10.029 page 7 of 19 http://dx.doi.org/10.11145/j.biomath.2021.10.029 j banasiak, r ouifki, w a woldegerima, some mathematical tools for modelling malaria: a subjective ... [20], [23], [57], [80], is given by ds h dt = fh (nh)+ρhrh−λhs h−gh (nh) s h, deh dt = λhs h − (νh + gh (nh)) eh, dih dt = νh eh − (δh + γh + gh (nh)) ih, drh dt = γh ih − (ρh + gh (nh)) rh, (1) ds v dt = fv (nv) − λvs v − gv (nv) s v, dev dt = λvs v − (νv + gv (nv)) ev, div dt = νv ev − gv (nv) iv, where λh and λv the forces of infections (infection rates) of humans and vectors, respectively and the total populations of humans and vectors at any time t≥0 is, respectively, nh (t) = s h (t) + eh (t) + ih (t) + rh (t) and nv (t) = s v (t) + ev (t) + iv (t). a. types of the force of infection when an infectious female anopheles mosquito bites a susceptible human, there is a probability that the parasite (in the form of sporozoites) will be injected into the human’s bloodstream and travel into liver cells to infect hepatocytes. the process when a susceptible human gets infected by an infectious mosquito is represented in the model by a function called the force of infection (of humans), denoted here by λh. the analogous process for the female mosquitoes when, after biting an infections human, the gametocytes form of the malaria parasite enters the mosquito’s midgut, is represented by a function called the force of infection of mosquitoes, denoted here by λv. the form of the force of infection has a significant impact on the dynamics of malaria, but it depends on many factors and should be carefully chosen for the problem. we shall briefly discuss the most common choices. we begin with a general form of the force of infection introduced in [20] without clear justification. here we shall derive this form based on the holling type argument, (holling, 1959). we begin with some introductory observations. the force of infection of humans, λh, is the product of the number of mosquito bites a human can have per unit of time, bh (nh, nv), the probability of the transmission of the disease from the mosquito to human, βhv and the probability that the biting mosquito is infected, ivnv . similarly, the force of infection of vectors, λv, is the product of the number of bites a susceptible mosquito can make per unit time on humans, bv (nh, nv), the probability of the transmission from an infectious human to vector and the probability that the bitten human is infectious, βvh ih nh +β̃vh rh nh . here we take into account that both ih and rh humans can be infectious with possibly different transmission probabilities, 0≤β̃vh < βvh < 1. therefore, the forces of infections of humans and vectors, respectively, are given by λh = bh (nh, nv) βhv iv nv , λv = bv (nh, nv) ( βvh ih nh + β̃vh rh nh ) . it remains to derive the formula for the total number of bites per unit time. it can be written as b (nh, nv) = bh (nh, nv) nh = bv (nh, nv) nv. to proceed, we define two parameters, σh and σv, that are, respectively, the constant average number of mosquito bites a human can receive (respectively, a mosquito can make on a human) per unit time. σh depends on the human’s exposed area, awareness, etc., while σv depends on the mosquito gonotrophic cycle, its preference for human blood and the time used for feeding. consider a certain period t that, for a mosquito, can be split as t = tna + ta, where tna is the time where the mosquito cannot bite and ta is the time available for biting. thus, in time t , the total number of bites received by all humans can be written as b (nh, nv) t = σhta nh. (2) now, if a mosquito can make only σv bites in a unit time, it means that it is not able to bite biomath 10 (2021), 2110029, http://dx.doi.org/10.11145/j.biomath.2021.10.029 page 8 of 19 http://dx.doi.org/10.11145/j.biomath.2021.10.029 j banasiak, r ouifki, w a woldegerima, some mathematical tools for modelling malaria: a subjective ... for 1/σv after each bite. besides, in time t it has σh nh ta nv meals. hence t = ta + σh nhta nvσv . (3) thus, plugging (3) into (2) and simplifying yields b (nh, nv) = σvσh nv nh σv nv + σh nh . (4) therefore, the force of infections are modelled by λh = σvσh σv nv + σh nh βhv iv, λv = σvσh σv nv + σh nh ( βvh ih + β̃vhrh ) . (5) as noted in [20], these formulae generalize several previously used expressions for the forces of infection. in particular, if nv/nh is small, then factoring out nh from the denominator and setting nv/nh to 0, we obtain λh = σvβhv iv nh , λv = σv βvh ih + β̃vhrh nh , (6) which are known as standard infection forces, (anderson, 2013), (allen, 2008), (martcheva, 2015), (hethcote, 2000). on the other hand, if nh/nv is small, we can write λh = bhβhv iv nv , λv = bvβvh ( ih nh + β̃hv rh nh ) , (7) where bh (nh, nv)≈σh and bv (nh, nv)≈nhσh/nv, which corresponds to the original mass action model of ross, that was written in terms of fractions of the population, [5], [3]. in particular, if the populations are constant, then this model is the usual mass action compartmental model. finally, we mention saturated infection rates, where the force of infection is given by a version of the holling ii functional response. it was first introduced to infectious disease modelling in [19] in their study of the cholera epidemic. we can adopt such a force of infection for malaria models with treatment or acquired immunity. then, assuming that only individuals from the infected class ih are infectious, the force of infection of humans by mosquitoes and the force of infection of mosquitoes will be given by, respectively, λh = σvβhv iv 1 + ηv iv , λv = σvβvh ih 1 + ηh ih , with an obvious modification if also rh individuals can contribute to the infections. the use of the holling type ii incidence function to model the infection process reflects the fact that the number of effective contacts between infective and susceptible individuals may saturate at high infective levels due to overcrowding, or due to the preventive measures applied in response to the disease, [42], [49], [58], [68], [81], [59]. b. choice of the demographic functions the demographic terms fh (nh) and gh (nh) can take several forms, depending on the population, and we list the common ones. we note that similar choices work for both fv (nv) and gv (nv) . we also assume that there is no vertical transmission of the infection, so all new newborns are susceptible. 1) there are births with a constant total birth rate πh and natural deaths with per capita natural death rate µh so that fh (nh) = πh and gh (nh) = µh. in this case, the vital dynamics (dynamics in the absence of infection) will be dnhdt = πh−µh nh. we note that this choice is used mostly for mathematical convenience as fitting this model to real populations produces unrealistic rates πh and µh, [52]. 2) the demography is governed by the malthusian law, that is, the total birth and death rates are proportional to the total population, [20], [57]. thus, fh (nh) = ψh nh and gh (nh) = µh and in this case the vital dynamics is given by dnhdt = ( ψh −µh ) nh. 3) the birth rate is directly proportional to the total human population, fh (nh) = ψh nh, while the death rate depends on the density of the population size in a nonlinear way. for instance, in addition to the intrinsic deaths, there may be additional deaths due to the overcrowding, which can be modelled as gh (nh) = µh + µ̃h nh, where µ̃h is biomath 10 (2021), 2110029, http://dx.doi.org/10.11145/j.biomath.2021.10.029 page 9 of 19 http://dx.doi.org/10.11145/j.biomath.2021.10.029 j banasiak, r ouifki, w a woldegerima, some mathematical tools for modelling malaria: a subjective ... (a) line plot of number of indigenous reported malaria cases (b) line plot of number of indigenous reported malaria caused deaths figure 2: line plots of the number of indigenous reported malaria cases and deaths in south africa for the years 2001 − 2018. the additional constant death rate. then the vital dynamics is given by dnhdt = ψh nh − nh ( µh + µ̃h nh ) , where we assume that ψh is sufficiently large so that the vital dynamics is logistic. we can use other types of birth functions, such as the ricker function, [65], the bevertonholt function, [13] or the maynard-smith-slatkin function, [72]. c. fitting the malaria model into the south african data according to the who report 2019 [78], south africa reported 9540 indigenous cases of malaria in 2018, with 69 deaths. however, the number of reported indigenous cases of malaria in 2017 was 22 064. in south africa, malaria is transmitted mainly in the border areas due to the crossborder movement of populations, including workers from neighbouring malaria-endemic countries and south africa residents travelling there. we also note here that according to the department of health of the republic of south africa, some parts of the country are endemic for malaria. at the same time, 10% of the population (approximately 4.9 million people) is at risk of contracting the disease, with p. falciparum being the dominant malaria species. a summary of the number of reported indigenous malaria cases and deaths in south africa for the years 2001-2018 is shown in figure 2. we plotted them in the python programming language using data from the who website, [79]. figure 3 and figure 4 depict curves fitted from model (1) using malaria data obtained from (who-gho, 2016) for the number of indigenous reported malaria cases and malaria caused deaths, respectively, in south africa in 2001−2018. here we use the forces of infections λh = bhβhv iv nv and λv = bvβvh ( ih +ζr rh + ζe eh nh ) . we set the initial conditions s h (2001) = 5 × 10 6, rh (2001) = 0 and let ih (2001) = to vary between 15, 000 and 30, 000 , since in 2001 the number of indigenous reported malaria cases was 26506. we then assumed eh (2001) = 50000, s v (2001) = 50000, ev (2001) = 5000 and iv (2001) = 4000. furthermore, as a guessed starting parameters we used the baseline values listed in table 1, obtained from literature except for the population projection of south africa in (worldometers, 2020), where we estimated µh = 0.0159 per year since the average biomath 10 (2021), 2110029, http://dx.doi.org/10.11145/j.biomath.2021.10.029 page 10 of 19 http://dx.doi.org/10.11145/j.biomath.2021.10.029 j banasiak, r ouifki, w a woldegerima, some mathematical tools for modelling malaria: a subjective ... lifespan of south africans in 2017 was 63 years and πh = 70000 per year since the annually approximately 7×105 people are added to the total population but only 10% are at risk of malaria. we use µv = 17.38 per year since the lifespan of an anopheles mosquito is 10-21 days. figure 3: a fitted curve for indigenous reported malaria cases in south africa , ih for model (1) using data from who-gho [79] for the years 2001 − 2018, and projections up to 2025. figure 4: a fitted curve for indigenous reported malaria caused deaths in south africa, for the model (1) using data from [79] for the years 20012018, and projections up to 2025. the method used to fit the model in both cases is lsqcurvefit with multi-start for global fit in matlab. we observe that the theoretical curves in figures 3 and 4 do not cater well for short term variations in the number of cases and deaths. this is since neither model considers seasonal or environmental variations that heavily affect malaria incidence. in particular, the spike in 2017 could be partly attributed to the higher than normal rainfall between the 2015 − 2016 and subsequent drought in southern africa. parameter uncertainty and the predicted uncertainty is important for qualifying a confidence in the solution, and for adjusting parameter values so that a correlation best fits data. as it is directly mentioned in here, “the 95% confidence bands enclose the area that you can be 95% sure contains the true curve. it gives you a visual sense of how well your data define the best-fit curve. as such, we extend the data fitting in figures 3 and 4 to include a 95% confidence bands, as can be observed in figure 5 and figure 6. figure 5: a fitted curve for indigenous reported malaria cases in south africa showing a 95% confidence interval, for the model (1) using data from (who-gho, 2016) for the years 2001-2018, and projections up to 2025. we note here that while fitting model (1) to the data using the baseline values as guessed starting parameter values with their corresponding lower and upper bounds in table i, we only let six parameters, namely, ch, ch,νh,νv and ρh to be unknown so that the algorithm will estimate their values with a 95% confidence interval. as can be seen in figure 5 or figure 6, several of data points lie below and above the fitted biomath 10 (2021), 2110029, http://dx.doi.org/10.11145/j.biomath.2021.10.029 page 11 of 19 http://dx.doi.org/10.11145/j.biomath.2021.10.029 j banasiak, r ouifki, w a woldegerima, some mathematical tools for modelling malaria: a subjective ... table i: parameters, their baseline values used for fitting, dimension, references and lower and upper bounds used. several of the ranges for the parameter values are directly taken from [21], where they used data for the high and low transmission areas. other parameter ranges are adapted from [17], [33] and some are estimated. parameter baseline value used dimension reference ranges (lower & upper bounds used) πh 70000 h×year−1 calculated from [82] fixed πv 5000 v×year−1 assumed [1 × 10 3, 5 × 105] cv = βvhb 0.2 h×v−1×year−1 [21] [0.01, 2] ch = βhvb 0.6 h×v−1×year−1 [21] [0.01, 2] νh 0.1 year−1 [21]) [0, 10] δh 0.008 year−1 calculated from [79] [0, 0.1] ρh 0.47 year−1 [21] [0.0005, 2] γh 0.275 year−1 assumed [0, 10] ζe 0.005 1 assumed [0, 1] ζr 0.001 1 assumed [0, 1] νv 0.083 year−1 [21] [0.005, 2] µv 17.38 year−1 [78] fixed µh 0.0159 year−1 calculated from [82] fixed figure 6: a fitted curve for indigenous reported malaria caused deaths in south africa showing a 95% confidence interval, for the model (1) for the years 2001-2018, and projections up to 2025. to fit the model into the malaria data we used a method in matlab called “lsqcurvefit” (leastsquare curve fit) with multi-start for global fit, and to obtain the confidence interval we used the method called “nlpredci”, which stands for a nonlinear prediction. it is a nonlinear regression prediction for confidence intervals. lines, and some of them are outside of the %95 confidence brand. clearly these residuals are not well-behaved and the residuals are not evenly distributed over the time series. so, we must emphasize here that the parameters of the model are not identifiable with best estimates, and they have large error ranges. this is because the model has 7 state variables and 15 parameters, but it is fitted to a time series of two observations, and thus it is overparameterized. its proper calibration and providing relevant statistical information such as confidence interval, margins of errors and parameter estimates require more data, not available at present. thus, we do not claim that the data fitting given in figure 3 and figure 4 or figure 5 and figure 6 are definitive. others may argue that it could have been better to use a much simpler model or simply to use an exponential decay function to statistically fit the given data for the number of malarai cases and deaths, and estimate the fewer parameters involved in defining an exponential decay function with best parameter fit and narrow confidence interval. however, we want to use a mechanistic model, as these model when correctly constructed and validated can have a strong predictive power, and can be used to test effectiveness of an intervention, as such interventions can in principle be include mechanistically in the model. the preceding discussion shows how imporbiomath 10 (2021), 2110029, http://dx.doi.org/10.11145/j.biomath.2021.10.029 page 12 of 19 http://dx.doi.org/10.11145/j.biomath.2021.10.029 j banasiak, r ouifki, w a woldegerima, some mathematical tools for modelling malaria: a subjective ... tant it is to validate the results obtained from mathematical models by comparing them with the field data. correctly done, statistical analysis of the results allows us to assess the model’s ability to simulate essential features of the transmission dynamics of malaria, that is, to validate the model. it also provides for extrapolation and interpolation of the data for future predictions and estimates of input parameters for more complex mathematical models [22], [46]. it is worth noting that though complex mechanistic models have more predictive power, the uncertainty in the parameters estimates strongly curtails this power. thus, an important part of modelling is balancing the model’s complexity with the availability of reliable data. iv. reducing complexity ofmodels bymultiple scale analysis one of the ways of reducing the complexity of a model is exploiting its structural features to aggregate functionally similar variables and thus simplify the model without losing salient features of is dynamics. there are various ways of models’ aggregation, [8], the most popular being the one based on the existence of multiple time scales in it. let us explain this approach using a malaria model as an example. we have observed that, due to the interplay of host and vector dynamics as well as the complex evolution of the vector population itself, see, e.g. [58], malaria models have indeed become increasingly involved, making their robust analysis difficult if not impossible. fortunately, biological phenomena often occur on time or size scales of widely different orders of magnitude. for instance, in malaria models, mosquitoes’ vital dynamics occurs on a much faster time scale (average lifespan of fewer than 21 days) than that of humans (average lifespan of around 65 years). because of this, malaria models can be considered as multi scale models, see, e.g. [30], which paves the way for their significant simplifications. showing that it is indeed possible and that the simplified models preserve essential features of the original dynamics requires a delicate mathematical analysis belonging to the field of singular perturbation theory, see, e.g. [9], [10], [66]. we shall illustrate this approach on the model given by (1) with fh (nh) = ψh nh, gh (nh) = µh nh, fv (nv) = ψv nv, and gv (nv) = ψv nv, that is, assuming a stable mosquito population. further, for computational simplicity, we discard the exposed classes so that ds h dt = (ψh −µh) s h + (ψh + ρh) rh + ψh ih −σvβhv iv s h + ih + rh s h, dih dt = σvβhv iv s h + ih +rh s h − (δh +γh +µh) ih, drh dt = γh ih − (ρh + µh) rh, (8) ds v dt = µv iv −σvβvh ih s h + ih + rh s v, div dt = σvβvh ih s h + ih + rh s v −µv iv, where we used nh = s h + ih + rh and nv = s v + iv. next, using parameter values directly from [20, table 4.1] in a time scale of years, we write system (8) as ds h dt = 1.26×10−2s h + 5.36rh + 2.8×10 −2 ih −4.38 iv s h + ih + rh s h, dih dt = 4.38 iv s h + ih + rh s h − 1.37ih, drh dt = 1.35ih − 5.34rh, (9) 10−3 ds v dt = 5.22×10−2 iv−1.82×10 −1 ih s h + ih +rh s v 10−3 div dt = 1.82×10−1 ih s h + ih +rh s v−5.22×10 −2 iv, with initial condition (s h(0),ih(0),rh(0),s v(0),iv(0)) = (s 0 h,i 0 h,r 0 h,s 0 v,i 0 v ). we can observe that µv = 5.22×10 1 per year, whereas µh = 1.58×10 −2 per year, that is, they differ by 3 orders of magnitude. the idea here is to replace the factor 10−3 by a small parameter � and try to approximate (9) by solutions of the biomath 10 (2021), 2110029, http://dx.doi.org/10.11145/j.biomath.2021.10.029 page 13 of 19 http://dx.doi.org/10.11145/j.biomath.2021.10.029 j banasiak, r ouifki, w a woldegerima, some mathematical tools for modelling malaria: a subjective ... (a) infected humans (b) infected mosquitoes figure 7: solution curves for the infected class for different values of �. the solution of the original system corresponds to � = 10−3 and solution to (10) corresponds to � = 0. simplified problem with � = 0, that is, ds̃ h dt = 1.26×10−2s̃ h + 5.36r̃h + 2.8×10 −2 ĩh −4.38 ĩv ñh s̃ h, dĩh dt = 4.38 ĩv ñh s̃ h − 1.37ĩh, (10) dr̃h dt = 1.35ĩh − 5.34r̃h, 0 = 5.22×10−2 ( nv − s̃ v ) − 1.82×10−1 ĩh ñh s̃ v, with the same initial conditions, where we used the fact that nv is constant. clearly, (10) is a lower-dimensional system and it follows that (subject to adding an initial layer corrector) its solutions approximate the solutions of (10), see [9], [10], [66]. we illustrate this result in figure 7, where we present numerical simulations of (10) with initial conditions s 0h = 1000, r0h = 0, i 0 h = 40, s 0 v = 100, i 0 v = 30, [20], and with 10−3 replaced by different values of �, so that with � = 10−3 we recover the solutions of system (8). the solution for (10) corresponds to � = 0. we observe that as � approaches to zero, the solutions get closer to the solution of (10). see, figure 7. v. current research towardsmalaria elimination and eradication even though antimalarial drugs are widely available and can be used at different cycles within a human, it is unlikely that they can eliminate malaria on their own and novel strategies are urgently needed. one of them is developing interventions to interrupt or completely block the malaria transmission by targeting the transmission of either the gametocytes to the vector or the sporozoites to humans. such transmission-blocking interventions (tbis) can be either transmission-blocking drugs (tbds) or transmission-blocking vaccines (tbvs), [25], [6]. then, [74], [70], tbds can be classified as drugs targeting the malaria parasite within the human host, drugs targeting the parasite in the vector or drugs targeting the vector itself. designing a drug whose primary objective is the transmission-blocking would be a game-changer leading to comprehensive malaria elimination, [14], [18]. during the drug development process, the central focus is on the ability of the compound to achieve one of the following goals: killing the malaria parasites during the liver stage or the blood biomath 10 (2021), 2110029, http://dx.doi.org/10.11145/j.biomath.2021.10.029 page 14 of 19 http://dx.doi.org/10.11145/j.biomath.2021.10.029 j banasiak, r ouifki, w a woldegerima, some mathematical tools for modelling malaria: a subjective ... figure 8: flow diagram showing the malaria transmission dynamics between human and mosquito populations with transmission-blocking drug treatment. stage of infection, blocking the formation and maturation of gametocytes, providing chemoprevention to high-risk groups or else preventing the parasite cycle within the mosquito. mathematical models are important at this stage to guide drug development by studying its pharmacokinetics (pk) and pharmacodynamics (pd). in the first case, the models can help us study the change of the drug concentration over time as determined by its absorption, distribution, metabolism and excretion. these are typically described by a set of differential equations representing the physical compartments, where the different processes occur. in the second case, mathematical models can describe the relationship between drug concentration and its killing efficacy. this can be shown as a ’doseresponse curve’ showing the efficacy as a function of the measured concentration of the drug in the blood, [38], [41], [63], [71]. still, there is a lack of insight about the population-level impact of different strategies of rolling out the tbds that can be obtained from mathematical modelling. in our recent work, [80], we have addressed this problem by proposing and providing a preliminary analysis of a population level mathematical model of human-mosquito interactions that take into account an intervention using tbds. our model extends (1) by including the class th of people under treatment with a tbd, and the class ph of people who have been successfully treated and are (at least temporarily) immune to malaria and (at least temporarily) does not transmit malaria. we allow the treatment to fail in which case the individuals from th move back to ih or rh (so that they remain infectious) or be successful whereby the individuals stop being infections. the model can be represented by the flow diagram in figure 8. our model is still preliminary and, in particular, it has not been tested on the real data that, due to the novelty of the field, are hard to obtain. nevertheless, we determined a threshold quantity that governs the spread of malaria under treatment by a tbd and, using this threshold, we investigated the impact and sensitivity of the model parameters that can be manipulated to drive the threshold quantity below unity. we also derived an expression that relates the drug efficacy and its treatment rate, and determined a critical treatment coverage rate, that is, the rate beyond which the tbd can eliminate malaria. on the other hand, also research for a malaria biomath 10 (2021), 2110029, http://dx.doi.org/10.11145/j.biomath.2021.10.029 page 15 of 19 http://dx.doi.org/10.11145/j.biomath.2021.10.029 j banasiak, r ouifki, w a woldegerima, some mathematical tools for modelling malaria: a subjective ... vaccine has been underway. in particular, significant progress has been made in this field with the development of the pre-erythrocytic vaccine, named rts,s. however, the use of malaria vaccines is not yet implemented well. as noted before, a limitation of such complex malaria models is that they can be overparameterized and it could be impossible to fit the parameters accurately on a given data set. this creates uncertainty in the simulation results. the main problem is that we require extensive data sets for estimating the parameter values with small error margins, as models of malaria are usually highly complex. vi. conclusion mathematical modelling is a process that uses mathematical tools to represent, analyze, make predictions, or otherwise provide insight into realworld phenomena. the value of mathematical models lies in that they help us to understand realworld phenomena by simplifying complex scenarios. for instance, analysis of epidemiological models informs us about their dynamics and thus enables predictions about the development of the disease. in this way, mathematical models offer a cheap alternative to expensive, impractical, or impossible field and experimental work. mathematical models also have their disadvantages and shortcomings. a model is just an approximation of a real-life phenomenon since the complexity of nature makes it impossible to exactly represent it by a manageable set of equations. a mathematical model is as good as the assumptions used to formulate it, and thus it will work only when these assumptions are satisfied. even after a model is formulated and analyzed, the results may be not entirely conclusive, since the theory on which we base it may be inadequate, or the available data do not suffice for its validation. in particular, for malaria, the data availability is often not sufficient for reliable identification of the parameters, and this uncertainty strongly limits the predictive power of many models. nevertheless, over the past century, mathematical models of malaria of various levels of complexity concerning the human, mosquito, and plasmodium parasite populations have been developed and studied to understand the malaria infection mechanisms and thus facilitate its control and, ultimately, elimination. moreover, mathematical modelling of malaria dynamics is important to guide the drug development, suggest a deployment strategy and quantify adequate treatment coverage. whenever possible, quantitative and qualitative results from mathematical models of malaria are compared with the observational data to identify the model’s strengths and weaknesses. in many cases, however, the scarcity of reliable data on the human behaviour, the life cycles and behaviour of both the vector and the parasite, proliferation and waning of immunity, age profiles of symptomatic and asymptomatic infections or parasite’s drug resistance limits the usefulness of many mathematical models in the policy-making processes. thus, the importance of models is often not so much quantitative but rather qualitative, as was already noted by ross and several other researchers. it is, therefore, crucial for mathematical modellers to collaborate closely with life and health scientists and public health workers to facilitate an informed and robust exchange leading to better models and knowledge-based decisions. vii. acknowledgement all authors acknowledge the financial support from the dst/nrf sarchi chair in mathematical models and methods in biosciences and bioengineering at the university of pretoria, grant no. 82770. 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[83] yang, h. m., and ferreira, m. u. assessing the effects of global warming and local social and economic conditions on the malaria transmission. revista de saude publica 34 (2000), 214–222. biomath 10 (2021), 2110029, http://dx.doi.org/10.11145/j.biomath.2021.10.029 page 19 of 19 https://www.who.int/malaria/publications/world-malaria-report-2018/en/ https://www.who.int/malaria/publications/world-malaria-report-2018/en/ https://apps.who.int/gho/data/node.main.a1362?lang=en https://apps.who.int/gho/data/node.main.a1362?lang=en https://www.worldometers.info/world-population/south-africa-population/ https://www.worldometers.info/world-population/south-africa-population/ http://dx.doi.org/10.11145/j.biomath.2021.10.029 malaria and its mathematical modelling a brief review of mathematical models of malaria history and background of mathematical models of malaria different types of mathematical models in malaria generic mathematical model of malaria types of the force of infection choice of the demographic functions fitting the malaria model into the south african data reducing complexity of models by multiple scale analysis current research towards malaria elimination and eradication conclusion acknowledgement conflict of interest references original article biomath 3 (2014), 1404212, 1–18 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum mathematical analysis of a size structured tree-grass competition model for savanna ecosystems valaire yatat1, yves dumont2, jean jules tewa1, pierre couteron3 and samuel bowong1 1ummisco, lirima project team grimcape, yaounde, cameroon email: yatatvalaire@yahoo.fr; tewajules@gmail.com; sbowong@gmail.com 2cirad, umr amap, montpellier, france 3ird, umr amap, montpellier, france email: yves.dumont@cirad.fr; pierre.couteron@ird.fr received: 23 october 2013, accepted: 21 april 2014, published: 29 may 2014 abstract—several continuous-time tree-grass competition models have been developed to study conditions of long-lasting coexistence of trees and grass in savanna ecosystems according to environmental parameters such as climate or fire regime. in those models, fire intensity is a fixed parameter while the relationship between woody plant size and fire-sensitivity is not systematically considered. in this paper, we propose a mathematical model for the tree-grass interaction that takes into account both fire intensity and size-dependent sensitivity. the fire intensity is modeled by an increasing function of grass biomass and fire return time is a function of climate. we carry out a qualitative analysis that highlights ecological thresholds that summarize the dynamics of the system. finally, we develop a non-standard numerical scheme and present some simulations to illustrate our analytical results. keywords-asymmetric competition, savanna, fire, continuous-time modelling, qualitative analysis, non-standard numerical scheme. i. introduction savannas are tropical ecosystems characterized by the durable co-occurrence of trees and grasses (scholes 2003, sankaran et al. 2005) that have been the focus of researches since many years. savanna-like vegetations cover extensive areas, especially in africa and understanding savannas history and dynamics is important both to understand the contribution of those areas to biosphereclimate interactions and to sustainably manage the natural resources provided by savanna ecosystems. at biome scale, vegetation cover is known to display complex interactions with climate that often feature delays and feed-backs. for instance any shift from savanna to forest vegetation not only means increase in vegetation biomass and carbon sequestration but also may translate into changes in the regional patterns of rainfall (scheffer et al. 2003, bond et al. 2005). in the face of the ongoing global change, it is therefore important to understand how climate along with local factors citation: valaire yatat, yves dumont, jean jules tewa, pierre couteron, samuel bowong, mathematical analysis of a size structured tree-grass competition model for savanna ecosystems, biomath 3 (2014), 1404212, http://dx.doi.org/10.11145/j.biomath.2014.04.212 page 1 of 18 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2014.04.212 v yatat et al., mathematical analysis of a size structured tree-grass... drive the dynamics of savannas ecosystems. in many temperate and humid tropical biomes, forest vegetation in known to recover quickly from disturbances and woody species are expected to take over herbaceous species. yet in the dry tropics, it is well-known that grassy and woody species may coexist over decades although their relative proportion may show strong variations (scholes 2003, sankaran et al. 2005, 2008). savanna-like ecosystems are diverse and explanations found in the literature about the longlasting coexistence of woody and grassy vegetation components therefore relate to diverse factors and processes depending on the location and the ecological context. several studies have pointed towards the role of stable ecological factors in shaping the tree to grass ratio along large-scale gradients of rainfall or soil fertility (sankaran et al. 2005, 2008). other studies have rather emphasized the reaction of vegetation to recurrent disturbances such as herbivory or fire (langevelde et al. 2003, d’odorico et al. 2006, sankaran et al. 2008, smit et al. 2010, favier et al. 2012 and references therein). those two points of view are not mutually-exclusive since both environmental control and disturbances may co-occur in a given area, although their relative importance generally varies among ecosystems. bond et al. (2003) proposed the name of climate-dependent for ecosystems that are highly dependent on climatic conditions (rainfall, soil moisture) and firedependent or herbivore-dependent for ecosystems which evolution are strongly dependent on fires or herbivores. in a synthesis gathering data from 854 sites across africa, sankaran et al. (2005) showed that the maximal observed woody cover appears as water-controlled in arid to semi-arid sites since it directly increase with mean annual precipitation (map) while it shows no obvious dependence on rainfall in wetter locations, say above c. 650 mm map where it is probably controlled by disturbance regimes. above this threshold, fire, grazing and browsing are therefore required to prevent tree canopy closure and allow the coexistence of trees and grasses. several models using a system of ordinary differential equations (odes) have been proposed to depict and understand the dynamics of woody and herbaceous components in savanna-like vegetation. a first attempt (walker et al. 1981) was orientated towards semiarid savannas and analyzed the effect of herbivory and drought on the balance between woody and herbaceous biomass. this model refers to ecosystems immune to fire due to insufficient annual rainfall. indeed, fires in savanna-like ecosystems mostly rely on herbaceous biomass that has dried up during the dry season. as long as rainfall is sufficient, fire can thus indirectly increase the inhibition of grass on tree establishment in a way far more pervasive than the direct competition between grass tufts and woody seedlings. more recently, several attempts have been made (see accatino et al. 2010, de michele et al. 2011 and references therein) to model the dynamics of fire-prone savannas on the basis of the initial framework of tilman (1994) that used coupled odes to model the competitive interactions between two kinds of plants. on analogous grounds, langevelde et al. (2003) have developed a model taking into account fires, browsers, grazers and walter’s (1971) hypothesis of niche separation by rooting zone depth. models relying on stochastic differential equations have also been used (baudena et al. 2010). notably, accatino et al. (2010) and de michele et al. (2011) focused on the domain of stability of tree-grass coexistence with respect to influencing ”biophysical” variables (climate, herbivory). however, fire was considered as a forcing factor independent of climate and vegetation, while woody cover was treated as a single variable with no distinction between seedling/saplings which are highly fire sensitive and mature trees which are largely immune to fire damages. the way in which the fundamental, indirect retroaction of grass onto tree dynamics is modeled is therefore to be questioned. in the present paper we therefore a model that differs in this respect. thus, to take into account the role of fire in biomath 3 (2014), 1404212, http://dx.doi.org/10.11145/j.biomath.2014.04.212 page 2 of 18 http://dx.doi.org/10.11145/j.biomath.2014.04.212 v yatat et al., mathematical analysis of a size structured tree-grass... savanna dynamics, we consider a tree-grass compartmental model with one compartment for grass and two for trees, namely fire-sensitive individuals (like seedlings, saplings, shrubs) and non-sensitive mature trees. based on field observations and experiments reported by scholes and archer (1997) and by scholes (2003), we develop a system of three coupled non-linear ordinary differential equations (odes), one equation per vegetation compartment that describes savanna dynamics. in addition, we model fire intensity (i.e. impact on sensitive woody plants) as an increasing function of grass biomass. compared to existing models, our model aims to properly acknowledge two major phenomena, namely the fire-mediated negative feedback of grasses onto sensitive trees and the negative feed-back of grown-up, fire insensitive trees on grasses. we therefore explicitly model the asymmetric nature of tree-grass competitive interactions in fire-prone savannas. after some theoretical results of the continuous fire model, though which we highlighted some ecological thresholds that summarize savanna dynamics and some interesting bistability, we present an appropriate non-standard numerical scheme (see anguelov et al. 2012, 2013, 2014 and dumont et al. 2010, 2012) for the model considered and we end with numerical simulations. we show that the fire frequency and the competition parameters are bifurcation parameters which allow the continuous fire model of asymmetric tree-grass competition to converge to different steady states. ii. the continuous fire model of asymmetric tree-grass competition (cofac) as we have mentioned before, we consider the class of sensitive tree biomass (ts ), the class of non-sensitive tree biomass (tns ) and the class of grass biomass (g). we model the fire intensity by an increasing function of grass biomass w(g). to built up our model, we consider the following assumptions. 1) the grass vs. sensitive-tree competition has a negative feedback on sensitive tree dynamics. 2) the grass vs. non sensitive-tree competition has a negative feedback on grass dynamics. 3) after an average time expressed in years, the sensitive tree biomass becomes non sensitive to fire. 4) fire only impacts grass and sensitive tree. we also consider the following parameters. • there exists a carrying capacity kt for tree biomass (in tons per hectare, t.ha−1). • there exists a carrying capacity kg for grass biomass (in tons per hectare, t.ha−1). • sensitive tree biomass is made up from non sensitive tree biomass with the rate γns (in yr−1) and from existing sensitive tree biomass with the rate γs (in yr−1). • sensitive tree biomass has a natural death rate µs (in yr−1). • non sensitive tree biomass has a natural death rate µns (in yr−1). • f is the fire frequency (in yr−1). • grass biomass has a natural death rate µg (in yr−1). • 1 ωs is the average time, expressed in year, that a sensitive tree takes to become non sensitive to fire. • 1 ωs + µs is the average time that a tree spends in the sensitive tree class without competition and fires. • σg is the competition rate, for light or/and nutrients, between sensitive tree and grass (in ha.t−1.yr−1). • σns is the competition rate, for light or/and nutrients, between non sensitive tree and grass (in ha.t−1.yr−1). • ηs is the proportion of sensitive tree biomass that is consumed by fire. • ηg is the proportion of grass biomass that is consumed by fire. remark 1. competition parameters σg and σns are asymmetric, indeed σg inhibits sensitive tree (ts ) growth and there is no reciprocal inhibition; likewise, σns inhibits grass (g) growth. based on these ecological premises, and taking biomath 3 (2014), 1404212, http://dx.doi.org/10.11145/j.biomath.2014.04.212 page 3 of 18 http://dx.doi.org/10.11145/j.biomath.2014.04.212 v yatat et al., mathematical analysis of a size structured tree-grass... into account the effect of fire as a forcing continuous in time, which is the classical approach, we propose a model for the savanna vegetation dynamics through a system of three interrelate non-linear equations. the cofac is given by  dts dt = (γs ts + γns tns ) ( 1 − ts + tns kt ) −ts (µs + ωs + σgg + f ηs w(g)), dtns dt = ωs ts −µns tns , dg dt = γg ( 1 − g kg ) g − (σns tns + f ηg + µg)g, (1) with ts (0) = ts 0 > 0, tns (0) = tns 0 ≥ 0 , g(0) = g0 > 0. (2) for this continuous fire model, the fire intensity function w is chosen as a sigmoidal function of grass biomass because we want first to investigate the ecological consequences of the nonlinear response of fire intensity to grass biomass, while nearly all published models using differential equations so far assumed a linear response. non linearity is justified since whenever grass biomass is low fires are virtually absent while fire impact increases rapidly with grass biomass before reaching saturation. thus, w(g) = g2 g2 + g20 , (3) where g0 = g20 is the value of grass biomass at which fire intensity reaches its half saturation (g0 in tons per hectare, t.ha−1). the feasible region for system (1) is the set ω defined by ω = {(ts , tns , g) ∈ r3+ | 0 ≤ ts + tns ≤ kt , 0 ≤ g ≤ kg}. iii. mathematical analysis a. existence of equilibria, ecological thresholds and stability analysis we set r01 = γs µns + γns ωs µns (µs + ωs ) and r02 = γg f ηg + µg . 1) existence of equilibria: setting the right hand-side of system (1) to zero, straightforward computations lead to the following proposition proposition 1. system (1) has four kinds of equilibria • the desert equilibrium point e0 = (0, 0, 0) which always exists. • the forest equilibrium point et = (t s ; t ns ; 0), with t s = kt µns ωs + µns 1 − 1 r01  and t ns = kt ωs ωs + µns 1 − 1 r01  which is ecologically meaningful whenever r01 > 1. • the point eg = (0, 0, g), with g = kg 1 − 1 r02  , is ecologically meaningful when r02 > 1 the point eg when it exists is the grassland equilibrium. • the savanna equilibrium point et g = (t∗s , t ∗ ns , g ∗), with t∗s , t ∗ ns and g ∗ given in appendix a, has an ecological significance whenever r 0 1 > 1, r 0 2 > 1 and 0 < g ∗ < kg 1 − 1 r02  . remark 2. the number of savanna equilibria depends on the form of the function w. • if w(g) = g, then the cofac has at most one savanna equilibrium. • if w(g) = g g + g0 (the holling type ii function), then the cofac has at most two savanna equilibria. biomath 3 (2014), 1404212, http://dx.doi.org/10.11145/j.biomath.2014.04.212 page 4 of 18 http://dx.doi.org/10.11145/j.biomath.2014.04.212 v yatat et al., mathematical analysis of a size structured tree-grass... • if w(g) = g2 g2 + g0 (the holling type iii function), then the cofac has at most three savanna equilibria. 2) ecological thresholds interpretation: the qualitative behaviors of the cofac depend on the following thresholds r01, r 0 2, rg1 = γs µns + γns ωs µns (µs + ωs + σgg + f ηs w(g)) , r t ns 2 = γg f ηg + µg + σns t ns , where • r01 is the sum of the average amount of biomass produced by a sensitive/young plant, without fires and competition with grass, and the average amount of biomass produced by a mature plant multiplied by the proportion of young plants which reach the mature stage. • rg1 is the sum of the average amount of biomass produced by a sensitive/young plant, in presence of fires and competition with grass, and the average amount of biomass produced by a mature plant multiplied by the proportion of young plants which reach the mature stage. • r02 is the average amount of biomass produced per unit of grass biomass during its whole lifespan in presence of fires and and free from competition with non-sensitive trees. • r t ns 2 is the average biomass produced per unit of grass biomass during its whole lifespan in presence of fires and experiencing competition from non-sensitive trees. remark 3. the following relations hold r g 1 < r 0 1, r t ns 2 < r 0 2. 3) stability analysis: let r = r(g∗) = γg (µns + ωs ) (γs µns + γns ωs ) kg kt µns ωs σns (σg + f ηs w′(g?)) . we have the following result: theorem 1. if r01 < 1 and r 0 2 < 1, then the desert equilibrium e0 is globally asymptotically stable. proof: see appendix b. theorem 2. if r01 > 1, then the forest equilibrium et exists. • if rt ns2 < 1, then the forest equilibrium et is locally asymptotically stable. • if r02 < 1, then the forest equilibrium et is globally asymptotically stable. • if r02 > 1, r t̄ns 2 < 1, r ḡ 1 > 1 and r < 1, then the forest equilibrium et is globally asymptotically stable. proof: see appendix c. furthermore, using the same approach as in the proof of theorem 2, we derive the following results theorem 3. suppose r02 > 1 so that the grassland equilibrium eg exists. • if rg1 < 1, then the grassland equilibrium eg is locally asymptotically stable. • if r01 < 1, then the grassland equilibrium eg is globally asymptotically stable. • if r01 > 1, r t̄ns 2 > 1,r ḡ 1 < 1 and r < 1, then the grassland equilibrium eg is globally asymptotically stable. theorem 4. suppose that r01 > 1, r 0 2 > 1 and r > 1. we have the following three cases: • the savanna equilibrium et g is locally asymptotically stable (las) when it is unique. • when there exists two savanna equilibria, one is las and the other is unstable. • when there exists three savanna equilibria, two are las and one is unstable. thus system (1) will converges to one of the two stable savanna equilibria depending on initial conditions. proof: see appendix d. 4) summary table of the qualitative analysis: the qualitative behavior of system (1) is summarized in the following table in which we present biomath 3 (2014), 1404212, http://dx.doi.org/10.11145/j.biomath.2014.04.212 page 5 of 18 http://dx.doi.org/10.11145/j.biomath.2014.04.212 v yatat et al., mathematical analysis of a size structured tree-grass... only the most realistic, from an ecological point of view, case i.e r01 > 1 and r 0 2 > 1. table i summary table of the qualitative analysis of system (1) thresholds e0 et eg et g r01 r 0 2 r ḡ 1 r t̄ns 2 > 1 r > 1 u u u l ? > > > r t̄ns 2 < 1 r > 1 u l u l 1 1 1 r < 1 u g u u rḡ1 r t̄ns 2 > 1 r > 1 u u l l < r < 1 u u g u 1 rt̄ns2 < 1 r > 1 u l l l r < 1 u l l u in table i, the notations u, l and g stand for unstable, locally asymptotically stable, globally asymptotically stable, respectively, while the notation l? means that we have the global stability if there are no periodic solutions. remark 4. from an ecological point of view, lignes 1 to 7 of table i are interesting because in these cases, one ton of grass biomass will produce during it lifespan at least one ton of grass biomass (r02 > 1) and simultaneously, one ton of tree biomass (sensitive and non sensitive) will produce during it lifespan at least one ton of tree biomass (r01 > 1). moreover, it is also in these cases that we have the most interesting situations of savanna dynamics, namely bistability cases (lines 2, 4, 7 in table 1) and a tristability case (line 6 in table 1). iv. numerical simulations compartmental models are usually solved using standard numerical methods, for example, euler or runge kutta methods included in software package such as scilab [18] and matlab [19]. unfortunately, these methods can sometimes present spurious behaviors which are not in adequacy with the continuous system properties that they aim to approximate i.e, lead to negative solutions, exhibit numerical instabilities, or even converge to the wrong equilibrium for certain values of the time discretization or the model parameters (see anguelov et al. 2012, dumont et al. 2010 for further investigations). for instance, we provided in appendix e some numerical simulations done with runge kutta schemes to illustrate some of its spurious behaviors. in this section, following anguelov et al. 2012, 2013, 2014 and dumont et al. 2010, 2012, we perform numerical simulations using an implicit nonstandard algorithm to illustrate and validate analytical results obtained in the previous sections. a. a nonstandard scheme for the cofac system (1) is discretized as follows: t k+1ns − t k ns φ(h) = ωs t k+1s −µns t k+1 ns , gk+1 − gk φ(h) = γg ( 1 − g k kg ) gk+1 −σns t kns g k+1 −(µg + f ηg)gk+1, t k+1s − t k s φ(h) = (γs − (µs + ωs ))t k+1s + γns t k+1 ns − γs kt t k+1s (t k s + t k ns ) − γns kt t kns t k+1 ns − ( γns kt t kns + (σgg k + f ηs w(gk)) ) t k+1s , (4) where the denominator function φ is such that φ(h) = h + o(h2), ∀h > 0. systems (1) − (2) can be written in the following matrix form: dx dt = a(x)x, x(0) = x0, (5) biomath 3 (2014), 1404212, http://dx.doi.org/10.11145/j.biomath.2014.04.212 page 6 of 18 http://dx.doi.org/10.11145/j.biomath.2014.04.212 v yatat et al., mathematical analysis of a size structured tree-grass... where x = (tns , g, ts ) ∈ r3+ and a(x) = (ai j)1≤i, j≤3 with a11 = −µns , a12 = 0, a13 = ωs , a21 = 0, a22(x) = γg ( 1 − gkg ) − (σns tns + f ηg + µg), a23 = 0, a31(x) = γns ( 1 − ts +tnskt ) , a32 = 0, a33(x) = γs ( 1 − ts +tnskt ) − (µs + ωs + σgg + f ηs w(g)). using (5), the numerical scheme (4) can be rewritten as follows: b(xk)xk+1 = xk, where b(xk) = (id3 −φ(h)a(x k)). (6) thus b(xk) = 1 + φ(h)µns 0 −ωs φ(h) 0 1 −φ(h)ak22 0 −φ(h)ak31 0 1 −φ(h)a k 33  it suffices now to choose φ(h) such that the matrix b(xk)) is an m-matrix for all h > 0, which implies that b−1(xk) is a nonnegative matrix, for all h > 0. in particular, choosing φ such that 1 −φ(h)(γg − (µg + f ηg)) ≥ 0 1 −φ(h)(γs − (µs + ωs )) ≥ 0, (7) lead to positive diagonal terms and nonpositive off diagonal terms. we need to show that b(xk) is invertible. obviously 1 − φ(h)ak22 is a positive eigenvalue. let us define n, a submatrix of matrix b(xk), as follows n = ( 1 + φ(h)µns −ωs φ(h) −φ(h)ak31 1 −φ(h)a k 33 ) . we already have trace(n) > 0. then, a direct computation shows that det(n) > 0 if φ(h) is choosen such that 1 −φ(h) ( γs + γns ωs µns − (µs + ωs ) ) ≥ 0. thus we have α(n) > 0, i.e. the eigenvalues have positive real parts, which implies that b(xk) is invertible. finally, choosing φ(h) = 1 − e−qh q , (8) with q ≥ max ( γg − (µg + f ηg),γs − (µs + ωs ) + γns ωs µns ) , (9) matrix b(xk) is an m-matrix. furthermore, assuming xk ≥ 0, we deduce xk+1 = b−1(xk)xk ≥ 0. lemma 1. using the expression of φ defined in (8), the numerical scheme (4) is positively stable ( i.e for xk ≥ 0, we obtain xk+1 ≥ 0). an equilibrium xe of the continuous model (1) verifies a(xe)xe = 0. multiplying the above expression by φ(h) and summing with xe yields (id3 −φ(h)a(xe))xe = xe, thus, we deduce that the numerical scheme (4) and the continuous model (1) have the same equilibria which are (assumed to be) hyperbolic. the dynamics of model (1) can be captured by any number q satisfying q ≥ max { |λ|2 2|re(λ)| } , (10) where λ ∈ sp(j) with ji j = ∂ai ∂x j . we also have the following result: lemma 2. if φ(h) is chosen as in eqs. (8), (9) and (10), then the numerical scheme (4) is elementary stable ( i.e local stability properties of equilibria are preserved). the proof of lemma 2 follows the proof of theorem 2 in dumont et al., 2010. b. numerical simulations and bifurcation parameters in literature we found the following parameters values biomath 3 (2014), 1404212, http://dx.doi.org/10.11145/j.biomath.2014.04.212 page 7 of 18 http://dx.doi.org/10.11145/j.biomath.2014.04.212 v yatat et al., mathematical analysis of a size structured tree-grass... table ii parameters values found in literature parameters values references f 0-1 langevelde et al. 2003 0-2 accatino et al. 2010 γg 0.4(1) − 4.6(2) (1) penning de vries 1982 (2) menaut et al. 1979 γs + γns 0.456-7.2 breman et al. 1995 µs + µns 0.03-0.3 accatino et al. 2010 0.4 langevelde et al. 2003 µg 0.9 langevelde et al. 2003 ηg 0.1(a)-1(b) (a) van de vijver 1999 (b) accatino et al. 2010 ηs 0.02-0.6 accatino et al. 2010 ωs 0.05-0.2 walkeling et al. 2011 we now provide some numerical simulations to illustrate the theoretical results and for discussions. 1) some monostability and bistability situations: • monostability. we choose γs γns γg ηs ηg 1 2 3.1 0.5 0.5 µs µns µg σns σg 0.1 0.3 0.3 0.3 0.05 g0 ωs kt kg f 2 0.05 50 12 0.5 yr−1 r01 r 0 2 r g 1 r t ns 2 r 8.8889 5.6364 1.5006 1.2644 2.9424 with the chosen parameters, the savanna equilibrium is stable, i.e. sensitive trees, nonsensitive trees and grasses coexist. figure 1 presents the 3d plot of the trajectories of system (1). it illustrates that the savanna equilibrium point is stable. figure 1 also illustrates the monostability situation presented in ligne 1 of table i. 0 10 20 30 0.511.5 22.53 3.54 0 5 10 t s (t) t ns (t) g (t ) e tg • • • • • • • • fig. 1. 3d plot of the trajectories of system (1) showing that the savanna equilibrium point et g point is stable. the red bullets represent different initial conditions. • bistability – bistability involving forest and grassland equilibria. the state trajectories of the model will converge to a state depending of initial quantity. we choose γs γns γg µs µns µg f 0.4 2 2.1 0.1 0.3 0.3 0.5 yr−1 ηs =0.5, ηg = 0.5, kt =50, kg = 12 r01 r 0 2 r g 1 r t ns 2 r σg σns 4.8889 3.8182 0.5731 0.9315 0.2958 0.1 0.3 the 3d plot of the trajectories of system (1) is depicted in figure 2. it clearly appears that the forest and grassland equilibria are stables. figure 2 illustrates the bistability situation presented in ligne 7 of table i. 0 5 10 15 20 25 30 35 5 10 15 20 0 5 10 t ns (t) t s (t) g (t ) only grasses (grassland) • • • • • • • • • • • • only trees (forest) • e t e g • • fig. 2. 3d plot of the trajectories of system (1) showing that the forest (et ) and grassland (eg ) equilibria are stable. the green bullets represent different initial conditions. – bistability involving forest and savanna equilibria. the state trajectories of the model will converge to a state depending on initial quantity. we choose γs γns γg µs µns µg f 0.6 2 2.1 0.1 0.3 0.3 0.5 yr−1 ηs =0.5, ηg = 0.5, kt =50, kg = 12 r01 r 0 2 r g 1 r t ns 2 r σg σns 6.2222 3.8182 1.1156 0.8942 1.2985 0.05 0.3 for these parameters, there exist two savanna equilibria but only one is stable as shown in figure 3. figure 3 also illustrates the bistability situation presented in ligne 2 of table i. biomath 3 (2014), 1404212, http://dx.doi.org/10.11145/j.biomath.2014.04.212 page 8 of 18 http://dx.doi.org/10.11145/j.biomath.2014.04.212 v yatat et al., mathematical analysis of a size structured tree-grass... 0 5 10 15 20 25 30 35 40 0 5 10 15 0 2 4 6 8 10 t ns (t) t s (t) g (t ) • • • • • • • • • • • • • forest savanna • fig. 3. 3d plot of the trajectories of system (1) showing that the forest (et ) and savanna (et g ) equilibria are stable. the green and red bullets represent different initial conditions. remark 5. note also that we didn’t observed periodic behaviors in the previous simulations, considering the set of parameters presented in table 2, while their existence cannot be completely ruled out by the analytical analysis. c. some bifurcation parameters in this section we emphasize on some bifurcation parameters of system (1) which are such that the cofac can converge to different steady state depending on the variation of these parameters. • the grass vs. sensitive-tree competition parameter σg is a bifurcation parameter. figure 4 presents how the system (1) changes from the savanna state to the grassland state as a function of the grass vs. sensitive-tree competition parameter σg. we choose γs γns γg µs µns µg ηs σns f 0.4 1 4 0.1 0.3 0.1 0.5 0.3 0.2 yr−1 ηg = 0.5, kt =45, kg = 10 for these parameters values, system (1) undergoes a transcritical bifurcation. indeed, we move from ligne 1 to ligne 5 of table i. from left to right, (r01 = 3.7778, r 0 2 = 20, r g 1 = 2.2865, rt ns2 = 2.4721, r = 99.3058) → (r01 = 3.7778, r 0 2 = 20, r g 1 = 1.2943, r t ns 2 = 2.4721, r = 5.6803) → (r 0 1 = 3.7778, r02 = 20, r g 1 = 0.9026, r t ns 2 = 2.4721). for the last case, the savanna equilibrium et g is undefined. 0 50 100 150 200 250 300 0 5 10 15 20 25 time (year) (a) sensitive tree non sensitive tree grass 0 50 100 150 200 250 300 0 5 10 15 20 25 time (year) (b) 0 50 100 150 200 250 300 0 5 10 15 20 25 time (year) (c) fig. 4. from savanna to grassland as a function of the grass vs. sensitive-tree competition parameter σg . from left to right, the fire period τ = 1f is fixed, while the grass vs. sensitive-tree competition parameter σg increases. in (a) (τ = 5, σg = 0), in (b) (τ = 5, σg = 0.02) and in (c) (τ = 5, σg = 0.04) • the fire period parameter τ = 1f is a bifurcation parameter. figure 5 presents a shift of the convergence of system (1) from the forest state to the grassland state as a function of the fire period τ. we choose table iii parameters values for figures 5 and 6 γs γns γg µs µns µg ηs σns f 0.4 2 2.1 0.1 0.3 0.3 0.5 0.3 yr−1 ηg = 0.5, kt =50, kg = 12 for these parameters values, system (1) undergoes a forward bifurcation. indeed, we move from ligne 3 to ligne 7 of table i. from left to right, (r01 = 4.8889, r 0 2 = 1.0678, r g 1 = 1.3025, rt ns2 = 0.5720) → (r 0 1 = 4.8889, r02 = 1.3548, r g 1 = 0.5446, r t ns 2 = 0.6453, r = 0.0983) → (r01 = 4.8889, r 0 2 = 1.6154, rg1 = 0.5679, r t ns 2 = 0.6989, r = 0.1201). for the first case, the savanna equilibrium et g is undefined. 0 50 100 150 200 0 5 10 15 20 25 30 35 time (year) (a) sensitive tree non sensitive tree grass 0 50 100 150 200 0 0.5 1 1.5 2 2.5 3 3.5 4 time (year) (b) 0 50 100 150 200 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time (year) (c) fig. 5. from forest to grassland as a function of the fire period τ. from left to right, the fire period τ increased, while the sensitive tree-grass competition parameter σg is fixed. in (a) (τ = 0.3, σg = 0.05), in (b) (τ = 0.4, σg = 0.05) and in (c) (τ = 0.5, σg = 0.05) biomath 3 (2014), 1404212, http://dx.doi.org/10.11145/j.biomath.2014.04.212 page 9 of 18 http://dx.doi.org/10.11145/j.biomath.2014.04.212 v yatat et al., mathematical analysis of a size structured tree-grass... suppose now that fire period is fixed and the grass vs. sensitive-tree competition parameter σg varies. figure 6 illustrates a shift of the convergence of system (1) from the forest state to the grassland state through a savanna state as a function of the grass vs. sensitivetree competition parameter σg. for the parameters values in table iii, system (1) exhibits to bifurcation phenomena: a pitchfork bifurcation and a transcritical bifurcation. we move from ligne 3 (figure 6 (a), -(b)) to ligne 7 (figure 6 (e), -(f)) through ligne 2 (figure 6 (c), -(d)) of table i. indeed, in figure 6 (a), -(b) et g is undefined, eg is unstable, et is stable. in in figure 6 (c), -(d), eg remains unstable but we have bistability between et g and et : it is a case of pitchfork bifurcation. in figure 6 (d), -(f), et g becomes unstable and we have a bistability between et and eg: it is a case of transcritical bifurcation. values of r01, r 0 2, r g 1 , r t ns 2 and r are given in appendix f. 0 50 100 150 200 0 10 20 30 40 g (t ), t s (t ), t n s (t ) (a). σ g =0 sensitive tree non sensitive tree grass 0 50 100 150 200 0 10 20 30 40 (b). σg =0.02 0 50 100 150 200 0 5 10 15 (c). σ g =0.04 0 50 100 150 200 250 300 0 2 4 6 8 time (year) g (t ), t s (t ), t n s (t ) (d). σ g =0.045 sensitive tree non sensitive tree grass 0 100 200 300 0 5 10 time (year) (e). σ g =0.05 0 100 200 300 0 5 10 time (year) (f). σ g =0.06 fig. 6. from forest to grassland, with a transition through a savanna state, as a function of the sensitive tree-grass competition parameter. from left to right, the fire period τ is fixed at 4, while the grass vs. sensitive-tree competition parameter σg increased. • the grass vs. non sensitive-tree competition parameter σns is a bifurcation parameter. a shift of the convergence of system (1) from the grassland state to the forest state as a function of the grass vs. non sensitive-tree competition parameter σns is depicted in fig. 7. we choose γs γns γg µs µns µg σg ηs f 0.4 1 4 0.1 0.3 0.1 0.05 0.5 yr−1 ηg = 0.5, kt =45, kg = 10 for these parameters values, system (1) exhibits a pitchfork bifurcation. we move from ligne 5 (figure 7 (a), -(b)) to ligne 7 (figure 7 (c), -(d)) of table i. indeed, in figure 7 (a), -(b) et g is undefined, et is unstable, eg is stable. in figure 7 (c), -(d), et g exits but it is unstable and we have bistability between eg and et : it is a case of pitchfork bifurcation. values of r01, r 0 2, r g 1 , r t ns 2 and r are given in appendix g. 0 100 200 300 0 5 10 15 20 25 time (year) g (t ), t s (t ), t n s (t ) (a). τ =0.5, σ ns =0.5 0 100 200 300 0 5 10 15 20 25 30 time (year) g (t ), t s (t ), t n s (t ) (b). τ =0.5, σ ns =0.6 0 50 100 150 0 5 10 15 20 25 30 time (year) g (t ), t s (t ), t n s (t ) (c). τ =0.5, σ ns =0.65 0 50 100 150 0 5 10 15 20 25 30 time (year) g (t ), t s (t ), t n s (t ) (d). τ =0.5, σ ns =0.7 sensitive tree non sensitive tree grass sensitive tree non sensitive tree grass sensitive tree non sensitive tree grass sensitive tree non sensitive tree grass fig. 7. from grassland to forest as a function of the grass vs. non sensitive-tree competition parameter σns . from left to right, the fire period τ is fixed, while the grass vs. non sensitive-tree competition parameter σns increases. v. conclusion and discussion in this work, we present and analyze a new mathematical model to study the interaction of tree and grass that explicitly makes fire intensity dependent on the grass biomass and distinguishes two levels of fire sensitivity within the woody biomass (implicitly relating to plant size and bark thickness). fire was considered as a timecontinuous forcing as in several existing models (langevelde et al. 2003, accatino et al. 2010, de michele et al. 2011 and reference therein) with a constant frequency of fire return that can be interpreted as mainly expressing an external forcing to the tree-grass system from climate and human practices. what is novel in our model is that fire impact on tree biomass is modeled as a non-linear function w of the grass biomass. using a non-linear function is to our knowledge only found in staver et al. 2011. but this latter model made peculiar assumptions and does not predict grassland and forest as possible equilibria (only desert and savanna). the advantage of a non-linear function is that it can account for the absence of fire at low biomass. as a consequence and biomath 3 (2014), 1404212, http://dx.doi.org/10.11145/j.biomath.2014.04.212 page 10 of 18 http://dx.doi.org/10.11145/j.biomath.2014.04.212 v yatat et al., mathematical analysis of a size structured tree-grass... although keeping the same modeling paradigm as in langevelde et al. 2003, accatino et al. 2010, de michele et al. 2011, we reached different results and predictions. distinguishing fire sensitive vs. fire insensitive woody biomass lead to three variables expressing fractions of the above ground phytomass, namely grass and both fire-sensitive and insensitive woody vegetation. it featured three coupled, non-linear ordinary differential equations.as several existing models (baudena et al. 2010, staver et al. 2011), our model acknowledges two major phenomena that regulate savanna dynamics, namely the fire-mediated negative feedback of grasses onto sensitive trees and the negative feedback of grown-up, fire insensitive trees on grasses. we therefore explicitly model the asymmetric nature of tree-grass competitive interactions in fireprone savannas. the analytical study of the model reveals three possible equilibria excluding tree-grass coexistence (desert, grassland, forest) along with equilibria for which woody and grassy components show durable coexistence (i.e. savanna vegetation). the number of such equilibrium points depends on the function used to model the increase of fire intensity with grass biomass(see remark 2); for our model, we can have at most three savanna equilibria. we identified four ecologically meaningful thresholds that defined in parameter space regions of monostability, bistability as in accatino et al. 2010, de michele et al. 2011 and tristability with respect to the equilibria. tristability of equilibria may mean that shifts from one stable state to another may often be less spectacular that hypothesized from previous models and that scenarios of vegetation changes may be more complex. the model features some parameters that have been analytically identified as liable to trigger bifurcations (i.e., the state variables of the model converges to different steady states), notably parameters σns and σg of asymmetric competition that embody the depressing influence of fire insensitive trees on grasses and of grasses on sensitive woody biomass respectively. since treegrass asymmetric competition is largely mediated by fire, this finding of the role of those two parameters is not intuitive and is the result of the modeling effort and of the analytical analysis. since such parameters that quantify direct interactions between woody and grassy components appear crucial to understand the tree-grass dynamics in savanna ecosystems and for enhanced parameter assessment, they could be the focus of straightforward field experiments that would not request manipulating fire regime. another bifurcation parameter is the fire frequency, f , (or fire period parameter τ = 1f ) which has been assumed to be an external forcing parameter that integrates both climatic and human influences. frequent fires preclude tree-grass coexistence and turn savannas into grasslands. in the wettest situations, or under subequatorial climates, very high fire frequencies (above one fire per year) seem to be needed to prevent the progression of forests over savannas (unpublished data of experiments carry out at la lopé national park in gabon). however, it is questionable to model fire as a continuous forcing that regularly removes fractions of fire sensitive biomass. indeed, several months can past between two successive fires, such that fire may be considered as an instantaneous perturbation of the savanna ecosystem. several recent papers have proposed to model fires as stochastic events while keeping the continuoustime differential equation framework (beckage et al. 2011) or using time discrete matrix models (accatino & de michele 2013). but in all those examples, fire characteristics remain mainly a linear function of grass biomass. another framework that we will explore in a forthcoming work in order to acknowledge the discrete nature of fire events is based on system of impulsive differential equations (lakshmikantham et al. 1989, bainov and simeonov 1993). references [1] f. accatino, c. de michele, humid savanna-forest dynamics: a matrix model with vegetation-fire interactions and seasonality. eco. mod. 265, pp. 170-179, 2013. biomath 3 (2014), 1404212, 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[36] h. walter, ecology of tropical and subtropical vegetation. oliver and boyd, edinburgh, uk., 1971. appendix a: expressions of t∗s , t ∗ ns and g ∗ after straightforward but long computation, we show that t∗ns = γg σns 1 − 1 r02 − g∗ kg  , t∗s = µns ωs t∗ns , 1 kt ( 1 + ωs µns ) t∗s = 1 − 1 r01 − µns (σgg∗ + f ηs w(g∗)) γs µns + γns ωs , where g∗ is solution of w(g) = ag + b = f(g), (11) with a = 1 f ηs ( γg(ωs + µns )(γs µns + γns ωs ) kg kt σns µns ωs −σg ) , b = (γs µns + γns ωs ) [ 1 − 1 r01 − γg (ωs +µns ) kt σns ωs ( 1 − 1 r02 )] f ηs µns . we summarize the problem of existence of solutions of equation (11) in the following table table iv existence of solutions of equation (11) a b number of solutions > 0 > 0 0 or 2 solutions < 0 1 or 3 solutions < 0 > 0 1 solution < 0 0 solution note that solutions g∗ of (11) that give rise to savanna equilibria must satisfy 0 < g∗ < kg 1 − 1 r02  . appendix b: proof of theorem 1 let r00 = γs µs + ωs + µns . in a matrical writing, system (1) reads as dx dt = a(x)x < amax(x)x, (12) with x = (ts , tns , g) ∈ r3+, a(x) = (ai j)1≤i, j≤3 with a11 = γs ( 1 − ts +tnskt ) − (µs + ωs + σgg + f ηs w(g)), a12 = γns ( 1 − ts +tnskt ) , a13 = 0, a21 = ωs , a22 = −µns , a23 = 0, a31 = 0, a32 = 0, a33 = γg ( 1 − gkg ) −(σns tns + f ηg +µg). and amax(x) =  γs −µs −ωs γns 0 ωs −µns 0 0 0 γg 1 − 1 r02   =( a b c d ) , with a = ( γs −µs −ωs γns ωs −µns ) , b =( 0 0 ) , c = ( 0 0 ) , and d = γg 1 − 1 r02  . matrix amax(x) is a metzler matrix ( i.e all its offdiagonal terms are nonnegative) and α(amax(x)) ≤ 0 if α(a) ≤ 0 and α(d) ≤ 0 where α denotes the stability modulus. moreover, for matrix d, α(d) ≤ 0 if r 0 2 < 1. (13) biomath 3 (2014), 1404212, http://dx.doi.org/10.11145/j.biomath.2014.04.212 page 13 of 18 http://dx.doi.org/10.1016/j.tree.2003.09.002 http://dx.doi.org/10.11145/j.biomath.2014.04.212 v yatat et al., mathematical analysis of a size structured tree-grass... for matrix a, α(a) ≤ 0 if trace(a) < 0 and det(a) > 0. trace(a) = γs −µs −ωs −µns = γs 1 − 1 r00  . (14) det(a) = µns (µs + ωs ) − (µns γs + ωs γns ) = µns (µs + ωs )(1 −r01). (15) furthermore, r00 < r 0 1 (16) thus, from relations (13), (14), (15) and (16) we deduce that the desert equilibrium (0; 0; 0) is globally asymptotically stable whenever r01 < 1 and r02 < 1. appendix c: proof of theorem 2 if r01 > 1, then the forest equilibrium et exists. • using the jacobian matrix of system (1) at et , one can prove that et is locally asymptotically stable if rt ns2 < 1. • the solution g of system (1) verify dg dt ≤ (γg − ( f ηg + µg))g, ≤ γg 1 − 1 r02  . (17) so, if r02 < 1, then lim t→+∞ g(t) = 0. (18) moreover, the solutions ts and tns of system (1) admit as a limit system, the system: dts dt = (γs ts + γns tns ) ( 1 − ts + tns kt ) −ts (µs + ωs ) = f1(ts , tns ), dtns dt = ωs ts −µns tns = f2(ts ; tns ). (19) now, let h(ts , tns ) = t−1s . then, one has ∂f1 h ∂ts + ∂f2 h ∂tns = −γns tns t 2s ( 1 − ts +tnskt ) − 1 kt ts (γs ts + γns tns ) −µns t−1s . furthermore, we have ∂f1h ∂ts + ∂f2h ∂tns < 0 in ω◦2 where ω2 = {(ts , tns ) ∈ r2+ | 0 ≤ ts + tns ≤ kt}, and by the bendixson-dulac theorem, we deduce that system (19) don’t admits a periodic solution in ω2. moreover, the equilibrium (t̄s , t̄ns ) exists if r01 > 1 and using the jacobian matrix of system (19), we deduce that (t̄s , t̄ns ) is locally asymptotically stable and then, globally asymptotically stable since there is no periodic solution. thus, if r02 < 1, then one has lim t→+∞ (ts , tns , g)(t) = et . • suppose that r01 = γs µns + γns ωs µns (µs + ωs ) > 1 and r02 = γg f ηg + µg > 1, then equilibria (t̄, t̄ns , 0), (0, 0, ḡ) and (t ?s , t ? ns , g ?) are defined. the jacobian matrix of system (1) at an arbitrarily equilibrium point is j =  j11 j12 j13 j21 j22 0 0 j32 j33  , where j11 = γs ( 1 − x+ykt ) − 1 kt (γs x + γns y ) −µs −ωs −σgz − f ηs w(z), j12 = γns ( 1 − x+ykt ) − 1 kt (γs x + γns y ), j13 = −σg x − x f ηs w′(z), j21 = ωs , j22 = −µns , j32 = −σns z, j33 = γg − 2 γg kg z −σns y − f ηg −µg. the second additive compound matrix of j is j[2] =  j11 + j22 0 −j13 j32 j11 + j33 j12 0 j21 j22 + j33  . (20) from the jacobian matrix of system (1), the equilibria (0, 0, ḡ) and (t ?s , t ? ns , g ?) are unstable if rḡ2 > 1 and r < 1. in the sequel, we suppose that rt̄ns2 < 1 to process with the discussion. biomath 3 (2014), 1404212, http://dx.doi.org/10.11145/j.biomath.2014.04.212 page 14 of 18 http://dx.doi.org/10.11145/j.biomath.2014.04.212 v yatat et al., mathematical analysis of a size structured tree-grass... the second additive compound matrix (20) at the equilibrium (t̄s , t̄ns , 0) is j[2](t̄s , t̄ns , 0) = j11 + j22 0 σt̄s 0 j11 + j33 j12 0 j21 j22 + j33  (t̄s ,t̄ns ,0) . let b = ( j11 + j33 j12 j21 j22 + j33 ) . then, a simple calculation gives (j11 + j22)(t̄s ,t̄ns ,0) = γs ( 1 − t̄s + t̄ns kt ) − 1 kt (γs t̄s + γns t̄ns ) −µs −ωs −µns . using the relations −µs −ωs = − ( γs + γns ωs µns ) ( 1 − t̄s + t̄ns kt ) and ( 1 − t̄s + t̄ns kt ) > 0, we have (j11 + j22)(t̄s ,t̄ns ,0) = − 1 kt t̄s ( γs + γns ωs µns ) −γns ωs µns ( 1 − t̄s + t̄ns kt ) −µns < 0. since j11 + j22 < 0 and r t̄ns 2 < 1, one has tr(b) = (j11 + j22 + 2j33)(t̄s ;t̄ns ;0), = j11 + j22 + 2γg 1 − 1 r t̄ns 2  < 0. also, if rt̄ns2 < 1, one has j11 j33 = ( − γg kt t̄s ( γs + γns ωs µns ) −γgγns ωs µns ( 1 − t̄s +t̄nskt )) × ( 1 − 1 r t̄ns 2 ) > 0, and j33(j22 + j33) = γg 1 − 1 r t̄ns 2  −µns + γg 1 − 1 r t̄ns 2   > 0. with this in mind, we have det(b) = (j11 + j33)(j22 + j33) − j12 j21, = j11 j22 − j21 j12 + j11 j33 +j33(j22 + j33), = µns kt t̄s ( γs + γns ωs µns ) + ωs kt (γs t̄s + γns t̄ns ) +j11 j33 + j33(j22 + j33) > 0. thus, if rt̄ns2 < 1, one has (j11 + j22)(t̄s ,t̄ns ,0) < 0, tr(b) < 0 and det(b) > 0. this implies that s(j[2](t̄s , t̄ns , 0)) < 0 where s denotes the stability modulus. following theorem 3.3 in li and wang 1998, we can deduce that there is no hopf bifurcation points for j(t̄s , t̄ns , 0). since r t̄ns 2 < 1, the equilibrium point (t̄s , t̄ns , 0) is locally asymptotically stable and one can conclude that this equilibrium point is globally asymptotically stable if rt̄ns2 < 1,r ḡ 2 > 1 and r < 1. this completes the proof. appendix d: proof of theorem 4 suppose that the savanna equilibrium et g exists. the jacobian matrix of system (1) at et g is j =  j11 j12 j13 j21 j22 0 0 j32 j33  , where j11 = γs ( 1 − ts +tnskt ) − 1 kt (γs ts + γns tns ) −µs −ωs −σgg − f ηs w(g), j12 = γns ( 1 − ts +tnskt ) − 1 kt (γs ts + γns tns ), j13 = −σgts − ts f ηs w′(g), j21 = ωs , j22 = −µns , j32 = −σns g, j33 = − γg kg g. let a1 = −j11 j22 j33, a2 = j21 j12 j33, a3 = −j21 j32 j13, c1 = −j11 − j22 − j33, c2 = a1 + a2 + a3, c3 = j11 j33 + j11 j22 − j21 j12 + j22 j33 − c2 c1 . biomath 3 (2014), 1404212, http://dx.doi.org/10.11145/j.biomath.2014.04.212 page 15 of 18 http://dx.doi.org/10.11145/j.biomath.2014.04.212 v yatat et al., mathematical analysis of a size structured tree-grass... note that, by the routh-hurwitz theorem, the savanna equilibrium et g is locally asymptotically stable if c1 > 0, c2 > 0 and c3 > 0. moreover, components of the savanna equilibrium et g satisfy tns = ωs µns ts , −µs −ωs −σgg − f ηs w(g) = − ( γs + γns ωs µns ) ×( 1 − ts +tnskt ) , thus, c1 = −j11 − j22 − j33, = 1kt (γs ts + γns tns ) + γns ωs µns ( 1 − ts +tnskt ) +µns + γg kg g, > 0. c3 = j11 j33 + j11 j22 − j21 j12 + j22 j33 − c2 c1 , = 1kt (µns + ωs )(γs ts + γns tns ) + γgg kg ( 1 kt (γs ts + γns tns ) +γns ωs µns ( 1 − ts +tnskt ) + µns ) − c2 c1 , = 1kt (µns + ωs )(γs ts + γns tns ) − γg g kg kt (µns +ωs )(γs ts +γns tns ) c1 + ωs σns g(σg ts f ηs w′(g)ts ) c1 + γgg kg ( 1 kt (γs ts + γns tns ) +γns ωs µns ( 1 − ts +tnskt ) + µns ) , = 1kt (µns + ωs )(γs ts + γns tns ) ( 1 − γg g kg c1 ) + ωs σns g(σg ts f ηs w′(g)ts ) c1 + γgg kg ( 1 kt (γs ts + γns tns ) +γns ωs µns ( 1 − ts +tnskt ) + µns ) , = 1kt c1 (µns + ωs )(γs ts + γns tns )×( 1 kt (γs ts + γns tns ) +γns ωs µns ( 1 − ts +tnskt ) + µns ) + ωs σns g(σg ts f ηs w′(g)ts ) c1 + γgg kg ( 1 kt (γs ts +γns tns ) + γns ωs µns ( 1 − ts +tnskt ) + µns ) , c3 > 0. c2 = a1 + a2 + a3, = γgg kg kt (µns + ωs )(γs ts + γns tns ) −ωs σns g(σgts + f ηs w′(g)), = ωs σns gts ( γg kg kt ωs σns (µns + ωs )×( γs + γns ωs µns ) −σg − f ηs w′(g) ) , = ωs σns gts (σg + f ηs w′(g))(r− 1). thus, c2 > 0 if and only if r > 1. finally, we deduce that the savanna equilibrium et g, when it is unique, is locally asymptotically stable if r = r(g) > 1. the first part of theorem 4 holds. one should note that c2 > 0 means that the slope of w (the sigmoidal function) is less than the slope of f where f is given by (11). furthermore, by using relation (11) we deduce part 2 and part 3 of theorem 4 graphically as follow fig. 8. there exist two savanna equilibria but one is stable and the other is unstable. fig. 9. there exist three savanna equilibria two are stable and one is unstable. thus system (1) will converge to one of the two stable equilibria depending on initial conditions. biomath 3 (2014), 1404212, http://dx.doi.org/10.11145/j.biomath.2014.04.212 page 16 of 18 http://dx.doi.org/10.11145/j.biomath.2014.04.212 v yatat et al., mathematical analysis of a size structured tree-grass... 0 5 10 15 20 25 30 35 40 45 50 −5 0 5 10 15 20 time (year) g (t ), t s (t ), t n s (t ) runge kutta order (2,3): ode23 sensitive tree non sensitive tree grass 0 5 10 15 20 25 30 35 40 45 50 0 2 4 6 8 10 12 14 16 18 20 time (year) g (t ), t s (t ), t n s (t ) nonstandard numerical scheme sensitive tree non sensitive tree grass fig. 10. for these figure, µg = 0.4. the ode’s routine which is a standard numerical algorithms shows a spurious negative solutions. appendix e: spurious behaviors of runge kutta methods to approximate solutions of system (1) for the following figures we choose γs γns γg ηs ηg 0.1 2 0.6 0.5 0.5 µs µns µg σns σg 0.3 0.3 0.02 0.05 g0 ωs kt kg f 2 0.05 50 12 0.5 other examples of spurious solutions given by standard methods are also given in (anguelov et al. 2009). appendix f: values ofr01, r 0 2, r g 1 , r t ns 2 andr in figure 6 • figure 6 (a): r01 r 0 2 r g 1 r t ns 2 4.8889 4.9412 2.6928 0.9863 • figure 6 (b): 0 50 100 150 −5 0 5 10 15 20 time (year) g (t ), t s (t ), t n s (t ) runge kutta order (4,5): ode45 sensitive tree non sensitive tree grass 0 50 100 150 0 2 4 6 8 10 12 14 16 18 20 time (year) g (t ), t s (t ), t n s (t ) nonstandard numerical scheme sensitive tree non sensitive tree grass fig. 11. for these figure, µg = 0.5. the ode’s routine which is a standard numerical algorithms shows again a spurious negative solutions. r01 r 0 2 r g 1 r t ns 2 4.8889 4.9412 1.5813 0.9861 • figure 6 (c): r01 r 0 2 r g 1 r t ns 2 r 4.8889 4.9412 1.1193 0.9861 1.3984 • figure 6 (d): r01 r 0 2 r g 1 r t ns 2 r 4.8889 4.9412 1.0431 0.9861 1.2980 • figure 6 (e): r01 r 0 2 r g 1 r t ns 2 r 4.8889 4.9412 0.9766 0.9861 0.5936 • figure 6 (f): r01 r 0 2 r g 1 r t ns 2 r 4.8889 4.9412 0.8662 0.9861 0.5746 biomath 3 (2014), 1404212, http://dx.doi.org/10.11145/j.biomath.2014.04.212 page 17 of 18 http://dx.doi.org/10.11145/j.biomath.2014.04.212 v yatat et al., mathematical analysis of a size structured tree-grass... appendix g: values ofr01, r 0 2, r g 1 , r t ns 2 andr in figure 7 • figure 7 (a): r01 r 0 2 r g 1 r t ns 2 3.7778 3.6364 0.3840 1.1549 • figure 7 (b): r01 r 0 2 r g 1 r t ns 2 3.7778 3.6364 0.3840 1.0162 • figure 7 (c): r01 r 0 2 r g 1 r t ns 2 r 3.7778 3.6364 0.3840 0.9587 0.2066 • figure 7 (d): r01 r 0 2 r g 1 r t ns 2 r 3.7778 3.6364 0.3840 0.9073 0.1453 biomath 3 (2014), 1404212, http://dx.doi.org/10.11145/j.biomath.2014.04.212 page 18 of 18 http://dx.doi.org/10.11145/j.biomath.2014.04.212 introduction the continuous fire model of asymmetric tree-grass competition (cofac) mathematical analysis existence of equilibria, ecological thresholds and stability analysis existence of equilibria ecological thresholds interpretation stability analysis summary table of the qualitative analysis numerical simulations a nonstandard scheme for the cofac numerical simulations and bifurcation parameters some monostability and bistability situations some bifurcation parameters conclusion and discussion references bibliographie original article biomath 3 (2014), 1404161, 1–16 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum mathematical modelling and optimal control of anthracnose david fotsa1, elvis houpa2, david bekolle2, christopher thron3 and michel ndoumbe4 1ensai, the university of ngaoundere, email: mjdavidfotsa@gmail.com 2faculty of science, the university of ngaoundere email: e-houpa@yahoo.com, bekolle@yahoo.fr 3texas a&m university, central texas email:thron@ct.tamus.edu 4irad, cameroon email: michel.ndoumbe@yahoo.com received: 19 november 2013, accepted: 16 april 2014, published: 28 may 2014 abstract—in this paper we propose two nonlinear models for the control of anthracnose disease. the first is an ordinary differential equation (ode) model which represents the within-host evolution of the disease. the second includes spatial diffusion of the disease in a bounded domain. we demonstrate the well-posedness of those models by verifying the existence of solutions for given initial conditions and positive invariance of the positive cone. by considering a quadratic cost functional and applying a maximum principle, we construct a feedback optimal control for the ode model which is evaluated through numerical simulations with the scientific software scilab r©. for the diffusion model we establish under some conditions the existence of a unique optimal control with respect to a generalized version of the cost functional mentioned above. we also provide a characterization for this optimal control. keywords— anthracnose modelling, nonlinear systems, optimal control. ams classification— 49j20, 49j15, 92d30, 92d40. i. introduction anthracnose is a phytopathology which attacks a wide range of commercial crops, including almond, mango, banana, blueberry, cherry, citrus, coffee, hevea and strawberry. the disease has been identified in such diverse areas as ceylon (1923), guadeloupe (1925), sumatra (1929), indochina (1930), costa rica (1931), malaysia (1932), java (1933), madagascar (1934), cameroon (1934), colombia (1940), salvador (1944), brazil (1946), nyassaland (1949), new caledonia (1954), and arabia (1956) [5]. anthracnose can affect various parts of the plant, including leaves, fruits, twigs and roots. possible symptoms include defoliation, fruit rot, fruit fall and crown root rot, which can occur before or after harvest depending on both pathogen and host [5], [28]. the anthracnose pathogen belongs to the colletotrichum species (acutatum, capsici, gloeosporioides, kahawae, lindemuthianum, musae, ...). colletotrichum is an ascomycete fungus. it can citation: david fotsa, elvis houpa, david bekolle, christopher thron, michel ndoumbe, mathematical modelling and optimal control of anthracnose, biomath 3 (2014), 1404161, http://dx.doi.org/10.11145/j.biomath.2014.04.161 page 1 of 16 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2014.04.161 d fotsa et al., mathematical modelling and optimal control of anthracnose fig. 1. symptoms of coffee berry disease (cbd) [4] reproduce either asexually or sexually, but sexual reproduction is rare in nature [28]. favourable growth conditions occur particularly in tropical zones. rainfall, wetness and altitude are all conducive to sporulation and conidia spreading [18], [23]. sources of inoculum are thought to be leaves, buds and mummified fruits. a. anthracnose pathosystem the process of infection by colletotrichum species can usually be divided into at least seven steps, depending on various factors including growth conditions, host tissues and involved species. conidia deposited on the host attach themselve on its surface. the conidia germinate after 12–48 hours, and appressoria are produced [5], [15]. severals studies on infection chronology show that appressoria production can occur between 3–48 hours following germination under favourable conditions of wetness and temperature [18], [28]. the pathogen then penetrates the plant epidermis, invades plant tissues, produces acevuli and finally sporulates. the penetration of plant epidermis is enabled by a narrow penetration peg that emerges from the appressorium base [7]. in some marginal cases penetration occurs through plant tissues’ stomata or wounds. once the cuticle is crossed, two infection strategies can be distiguished: intracellular hemibiotrophy and subcuticular intramural necrotrophy, as shown in figure 2. invasion of the host is led through formation of hyphae which narrow as the infection progresses. colletotrichum produce enzymes that degrade carbohydrates, dissolve cell walls, and hydrolyze cuticle. some of those enzymes are polyglacturonases, pectin lyases and proteases. some hosts may employ various biochemical strategies to counter the pathogen. for example, the peel of unripe avocados has been found in vitro to contain a preformed antifungal diene (cis, cis-1-acetoxy2-hydroxy-4-oxo-heneicosa-12, 15-diene) that inhibits the growth of colletotrichum gloeosporioides when present above a certain concentration [28]. fig. 2. infection strategies. (a)=apressorium (c)=conidium (cu)=cuticle (e)=epidermal (ils)=internal light spot (m)=mesophyl cell (n)=necrotrophic (ph)=primary hyphae (pp)=penetration peg (sch)=subcuticular and intramural hyphae (sh)=secondary hyphae [28] b. models in the literature most previous mathematical studies on colletotrichum-host pathosystem have focused on forecasting disease onset based on environmental factors affecting host sensitivity. danneberger et al. in [9] have developed a forecasting model for the annual bluegrass anthracnose severity index, using weather elements such as temperature and wetness. their model is a quadratic regression asi = a0 + a0,1w + a1,0t + a1,1t ×w + a0,2t 2 + a2,0w 2 where asi is the anthracnose severity index, t is the daily average temperature and w is the average number of hours of leaves’ wetness per biomath 3 (2014), 1404161, http://dx.doi.org/10.11145/j.biomath.2014.04.161 page 2 of 16 http://dx.doi.org/10.11145/j.biomath.2014.04.161 d fotsa et al., mathematical modelling and optimal control of anthracnose day. dodd et al. in [10] have studied the relationship between temperature (t), relative humidity (h), incubation period (t) and the percentage (p) of conidia of colletotrichum gloeosporioides producing pigmented appressoria on one month old mangoes. they used the following logistic model: ln (p/ (1 −p)) = a0 + a0,1h + a1,0t + a0,2h2 + a2,0t 2 + b ln (t) duthie in [12] examines the parasite’s response (r) to the combined effects of temperature (t) and wetness duration (w). that response could be the rate of germination, infection efficiency, latent period, lesion density, disease incidence or disease severity. several models are discussed, the two principal being r (t,w) = f (t) [ 1 − exp ( − [b (w − c)]d )] and r (t,w) = a [ 1 − exp ( − [f (t) (w − c)]d )] , where f (t) = e (1 + h) h h 1+h (1 + exp (g [t −f])) exp ( g [t −f] 1 + h ) and a > 0, b > 0, w ≥ c ≥ 0, d > 0, e > 0, f ≥ 0, g > 0, h > 0. mouen et al. attempt in [17] to develop a spatiotemporal model to analyse infection behaviour with respect to the time, and identify potential foci for disease inoculum. logistic regression and kriging tools are used used. in addition to these references, there are several other statistical models in literature [12], [17], [18], [19], [20], [21], [28]. c. controlling anthracnose there are many approaches to controlling anthracnose diseases. the genetic approach involves selection or synthesis of more resistant cultivars [3], [4], [5], [14], [27]. several studies have demonstrated the impact of cultivational practices on disease dynamics [19], [20], [21], [28]. other tactics may be used to reduce predisposition and enhance resistance, such as pruning old infected twigs, removing mummified fruits, and shading [5]. biological control uses microorganisms or biological substrates which interact with pathogen or induce resistance in the host [11]. finally there is chemical control, which requires the periodic application of antifungal compounds [5], [22], [24]. this seems to be the most reliable method, though relatively expensive. the best control policy should schedule different approaches to optimize quality, quantity and cost of production. note that inadequate application of treatments could induce resistance in the pathogen [26]. d. organization of the paper the remainder of this paper is organized as follows. in section ii we propose and study a within-host model of anthracnose. we present that model and give parameters meaning in subsection ii-a. throughout subsection ii-b we establish the well-posedness of the within-host model both in mathematical and epidemiological senses. the optimal control of the model is surveyed in subsection ii-c and numerical simulations are performed in the last subsection ii-d. we make a similar study on a spatial version of the model includind a diffusion term in section iii. that last model is presented in subsection iii-a. studies on its well-posedness and its optimal control are made respectively in subsections iii-b and iii-c. finally, in section iv we discuss our modelling and some realistic generalizations which could be added to the model. ii. a within-host model a. specification of the within-host model the detrimental effects of colletotrichum infection on fruit growth are closely related to its life cycle. it is mathematically convenient to express these effects in terms of the effective inhibition rate (denoted by θ), which is a continuous function of time. the effective inhibition rate is defined such that the maximum attainable fruit volume is reduced by a factor 1 − θ if current infection conditions are maintained. in addition to θ, the other time-dependent variables in the model are biomath 3 (2014), 1404161, http://dx.doi.org/10.11145/j.biomath.2014.04.161 page 3 of 16 http://dx.doi.org/10.11145/j.biomath.2014.04.161 d fotsa et al., mathematical modelling and optimal control of anthracnose host fruit total volume and infected volume, denoted by v and vr respectively. we have on the set s = r+ \{1}×r∗+ ×r+ the following equations for the time-evolution of the variables (θ,v,vr):   dθ/dt = α (t,θ) (1 −θ/ (1 −θ 1 u (t))) dv/dt = β (t,θ) (1 −vθ 2 / ((1 −θ) η (t) vmax)) dvr/dt = γ (t,θ) (1 −vr/v) (1) the parameters in (1) have the following practical interpretations: • α,β,γ characterize the effects of environmental and climatic conditions on the rate of change of inhibition rate, fruit volume, and infected fruit volume respectively. these are all positive functions of the time t and inhibition rate θ. • γ is an increasing function with respect to θ and satisfies γ (t, 0) = 0, ∀t ≥ 0. • u is a measurable control parameter which takes values in the set [0, 1]. • 1 − θ 1 ∈ [0, 1] is the inhibition rate corresponding to epidermis penetration. once the epidermis has been penetrated, the inhibition rate cannot fall below this value, even under maximum control effort. in the absence of control effort (u(t) = 0), the inhibition rate increases towards 1. • η is a function of time that characterizes the effects of environmental and climatic conditions on the maximum fruit volume. its range is the interval ]0,θ 2 ]. • vmax represents the maximum size of the fruit. • 1 − θ 2 ∈ [0, 1] is the value of inhibition rate θ that corresponds to a limiting fruit volume of ηvmax. according to the second equation in (1), the limiting volume size is ηvmax (1 −θ) /θ2 ≤ vmax. when the volume is less than this value, it increases (but never passes the limiting value); while if the volume exceeds this value, then it decreases. this limiting value for v is less than ηvmax when θ > 1 −θ 2 (note η ≤ θ2 ≤ 1). note that equations (1) are constructed so that v ≤ vmax and vr ≤ v as long as initial conditions satisfy these inequailities. with the definitions a ≡   −α(t,θ) (1−θ1u(t)) 0 0 0 − θ2β(t,θ) ((1−θ)η(t)vmax) 0 0 0 −γ(t,θ) v   , b ≡ [ α (t,θ) β (t,θ) γ (t,θ) ]t , and x ≡ [ θ v vr ]t , then model (1) can be reformulated as dx/dt = f (t,x) , (2) where f (t,x) ≡ a (t,x,u) x + b (t,x) . (3) as indicated above, model (1) is an exclusively within-host model, and as such does not include the effects of spreading from host to host. (in section iii we propose a diffusion model for between-host spreading.) such a model has several practical advantages. in practice, monitoring of the spreading of the fungi population is difficult. furthermore, conidia sources and spreading mechanisms are not well-understood, although the literature generally points to mummified fruits, leaves and bark as sources of inoculum. instead of controlling the host-to-host transmission, an alternative control method is to slow down the withinhost fungi evolution process. such an approach enables the use of statistical methods, since large samples of infected hosts may easily be obtained [15]. b. well-posedness of the within-host model in the following discussion, we demonstrate that model (1) is well-posed both mathematically and epidemiologically, under the following standard technical assumptions: (h1) the control parameter u is measurable. (h2) the function f is continuous with respect to the variable x. biomath 3 (2014), 1404161, http://dx.doi.org/10.11145/j.biomath.2014.04.161 page 4 of 16 http://dx.doi.org/10.11145/j.biomath.2014.04.161 d fotsa et al., mathematical modelling and optimal control of anthracnose (h3) for every compact subset k ⊂ s, there is an integrable map mk : r+ → r+ such that for every x in k and t in r+, ‖f (t,x)‖s ≤ mk (t). existence of a solution is guaranteed by the following proposition, which follows from a simple application of the carathéodory theorem. proposition 1. for every initial condition (t0,x0) in r+×s there is a function x (t0,x0, t) which is absolutely continuous and satisfies (2) for almost any time t ∈ r+. uniqueness and smoothness of the solution may be established using the cauchy-lipschitz theorem, based on properties (h2) and (h3) of the function f . next we will etablish positive invariance of the set s, and the positive invariance of a bounded subset bs. these results are needed to show consistency of the biological interpretation of the solution, as explained below. with the definitions a1 ≡  − α(t,θ) (1−θ1u(t)) 0 0 0 − θ2 ((1−θ)η(t)vmax) 0 0 0 −1 v   , a2 ≡  α (t,θ) 0 00 β (t,θ) 0 0 0 γ (t,θ)   , b1 ≡ [ 1 1 1 ]t , and x as defined above, then model (1) can be reformulated as dx/dt = a2 (a1x + b1) . (4) theorem 2. the set s is positively invariant for the system (4). proof: a solution to (4) satisfies for every time t ≥ 0, x (t) = exp [∫ t 0 a2 (s) ·a1 (s) ds ] x (0) + ∫ t 0 exp [∫ t s a2 (ξ) a1 (ξ) dξ ] a2 (s) b1ds since −a2 (s) a1 (s) is a m−matrix for every time s ≥ 0, exp [∫ t s a2 (ξ) a1 (ξ) dξ ] is a positive matrix. moreover, since b1 is nonnegtive, one can conclude that x remain nonnegative when x (0) is taken nonnegative. theorem 3. let bs be the subset of s defined such as bs = { (θ,v,vr) ∈ r3; 0 ≤ θ < 1, 0 < v ≤ vmax, 0 ≤ vr ≤ v } then bs is positively invariant for system (4). proof: we will show that at each point of the boundary of bs, the system (4) returns into bs. we prove this by showing that the scalar product of the system time derivative with the normal vector n at each boundary point is nonpositive. it has been already shown that positive orthant is positively invariant. let f1 ≡{(θ,v,vr) ∈ bs; θ = 1} f2 ≡{(θ,v,vr) ∈ bs; v = vmax} f3 ≡{(θ,v,vr) ∈ bs; vr = v} for all points on f1, n can be choosen as (1, 0, 0). since the control u takes its value in [0, 1] which also contains θ1, dθdt is negative and the result is obtained. for all points on f2, n can be choosen as (0, 1, 0). thanks to definition of θ2 and η, dvdt is negative and the result is obtained. for all points on f3, n can be choosen as (0,−1, 1). dvrdt is zero, and consequently f3 is positively invariant. the invariance of the set f3 is biologically plausible, since once the fruit is totally rotten it remain definitely in that state, the fruit is lost. the set bs is also reasonable for biological reasons: the inhibition rate is bounded, the rotten volume is no larger than the total volume, and fungus attack reduces the size of a mature fruit. c. optimal control of the within-host model in this subsection we apply control to model (1), which we repeat here for convenience:  dθ/dt = α (t,θ) (1 −θ/ (1 −θ 1 u (t))) dv/dt = β (t,θ) (1 −vθ 2 / ((1 −θ) η (t) vmax)) dvr/dt = γ (t,θ) (1 −vr/v) (5) biomath 3 (2014), 1404161, http://dx.doi.org/10.11145/j.biomath.2014.04.161 page 5 of 16 http://dx.doi.org/10.11145/j.biomath.2014.04.161 d fotsa et al., mathematical modelling and optimal control of anthracnose for the control problem we focus on the first equation. this equation is controllable in ]0, 1[ since θ is continuous and 1−θ 1 u (t) is an asymptotic threshold which can be set easily. giving a time t > 0 (for example the annual production duration) we search for u in l2loc (r+, [0, 1]) such that the following functional (previously used in [1], [13]) is minimized: jt (u) = ∫ t 0 ( ku2 (t) + θ2 (t) ) dt + f (θ (t)) , where k > 0 can be interpreted as the cost ratio related to the use of control effort u. this functional reflects the fact that reducing inhibition rate θ will lead to increased fruit production (larger volumes with a relatively lower level of infection), while minimizing u will reduce financial and environmental costs. we use the squares of u and θ in the integrand because this choice facilitates the technical calculations required for minimization. we note in passing that we could had tried to minimize the more practically relevant expression : ∫ t 0 (ku (t) + θ (t) + (vmax −v (t)) + vr (t)) dt + θ (t) + (vmax −v (t)) + vr (t) however, an exact computation of this functional would require precise expressions for α,β,γ,η in the system (1). as far as the authors know, there is no previous study which gives those parameters. it seemed more advantageous to us to limit the random choice of parameters, so that we could perform representative simulations. we define the set uk ≡ { u ∈ c ([0,t] ; [0, 1]) ;∀t,s ∈ [0,t] , |u (t) −u (s)| ≤ k |t−s| } which is nonempty for every k ≥ 0. theorem 4. let k ≥ 0. there is a control u∗ ∈ uk which minimizes the cost jt . proof: since jt ≥ 0 it is bounded below. let the infinimum be j∗, and let (un)n∈n be a sequence in uk such that (jt (un))n∈n converges to j∗. the definition of uk implies that (un)n∈n is bounded and uniformly equicontinuous on [0,t]. by the ascoli theorem, there is a subsequence (unk ) which converges to a control u ∗. since the cost function is continuous with respect to u it follows that jt (u∗) = j∗. theorem 5. suppose that α depends only on time. if there is an optimal control strategy u which minimizes jt , then u is unique and satisfies u (t) = { 1 when 27αθ2 1 θp ≥ 8k w3(t)−1 θ 1 w3(t) when 27αθ2 1 θp < 8k (6) where w3 (t) is the element of [1, min{3/2, 1/ (1 −θ 1 )}] which is the nearest to the smallest nonnegative solution of the equation αθ2 1 θpw3 − 2kw + 2k = 0 and  dθ/dt = α (t,θ) (1 −θ/ (1 −θ 1 u (t))) dp/dt = α (t) p (t) / (1 −θ 1 u (t)) − 2θ θ (0) = θ0, θ (t) = θt , p(t) = f ′ (θt ) (7) proof: according the maximum principle, minimizing jt is equivalent to minimizing the hamiltonian functional h (t,θ,u) = ku2 (t) + θ2 (t) + f (θ (t)) +α (t) p (t) (1 −θ 1 u (t) −θ) / (1 −θ 1 u (t)) where the adjoint state p is the solution to the following problem{ dp/dt = α (t) p/ (1 −θ 1 u (t)) − 2θ p(t) = f ′ (θ (t)) (8) to simplify the expression, let w ≡ 1/ (1 −θ 1 u) ∈ [1, 1/ (1 −θ 1 u)] . (9) then the new equivalent functional to minimize is j1t (w) = ∫t 0 ( k ( w(t)−1 θ 1 w(t) )2 + θ2 (t) ) dt + f (θ (t)) ∂wh = 0 if and only if αθ2 1 θpw3 − 2kw + 2k = 0. (10) this equation has a unique nonpositive solution when 27αθ2 1 θp ≥ 8k. it has at least one nonnegative solution when 27αθ2 1 θp < 8k. we can choose w in the following way: w (t) = { 1 1−θ 1 when 27αθ2 1 θp ≥ 8k w3 (t) when 27αθ21 θp < 8k biomath 3 (2014), 1404161, http://dx.doi.org/10.11145/j.biomath.2014.04.161 page 6 of 16 http://dx.doi.org/10.11145/j.biomath.2014.04.161 d fotsa et al., mathematical modelling and optimal control of anthracnose where w3 (t) is the element of [1, min{3/2, 1/ (1 −θ 1 )}] that is the nearest to the smallest nonnegative solution of (10). it follows from the definition of w in (9) and algebraic rearrangement that the optimal control u is given by (6), where (p,θ) is a solution to the system (7). the uniqueness of u follows from the uniqueness of the solution of the system (7). d. computer simulations of the controlled withinhost model we performed simulations in order to demonstrate the practical controllability of the system, for these simulations we used an inhibition pressure of the form α (t) = a (t− b)2 (1 − cos (2πt/c)) , with b and c in [0, 1]. this function reflects the seasonality of empirically-based severity index models found in the literature [9], [10], [12]. the particular values used in the simulation were a = 4,b = 0.75, c = 0.2 and k = 1. we also took f (θ (t)) = θ (t), so that p (t) = f ′ (θ (t)) = 1. in this case and the shooting method can be used to numerically estimate the value of p0 which produces a solution to (8). fig1: optimum control effort over a one-year period θ0 = 0.2,θ1 = 1 − 0.4. fig2: evolution of inhibition rate over a one-year period with θ0 = 0.2,θ1 = 1 − 0.4. fig3: optimum control effort over a one-year period θ0 = 0.5,θ1 = 1 − 0.4. fig4: evolution of inhibition rate over a one-year period θ0 = 0.5,θ1 = 1 − 0.4. the above figures show how the control strategy adapts itself in response to inhibition pressure represented by α. figures 1-2 correspond to the case biomath 3 (2014), 1404161, http://dx.doi.org/10.11145/j.biomath.2014.04.161 page 7 of 16 http://dx.doi.org/10.11145/j.biomath.2014.04.161 d fotsa et al., mathematical modelling and optimal control of anthracnose of low initial effective inhibition rate (θ0 = 0.2, corresponding to a low initial level of infection), while figures 3-4 correspond to the case of high initial effective inhibition rate (θ0 = 0.5). the simulations also show in different cases the effectiveness of the optimal strategy as compared to taking no control action. regardless of whether the initial inhibition rate is above or below the threshold 1−θ1, the dynamics are sensitive to the control effort. iii. a diffusion model in this section, we will model the geographical spread of the disease via diffusive factors such as the movement of inoculum through ground water and wind. a. specification of the diffusion model we include the effect of diffusive factors on the spread of infection by adding a diffusion term to the within-host equation for θ from system (1). together with boundary conditions, the model is ∂θ/∂t = α (t,x,θ) (1 −θ/ (1 −θ 1 u (t,x))) + div (a (t,x,θ)∇θ) on r∗+ × ω, (11) 〈a (t,x,θ)∇θ,n〉 = 0 on r∗+ ×∂ω, (12) θ (0,x) = θ0 (x) ≥ 0 x ∈ ω, (13) where ω is an open bounded subset of r3 with a continuously differentiable boundary ∂ω, and θ 1 ∈ [0, 1[. for a given element (t,x,θ), a is a 3×3-matrix (aij (t,x,θ)). the functions α and aij are assumed to be nonnegative. since θ depends on (t,x), the functions α and aij can be identified with elements of the set c ( [0,t] ; h1 (ω) ) . the function u ∈ c ( [0,t] ; h1 (ω) ) designates the control, which takes its values in the set [0, 1]. (12) may be interpreted as a dependence of the flux of inoculum with respect to diffusion factors. in particular, when a is the identity matrix (14) could be interpreted as there is no flux between exterior and interior of the domain ω. practically, time can be subdivided into intervals on which parameters are approximately constant. we may thus study the system on each interval separately, and assume that all parameters are constant. we also assume that functions α and a do not depend on θ. this leads to the following simplified model, ∂θ/∂t = α (x) (1 −θ/ (1 −θ 1 u (x))) + div (a (x)∇θ) on ]0,t[ × ω, (14) 〈a (x)∇θ,n〉 = 0 on ]0,t[ ×∂ω, (15) θ (0,x) = θ0 (x) ≥ 0 x ∈ ω, (16) in order to formalize the model, we define the hilbert space e = { θ ∈ h2 (ω) ; θ satisfies (15) } provided with the inner product 〈f,g〉e = ∫ u (fg + 〈∇f,∇g〉 + ∆f.∆g) dx define also the linear unbounded operator £ : d (£) = e ⊂ l2 (ω) → l2 (ω) as £θ = αθ 1 −θ 1 u − div (a (x)∇θ) then equation (14) takes the following form ∂θ/∂t + £θ = α. (17) we also introduce the following condition, which we will use to ensure that the system has realistic solutions: (h4) there exists a constant c > 0 such that for almost every x ∈ ω, a (x) is symmetric, positive definite and 〈v,a (x) v〉≥ c 〈v,v〉 , ∀v. b. well-posedness of the diffusion model we are now ready to prove that our model has been a mathematically and epidemiologically wellposed. in other words, we show that exists a unique solution 0 ≤ θ(t,x) ≤ to the system (14) − (16). biomath 3 (2014), 1404161, http://dx.doi.org/10.11145/j.biomath.2014.04.161 page 8 of 16 http://dx.doi.org/10.11145/j.biomath.2014.04.161 d fotsa et al., mathematical modelling and optimal control of anthracnose this shall follow from the hille-yosida theorem1: but first we need the following proposition. proposition 6. if a (x) is a positive semidefinite bilinear form for almost every x ∈ ω, then the linear operator £ is monotone on e. moreover, if a (x) satisfies (h4), then £ is maximal. proof: (ii) to show £ is monotone, we let θ ∈ e and compute:∫ ω £θ ×θdx = ∫ ω ( αθ2/ (1 −θ 1 u) ) dx− ∫ ω div (a (x)∇θ) θdx = (1/ (1 −θ 1 u))‖θ √ α‖2l2(ω) − ∫ ω div (a (x)∇θ) θdx ≥− ∫ ∂ω 〈a (x)∇θ,n (x)〉θdx + ∫ ω 〈a (x)∇θ,∇θ〉dx ≥ 0. (ii) to show £ is maximal , we let f ∈ l2 (ω) and seek θ ∈ e such that θ+£θ = f. given ϕ ∈ e, we have∫ ω (θ + £θ) ×ϕdx = ∫ ω ϕθ (1 + α−θ 1 u) / (1 −θ 1 u) dx − ∫ ω div (a (x)∇θ) ϕdx = ∫ ω ϕθ (1 + α−θ 1 u) / (1 −θ 1 u) + ∫ ω 〈a (x)∇θ,∇ϕ〉dx − ∫ ∂ω 〈a (x)∇θ,n (x)〉ϕdx = ∫ ω ϕθ (1 + α−θ 1 u) / (1 −θ 1 u) + ∫ ω 〈a (x)∇θ,∇ϕ〉dx ≡ p (θ,ϕ) , where p is a symmetric continuous and coercive bilinear form on h1 (ω). the laxmilgram theorem2 implies that there is a unique θ ∈ h1 (ω) such that θ + £θ = f. 1see [6] p 185. 2see [6] using regularization methods similar those used in theorem 9.26 of [6], it follows that θ ∈ e. given that the linear operator £ is maximal monotone and θ0 is in e, then by the hilleyosida theorem there is a unique function θ ∈ c1 ( [0,t] ; l2 (ω) )⋂ c ([0,t] ; e) which satisfies (14)−(16), and ∀(t,x) ∈ [0,t]×ω we have θ (t,x) = (s£ (t) θ0) (x) + ∫ t 0 (s£ (t−s) α) (x) ds, where s£ (t) is the contraction semigroup generated by −£. now that we have established existence and uniqueness of the solution θ, we now prove that 0 ≤ θ(t,x) ≤ 1 for all (t,x) in the domain. assuming that a(x) satisfies the condition (h4). we define m ≡ inf ∂ω θ0, m ≡ max { sup ∂ω θ0, sup ω (1 −θ 1 u) } , v ≡ 1/ (1 −θ 1 u) . note that m ≤ 1 as long as θ0 ≤ 1 and 0 ≤ θ1u ≤ 1. let e+ designate the set of elements in e which are nonnegative almost everywhere on ω. the following theorem gives sufficient conditions under which the solution θ of (14) − (16) is bounded by m ≤ 1 and the positive cone e+ of e is positively invariant. theorem 7. if a (x)∇etvα = 0 for every (t,x) ∈ [0,t] × ω then for almost every x in ω, m ≤ etvαθ (t,x) , t ∈ [0,t] . (18) moreover if a (x)∇v = 0 for every (t,x) ∈ [0,t] × ω, then θ (t,x) ≤ m (19) in particular, (18) and (19) hold when α and u do not depend on the space variable x. proof: let g ∈ c1 (r) such that (i) g (s) = 0, ∀s ≤ 0, and biomath 3 (2014), 1404161, http://dx.doi.org/10.11145/j.biomath.2014.04.161 page 9 of 16 http://dx.doi.org/10.11145/j.biomath.2014.04.161 d fotsa et al., mathematical modelling and optimal control of anthracnose (ii) 0 < g′ (s) ≤ c, ∀s > 0. define h (s) ≡ ∫ s 0 g (σ) dσ,∀s ∈ r, ϕ1 (t) ≡ ∫ ω h ( m−etvαθ (t,x) ) dx, ϕ2 (t) ≡ ∫ ω h ( etvα (θ (t,x) − 1/v) ) dx. we observe that ϕ1,ϕ2 ∈ c ([0,t] ; r) ⋂ c1 (]0,t] ; r) , ϕ1,ϕ2 ≥ 0 on [0,t] , ϕ1 (0) = ϕ2 (0) = 0. we may also compute ϕ′1 (t) = − ∫ ω etvαg ( m−etvαθ ) (vαθ + ∂θ/∂t) dx = − ∫ ω etvαg ( m−etvαθ ) (α−£θ + vαθ) dx = − ∫ ω αg ( m−etvαθ ) dx + ∫ ω 〈 a (x)∇θ,∇etvαg ( m−etvαθ )〉 dx = − ∫ ω αg ( m−etvαθ ) dx − ∫ ω e2tvαg′ ( m−etvαθ ) 〈a (x)∇θ,∇θ〉dx + ∫ ω ( g ( m−etvαθ ) −etvαθg′ ( m−etvαθ )) × 〈 a (x)∇θ,∇etvα 〉 dx ≤ 0. since ϕ′1 ≤ 0 on r ∗ +, ϕ1 is identically zero on r+ and therefore almost everywhere in ω. m ≤ etvαθ (t,x) if a (x)∇v = 0 for every (t,x) ∈ [0,t]×ω, then ϕ′2 (t) = ∫ ω g ( etvα (θ − 1/v) ) (−α + vαθ + ∂θ/∂t) dx = ∫ ω etvαg ( etvα (θ − 1/v) ) (−£θ + vαθ) dx = − ∫ ω 〈 a (x)∇θ,∇etvαg ( etvα (θ − 1/v) )〉 dx = − ∫ ω e2tvαg′ ( etvα (θ − 1/v) ) 〈a (x)∇θ,∇θ〉dx − ∫ ω ( e2tvα/v2 ) g′ ( etvα (θ − 1/v) ) 〈a (x)∇θ,∇v〉dx − ∫ ω g ( etvα (θ − 1/v) )〈 a (x)∇θ,∇etvα 〉 dx − ∫ ω etvα (θ − 1/v) g′ ( etvα (θ − 1/v) ) × 〈 a (x)∇θ,∇etvα 〉 dx ≤ 0. since ϕ′2 ≤ 0 on r ∗ +, ϕ2 is identically zero on r+ and therefore almost everywhere in ω θ (t,x) ≤ m. the following theorem proves boundedness of θ under more general conditions. theorem 8. suppose that v and αv are increasing functions h and g of the state θ, and there is a constant k > 0 such that ag′ (θ) ≤ k ( 1 + ag′ (θ) ) exp (ag (θ)) , ∀a ≥ 0. (20) then for every time t ∈ [0,t] and almost every x in ω, m ≤ etvαθ (t,x) (21) and θ (t,x) ≤ m. (22) proof: let g ∈ c1 (r) such that (i) g (s) = 0, ∀s ≤ 0, and (ii) kg (s) ≤ g′ (s) ≤ c, ∀s > 0. using (20) and the fact that operator a is monotone, we have 〈a (x)∇θ,∇v〉 = 〈a (x)∇θ,∇h (θ)〉 = h′ (θ)〈a (x)∇θ,∇θ〉 ≥ 0, biomath 3 (2014), 1404161, http://dx.doi.org/10.11145/j.biomath.2014.04.161 page 10 of 16 http://dx.doi.org/10.11145/j.biomath.2014.04.161 d fotsa et al., mathematical modelling and optimal control of anthracnose〈 a (x)∇θ,∇etwα 〉 = 〈a (x)∇θ,∇exp (tg (θ))〉 = tg′ (θ) exp (tg (θ))〈a (x)∇θ,∇θ〉 ≥ 0, and〈 a (x)∇θ,∇etvαg ( m−etvαθ )〉 = 〈a (x)∇θ,∇exp (tg (θ)) g (m−θ exp (tg (θ)))〉 = tg′ (θ) exp (tg (θ)) g (m−θ exp (tg (θ)))〈a (x)∇θ,∇θ〉 −exp (2tg (θ)) (1 + tg′ (θ)) g′ (m−θ exp (tg (θ)))〈a (x)∇θ,∇θ〉 ≤ (tg′ (θ) −k (1 + tg′ (θ)) exp (tg (θ))) exp (tg (θ)) ×g (m−θ exp (tg (θ)))〈a (x)∇θ,∇θ〉 ≤ 0. define h (s) ≡ ∫ s 0 g (σ) dσ,∀s ∈ r, ϕ1 (t) ≡ ∫ ω h ( m−etvαθ (t,x) ) dx, ϕ2 (t) ≡ ∫ ω h ( etvα (θ (t,x) − 1/w) ) dx. note that as in theorem 7 we have ϕ1,ϕ2 ∈ c ([0,t] ; r) ⋂ c1 (]0,t] ; r) , ϕ1,ϕ2 ≥ 0 on [0,t] , ϕ1 (0) = ϕ2 (0) = 0. as in theorem 7 we may compute ϕ′1 (t) = − ∫ ω etvαg ( m−etvαθ ) (wαθ + ∂θ/∂t) dx = − ∫ ω etvαg ( m−etvαθ ) (α−£θ + vαθ) dx = − ∫ ω αg ( m−etvαθ ) dx + ∫ ω 〈 a (x)∇θ,∇etvαg ( m−etvαθ )〉 dx ≤ ∫ ω 〈 a (x)∇θ,∇etvαg ( m−etvαθ )〉 dx ≤ 0. since ϕ′1 ≤ 0 on r ∗ +, ϕ1 is identically null on [0,t] and therefore almost everywhere in ω m ≤ etvαθ (t,x) . we also have ϕ′2 (t) = ∫ ω g ( etvα (θ − 1/v) ) (−α + vαθ + ∂θ/∂t) dx = ∫ ω etvαg ( etvα (θ − 1/v) ) (−£θ + vαθ) dx = − ∫ ω 〈 a (x)∇θ,∇etvαg ( etvα (θ − 1/v) )〉 dx = − ∫ ω e2tvαg′ ( etvα (θ − 1/v) ) 〈a (x)∇θ,∇θ〉dx − ∫ ω ( e2tvα/v2 ) g′ ( etvα (θ − 1/v) ) 〈a (x)∇θ,∇v〉dx − ∫ ω g ( etvα (θ − 1/v) )〈 a (x)∇θ,∇etvα 〉 dx − ∫ ω etvα (θ − 1/w) g′ ( etvα (θ − 1/v) ) × 〈 a (x)∇θ,∇etvα 〉 dx ≤ 0. since ϕ′2 ≤ 0 on [0,t]\{0}, ϕ2 is identically null on [0,t] and therefore almost everywhere in ω θ (t,x) ≤ m. condition (20) of theorem 8 is satisfied in particular when g ≥ 0. using the same arguments as in the proof of proposition 6, there is a unique equilibrium θ∗, for the system (14) − (16). θ∗ is asymptotically stable if and only if all the eigenvalues of the linear operator £ have nonnegative real parts. stability of the equilibrium θ∗ has the advantage that the disease inhibition is maintained in its neighborhood, which enables easier control strategies. in particular, the norm of θ∗ is a decreasing function of the control u. proposition 9. the real number λ is not an eigenvalue of £ if at least one of the following conditions is satisfied: (i) α ≥ λ (1 −θ 1 u) almost everywhere in ω and that inequality is strict on an nonnegligible subset of ω. (ii) there exists a real k ≥ 0 such that for every θ ∈ e∫ ω 〈a (x)∇θ,∇θ〉dx ≥ k‖θ‖h2(ω) , and (α−λ (1 −θ 1 u)) / (1 −θ 1 u) > −k almost everywhere in ω. biomath 3 (2014), 1404161, http://dx.doi.org/10.11145/j.biomath.2014.04.161 page 11 of 16 http://dx.doi.org/10.11145/j.biomath.2014.04.161 d fotsa et al., mathematical modelling and optimal control of anthracnose proof: let θ,ϕ ∈ e. then we may compute∫ ω (£θ −λθ) ×ϕdx = ∫ ω ϕθ (α−λ (1 −θ 1 u)) / (1 −θ 1 u) dx − ∫ ω div (a (x)∇θ) ϕdx = ∫ ω ϕθ (α−λ (1 −θ 1 u)) / (1 −θ 1 u) + 〈a (x)∇θ,∇ϕ〉dx ≡ p1 (θ,ϕ) . if either of the two conditions of the proposition is satisfied, we may use the lax-milgram theorem to obtain the desired result. corollary 10. the principal spectrum of −£ is contained in d∗0 ≡{λ ∈ c ∗; re (λ) ≤ 0}. proof: from assumption (h4), £ is maximal monotone and s£ is a contraction semigroup. since s£ is a contraction semigroup, the resolvant set ρ (−£) of −£ contains r+ � {0}3 and ‖s£ (t)‖ ≤ 1, ∀t ∈ [0,t]. therefore, the spectral radius of s£ (t) is less than one. on the other hand 0 /∈ exp (tσp (−£)) ⊆ σp (s£ (t)) ⊆ {0} ⋃ exp (tσp (−£)) , ∀t ∈ [0,t]. clearly, if λ = re (λ) + i im (λ) is an element of the principal spectrum of −£ then exp (λt) is an element of the principal spectrum of s£ (t) and |exp (λt)| = exp (re (λ) t)‖s£ (t)‖ ≤ 1. it follows that re (λ) ≤ 0. corollary 11. the equilibrium θ∗ is stable. moreover if all the complex eigenvalues λ of the operator θ 7→ div (a (x)∇θ) satisfy α ≥ (1 −θ 1 u) re (λ) almost everywhere in ω, then θ∗ asymptotically stable. c. optimal control of the diffusion model in the previous section we have seen that the equilibrium of system (14) − (16) was conditionally asymptotically stable. whether or not the equilibrium θ∗ is asymptotically stable, the disease progression shall be contained with respect to some criteria given in terms of costs. the aim 3see theorem 3.1 in [25], p8. of this section is to control the system such that the following cost functional is minimized: j3t (u) = ∫t 0 ∫ ω ( θ2 + k1 (x) u 2 ) dxdt + ∫ ω k2 (x) θ 2 (t,x) dx, where k1 > 0,k2 ≥ 0 are bounded penalization terms. the function k1(x) can be interpreted as the cost ratio related to the use of control effort u; while k2 is the cost ratio related to the magnitude of the final inhibition rate θ (t, ·). in practice, k1 reflects the spatial dependence of environmental sensitivity to control means, while k2 reflects geographical variations in the cost of the inhibition rate of colletotrichum at the end of the control period. in order to establish the optimal control, we will first need to define uk,c as the set of controls u ∈ c ( [0,t] ; h1 (ω; [0, 1]) ) such that for every t,s ∈ [0,t] , ‖u (t, ·) −u (s, ·)‖h1(ω) ≤ k |t−s| and ‖∇u (t, ·)‖l2 ≤ c. for every k,c ≥ 0, u k,c is nonempty. theorem 12. let k,c ≥ 0. then there is a control v ∈ uk,c which minimizes the cost j3t . proof: since j3t is greater than zero it is bounded below. let that infinimum be j∗. there is a sequence (un)n∈n such that the sequence( j3t (un) ) n∈n converges to j ∗. using definition of uk,c the (un)n∈n is bounded and uniformly equicontinuous on [0,t]. by the ascoli theorem, there is a subsequence (unk ) which converges to a control v. since the cost function is continuous with respect to u it follows that j3t (v) = j ∗. we first look the linearized system in the neighborhood of (θ,u) = (ε, 0), where ε depends on x. indeed, this case is of practical significance since the monitoring is assumed to be continuous year-round, and the endemic period corresponds to particular conditions. thus the outbreak of the disease is ”observable” at the moment of onset. the linearized version of (14) is ∂θ/∂t = α−αθ −αεuθ 1 + div (a (x)∇θ) , on ]0,t[ × ω (23) note that if ε = 0 the linearized system is not controllable. biomath 3 (2014), 1404161, http://dx.doi.org/10.11145/j.biomath.2014.04.161 page 12 of 16 http://dx.doi.org/10.11145/j.biomath.2014.04.161 d fotsa et al., mathematical modelling and optimal control of anthracnose let £1θ = −αθ + div (a (x)∇θ). equation (23) becomes ∂θ/∂t = £1θ −αεuθ1 + α, on ]0,t[ × ω theorem 13. the linearized version of (14)−(16) has an optimal control in c ( [0,t] ; l2 (ω) ) given by u (t, ·) = (1/k1) bp (t − t, ·) θ (t, ·) + 1/(εθ1 ), t ∈ [0,t] where the linear operator p is solution to the following riccati equation: · p = £1p+p£1−(1/k1) pb2p+i, p (0) = k2i. in that equation i is the identity linear operator and b is the linear operator αεθ 1 i. proof: (sketch) if we set v = u − 1/ (εθ 1 ) then equation (23) becomes ∂θ/∂t = £1θ −αεvθ1, on ]0,t[ × ω. the rest of the proof is similar to the proof in [29] concerning linear regulators. if s£1 is the contraction semigroup generated by £1, then we have ∀t ∈ [0,t] p (t) f = s£1 (t) p (0) s£1f + ∫ t 0 s£1 (t−s) ( i − (1/k1) pb2p ) s£1 (t−s) fds let now consider the nonlinear equation (14). let £u be the operator £ corresponding to control strategy u and let s£u be the contraction semigroup generated by −£u. let u ≡ { u ∈ c ( [0,t] ; h1 (ω; [0, 1]) ) ; ∀t ∈ [0,t] , s£u (t) is invertible } . some necessary and sufficient conditions for a semigroup of operators to be embedded in a group of operators are given in [25]. theorem 14. assume that there is a bounded admissible control u∗ ∈ u which minimizes the cost function j3t . let θ̃ be the absolutely continuous solution of (14) − (16) associated with u∗. then ∫ ω (( θ̃ (t0,x) )2 + k1 (x) (u ∗ (t0,x)) 2 −p (t) £u∗θ̃ (t,x) ) dx ≤ ∫ ω (( θ̃ (t0,x) )2 + k1 (x) (u (t0,x)) 2 −p (t) £uθ̃ (t,x) ) dx, where p is the absolutely continuous solution on [0,t] of the adjoint state problem  ∂p/∂t = £u∗p− 2θ̃, (t,x) ∈ r∗+ × ω 〈a (x)∇p,n〉 = 0, on r∗+ ×∂ω p (t) = 2k2θ̃ (t, ·) (24) proof: we give a proof following the maximum principle proof in [29]. for an arbitrary control w and sufficiently small h ≥ 0, define the needle variation of u∗ as uh (t) =   u∗ (t) , t ∈ [0, t0 −h] w, t ∈ ]t0 −h,t0[ u∗ (t) , t ∈ [t0,t] let θh be the output corresponding to uh. since u∗ minimizes j3t , j 3 t ( θh ) > j3t ( θ0 ) and ∂+j3t ( θ0 ) /∂h > 0. ∂+θ0 (t0, ·) /∂h = lim h→0+ 1 h [ θh (t0, ·) −θ0 (t0, ·) ] = lim h→0+ 1 h ∫ t0 t0−h ( £u0θ̃ (s, ·) −£uhθh (s, ·) ) ds = (£u0 −£w) θ0 (t0, ·) since vh (t) = u∗ (t) on [t0,t], for almost t in [t0,t], ∂θh/∂t = α−£u∗θh. ∂ ( ∂+θ0 (t, ·) /∂h ) /∂t = ∂ ( ∂+θh (t, ·) /∂h ) /∂t ∣∣∣ h=0 = ∂+ ( ∂θh (t, ·) /∂t ) /∂h ∣∣∣ h=0 = ∂+ ( α−£u∗θh ) /∂h ∣∣∣ h=0 = −£u∗ ( ∂+θh/∂h )∣∣∣ h=0 = −£u∗ ( ∂+θ0/∂h ) therefore, ∂+θ0 (t, ·) /∂h = s£u∗ (t) (s£u∗ (t0)) −1 (£u0 −£w) θ0 (t0, ·) biomath 3 (2014), 1404161, http://dx.doi.org/10.11145/j.biomath.2014.04.161 page 13 of 16 http://dx.doi.org/10.11145/j.biomath.2014.04.161 d fotsa et al., mathematical modelling and optimal control of anthracnose consequently, ∂+ (∫ ω k2 (x) ( θ0 (t,x) )2 dx ) /∂h = 2 ∫ ω k2 (x) θ 0 (t,x) ∂+θ0 (t,x) /∂hdx = 2 ∫ ω k2 (x) θ 0 (t,x) s£u∗ (t) (s£u∗ (t0)) −1 × (£u0 −£w) θ0 (t0,x) dx = 2 ∫ ω (s£u∗ (t0)) −1 s£u∗ (t) k2 (x) θ 0 (t,x) × (£u0 −£w) θ0 (t0,x) dx, and in the same manner ∂+ (∫ t 0 ∫ ω ( θ0 (t,x) )2 dxdt ) /∂h = ∫ t t0 ∫ ω ( ∂+ ( θ0 (t,x) )2 /∂h ) dxdt = 2 ∫ t t0 ∫ ω (s£u∗ (t0)) −1 s£u∗ (t) θ 0 (t,x) × (£u0 −£w) θ0 (t0,x) dxdt. since ∂+j3t ( θ0 ) /∂h ≥ 0, we have 2 ∫t t0 ∫ ω (s£u∗ (t0)) −1 s£u∗ (t) θ 0 (t,x) £u0θ 0 (t0,x) dxdt + 2 ∫ ω (s£u∗ (t0)) −1 s£u∗ (t) k2 (x) θ 0 (t,x) £u0θ 0 (t0,x) dx − ∫ ω (( θ̃ (t0,x) )2 + k1 (x) (u ∗ (t0,x)) 2 ) dx ≥ 2 ∫t t0 ∫ ω (s£u∗ (t0)) −1 s£u∗ (t) θ 0 (t,x) £wθ 0 (t0,x) dxdt + 2 ∫ ω (s£u∗ (t0)) −1 s£u∗ (t) k2 (x) θ 0 (t,x) £wθ 0 (t0,x) dx − ∫ ω (( θ̃ (t0,x) )2 + k1 (x) (w (t0,x)) 2 ) dx. note that t0 has been chosen arbitrarily. let p be the solution of the adjoint state problem (24). then for every time t ∈ [0,t],∫ ω (( θ̃ (t0,x) )2 + k1 (x) (u ∗ (t0,x)) 2 −p (t) £u∗θ̃ (t,x) ) dx ≤ ∫ ω (( θ̃ (t0,x) )2 + k1 (x) (u (t0,x)) 2 −p (t) £uθ̃ (t,x) ) dx. this theorem shows that the optimal control u∗ minimizes the following hamiltonian. h ( θ̃,p,u ) = ∫ ω ( θ̃2 + k1u 2 −p£uθ̃ ) dx as a result, we have the following necessary condition corresponding to ∂h/∂u ( θ̃,p,u∗ ) = 0: ∫ ω ( 2k1u ∗ −αθ1θ̃p/(1 −θ1u∗)2 ) dx = 0. (25) condition (25) is satisfied in particular if 2k1u ∗(1 −θ1u∗)2 = αθ1θ̃p, (26) which is analogous to (10) for the within-host model . then we can adopt the following corresponding strategy u∗ (t) = { 1 when 27αθ2 1 θp ≥ 8k1, w3 (t) when 27αθ21 θp < 8k1, where w3 (t) is the element of [ 0, min { 1 3θ 1 , 1 }] which is the nearest to the smallest nonnegative solution of the equation (26). iv. conclusion in this paper two models of anthracnose control have been surveyed. these models both have the general form ∂θ/∂t = f (t,θ,u) + g (t) , where f is linear in the state θ but not necessarily in the control u. as far as the authors know, this type of control system has not been extensively studied. this may be due to the fact that physical control problems usually do not take this form. the majority of such problems tend to use ”additive” controls (see [8], [16] for literature on models). but in models of population dynamics, ”mutiplicative” control are often more realistic. our first model characterizes the within-host behaviour of the disease. we were able to explicitly calculate an optimal control strategy that effectively reduces the inhibition rate compared to the case where no control is used. in our second model we take into account the spatial spread of the disease by adding a diffusion term. that makes the model more interesting but considerably more difficult to analyze. moreover, visual evaluation appears more difficult because in this case the state of the system is a function of three spatial variables biomath 3 (2014), 1404161, http://dx.doi.org/10.11145/j.biomath.2014.04.161 page 14 of 16 http://dx.doi.org/10.11145/j.biomath.2014.04.161 d fotsa et al., mathematical modelling and optimal control of anthracnose plus time. although we have provided equations satisfied by the optimal control (for the linearized system), in this paper we do not give a practical method for computing the optimal control. it is possible that adapted gradient methods may be used [2]: this is a subject of ongoing research. our models seems quite theoretical, but could be used for practical applications if the needed parameters were provided. indeed, in the literature [9], [10], [12], [17] there are several attempts to estimate these parameters. the principal advantage of our abstract approach is that it can be used to set automatic means to control the disease which are able to adapt themselves with respect to the host plant and to the parameters values. obviously our models can be improved. in particular, several results are based on some conditions of smoothness of parameters, and the control strategy is also very regular. in practice parameters are at most piecewise continuous, and some control strategies are discontinuous. for instance, cultural interventions in the farm are like pulses with respect to a certain calendar. the application of antifungal chemical treatments are also pulses, and the effects of these treatments though continuous are of limited duration. we are currently investigating a more general model that takes into account those irregularities. acknowledgments the authors thank professor sebastian anita for discussion with the first author on modelling diseases spread with diffusion equations. that have been possible through a stay at the university alexandru ioan cuza in iasi (romania) in the frame of the eugen ionescu scholarship program. references [1] aldila d., götz t., soewono e., an optimal control problem arising from dengue disease transmission mode, math. biosci., 242, p9-16, 2013. [2] anita s., arnautu v., capasso v., an introduction to optimal control problems in life scienes and economics, springer science+business media, new york, 2011. [3] bella-manga, bieyss d., mouen b., nyass s., berry d., elaboration d’une stratégie de lutte durable et efficace contre l’anthracnose des baies du caféier arabica dans les hautes terres de l’ouest-cameroun: bilan des connaissances acquises et perspectives, colloque scientifique international sur le café, 19, trieste, mai 2001. [4] bieysse d., bella-manga d., mouen b., ndeumeni j., roussel j., fabre v. and berry d., l’anthracnose des baies une menace potentielle pour la culture mondiale de l’arabica. plantations, recherche, développement, pp 145-152, 2002. [5] boisson c., l’anthracnose du caféier, revue de mycologie, 1960. [6] brezis h., functional analysis, sobolev spaces and partial differential equations, springer science+business media, new york, 2011. 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[29] zabczyk j., mathematical control theory: an introduction, birkhäuser, boston, 1995. biomath 3 (2014), 1404161, http://dx.doi.org/10.11145/j.biomath.2014.04.161 page 16 of 16 http://dx.doi.org/10.11145/j.biomath.2014.04.161 introduction anthracnose pathosystem models in the literature controlling anthracnose organization of the paper a within-host model specification of the within-host model well-posedness of the within-host model optimal control of the within-host model computer simulations of the controlled within-host model a diffusion model specification of the diffusion model well-posedness of the diffusion model optimal control of the diffusion model conclusion references original article biomath 4 (2014), 1407121, 1–10 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum on the mathematical modelling of eps production by a thermophilic bacterium nadja radchenkova1, margarita kambourova1, spasen vassilev1, rene alt2, svetoslav markov3 1 institute of microbiology, bulgarian academy of sciences 2 sorbonne universities, lip6, upmc, cnrs umr7606 3 institute for mathematics and informatics, bulgarian academy of sciences received: 24 january 2014, accepted: 12 july 2014, published: 23 july 2014 abstract—this paper presents experimental data coming from a batch fermentation process and theoretical models aiming to explain various aspects of these data. the studied process is the production of exopolysaccharides (eps) by a thermophilic bacterium, aeribacillus pallidus 418, isolated from the rupi basin in south-west bulgaria. the modelling approach chosen here is: first, biochemical reaction schemes are formulated, comprising several reaction steps; then the reaction schemes are translated into systems of ordinary differential equations (ode) using the mass action law; then the ode systems are studied by means of numerical simulations. the latter means that the ode systems are parametrically identified in order to possibly fit the experimental data. a main peculiarity of the proposed reaction schemes, resp. models, is the assumption that the cell biomass consist of two dynamically interacting cell fractions (dividing and non-dividing cells). this assumption allows us to implement certain modelling ideas borrowed from enzyme kinetics. the proposed models are compared to a classical model used as reference. it is demonstrated that the introduction of the two cell fractions allows a much better fit of the experimental data. moreover, our modelling approach allows to draw conclusions about the underlying biological mechanisms, formulating the latter in the form of simple biochemical reaction steps. keywords-batch fermentation processes, thermophilic bacterium, reaction schemes, dynamic models, numerical simulations i. introduction an increasing interest towards microbial exopolysaccharides (eps) is determined by the wide variety of their properties as a result of diversity in their composition. the biodegradability of eps has an impact on environmentally friendly processes. thermophilic microorganisms offer short fermentation processes, better mass transfer, decreased viscosity of synthesized polymer and of the corresponding culture liquid. eps from thermophilic microorganisms are of special interest due to the advantages of the thermophilic processes and the non-toxic nature of the polymer allowing applications in food and pharmaceutical industries [4]. low eps production in thermophilic processes determines the importance of mass transfer optimization for further development of thermophilic processes in an industrial citation: nadja radchenkova, margarita kambourova, spasen vassilev, rene alt, svetoslav markov, on the mathematical modelling of eps production by a thermophilic bacterium, biomath 4 (2014), 1407121, http://dx.doi.org/10.11145/j.biomath.2014.07.121 page 1 of 10 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2014.07.121 n. radchenkova et al., on the mathematical modelling of eps production ... scale. the aim of the current work is to describe and study the dynamics of microbial growth and product synthesis by means of suitable mathematical models, which may give more understanding for the underlying biochemical mechanisms. our main assumption from a biological perspective is the existence of two cell fractions: dividing and nondividing cells, as proposed in [10]. based on comparisons with a classical model it is demonstrated that the introduction of the two cell fractions allows a much better fit of the experimental data. in addition, our modelling approach allows to formulate the underlying biological mechanisms in the form of simple biochemical reaction steps. to achieve our aim we propose several biochemical reaction schemes that conform with the basic assumption for the existence of two cell fractions. applying the mass action law the reaction schemes lead to dynamical models that are parametrically identified in order to fit the observed data. we demonstrate that the proposed models are compatible with classical monod fermentation models which are based on different principles. in order to visualize the numerically computed theoretical results, the latter are graphically compared to experimental data. the graphics show that the proposed models reflect specific features of the mechanism of the fermentation process, which may suggest further experimental and theoretical work. we believe that using the proposed approach one can study the basic mechanisms underlying the dynamics of cell growth, substrate uptake and product synthesis. finally, we hope that the present study will contribute to the optimization of mass transfer and an enhancement of eps yield by aeribacillus pallidus 418. the paper is structured as follows. the methods for the experiments are described in section ii. section iii presents the experimental results and section iv outlines the modelling approach. section v is devoted to several theoretical models whose solutions fit the experimental measurements. the underlying idea is that such a theoretical approach can help biologists in choosing or rejecting a possible mechanism for the dynamics of microbial growth and eps synthesis. the numerical solutions obtained with each proposed model are compared to the experimental data and commented. the conclusion explains the merits of the proposed approach and comments on the suggested biological assumptions. it is demonstrated that the proposed models based on the assumption of existence of two cell fractions (dividing and non-dividing cells) are compatible with classical monod fermentation models. ii. materials and methods a. strain, medium and cultivation aeribacillus pallidus 418 was isolated from a hot spring at rupi basin, south-west bulgaria, and selected as an eps producer among 38 thermophilic bacterial polymer producers [12]. b. experimental set-up and operation mode a 1.5 l jacketed glass reactor (ak-02, russia) equipped with a hydrofoil impeller narcissus (0.05 m diameter) was filled with l l of msm and 35 ml of 18 h strain culture was added as an inoculum. fermentation parameters such as temperature, ph, foaming and aeration were kept constant during the whole processes performed. temperature was 550c; ph value was maintained at 7.0 due to buffer properties of the medium; air flow 0.8 vvm (volume per volume medium) was continuously supplied. c. determination of growth and eps production the influence of agitation on the growth and eps production was followed at 100, 400, 500, 600 and 800 rpm. growth was determined by measuring of turbidity at 660 nm. a correlation curve reflecting the proportionality between turbidity and dry weight was obtained for the chosen strain. one unit od corresponds to 1.05 mg ml−1 dry cells of aeribacillus pallidus 418. eps was recovered from the culture supernatant samples as previously described [4]. biomath 4 (2014), 1407121, http://dx.doi.org/10.11145/j.biomath.2014.07.121 page 2 of 10 http://dx.doi.org/10.11145/j.biomath.2014.07.121 n. radchenkova et al., on the mathematical modelling of eps production ... iii. experimental results for batch cultivation a. influence of agitation speed comparison of growth curves at different agitation conditions show that a longest exponential phase (8 h) and lowest biomass yield reached at stationary phase is observed at a lowest speed of stirring (100 rpm) (figure 1). exponential phase duration of 4–5 h is observed for all other agitation speeds. an increase in biomass accumulation and polymer production is observed with increase of stirring speed being highest at 500 rpm for growth and 600 rpm for eps, after that it decreased. the lowest speed of stirring (100rpm) unfavorable for growth and eps production and the speed that provide best growth (500 rpm) and best product synthesis (600 rpm) have been chosen for further modelling. investigations on the influence of agitation speed on eps production (figure 2) reveal that the observed polymer production is connected with the biomass accumulation. first quantities are registered in the early exponential phase and highest levels in the stationary phase. the concentration of the measured polymer is highest at 600 rpm (124 µg ml−1), it is five fold higher than that at 100 rpm (24 µg ml−1). 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 5 10 15 20 b io m a ss m g /m l time in hrs fermentor 1litre; experimental biomass, various rpm, air 0.8:1 100 rpm 400 rpm 500 rpm 600 rpm 800 rpm fig. 1. experimental biomass for different rpm 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 5 10 15 20 e p s m g /m l time in hrs fermentor 1litre; experimental eps, various rpm, air 0.8:1 100 rpm 400 rpm 500 rpm 600 rpm 800 rpm fig. 2. experimental eps for different rpm the experimentally obtained average values are presented in figure 1 for the biomass and in figure 2 for eps. the figures show clearly that the best production of eps appears for a speed of 600 revolutions per minute. the experimental results corresponding to maximum eps yield (at 600 rpm) are presented in table i. the measurement error is estimated to be ≤ 15% for the biomass data (≤ 18% for the last three data) and ≤ 5% for the eps data. taking into account the measurement error, the experimental data of table 1 are used in the modelling process as interval data [9]. for example, biomass data for hour seven is 1.0, hence the resp. interval value is 1.0±15% = [0.85,1.15]. the interval data are visualized as vertical segments in the presented figures. time (hours) 0 1 2 3 4 biomass (mg/ml) 0 0 0 0 0 time (hours) 5 6 7 8 12 biomass (mg/ml) 0.16 0.45 1.0 1.3 1.2 eps (mg/ml) 0.006 0.037 0.07 0.112 0.117 time (hours) 14 16 18 21 24 biomass (mg/ml) 1.2 1.2 1.1 1.1 1.1 eps (mg/ml) 0.119 0.120 0.124 0.124 0.124 table i experimental values of the biomass and eps biomath 4 (2014), 1407121, http://dx.doi.org/10.11145/j.biomath.2014.07.121 page 3 of 10 http://dx.doi.org/10.11145/j.biomath.2014.07.121 n. radchenkova et al., on the mathematical modelling of eps production ... iv. modelling approach the experimental measurements for the cell growth and eps production together with the estimated measurement errors have been used for testing and studying several dynamical models aiming to explain the nature of the involved metabolic reactions. to this end an original modelling approach similar to the one used in (bio)chemistry has been used. instead of starting the construction of the model with formulating systems of ordinary differential equations (ode) and specifying reaction terms in the right-hand side of the system’s equations as usually done in classical cell growth models, we start with proposing suitable reaction schemes. the origin of such an approach is seen in the works [2], [6], where reaction steps involving variable reaction rate coefficients are admitted. our reaction schemes differ from the ones used in these works in that all reaction rate coefficients are numerical constants—as (commonly) in biochemistry and more specifically in henri-michaelis-menten enzyme kinetics [10], [14], [16]. the proposed reaction schemes are then translated into systems of ode’s using the mass action law, and the ode systems are studied by means of numerical simulations. another peculiarity of our modelling approach is the partitioning of the cell biomass into two dynamically interacting fractions of dividing and non-dividing cells. this allows to implement ideas borrowed from enzyme kinetics—there enzymes are subdivided into two fractions: free and bound. bio-reactor models using fractions (compartments) for the biomass are known in the literature [6], [7], [17]. the proponents of the modelling approach using fractions of the cell biomass note that such structured models allow a better fit of the cell growth dynamics in the lag phase in comparison to the one obtained with classical monod type models [17]. in our proposed models the cell population is conditionally partitioned into two fractions: dividing and non-dividing cells. this allows to relax the rather restrictive assumption for synchronized states of individual cells that follows from (or is incorporated in) classical models using only one variable for the cell biomass. thus structured models allow us to assume that cells are not all simultaneously in a specific state—something that is observed in reality. the terms dividing and non-dividing states refer here to individual cells as opposite to the terms lag, log and stationary phases that refer to the cell community. classical monod type models consider a population of a microbial species as being in one of the mentioned phases at any given moment. we consider the dividing cell property as a property of the individual cell which may be inherited from the mother cell [5], [13]. this property may change in time from dividing to non-dividing or vice versa depending on the environmental conditions but also on individual cell properties. individual cell states are close to the biomass phases but do not coincide. if a cell population has been in a lag or stationary phase for a long time, then most (or all) of the individual cells will be in non-dividing state, conversely, if the cell community is in an established log phase, then most (or all) of the individual cells are in a dividing state. however, at the time when the environmental conditions change (from favorable to unfavorable or vice versa), then individual cells do not change their state simultaneously and there is a time interval when cells of both states coexist. thus whenever the cell community passes from a lag or stationary phase to a log phase (or conversely) there are cells in dividing state, and others in non-dividing state. as will be demonstrated in this work structured models based on the dividing property effectively contribute to overcoming the deficiency of classical methods to adequately describe the bioprocesses during the intermediate time intervals when biomass phases change; we further refer to this problem as “intermediate-phase-deficiency” problem. one more peculiarity of our modelling approach is that we fit the theoretical solutions of the proposed models into the interval experimental data (data ± upper bounds for the estimated measurement errors), which contributes to the model verification process [9]. in other words, our aim biomath 4 (2014), 1407121, http://dx.doi.org/10.11145/j.biomath.2014.07.121 page 4 of 10 http://dx.doi.org/10.11145/j.biomath.2014.07.121 n. radchenkova et al., on the mathematical modelling of eps production ... in the parameter identification process is to make the theoretical solutions pass through the interval experimental data. appropriate parameters for the various models have been obtained using an optimization method for minimizing the sum of the squared differences between the observed experimental data and the computed theoretical solution. but one cannot assert that these parameters are optimal as there are several local minima and different sets of parameters may lead to almost identical solutions. that is why we chose“manually” appropriate sets of parameters with the additional requirement to make the theoretical solutions pass within as many as possible interval experimental data (measurements plus/minus estimated upper bound for the measurement errors). v. mathematical modelling several mathematical models are proposed in the sequel and their parameters are numerically identified in order to fit with the observed experimental results as given in table i. the first model is a classical monod type model, whereas the other models conform with the idea of a structured bacterial biomass and the reaction-scheme approach as proposed in [10]. classical model 1 is used as a reference in order to be compared to the remaining structured models following the reaction-scheme approach. a. model 1 model 1 is a classical monod type cell growth model [11]. the model is a modification of a model for cgtase production by bacteria of the species b. circulans atcc21783 [3]. the model is mathematically represented by the following differential system: dx/dt = (µ−γ) x ds/dt = −(µ/δ) x dp/dt = (α µ + β) x, (1) wherein: µ(s) = (µmax s/(ks + s + kis 2), with initial conditions s(0) = s0, x(0) = x0, p(0) = p0. typically model (1) makes use of a growth rate parameter µ which is not a constant but a function; in this case µ is a function on the substrate concentration s, namely the familiar andrews/haldane specific growth rate function µ(s). the meaning of the numeric parameters in model (1) is given in [3]. computer simulations provide the following set of parameters: µmax = 5, α = 93, β = 0, γ = 0.005, ks = 0.25, ki = 30, δ = 13 with initialization: x0 = 0.05, s0 = 0.1, p0 = 0. the experimental data and the computed theoretical solutions of model (1) are visualized in figure 3. the vertical intervals correspond to the experimental values as given in table 1 together with the measurement errors in the data (the same pattern is followed in the remaining figures). the aim has been to identify the parameters of system (1) in such a way that the computed solutions fit the experimental data in the sense that they possibly pass through the experimental interval data. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 5 10 15 20 25 30 b io m a ss a n d e p s time in hrs aeribacillus pallidus 418, fermentor1; 1 l; 600 rpm,air 0.8:1 theor. biomass theor. product eps . fig. 3. solution of model (1), 600 rpm monod type models describe quite well bioprocesses under favorable conditions and hence allow a good fit of the log (exponential) phase. however, such models do not generally allow good fit for the intermediate phases, e. g. between the biomath 4 (2014), 1407121, http://dx.doi.org/10.11145/j.biomath.2014.07.121 page 5 of 10 http://dx.doi.org/10.11145/j.biomath.2014.07.121 n. radchenkova et al., on the mathematical modelling of eps production ... lag and log phases, as is observed on fig 3. note that all experimental values for the biomass in the first four hours are zero whereas the computed theoretical solution for the biomass increases quite rapidly and cannot be fitted well for any values of the model parameters to the experimental interval data for t ≤ 5. thus model (1) demonstrates the inherent “intermediate-phase-deficiency” of classical monod type models. a characteristic feature of monod models is the use of the specific growth rate function µ = µ(s). some authors formulate reaction schemes that make use of such a reaction rate function (instead of coefficient!) [2], [6]. it has been demonstrated that monod type models cannot be induced from reaction schemes with constant reaction rate coefficients by means of the mass action law [10]. it has been also discussed in [10] that monod type models are special cases of appropriate structured models under specific restrictions (providing exponential growth). in order to overcome the “intermediate-phasedeficiency” problem and to obtain a better fit for the experimental interval data two original models are proposed and numerically tested using the same experimental data. these models follow the modelling approach of building biochemical reaction schemes with constant reaction rate coefficients [10]. the proposed models are expected to throw some light on the related cell growth mechanisms; both proposed models are based on the idea of structured biomass and the use of an enzyme-kinetics-like reaction scheme as a starting point. b. model 2 here the total biomass has been split into two fractions of non-dividing x-cells and dividing y cells. according to the reaction-scheme modelling methodology we first formulate suitable reaction steps underlying the biological (biochemical) mechanism behind the cell growth and cell production processes. the corresponding reaction scheme involves six reaction steps: a double growth step (rsg), two reproduction steps (rsr), a producing step (rsp) and a decay step (rsd). as shown in [10] the growth step (rsg) reflects the transition of x-cells into y -cells and substitutes the use of the “specific growth rate function” µ(s) proposed by monod [11]. this step also reflects the conversion of nutrient substrate s into metabolic products p used for the growth of the cells and for their preparation to pass into dividing y -state. note that p represents the total amount of metabolites obtained as result of the transformation of the nutrient substrate s. the two reproduction steps (rsr) describe the transfer of nutrient metabolite p partially into xand y -cells. the production step (rsp) describes the transition of the intermediate metabolites p into (useful) product p1 (eps). the decay step (rsd) reflects the decay of x-cells assuming that the disintegrated cells transform into a waste product q. s + x k1−→ y k2−→ p + x, (rsg) p + y β −→ 2y, (rsr) p + y α−→ x + y, (rsr) p + y γ −→ y + p1, (rsp) x kd−→ q, (rsd) applying the mass action law to the above reaction steps as in chemical and enzyme kinetics we obtain the following system of ode’s where p1 represents the product eps: ds/dt = −k1xs dx/dt = −k1xs + k2y + αpy −kdx dy/dt = k1xs−k2y + βpy dp/dt = k2y − (α + β + γ)py dp1/dt = γpy. (2) biomath 4 (2014), 1407121, http://dx.doi.org/10.11145/j.biomath.2014.07.121 page 6 of 10 http://dx.doi.org/10.11145/j.biomath.2014.07.121 n. radchenkova et al., on the mathematical modelling of eps production ... 0 0.5 1 1.5 2 0 5 10 15 20 b io m a ss a n d e p s , m g /m l time in hrs fermentor 1litre, 600 rpm, air 0.8:1 interval experim.eps x cells y cells product eps total biomass fig. 4. solution of model (2), 600 rpm the values of the parameters numerically identified in model (2) are as follows: k1 = 5.8,k2 = 2.8,kd = 0.005,α = 9,β = 9, γ = 1.7,s(0) = 0.9,x(0) = 0.011,y(0) = p(0) = p1(0) = 0. the graphs of the solutions fitting the experimental data are visualized in fig. 4. a very good fit of the biomass, as well as of the product eps, can be observed. model (2) has been fitted for the measurement data obtained with other agitation rates. the results are always good within the accuracy of the experimental measurements. figures 5 and 6 are examples of the results obtained for 500 rpm and 100 rpm. 0 0.5 1 1.5 2 0 5 10 15 20 b io m a ss a n d e p s m g /m l time in hrs fermentor 1litre, 500 rpm, air 0.8:1 interval experim. biomass interval experim.eps x cells y cells product eps total biomass fig. 5. solution of model (2), 500 rpm 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 5 10 15 20 b io m a ss a n d e p s m g /m l time in hrs fermentor, 1litre; 100 rpm, air 0.8:1 interval experim. biomass interval experim.eps x cells y cells product eps total biomass fig. 6. solution of model (2), 100 rpm the coefficients obtained for the best fit at various agitation rates reported in figure 1 are given in table ii. rpm k1 k2 α β γ s0 x0 100 83 5.5 17 17 2.7 0.32 0.002 400 13 2.8 9 9 1.7 0.6 0.01 500 5.5 2.7 9 9 1.5 0.9 0.01 600 5.8 2.8 9 9 1.7 0.9 0.01 800 9 2.8 9 9 2.5 0.6 0.01 table ii coefficients of model (2) for various agitation rates it can be seen from table ii that the two coefficients α and β are always equal. an interpretation could be that newborn cells belong wth equal probability to one of the two fractions (of dividing or non-dividing cells). concerning the other coefficients k1,k2,γ, they have minimal values for 500– 600 rpm. note that these are the agitation rates corresponding to a maximum product synthesis. for model (2) another “good” set of parameters is found to be the following k1 = 3,k2 = 1.4,α = 6.5,β = 6.5,γ = 1.2,kd = 0.005,s0 = 1.65,x0 = 0.011,y0 = p0 = 0. the explanation of this phenomenon is the fact that the fitting of the mathematical solutions to the experimental data has been done with an optimization method biomath 4 (2014), 1407121, http://dx.doi.org/10.11145/j.biomath.2014.07.121 page 7 of 10 http://dx.doi.org/10.11145/j.biomath.2014.07.121 n. radchenkova et al., on the mathematical modelling of eps production ... which minimizes the sum of the squared differences between the experimental values and the mathematical values provided by the model (least square approximation) and the objective function has several local minima leading to several sets of parameters. to conclude, model (2) provides a very good representation of the experimental data. the model is a modification of an ode model proposed in [1]. c. model 3 model 3 is a particular case of model (2). here we assume that the rate constant α = 0, that is the first reproduction reaction step p +y α−→ x+y is omitted. consider then the following reaction scheme: s + x k1−→ y k2−→ p + x, (rsg) p + y β −→ 2y, (rsr) p + y γ −→ y + p1, (rsp) x kd−→ q, (rsd) it implies the following system of ode’s: ds/dt = −k1xs dx/dt = −k1xs + k2y −kdx dy/dt = k1xs−k2y + βpy dp/dt = k2y − (β + γ)py dp1/dt = γpy (3) the values of the parameters in model (3) are identified as follows: k1 = 20,k2 = 5,β = 1.5,γ = 0.15,kd = 0.005;s0 = 2.2,x0 = 0.02,y0 = 0,p0 = p1(0) = 0. the graphs of the solutions fitting real experimental data are visualized in figure 7. the fit of the biomass and eps to the experimental data is almost as good as in model (2). this shows that the reaction step p+y α−→ x+y does not influence much the final solutions and can be omitted. the omission of this step can be interpreted as postulating that newborn cells are always in dividing state, which is biologically relevant [5]. of course the division process can be inhibited in case that meanwhile environmental conditions become unfavorable. model (3) is a modification of a model proposed in [10]. 0 0.5 1 1.5 2 0 5 10 15 20 b io m a ss a n d p ro d u ct m g /m l time in hrs aeribacillus pallidus 418, fermentor 1 litre x -cells y cells product eps total biomass x+y fig. 7. solution of model (3), 600 rpm vi. conclusion we described the dynamics of microbial growth and eps synthesis using several mathematical models. the first model is a classical one, while models 2 and 3 are original structured models formulated in terms of reaction schemes, thereby the reaction growth step is borrowed from henrimichaelis-menten enzyme kinetics. as shown in [10] a reaction scheme using such a reaction step generalizes classical models like model 1. the suggested models differ in the choice of the reaction steps. compared to classical model 1 the proposed models produce a better fit for the cell biomass experimental data. to explain this, note that classical monod type models make use of the so-called “specific growth rate function” µ(s). recall that this function coincides (up to a multiplier) with the nutrient substrate uptake rate µ(s) in michaelis-menten enzyme kinetics. however, michaelis-menten substrate uptake function µ(s) is an approximation of the substrate rate function induced by the henrimichaelis-menten mechanism under the assumption that the enzyme/substrate ratio is small. the latter may not be true in vivo, as this ratio is not small in the cell cytoplasm [14], [15]. note that our proposed models keep close to henri-michaelismenten mechanism, which is independent on the biomath 4 (2014), 1407121, http://dx.doi.org/10.11145/j.biomath.2014.07.121 page 8 of 10 http://dx.doi.org/10.11145/j.biomath.2014.07.121 n. radchenkova et al., on the mathematical modelling of eps production ... value of the enzyme/substrate ratio. bacterial cells are metabolically active not only in the dividing state but in the non-dividing state as well [13]. hence, when formulating reaction steps corresponding to these two states one should take into account such metabolic activities. our dynamical models based on reaction schemes are first steps in this direction. we hope that the proposed models throw some light on the related cell growth mechanisms. indeed, the reaction steps used in our models presume metabolic activity of the cell in its non-dividing state. our proposed models conform with the hypothesis from [10] saying that the reaction terms in cell growth dynamical systems can be deduced from appropriate biochemical reaction schemes with constant reaction rates via mass action law. such an approach contributes to the interpretation of the underlying biological mechanism of the cell growth phenomena. the biological experiments have been performed with highest possible precision in order to serve further for accurate numerical simulations. the overall goal is to verify as much as possible the reaction scheme approach for cell growth and product synthesis modelling and possibly determine a set of adequate reaction steps. a most useful feature of the proposed reaction scheme approach is its methodological value. this approach allows the biologists to focus on the model formulation in terms of reaction schemes as done in biochemistry; thus biologists may not need to formulate their models directly in terms of differential equations. once the model is formulated in terms of reaction schemes, then the remaining part of the modelling process can be rather automated. of course a theoretical (mathematical) study of the model, e.g. with respect to stability, may also be of scientific interest. a most important conclusion (in our opinion) is that the modelling approach based on the subdivision of the cell population into two fractions (of dividing and non-dividing cells) proves as practically useful. this biological paradigm is different from the classical one based on the biomass phases (lag, log, etc) which implies simultaneous transitions of all cells from one biomass phase to the other. the classical approach is oriented towards properties of the whole bio-population, whereas our paradigm is oriented towards the individual cell cycles. individual cell cycles are considered in [6], [7] under the hypothesis that a certain part of the cell lifetime is devoted to maturing and another part to division. this interpretation is in accord with recent findings of cell biology [5] and is confirmed by the proposed models. we hope that our models contribute to a clarification of the biological mechanisms for the transition of an individual cell from a non-dividing to a dividing state (or conversely). assume that the biomass has been inhibited or stressed for a sufficiently long period and (almost) all cells are in non-dividing state. when the environmental conditions become favorable, the cells start to divide, however this process does not happen simultaneously in time for all cells. the precise mechanism for this transition is not known; it is suggested that cell signalling (quorum sensing) plays an important role in this process. some cells are able to quickly respond to environmental changes and then transmit signals to other cells. in certain time periods there exist significant numbers of both cell populations and this biologically realistic assumption is confirmed by the proposed models involving two cell fractions. future work. future work is foreseen in the following directions: a. to study the biological relevance of the proposed reaction steps and the solutions of the corresponding dynamic systems; b. to consider continuous fermentation experimental data and appropriate modelling within the outlined methodology; c. to provide suitable fermentation experimental data for studying mathematically inhibition phenomena; d. to study mathematically the proposed dynamic systems with respect to stability. biomath 4 (2014), 1407121, http://dx.doi.org/10.11145/j.biomath.2014.07.121 page 9 of 10 http://dx.doi.org/10.11145/j.biomath.2014.07.121 n. radchenkova et al., on the mathematical modelling of eps production ... acknowledgment the authors are grateful to the national fund for scientific research, bulgaria, for financial support of this work (contract dtk 02/46). they are also extremely grateful to both anonymous reviewers for their deep, detailed, very competent, constructive and beneficial comments and critical remarks. references [1] alt, r., s. markov, theoretical and computational studies of some bioreactor models, computers and mathematics with applications 64 (2012), 350–360. http://dx.doi.org/10.1016/j.camwa.2012.02.046 [2] bastin g., d. dochain, on-line estimation and 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http://dx.doi.org/10.1128/jb.06112-11 http://dx.doi.org/10.1006/bulm.1999.0163 http://dx.doi.org/10.1080/08948550390206768 http://dx.doi.org/10.1016/0307-904x(86)90006-5 http://dx.doi.org/10.11145/j.biomath.2014.07.121 introduction materials and methods strain, medium and cultivation experimental set-up and operation mode determination of growth and eps production experimental results for batch cultivation influence of agitation speed modelling approach mathematical modelling model 1 model 2 model 3 conclusion references www.biomathforum.org/biomath/index.php/biomath original article accounting for multi-delay effects in an hiv-1 infection model with saturated infection rate, recovery and proliferation of host cells debadatta adak1, nandadulal bairagi2, robert hakl3 1department of applied mathematics maharaja bir bikram university, agartala, india dev.adak.math95325@gmail.com 2centre for mathematical biology and ecology department of mathematics, jadavpur university, kolkata, india nbairagi@yahoo.co.in 3institute of mathematics czech academy of sciences, branch in brno, brno, czech republic hakl@drs.ipm.cz received: 11 november 2019, accepted: 29 december 2020, published: 31 december 2020 abstract— biological models inherently contain delay. mathematical analysis of a delay-induced model is, however, more difficult compare to its non-delayed counterpart. difficulties multiply if the model contains multiple delays. in this paper, we analyze a realistic hiv-1 infection model in the presence and absence of multiple delays. we consider self-proliferation of cd4+t cells, nonlinear saturated infection rate and recovery of infected cells due to incomplete reverse transcription in a basic hiv-1 in-host model and incorporate multiple delays to account for successful viral entry and subsequent virus reproduction from the infected cell. both of delayed and non-delayed system becomes disease-free if the basic reproduction number is less than unity. in the absence of delays, the infected equilibrium is shown to be locally asymptotically stable under some parametric space and unstable otherwise. the system may show unstable oscillatory behaviour in the presence of either delay even when the non-delayed system is stable. the second delay further enhances the instability of the endemic equilibrium which is otherwise stable in the presence of a single delay. numerical results are shown to be in agreement with the analytical results and reflect quite realistic dynamics observed in hiv-1 infected individuals. keywords-hiv model, saturated incidence, selfproliferation, recovery, multiple delays, stability, bifurcation copyright: © 2020 adak et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: debadatta adak, nandadulal bairagi, robert hakl, accounting for multi-delay effects in an hiv-1 infection model with saturated infection rate, recovery and proliferation of host cells, biomath 9 (2020), 2012297, http://dx.doi.org/10.11145/j.biomath.2020.12.297 page 1 of 20 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2020.12.297 d. adak, n. bairagi, r. hakl, accounting for multi-delay effects in an hiv-1 infection model with ... i. the model human immune deficiency virus-type 1 (hiv1 or simply hiv) is a retrovirus which preferably infects cd4+t lymphocytes and supposed to be the causative agent of aids (acquired immune deficiency syndrome). with the help of t cell receptor, cd4+t cells recognize the pathogen and interact with other lymphocytes like cd8+t cells to respond efficiently against the virus so that virus can be either removed or killed. the entry of hiv into a host cell is a complex process [1], [2]. it begins with the adhesion of virus to the host cells and ends with the fusion of its genome into the host cell’s cytoplasm. the outer envelope (env) of hiv contains spikes made up of gp120 and gp41 glycoprotein. the envelope glycoprotein gp120 contains the receptor-binding region and is attached with the receptor of cd4+t cells. after gp120 binding on cd4+t molecule, a conformational change occurs in env followed by dissociation of env from the viral membrane. the surface of gp120 contains five variable loops which (particularly v3) plays crucial roles in immune evasion and coreceptor binding. due to dissolution from the viral membrane, amino-terminal hydrophobic domains of gp41 are exposed. this initiates the fusion and hiv enters into cd4+t cells through six-helix bundle formation [3], [4]. after successful entry of virus in the cell’s cytoplasm, information coated in the viral rna is transcribed into dna (called provirus) with the help of reverse transcriptase enzyme of virus [5]. reverse transcriptase, however, is not a good copier [6]. some observations report that when the virus enters into a cd4+t cell, viral rna may not be completely reverse transcribed into dna [7], [8] and the success depends on the completion time of transcription process [9]. if the cell is activated shortly after entry of the virus into the cell, successful completion of reverse transcription is highly expected [7]. however, if the time is long enough then the partial dna transcripts may be degraded and the cell may be infection-free [8], [9]. after reverse transcription, provirus moves to the nucleus of cd4+t cell and integrates with the dna of the cell with the help of a viral enzyme called integrase and controls over the host cell’s machinery [10]. when the infected cd4+t is activated, the viral genome is transcribed back into rna, resulting in the production of multiple copies of viral rna. these rnas are translated into proteins that require a viral protease to cleave them into active forms. finally, the mature proteins assemble with the viral rna to produce new virus particles that bud from the infected cell [11]. let x(t) be the concentration of susceptible cd4+t cells in the blood plasma at any time t and y(t) be the concentration of cd4+t cells in which virus penetration has occurred successfully. we call this later class as infected cd4+t cells. assuming that a proportion of infected cd4+t cells can be reverted to the uninfected class and v(t) be the concentration of free virus particle in the blood plasma at time t, following mathematical models have been proposed and analyzed [11], [12], [13], [14], [15]: ẋ = f1(x) −f2(x,v) + py, ẏ = f2(x,v) −d1y −py, (1) v̇ = cy −d2v, here f1 is the demography function of the uninfected cd4+t cells and f2 is the incidence function. the parameters d1,c and d2 are the death rate of infected cells, virus reproduction rate and virus clearance rate, respectively. p is the rate at which infected cells revert to the uninfected class due to unsuccessful reverse transcription. in basic hiv models [16], [17], [18], [19], f1 and f2 are considered as f1 = s − µx and f2(x,v) = βxv, where s is the constant input rate of cd4+t cells, µ is its death rate and β is the disease transmission coefficient. it is a well-established fact that cd4+t cells proliferate upon exposer to hiv antigen through a number of mechanisms, like antigenic stimulation, direct activation by the virus or its products, and homeostatic t cell proliferation as a response to cd4+t cell depletion [20], [21]. in such a case, density-dependent demography function f1(x) = s − µx + rx(1 − x/k) is assumed to be a more realistic one [22], biomath 9 (2020), 2012297, http://dx.doi.org/10.11145/j.biomath.2020.12.297 page 2 of 20 http://dx.doi.org/10.11145/j.biomath.2020.12.297 d. adak, n. bairagi, r. hakl, accounting for multi-delay effects in an hiv-1 infection model with ... [23]. the parameter r represents the maximum proliferation rate of cd4+t cells and k is the value where cd4+t cells proliferation ceases to happen. one of the arguable issues in constructing infection-related model is the representation of the incidence rate. though most of the epidemic models consider bilinear/mass action incidence, there are many reasons for which this incidence function needs modification. may and anderson [24] argued that nonlinear incidence should be considered in case of the complex transformation process of parasite infections with indirect life cycles and the case is similar here. different drawbacks of bilinear/mass action law have also been pointed out, e.g., it does not address the crowding effect when the density of either host cells or virus particle is high [25], [26]. in fact, the incidence becomes unbounded whenever either of x and v becomes large. many researchers have considered the saturation effect of virus particles only and considered f2(x,v) = βxv 1+bv , where β and b are positive constants. considering virus as predator and cd4+t cells as its prey, de boer [27] considered saturation on cd4+t cells and used f2(x,v) = βxv a+x , a > 0 is the half-saturation constant, as the incidence function. bairagi and adak [28], [29] considered more general incidence rate of the form f2(x,v) = βxnv an+xn ,n ≥ 1. note that this incidence function does not consider the saturation effect on virus particle. here we consider a more general incidence function of the form f2(x,v) = βxv 1+ax+bv (a,b > 0). it is worth mentioning that this function considers the saturation effect of both cd4+t cells and virus particle. a similar function has also been used to study the dynamics of other viral infection models [30], [31]. with these assumptions, the model system (1) becomes dx dt = s−µx+rx ( 1− x k ) − βxv 1+ax+bv +py, dy dt = βxv 1 + ax + bv −d1y −py, (2) dv dt = cy −d2v. we consider the initial condition x(0) > 0, y(0) ≥ 0, v(0) ≥ 0. (3) the above model generalizes a large number of hiv-1 infection model. for example, the basic hiv model studied in [16], [17], [18], [19] can be obtained from (2) by setting zero to r,a,b.p. one can also deduce the basic hiv model with recovery studied in [11], [12] for a = b = r = 0 and the models studied in [14], [15] can be deduced for a = b = 0. researchers frequently use delay in the model to take into account various time-consuming events in the biological processes. a single delay has been used in hiv models to represent the incubation delay [25], [26], [32], [33] and virus reproduction delay [34]. an hiv infection model may be more interesting if it considers both delays simultaneously. we here consider a time lag τ1 between the events a host cell is contacted by a virus particle and the contacted cell becomes infected after various successful biochemical events. we consider another time lag τ2 which represents the time taken by the virus for completion of its life cycle within the infected cell and subsequent release of the new virus through cell lysis. local and global stability of the model proposed in (1) has been studied in presence of a single delay τ1 in [12] with f1 = s−µx and f2 = βxv. it has been shown that this delay does not affect the system dynamics and the endemic equilibrium is globally asymptotically stable if the basic reproduction number is greater than 1. sun and min [35] analyzed a non-delayed system with f1 = s − µx and f2 = βxvx+v and concluded similarly. zhou et al. [14] has studied the model (1) with f1 = s−µx + rx(1 − xk ) and f2 = βxv. it is proved that infection is cleared if the basic reproduction number is less than unity. in the opposite condition, the chronic equilibrium is globally asymptotically stable. there may exist a stable limit cycle around the endemic equilibrium if some additional condition is satisfied. a single delay τ1 was considered in the model studied by zhou et al. [14] and instability effect of the delay was observed [15]. though there exists lots of biomath 9 (2020), 2012297, http://dx.doi.org/10.11145/j.biomath.2020.12.297 page 3 of 20 http://dx.doi.org/10.11145/j.biomath.2020.12.297 d. adak, n. bairagi, r. hakl, accounting for multi-delay effects in an hiv-1 infection model with ... table i: parameter descriptions with their values and sources. parameter description reported range references default value s constant inpute 0-10 cells mm−3 [50], [51] 10 rate of cd4+t cells µ death rate of 0.0014-0.03 day−1 [54], [53] 0.01 susceptible cd4+t cells r proliferation rate 0.03-3 day−1 [50], [52] 3 of cd4+t cells k cd4+t cell density 600-1600 cells mm−3 [52], [23] 1500 where proliferation stops β disease transmission 0.00025-0.5 [50], [52] 0.0006 coefficient mm−3 virion day−1 a positive constant ... ... 0.007 b positive constant ... ... 0.00001 p recovery rate of 0.01-1.3 day−1 [55], [56] 0.01 infected cd4+t cells d1 removal rate of 0.16-1 day−1 [57], [12] 1 infected cd4+t cells c virus production rate constant 26-4000 virion day−1 [56], [57] 800 d2 removal rate of virus 2-5 day −1 [52], [57], [53] 2.5 τ1 virus transmission delay 1 day [51] variable τ2 virus reproduction delay 2 days variable literature with a single delay, there are very few works [13], [36], [37] which consider multi delays in a basic hiv model because more than one delay increases mathematical complexity significantly. incorporating two delays τ1 & τ2, our system (2) then reads dx(t) dt =s−µx(t) + rx(t)(1 − x(t) k ) − βx(t)v(t) 1 + ax(t) + bv(t) + py(t), dy(t) dt = βx(t− τ1)v(t− τ1) 1 + ax(t− τ1) + bv(t− τ1) −d1y(t) −py(t), dv(t) dt =cy(t− τ2) −d2v(t). (4) the model (4) will be analyzed with the initial conditions x(θ) = %1(θ),y(θ) = %2(θ),v(θ) = %3(θ), θ ∈ [−τ, 0], (5) where τ = max{τ1,τ2}, τ > 0, and % = (%1,%2,%3) ∈ r3+ with %1(θ) ≥ 0, %2(θ) ≥ 0, %3(θ) ≥ 0. all parameters are assumed to be positive. parameter descriptions and their values are presented in table 1. adak and bairagi [13] studied the role of immune response in a hiv model when f2(x,v) = βxnv 1+axn , p = 0 in (1). srivastava et al. [36] considered two delays in a four-dimensional hiv model with f1 = s−µx and f2 = βxv,p = 0. they showed that the delay may have both stable and unstable effect on the system dynamics. pawelek et al. [37] considered the same model and showed the global stability of the interior equilibrium point under some restrictions. they ignored the proliferation character of cd4+t cells in the presence of antigenic infection, saturation effect of the incidence and recovery of some infected cd4+t cells due to incomplete reverse transcription. alshorman et al. [38] considered a two multi-delayed hiv models with f1 = s−µx and f2 = βxv,p = 0. one delay is used to represent the time between cell infection and viral production, and the other delay is used to incorporate the time needed for the adaptive immune response biomath 9 (2020), 2012297, http://dx.doi.org/10.11145/j.biomath.2020.12.297 page 4 of 20 http://dx.doi.org/10.11145/j.biomath.2020.12.297 d. adak, n. bairagi, r. hakl, accounting for multi-delay effects in an hiv-1 infection model with ... to emerge. it is shown that intracellular delay does not change the stability results but immune delay can generate rich dynamics. multiple delay has also been considered in the fractional-order hiv model [39]. the issue we address here is the role of self proliferation of cd4+t cells in an in-host hiv-1 ode model with more general infection rate and recovery of infected cells. the objective is to find how the dynamics change in the presence of multiple delays. in the absence of delays, we show that the disease-free state is locally asymptotically stable if the basic reproduction number is less than unity. in the opposite case, however, the infection always persists. the infected equilibrium is locally asymptotically stable under some parametric space and unstable for some other parametric space. the system may show unstable oscillatory behavior in the presence of either delay even when the nondelayed system is stable. it is further observed that instability is augmented in the presence of multiple delays. the paper is arranged in the following sequence. in the next section, we analyze the non-delayed system. we show positivity, boundedness and permanence of the system along with the stability results of different equilibrium points. in section 3, we study the delay-induced system and give local stability results in the presence of single and multiple delays. numerical results to validate analytical findings are presented in section 4. the paper ends with a discussion in section 5. ii. analysis of the non-delayed system a. positivity and boundedness of solutions unless stated otherwise, we always assume that self-proliferation rate of healthy cd4+t cells is greater than its natural death rate, i.e. r > µ. it is also assumed that µk > s, so that cd4+t cells count decreases if it ever reaches k [40]. proposition ii.1 all solutions of the system (2) are positively invariant and uniformly bounded in γl when t is large, where γl = { (x,y,v) ∈ r3+ | 0<x≤ s0 d1 , 0≤y ≤ s0 d1 , 0 ≤ v ≤ cs0 d2d1 } (6) with s0 = s + k[(r−µ)+d1]2 4r , µk > s and r > µ. proof: from the second and third equations of the system (2), it follows that if y(0) = 0 and v(0) = 0 then y(t) = 0 and v(t) = 0 for t ∈ r+. we therefore assume that y(0) + v(0) > 0. (7) then from the second equation of (2), we obtain y(t) = e−(d1+p)ty(0) + e−(d1+p)t ∫ t 0 e(d1+p)s βx(s)v(s) 1 + ax(s) + bv(s) ds for t > 0, (8) which together with (7) implies that there exists t > 0 such that y(t) > 0 for t ∈ (0,t). consequently, the third equation of (2) yields v(t) > 0 for t ∈ (0,t). we will show that x(t) > 0 for t ∈ (0,t). (9) indeed, assume that x has a zero in (0,t) and let t0 ∈ (0,t) be such that x(t0) = 0, x(t) > 0 for t ∈ [0, t0). then, obviously, ẋ(t0) ≤ 0. on the other hand, the first equation in (2) yields ẋ(t0) = s + py(t0) > 0, a contradiction. thus (9) holds. now let t0 > 0 be such that y(t0) = 0, y(t) > 0 for t ∈ (0,t0). then, from the above-proven, we have x(t) > 0, v(t) > 0 for t ∈ (0,t0) and so, from (8) we get y(t0) > 0, a contradiction. consequently, y(t) > 0 for t > 0 and therefore also x(t) > 0 and v(t) > 0 for t > 0. we below prove the boundedness of the solutions for all large t. let v1(t) = x(t) + y(t). differentiating v1(t) along the solutions of (2), we biomath 9 (2020), 2012297, http://dx.doi.org/10.11145/j.biomath.2020.12.297 page 5 of 20 http://dx.doi.org/10.11145/j.biomath.2020.12.297 d. adak, n. bairagi, r. hakl, accounting for multi-delay effects in an hiv-1 infection model with ... parameters s r k a b p c p r c c -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 x y v µ β d 1 d 2 fig. 1: sensitivity analysis of system parameters using partially ranked correlation coefficients (prcc) in absence of delays with p-value < 0.001. lengths of the bar against each parameter measure its sensitivity on the state variables. this diagram shows that c and β are the most sensitive parameters. all parameters have been varied in their reported range mentioned in the table 1. have v̇1(t) = ẋ(t) + ẏ(t) = s + [(r −µ) + d1]x(t) − rx2(t) k −d1[x(t) + y(t)] ≤ s0 −d1v1(t), where s0 = s + k[(r −µ) + d1]2 4r . therefore, lim sup t→+∞ v1(t) ≤ s0d1 , implying that solutions x(t; x(0)) and y(t; y(0)) are uniformly bounded for all large t. finally, from the third equation of (2), one has lim sup t→+∞ v(t) ≤ cs0 d2d1 . hence the proposition. b. equilibria, stability and permanence system (2) has two equilibrium points. the disease-free equilibrium e1(x0, 0, 0) always exists with x0 = k2r [(r − µ) + √ (r −µ)2 + 4sr k ]. for simplicity, we write n(x) = s−µx+rx ( 1− x k ) . then the infected or endemic equilibrium is given by e∗ = (x∗,y∗,v∗), where v∗ = cy∗ d2 , y∗ = n(x∗) d1 and x∗ is the positive root of a1x 2 + a2x + a3 = 0, where a1 = rbc d1d2k (> 0), a2 = βc d2(d1 + p) − (r −µ)bc d1d2 −a, a3 = − ( sbc d1d2 + 1 ) (< 0). thus the unique positive root of the quadratic equation is determined as x∗ = −a2 + √ a22 − 4a1a3 2a1 > 0. biomath 9 (2020), 2012297, http://dx.doi.org/10.11145/j.biomath.2020.12.297 page 6 of 20 http://dx.doi.org/10.11145/j.biomath.2020.12.297 d. adak, n. bairagi, r. hakl, accounting for multi-delay effects in an hiv-1 infection model with ... c0 200 400 600 β 0.00025 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 unstable e * stable e * stable e * stable e 1 fig. 2: stability regions of the disease-free equilibrium e1 and infected equilibrium e∗ in c−β plane. here s = 10, µ = 0.01, r = 3, k = 1500, a = 0.007, b = 0.00001, p = 0.01, d1 = 1, d2 = 2.5, τ1 = τ2 = 0. if y∗ is also positive then e∗ will exist uniquely. note that, in view of y∗ = n(x ∗) d1 , we have y∗ > 0 iff n(x∗) > 0. this last inequality is satisfied if x∗ < x0. thus, the system (2) possesses a unique interior equilibrium e∗ whenever x∗ < x0. c. basic reproductive number we determine the basic reproductive number using the next generation matrix method [42]. the jacobian of (2) at e1 is given by j0 =   r −µ− 2rx0 k p − βx0 1+ax0 0 −(d1 + p) βx01+ax0 0 c −d2   . (10) the infected sub-matrix is j1 = ( −(d1 + p) βx01+ax0 c −d2 ) . one can express j1 as j1 = ( 0 βx0 1+ax0 0 0 ) − ( d1 + p 0 −c d2 ) = m1 −m2. then the basic reproduction number r0 is determined by the spectral radius of the next generation matrix m1m −1 2 [42] and evaluated as r0 = βcx0 d2(1 + ax0)(d1 + p) . (11) at e∗, we have βcx∗v∗ 1 + ax∗ + bv∗ = (d1 + p)y ∗ or, equivalently, βcx∗ d2(d1+p) [ 1+ax∗+ bc d1d2 ( s−µx∗+rx∗(1−x∗ k ) )] = 1. biomath 9 (2020), 2012297, http://dx.doi.org/10.11145/j.biomath.2020.12.297 page 7 of 20 http://dx.doi.org/10.11145/j.biomath.2020.12.297 d. adak, n. bairagi, r. hakl, accounting for multi-delay effects in an hiv-1 infection model with ... define m(x) = βcx d2(d1+p) [ 1+ax+ bc d1d2 ( s−µx+rx(1− x k ) )], p(x) = 1 m(x) , ∀x > 0, (12) so that m(0) = 0, m(x∗) = 1, m(x0) = r0 and p ′(x) = −d2(d1+p) βc ( bcs d1d2 1 x2 + 1 x2 + rbc d1d2k ) < 0 for all x > 0. thus, p(x) is monotonically decreasing for all x > 0, which in turn implies that m(x) is monotonically increasing for all x > 0. therefore, x0 > x∗ ⇒ m(x0) > m(x∗) ⇒ r0 > 1. we, therefore, state the following proposition. remark ii.1 the endemic equilibrium e∗(x∗,y∗,v∗) of system (2) exists uniquely if x∗ < x0, or equivalently r0 > 1. d. local stability of disease-free equilibrium theorem ii.1 the disease-free equilibrium e1 is locally asymptotically stable if r0 < 1. proof: the jacobian of (2) at e1 is given by (10). it can be easily verified by a direct calculation that all the eigenvalues of j0 are negative iff r0 < 1. hence the proposition follows. e. permanence of the system definition ii.1 system (2) is said to be uniformly persistent if there exists ς > 0 (independent of initial conditions) such that every solution (x(t),y(t),v(t)) with initial condition (3) satisfies lim inf t→+∞ x(t)≥ς, lim inf t→+∞ y(t)≥ς, lim inf t→+∞ v(t)≥ς. definition ii.2 system (2) is said to be permanent if there exists a compact region γ0 ∈ int γl such that every solution of system (2) with initial condition (3) will eventually enter and remain in region γ0. we further define metzler matrix and state perron-frobenius theorem following gantmacher [45]. definition ii.3 a square matrix is said to be a metzler matrix if all its off-diagonal elements are non-negative. theorem ii.2 (perron-frobenius theorem) let f be an irreducible metzler matrix. then λf, the eigenvalue of f of largest real part is real and the elements of its associated eigenvector νf are positive. moreover, any eigenvector of f with nonnegative elements belongs to span of νf. with these results and the concept of persistence in infinite-dimensional system given by hale and waltman [44], one can prove the following theorem as in [14]. theorem ii.3 model system (2) is permanent if r0 > 1, i.e., whenever e∗ exists. f. local stability of the interior equilibrium after linearizing about e∗, one can express system (2) as dx dt = m3x(t), (13) where x(t) = [x(t), y(t), v(t)]t , m3 =   m11 m12 m13n21 m22 n23 0 l32 m33   , and  m11 = r−µ− 2rx∗ k − βv∗(1+bv∗) (1+ax∗+bv∗)2 , m12 = p, m13 = − βx∗(1 + ax∗) (1 + ax∗ + bv∗)2 , n21 = βv∗(1 + bv∗) (1 + ax∗ + bv∗)2 , m22 = −(d1 + p) n23 = βx∗(1 + ax∗) (1 + ax∗ + bv∗)2 , l32 = c, m33 = −d2. (14) theorem ii.4 let r0 > 1. then the interior equilibrium e∗ of (2) is locally asymptotically stable if b1 > 0, b3 > 0 and b1b2 − b3 > 0, where bi (i = 1, 2, 3) are defined bellow. biomath 9 (2020), 2012297, http://dx.doi.org/10.11145/j.biomath.2020.12.297 page 8 of 20 http://dx.doi.org/10.11145/j.biomath.2020.12.297 d. adak, n. bairagi, r. hakl, accounting for multi-delay effects in an hiv-1 infection model with ... 0 500 1000 1,000 1,200 1,400 (ii) 0 500 1000 0 200 400 600 800 1000 1200 0 500 1000 0 5000 10000 15000 t 0 100 200 0 100 200 300 400 (iii) 0 100 200 0 200 400 600 800 1000 0 100 200 0 1 2 3 4 x 10 4 t 0 200 400 0 10 20 30 40 (iv) 0 200 400 0 20 40 60 0 200 400 0 5000 10000 15000 t 0 500 1000 1200 1400 1600 1500 x (i) 0 1000 2000 0 0.01 0.02 0.03 0.04 0.05 y 0 1000 2000 0 0.5 1 1.5 2 t v c = 34 c = 250 c = 800c = 30 fig. 3: time series solutions of system (2) for some particular values of c. the first column shows that infection is eradicated if virus reproduction is too low (c = 30). subsequent columns demonstrate stable coexistence (c = 34), oscillatory coexistence (c = 250) and again stable coexistence of infection with increasing virus reproduction (c = 800). here β = 0.0006 and other parameters are as in fig. 2. proof: the characteristic equation corresponding to m3 is ξ3 + b1ξ 2 + b2ξ + b3 = 0, (15) where b1 = −(m11 + m22 + m33), b2 = m11m22 + m22m33 + m33m11 −n23l32 −m12n21, b3 = (m11n23 −m13n21)l32 +(m12n21 −m11m22)m33. (16) we have b1b2−b3 =−m211m22−m 2 11m33−m 2 22m11 −m222m33 −m 2 33m11 −m 2 33m22 +m11m12n21 + m22n23l32 + m22m12n21 +m33n23l32 + m13n21l32 − 2m11m22m33. following routh-hurwitz conditions, the necessary and sufficient conditions for e∗ to be locally asymptotically stable are b1 > 0, b3 > 0, b1b2 −b3 > 0. hence the theorem. g. hopf bifurcation analysis of the endemic equilibrium suppose e∗ is locally asymptotically stable. we want to know whether e∗ will lose its stability when one of the system parameters is smoothly varied. noting that the virus production rate (c) as one of the most sensitive parameters, we perform this study with respect to c. one can, however, choose another parameter, say β, as the bifurcation parameter. biomath 9 (2020), 2012297, http://dx.doi.org/10.11145/j.biomath.2020.12.297 page 9 of 20 http://dx.doi.org/10.11145/j.biomath.2020.12.297 d. adak, n. bairagi, r. hakl, accounting for multi-delay effects in an hiv-1 infection model with ... theorem ii.5 if c crosses a critical value c∗ then there exists a hopf bifurcation around e∗ when the following conditions hold: (i) b1(c∗) > 0, b3(c∗) > 0, (ii) b1(c∗)b2(c∗) −b3(c∗) = 0. (iii) b ′ 3(c∗)−b ′ 1(c∗)b2(c∗)−b1(c∗)b ′ 2(c∗) 6= 0. proof: assume that there exists a critical value c∗ such that the conditions (i) − (iii) hold. for a hopf bifurcation to occur at c = c∗, the characteristic equation (15) satisfies [ξ2(c∗) + b2(c∗)][ξ(c∗) + b1(c∗)] = 0. (17) this equation has three roots ξ1(c∗) = i √ b2(c∗), ξ2(c∗) = −i √ b2(c∗) and ξ3(c∗) = −b1(c∗) < 0. for the existence of hopf bifurcation at c = c∗, one needs to verify the transversality condition[ re(ξ(c)) dc ] c=c∗ 6= 0. note that, for all c, the roots of (15) are in general of the form ξ1(c) = l(c) + i m(c), ξ2(c) = l(c) − i m(c) and ξ3 = −b1(c). substituting ξj(c) = l(c) ± i m(c), (j = 1, 2) in (15) and calculating the derivative, one obtains k̂(c∗) l ′ (c) − l̂(c) m ′ (c) + m̂(c) = 0, l̂(c∗) l ′ (c) + k̂(c) m ′ (c) + n̂(c) = 0, (18) where k̂(c) = 3l2(c) + 2b1(c)l(c) + b2(c) − 3m2(c), l̂(c) = 6l(c)m(c) + 2b1(c)m(c), m̂(c) = l2(c)b ′ 1(c) + b ′ 2(c)l(c) + b ′ 3(c) −b ′ 1(c)m 2(c), n̂(c) = 2l(c)m(c)b ′ 1(c) + m(c)b ′ 2(c). since l(c∗) = 0 and m(c∗) = √ b2(c∗), we have k̂(c∗) = −2b2(c∗), l̂(c∗) = 2b1(c∗) √ b2(c∗), m̂(c∗) = b ′ 3(c∗) −b ′ 1(c∗)b2(c∗), n̂(c∗) = b ′ 2(c∗) √ b2(c∗). solving for l ′ (c∗) from (18) and noting the condition (iii) of theorem ii.5, we have [ re(ξ1,2(c)) dc ] c=c∗ = l ′ (c∗) = − l̂(c∗)n̂(c∗) + k̂(c∗)m̂(c∗) k̂2(c∗) + l̂2(c∗) = b ′ 3(c∗) −b ′ 1(c∗)b2(c∗) −b1(c∗)b ′ 2(c∗) 2[b22(c∗) + b 2 1(c∗)] 6= 0. thus, the transversality condition is satisfied. therefore, the equilibrium point e∗ switches its stability (stable to unstable) through hopf bifurcation at c = c∗ when the conditions of theorem ii.5 hold. hence the theorem. iii. analysis of the delay-induced system it is worth mentioning that the system (4) has the same equilibrium points and same expression for the basic reproduction number r0 as in the case of non-delayed system (2). it is easy to check that for all τ1 ≥ 0 and τ2 ≥ 0, e1 is stable when r0 < 1 and therefore omitted. we are therefore interested to study the case r0 > 1, i.e., when the interior equilibrium e∗ exists, in presence of both delays. consider the perturbed variables as x̄ = x − x∗, ȳ = y − y∗, v̄ = v − v∗. dropping the bars for convenience, system (4) then reads dx dt =m11x(t) + m12y(t) + m13v(t) + f1, dy dt =n21x(t− τ1) + m22y(t) + n23v(t− τ1) + f2, dv dt =l32y(t− τ2) + m33v(t) + f3, (19) biomath 9 (2020), 2012297, http://dx.doi.org/10.11145/j.biomath.2020.12.297 page 10 of 20 http://dx.doi.org/10.11145/j.biomath.2020.12.297 d. adak, n. bairagi, r. hakl, accounting for multi-delay effects in an hiv-1 infection model with ... τ 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 c 0 200 400 600 800 1,000 1,200 stable e * unstable e * stable e 1 fig. 4: stability regions of the disease-free equilibrium e1 and infected equilibrium e∗ in τ2−c plane. parameters are s = 10, µ = 0.01, r = 3, k = 1500, a = 0.007, b = 0.00001, p = 0.01, d1 = 1, d2 = 2.5, τ1 = 0 with β = 0.0006. where the coefficients are as in (14) and f1 =α21x 2(t) + α22v 2(t) + α23x(t)v(t) + α24x 3(t) + α25v 3(t) + α26x 2(t)v(t) + ..... , f2 =β21x 2(t− τ1) + β22v2(t− τ1) + β23x(t− τ1)v(t− τ1) + β24x 3(t− τ1) + β25v3(t− τ1) + β26x 2(t− τ1)v(t− τ1) + ..... , f3 =0, (20) with α21 = − r k + aβv∗(1 + bv∗) (1 + ax∗ + bv∗)3 , β21 = − aβv∗(1 + bv∗) (1 + ax∗ + bv∗)3 , α22 = bβx∗(1 + ax∗) (1 + ax∗ + bv∗)3 = −β22, α23 = − (1 + ax∗ + bv∗ + 2abx∗v∗)β (1 + ax∗ + bv∗)3 =−β23, α24 = − a2βv∗(1 + bv∗) (1 + ax∗ + bv∗)4 = −β24, α25 = − b2βx∗(1 + ax∗) (1 + ax∗ + bv∗)4 = −β25, α26 = aβ(1 + ax∗ + 2abx∗v∗ − b2v∗2) (1 + ax∗ + bv∗)4 =−β26. corresponding linear system is then given by dx dt = m11x(t) + m12y(t) + m13v(t), dy dt = n21x(t− τ1) + m22y(t) + n23v(t− τ1), dv dt = l32y(t− τ2) + m33v(t), (21) and the corresponding characteristic equation is expressible as (ξ3 + āξ2 + b̄ξ + c̄) + (d̄ξ + ē)e−ξτ1 + (f̄ξ + ḡ)e−(τ1+τ2) = 0, (22) biomath 9 (2020), 2012297, http://dx.doi.org/10.11145/j.biomath.2020.12.297 page 11 of 20 http://dx.doi.org/10.11145/j.biomath.2020.12.297 d. adak, n. bairagi, r. hakl, accounting for multi-delay effects in an hiv-1 infection model with ... where ā = −(m11 + m22 + m33), b̄ = m11m22 + m22m33 + m33m11, c̄ = −m11m22m33, d̄ = −m12n21, ē = m12n21m33, f̄ = −n23l32, ḡ = (m11n23 −m13n21)l32. (23) one can discuss the following cases in sequel. case 1. τ1 = τ2 = 0 theorem iii.1 the interior equilibrium e∗ is locally asymptotically stable in absence of delays if r0 > 1, b1 > 0, b3 > 0 and b1b2 −b3 > 0, where b1, b2, b3 are given by (16). proof: in this case, (22) takes the form ξ3 +āξ2 +(b̄+d̄+f̄)ξ+(c̄+ē+ḡ) = 0. (24) one can notice that (23) and (16) are identical as ā = −(m11 + m22 + m33) = b1, b̄ + d̄ + f̄ = m11m22 +m22m33 +m33m11−n23l32−m12n21 = b2 and c̄ + ē + ḡ = (m11n23 − m13n21)l32 + (m12n21 − m11m22)m33 = b3. hence the result follows. case 2. τ2 > 0, τ1 = 0 theorem iii.2 assume that the conditions of theorem iii.1 are satisfied and the interior equilibrium e∗ is stable when τ1 = 0 = τ2. then the following results are true. (i) e∗ is stable for all τ2 ≥ 0 if (28) has no positive root. (ii) if (28) has one positive root then there exists a critical value τ∗2 of τ2 such that e ∗ is stable for 0 < τ2 < τ∗2 and unstable for τ2 > τ ∗ 2 . system undergoes a hopf bifurcation around e∗ at τ2 = τ ∗ 2 . (iii) if (28) has k positive roots, k = 2, 3, then there exists k critical values of τ2 given by τ∗2,j, 1 ≤ j ≤ k ; j ∈ n, where e ∗ will change its stability and a hopf bifurcation will occur at each of these critical points. proof: if τ2 > 0, τ1 = 0 then (22) has the form ξ3 + āξ2 + (b̄ + d̄)ξ + (c̄ + ē) + (f̄ξ + ḡ)e−ξτ2 = 0. (25) we investigate if (25) has a pair of purely imaginary roots of the form ξ = ±iω2, ω2 > 0, for some parametric conditions. putting ξ = iω2, we obtain β1 cos(ω2τ2) + β2 sin(ω2τ2) −β3 + i (β2 cos(ω2τ2) −β1 sin(ω2τ2) −β4) = 0, where β1 = ḡ, β2 = f̄ω2, β3 = āω 2 2 − (c̄ + ē), β4 = ω 3 2 −ω2(b̄ + d̄). (26) separating real and imaginary parts, we get β1 cos(ω2τ2) + β2 sin(ω2τ2) = β3, β2 cos(ω2τ2) −β1 sin(ω2τ2) = β4. eliminating τ2, these equations yield f2(ω2) = ω62 + k1 ω 4 2 + k2 ω 2 2 + k3 = 0, (27) where k1 = ā2 − 2(b̄ + d̄),k2 = (b̄ + d̄)2 − 2ā(c̄ + ē) − f̄2 and k3 = (c̄ + ē)2 − ḡ2. we reduce the degree of eq. (27) by setting h2 = ω22 and obtain f2(h2) = h32 + k1 h 2 2 + k2 h2 + k3 = 0. (28) if f2(h2) = 0 has no positive root then no change in stability will occur for τ2 > 0. since e∗ was stable at τ2 = 0, it will remain stable for all τ2 > 0. if f2(h2) has one positive root h2 = h+2 (say) and the corresponding positive ω2 is ω∗2 = √ h+2 , then eq. (25) will have a pair of purely imaginary roots of the form ξ = ±iω∗2 and the equilibrium e∗ will undergo a hopf bifurcation at τ2 = τ∗2 , i.e., a family of periodic solutions will bifurcate from e∗ as τ2 crosses the value τ∗2 , where τ∗2 = 1 ω∗2 arccos ( β1β3 + β2β4 β21 + β 2 2 ) . (29) here α1, α2, α3, α4 will be calculated from (26) with ω2 = ω∗2. moreover, the transversality condition is given by[ re ( dξ dτ2 )−1] ξ=iω∗2, τ2=τ ∗ 2 = f ′ 2(ω ∗2 2 ) ω∗22 (ḡ 2 + f̄2ω∗22 ) 6= 0. (30) biomath 9 (2020), 2012297, http://dx.doi.org/10.11145/j.biomath.2020.12.297 page 12 of 20 http://dx.doi.org/10.11145/j.biomath.2020.12.297 d. adak, n. bairagi, r. hakl, accounting for multi-delay effects in an hiv-1 infection model with ... 0 100 200 300 400 500 0 10 20 30 40 50 (ii) 0 100 200 300 400 500 0 50 100 0 100 200 300 400 500 0 1 2 3 4 x 10 4 t 0 200 400 600 800 1000 0 5 10 15 20 x (i) 0 200 400 600 800 1000 0 20 40 60 y 0 200 400 600 800 1000 0 0.5 1 1.5 2 x 10 4 t v τ 2 = 1.1τ 2 = 0.95 fig. 5: time evolutions of system (4) for different values of τ2. first panel shows stable behavior of the system populations when τ2 = 0.95(< τ∗2 ) and the second panel depicts the unstable behavior for τ2 = 1.1(> τ ∗ 2 ). parameters are as in fig. 4 with β = 0.0006, c = 800 and τ ∗ 2 = 0.9877 . if f2(h2) has k positive roots, k = 2, 3, given by h2 = h + 2,j, 1 ≤ j ≤ k ; j ∈ n, we will get k positive values of ω2 given by ω∗2,j = √ h+2,j , 1 ≤ j ≤ k ; j ∈ n. for each of these critical values of ω∗2, eq. (25) will have a pair of purely imaginary roots of the form ξ = ±iω∗2,j and the equilibrium e∗ will undergo a hopf bifurcation at τ2 = τ ∗ 2,j , 1 ≤ j ≤ k ; j ∈ n, where τ∗2,j = 1 ω∗2,j arccos ( β1β3 + β2β4 β21 + β 2 2 ) , 1 ≤ j ≤ k ; j ∈ n, k = 2, 3. (31) as before, β1, β2, β3, β4 are calculated from (26) with ω2 = ω∗2,j , 1 ≤ j ≤ k ; j ∈ n. the transversality condition is given by[ re ( dξ dτ2 )−1] ξ=iω∗2,j, τ2=τ ∗ 2,j = g ′ 1(ω ∗2 2,j) ω∗22,j(ē + ḡ) 2 + ω∗41,j(d̄ + f̄) 2 6= 0, 1 ≤ j ≤ k; j ∈ n,k = 2, 3. this completes the proof. case 3. τ1 > 0, τ2 > 0 theorem iii.3 assume that conditions of theorem iii.1 and either of theorem iii.2 (ii) or iii.2 (iii) are satisfied. corresponding to each positive root ω = ω̂1 of eq. (35), there exists a critical value τ∗1 (τ2) of τ1 such that e ∗ will change its biomath 9 (2020), 2012297, http://dx.doi.org/10.11145/j.biomath.2020.12.297 page 13 of 20 http://dx.doi.org/10.11145/j.biomath.2020.12.297 d. adak, n. bairagi, r. hakl, accounting for multi-delay effects in an hiv-1 infection model with ... 0 100 200 300 400 500 600 700 800 900 1000 0 200 400 600 800 1000 1200 1400 1600 c y τ 2 = 0.6 fig. 6: bifurcation diagram of the system (4) with c as the bifurcation parameter when τ2 = 0.6. other parameters are as in fig. 4. it demonstrates that infection can not persist if c < 38 and persists in oscillatory state for 38 < c < 663 and in stable state for c > 663. stability at each τ1 = τ∗1 (τ2), where τ2 is fixed and taken from the interval where e∗ was stable in the case τ2 > 0, τ1 = 0. a hopf bifurcation will occur at τ1 = τ∗1 (τ2). proof: putting ξ = iω in (22) and equating real and imaginary parts, we have ē cos(ωτ1) + d̄ω sin(ωτ1) + ḡ cos(ω(τ1 + τ2)) + f̄ω sin(ω(τ1 + τ2)) + ( c̄ − āω2 ) = 0, d̄ω cos(ωτ1) − ē sin(ωτ1) + f̄ω cos(ω(τ1 + τ2)) − ḡ sin(ω(τ1 + τ2)) + ( b̄ω −ω3 ) = 0. (32) here τ2 is fixed in any of its stable range stated in theorem iii.2 (ii) or iii.2 (iii), and τ1 is considered as a free parameter. from (32), we then get κ1 cos(ωτ1) + κ2 sin(ωτ1) = κ3, κ2 cos(ωτ1) −κ1 sin(ωτ1) = κ4, (33) where  κ1 = ē + ḡ cos(ωτ2) + f̄ω sin(ωτ2), κ2 = d̄ω − ḡ sin(ωτ2) + f̄ω cos(ωτ2), κ3 = (āω 2 − c̄), κ4 = ω 3 − b̄ω. (34) eq. (33) yields j(ω) =ω6 + g1ω4 + g2ω2 + g3 + 2g4 sin(ωτ2) + 2g5 cos(ωτ2) = 0, (35) biomath 9 (2020), 2012297, http://dx.doi.org/10.11145/j.biomath.2020.12.297 page 14 of 20 http://dx.doi.org/10.11145/j.biomath.2020.12.297 d. adak, n. bairagi, r. hakl, accounting for multi-delay effects in an hiv-1 infection model with ... where   g1 = ā 2 − 2b̄, g2 = b̄ 2 − d̄2 − f̄2 − 2āc̄, g3 = c̄ 2 − (ē2 + ḡ2), g4 = (d̄ḡ− ēf̄)ω, g5 = −ēḡ− d̄f̄ω2. one can note that if c̄2 < (ē + ḡ)2 then eq. (35) will have at least one positive root ω̂1. for this ω̂1, (22) will have a pair of purely imaginary roots of the form ξ = ±iω̂1 and e∗ will change its stability. a hopf bifurcation will occur at τ1 = τ∗1 (τ2), where τ∗2 (τ1) = 1 ω̂1 arccos ( κ1κ3 + κ2κ4 κ21 + κ 2 2 ) , (36) κ1,κ2,κ3,κ4 have to be calculated from (34) with ω = ω̂1. the transversality condition is also verified as re [( dξ dτ2 )−1] ω=ω̂1, τ1=τ ∗ 1 (τ2) = �7(�1 + �3 + �5) + �8(�2 + �4 + �6) �27 + � 2 8 6= 0, where  �1 = b̄ − 3ω2, �2 = 2āω, �3 = (d̄ − ēτ1) cos(ωτ1) −d̄ωτ1 sin(ωτ1), �4 = −(d̄ − ēτ1) sin(ωτ1) −d̄ωτ1 cos(ωτ1), �5 = (f̄ − ḡ(τ1 + τ2)) cos(ω(τ1 + τ2)) −f̄ω(τ1 + τ2) sin(ω(τ1 + τ2)), �6 = −(f̄ − ḡ(τ1 + τ2)) sin(ω(τ1 + τ2)) −f̄ω(τ1 + τ2) cos(ω(τ1 + τ2)), �7 = ēω sin(ωτ1) − d̄ω2 cos(ωτ1) +ḡ cos(ω(τ1 +τ2))+f̄ω sin(ω(τ1 +τ2)), �8 = ēω sin(ωτ1) + d̄ω 2 sin(ωτ1) −ḡ sin(ω(τ1 +τ2))+f̄ω cos(ω(τ1 +τ2)). sign of the transversality condition will indicate the direction of hopf bifurcation. hence the theorem. iv. numerical simulations we considered the default values of the system parameters from their reported range (see table 1) and simulated the system (2). we first performed a sensitivity analysis using latin hypercube sampling (lhs) method and partial ranked correlation coefficient (prcc) technique [58] to determine the most sensitive parameters of the model. as the complexity of the biological models is related to a high degree of uncertainty in estimating the values of various system parameters, uncertainty and sensitivity analysis are essential to interpret the dynamics of biological models. lhs estimates the uncertainty for each input parameter by considering it as a random variable only once in the analysis and by defining probability distribution functions, marginal distributions etc. corresponding to each parameter. prcc analysis is performed for each input parameter sampled by the lhs scheme and each outcome variable. the bar length against each parameter (see fig. 1) indicates its sensitivity on the system and the level of significance of the test is given by the p− value, which is obtained to be less than 0.001 implying high accuracy of the lhs-prcc analysis. it shows that the parameters c and β are the most sensitive parameters out of eleven parameters (excluding delay parameters) of the system. it is noticeable that the values of these two parameters can be changed by antiretroviral drugs used in the treatment of hiv infected patients and therefore it makes sense to be considered as important parameters. based on this sensitivity analysis, we performed our simulations with respect to these two parameters. all the numerical simulations have been performed using matlab r2015a. the time series and bifurcation diagrams have been drawn using solvers ode45 (for the deterministic system) and dde23 (for the delayed system). in fig. 2, we presented the stability region of different equilibrium points when c and β varied simultaneously. this figure shows that, for any given value of β, the system becomes infection-free when virus reproduction is low and does not cross some critical value of c for which r0 = 1. infection, however, persists biomath 9 (2020), 2012297, http://dx.doi.org/10.11145/j.biomath.2020.12.297 page 15 of 20 http://dx.doi.org/10.11145/j.biomath.2020.12.297 d. adak, n. bairagi, r. hakl, accounting for multi-delay effects in an hiv-1 infection model with ... 0 200 400 600 800 1000 0 5 10 15 20 x (i) 0 200 400 600 800 1000 0 20 40 60 y 0 200 400 600 800 1000 0 0.5 1 1.5 2 x 10 4 t v 0 100 200 300 400 500 0 5 10 15 20 (ii) 0 100 200 300 400 500 0 20 40 60 0 100 200 300 400 500 0 0.5 1 1.5 2 x 10 4 t τ 1 = 0.45 τ 2 = 0.6 τ 1 = 0.37 τ 2 = 0.6 fig. 7: time series representation of populations of the system (4) for τ1 = 0.37(< τ∗1 ), depicting stability of e∗ (first panel) and the same for τ1 = 0.45(> τ∗1 ), depicting instability of e ∗. here β = 0.0006, c = 800, τ2 = 0.6 and other parameters are as in fig. 4. for all higher values once c crosses the critical value. persistence of host cells and virus particles in a stable state or oscillatory state depends on the value of virus reproduction rate. populations coexist in a stable state for a small range of c after crossing the critical value and coexist in the oscillatory state for a longer intermediate range of c. all populations, however, coexist in a stable state when c is significantly large. such behavior of the solutions for some fixed values of c are presented in fig. 3. to demonstrate the effect of single delay τ2 on the stability of the system, we plotted the stability and instability regions of different equilibrium points in τ2 − c plane (fig. 4). it shows the interdependency of this delay with the virus reproduction rate in the dynamic behavior of the system. from this figure, one can determine the critical length of delay τ2 for some given value of c for which the system will remain in ae state and if it crosses the critical length then the system will be in an unstable state. on the other hand, if one knows the length of delay τ2 then the value of c can be estimated for obtaining an infection-free system or an infected system with fluctuating or stable populations. to illustrate, we consider c = 800 and compute the critical length of τ2 as τ∗2 = 0.9877, following theorem iii.2. therefore, all populations will remain in stable state for τ2 < 0.9877 and in unstable state if τ2 > 0.9877 (fig. 5). in contrast, one can fixed τ2 (say, τ2 = 0.6) and then plot a bifurcation diagram to show how the system biomath 9 (2020), 2012297, http://dx.doi.org/10.11145/j.biomath.2020.12.297 page 16 of 20 http://dx.doi.org/10.11145/j.biomath.2020.12.297 d. adak, n. bairagi, r. hakl, accounting for multi-delay effects in an hiv-1 infection model with ... behavior changes with respect to c (fig. 6). this figure demonstrates that infection cannot persist if virus production is too low, c < 38. infection persists in oscillatory state for 38 < c < 663 and in stable state for c > 663. to observe the joint effect of both delays on the system dynamics, following theorem iii.3, we pick up any value of τ2, say τ2 = 0.6, from its stable range 0 < τ2 < τ∗2 and then compute the critical value of τ1 as τ∗1 (τ2) = 0.415. thus, e∗ will be stable for τ1 < 0.415 and unstable for τ1 > 0.415 for the fixed value of τ2 = 0.6. this behavior of the system is represented by time series solutions (fig. 7) for τ1 = 0.37 and τ1 = 0.45. one can find all such critical values corresponding to the entire stable range of τ2 ∈ [0,τ∗2 = 0.9877) and obtain the stability region of e∗ in τ1 −τ2 parametric plane (fig. 8). this figure is important to understand the combined effect of two delays on the system dynamics. it shows that one delay is inversely related to the other. it is worth mentioning that the equilibrium e∗ loses its stability and population densities fluctuate around it when the value of (τ1, τ2) lies above the bifurcation line and it remains stable in the opposite case. v. discussion in this paper, we studied a more general model for the basic hiv-1 in-host infection in presence and absence of multiple delays. we considered self-proliferation of cd4+t cells, nonlinear saturated infection rate and recovery of infected cells due to incomplete reverse transcription in a basic hiv model. various cell signalling and biochemical processes are required for a virus to enter into the host cell after attachment to the cell surface and a considerable time is elapsed in accomplishing these activities. we, therefore, incorporated one delay (τ1) in the model system to account for this time. after successful entry, a virus goes through different steps (like uncoating, reverse transcription, circularization, transcription, translation, core particle assembly, final assembly and budding) before producing new virus through cell lysis. to encompass this time, we added another delay (τ2) in the virus growth equation. the objective was to find how the dynamics of a generalized hiv-1 model change in the presence of multiple delays. whether infection persists or not it depends on the basic reproduction number of the system which we have determined from the second generation matrix. in particular, we have shown that elimination of infection is possible if the basic reproduction number is less than unity. on the other hand, the infection persists if the basic reproduction number is greater than unity. local stability of the infected equilibrium of the non-delayed system has been proved under some parametric restrictions. the novelty of this work lies in the fact that it can reciprocate the experimental observations of many studies [35], [14], [15]. it is reported that virus load in infected individuals may be in steadystate or in oscillatory state [59], [60], [61]. here we have shown that the endemic equilibrium e∗ of the non-delayed system may undergo a hopf bifurcation if the virus production rate, c, crosses a critical value, c∗. this implies that the state variables will show stable behaviour if c < c∗ and oscillatory behaviour if c > c∗. thus, both the steady and fluctuating levels of virus count in an hiv patient may be explained once the hopf bifurcation point is known. furthermore, if the non-delayed system is stable, either delay can destabilize the system through hopf bifurcation if the length of delay crosses some critical value. these observations imply that not only the virus production rate but also the disease transmission delay (τ1) and virus maturation delay (τ2) may be the cause of intra-patient variability of cd4+t cells and virus load in an hiv patient. our simulation results showed four types of dynamic behaviors of the system with respect to the virus reproduction rate. if the virus reproduction is too small then the infection can not establish and the system becomes infection-free. infection always persists as the virus production crosses the threshold value. the stable coexistence of all population may be possible either for low or for very high virus production rate. for intermediate virus production rate populations biomath 9 (2020), 2012297, http://dx.doi.org/10.11145/j.biomath.2020.12.297 page 17 of 20 http://dx.doi.org/10.11145/j.biomath.2020.12.297 d. adak, n. bairagi, r. hakl, accounting for multi-delay effects in an hiv-1 infection model with ... τ 1 0 0.2 0.4 0.6 0.8 1.0414 1.2 τ 2 0 0.2 0.4 0.6 0.8 0.9877 1.1 unstable e * stable e * fig. 8: stability and instability regions of the infected equilibrium e∗ in τ1−τ2 plane. the bifurcation line separates the plane into two stability zones. parameters are as in fig. 4 with c = 800 and β = 0.0006. coexist in an oscillatory state. similar complex behavior was observed by srivastava et al. [36] in a four-dimensional hiv-1 model in the presence of delay. we, however, observed similar behavior in a more generalized three dimensional model in the absence of delay. de boer [27], bairagi and adak [28] and xu [26] considered two different types of incidence rates with intracellular delay. however, their models could not explain the periodic fluctuations of viral load in hiv-1 patients even in the presence of delay. the model proposed by zhou et al. [14] shows fluctuations of the endemic equilibrium only in the presence of a delay. whereas, our model shows oscillations for the variation in a virus production number, even when there is no delay. moreover, as we have considered a generalized incidence function, our model is able to reproduce the dynamics of various other hiv infection in-host models as a special case of our model. to summarize, we have been able to show that introduction of the single delay makes a system unstable which might be stable in the absence of delay. we have shown an interdependency between virus reproduction delay and virus production rate. range of virus production for which stable persistence is feasible decreases with the increasing length of virus production delay. a second delay further enhances the instability of the system. the system, which was stable in the presence of a single delay, maybe unstable in the presence of double delay. thus a basic hiv model that considers nonlinear incidence, self-proliferation of host cells and recovery of infected host cells may show a variety of dynamics even in the absence of delay. introduction of delay in such a system further increases the complexity in the dynamics. these observations suggest that virus load in infected individuals may be in steady-state or in an oscillatory state. furthermore, our analysis demonstrates that intra-and inter-patients variabilities of cd4+t cells and virus particles in case of hiv infecbiomath 9 (2020), 2012297, http://dx.doi.org/10.11145/j.biomath.2020.12.297 page 18 of 20 http://dx.doi.org/10.11145/j.biomath.2020.12.297 d. adak, n. bairagi, r. hakl, accounting for multi-delay effects in an hiv-1 infection model with ... tion model may be observed through a number of mechanisms and reciprocate the experimental results that there exist considerable fluctuations in the counts of cd4+t cells and virus particles in the blood of hiv-1 infected individuals. acknowledgements the research of nandadulal bairagi was supported by the csir, ref. no.: 25(0294)/18/emrii. the research of robert hakl was supported by rvo 67985840. references [1] c. b. wilen, j. c. tilton, and r. w. doms, hiv: cell binding and entry, cold spring harb perspect med doi: 10.1101/cshperspect.a006866, 2012. 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[61] a. c. crampin et al., normal range of cd4 cell counts and temporal changes in two hiv negative malawian populations, open aids j., vol. 5, pp. 74-79, 2011. biomath 9 (2020), 2012297, http://dx.doi.org/10.11145/j.biomath.2020.12.297 page 20 of 20 http://dx.doi.org/10.11145/j.biomath.2020.12.297 the model analysis of the non-delayed system positivity and boundedness of solutions equilibria, stability and permanence basic reproductive number local stability of disease-free equilibrium permanence of the system local stability of the interior equilibrium hopf bifurcation analysis of the endemic equilibrium analysis of the delay-induced system numerical simulations discussion references original article biomath 2 (2013), 1312311, 1–10 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum a class of mathematical models describing processes in spatially heterogeneous biofilm communities stefanie sonner bcam basque center for applied mathematics, mazarredo, 14 e48009 bilbao, basque country spain ssonner@bcamath.org received: 11 october 2013, accepted: 31 december 2013, published: 8 january 2014 abstract—we present a class of deterministic continuum models for spatially heterogeneous biofilm communities. the prototype is a single-species biofilm growth model, which is formulated as a highly non-linear system of reaction-diffusion equations for the biomass density and the concentration of the growth controlling substrate. while the substrate concentration satisfies a standard semi-linear reaction-diffusion equation the equation for the biomass density comprises two non-linear diffusion effects: a porous medium-type degeneracy and super diffusion. when further biofilm processes are taken into account equations for several substrates and multiple biomass components have to be included in the model. the structure of these multi-component extensions is essentially different from the mono-species case, since the diffusion operator for the biomass components depends on the total biomass in the system and the equations are strongly coupled. we present the prototype biofilm growth model and give an overview of its multi-component extensions. moreover, we summarize analytical results that were obtained for these models. keywords-biofilm; antibiotic disinfection; probiotic control; quorum-sensing; degenerate reaction-diffusion systems; i. introduction the dominant mode of microbial life in aquatic ecosystems are biofilm communities rather than planktonic cultures ([1]). biofilms are dense aggregations of microbial cells encased in a slimy extracellular matrix forming on biotic or abiotic surfaces (called substrata) in aqueous surroundings. such multicellular communities are a very successful life form and able to tolerate harmful environmental impacts that would eradicate free floating individual cells ([3], [17]). whenever environmental conditions allow for bacterial growth, microbial cells can attach to a substratum and switch to a sessile life form. they start to grow and divide and produce a gel-like layer of extracellular polymeric substances (eps) often forming complex spatial structures. the selfproduced eps yields protection and allows survival in hostile environments. for instance, the mechanisms of antibiotic resistance in biofilm cultures are essentially different from those of free swimming cells, which makes it difficult to eradicate bacterial biofilm infections. the eps retards diffusion of antibiotics and the antibiotic agents fail to penetrate into the inner cores of the biofilm ([3], [17], [4]). biofilms play a significant role in various fields. they are beneficially used in environmental engineering technologies for groundwater protection and wastewater treatment. however, in most occurrences biofilm formations have negative effects. if they form on implants and natural surfaces in the human body they can provoke bacterial infections such as dental caries and otitis media ([3]). biofilm contamination can lead to health risks in food processing environments, and biofouling of industrial equipment or ships can cause severe economic citation: stefanie sonner, a class of mathematical models describing processes in spatially heterogeneous biofilm communities, biomath 2 (2013), 1312311, http://dx.doi.org/10.11145/j.biomath.2013.12.311 page 1 of 10 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2013.12.311 s sonner, models for processes in spatially heterogeneous biofilm communities defects for the industry ([4], [21]). mathematical models of biofilms have been studied for several decades. they range from traditional onedimensional models that describe biofilms as homogeneous flat layers, to more recent twoand threedimensional biofilm models that account for the spatial heterogeneity of biofilm communities. a variety of mathematical modeling concepts has been suggested, including discrete stochastic particle based models and deterministic continuum models, that are based on the description of the mechanical properties of biofilms ([7], [21]). we are concerned with the latter, where biofilm and liquid surroundings are assumed to be continua, and its time evolution is governed by systems of deterministic pdes. the first continuum model [22] was a one-dimensional biofilm growth model and essentially based on the assumption that biofilms are homogeneous flat layers. such models serve well for engineering applications on the macro-scale (larger than 1cm) are, however, not capable to predict the often highly irregular spatial structure of microbial populations and the behavior of biofilms on the meso-scale (between 50µm and 1mm), the actual length scale of mature biofilms ([7]). biofilms can form mushroom-like caps and contain clusters and channels, where substrates can circulate. cells in different regions of the biofilm live in diverse micro-environments and exhibit differing behavior ([3]). to capture the spatial heterogeneity of biofilms a higher dimensional biofilm growth model was proposed in [6], which is based on the interpretation of a biofilm as a continuous, spatially structured microbial population. the essential difficulty is to model the spatial spreading mechanism of biomass and to reproduce the growth characteristics of biofilms that have been observed in experiments ([6]): • biofilm and aqueous surroundings are separated by a sharp interface. • the biomass density is bounded by a known maximum value. • spatial spreading only takes place where the local biomass density approaches values close to its maximum possible value, while it does not occur in regions where the biomass density is low. the mathematical model [6] is formulated as system of highly non-linear reaction-diffusion equations for the biomass density and the concentration of a growth limiting nutrient, and is the prototype of the biofilm models we consider. while the substrate concentration satisfies a standard semi-linear reaction-diffusion equation the governing equation for the biomass density exhibits two non-linear diffusion effects. the biomass diffusion coefficient degenerates like the porous medium equation and shows super diffusion, which causes difficulties in the mathematical analysis of the model. it was shown by numerical experiments that the model is capable of predicting the heterogeneous spatial structure of biofilms and is in good agreement with experimental findings ([6]). in [10] and [9] the model was studied analytically. in particular, the well-posedness of the model and the existence of a compact global attractor was shown. the prototype single-species single-substrate model was extended to model biofilms which consist of several types of biomass and account for multiple dissolved substrates. the model introduced in [4] describes the diffusive resistance of biofilms against the penetration by antibiotics. in [13] an amensalistic biofilm control system was modelled, where a beneficial biofilm controls the growth of a pathogenic biofilm. the structure of these multi-species models differs essentially from the mono-species model, and the analytical results for the prototype model could not all be carried over to the more involved multi-species case. in both articles, existence proofs for the solutions were given, and numerical studies presented, but the question of uniqueness of solutions remained unanswered in [4] and [13]. recently, another multi-component biofilm model was proposed in [11], which describes quorum-sensing in growing biofilm communities. quorum-sensing is a cell-cell communication mechanism used by bacteria to coordinate behavior in groups. the model behavior was studied by numerical experiments in [11] and [21], analytical questions were addressed in [21]. compared to the multicomponent biofilm models [4] and [13], the particularity of the quorum-sensing model is, that adding the governing equations for the involved biomass components we recover exactly the monospecies biofilm growth model. taking advantage of the known results for the prototype model the existence and uniqueness of solutions and the continuous dependence of solutions on initial data could be established in [21]. it is the first uniqueness result for multi-species reaction-diffusion models of biofilms that extend the single-species model [6]. in section ii we introduce the prototype biofilm growth model. multicomponent extensions are addressed in section iii. in section iv we give an overview of the analytical results obtained for these models. for numerical simulations and further details we refer to [6], biomath 2 (2013), 1312311, http://dx.doi.org/10.11145/j.biomath.2013.12.311 page 2 of 10 http://dx.doi.org/10.11145/j.biomath.2013.12.311 s sonner, models for processes in spatially heterogeneous biofilm communities [10], [13], [4], [21] and [8]. in [8] also extensions in other directions of the prototype biofilm growth model are discussed, e.g., models that take the effect of hydrodynamics into account. ii. prototype biofilm growth model the multi-dimensional biofilm growth model proposed in [6] is formulated as a non-linear reaction-diffusion system for the biomass density and the concentration of the growth controlling nutrient in a bounded domain ω ⊂ rn, n = 1, 2, 3, where the boundary of the domain ∂ω is piecewise smooth. in dimensionless form the substrate concentration s is scaled with respect to the bulk concentration, and the biomass density is normalized with respect to the maximal bound for the cell density. consequently, the dependent model variable m represents the volume fraction occupied by biomass. the eps is implicitly taken into account, in the sense that the biomass volume fraction m describes the sum of biomass and eps assuming that their volume ratio is constant. both unknown functions depend on the spatial variable x ∈ ω and time t ≥ 0, and satisfy the parabolic system ∂ts = ds∆s −k1 sm k2 + s , (1) ∂tm = do · (d(m)om) + k3 sm k2 + s −k4m, m|∂ω = 0, s|∂ω = 1, m|t=0 = m0, s|t=0 = s0, where the constants d,ds and k2 are positive, the constants k1,k3 and k4 are non-negative. furthermore, ∂t denotes the partial derivative with respect to time t > 0, ∆ the laplace operator and o the gradient with respect to the spatial variable x ∈ ω and · the inner product in rn. the solid region occupied by the biofilm as well as the liquid surroundings are assumed to be continua. the actual biofilm is described by the region ω1(t) := {x ∈ ω | m(x,t) > 0}, and the liquid area by ω2(t) := {x ∈ ω | m(x,t) = 0}. the substratum, on which the biofilm grows, is part of the boundary ∂ω. the constants in system (1) have the following meaning, for further details and their typical values in applications we refer to [6]. fig. 1. biofilm domain ds substrate diffusion coefficient d biomass motility constant k1 maximum specific consumption rate k2 monod half saturation constant k3 maximum specific growth rate k4 biomass decay rate a,b biomass spreading parameters biomass is produced due to the consumption of nutrients, which is described the monod reaction functions −k1 sm k2 + s and k3 sm k2 + s . natural cell death is also included in the model and given by the lysis rate k4 in the equation for the biomass fraction. while the nutrient is dissolved in the domain and the substrate concentration satisfies a standard semilinear reaction-diffusion equation, the spatial spreading of biomass is determined by the density-dependent diffusion coefficient d(m) = ma (1 −m)b a,b ≥ 1. the biomass motility constant d is small compared to the diffusion coefficient ds of the dissolved substrate, which reflects that the cells are to some extent immobilized in the eps matrix. accumulation of biomass leads to spatial expansion of the biofilm. we observe that the biomass diffusion coefficient vanishes when the total biomass approaches zero and blows up when the biomass density tends to its maximum value. the polynomial degeneracy ma is well-known from the porous medium equation and guarantees that spatial spreading is negligible for low values of m. moreover, it yields the separation of biofilm and liquid phase, that is, a finite speed of interface propagation. spreading of biomass takes place when and where the biomass fraction takes values close to its maximal value. when m = 1 instantaneous spreading occurs, which is known as the effect of super diffusion. this singularity at biomath 2 (2013), 1312311, http://dx.doi.org/10.11145/j.biomath.2013.12.311 page 3 of 10 http://dx.doi.org/10.11145/j.biomath.2013.12.311 s sonner, models for processes in spatially heterogeneous biofilm communities m = 1 of the biomass diffusion coefficient ensures the maximal bound for the biomass density. it was shown in numerical experiments that the model (1) is in good agreement with experimental findings and is capable to reproduce the irregular, heterogeneous spatial structure of biofilms observed on the mesoscale ([6], [10]). more precisely, the simulations show that the biofilm develops a rather regular, homogeneous structure if nutrients are nowhere limited in the system. on the other hand, when the nutrient supply is not symmetric and nutrients become limited the colonies grow in the direction of higher nutrient concentrations, which can lead to cluster and channel morphologies and mushroomshaped architectures. iii. multicomponent biofilm models the prototype biofilm growth model (1) was extended to incorporate further biofilm processes. it requires to distinguish different types of biomass and dissolved substrates and to include governing equations for these multiple biomass fractions and dissolved substrates in the model. the pattern of the multi-component biofilm models is essentially different from the prototype model, the equations for the biomass components are strongly coupled through the diffusion operators. in this section we discuss multi-component biofilm models, that were proposed and studied in [4], [5], [11], [13], [21]. a. antibiotic disinfection of biofilms the first multi-species multi-substrate generalization of the prototype model (1) was suggested in [5]. in [4] existence results for the solutions were established and numerical simulations presented. the model describes a growing biofilm community and its disinfection by antimicrobial agents. bacteria in biofilm populations are better protected than free floating cells and behave essentially different under antibiotic treatment. the eps retards diffusion of antimicrobial agents into the biofilm, cells in the outer layers are attacked first while bacteria in the inner cores are well protected and continue to grow. the dependent model variables are: s nutrient concentration b concentration of the antimicrobial agent x volume fraction occupied by active biomass y volume fraction occupied by inert biomass the dissolved nutrient s controls the growth of the biomass, and the antimicrobial agent b regulates the disinfection process. as previously, the eps is implicitly taken into account, and the total biomass fraction m := x+y is normalized with respect to the maximum bound for the cell density. in dimensionless form the model is represented by the parabolic system ∂ts = ds∆s −k1 sx k2 + s , (2) ∂tb = db∆b − ζ1bx, ∂tx = do · (d(m)ox) + k3 sx k2 + s −k4x − ζ2bx, ∂ty = do · (d(m)oy ) + ζ2bx, with non-negative and bounded initial and boundary data x|∂ω = 0, y |∂ω = 0, s|∂ω = sr, b|∂ω = br, x|t=0 = x0, y |t=0 = y0, s|t=0 = s0, b|t=0 = b0, where we use the same notations as in (1). the additional constants ζ1,ζ2 and db in (2) are positive and have the following meaning: db diffusion coefficient of antibiotics ζ1 antibiotics consumption rate ζ2 inert biomass production rate apart from the diffusion of the dissolved substrates and the death, growth and spatial spreading of biomass the disinfection mechanism is included in the model. during this process antibiotic agents are consumed and active biomass is directly converted into inert biomass, which is determined by the disinfection parameters ζ1 and ζ2. like in the mono-species model, the production of active biomass due to the consumption of nutrients is described by monod reaction functions. in the absence of antimicrobial agents and inert biomass, the model reduces to the single species biofilm growth model (1). in [4] numerical simulations were presented to illustrate the model behavior and to analyze the efficiency of different disinfection strategies. the numerical experiments show that cells in the outer layers of the biofilm are attacked first while cells in the inner scores remain protected and survive longer. b. amensalistic biofilm control system the model of an amensalistic biofilm control system [13] extends the single-species probiotic model [14] and possesses a similar structure as the model of antibiotic biomath 2 (2013), 1312311, http://dx.doi.org/10.11145/j.biomath.2013.12.311 page 4 of 10 http://dx.doi.org/10.11145/j.biomath.2013.12.311 s sonner, models for processes in spatially heterogeneous biofilm communities disinfection. in [13] existence results for the solutions were established and numerical simulations presented. the model describes how a beneficial biofilm controls the growth of a pathogenic biofilm community by alternating the environmental conditions. the probiotic biofilm modifies the local concentration of protonated lactic acids, which decreases the ph concentration and deteriorates the growth conditions for the pathogens, while the controlling bacteria are more tolerant to these changes. the dependent model variables are: c concentration of protonated lactic acids p concentration of hydrogen ions x volume fraction occupied by pathogens y volume fraction occupied by probiotics z volume fraction occupied by inert biomass in dimensionless form the model is represented by the parabolic system ∂tc = dc∆c + α1x(ζ1 −c) + α2y (ζ1 −c), (3) ∂tp = dp ∆p + α3c(ζ2 −p), ∂tx = do · (d(m)ox) + µ1ψ1(c,p)x, ∂ty = do · (d(m)oy ) + µ2ψ2(c,p)y, ∂tz = do · (d(m)oz) − min{0,µ1ψ1(c,p)x} − min{0,µ2ψ2(c,p)y}, with non-negative and bounded initial and boundary data c|∂ω = cr, p |∂ω = pr, x|∂ω = 0, y |∂ω = 0, z|∂ω = 0, c|t=0 = c0, p |t=0 = p0, x|t=0 = x0, y |t=0 = y0, z|t=0 = z0, where we use the same notations as in (1). the constants dc,dp ,α1,α2,α3,µi and ζi, i = 1, 2, are positive and have the following meaning: dc diffusion coefficient of protonated lactic acids dp diffusion coefficient of hydrogen ions α1 acid production rate by pathogens α2 acid production rate by probiotics α3 hydrogen ions production rate µ1 maximum growth rate of pathogens µ2 maximum growth rate of probiotics ζ1 acid saturation level ζ2 hydrogen ion saturation level ζ1i pathogen growth kinetics, i = 1, . . . , 4 ζ2i probiotics growth kinetics, i = 1, . . . , 4 inert probiotics and pathogens are not distinguished in the model. as previously, the eps is implicitly taken into account, and the total biomass fraction m := x +y +z is normalized with respect to the maximum bound for the cell density. protonated lactic acids c are produced by both bacterial species until a saturation level is reached. the hydrogen ion concentration p is related to the local ph value by ph = − log p. it increases, facilitated by the protonated lactic acids, until a threshold value is archived. the growth and inhibition functions ψ1 and ψ2 are piecewise linear such that they are positive if c and p are small, and negative if c or p becomes large. between the growth and inhibition range there is an extended neutral range. more precisely, the functions ψi are given by ψi(c,p) = min { 1 − c hi1(c) , 1 − p hi2(p) } , i = 1, 2, where hi1 and h i 2 are defined as hi1(c) = ζ i 1h(ζ i 1 −c) + ch(c − ζ i 1)h(ζ i 2 −c) + h(c − ζi2), hi2(p) = ζ i 3h(ζ i 3 −p) + ph(p − ζ i 3)h(ζ i 4 −p) + h(p − ζi4), for i = 1, 2. moreover, the function h is given by h(s) :=   1 s > 0, 1 2 s = 0, 0 s < 0, s ∈ r, and the constants ζ1j and ζ 2 j , j = 1, . . . , 4, are positive with ζi1 < ζ i 2, ζ i 3 < ζ i 4, i = 1, 2. for the probiotic strategy to be effective we require that ζ2j ≥ ζ 1 j , j = 1, . . . , 4 ([13]). the mechanism of probiotic control is different from traditional antibiotic control strategies of biofilms, where the inner layers of the film are protected by the outer layers and the antibiotics fail to fully penetrate the biofilm. the numerical experiments for the probiotic biofilm model in [13] show that pathogens in the core of the biofilm, close to the substratum, are eradicated first. c. quorum-sensing in patchy biofilm communities a model for quorum-sensing in growing biofilm communities was proposed and studied by numerical simulations in [11]. it extends the prototype biofilm growth biomath 2 (2013), 1312311, http://dx.doi.org/10.11145/j.biomath.2013.12.311 page 5 of 10 http://dx.doi.org/10.11145/j.biomath.2013.12.311 s sonner, models for processes in spatially heterogeneous biofilm communities model (1) and combines it with the model for quorumsensing in planktonic cultures in [16]. analytical aspects of the quorum-sensing model [11] were addressed in [21]. quorum-sensing is a cell-cell communication mechanism used by bacteria to coordinate gene expression and behavior in groups. bacteria constantly produce low amounts of signaling molecules that are released into the environment. accumulation of autoinducers triggers a response by the cells and since the producing cells respond to their own signals the molecules are also called autoinducers ([16], [12]). when the concentration of autoinducers locally passes a certain threshold, the cells are rapidly induced, and switch from a so-called downregulated to an up-regulated state. in an up-regulated state they typically produce the signaling molecule at a highly increased rate ([11]). the dependent model variables are: s concentration of growth controlling substrate a concentration of autoinducers x volume fraction occupied by down-regulated cells y volume fraction occupied by up-regulated cells in dimensionless form the model is represented by the parabolic system ∂ts = ds∆s −k1 sm k2 + s , (4) ∂ta = da∆a−γa + αx + (α + β)y, ∂tx = do · (d(m)ox) + k3 xs k2 + s −k4x −k5|a|mx + k5|y |, ∂ty = do · (d(m)oy ) + k3 y s k2 + s −k4y + k5|a|mx −k5|y |, with non-negative and bounded initial and boundary data s|∂ω = 1, a|∂ω = 0, x|∂ω = 1, y |∂ω = 0, s|t=0 = s0, a|t=0 = a0, x|t=0 = x0, y |t=0 = y0, where we use the same notations as in (1) and |·| denotes the absolute value. the constants da and γ are positive, m ≥ 1, and α,β and k5 are non-negative. moreover, we require that α + β > γ. apart from the constants in the prototype model (1) the constants in (4) have the following meaning: da diffusion coefficient of autoinducers k5 up-regulation rate α autoinducer production rate of down-regulated cells β increased autoinducer production rate of up-regulated cells γ abiotic decay rate of autoinducers m polymerization exponent the total biomass density m = x + y is normalized with respect to the maximum bound for the cell density and the eps is implicitly taken into account. assuming that induction switches the cells between downand upregulated states without changing their growth behavior we can assume that the spatial spreading of both biomass fractions is described by the same diffusion operator. the biomass motility constant d is small compared to the diffusion coefficients ds and da of the dissolved substrates. like in the mono-species biofilm growth model, biomass is produced due to the consumption of nutrients, which is described by the monod reaction functions k3 xs k2 + s and k3 y s k2 + s in the equations for the biomass fractions x and y . natural cell death is included in the model and determined by the lysis rate k4. if we do not distinguish between down-regulated and up-regulated cells in the model (4), we recover the prototype biofilm growth model (1) for the total biomass m = x + y and the growth controlling nutrient s. the autoinducer concentration a is normalized with respect to the threshold concentration for induction. down-regulated cells produce the signaling molecule at rate α, while up-regulated cells produce it at the increased rate α +β, where β is one order of magnitude larger than α. due to an increase of the autoinducer concentration a, down-regulated cells are converted into up-regulated cells at rate k5am. in applications, typical values for the degree of polymerization are 2 < m < 3 ([11], [21]). up-regulated cells are converted back into down-regulated cells at constant rate k5. if the molecule concentration a < 1 the latter effect dominates, if a > 1 up-regulation is super-linear. moreover, abiotic decay of signaling molecules is taken into account in the model and determined by the constant rate γ. the numerical simulations for the model in [21] indicate that quorum-sensing in spatially structured biofilm populations does not only depend on the local cell biomath 2 (2013), 1312311, http://dx.doi.org/10.11145/j.biomath.2013.12.311 page 6 of 10 http://dx.doi.org/10.11145/j.biomath.2013.12.311 s sonner, models for processes in spatially heterogeneous biofilm communities density of the population but also on mass transfer effects. namely, the location of the cell colonies relative to each other and on the prescribed boundary conditions for the substrates. iv. analytical results in this section we give an overview of the analytical results obtained for the prototype biofilm growth model (1) and the multicomponent models (2), (3) and (4) in [10], [4], [13], [21]. a. prototype biofilm growth model a solution theory for the prototype model (1) was developed in [10]. in particular, the well-posedness was established and the existence of a compact global attractor was shown. we recall the main results in [10] regarding the wellposedness of the model (1). the following theorem yields the existence and regularity results for the solutions (theorem 3.1, [10]). theorem 1. we assume the initial data satisfies s0 ∈ l∞(ω) ∩h1(ω), s0|∂ω = 1, m0 ∈ l∞(ω), f(m0) ∈ h10 (ω), 0 ≤ s0 ≤ 1, 0 ≤ m0 in ω, ‖m0‖l∞(ω) < 1, where the function f(v) := ∫ v 0 za (1−z)b dz, for 0 ≤ v < 1. then, there exists a solution (s,m) satisfying system (1) in the sense of distributions, and the solution belongs to the class m,s ∈ l∞(ω × (0,∞)) ∩c([0,∞); l2(ω)), f(m),s ∈ l∞((0,∞); h1(ω)) ∩c([0,∞); l2(ω)), 0 ≤ s,m ≤ 1 in ω × (0,∞), ‖m‖l∞(ω×(0,∞)) < 1. moreover, it was shown that the solutions are l1(ω)lipschitz continuous with respect to initial data, which implies its uniqueness. the following result recalls theorem 3.2 in [10]. proposition 1. let (s,m) and (s̃,m̃) be two solutions of system (1) corresponding to initial data (s0,m0), (s̃0,m̃0) respectively, and the initial data satisfy the assumptions of the previous theorem. then, the following estimate holds ‖s(t) − s̃(t)‖l1(ω) + ‖m(t) −m̃(t)‖l1(ω) ≤ ect ( ‖s0 − s̃0‖l1(ω) + ‖m0 −m̃0‖l1(ω) ) for t ≥ 0 and some constant c ≥ 0. we shortly indicate the main ideas of the proofs, for all details and further results we refer to [10]. the solutions for the degenerate system (1) are obtained as limits of the solutions of smooth regular approximations. more precisely, the non-degenerate auxiliary systems for the single-species model are given by ∂ts = ds∆s −k1 sm k2 + s , (5) ∂tm = do · (d�(m)om) + k3 sm k2 + s −k4m, m|∂ω = 0, s|∂ω = 1, m|t=0 = m0, s|t=0 = s0, where the diffusion coefficient in the equation for the biomass fraction in (1) is replaced by the regularized function d�(m) :=   �a m < 0, (m+�)a (1−m)b 0 ≤ m ≤ 1 − �, 1 �b m ≥ 1 − �, m ∈ r, for small � > 0. first, the non-negativity of the approximate solutions (s�,m�) is shown and uniform a-priori l∞(ω)-bounds are established for all sufficiently small � > 0. this is archived by comparison principles and the construction of appropriate barrier functions (proposition 1, [10]). once a-priori bounds for the solutions are known the existence of solutions of the regular auxiliary systems (5) follows by standard arguments ([15]). we further remark that the fast diffusion effect prevents the biomass density from attaining the singular value. more precisely, if the initial data satisfies ‖m0‖l∞(ω) = 1 − δ for some 0 < δ < 1, then, there exists 0 < η < 1 such that the solutions of the nondegenerate approximations (5) satisfy ‖m�(t)‖l∞(ω) ≤ 1 −η, t ≥ 0, for all sufficiently small � > 0 (proposition 6, [10]). having established uniform a-priori bounds we can pass to the limit � tends to zero in the distributional formulation of the equation for the nutrient concentration in (5) and obtain a solution s of the original degenerate problem (1). in the governing equation for the biomass density the passage to the limit in the reaction terms is also immediate, the difficulty is to justify the limit for the biomass diffusion terms (see the proof of theorem 3.1, [10]). the lipschitz continuity with respect to initial data in l1(ω)-norm can be deduced from kato’s inequality and is shown in proposition 2 and theorem 3.2, [10]. biomath 2 (2013), 1312311, http://dx.doi.org/10.11145/j.biomath.2013.12.311 page 7 of 10 http://dx.doi.org/10.11145/j.biomath.2013.12.311 s sonner, models for processes in spatially heterogeneous biofilm communities b. multicomponent biofilm models the pattern of the multi-component biofilm models is essentially different from the prototype model, the equations are strongly coupled through the diffusion operators and the analytical results for the single-species model could not all be carried over. in [4] and [13] the behavior of solutions was studied in numerical simulations and the existence of solutions was established, but the question of uniqueness of solutions remained unanswered in both cases. the first uniqueness result for multispecies models was obtained in [21]. 1) antibiotics and probiotics model: the following existence result for solutions of the antibiotics model (2) was shown in [4] (theorem 2.3). theorem 2. we assume the functions br and sr are nonnegative and belong to the class l∞(∂ω). moreover, if the initial data x0,y0,s0,b0 are non-negative, belong to l∞(ω) and satisfy 0 ≤ s0 ≤ 1 in ω, ‖x0 + y0‖l∞(ω) < 1, then, there exists a global solution of the antibiotics model, the functions s,b,x and y belong to the space l∞(ω × (0,∞)), are non-negative and satisfy system (2) in distributional sense. it was shown in [4] that the limit of solutions of non-degenerate approximations yields a solution of the degenerate problem (2). the smooth regular approximations for the antibiotics model are obtained from the system of equations (2) by replacing the diffusion coefficient d(m) in the equations for the biomass components by the regularized function d�(m), ∂tx = do · (d�(m)ox) + k3 sx k2 + s −k4x − ζ2bx, ∂ty = do · (d�(m)oy ) + ζ2bx. the non-negativity and uniform boundedness of the solutions (s�,b�,x�,y�) of the auxiliary systems can be deduced from comparison principles and by constructing suitable barrier functions. once a-priori l∞(ω)-bounds are known, the existence of solutions of the regular approximations follows by standard arguments ([15]). using the uniform a-priori bounds for the solutions of the non-degenerate approximations we can pass to the limit � tends to zero in the distributional formulation of the equations for the dissolved substrates. the difficulty is to justify the limit for the diffusion terms in the governing equations for the biomass fractions. since the equations are strongly coupled, the proof requires different arguments than the ones applied for the single-species model in [10]. for all details and the proof of theorem 2 we refer to [4]. the structure of the probiotics model (3) is similar to the structure of the antibiotics model (2) and the existence of solutions was shown by similar arguments (theorem 3.3, [13]). theorem 3. we assume the functions cr and pr belong to the class l∞(∂ω) and satisfy 0 ≤ cr ≤ ζ1, 0 ≤ pr ≤ ζ2. moreover, if the initial data c0,p0,x0,y0,z0 are nonnegative, belong to l∞(ω) and satisfy 0 ≤ c0 ≤ ζ1, 0 ≤ p0 ≤ ζ2 in ω, ‖x0 + y0 + z0‖l∞(ω) < 1, then, there exists a global solution of the probiotics model, the functions c,p,x,y and z belong to l∞(ω × (0,∞)), are non-negative and satisfy system (3) in distributional sense. for the proof of theorem 3 we refer to [13]. further analytical results were not obtained for the models (2) and (3). in particular, the question of uniqueness of solutions remained unanswered in [4] and [13]. 2) quorum-sensing model: a different approach was developed for the quorum-sensing model (4) in [21], which led to a uniqueness result for the solutions. the following theorem states the well-posedness of the model (theorem 3.5 and theorem 3.11, [21]). theorem 4. let the initial data satisfy x0,y0,a0 ∈ h10 (ω), s0 ∈ h 1(ω) such that s0|∂ω = 1, and 0 ≤ s0,x0,y0,a0 ≤ 1 in ω, ‖x0 + y0‖l∞(ω) < 1. then, there exists a unique global solution of the quorum-sensing model (4), a,s,x,y ∈ c([0,∞); l2(ω)) ∩l∞(ω × [0,∞)), a,s ∈ l2((0,∞); h1(ω)), d(m)ox, d(m)oy ∈ l2((0,∞); l2(ω; rn)), the functions a,s,x and y are non-negative and satisfy system (4) in distributional sense. the particularity of the quorum-sensing model (4) is that adding the equations for the biomass fractions x and y we recover exactly the prototype biofilm growth model (1) for the total biomass m = x + y and the growth limiting nutrient s. hence, we may regard m biomath 2 (2013), 1312311, http://dx.doi.org/10.11145/j.biomath.2013.12.311 page 8 of 10 http://dx.doi.org/10.11145/j.biomath.2013.12.311 s sonner, models for processes in spatially heterogeneous biofilm communities and s as known functions, and the problem reduces to show the well-posedness of the semi-linear degenerate problem for the biomass fraction x and the signaling molecule concentration a, ∂ta = da∆a−γa + αx + (α + β)(m −x), ∂tx = do · (d(m)ox) + k3 xs k2 + s −k4x −k5|a|mx + k5|m −x|. like for the antibiotics and probiotics model the existence of solutions was established by considering non-degenerate approximations, where the diffusion coefficient d(m) in the equations for the biomass fractions in (4) is replaced by the regularized function d�(m), ∂tx = do · (d�(m)ox) + k3 xs k2 + s −k4x −k5|a|mx + k5|y |, ∂ty = do · (d�(m)oy ) + k3 y s k2 + s −k4y + k5|a|mx −k5|y |. moreover, using the results obtained for the solutions of the single-species model in [10], further regularity results for the solutions could be established, which led to a uniqueness result. for further details and the proofs we refer to [21]. v. concluding remarks the existence results in the previous section are formulated assuming homogeneous dirichlet boundary conditions for the biomass components. this situation resembles growing biofilms without substratum, which are commonly called microbial flocs. such bacterial aggregates enclosed in an eps matrix are used in the industry for wastewater treatment and also occur in natural settings ([18]). boundary conditions of mixed type are, however, often more appropriate in applications. typically, dirichlet conditions are specified on some part of the boundary, while neumann or robin conditions are imposed on the other parts. in particular, the substratum, on which the biofilm grows is impermeable for all dependent variables, which is described by homogeneous neumann boundary values. all existence proofs of the previous section carry over to these more general situations as long as homogeneous dirichlet conditions are imposed for the biomass fractions on one part of the boundary (for details see theorem 4.1, [10]). on the other hand, if homogeneous neumann conditions are assumed for all biomass components and constant dirichlet conditions for the nutrient concentration, which reflects the situation that no biomass can leave the system and nutrients are constantly added, it was shown that the biomass density reaches the singular value in finite time (proposition 7 and proposition 8, [10]). we finally remark that the solution theory for the single-species model was also extended to less regular initial data. it suffices to assume that the total biomass initially fulfils m0 ∈ l∞(ω) and 0 ≤ m0 ≤ 1. if the biomass components satisfy homogeneous dirichlet conditions on one part of the boundary, the biomass spreading mechanism is strong enough to prevent the solution from attaining the singular value (see theorem 3.5, [10]). acknowledgment the author would like to thank messoud a. efendiev and the anonymous referees for their valuable comments and remarks. references [1] battin t.j., sloan w.t., kjelleberg s., daims h., head i.m., curtis t.p., eberl l., microbial landscapes: new paths to biofilm research, nature reviews microbiology 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[17] o’toole g.a., stewart p.s., biofilms strike back, nature biotechnology 23(11), pp.1378–1379, 2005. http://dx.doi.org/10.1038/nbt1105-1378 [18] parsek m.r., greenberg e.p., sociomicrobiology: the connections between quorum sensing and biofilms, trends in microbiology 13(1), pp.27–33, 2005. http://dx.doi.org/10.1016/j.tim.2004.11.007 [19] renardy m., rogers r.c., an introduction to partial differential equations, second edition, springer-verlag, new york, 2004. [20] smoller j., shock waves and reaction-diffusion equations, second edition, springer-verlag, new york, 1994. http://dx.doi.org/10.1007/978-1-4612-0873-0 [21] sonner s., efendiev m.a., eberl h.j., on the well-posedness of a mathematical model of quorum-sensing in patchy biofilm communities, mathematical methods in the applied sciences 34(13), pp.1667–1684, 2011. http://dx.doi.org/10.1002/mma.1475 [22] wanner o., gujer w., a multispecies biofilm model, biotechnology and bioengineering 28(3), pp.314–328, 1986. biomath 2 (2013), 1312311, http://dx.doi.org/10.11145/j.biomath.2013.12.311 page 10 of 10 http://dx.doi.org/10.1038/nbt1105-1378 http://dx.doi.org/10.1016/j.tim.2004.11.007 http://dx.doi.org/10.1007/978-1-4612-0873-0 http://dx.doi.org/10.1002/mma.1475 http://dx.doi.org/10.11145/j.biomath.2013.12.311 introduction prototype biofilm growth model multicomponent biofilm models antibiotic disinfection of biofilms amensalistic biofilm control system quorum-sensing in patchy biofilm communities analytical results prototype biofilm growth model multicomponent biofilm models antibiotics and probiotics model quorum-sensing model concluding remarks references www.biomathforum.org/biomath/index.php/biomath review article a survey of adaptive cell population dynamics models of emergence of drug resistance in cancer, and open questions about evolution and cancer jean clairambault∗, camille pouchol† ∗inria paris & sorbonne université, umr 7598, ljll, bc 187, 75252 paris cedex 05, france jean.clairambault@ljll.math.upmc.fr †department of mathematics kth, brinellvägen 8, 114 28 stockholm, sweden pouchol@kth.se received: 9 march 2019, accepted: 14 may 2019, published: 24 may 2019 abstract—this article is a proceeding survey (deepening a talk given by the first author at the biomath 2019 international conference on mathematical models and methods, held in będlewo, poland) of mathematical models of cancer and healthy cell population adaptive dynamics exposed to anticancer drugs, to describe how cancer cell populations evolve toward drug resistance. such mathematical models consist of partial differential equations (pdes) structured in continuous phenotypes coding for the expression of drug resistance genes; they involve different functions representing targets for different drugs, cytotoxic and cytostatic, with complementary effects in limiting tumour growth. these phenotypes evolve continuously under drug exposure, and their fate governs the evolution of the cell population under treatment. methods of optimal control are used, taking inevitable emergence of drug resistance into account, to achieve the best strategies to contain the expansion of a tumour. this evolutionary point of view, which relies on biological observations and resulting modelling assumptions, naturally extends to questioning the very nature of cancer as evolutionary disease, seen not only at the short time scale of a human life, but also at the billion year-long time scale of darwinian evolution, from unicellular organisms to evolved multicellular organs such as animals and man. such questioning, not so recent, but recently revived, in cancer studies, may have consequences for understanding and treating cancer. copyright: c©2019 clairambault et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: jean clairambault, camille pouchol, a survey of adaptive cell population dynamics models of emergence of drug resistance in cancer, and open questions about evolution and cancer, biomath 8 (2019), 1905147, http://dx.doi.org/10.11145/j.biomath.2019.05.147 page 1 of 23 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2019.05.147 j. clairambault, c. pouchol, a survey of adaptive cell population dynamics models of emergence ... some open and challenging questions may thus be (non exhaustively) listed as: • may cancer be defined as a spatially localised loss of coherence between tissues in the same multicellular organism, ‘spatially localised’ meaning initially starting from a given organ in the body, but also possibly due to flaws in an individual’s epigenetic landscape such as imperfect control of differentiation genes? • if one assumes that “the genes of cellular cooperation that evolved with multicellularity about a billion years ago are the same genes that malfunction in cancer.” (davies and lineweaver, 2011), how can these genes be systematically investigated, looking for zones of fragility that depend on individuals in the ‘tinkering’ (f. jacob, 1977) evolution is made of, tracking local defaults of coherence? • what is such coherence made of and to what extent is the immune system responsible for it (the self and differentiation within the self)? related to this question of self, what parallelism can be established between the development of multicellularity in different species proceeding from the same origin and the development of the immune system in these different species? keywords-cell population dynamics; structured models; darwinian evolution; druginduced drug resistance; cancer therapeutics; optimal control i. introduction: motivation from and focus on drug resistance in cancer slow genetic mechanisms of ‘the great evolution’ that has designed multicellular organisms, together with fast reverse evolution on smaller time windows, at the scale of a human disease, may explain transient or established drug resistance. this will be developed around the socalled atavistic hypothesis of cancer. plasticity in cancer cells, i.e., epigenetic [30], [64] (much faster than genetic mutations, and reversible) propension to reversal to a stemlike, de-differentiated phenotypic status, resulting in fast adaptability of cancer cell populations, makes them amenable to resist abrupt drug insult (high doses of cytotoxic drugs, ionising radiations, very low oxygen concentrations in the cellular medium) as response to cellular stress. intra-tumour heterogeneity with respect to drug resistance potential, meant here to model between-cell phenotypic variability within cancer cell populations, is a good setting to represent continuous evolution towards drug resistance in tumours. this is precisely what is captured by mathematical (pde) models structuring cell populations in relevant phenotypes, relevant here meaning adapted to describe an environmental situation that is susceptible to abrupt changes, such as introduction of a deadly molecule in the environment. beyond classical (in ecology) viability and fecundity, reversible plasticity for cancer cell populations may also be set as one of such phenotypes. such structured pde models have the advantage of being compatible with optimal control methods for the theoretical design of optimised therapeutic protocols involving combinations of cytotoxic and cytostatic (and later possibly epigenetic [89]) treatments. the objective function of such optimisation procedure being chosen as minimising a cancer cell population number, the constraints will consist of minimising unwanted toxicity to healthy cell populations. the innovation in this point of view is that success or failure of therapeutic strategies may be evaluated by a mathematical looking glass into the hidden core of the cancer cell population, in its potential of adaptation to cellular stress. the poor understanding of the determinants of drug resistance in cancer at the epigenetic level thus far, and the unexplained failure or partial failure of initially promising treatments such as targeted therapies and immunotherapies make it mandatory, from our point of view, to examine cancer, its evolution and its treatment at the level of a whole multicellular organism that locally, to begin with progressively lacks biomath 8 (2019), 1905147, http://dx.doi.org/10.11145/j.biomath.2019.05.147 page 2 of 23 http://dx.doi.org/10.11145/j.biomath.2019.05.147 j. clairambault, c. pouchol, a survey of adaptive cell population dynamics models of emergence ... its withinand between-tissue cohesion. ii. biological background a. the many facets of drug resistance in cancer drug resistance is a phenomenon common to various therapeutic situations in which an external pathogenic agent is proliferating at the expense of the resources of an organism: antibiotherapy, virology, parasitology, target populations are able to develop drug resistance mechanisms (e.g., expression of β-lactamase in bacteria exposed to amoxicillin). in cancer, there is in general no external pathogenic agent (even though one may have favoured the disease) and the target cell populations share much of their genome with the host healthy cell population, making overexpression of natural defence phenomena easy (e.g., abc transporters in cancer cells). note that drug resistance (and resistance to radiotherapy) is one of the many forms of fast resistance to cellular stress, possibly coded in ‘cold’, i.e., strongly preserved throughout evolution, rather than in ‘hot’, i.e., mutation-prone, genes [107]. at the molecular level in a single cell (that is per se insufficient to explain the emergence of drug resistance), overexpression of abc transporters, of drug processing enzymes, decrease of drug cellular influx, etc. [38] are relevant to describe endpoint molecular resistance mechanisms. at the cell population level, representing drug resistance by an abstract continuous variable x standing for the level of expression of a resistance phenotype (in evolutionary game theory [4]: a strategy of the population) is adapted to describe continuous evolution from total sensitivity (x = 0) towards total resistance (x = 1). is such evolution towards drug resistance due to sheer darwinian selection of the fittest by mutations in differentiation at cell division or, at least partially, due to phenotype adaptation in individual cells? this is by no means clear from biological experiments. in particular, it has been shown in [88] that emergence of drug resistance may be totally reversible, and, furthermore, that it may be completely dependent on the expression and activity of epigenetic control drugs (dna methyltransferases). this has been completed by molecular studies of the role of repeated sequences in drug tolerance in [42]. b. ecology, evolution and cancer in cell populations “nothing in biology makes senses except in the light of evolution.” (theodosius dobzhansky, 1964 [27]) the animal genome (of the host to cancer) is rich and amenable to adaptation scenarios that, especially under deadly environmental stress, may recapitulate salvaging developmental scenarios in particular blockade of differentiation or dedifferentiation, allowing better adaptability but resulting in insufficient cohesion of the ensemble that have been abandoned in the process of the great evolution [45], [46] from unicellular organisms (aka protozoa) to coherent metazoa [23] (aka multicellular organisms). in cancer populations, enhanced heterogeneity with enhanced proliferation and poor differentiation results in a high phenotypic or genetic diversity of immature proliferating clonal subpopulations, so that drug therapy may be followed, after initial success, by relapse due to selection of one or more resistant clones [25]. as regards ecology and evolution, genetics and epigenetics: ecology is concerned with thriving or dying of living organisms in populations in the context of their trophic environment. evolution is concerned by the somatic changes, either inscribed by genetic mutations of base pairs in the marble of their dna, or only and sometimes geneticists in that case dismiss the term evolution, preferring adaptation reversibly (however transmissible to the next generations) modified by silencing or re-expressing genes by means of grafting methyl or acetyl radicals on on the base pairs of the dna or on the aminoacids that constitute the chromatin (i.e., histones) around which the dna is wound. in the latter case, such evolution is determined by epigenetic mechanisms, i.e., mechanisms that biomath 8 (2019), 1905147, http://dx.doi.org/10.11145/j.biomath.2019.05.147 page 3 of 23 http://dx.doi.org/10.11145/j.biomath.2019.05.147 j. clairambault, c. pouchol, a survey of adaptive cell population dynamics models of emergence ... do not change the sequence of the base pairs, but only locally modify their transcriptional function. at the level of a same multicellular organism, such so-called epimutations govern the succession of events that physiologically result in cell differentiations. at this stage, one must clearly distinguish three different meanings of the words evolution and differentiation/maturation: 1) at the shortterm time scale of a cancer cell population, evolution by (plastic) adaptation of phenotype or by mutations both phenomena may be encountered to a changing environment such as infusion of an anti-cancer drug [32]; 2) at the mid-term time scale of a developing animal or human organism, programmed succession of cell differentiations, leading from one original cell to the circa 200-400 different cell types (in the human case) that constitute us as multicellular organisms (development): no mutations, only differentiations (aka maturations), i.e., epigenetic changes within the same genome until each cell reaches its completely physiologically mature state (an example of an ode model of differentiation may be found, e.g., in [44]); 3) at the very long-term (billions of years) time scale of darwinian evolution of species, succession of mutations from protozoa until evolved metazoa. epigenetic changes in a cell population are not rare events and may be fast, operating under environmental pressure by means of epigenetic control enzymes (methylases and acetylases, dna methyltransferases, etc.), and they are reversible, however likely less quickly than they have occurred [88]. it is also likely, and indeed this has been shown in some cases, that following such reversible epigenetic events, rare events that are mutations (not so rare in the context of genetic instability that often characterises cancer cell divisions) stochastically happen, fixing in the dna an acquired advantage in the context of a changing environment. conversely, mutations in parts of the genome that code for epigenetic control enzymes may determine epigenetic changes, metaphorically representable in the waddington epigenetic landscape [45], [46], [106]. such genetic to epigenetic modifications and vice versa are discussed in [14], [36], [37]. also note that the relationships between ecology, evolution and cancer are extensively developed in the book [102]. c. the atavistic theory of cancer there has been some debate in the past 20 years about two opposed views of cancer, the classic one, advocated by p. nowell [71] states that cancer starts from a single “renegade” cell (a cheater), that by a succession of mutations initiates cancer (somatic mutation theory, smt), being followed by strict darwinian selection of the fittest, while the less admitted tissue organisational field theory (toft), advocated in particular by a. soto and c. sonnenschein [90], contends that cancer is a matter of deregulated ecosystem, amenable to eradication by changing the tumour ecosystem. the atavistic theory of cancer is completely different from those two, in as much as it relates cancer to a regression in the billion year-long evolution of multicellularity. the idea that cancer is a form of backward evolution from organised multicellularity toward unicellularity, stalled at the poorly organised forms of multicellularity tumours consist of (as cancer cell populations, escaping the collective control present in delicate organismic organisations that constitute coherent multicellularity, continuously reinvents the wheel of multicellularity, starting from scratch for their own sake) is not new and it has at least been proposed by t. boveri in 1929 [7] and l. israel in 1996 [47]. however, it has regained visibility thanks to the documented and simultaneous studies by physicists p.c.w. davies and c.h. lineweaver [23], [55] and oncologist m. vincent [103], [104], followed by various subsequent studies, constituting a new body of knowledge [11], [19], [97], [98], [108] under the name of atavistic theory of cancer. this theory, or hypothesis, postulates that, although cancer reinvents the wheel of multicellularity, it has at its disposal for this task “an ancient biomath 8 (2019), 1905147, http://dx.doi.org/10.11145/j.biomath.2019.05.147 page 4 of 23 http://dx.doi.org/10.11145/j.biomath.2019.05.147 j. clairambault, c. pouchol, a survey of adaptive cell population dynamics models of emergence ... toolkit of pre-existing adaptations” that makes it fundamentally differ from classical darwinian evolution [23]. in this respect, cancer is clearly “more an archeoplasm than a neoplasm” (mark vincent [103]). what is relatively new in this theory, compared with the previously cited ones, is the idea that an intermediate set of coarse forms of multicellularity (which they call “metazoa 1.0”), lacking coherent control of intercellular and between cell populations cooperativity and proliferation, qualified a “robust toolkit for the survival , maintenance and propagation of non-differentiated or weakly-differented cells” is a safety state to which “metazoa 2.0” (us, in particular) revert when our sophisticated form of multicellularity goes astray in cancer. such events are due to failures in control of evolved cooperativity genes, and this incoherent, chaotic, poorly organised “metazoa 1.0” system endows the tissues in which it is installed with high phenotype adaptability, aka cancer plasticity, on which tumour development relies. such plasticity makes tumour cells in particular able to exploit for their own sake, plasticity resulting in resistance to cytotoxic drugs, epigenetic enzymes that were physiologically designed to control finely tuned cell differentiations, in acquired resistance to cancer treatments. this illuminating view of cancer, according to which the genes that malfunction are precisely “the genes of cellular cooperation that evolved with multicellularity about a billion years ago” (paul davies and charles lineweaver [23]), and the reason of resistance is to be found in ancient, well preserved, genes of our dna, is however quite often not admitted by many biologists of cancer who strongly believe in the strictly darwinian nature of evolution in cancer cell populations [33], [34], [39], [40], without any kind of such “genomic memory”. compatible with the atavistic hypothesis that postulates such backward evolution, a possible scenario suggests that cancer may start with a local deconstruction of the epigenetic control of cell differentiation (that is an essential piece of the coherence of multicellularity, e.g., in haematopoiesis, by genes tet2, dnmt3a, asxl1), followed by deregulation of cooperativity between cell populations (essential to division of work in a multicellular organism) initiated by disruption of transcription factors responsible for differentiation (e.g., by genes runx1, cebpα, npm1) and finally deregulation of the determinants of the strongest and most ancient bases of multicellularity, proliferation and apoptosis (e.g., by genes flt3, kit, and genes of the ras pathway). even though many cancer biologists are reluctant to endorse this scenario, biological observations exist, showing that a scenario of successions of mutations may be found in fresh blood samples of patients with acute myeloid leukaemia, phylogenetically recapitulating such hierarchically ordered deconstruction of the multicellular haematopoietic structure, from the finest (epigenetic) to the coarsest (proliferation and apoptosis) elements of the construction of multicellularity [43]. from the point of view of therapeutic applications, the atavistic theory of cancer has the consequence that, even though those genes of cooperativity that are altered in cancer (the “multicellularity gene toolkit of metazoa 1.0”) have taken one billion years of darwinian evolution to achieve (by ‘tinkering’ [49]) coherent evolved multicellular organisms, they are nevertheless in finite number and can be systematically investigated, as has been initiated in phylostratigraphic studies led by tomislav domazet-lošo and diethard tautz [28], [29]. such systematic between-species phylogenetic biocomputer studies should open observation windows onto altered genes in patients and their possible correction in the future. from the point of view of mathematical modelling, the fact that ancient genes of survival have been developed in the course of evolution to make individual cells, and later coherently heterogeneous and nevertheless communicating biomath 8 (2019), 1905147, http://dx.doi.org/10.11145/j.biomath.2019.05.147 page 5 of 23 http://dx.doi.org/10.11145/j.biomath.2019.05.147 j. clairambault, c. pouchol, a survey of adaptive cell population dynamics models of emergence ... together (failures in intercellular communications [99], [100], [101] incidentally being also a possible source of default of cooperativity in cell populations), and may be conserved as silent capacities in our genome, only waiting to be unmasked by epigenetic enzymes [88] put on the service of survival in highly plastic cancer cells [107], gives reasonable biological support to the notion of cell populations structured in phenotype of survival and of drug resistance. how such ancient genes (‘cold genes’ [107]) have been preserved in our genome while serving in rare and extreme environmental conditions only is not clear (in principle, genes that are not expressed are prone to disappear), however observations reported in [107] propose that ancient genes evolve more slowly than younger ones. hence, preserved in the genomic memory as survival genes, revivable in plastic cancer cell populations (plastic here meaning that they have easy access to epigenetic enzymes to change their phenotype under environmental pressure), their level of expression may offer a basis for evolvability and reversibility, under environmental pressure, of continuous phenotypes structuring the heterogeneity (aka biological variability) of cancer cell populations that is developed in mathematical models of adaptive cell population dynamics. iii. models of adaptive dynamics a. models structured in resistance phenotype the simplest model of a resistance phenotypestructured cell population may be described by a nonlocal lotka-volterra-like integro-differential equation, here x ∈ [0, 1] representing a continuous resistance phenotype, from x = 0, total sensitivity to the drug, until x = 1, total insensitivity: ∂n ∂t (t,x) = ( r(x) −d(x)ρ(t) ) n(t,x), with ρ(t) := ∫ 1 0 n(t,x) dx and n(0,x) = n0(x). note that this simple integro-differential equation may, when this makes biological sense, be generalised to a reaction-diffusion-advection (rda) one written as ∂n ∂t (t,x) + ∂ ∂x {v(x)n(t,x)} = β ∂2 ∂x2 n(t,x) + {r(x) −d(x)ρ(t)}n(t,x). we assume reasonable (l∞) hypotheses on r and d, and n0 ∈ l1([0, 1]). phenotypedependent functions r and d stand for intrinsic proliferation rate and intrinsic death rate due to within-population competition for space and nutrients, respectively. note that space is represented here only in the abstract nonlocal logistic term d(x)ρ(t). it is nevertheless possible to mix phenotype and actual cartesian space variables to structure the population, as will be shown later. one can then prove for the simple integrodifferential model the asymptotic behaviour theorem: theorem 1. [24], [48], [74] (i) ρ converges to ρ∞, the smallest value ρ such that r(x) − d(x)ρ ≤ 0 on [0, 1] (i.e., ρ∞ = max[0,1] rd). (ii) the population n(t, ·) concentrates on the phenotype set { x ∈ [0, 1], r(x) −d(x)ρ∞ = 0 } . (iii) furthermore, if this set is reduced to a singleton x∞, then n(t, ·) ⇀ ρ∞δx∞ in m1(0, 1). (the measure space m1(0, 1) being the dual for the supremum norm of the space of continuous real-valued fonctions on [0, 1]; note the dirac mass on the rhs, convergence is here meant in the sense of measures.) although in the one-population case, as stated above, a direct proof of convergence based on proving that ρ(t) is bv on the half-line, from which concentration easily follows from exponential growth, it is interesting to note, as this argument can be used in the case of two interacting populations, that a global proof based on the design of a lyapunov function gives at the same time convergence and concentration: biomath 8 (2019), 1905147, http://dx.doi.org/10.11145/j.biomath.2019.05.147 page 6 of 23 http://dx.doi.org/10.11145/j.biomath.2019.05.147 j. clairambault, c. pouchol, a survey of adaptive cell population dynamics models of emergence ... choosing any measure n∞ on [0, 1] with support in argmax r d such that∫ 1 0 n∞(x) dx = ρ∞ = max [0,1] r d , and for an appropriate weight w(x) (in fact 1 d(x) )setting v(t)= ∫ 1 0 w(x){n(t,x)−n∞(x)−n∞(x) ln n(t,x)}dx, one can show that dv dt = −(ρ(t) −ρ∞)2 + ∫ 1 0 w(x){r(x)−d(x)ρ∞}n(t,x) dx, which is always nonpositive, tends to zero for t → ∞, thus making v a lyapunov function, and showing at the same time convergence and concentration. indeed, in this expression, the two terms are nonpositive and their sum tends to zero; the zero limit of the first one accounts for convergence of ρ(t), and the zero limit of the second one accounts for concentration in x (on a zeromeasure set) of lim t→+∞ n(t,x). starting from this simple model, one can generalise it to the case of two interacting cell populations, cancer (nc) and healthy (nh), again using a nonlocal lotka-volterra setting, with two different drugs, u1, cytotoxic (= cell-killing drug, towards which resistance evolves according to the continuous phenotype x ∈ [0, 1]) and u1, cytostatic (only thwarting proliferation without killing cells): ∂ ∂t nh(t,x) =[ rh(x) 1+khu2(t) −dh(x)ih(t)−u1(t)µh(x) ] nh(t,x), ∂ ∂t nc(t,x) =[ rc(x) 1+kcu2(t) −dc(x)ic(t)−u1(t)µc(x) ] nc(t,x). (1) the environment in the logistic terms is defined by: ih(t) = ahh.ρh(t) + ahc.ρc(t), ic(t) = ach.ρh(t) + acc.ρc(t), with ahh > ahc, acc > ach (higher withinspecies than between-species competition) and ρh(t) = ∫ 1 0 nh(t,x) dx, ρc(t) = ∫ 1 0 nc(t,x) dx. the cytotoxic drug terms, tuned by drug sensitivity functions µc and µh, act as added death terms to the logistic term, whereas the cytostatic drug terms act by inhibiting the intrinsic proliferation rates rc and rh. functions µc and µh obviously have to be decreasing functions of x, and so, less obviously, but representing a trade-off between survival and proliferation (“cost of resistance”), have to be rc and rh. as regards dc and dh, no modelling choice imposes itself; however, in order to make the function r d globally decreasing and thus, in the absence of drug, obtain its maximum around zero, it was assumed in this study that it is a nondecreasing function of x. biologically, this means that the more resistant a cell is, the stronger opposition to its proliferation it encounters in its own species, cancer or healthy, which is another way, coherent with the modelling choice made on rc and rh, to express a cost of resistance. in this 2-population case, following an argument by pierre-emmanuel jabin and gaël raoul [48], one can prove, as in the 1-population case, at the same time convergence and concentration by using a lyapunov functional of the form∫ w(x){n(t,x) −n∞(x) −n∞(x) ln n(t,x)} dx. we have also in this case the asymptotic behaviour theorem: theorem 2. [81], [83] assume that u1 and u2 are constant: u1 ≡ ū1, and u2 ≡ ū2. then, for any positive initial population of healthy and of tumour cells, (ρh(t),ρc(t)) converges to the biomath 8 (2019), 1905147, http://dx.doi.org/10.11145/j.biomath.2019.05.147 page 7 of 23 http://dx.doi.org/10.11145/j.biomath.2019.05.147 j. clairambault, c. pouchol, a survey of adaptive cell population dynamics models of emergence ... equilibrium point (ρ∞h ,ρ∞c ), which can be exactly computed as follows: let a1 ≥ 0 and a2 ≥ 0 be the smallest nonnegative real numbers such that rh(x) 1 + αhū2 − ū1µh(x) ≤ dh(x)a1 and rc(x) 1 + αcū2 − ū1µc(x) ≤ dc(x)a2. then (ρ∞h ,ρ∞c ) is the unique solution of the invertible (ahh.acc > ach.ahc) system i∞h = ahhρ ∞ h + ahcρ ∞ c = a1, i∞c = achρ ∞ h + accρ ∞ c = a2. let ah ⊂ [0, 1] (resp., ac ⊂ [0, 1]) be the set of all points x ∈ [0, 1] such that equality holds in the inequalities above. then the supports of the probability measures νh(t) = nh(t,x) ρh(t) dx and νc(t) = nc(t,x) ρc(t) dx converge respectively to ah and ac as t tends to +∞. in [81], this result is complemented with numerical simulations which show the failure of constant administration of high doses of both drugs. the theorem explains the phenomenon: such a strategy makes the cancer cell density concentration on a very resistant phenotype near x = 1. once most of the mass is close to x = 1, further treatment is hopeless as the tumour has become mostly resistant, and it starts increasing again after having first decreased. this can be interpreted as relapse. note that numerical studies based on a similar model of adaptive dynamics in a reactiondiffusion version, dealing with the question of relapse, can be found in [16], [17], [56], [57], see also [74], [75] for more theoretical considerations. this result extends to two competitively interacting populations the result of convergence and concentration for nonlocal lotka-volterra phenotype-structured models previously published in [24], [48], [74]. note that it assumes the invertibility of the square matrix [aij] where i,j ∈{h,c}. a natural question then arises: is it possible to extend this result to n > 2 interacting populations? this is the object of the study [82], in which the following n-dimensional nonlocal lotka-volterra system is set, for which one can look for coexistence of positive steady states (i.e., persistence of all species): ∂ ∂t ni(t,x) =  ri(x) + di(x) n∑ j=1 aijρj(t)  ni(t,x), in which x stands for all xi ∈ xi for simplicity, each xi being a compact subset of some rpi, ri,di smooth enough, and as usual ρi(t) = ∫ xi ni(t,x) dx. this system generalises to a nonlocal setting classical lotka-volterra models (for 2 populations in an ode setting, see, e.g., britton [8] or murray [69]) with ecological cases: mutualistic if aij > 0 and aji > 0, competitive if aij < 0 and aji < 0 , predator-prey-like if aijaji < 0, for the interaction matrix a = [aij] in [82], to which the reader is sent for more details, it is proved that a coexistent positive steady state ρ∞ = [ρ∞1 , . . . ,ρ ∞ i , . . . ,ρ ∞ n ] t exists in rn if and only if, setting i∞ = [i∞1 , . . . ,i ∞ i , . . . ,i ∞ n ] t, where each i∞i = max x∈xi ri(x) di(x) , the equation aρ + i∞ = 0 has a solution ρ∞ ∈ rn. then, under some precise conditions on a, it can be proved, again using the same kind of lyapunov function as in [81], that the solution to the n-dimensional nonlocal lotka-volterra system exists and is globally defined; furthermore, the solution ρ∞ to the equation aρ + i∞ = 0 in rn is then unique and globally asymptotically stable. as in the 1and 2-dimensional cases, a result of concentration in phenotype follows, with moreover an estimation of the speeds of convergence and of concentration. biomath 8 (2019), 1905147, http://dx.doi.org/10.11145/j.biomath.2019.05.147 page 8 of 23 http://dx.doi.org/10.11145/j.biomath.2019.05.147 j. clairambault, c. pouchol, a survey of adaptive cell population dynamics models of emergence ... b. modelling mutualistic tumour-stroma interactions noting that breast and prostate tumours are accompanied in their stroma by so-called cancer-associated adipocytes (caas) or cancerassociated fibroblasts (cafs) [18], [26], [60], which favour cancer growth, likely by exchanging bidirectional messenger molecules, one can model such mutualistic interactions in a nonlocal lotka-volterra way: ∂ ∂t na(t,x) =[ra(x)−daρa(t)+sa(x)ϕc(t)]na(t,x) ∂ ∂t nc(t,y) =[rc(y)−dcρc(t)+sc(y)ϕa(t)]nc(t,y) with ρa(t) = ∫ na(t,x) dx, ρc(t) = ∫ nc(t,y) dy, ϕa(t) = ∫ ψa(x)na(t,x) dx and ϕc(t) = ∫ ψc(y)nc(t,y) dy, for some weight functions ψa and ψc (that in the absence of known data may be chosen as simply affine), and some given initial conditions na(0,x) = n0a(x),nc(0,y) = n0c(y) for all (x,y) in [0, 1]2, x standing for transformation towards a caa or caf state in the adipocyte population and y standing for strength of malignancy in the cancer cell population. this model is studied, theoretically and numerically, in [80] in its generalised reaction-diffusion-advection form (see above) with explicit functions and initial functions n0a,n0c assumed to be gaussian. in a setting in which mutualistic interactions between two cell species, one of them being initially healthy, but susceptible to become cancerous, namely proliferating haematopoietic stem cells and early progenitors nh, in the mandatory presence of the other species ns, supporting stromal cells, a model closely related to the previous one is presented [70]. it writes  ∂tnh(t,x) = [ rh(x)−ρh(t)−ρs(t)+α(x)σs(t) ] nh, ∂tns(t,y) = [ rs(y)−ρh(t)−ρs(t)+β(y)σh(t) ] ns. this system is completed with initial data nh(0,x) = nh0(x) ≥ 0, ns(0,y) = ns0(y) ≥ 0. here the assumptions and notations are • ρh(t) := ∫ b a nh(t,x)dx, ρs(t) := ∫ d c ns(t,y)dy are the total populations of hscs and their supporting stromal cells mscs, respectively, x representing a malignancy potential in haematopoietic cells and y a trophic potential in stromal cells. • the functions σh(t) := ∫ b a ψh(x)nh(t,x) dx, and σs(t) := ∫ d c ψs(y)ns(t,y) dy denote an assumed chemical signal (σh) from the hematopoietic immature stem cells (haematopoietic stem cells, hscs) to their supporting stroma (mesenchymal stem cells, mscs), i.e., “call for support” and conversely, a trophic message (σs) from mscs to hscs. the weight functions ψh,ψs are nonnegative and defined on (a,b) and (c,d), intervals of the real line. • the function rh ≥ 0 represents the intrinsic (i.e., without contribution from trophic messages from mscs) proliferation rate of hscs. assume that rh is non-decreasing, rh(a) = 0 and rh(b) > 0. • the function α ≥ 0, satisfying α′ ≤ 0 and α(b) = 0, is the sensitivity of hscs to the trophic messages from supporting cells. • for the function rs ≥ 0, it is assumed that r′s(y) ≤ 0. the function β(y) ≥ 0 with β′(y) ≥ 0 represents the sensitivity of the stromal cells mscs to the (call for support) message coming from hscs. some examples for rh, α are given by rh = r∗h(x−a) or rh = r ∗ h(x−a) 2, α(x) = α∗(b−x) with positive constants r∗h,α ∗, ψs(y) = y and ψh(x) = x. the reader is sent to [70] for a detailed study of this model. in particular, theoretical conditions for extinction, invasion or possible stable biomath 8 (2019), 1905147, http://dx.doi.org/10.11145/j.biomath.2019.05.147 page 9 of 23 http://dx.doi.org/10.11145/j.biomath.2019.05.147 j. clairambault, c. pouchol, a survey of adaptive cell population dynamics models of emergence ... coexistence of a leukaemic clone emerging in an initial healthy hsc population together with a maintained healthy fraction of it, with numerical simulations, are given in this study. they are related to convexity or concavity properties of functions of the model describing proliferation of the population, rh and α, the same kind of evolution being possible in the stromal cell population. c. models structured in phenotype and space although purely space-structured models lack the necessary heterogeneity in phenotype to take into account continuous evolution towards drug-induced drug resistance, purely phenotypestructured models lack the possibility to examine possible heterogeneities due to extension of tumours in cartesian space, in particular due to diffusion of molecules (anticancer drugs and nutrients) in the medium. hence, provided that something is known of the geometry of the space occupied by a cancer cell population, and this is indeed the case with initial tumours that spontaneously thrive in spheroids, mixing space with phenotype to structure a model of a cancer cell population under drug exposure, to study its behaviour with respect to drug resistance, is a natural way to proceed. relying on modelling principles developed in [52], [58], integrated in spheroid-like space, such a model is studied in [59]: ∂tn(t,r,x) = [ p(x) 1+µ2c2(t,r) s(t,r) −d(x)%(t,r)−µ1(x)c1(t,r) ] n(t,r,x), −σs∆s(t,r)+ [ γs+ ∫ 1 0 p(x)n(t,r,x)dx ] s(t,r) = 0, −σc∆c1(t,r)+ [ γc+ ∫ 1 0 µ1(x)n(t,r,x)dx ] c1(t,r) = 0, −σc∆c2(t,r)+ [ γc+µ2 ∫ 1 0 n(t,r,x)dx ] c2(t,r) = 0, with zero neumann conditions at r = 0 (spheroid centre) coming from radial symmetry and dirichlet boundary conditions at r = 1 (spheroid rim): s(t,r = 1) = s1, ∂rs(t,r = 0) = 0, c1,2(t,r = 1) = c1,2(t), ∂rc1,2(t,r = 0) = 0, where: • the function p(x) is the intrinsic (i.e., independently of cell death) proliferation rate of cells expressing resistance level x due to the consumption of resources. the factor 1 1 + µ2c2(t,r) mimics the effects of cytostatic drugs, which act by slowing down cellular proliferation, rather than by killing cells. the parameter µ2 models the average cell sensitivity to these drugs. • the function d(x) models the death rate of cells with resistance level x due to the competition for space and resources with the other cells. • the function µ1(x) denotes the destruction rate of cells due to the consumption of cytotoxic drugs, whose effects are here summed up directly on mortality. • parameters σs and σc model, respectively, the diffusion constants of nutrients and cytotoxic/cytostatic drugs. • parameters γs and γc represent the decay rate of nutrients and cytotoxic/cytostatic drugs, respectively. the model can be recast in the equivalent form ∂tn(t,r,x) =r ( x,%(t,r),c1,2(t,r),s(t,r) ) n(t,r,x), in order to highlight the role played by the net growth rate of cancer cells, which is described by r ( x,%(t,r),c1,2(t,r),s(t,r) ) := p(x) 1 + µ2c2(t,r) s(t,r) −d(x)%(t,r) −µ1(x)c1(t,r). the following considerations and hypotheses are assumed to hold: • with the aim of translating into mathematical terms the idea that expressing cytotoxic resistant phenotype implies resource reallocation biomath 8 (2019), 1905147, http://dx.doi.org/10.11145/j.biomath.2019.05.147 page 10 of 23 http://dx.doi.org/10.11145/j.biomath.2019.05.147 j. clairambault, c. pouchol, a survey of adaptive cell population dynamics models of emergence ... (‘cost of resistance’, i.e., redistribution of energetic resources from proliferation-oriented tasks toward development and maintenance of drug resistance mechanism, such as higher expression or activity of abc transporters in individual cells), p is assumed to be decreasing p(·) > 0, p′(·) < 0. as regards function d, one can note that in this study [59], the advocated modelling choice (d′(·) < 0) is the opposite of the one that was made in [58], a study nevertheless published by the same authors. in [58], the underlying biological reason is possibly that ‘the more resistant a cell is, the stronger opposition to its proliferation it encounters in its own species, cancer or healthy, which is another way, coherent with the modelling choice made on rc and rh, to express a cost of resistance’ (see this argument developed above in subsection iii-a). as a matter of fact, the simulations shown in this study [59] always use a constant value for d, and, contrary to [58], no theorem is proposed to the reader of [59], which should induce to actually choose d′(·) ≥ 0. • the effects of resistance to cytotoxic therapies are modeled by the obvious condition that the drug sensitivity function µ1 is non-increasing: µ1(·) > 0, µ′1(·) ≤ 0. the interesting results of this model consist of simulations, illustrated by figures to which the interested reader is referred. d. models structured in cell-functional variables a puzzling observation on an in-vitro aggressive cancer cell culture (pc9, a variant of nsclc cells) exposed to high doses of anticancer drug, experiment reported in [88], is that: 1) even though 99.7% of cells quickly die when exposed to the drug, sparse and tiny subpopulations (0.3%) survive, named drug-tolerant persisters (dtps), and for some time just survive, exposed to the same very high concentration of drug; 2) after some time (not precisely defined in the paper), these surviving cells change their phenotype as expressed by membrane markers, and proliferate again, then named drug-tolerant expanded persisters (dteps), unabashed in the maintained high drug dose; 3) when the drug is washed out from the cell culture, the cell population reverts to initial drug sensitivity, and such resensitisation occurs ten times more slowly at the dtep stage than at the dtp stage; 4) if the cell culture is exposed to an inhibitor of the epigenetic enzyme kdm5a together with the drug, be it at the dtp or dtep stage, dtps or dteps die. such clearly epigenetic and completely reversible mode of resistance, developed in two stages, called for designing a cell population dynamic model structured, not as previously, monotonically in drug resistance gene expression level, but in phenotypes linked to the cell fate, which in cell populations always may be reduced to proliferation, death or differentiation (senescence being a version of delayed death). in the modelling and numerical study [13], 2 phenotypes are thus chosen to take into account the cell population heterogeneity relevant for the experiment: survival potential under extreme environmental conditions (called by ecology theoreticians viability), x, and proliferation potential (called fecundity), y. the resulting model is described by the reactiondiffusion-advection equation, that describes the behaviour of a very plastic cell population under exposure to a high dose of anticancer drug: ∂n ∂t (x,y,t) + ∂ ∂y ( v(x,c(t); v̄)n(x,y,t) ) ︸ ︷︷ ︸ stress-induced adaptation of the proliferation level = β∆n(x,y,t)︸ ︷︷ ︸ non-genetic phenotype instability + [ p(x,y,%(t))−d(x,c(t)) ] n(x,y,t)︸ ︷︷ ︸ non local lotka-volterra selection • %(t)= ∫ 1 0 ∫ 1 0 n(x,y,t) dx dy, p(x,y,%(t))=(a1+a2y+a3(1−x)) ( 1− %(t) k ) and d(x,c) = c(b1 + b2(1 −x)) + b3 biomath 8 (2019), 1905147, http://dx.doi.org/10.11145/j.biomath.2019.05.147 page 11 of 23 http://dx.doi.org/10.11145/j.biomath.2019.05.147 j. clairambault, c. pouchol, a survey of adaptive cell population dynamics models of emergence ... • the global population term %(t) = ∫ 1 0 ∫ 1 0 n(x,y,t) dx dy occurs in p as a logistic environment limiting term (availability of space and nutrients). • the drift term w.r.t. proliferation potential y represents possible (if v 6= 0) ‘lamarckian-like’, epigenetic and reversible, adaptation from pc9s to dtps; switching from v ≥ 0 to v = 0 here means switching from a possible adaptation scenario to a strictly darwinian one (it is biologically impossible to decide between the two scenarios). • v(x,c(t); v̄) = −v̄c(t)h(x∗ − x) where t 7→ c(t) is the drug infusion function, and x∗ is a fixed viability threshold. • no-flux boundary conditions. of note, another, individual-based, model (ibm) yielding the same simulation results (no theorem) is proposed in a complementary way to the interested reader, sent to [13]. the simulation results firstly show total reversibility to drug sensitivity when the drug is withdrawn, and also allow to study the evolution of the two phenotypes in the absence of drug, under drug exposure, and when the drug is withdrawn. furthermore, the model was put at stake by asking 3 questions: q1. is non-genetic instability (laplacian term) crucial for the emergence of dteps? q2. what can we expect if the drug dose is low? q3. could genetic mutations, i.e., an integral term involving a kernel with small support, to replace both adapted drift (advection) and non-genetic instability (diffusion), yield similar dynamics? consider c(·) = constant and two scenarios: (i) (‘lamarckian’ scenario (a): the outlaw) only pc9s initially, adaptation present (v 6= 0) (ii) (‘darwinian’ scenario (b): the dogma) pc9s and few dtps initially, no adaptation (v = 0) to make a long story short [13], • q1. always yes! whatever the scenario. • q2. low doses result in dteps, but no dtps. • q3. never! whatever the scenario. can such cell-functional models be used to actually manage drug resistance in the clinic? an idea would be to counter the plastic adaptation that cancer cell populations show in the presence of high doses of drugs by infusing at the same time as cytotoxic drugs inhibitors of epigenetic enzymes such as kdm5a in [88]. however, even though epigenetic drugs are the object of active research in the pharmaceutic industry [89], the importance of epigenetic control of physiological processes (all differentiation is epigenetic!) and the role of impaired epigenetic controls in impaired cell differentiation, which is a characteristic of cancer, has been stressed [31] makes them delicate to manage in the clinic so far. iv. optimisation and optimal control a. ode models and their optimal control in cancer to go beyond the administration of constant doses, one is led to let drug infusion rates vary in time and try to find the best such rates to minimise a given criterion, such as the number of cancer cells at the end of a given time-window. this is the purpose of the mathematical field of optimisation (see, e.g., in another framework [5], [20]) and optimal control, with all its available theoretical and numerical tools. biomath 8 (2019), 1905147, http://dx.doi.org/10.11145/j.biomath.2019.05.147 page 12 of 23 http://dx.doi.org/10.11145/j.biomath.2019.05.147 j. clairambault, c. pouchol, a survey of adaptive cell population dynamics models of emergence ... at this stage, it is noteworthy that the discretisation of the phenotype-structured pde models introduced so far leads to odes, usually of lotka-volterra type. to illustrate the idea, let us go back to the prototype integro-differential model ∂n ∂t (t,x) = ( r(x) −d(x)ρ(t) ) n(t,x). if one discretises the phenotype space into nx + 1 equidistant phenotypes through xi = i∆x, ∆x = 1 nx the above equation is approximated by the ode system y ′ i(t) = ( r(xi) −d(xi)ρ(t) ) yi(t), i = 0, . . . ,n where yi(t) ≈ ∆xn(t,xi) and ρ(t) = ∑n j=0 yi(t). this remark is general and applies to the numerical simulation of phenotype-structured pde models (this is nothing but a semidiscretisation of the corresponding pde). this point of view also makes the link between ode models where resistance is represented by a binary variable, or more generally, a discrete variable. with a coarser discretisation, the ode model has few equations and is more amenable to parameter identification, quick numerical simulation, but is also less accurate in representing resistance. when it comes to optimal control, ode models with a discrete representation of resistance have long been studied, either theoretically or numerically [91], [22], [50]. this is one aspect of the rich literature on optimal control for cancer modelling, see the reference book [86]. note that these ode models can be made richer, as they may additionally model healthy cells, cells in different compartments of the cell cycle, immune cells, etc. independently of the number of equations, the investigation of the optimal control problem typically leads to optimal strategies being the concatenation of bang-bang and singular arcs. bang-bang arcs correspond to drugs being either given at the maximum tolerated dose or not at all, whereas singular arcs correspond to intermediate doses which can be computed in feedback form from the yi’s. these results are obtained either numerically, or theoretically by applying the pontryagin maximum principle, possibly with higher order criteria (such as the legendre-clebsch criterion) and/or with geometric optimal control techniques. the usual clinical practice is to use maximum tolerated doses, a strategy which has been called into question as it can lead to an initial drop in tumour size before regrowth due to acquired resistance [25]. this corresponds to bang-bang controls. instead, alternative and more recent strategies advocate for the infusion of intermediate doses [35], [73], [93]. thus, as explained in [53], understanding whether the optimal controls do contain singular arcs is of paramount importance, and might depend from the parameters governing the cost. the recent work [12], where resistance is modelled to be binary, also features parameter regions leading to singular arcs which follow a first arc with maximum tolerated doses. this naturally poses the question of optimal scheduling for pde models of resistance. the corresponding optimal control problems are then significantly harder to solve. numerically, this is because a fine discretisation leads to computationally demanding algorithms. theoretically, as is already the case for a high dimensional ode, it becomes more difficult to obtain precise results on the optimal control strategy, even from an (infinite-dimensional) pontryagin maximum principle. b. optimal control of phenotype-structured pde models these difficulties might explain why there are up to date few optimal control results on phenotype-structured pdes for resistance. most studies are restricted to constant doses and the optimisation is then performed on the resulting scalar parameters. this can be done by numerical investigation of the parameter space as in [16], [58] or theoretically for cases in which explicit solutions are available [3]. other nonconstant infusion strategies mimicking popular biomath 8 (2019), 1905147, http://dx.doi.org/10.11145/j.biomath.2019.05.147 page 13 of 23 http://dx.doi.org/10.11145/j.biomath.2019.05.147 j. clairambault, c. pouchol, a survey of adaptive cell population dynamics models of emergence ... protocols are sometimes also tested in the aforementioned works. finally, in [3], restriction to gaussian solutions allows the authors to reduce the pde to a system of three odes (for the total mass, the mean and the standard deviation). all these studies conclude that the continuous administration of maximum tolerated doses might lead to relapse, and that alternative strategies with lower infusion of drugs might be preferable. there is, up to our knowledge, very little work in the direction of tackling a full optimal control problem for the phenotype-structured pdes models, without such simplifications as above. the two works we are aware of are [72] and [81], both concerned with the model (1) (see section iii-a). in [72], the model is more complex since genetic instability is introduced in the pde, modelled by diffusion terms. the goal is to minimise the total number of cancer cells ρc(t), and the overall model is complemented with the constraints • maximum tolerated doses: 0 ≤ u1(t) ≤ umax1 , 0 ≤ u2(t) ≤ u max 2 , • control of the tumour size: ρh(t) ρh(t) + ρc(t) ≥ θhc, (2) • control of the toxic side-effects: ρh(t) ≥ θhρh(0), (3) where 0 < θhc,θh < 1. in order to solve the problem numerically, the approach consists in discretising the whole problem in phenotype and time, thus using a so-called direct method in numerical optimal control [96]. this is equivalent to discretising in time an ode system which has as many equations as there are discretised phenotypes. the optimal control problem then becomes a high finite-dimensional optimisation problem, which can be handled, for example, by interior point methods. as is common to most numerical optimisation problems, the biggest difficulty lies in choosing the initial guess for the algorithm. the approach of [81] is to solve the optimisation problem with a very coarse discretisation (few unknowns) before scaling the problem up progressively to a fine discretisation. for the generalised model with mutations, such a strategy fails because of the computational cost of laplacians. to circumvent this, the numerical strategy introduced in [72] is to simplify the pdes by setting some coefficients to zero, so that the resulting optimal control problem can be solved by a pontryagin maximum principle. although this problem is non-realistic from the applicative a point of view, it provides an excellent starting point for a homotopy procedure which allows to go all the way back to the original more complicated problem, with a very accurate discretisation. an optimal strategy clearly emerges from these two works, when the initial tumour is heterogeneous (as a result of a first standard administration of cytotoxic drugs). the idea is to let the tumour density evolve to a sensitive phenotype by using no cytotoxic drugs and intermediate (constant) doses of cytostatic drugs for a long phase, during which the constraint on tumour size saturates. only then one takes profit of a sensitive tumour by using the maximum tolerated doses, up until the sideeffects constraint saturates. it is then possible to further reduce the tumour size by lowering the cytotoxic dosage. the asymptotic analysis comes in handy in understanding the optimality of such a strategy: the first long phase leads to the convergence of the cancer cell density onto a dirac mass located on a sensitive phenotype (or a smoothed version of such a dirac mass when there is a diffusion term). this property allows the authors of [81] to perform a theoretical study of the optimal control problem in a reduced control set where the controls are forced to take constant values during a first long phase. the strategy obtained numerically is then proved to be optimal in a theorem, informally given below. biomath 8 (2019), 1905147, http://dx.doi.org/10.11145/j.biomath.2019.05.147 page 14 of 23 http://dx.doi.org/10.11145/j.biomath.2019.05.147 j. clairambault, c. pouchol, a survey of adaptive cell population dynamics models of emergence ... theorem 3. [81] when the final time t is large, the optimal solution is such that 1) at the end of the first phase, the density of cancer cells has concentrated on a sensitive phenotype, 2) the optimal strategy is then the concatenation of three arcs • an arc with saturation of the constraint on ρh ρh +ρc . • a free arc with maximum tolerated doses, namely u1 = umax1 and u2 = umax2 , • an arc with saturation of the constraint on ρh and u2 = umax2 . we insist that this result is proved only in the absence of diffusion. the proof relies on the fact that dirac mass concentration at the end of the first phase allows to replace the pde system by an 2x2 ode system, up to an error becoming arbitrarily small as the length of the phase increases. the resulting optimal control problem can then be handled with a pontryagin maximum principle with state constraints. c. future prospects in optimal control applying the strategy advocated in [72], [81] requires thinking it in a quasi-periodic manner, and as a strategy relevant after the traditional admnistration of the first dose, which usually induces resistance. the idea would then be to alternate between: 1. a long phase with cytostatic doses and no cytotoxic doses (a drug holiday) to resensitise the tumour, 2. a short phase with maximum tolerated doses until the toxicity is considered to have reached its limit, with a possible subsequent switch in dose for the cytotoxic drugs to keep diminishing the tumour size. such a protocol requires to assess the level of resistance in order to decide when to switch from 1. to 2. and determine when damage to healthy tissue justifies switching back to 1. a major difficulty is of course the scarce availability of biological markers, which critically depends on each particular cancer. for instance, in prostate cancer, a regrowth of the plasmatic level of psa, routinely available to clinical measures for quite a long time, after some stagnation time under treatment may indicate the emergence of resistance. in the same way, for colorectal cancer, it has been advocated that circulating tumoral dna detection may be used for clinical management [54], and the same is true of circulating tumour cells [10]; however these techniques are far from being clinical routine. as regards damages to healthy tissue, they are numerous (e.g., for 5-fu and other cytotoxic drugs, classical handfoot syndrome, mouth sores, neutropenia, that often lead to treatment interruption), depending on each molecule and most of all on the evaluation of their severity by the oncologist in the clinic, given the health status of the patient under treatment (in the case of laboratory animals, weight loss is a common indicator of toxicity). taking advantage of the models introduced in section iii, there are several directions for analysing such types of optimal control but in a slighly different or generalised setting. this would both test the robustness of the strategy presented above, and possibly lead to alternative ones depending on the context. the addition of an advection term would help modulate the speed at which emergence and resensitisation occur. modelling how such terms would depend (or not) on a given drug is already an issue. however, it is likely that the addition of such a term will not jeopardise the numerical computation of optimal controls with direct methods refined with homotopies. of course, considering higher-dimensional phenotype variables or adding a space variable will inevitably lead to an explosion in complexity. this is why the numerical optimal control of phenotype-structured pdes will benefit from state-of-the-art methods in that field, which in turn highly rests on the quality of optimisation solvers. in other words, expert numerical methods will undoubtedly be at the core of any attempt at solving these complex infinitebiomath 8 (2019), 1905147, http://dx.doi.org/10.11145/j.biomath.2019.05.147 page 15 of 23 http://dx.doi.org/10.11145/j.biomath.2019.05.147 j. clairambault, c. pouchol, a survey of adaptive cell population dynamics models of emergence ... dimensional problems. the theoretical aspects are more exploratory. even for the integro-differential system of [81], the optimal control had to be solved in a restricted class of controls, and a complete understanding of the interplay between concentration phenomena and optimal control is yet to emerge. with few tools available for the control and optimal control of non-local pdes (an active area of research), a theoretical analysis of optimal controls of pde structured models is at this stage a real hurdle. v. open and challenging questions a. conflicting phenotypes and multicellularity the question of emergence of drug-resistant clones in a possibly totally genetically homogeneous cancer cell population (as is likely the case of the observations reported in [76], [77], [88]) under environmental pressure, here drug exposure, is related to the emergence of multicellularity in unicellular organisms. this question has been the object of many studies by evolutionary biologists [1], [63], [65], [66], [67], and they hypothesise that, confronted with a challenging, possibly deadly, environmental pressure, an already existing, without specialisation, multicellular aggregate (this had to occur after the beginning of massive oxygenation of the ocean and atmosphere, about one billion years ago, as, to stick together, cells need some glue of collagen family, which is synthesised only in the presence of free oxygen [94], [95]) had to specialise to survive. the proposed paradigmatic scenario is in [63], [65] the conflict between proliferation or fecundity, with adhesivity, to maintain against predators a colony of replicating cells on a good environmental trophic niche and motility to make the aggregate able to change its location, to leave for a more favourable one when resources are become scarce or when predators are threatening. the solution of such conflict is found in specialisation in two phenotypes, later to be refined, likely by bifurcations in more than two if the environmental pressure is diversified. this question has been tackled by yannick viossat together with richard michod in a simple setting [67], from which one finds that according to the convexity or concavity of a level set on which an optimum of fitness is to be found, there may be coexistence of two phenotypes or predominance of a single one. such a situation is encountered in an adaptive dynamics framework in [70] (see section iii-b) for the possible invasion, or coexistence with healthy cells, of a leukaemic cell clone. however, how to model such specialisation and cooperativity in general, or in the particular case of viability vs. fecundity for cancer cells under drug pressure, still escapes our efforts. b. coherence in an organism and its control the question of coherence and of withinand between-tissue cohesion of a whole multicellular organism with so many diverse and specialised subpopulations is seldom posed, and except in [78], it is generally ignored. matej plankar and co-authors ask the main question that is so often dodged when speaking of cancer as a developmental disorder: ‘what exactly is disorganised that was previously organised?’ and they propose that the biophysical base of such coherence resides in the coherence, in the physical sense, of oscillatory signals between cells that might be of electromagnetic or quantum nature, transporting energy and information, that could originate from, or be transmitted by, microtubules working like antennas, to likely too vaguely and unfaithfully sum up their hypotheses. how are such signals synchronised? is there a forcing signal originating from an organisational centre, or is it based on some sort of multilevel system of phase-locked loops? a possible candidate for such an organising system is the circadian system, that is made of circadian clocks [84], [87], existing in all nucleated cells (and even, so it seems, in red blood cells by different mechanisms), and that consist of oscillators based on gene networks that exist in all, at least, animal cells (but have also been individuated in some plants). such oscillators biomath 8 (2019), 1905147, http://dx.doi.org/10.11145/j.biomath.2019.05.147 page 16 of 23 http://dx.doi.org/10.11145/j.biomath.2019.05.147 j. clairambault, c. pouchol, a survey of adaptive cell population dynamics models of emergence ... have firstly been evidenced in fruitflies [51], and later in all mammals [105] where they have been searched for. they have the general property to be daily reset by the sun (or by routine social activities when the sun does not shine its rays), and they date back to a very ancient past of our planet. there exists a central control centre, the circadian pacemaker located in the suprachiasmatic nuclei of the hypothalamus in mammals, that receives synchronising electric signals from external light via the retinohypothalamic tract and physiologically sends synchronising messages to all peripheral cells via hormones and the autonomic nervous system. the activity of the central circadian pacemaker, that is in particular reflected by body temperature oscillations and by oscillations of corticosteroids in the surrenal gland, is known to be disrupted in cancer [85], and this all the more so as cancer is more advanced. although the authors of [78] do not mention this synchronising system, it is coherent with their view. is the circadian the synchronising system? or is it a dubbing system, under the dependence of an electromagnetic or quantum signalling system advocated in [78]? more hidden than the circadian system, could some organising coded plan, progressively established together with the immune system in the development of multicellularity, be the glue and control on which all metazoan betweenand withintissue coherence relies? in other words, could there exist a set of genes, already present in early metazoa, likely related to epigenetic control, that would on the one hand define, in a mhc (major histocompatibility complex)-like way, in its fixed part a species and, at the individual level, an individual within a species, and on the other hand a variable part within a species that would give rise to the different cell phenotypes that make a coherent multicellular organism (in limited number, 200 to 400 cell types or so, from enterocytes to neurons in a human). would this be the case, then one could imagine that tumours as metazoa 1.0, according to the atavistic hypothesis of cancer might have developed a sort of primitive, failed, immune response system whose main failure and difference with respect to the host normal immune system would be a strong tolerance to plasticity, i.e., to lack of belonging to a well-differentiated cell class. in the metaphoric waddingtonian view, this would imply an ablated, flattened epigenetic landscape, with plenty of room for dedifferentiation and transdifferentiation between cell phenotypes. could such flattened waddington’s epigenetic barriers in return be interpreted in terms of undecided, empty, spins borne on cell antigens that should normally, to avoid recognition as foe by antigen presenting cells, be coded as either 1 (differentiated) or 0 (open to further differentiation in a well-defined cell fate), but not blank? some support to these speculations may be found in a study dedicated to the origin of the metazoan immune system [68]. c. intra-tumour cooperativity, plasticity, bet hedging if tumours, as metazoa 1.0, have developed some internal cooperativity that makes them able to survive as a whole to cytotoxic stress and to friend-or-foe recognition by the immune system, what does such cooperativity consist of? experimental evidence exists that such cooperativity exists [21], [79], [92] and is necessary for a tumour to thrive, while some studies focus on evidencing tumour genetic or only phenotypic heterogeneity [61], [62] without necessarily proposing a rationale for such heterogeneity. however, as pointed by mark vincent, “heterogeneity, even though it might in some superficial way, ‘explain’ differential drug sensitivity, is not in itself an explanation of cancer; rather, it is the heterogeneity itself that requires explanation.” [104]. indeed, focusing only on drug resistance, we might satisfy ourselves with the plain observation of tumour heterogeneity to explain the variety of drug resistance mechanisms and why it is so uneasy to eradicate cancer. but trying to understand its determinants is more difficult. leaving aside the obvious fact that spatial isolation of cells inside a tumour may lead under biomath 8 (2019), 1905147, http://dx.doi.org/10.11145/j.biomath.2019.05.147 page 17 of 23 http://dx.doi.org/10.11145/j.biomath.2019.05.147 j. clairambault, c. pouchol, a survey of adaptive cell population dynamics models of emergence ... different forms of environmental pressure to different (phenotypically or genetically) clones that may have nothing to do with cooperativity, we may wonder why different phenotypes may be found in the same spatial niche. to begin with, does cooperativity exist with a fixed repartition of phenotypes inducing some division of labour in a tumour, or is it not a transient state that is observed only when a cancer cell population is put at stake under cellular stress?... or in artificial lab conditions [21], [79], [92]? can we consider that no actual cooperativity, in the sense of division of labour in an integrated structure, exists within a cancer cell population, all the more so as cell differentiations are impaired in cancer, but that plasticity of cancer cells (and not only of cancer populations) is so high within a preestablished metazoa 1.0 plan, i.e., it is not infinite, but takes advantage of many, but finitely many, stress response mechanisms inscribed in their genome and easily reactivatable that tumours can react to deadly insults by different resistance mechanisms, the simplest one being enhanced proliferation out of control, making them winners in all (known to their genome) cases? the sole idea of cancer cooperativity should be examined with care, if one admits that cancer cells are fundamentally cheaters, or otherwise said, defectors in the evolutionary game of multicellularity [2]. however, primitive metazoa, such as sponges [68] or algae show some cooperativity, in particular as regards immunity to invasion by pathogens. have successful tumours regressed in evolution at an earlier stage than sponges? likely yes, as multicellularity in sponges is well controlled. following the theme of cancer cell plasticity, an interesting notion has recently emerged, the so-called bet hedging fail-safe strategy of cancer cell populations [9], [41]. according to this hypothesis, cancer cells or cancer cell populations are so plastic that they can adapt their phenotypes to sustain different insults involving critical cell stress by developing different adapted subpopulations. it has also been observed that some cancer cells may express very ancient genes (so-called ‘cold genes’, i.e., that are conserved throughout evolution, being protected from evolution due to their essential role in facing unpredictable, but already met in a remote past of evolution, deadly insults) in case of exposure to chemotherapies [107]. one could speculate that some sentinel cells, expressing these ‘cold genes’, might send various resistance messages to other cells, or that they could themselves, being extremely plastic, differentiate into diverse categories of cell subpopulations, each one developing one of the resistance mechanisms elaborated in the course of evolution from a protozoan state, and then sheer darwinian selection would prevail. whether cells themselves are plastic and can adapt in a sort of lamarckian (necessarily epigenetic) way or only cell populations are plastic, constituted by preexisting (prior to any insult) genetically well-defined subpopulations is not easy to decide, and in [13], both scenarios were challenged by a reactiondiffusion-advection model (see section iii-d). however, the very fact that cell differentiations are always to some extent impaired in cancer cells, the fact that inhibitors of epigenetic enzymes have been shown in some cases to annihilate drug resistance in very aggressive forms of cancer (nsclc cells in culture in [88]) induce us to propose plasticity as a distinctive and continuous trait of cancer cells. with respect to the class of cell-functional models proposed in section iii-d, i.e., structured in the conflicting continuous traits named viability (x, potential of survival in extreme conditions, opposed to apoptosis) and fecundity (y, proliferation potential), one could then add a plasticity trait (θ, opposed to differentiation), characterising, together with the first two, each cell in its relevant variability in a heterogeneous cancer cell population (which would not give an explanation of such heterogeneity, but might help understand its evolution under cellular stress). a general class of cell population adaptive dynamics models that would be structured in biomath 8 (2019), 1905147, http://dx.doi.org/10.11145/j.biomath.2019.05.147 page 18 of 23 http://dx.doi.org/10.11145/j.biomath.2019.05.147 j. clairambault, c. pouchol, a survey of adaptive cell population dynamics models of emergence ... (x,y,θ) could then, following an idea popularised in [6] for the so-called cane toad equation (that describes the invasion of cane toads in australia by using an equation than cannot be of the classical reaction-diffusion type yielding travelling waves), be described by nt + ∇·{v (x,y,θ,d) n} = α(θ)nxx + β(θ)nyy + γ(θ)nθθ +n { r(x,y,θ) −µ(x,y,θ,d) − ρ(t) c(x,y) } , where r(x,y,θ) is the intrinsic (i.e., in the absence of any limitation) growth rate of the population, µ is an added death term due to the drug dose d, condition c(x,y) ≤ k represents an environmental constraint, v an optional advection function standing for abrupt modifications of the environment (such as the major cell stress-inducing delivery of high doses of drugs as in [13]) and ρ(t) = ∫ [0,1]3 n(x,y,θ,t) dxdy dθ is the total cell population at time t, put as usual in lotka-volterra settings in a logistic position to represent competition, e.g., for nutrients, hence growth limitation, within the cell population. how such class of models might lead to represent the emergence of different cell subpopulations under environmental pressure is still work underway. vi. conclusion in this review of models of adaptive dynamics dedicated to represent, analyse and control drug-induced drug resistance in cancer, we have firstly proposed a brief description of the biological background of cancer evolution, describing in particular the atavistic hypothesis of cancer, which to our meaning illuminates the scenery of the cancer disease, with more and more observation facts to support it. then (neglecting classical compartmental ode models that cannot claim to represent adaptive phenomena), we presented different cell population dynamics models that belong to the mathematical category of adaptive dynamics, i.e., integro-differential or partial differential equations structured by continuous traits describing the heterogeneity of cancer cell populations and their evolution under drug exposure. in a third part, we showed how optimal control methods can be applied to such equations of adaptive dynamics and used to design theoretical optimal therapeutic control strategies. such strategies, even though methods for the identification of the parameters and functions of the models still remain to be found, are amenable to predict qualitative behaviour of cancer cell populations under optimised timescheduled drug exposure. finally, we presented some challenging questions, addressed to evolutionary biologists and ecologists of cancer, to oncologists, and to mathematicians to accurately represent, analyse and control the behaviour of cancer cell populations. acknowledgments the authors are gratefully indebted towards their colleagues luis almeida, rebecca chisholm, tommaso lorenzi, alexander lorz, benoît perthame, and emmanuel trélat, coauthors with them of the mathematical papers on structured cell population models of evolution towards drug-induced drug resistance, whose main results have been presented in this review. c.p. acknowledges support from the swedish foundation of strategic research grant am13-004. references [1] a. aktipis, et al., life history trade-offs in cancer evolution, nature rev. cancer (2013), 13:883-892. 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(2018), 4:025002. biomath 8 (2019), 1905147, http://dx.doi.org/10.11145/j.biomath.2019.05.147 page 23 of 23 http://dx.doi.org/10.11145/j.biomath.2019.05.147 introduction: motivation from and focus on drug resistance in cancer biological background the many facets of drug resistance in cancer ecology, evolution and cancer in cell populations the atavistic theory of cancer models of adaptive dynamics models structured in resistance phenotype modelling mutualistic tumour-stroma interactions models structured in phenotype and space models structured in cell-functional variables optimisation and optimal control ode models and their optimal control in cancer optimal control of phenotype-structured pde models future prospects in optimal control open and challenging questions conflicting phenotypes and multicellularity coherence in an organism and its control intra-tumour cooperativity, plasticity, bet hedging conclusion references original article biomath 2 (2013), 1312051, 1–14 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum backward bifurcation in sivs model with immigration of non-infectives diána h. knipl mta–szte analysis and stochastics research group bolyai institute, university of szeged szeged, aradi v. tere 1, 6720, hungary email: knipl@math.u-szeged.hu gergely röst bolyai institute university of szeged szeged, aradi v. tere 1, 6720, hungary email: rost@math.u-szeged.hu received: 28 september 2013, accepted: 5 december 2013, published: 6 january 2014 abstract—this paper investigates a simple sivs (susceptible–infected–vaccinated–susceptible) disease transmission model with immigration of susceptible and vaccinated individuals. we show global stability results for the model, and give an explicit condition for the existence of backward bifurcation and multiple endemic equilibria. we examine in detail how the structure of the bifurcation diagram depends on the immigration. keywords-vaccination model with immigration; backward bifurcation; stability analysis i. introduction the basic reproduction number r0 is a central quantity in epidemiology as it determines the average number of secondary infections caused by a typical infected individual introduced into a wholly susceptible population. in epidemic models describing the spread of infectious diseases, the reproduction number works as a threshold quantity for the stability of the disease-free equilibrium. the usual situation is that for r0 < 1 the dfe is the only equilibrium and it is asymptotically stable, but it loses its stability as r0 increases through 1, where a stable endemic equilibrium emerges, which depends continuously on r0. such a transition of stability between the disease-free equilibrium and the endemic equilibrium is called forward bifurcation. however, it is possible to have a very different situation at r0 = 1, as there might exist positive equilibria also for values of r0 less than 1. in this case we say that the model undergoes a backward bifurcation at r0 = 1, when for values of r0 in an interval to the left of 1, multiple positive equilibria coexist, typically one unstable and one stable. the behavior in the change of stability is of particular interest from the perspective of controlling the epidemic: considering r0 > 1, in order to eradicate the disease it is sufficient to decrease r0 to 1 if there is a forward bifurcation at r0 = 1, nevertheless it is necessary to bring r0 well below 1 to eliminate the infection in case of a backward bifurcation. this also implies that the qualitative behavior of a model with backward bifurcation is more complicated than that of a model which undergoes forward bifurcation at r0 = 1, since in the latter case the infection usually does not persist if r0 < 1, although with backward bifurcation the presence of a stable endemic equilibrium for r0 < 1 implies that, even for values of r0 less than 1, the epidemic can sustain itself if enough infected individuals are present. backward bifurcation has been observed in several studies in the recent literature (for an overview see, for instance, [6] and the references therein). the well known works [4], [7], [8] consider multi-group epidemic models with asymmetry between groups or multiple interaction mechanisms. some simple epidemic models of disease transmission in a single population with vaccination of susceptible individuals are presented and analyzed in [1], [2], [9], [10]. a basic model, which exhibits backward citation: diána h. knipl, gergely röst, backward bifurcation in sivs model with immigration of non-infectives, biomath 2 (2013), 1312051, http://dx.doi.org/10.11145/j.biomath.2013.12.051 page 1 of 14 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2013.12.051 diána h. knipl et al., backward bifurcation in sivs model... bifurcation, can be described by the following system of ordinary differential equations: s′(t) =λ(n(t)) −β(n(t))s(t)i(t) − (µ + φ)s(t) + γi(t) + θv (t), i′(t) =β(n(t))s(t)i(t) + σβ(n(t))v (t)i(t) − (µ + γ)i(t), v ′(t) =φs(t) −σβ(n(t))v (t)i(t) − (µ + θ)v (t), (1) where s(t), i(t), v (t) and n(t) denote the number of susceptible, infected, vaccinated individuals and the total population, respectively, at time t. λ represents the birth function into the susceptible class and µ is the natural death rate in each class. disease transmission is modeled by the infection term β(n)si, φ and γ stand for the vaccination rate of susceptible individuals and the recovery rate of infected individuals. it is assumed that vaccination loses effect at rate θ, moreover 0 ≤ σ ≤ 1 is introduced to model the phenomenon that vaccination may reduce but not completely eliminate susceptibility to infection. with certain conditions on the birth function λ, system (1) can be reduced to a twodimensional system, of which a complete qualitative analysis including a condition for the existence of backward bifurcation has been derived in [1]. the aim of this paper is to describe and analyze an epidemic model in which demographic effects, such as immigration of non-infected individuals are included into a single population. the model we study generalizes the above presented vaccination model (1) by incorporating the possibility of immigration, and we investigate how immigration changes the bifurcation behavior. the paper is organized as follows. a threedimensional ode model is given in section ii, which we reduce to two dimensions by means of the theory of asymptotically autonomous systems. some fundamental properties of the two-dimensional system –as positivity and boundedness of solutions and stability of the diseasefree equilibrium– are discussed in section iii, then section iv concerns with the existence of endemic equilibria and conditions for the forward / backward bifurcation. we obtain our results by algebraic means, without using center manifold theory and normal forms. in section v a complete qualitative analysis has been carried out for the two-dimensional system, furthermore we analyze how immigration deforms the bifurcation curve in section vi. finally, in section vii we return to the original threedimensional model, then discuss our findings in the last section. ii. sivs model with immigration a general vaccination model with immigration of noninfected individuals can be described by the system s′(t) =λ(n(t)) −β(n(t))s(t)i(t) − (µ + φ)s(t) + γi(t) + θv (t) + η, i′(t) =β(n(t))s(t)i(t) + σβ(n(t))v (t)i(t) − (µ + γ)i(t), v ′(t) =φs(t) −σβ(n(t))v (t)i(t) − (µ + θ)v (t) + ω, (2) where we assume that immigration of susceptible and vaccinated individuals occurs with constant rate η and ω, respectively. the other parameters of the model have been described in section i, and for the total population n(t) we obtain n ′(t) = λ(n(t)) −µn(t) + η + ω. (3) the proof of the following proposition is obvious and thus omitted. proposition ii.1. if for the birth function λ it holds that λ(0) = 0, λ′(0) > µ and there exists an x∗ > 0 such that λ′(x∗) < µ, moreover λ′(x) > 0 and λ′′(x) < 0 for all x > 0, then for any η,ω ≥ 0 there exists a unique positive solution of λ(x) = µx−η −ω. we define the population carrying capacity k = k(λ,µ,η,ω) as the unique solution of λ(x) = µx − η −ω. note that from λ(k) = µk −η −ω it follows that µk − η − ω > 0. we can rewrite equations (2)2 and (2)3 in terms of n(t), i(t) and v (t) using s(t) = n(t) − i(t) − v (t) and consider this system as a system of non-autonomous differential equations with non-autonomous term n(t), which is governed by system (3). then, by limt�∞n(t) = k we find that system (2) is asymptotically autonomous with the limiting system i′(t) =β(k − i(t) − (1 −σ)v (t))i(t) − (µ + γ)i(t), v ′(t) =φ(k − i(t)) −σβv (t)i(t) − (µ + θ + φ)v (t) + ω, (4) where β = β(k). in what follows we focus on the mathematical analysis of system (4), then we use the theory of asymptotically autonomous systems [11], [12], biomath 2 (2013), 1312051, http://dx.doi.org/10.11145/j.biomath.2013.12.051 page 2 of 14 http://dx.doi.org/10.11145/j.biomath.2013.12.051 diána h. knipl et al., backward bifurcation in sivs model... [13] to obtain information on the long-term behavior of solutions of (2). iii. fundamental properties of the system the existence and uniqueness of solutions of system (4) follows from fundamental results for odes. since k was defined as the carrying capacity of the population, it is biologically meaningful to assume that for the initial conditions of system (4) it is satisfied that 0 ≤ i(0),v (0),i(0) + v (0) ≤ k. proposition iii.1. if 0 ≤ i(0),v (0),i(0) + v (0) ≤ k, then 0 ≤ i(t),v (t),i(t) + v (t) ≤ k is satisfied for all t > 0. proof: if i(t) = 0 then i′(t) = 0, which yields that for nonnegative initial conditions i never goes negative. if v (t) = 0 when 0 ≤ i(t) ≤ k, then v ′(t) ≥ ω ≥ 0, thus solutions never cross the line v = 0 from the inside of the region r : 0 ≤ i,v,i + v ≤ k. if i(t) + v (t) = k when i(t),v (t) ≥ 0, then summing (4)1 and (4)2 gives i′(t) + v ′(t) = −µk −γi(t) −θv (t) + ω, which is negative since ω − µk is non-positive, thus i(t) + v (t) > k is impossible. the disease-free equilibrium of system (4) can be obtained as v̄ = φk + ω µ + θ + φ . in the initial stage of the epidemic, we can assume that system (4) is near the equilibrium (0, v̄ ) and approximate the equation of class i with the linear equation y′(t) = (β(k − (1 −σ)v̄ ) − (µ + γ))y(t), (5) where y : r � r. the term β(k−(1−σ)v̄ ) corresponds to the production of new infections, and µ + γ is the transition rate describing changes in state, hence with the formula for the disease-free equilibrium v̄ we can define the basic reproduction number as r0 = β(k − (1 −σ)v̄ ) µ + γ = β µ + γ ( k(µ + θ + σφ) µ + θ + φ − (1 −σ)ω µ + θ + φ ) . (6) the following proposition shows that r0 works as a threshold quantity for the stability of the disease-free equilibrium of system (4). proposition iii.2. the disease-free equilibrium of system (4) is asymptotically stable if r0 < 1 and unstable if r0 > 1. proof: the stability of the zero steady-state of system (5) is determined by the sign of β(k − (1 − σ)v̄ )−(µ+γ), which coincides with the sign of r0−1. this means that the zero solution of (5) is asymptotically stable if r0 < 1 and unstable if r0 > 1. this statement extends to the nonlinear system (4) by the principle of linearized stability. iv. endemic equilibrium the problem of finding equilibrium (î, v̂ ) for system (4) yields the two dimensional system 0 = β(k − î − (1 −σ)v̂ )î − (µ + γ)î, 0 = φ(k − î) −σβv̂ î − (µ + θ + φ)v̂ + ω. (7) the existence of a unique disease-free equilibrium has been proved, so now we focus on finding endemic equilibria (î, v̂ ) with î > 0. from (7)1 we obtain the formula v̂ = β(k − î) − (µ + γ) β(1 −σ) , (8) then by substituting v̂ into (7)2 it follows from straightforward computations that aî2 + bî + c = 0 (9) should hold for î, where a =σβ, b =(µ + θ + σφ) + σ(µ + γ) −σβk, c = (µ + γ)(µ + θ + φ) β − (µ + θ + σφ)k + (1 −σ)ω. (10) we note that βc = (1− r0)(γ + µ)(µ + φ + θ) and we characterize the number of solutions of the equilibrium condition (9). proposition iv.1. if r0 > 1 then there exists a unique positive equilibrium î = −b+ √ b2−4ac 2a . proof: if c < 0, or equivalently, r0 > 1, then the equilibrium condition (9) has a unique positive solution, which can be obtained as î = −b+ √ b2−4ac 2a . at r0 = 1 it holds that a > 0 and c = 0, so there exists a unique nonzero solution î = −b/a of (9), which is positive (and thus, biologically relevant) if and only if b < 0. let us now assume that b is negative biomath 2 (2013), 1312051, http://dx.doi.org/10.11145/j.biomath.2013.12.051 page 3 of 14 http://dx.doi.org/10.11145/j.biomath.2013.12.051 diána h. knipl et al., backward bifurcation in sivs model... at r0 = 1, which also implies that b2−4ac = b2 > 0. then there is a positive root of the equilibrium condition at r0 = 1, and due to the continuous dependence of the coefficients a, b and c on β there must be an interval to the left of r0 = 1 where b < 0 and b2 − 4ac > 0 still hold. since c > 0 whenever r0 < 1, it follows that on this interval there exist exactly two positive solutions of (9) and thus, two endemic equilibria of system (4). we denote these equilibria by ĭ1 = −b − √ b2 − 4ac 2a , ĭ2 = −b + √ b2 − 4ac 2a , and with the aid of formula (8) we can derive the v̂ components to get the equilibria (ĭ1, v̆1) and (ĭ2, v̆2). with other words, if b < 0 when r0 = 1 then system (4) has a backward bifurcation at r0 = 1 since besides the zero equilibrium and the positive equilibrium ĭ2 = −b+ √ b2−4ac 2a , which also exist for r0 > 1, another positive equilibrium emerges when r0 passing through 1 from the right to the left. theorem iv.2. if the condition (1 −σ)ω k > (θ + µ + σφ)2 −σ(µ + γ)(1 −σ)φ (θ + µ + σφ) + σ(µ + γ) (11) holds then there is a backward bifurcation at r0 = 1. proof: the condition for the backward bifurcation is that b < 0 when β satisfies r0 = 1. this can be obtained as an explicit criterion of the parameters: as b < 0 yields σβk > (µ + θ + σφ) + σ(µ + γ), moreover from c = 0 we derive βk = (µ + γ)(µ + θ + φ) (θ + µ + σφ) − (1−σ)ω k , we get σ(µ + γ)(µ + θ + φ) (θ + µ + σφ) − (1−σ)ω k >(µ + θ + σφ) + σ(µ + γ), σ(µ + γ)(µ + θ + φ) (θ + µ + σφ) + σ(µ + γ) >(θ + µ + σφ) − (1 −σ)ω k , (1 −σ)ω k >(θ + µ + σφ), − σ(µ + γ)(µ + θ + φ) (θ + µ + σφ) + σ(µ + γ) , (1 −σ)ω k > (θ + µ + σφ)2 (θ + µ + σφ) + σ(µ + γ) , − σ(µ + γ)(1 −σ)φ (θ + µ + σφ) + σ(µ + γ) , where we used that µk −ω > 0. theorem iv.3. if condition (11) does not hold, then system (4) undergoes a forward bifurcation at r0 = 1. in this case there is no endemic equilibrium for r0 ∈ [0, 1]. proof: we proceed similarly as in the proof of theorem iv.2 to find that if (1 −σ)ω k ≤ (θ + µ + σφ)2 −σ(µ + γ)(1 −σ)φ (θ + µ + σφ) + σ(µ + γ) , then b ≥ 0 when c = 0, or equivalently, when β is set to satisfy r0 = 1. for r0 < 1 it holds that a,c > 0, moreover b is also positive because b is decreasing in β, these imply that there is no endemic equilibrium on r0 ∈ [0, 1). at r0 = 1 the equilibrium condition (9) becomes aî2 +bî = 0, and a > 0, b ≥ 0 give that (9) has only non-positive solutions. however, we know from proposition iv.1 that there is a positive solution of (9) for r0 > 1, thus we conclude that if the condition (11) does not hold then system (4) undergoes a forward bifurcation at r0 = 1, where a single endemic equilibrium emerges when r0 exceeds 1. if (11) is satisfied, then there is an interval to the left of r0 = 1 where there exist positive equilibria. in what follows we determine the left endpoint of this interval. let us assume that there is a backward bifurcation at r0 = 1. we define u = (θ + µ + σφ) − (1 −σ)ω k , x = (1 −σ)ω k + σ(µ + γ), w = −x + σ (γ + µ)(µ + φ + θ) u . (12) note that x and u are positive since µk − ω > 0 by assumption. the condition for the backward bifurcation can be obtained as w > u, (13) which also yields the positivity of w . we let rc = x−u + 2 √ uw (µ + γ)σ · u µ + θ + φ (14) and claim that it defines the critical value of the reproduction number for which there exist endemic equilibria on the interval [rc, 1]. proposition iv.4. let us assume that there is a backward bifurcation at r0 = 1. with rc defined in (14) only the biomath 2 (2013), 1312051, http://dx.doi.org/10.11145/j.biomath.2013.12.051 page 4 of 14 http://dx.doi.org/10.11145/j.biomath.2013.12.051 diána h. knipl et al., backward bifurcation in sivs model... disease-free equilibrium exists if r0 < rc, a positive equilibrium emerges at r0 = rc, and on (rc, 1) there exist two distinct endemic equilibria. there also exists a positive equilibrium at r0 = 1. proof: the last statement follows from the fact that at r0 = 1 (c = 0) the single non-zero solution î = −ba of (9) of is positive since b < 0. the necessary and sufficient conditions b < 0 and b2 − 4ac > 0 for the existence of two positive distinct equilibria hold on an interval to the left of r0 = 1. b = 0 automatically yields b2 − 4ac < 0 if r0 < 1, hence it is clear that the condition b2 − 4ac = 0 determines the value of r0 for which the positive equilibria disappear. first, we derive the critical value βc of the transmission rate from this equation, then substitute β = βc into the formula of r0 (6) to give the critical value of the reproduction number. using notations u,x and w introduced in (12), we reformulate b as b = u + x − σβk and c as c = (µ+γ)(µ+θ+φ) β −uk. the condition b2−4ac = 0 becomes u2 + 2u(x−βkσ) + (x−βkσ)2 − 4σ(µ + γ)(µ + θ + φ) + 4σβku =u2 − 2u(x−βkσ) + (x−βkσ)2 + 4ux − 4σ(µ + γ)(µ + θ + φ) =u2 − 2u(x−βkσ) + (x−βkσ)2 − 4uw = 0, so we obtain the roots (x−βkσ)1,2 = 2u ± √ 4u2 − 4u2 + 16uw 2 = u ± 2 √ uw. for the positive root (x−βkσ)2 we get b = u + (x− βkσ)2 > 0, but we require b < 0 thus we derive from x−βkσ = u − 2 √ uw that βc = x−u + 2 √ uw kσ . (15) substituting βc into (6) gives r0(βc) = βc µ + γ ( k(µ + θ + σφ) µ + θ + φ − (1 −σ)ω µ + θ + φ ) = x−u + 2 √ uw (µ + γ)σ · u µ + θ + φ , which is indeed equal to rc defined in (14). the condition r0 = 1 reformulates as σβk = w +x, so with the aid of (13) and the computations 0 < (√ u − √ w )2 , 2 √ uw < u + w, x−u + 2 √ uw < w + x, it is easy to verify that rc < 1. the positivity of βc, and hence, the positivity of rc follows from the fact that at β = βc it should hold that b < 0, which is only possible if β > 0. we wish to draw the graph of î as a function of β to obtain the bifurcation curve. by implicitly differentiating the equilibrium condition (9) with respect to β we get (2aî + b) dî dβ = − ( da dβ î2 + db dβ î + dc dβ ) , (2aî + b) dî dβ =σî(k − î) + (γ + µ)(µ + φ + θ) β2 . the positivity of the right hand side follows from k ≥ î, which implies that the term 2aî + b has the same sign as dî dβ . if r0 > 1 then there exists the equilibrium ĭ2 = −b+ √ b2−4ac 2a , and we obtain that 2aĭ2 + b > 0 hence for r0 > 1 the curve has positive slope. if there is a backward bifurcation at r0 = 1, then on (rc, 1) there exists two positive equilibria ĭ2 and ĭ1 = −b− √ b2−4ac 2a with ĭ2 > ĭ1, and since it holds that 2aĭ1 + b < 0, we conclude that on (rc, 1) the bifurcation curve has negative slope for the smaller endemic equilibrium and positive slope for the larger one. as a matter of fact, the unstable equilibrium is a saddle point, and thus the system experiences a saddle-node bifurcation. v. stability and global behavior the stability of the disease-free equilibrium has been examined in section iii, so now we derive local stability analysis of endemic equilibria. the jacobian of the linearization of system (4) at (î, v̂ ) gives j = ( −βî −(1 −σ)βî −(φ + σβv̂ ) −(µ + θ + φ + σβî) ) , where we used the identity β(k−î−(1−σ)v̂ ) = µ+γ from (7), hence the characteristic equation has the form a2λ 2 + a1λ + a0 = 0 biomath 2 (2013), 1312051, http://dx.doi.org/10.11145/j.biomath.2013.12.051 page 5 of 14 http://dx.doi.org/10.11145/j.biomath.2013.12.051 diána h. knipl et al., backward bifurcation in sivs model... with a2 = 1, a1 = βî + (µ + θ + φ + σβî), a0 = βî(µ + θ + φ + σβî) − (1 −σ)βî(φ + σβv̂ ). theorem v.1. the endemic equilibrium (î, v̂ ) for which î = ĭ2 is locally asymptotically stable where it exists: on r0 ∈ (1,∞), and also on r0 ∈ (rc, 1] in case there is a backward bifurcation at r0 = 1. the endemic equilibrium (î, v̂ ) for which î = ĭ1 is unstable where it exists: on r0 ∈ (rc, 1) in case there is a backward bifurcation at r0 = 1. proof: the routh-hurwitz stability criterion (for a reference see, for example, [5]) states that for all the solutions of the characteristic equation to have negative real parts, all coefficients must have the same sign. a2 and a1 are positive, hence the sign of a0 determines the stability. for that it holds that a0 =βî(µ + θ + φ + σβî) − (1 −σ)βî(φ + σβv̂ ) =βî(µ + θ + σφ + 2σβî −σβ(î + (1 −σ)v̂ ), so using −β(î + (1 −σ)v̂ ) = µ + γ −βk we derive a0 =βî(µ + θ + σφ + 2σβî + σ(µ + γ −βk)) = βî(2aî + b). for r0 > 1 the only endemic equilibrium is ĭ2 = −b+ √ b2−4ac 2a , for which 2aĭ2 + b > 0 holds and thus a0 > 0 yields its stability. if there is a backward bifurcation at r0 = 1, then endemic equilibria exists on (rc, 1] as well; here ĭ2 is again stable for the same reason as above, however ĭ1 = −b− √ b2−4ac 2a is unstable since a0 = βĭ1(aĭ1 + b) < 0. with the next theorem we describe the global behavior of solutions of system (4). theorem v.2. if there exists no endemic equilibrium, that is, if r0 < 1 in case of a forward bifurcation and if r0 < rc in case of a backward bifurcation, then every solution converges to the disease-free equilibrium. for r0 > 1, the unique endemic equilibrium is globally attracting. if there is a backward bifurcation at r0 = 1 then on (rc, 1) there is no globally attracting equilibrium, though every solution approaches an equilibrium. proof: we first show that every solution of system (4) converges to an equilibrium. in section iii we have proved that the region r : 0 ≤ i,v,i + v ≤ k is positively invariant for the solutions of system (4). we take the c1 function ϕ(i,v ) = 1/i, which does not change sign on r to show that system (4) has no periodic solutions lying entirely within the region r. the computation ∂ ∂i β(k − i − (1 −σ)v )i − (µ + γ)i i + ∂ ∂v φ(k − i) −σβv i − (µ + θ + φ)v + ω i = −β −σβ − µ + θ + φ i < 0 yields the result by means of the dulac criterion [3]. we use the well known poincaré-bendixson theorem to conclude that every solution of (4) approaches an equilibrium. the first statement of the theorem immediately follows from the fact that every solution of (4) approaches an equilibrium. if r0 > 1, then besides the disease-free equilibrium, which is unstable according to theorem v.1, there exists a single locally stable endemic equilibrium ĭ2. we show that no solution can converge to the diseasefree equilibrium. if limt�∞i(t) = 0 when i(0) > 0, then it follows from (4)2 that limt�∞v (t) = φk+ω µ+θ+φ . then for every � > 0 there exists a t∗(�) such that i(t) < � and v (t) < φk+ω µ+θ+φ + � for t > t∗. using (4)1 we get i′(t) ≥β ( k − �− (1 −σ) ( φk + ω µ + θ + φ + � )) i(t) − (µ + γ)i(t) =β ( k(µ + θ + σφ) µ + θ + φ − (1 −σ)ω µ + θ + φ ) i(t) + (−2� + σ�− (µ + γ)) i(t) (16) for t > t∗, moreover r0 = β µ+γ ( k(µ+θ+σφ) µ+θ+φ − (1−σ)ω µ+θ+φ ) > 1 implies that there exists an �1 small enough such that β ( k(µ + θ + σφ) µ + θ + φ − (1 −σ)ω µ + θ + φ ) + (−2�1 + σ�1 − (µ + γ)) > 0. with the choice of � = �1 the right hand side of (16) is linear in i(t) with positive multiplier, which implies that i(t) increases for t∗(�1) > t and thus, cannot converge to 0. we conclude that no solution of (4) with positive initial conditions converges to the disease-free equilibrium, so the endemic equilibrium indeed attracts every solution. if there is a backward bifurcation at r0 = 1 then besides the disease-free equilibrium there exist two biomath 2 (2013), 1312051, http://dx.doi.org/10.11145/j.biomath.2013.12.051 page 6 of 14 http://dx.doi.org/10.11145/j.biomath.2013.12.051 diána h. knipl et al., backward bifurcation in sivs model... 0 2 4 6 8 10 12 0 20 40 60 80 100 t ih tl (a) solutions of system (4). 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 i v (b) stream plot of system (4) on r : 0 ≤ i,v,i + v ≤ k. fig. 1: solutions of system (4) in case there is a backward bifurcation at r0 = 1 and rc < r0 < 1. we let λ(x) = x c+dx and choose parameter values as µ = 0.1, γ = 12, θ = 0.5, σ = 0.2, φ = 16, c = 1, d = 1.8, β = 0.33, η = 5, ω = 5, which makes k = 153.6 and r0 = 0.95. endemic equilibria (ĭ1, v̆1) = (8.6, 135.4) and (ĭ2, v̆2) = (50.7, 82.8) are represented as (a) red-dashed and blue-dashed lines, (b) red and blue points, respectively. on (b) the green point denotes the unique disease-free equilibrium (0, 148.4). solutions with initial values (i(0),v (0)) = (9, 120) – red curve, (18, 130) – blue curve and (100, 50) – black curve converge to (ĭ2, v̆2), however for (i(0),v (0)) = (5, 140) the curve of i – here, green – approaches the dfe. endemic equilibra on (rc, 1), one locally stable and one unstable (see again theorem v.1). as the dfe is locally stable when r0 < 1, we experience bistability on (rc, 1), which implies the third statement of the theorem. we present figure 1 to illustrate the statements of this section. the values of the model parameters were set to ensure that system (4) undergoes a backward bifurcation at r0 = 1, moreover we chose the value of β such that there exist two endemic equilibria. the plots of the figure support our results about the long-term behavior of solutions and the local stability of equilibria; solutions starting near the unstable saddle point (ĭ1, v̆1) approach another equilibrium, however (ĭ2, v̆2) seems to attract every solution with i(0) ≥ ĭ1 for the particular set of parameter values indicated in the caption of the figure. vi. the influence of immigration on the backward bifurcation in this section, we would like to investigate the effect of parameters η and ω on the bifurcation curve. in section iv we gave the condition (11) (1 −σ)ω k > (θ + µ + σφ)2 −σ(µ + γ)(1 −σ)φ (θ + µ + σφ) + σ(µ + γ) for the existence of backward bifurcation at r0 = 1; in what follows we analyze this inequality in terms of the immigration parameters. we keep in mind that if there is no backward bifurcation at r0 = 1 then there is forward bifurcation, i.e., there always exists an endemic equilibrium for r0 > 1. first we present results about how the existence of backward bifurcation depends on η and ω. the nonnegativity of ω and k immediately yields the following proposition. proposition vi.1. if (θ +µ+σφ)2 < σ(µ+γ)(1−σ)φ, then for all η and ω there is a backward bifurcation at r0 = 1. the special case of ω = 0 automatically makes the left hand side of inequality (11) zero, hence in this case there is a backward bifurcation if and only if the right hand side is negative; note that the right hand side is independent of η. proposition vi.2. if ω = 0, then there is a backward bifurcation at r0 = 1 if and only if (θ + µ + σφ)2 < σ(µ + γ)(1 −σ)φ. this also means that in this case η has absolutely no effect on the existence of a backward bifurcation. biomath 2 (2013), 1312051, http://dx.doi.org/10.11145/j.biomath.2013.12.051 page 7 of 14 http://dx.doi.org/10.11145/j.biomath.2013.12.051 diána h. knipl et al., backward bifurcation in sivs model... 0.0 0.2 0.4 0.6 0.8 1.0 0 50 100 150 200 250 300 β i` (a) η = 5,ω = 0,1, . . .19.. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 100 200 300 400 500 β i` (b) ω = 0,η = 10,12, . . .48. fig. 2: bifurcation diagrams for 20 different values of (a) ω and (b) η in the case when (θ + µ + σφ)2 < σ(µ + γ)(1 −σ)φ. proposition vi.1 implies that for all η and ω there is a backward bifurcation at r0 = 1. the curves move to the left as the immigration parameter increases. we let λ(x) = x c+dx and choose parameter values as µ = 0.1, γ = 12, θ = 0.5, σ = 0.2, φ = 16, c = 1, d = 1.8. figure 2 shows how the bifurcation curve deforms as we increase (a) ω and (b) η. parameter values µ = 0.1, γ = 12, θ = 0.5, σ = 0.2, φ = 16 were chosen so that the condition (θ + µ + σφ)2 < σ(µ + γ)(1 −σ)φ holds (14.44 < 30.976). after all this, the following question arises naturally: is it possible to have backward bifurcation at r0 = 1 when (θ + µ + σφ)2 ≥ σ(µ + γ)(1−σ)φ, i.e., when the right hand side of condition (11) is nonnegative? recall that if ω = 0 then (θ + µ + σφ)2 ≥ σ(µ + γ)(1 −σ)φ means forward bifurcation. note that the right hand side of (11) is independent of η and ω; however, k depends on both of these parameters, µ and the birth function λ. as we did not define λ explicitly (in section ii, we only gave conditions to ensure that for each η,ω ≥ 0 the population carrying capacity k > 0 can be defined uniquely), it is not clear how the left hand side of (11) depends on the immigration parameters. in the sequel, we use the general form λ(x) = x c + dx (17) for the birth function with parameters 0 < c < 1/µ and d > 0; it is not hard to see that with this definition all the conditions made in section ii for λ are satisfied. the carrying capacity k(µ,η,ω) then arises as the solution of λ(x) = µx−η −ω, which with our above definition (17) gives the secondorder equation x2µd + x(−1 + cµ−d(η + ω)) − c(η + ω) =0. the unique positive root yields k as k(µ,η,ω) = 1 − cµ + d(η + ω) 2µd + √ (1 − cµ + d(η + ω))2 + 4µdc(η + ω) 2µd . (18) our assumption c < 1/µ implies 1 − cµ > 0, hence k ω = 1 2µd ( 1 − cµ + dη ω + d + √( 1 − cµ + dη ω + d )2 + 4µdcη ω2 + 4µdc ω   > 1 2µd ( 1 − cµ + dη ω + d + 1 − cµ + dη ω + d ) > 1 2µd 2d = 1 µ and thus (1 −σ)ω k < (1 −σ)µ. (19) it also follows from the above computations that limω�∞ (1−σ)ω k = (1 −σ)µ, i.e., although the left hand side of (11) is always less than (1−σ)µ, the expression gets arbitrary close to this limit as ω approaches ∞. next we fix every model parameter but η and ω and obtain two propositions as follows. for the proofs, see the appendix. proposition vi.3. let us assume that (θ + µ + σφ)2 ≥ σ(µ + γ)(1 −σ)φ holds. if the condition (θ + µ + σφ) (θ + σµ + σφ) < σ(1 −σ)(µ + γ)(µ + φ) biomath 2 (2013), 1312051, http://dx.doi.org/10.11145/j.biomath.2013.12.051 page 8 of 14 http://dx.doi.org/10.11145/j.biomath.2013.12.051 diána h. knipl et al., backward bifurcation in sivs model... is satisfied, then for any η there is an ωc such that for any ω ∈ (ωc,∞) there is a backward bifurcation at r0 = 1, and for any ω ∈ [0,ωc] there is a forward bifurcation at r0 = 1. in case the above condition does not hold, then for any η and ω there is a forward bifurcation at r0 = 1. with other words, for parameter values satisfying the assumption and condition of proposition vi.3, a unique critical value ωc can be defined which works as a threshold of ω for the backward bifurcation: there is no backward bifurcation if ω ≤ ωc, and once ω is large enough so that a backward bifurcation is established at r0 = 1, it can not happen that for any larger values of ω the system undergoes forward bifurcation again. with certain conditions, such threshold also exists for η as we show it in the following proposition. proposition vi.4. we assume that (θ + µ + σφ)2 ≥ σ(µ + γ)(1 −σ)φ holds, and fix ω. if ω is such that (1 −σ)ω k(µ, 0,ω) > (θ + µ + σφ)2 −σ(µ + γ)(1 −σ)φ (θ + µ + σφ) + σ(µ + γ) then there exists ηc > 0 such that there is a backward bifurcation at r0 = 1 for η < ηc, and the system undergoes a forward bifurcation for η ≥ ηc. if the above inequality does not hold then there is a forward bifurcation at r0 = 1. we illustrate propositions vi.3 and vi.4 with figure 3. with parameter values µ = 1, γ = 7.5, θ = 0.5, σ = 0.02, φ = 16, c = 0.1, d = 0.03 and η = 10 used for figure 3 (a), the condition in proposition vi.3 becomes 1.5288 < 2.8322. in case of figure 3 (b), the parameters µ = 1.5, γ = 11, θ = 0.5, σ = 0.02, φ = 16, c = 1/15, d = 9/300 and ω = 60 give (1−σ)ω k(µ,0,ω) = 0.956928 and (θ+µ+σφ)2−σ(µ+γ)(1−σ)φ (θ+µ+σφ)+σ(µ+γ) = 0.569027, so the condition in proposition vi.4 is satisfied. it is easy to check that the assumption (θ + µ + σφ)2 ≥ σ(µ + γ)(1 − σ)φ holds in both cases since (a) 3.3124 ≥ 2.6656 and (b) 5.3824 ≥ 3.92. proposition vi.1 states that for any values of η and ω the condition (θ + µ + σφ)2 < σ(µ + γ)(1 − σ)φ is sufficient for the existence of a backward bifurcation at r0 = 1; moreover we know from proposition vi.2 that it is also necessary in the special case of ω = 0. we remark that backward bifurcation is possible for any η ≥ 0 and ω > 0, even if (θ + µ + σφ)2 ≥ σ(µ + γ)(1−σ)φ. let us choose η ≥ 0 and ω > 0 arbitrary, fix parameters µ, σ, φ, and choose θ and γ such that (θ + µ + σφ)2 = σ(µ + γ)(1 − σ)φ holds. as now the right hand side of condition (11) is 0 and ω,k > 0, there is a backward bifurcation, moreover it is easy to see that the right hand side is increasing in θ. thus, due to the continuous dependence of the right hand side on θ, there is an interval for θ (with all the other parameters fixed) where condition (11) still holds, though (θ + µ + σφ)2 > σ(µ + γ)(1 − σ)φ since the quadratic term increases in θ. next, we investigate how immigration deforms the bifurcation curve. let us denote by β0 the value of the transmission rate for which r0 = 1 is satisfied, using (6) it arises as β0 = (µ + θ + φ)(µ + γ) k(µ + θ + σφ) − (1 −σ)ω . (20) we recall that endemic equilibria ĭ1 and ĭ2 were defined as ĭ1 = −b − √ b2 − 4ac 2a , ĭ2 = −b + √ b2 − 4ac 2a , with a,b and c given in (10). obviously −b −√ b2 − 4ac > 0 where ĭ1 exists and −b +√ b2 − 4ac > 0 where ĭ2 exists. we prove the following two propositions in the appendix. proposition vi.5. it holds that β0 decreases in both ω and η. proposition vi.6. for the endemic equilibrium ĭ2 it holds that ∂ ∂ω ĭ2, ∂ ∂η ĭ2 > 0, the inequalities ∂ ∂ω ĭ1, ∂ ∂η ĭ1 < 0 are satisfied for the endemic equilibrium ĭ1. the equilibrium ĭ1 = ĭ2 = −b 2a increases in both ω and η. these results give us information about how the bifurcation curve changes when the immigration parameters increase. if there is a forward bifurcation at r0 = 1, the curve moves to the left since β0 decreases in η and ω, and the curve expands because ∂ ∂ω ĭ2, ∂ ∂η ĭ2 > 0. in case there is a backward bifurcation at r0 = 1, β0 again moves to the left, and ∂∂ω ĭ1, ∂ ∂η ĭ1 < 0 and ∂ ∂ω ĭ2, ∂ ∂η ĭ2 > 0 imply that for each fixed β the two equilibria move away from each other in the region where they coexist, moreover ĭ2 increases when it is the only endemic equilibrium. the singular point of the bifurcation curve, where the equilibrium is −b/2a, moves upward as η and ω increase, this together with biomath 2 (2013), 1312051, http://dx.doi.org/10.11145/j.biomath.2013.12.051 page 9 of 14 http://dx.doi.org/10.11145/j.biomath.2013.12.051 diána h. knipl et al., backward bifurcation in sivs model... 0.0 0.5 1.0 1.5 2.0 2.5 0 20 40 60 80 100 120 β i` (a) η = 10,ω = 1,6, . . .96. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 20 40 60 80 100 β i` (b) ω = 60,η = 1,6, . . .96. fig. 3: bifurcation diagrams for 20 different values of (a) ω and (b) η in the case when (θ + µ + σφ)2 ≥ σ(µ + γ)(1 −σ)φ. the curves move to the left as the immigration parameter increases. we let λ(x) = x c+dx and choose parameter values as (a) µ = 1, γ = 7.5, θ = 0.5, σ = 0.02, φ = 16, c = 0.1, d = 0.03, (b) µ = 1.5, γ = 11, θ = 0.5, σ = 0.02, φ = 16, c = 1/15, d = 9/300. βckσ =x−u + 2 √ uw =σ(µ + γ) − (θ + µ + σφ) + 2 √ −(θ + µ + σφ)σ(µ + γ) + σ(γ + µ)(µ + φ + θ) =σ(µ + γ) − (θ + µ + σφ) + 2 √ σ(µ + γ)φ(1 −σ) (21) the above described behavior of ĭ1 and ĭ2 imply that the left-most equilibrium cannot move to the right, or equivalently, the corresponding value of the transmission rate βc decreases if we increase η and ω. we give the last statement of the above discussion in the form of a proposition. see figures 2 and 3 for visual proof of the results of this section. proposition vi.7. in case there is a backward bifurcation at r0 = 1, βc decreases in both ω and η. actually, using (20) it is easy to see that β0 converges to 0 as any of the immigration parameters approaches infinity: for any fixed ω (η), the carrying capacity k reaches arbitrary large values if we increase η (ω), moreover µk −ω is positive by assumption, hence lim ω�∞ (k(µ + θ + σφ) − (1 −σ)ω) = = lim ω�∞ (k(θ + σφ) + σω + µk −ω) = ∞. βc < β0 implies that βc also goes to 0 as ω � ∞ or η � ∞. we can also show that in the special case of ω = 0, increasing η decreases the region where two endemic equilibria exist. the equation (15) for βc then reformulates as (21), thus for β0 −βc we have (β0 −βc)kσ = σ(µ + θ + φ)(µ + γ) (µ + θ + σφ) −σ(µ + γ) − ( (θ + µ + σφ) + 2 √ σ(µ + γ)φ(1 −σ) ) . the right hand side is independent of η, moreover k increases monotonically as η increases, so the length of the interval (βc,β0) decreases as η increases. in the light of the results of this section we conclude that, although sivs models without immigration can also exhibit backward bifurcation [1], incorporating the possibility of the inflow of non-infectives may significantly influence the dynamics: under certain conditions on the model parameters, increasing ω just as decreasing η can drive a system with forward bifurcation into backward bifurcation and the existence of multiple endemic equilibria. nevertheless we showed that including immigration moves the left-most point of the bifurcation curve to the left, which means that the larger the values of the immigration parameters the smaller the threshold for the emergence of endemic equilibria. vii. revisiting the three-dimensional system based on our results for system (4), we draw some conclusions on the global behavior of the original model (2). given that n(t) converges, and substituting s(t) = n(t) − i(t) − v (t), (2)2 and (2)3 together can be considered as an asymptotically autonomous system with limiting system (4). we use the theory from [13]. theorem vii.1. all nonnegative solutions of (2) converge to an equilibrium. in particular, if r0 > 1, biomath 2 (2013), 1312051, http://dx.doi.org/10.11145/j.biomath.2013.12.051 page 10 of 14 http://dx.doi.org/10.11145/j.biomath.2013.12.051 diána h. knipl et al., backward bifurcation in sivs model... then the endemic equilibrium is globally asymptotically stable. if there is a forward bifurcation for (4) and r0 ≤ 1, or there is a backward bifurcation for (4) and r0 < rc, then the disease-free equilibrium is globally asymptotically stable. proof: theorem v.2 excluded periodic orbits in the limit system by a dulac-function, hence we can apply corollary 2.2. of [13] and conclude that all solutions of (2)2 − (2)3 converge. as i(t),v (t) and n(t) converge, s(t) converges as well for system (2). now consider the case r0 > 1. then the endemic equilibrium is globally asymptotically stable for (2) (see theorem v.2), and its basin of attraction is the whole phase space except the disease-free equilibrium. we can proceed analogously as in (16) to show that no positive solutions of (2)2 − (2)3 can converge to (0, v̄ ) when r0 > 1, since n(t) > k − � holds for sufficiently large t. thus, the ω-limit set of any positive solution of (2)2 − (2)3 intersects the basin of attraction of the endemic equilibrium in the limit system, and then by theorem 2.3 of [13] we conclude that the positive solutions of (2)2−(2)3 converge to the endemic equilibrium. when the disease-free is the unique equilibrium of (4), (i.e., when r0 ≤ 1 in the case of forward, or r0 < rc in the case of backward bifurcation), then it is globally asymptotically stable for (4) (see theorem v.2) with the basin of attraction being the whole space, thus theorem 2.3 of [13] ensures that the dfe is globally asymptotically stable for (2)2 − (2)3 as well. viii. conclusion we have examined a dynamic model which describes the spread of an infectious disease in a population divided into the classes of susceptible, infected and vaccinated individuals, and took the possibility of immigration of non-infectives into account. such an assumption is reasonable if there is an entry screening of infected individuals, or if the disease is so severe that it inhibits traveling. after obtaining some fundamental, but biologically relevant properties of the model, we investigated the possible equilibria and gave an explicit condition for the existence of backward bifurcation at r0 = 1 in terms of the model parameters. our analysis showed that besides the disease-free equilibrium – which always exists – there is a unique positive fixed point for r0 > 1, moreover in case of a backward bifurcation there exist two endemic equilibria on an interval to the left of r0 = 1. an equilibrium is locally asymptotically stable if and only if it corresponds to a point on the bifurcation curve where the curve is increasing, moreover it is also globally attracting if r0 > 1. we investigated how the structure of the bifurcation curve depends on η and ω (the immigration parameter for susceptible and vaccinated individuals, respectively), when other model parameters are fixed. as discussed in propositions vi.1 and vi.3, two regions can be characterized in the parameter space where for any values of the immigration parameters the system experiences a backward or forward bifurcation, respectively. nevertheless, under certain conditions described in propositions vi.3 and vi.4, modifying the value of ω and η has a significant effect on the dynamics: critical values ωc and ηc can be defined such that the bifurcation behavior at r0 = 1 changes from forward to backward when we increase ω through ωc and/or we decrease η through ηc. however, propositions vi.2 and vi.4 yield that in some cases ω can be chosen such that, independently from the value of η, backward bifurcation is impossible. we also showed that immigration decreases the value of the transmission rate for which endemic equilibria emerge, furthermore increasing ω and/or η moves the branches of the bifurcation curve apart which implies that the stability region of the disease-free equilibrium shrinks (see figures 2 and 3). last, we wish to point out that, as it follows from the discussion after proposition vi.4, backward bifurcation is possible for any values of ω and η, so when one’s aim is to mitigate the severity of an outbreak it is desirable to control the values of other model parameters, for example, the vaccination rate in a way that such scenario is never realized. appendix for readers’ convenience here we recall propositions vi.3, vi.4, vi.5 and vi.6, and present their proofs. proposition vi.3. let us assume that (θ+µ+σφ)2 ≥ σ(µ + γ)(1 −σ)φ holds. if the condition (θ + µ + σφ) (θ + σµ + σφ) < σ(1 −σ)(µ + γ)(µ + φ) is satisfied, then for any η there is an ωc such that for any ω ∈ (ωc,∞) there is a backward bifurcation biomath 2 (2013), 1312051, http://dx.doi.org/10.11145/j.biomath.2013.12.051 page 11 of 14 http://dx.doi.org/10.11145/j.biomath.2013.12.051 diána h. knipl et al., backward bifurcation in sivs model... k −ω ∂k ∂ω = 1 − cµ + d(η + ω) + √ (1 − cµ + d(η + ω))2 + 4µdc(η + ω) 2µd −ωd 1 2µd ( 1 + 1 − cµ + d(η + ω) + 2µc√ (1 − cµ + d(η + ω))2 + 4µdc(η + ω) ) = 1 − cµ + dη 2µd + (1 − cµ + d(η + ω))2 + 4µdc(η + ω) 2µd √ (1 − cµ + d(η + ω))2 + 4µdc(η + ω) − ωd(1 − cµ + d(η + ω) + 2µc) 2µd √ (1 − cµ + d(η + ω))2 + 4µdc(η + ω) = 1 − cµ + dη 2µd + (1 − cµ + d(η + ω))(1 − cµ + dη) + 4µdcη + 2µdcω 2µd √ (1 − cµ + d(η + ω))2 + 4µdc(η + ω) > 0 (22) at r0 = 1, and for any ω ∈ [0,ωc] there is a forward bifurcation at r0 = 1. in case the above condition does not hold, then for any η and ω there is a forward bifurcation at r0 = 1. proof of proposition vi.3: if (θ + µ + σφ) (θ + σµ + σφ) ≥σ(1 −σ)· · (µ + γ)(µ + φ), (θ + µ + σφ) ( θ + µ + σφ 1 −σ −µ ) ≥σ(µ + γ)(µ + φ), (θ + µ + σφ)2 1 −σ −σ(µ + γ)φ ≥µ(θ + µ + σφ) + µσ(µ + γ)), (θ + µ + σφ)2 −σ(µ + γ)(1 −σ)φ (θ + µ + σφ) + σ(µ + γ) ≥(1 −σ)µ, then it follows from (19) that backward bifurcation is not possible at r0 = 1 since the right hand side of condition (11) is always greater than or equal to the left hand side. next let us consider the case when (θ + µ + σφ) (θ + σµ + σφ) <σ(1 −σ)· · (µ + γ)(µ + φ), (θ + µ + σφ)2 −σ(µ + γ)(1 −σ)φ (θ + µ + σφ) + σ(µ + γ) <(1 −σ)µ. we show that (1−σ)ω k is monotone increasing in ω; if so, then, following relation (19) and the discussion afterwards, the formulas (1−σ)·0 k(µ,η,0) = 0 and limω�∞ (1−σ)ω k(µ,η,ω) = (1 −σ)µ imply that ωc can be defined uniquely by (1 −σ)ωc k(µ,η,ωc) = (θ + µ + σφ)2 −σ(µ + γ)(1 −σ)φ (θ + µ + σφ) + σ(µ + γ) , and from the monotonicity it follows that the condition for the backward bifurcation (11) is satisfied if and only if ω > ωc. we obtain the derivative ∂ ∂ω ( ω k ) = k −ω∂k ∂ω k2 , which implies that (1−σ)ω k increases in ω if and only if k −ω∂k ∂ω is positive. with our assumption 1 − cµ > 0 the computations in (22) yield the result. proposition vi.4. we assume that (θ + µ + σφ)2 ≥ σ(µ + γ)(1 −σ)φ holds, and fix ω. if ω is such that (1 −σ)ω k(µ, 0,ω) > (θ + µ + σφ)2 −σ(µ + γ)(1 −σ)φ (θ + µ + σφ) + σ(µ + γ) then there exists ηc > 0 such that there is a backward bifurcation at r0 = 1 for η < ηc, and the system undergoes a forward bifurcation for η ≥ ηc. if the above inequality does not hold then there is a forward bifurcation at r0 = 1. proof of proposition vi.4: first we note that k(µ,η,ω) (defined in (18)) is an increasing function of η and it attains its minimum at η = 0. this implies that (1 −σ)ω k(µ,η,ω) ≤ (1 −σ)ω k(µ, 0,ω) for all η, hence the condition for the backward bifurcation (11) cannot be satisfied if (1 −σ)ω k(µ, 0,ω) ≤ (θ + µ + σφ)2 −σ(µ + γ)(1 −σ)φ (θ + µ + σφ) + σ(µ + γ) . on the other hand, k(µ,η,ω) takes arbitrary large values, and hence (1−σ)ω k(µ,η,ω) converges to zero monotonically as η increases, so if (1 −σ)ω k(µ, 0,ω) > (θ + µ + σφ)2 −σ(µ + γ)(1 −σ)φ (θ + µ + σφ) + σ(µ + γ) , biomath 2 (2013), 1312051, http://dx.doi.org/10.11145/j.biomath.2013.12.051 page 12 of 14 http://dx.doi.org/10.11145/j.biomath.2013.12.051 diána h. knipl et al., backward bifurcation in sivs model... ∂ ∂ω (√ b2 − 4ac −b ) = 2b∂b ∂ω − 4(−σβ(µ + θ + σφ)∂k ∂ω + σβ(1 −σ)) 2 √ b2 − 4ac − ∂b ∂ω , = ∂b ∂ω ( b − √ b2 − 4ac ) √ b2 − 4ac + 2σβ((µ + θ + σφ)∂k ∂ω − (1 −σ)) √ b2 − 4ac , ∂ ∂η (√ b2 − 4ac −b ) = 2b∂b ∂η − 4(−σβ(µ + θ + σφ)∂k ∂ω ) 2 √ b2 − 4ac − ∂b ∂η , = ∂b ∂η (b − √ b2 − 4ac) √ b2 − 4ac + 2σβ(µ + θ + σφ)∂k ∂η√ b2 − 4ac . (23) ∂ ∂ω (√ b2 − 4ac + b ) = ∂b ∂ω ( b + √ b2 − 4ac ) √ b2 − 4ac + 2σβ((µ + θ + σφ)∂k ∂ω − (1 −σ)) √ b2 − 4ac > 0, ∂ ∂η (√ b2 − 4ac + b ) = ∂b ∂η ( b + √ b2 − 4ac ) √ b2 − 4ac + 2σβ(µ + θ + σφ)∂k ∂η√ b2 − 4ac > 0. (24) then there is a unique ηc > 0 which satisfies (1 −σ)ω k(µ,ηc,ω) = (θ + µ + σφ)2 −σ(µ + γ)(1 −σ)φ (θ + µ + σφ) + σ(µ + γ) , and the monotonicity of k in η yields that for η < ηc (η ≥ ηc) the condition for the backward bifurcation (11) holds (does not hold). thus it is clear that ηc is a threshold for the existence of backward bifurcation. note that if (θ + µ + σφ)2 = σ(µ + γ)(1−σ)φ then ηc = ∞, i.e., for each value of η there is a backward bifurcation if ω > 0. the proof is complete. proposition vi.5. it holds that β0 decreases in both ω and η. proof of proposition vi.5: using (20) we see that β0 decreases as η increases since ∂ ∂η (k(µ + θ + σφ) − (1 −σ)ω) = ∂k ∂η (µ + θ + σφ) > 0. on the other hand, β0 decreases in ω if and only if ∂ ∂ω (k(µ + θ + σφ) − (1 −σ)ω) = ∂k ∂ω (µ + θ + σφ) − (1 −σ) > 0. first, ∂k ∂ω > 1 µ since 1 − cµ + d(η + ω) + 2µc√ (1 − cµ + d(η + ω))2 + 4µdc(η + ω) > 1 ∂k ∂ω = 1 2µ ( 1 + 1 − cµ + d(η + ω) + 2µc√ (1 − cµ + d(η + ω))2 + 4µdc(η + ω) ) > 1 µ , second, from θ + σφ > −µσ µ + θ + σφ > µ(1 −σ) we have 1 µ > 1−σ µ+θ+σφ . we conclude that ∂k ∂ω > 1 µ > 1 −σ µ + θ + σφ (25) and hence β0 decreases as ω increases. proposition vi.6. for the endemic equilibrium ĭ2 it holds that ∂ ∂ω ĭ2, ∂ ∂η ĭ2 > 0, the inequalities ∂ ∂ω ĭ1, ∂ ∂η ĭ1 < 0 are satisfied for the endemic equilibrium ĭ1. the equilibrium ĭ1 = ĭ2 = −b 2a increases in both ω and η. proof of proposition vi.6: as ∂ac ∂ω = −σβ(µ + θ + σφ) ∂k ∂ω + σβ(1 −σ), ∂ac ∂η = −σβ(µ + θ + σφ) ∂k ∂η , biomath 2 (2013), 1312051, http://dx.doi.org/10.11145/j.biomath.2013.12.051 page 13 of 14 http://dx.doi.org/10.11145/j.biomath.2013.12.051 diána h. knipl et al., backward bifurcation in sivs model... we derive (23), moreover it follows from (25), ∂b ∂ω = −σβ∂k ∂ω < 0, ∂b ∂η = −σβ∂k ∂η < 0 and b −√ b2 − 4ac < 0 that ∂ ∂ω (√ b2 − 4ac −b ) > 0, ∂ ∂η (√ b2 − 4ac −b ) > 0. similarly, using b + √ b2 − 4ac < 0 we get (24) and thus ∂ ∂ω ĭ1 = − ∂ ∂ω (√ b2 − 4ac + b ) 2a < 0, ∂ ∂η ĭ1 = − ∂ ∂η (√ b2 − 4ac + b ) 2a < 0, moreover ∂ ∂ω ĭ2 = ∂ ∂ω (√ b2 − 4ac −b ) 2a > 0, ∂ ∂η ĭ2 = ∂ ∂η (√ b2 − 4ac −b ) 2a > 0. the equilibrium ĭ1 = ĭ2 = −b2a is increasing in both ω and η since a is independent of these parameters and ∂b ∂ω < 0, ∂b ∂η < 0. acknowledgment dhk acknowledges support by the european union and the state of hungary, co-financed by the european social fund in the framework of támop 4.2.4. a/211-1-2012-0001 ’national excellence program’. rg was supported by the european union and the european social fund through project futurict.hu (grant támop– 4.2.2.c-11/1/konv-2012-0013), by the european research council stg nr. 259559, and hungarian scientific research fund otka k109782. the authors are grateful to the reviewers for their valuable comments that improved the manuscript. references [1] f. brauer, backward bifurcations in simple vaccination models, j. math. anal. appl. 298 (2004) pp. 418-431. http://dx.doi.org/10.1016/j.jmaa.2004.05.045 [2] f. brauer, backward bifurcations in simple vaccination/treatment models, journal of biological dynamics, 5:5 (2011) pp. 410-418. http://dx.doi.org/10.1080/17513758.2010.510584 [3] h. dulac, points singuliers des equations differentielles, mem. sci. math., vol. 61, gauthier-villars (1934) [4] j. dushoff, w. huang, c. castillo-chavez, backwards bifurcations and catastrophe in simple models of total diseases, j. math. biol. 36 (1998) pp. 227-248. http://dx.doi.org/10.1007/s002850050099 [5] l. edelstein-keshet mathematical models in biology vol. 46. siam (1988) [6] a. b. gumel, causes of backward bifurcations in some epidemiological models, j. math. anal. appl., 395(1) (2012) pp. 355-365. http://dx.doi.org/10.1016/j.jmaa.2012.04.077. [7] k. p. hadeler, c. castillo-chavez, a core group model for disease transmission, math. biosci. 128 (1995) pp. 41-55. http://dx.doi.org/10.1016/0025-5564(94)00066-9 [8] k. p. hadeler, p. van den driessche, backward bifurcation in epidemic control, math. biosci. 146 (1997) pp. 15-35. http://dx.doi.org/10.1016/s0025-5564(97)00027-8 [9] c. m. kribs-zaleta, center manifolds and normal forms in epidemic models, in: c. castillo-chavez, s. blower, d. kirschner, p. van den driessche, a. a. yakubu (eds.), mathematical approaches for emerging and re-emerging infectious diseases: an introduction, springer-verlag (2001) pp. 269-286. [10] c. m. kribs-zaleta, j. x. velasco-hernández, a simple vaccination model with multiple endemic states, math. biosci. 164 (2000) pp. 183-201. http://dx.doi.org/10.1016/s0025-5564(00)00003-1 [11] l. markus, asymptotically autonomous differential systems, in: s. lefschetz (ed.), contributions to the theory of nonlinear oscillations iii, in: ann. math. stud., vol. 36, princeton univ. press (1956) pp. 17-29. [12] h. r. thieme, asymptotically autonomous differential equations in the plane, rocky mountain j. math. 24 (1994) pp. 351-380. [13] h. r. thieme, c. castillo-chavez, asymptotically autonomous epidemic models, in: o. arino, d. axelrod, m. kimmel, m. langlais (eds.), mathematical population dynamics: analysis of heterogeneity, vol. i, theory of epidemics, wuerz (1995) pp. 33-50. biomath 2 (2013), 1312051, http://dx.doi.org/10.11145/j.biomath.2013.12.051 page 14 of 14 http://dx.doi.org/10.1016/j.jmaa.2004.05.045 http://dx.doi.org/10.1080/17513758.2010.510584 http://dx.doi.org/10.1007/s002850050099 http://dx.doi.org/10.1016/j.jmaa.2012.04.077. http://dx.doi.org/10.1016/0025-5564(94)00066-9 http://dx.doi.org/10.1016/s0025-5564(97)00027-8 http://dx.doi.org/10.1016/s0025-5564(00)00003-1 http://dx.doi.org/10.11145/j.biomath.2013.12.051 introduction sivs model with immigration fundamental properties of the system endemic equilibrium stability and global behavior the influence of immigration on the backward bifurcation revisiting the three-dimensional system conclusion references communication/review biomath 1 (2012), 1209307, 1–1 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum in memoriam antony popov (1962–2012) we are deeply saddened by the sudden death at the age of 49 of our colleague associate professor dr. antony popov, member of the department ”information technologies”, faculty of mathematics and informatics, sofia university. tony was a very good mathematician, specializing in the area of pattern recognition. he was able to master and apply techniques from several important subfields of this broad area: mathematical morphology, interval analysis, and fuzzy sets. dr. popov pioneered the methods and theory of mathematical morphology in bulgaria. this particular research filed requires solid foundation and understanding of geometry, set theory, topology, analysis, and statistics. tony was an example of a true professional who combined the knowledge from all of these fields to produce several excellent papers on important mathematical morphology topics. many of his colleagues from the ”biomathematics” group had been active participants in the m.s. program ”biomedical informatics”. this graduate program was created under tony’s leadership about ten years ago. he put a lot of effort to build the program up, and found a way to involve his colleagues from various departments: mathematics, biophysics, bioengineering, and biology. many students got enrolled and graduated from the program producing quality theses. tony always was encouraging them and was really happy for their success. dr. popov was also actively involved in organizing the regular bioinformatics & computational biology seminar supported jointly by the agrobio institute (abi) and the faculty of mathematics and informatics. the seminar has served as an incubator for many new ideas, attracted new ph.d students, facilitated the establishment of international collaborative projects, publications, and software development and implementation. tony was instrumental in organizing the school for young scientists as a workshop held at the yearly biomath conference. he was pretty much the sole organizer of the last year workshop, and planned the entire scientific program for the current 2012 one. during the last several years, dr. popov was actively involved in the research project ”computer simulation and innovative model-based study of bioprocesses” supported by the national science fund established by the bulgarian ministry of education. several of his ph.d. students became a part of the research team. tony will be always remembered as a very helpful, modest, and trusted colleague with a big hart. he was and will be an example of a real gentleman for all of us. he will be dearly missed! from: tony’s colleagues; department of biomathematics, institute of mathematics and informatics, bulgarian academy of science; faculty of mathematics and informatics, sofia university ”st. kliment ohridski”; bioinformatics group at the agrobio institute, sofia; genomic signal processing laboratory, texas a&m university. http://dx.doi.org/10.11145/j.biomath.2012.09.307 page 1 of 1 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.09.307 editorial biomath 2 (2013), 1312319 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum chalenges and opportunities in mathematical and theoretical biology and medicine: foreword to volume 2 (2013) of biomath the challenges faced today at the interface of the computational, life, social, mathematical and data sciences expand as the world’s population approaches the seven billion mark. today, we are asked to address questions over multiple temporal and spatial scales as well as across various levels of organization. in the process, the world has placed mathematical scientists, broadly defined to include computational, computer, statistics and data science experts, at the heart of trans-disciplinary and interdisciplinary efforts aimed primarily at assessing arguably the most important technical challenge of our times, namely, how to model and quantify uncertainty. twenty first century research is driven to a great extent by questions involving the study of the dynamics of biological process. and so, we have been forced to account for the impact of social systems, bringing the role of human-decisions to the forefront of most efforts to put the power of the quantitative sciences into the hands of policy and decision makers. the challenges faced by mathematical scientist include the development and implementation of frameworks and methods aimed at assessing and mitigating the impact of health disparities; the identification, understanding and management of sustainability concepts and systems; the development and systematic implementation of holistic approaches, a key to the study of the dynamics and control of biological systems; the development of methods and approaches that help evaluate uncertainty in current studies of climate change, including global warming; the need to find scalable solutions that address issues of food security and world hunger; and more. a prototypical and timely global challenge, that cuts across issues of sustainability, evolution and disease dynamics, is briefly discussed in this introduction as a way of highlighting the challenges posed by the study of phenomena across multiple temporal and spatial scales and levels of organization. on march 11, 2013, the british chief medical officer dame sally davies the independent, tuesday 12 march 2013 observed that “the problem of microbes becoming increasingly resistant to the most powerful drugs should be ranked alongside terrorism and climate change on the list of critical risks to the nation. ... yet while antibiotic use is rising not least in agriculture for farmed animals and fish ... resistance is steadily growing and the pipeline is drying up ... of new drugs which can replace those becoming useless. [in fact] no new classes of antibiotics have been developed since 1987, and none are in the pipeline.” professor nigel brown, president of the society for general microbiology remarks that immediate action by scientists is required if we are going to identify and mass produce new antibiotics; the kind of effort needed to tackle the problem of antimicrobial resistance and its transmission, particularly in the context of nosocomial (hospital) infections. (ibid.) the head of the who while addressing a meeting of infectious disease experts in copenhagen, highlighted the global crisis in antibiotics, the result of “rapidly evolving resistance among microbes responsible for common infections that threaten to turn them into untreatable diseases ... every antibiotic ever developed was at risk of http://dx.doi.org/10.11145/j.biomath.2013.12.319 page 1 of 2 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2013.12.319 becoming useless. a post-antibiotic era means, in effect, an end to modern medicine as we know it. things as common as strep throat or a child’s scratched knee could once again kill ... antimicrobial resistance is on the rise in europe, and elsewhere in the world. we are losing our first-line antimicrobials.” (the independent, friday 16 march 2012). the issues of the persistence, evolution and the expansion of resistance to antimicrobials are important because the number of drugs available is limited with no new ones being developed for the past three decades. the magnitude of the catastrophes predicted by the british chief medical officer dame sally davies and by the director general of the world health organization, dr. margaret chan on the global challenges posed by antibiotic resistance, must therefore be assessed and monitored and mathematical scientists have the tool kits needed to make a difference. my own experiences as a past participant, in a variety of roles nearly three decades ago, in the schools of ecology organized at the international centre for theoretical physics (ictp), trieste italy, led for several years by tom hallam, lou gross and simon levin, brought me head on for the first time in my life with the most important resource that we have to address the grand global challenges that dominate the twenty-first century, namely, the talent, energy, passion, foresight and commitment of young researchers across the world, an experience repeated in 2013 in sofia, bulgaria. the series biomath conferences carried, unfortunately not continuously, for over a couple of decades, provide the international environment needed for mentoring and promoting the career of young individuals, talented mathematical scientists, who are passionate about issues of national and international impact. biomath conferences are “devoted to recent research in biosciences based on applications of mathematics as well as mathematics applied to or motivated by biological applications.” its commitment to building multi-national environments that foster interdisciplinary and trans-disciplinary research is expressed explicitly in its chart that identifies this venue as “a multidisciplinary meeting forum for researchers who develop and apply mathematical and computational tools to the study of phenomena in the broad fields of biology, ecology, medicine, biotechnology, bioengineering, environmental science, etc.” this volume includes selected papers from the presentations at the 2013 biomath conference. they provide a sample of the richness, talent and diversity of the community of mathematical scientists that participated in this event. the articles involve applications to cell biology, epidemiology, immunology and ecology as well as methodological work in numerical methods, optimal control, statistics and network theory. i was proud to be asked to participate at the 2013 biomath conference held in sofia, bulgaria, honored to be asked to write a foreword for volume 2 of journal biomath, blessed by the opportunity to listen and learn from talented individuals and develop new friendships. carlos castillo-chavez arizona state university math., comp., and model. sciences ctr. tempe, az, usa email: ccchavez@asu.edu biomath 2 (2013), 1312319, http://dx.doi.org/10.11145/j.biomath.2013.12.319 page 2 of 2 http://dx.doi.org/10.11145/j.biomath.2013.12.319 editorial biomath 1 (2012), 1211113 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum from the guest-editor the biomath 2012 international conference on mathematical methods and models in biosciences was held at the bulgarian academy of sciences in sofia, in june 17–22, 2012, http://www.biomath.bg/2012/. we were happy to meet more than 70 participants from twenty different countries. more than 40 contributions were submitted for publication in the present biomath proceedings. all submitted papers have been peer-reviewed by at least two independent anonymous reviewers. twelve selected papers are published in the first issue of this journal. this second issue contains another ten selected contributions which will be published continuously in the electronic version of the journal. we are grateful for the support provided by several academic units and universities and to all members of the program and organizing committees for their active help in the organization of the biomath 2012 international conference. we thank also all participants for their contribution to the success of this conference. we are especially grateful to all reviewers for their time and efforts. with deep sorrow we announce at the end of the issue the sudden deaths of two dear colleagues of us who were very helpful in the organization of the biomath conferences: stefan dodunekov and anthony popov. academician stefan dodunekov, president of the bulgarian academy of sciences, was chairman of the biomath 2011 and biomath 2012 conferences. dr. anthony popov, from the faculty of mathematics and informatics at the university of sofia “kl. ohridski”, was the main organizer of the school for young scientists at the biomath 2011 and biomath 2012 conferences. svetoslav markov http://dx.doi.org/10.11145/j.biomath.2012.11.113 page 1 of 1 http://www.biomathforum.org/biomath/index.php/biomath http://www.biomath.bg/2012/ http://dx.doi.org/10.11145/j.biomath.2012.11.113 editorial biomath 1 (2012), 1210113 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum from the guest-editor the biomath 2012 international conference on mathematical methods and models in biosciences was held at the academy of sciences in sofia, bulgaria, in june 17–22, 2012, http://www.biomath.bg/2012/. we were happy to meet more than 70 participants from twenty different countries. more than 40 contributions were submitted for publication in the present biomath proceedings. all papers have been peer-reviewed by at least two independent anonymous reviewers and are going to be published in two consecutive issues of this journal. this first issue contains 12 selected contributions. a special session of biomath 2012 was dedicated to blagovest sendov on the occasion of his 80th birthday. he is one of the pioneers of biomathematics in bulgaria. we are grateful to the support provided by several academic units and universities and to all members of the program and organizing committees for their active help in the organization of the biomath 2012 international conference. we thank also all participants for their contribution to the success of this conference. we are especially grateful to all reviewers for their time and efforts. svetoslav markov http://dx.doi.org/10.11145/j.biomath.2012.10.113 page 1 of 1 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.10.113 communication/review biomath 1 (2012), 1209305, 1–1 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum in memoriam stefan dodunekov (1945–2012) the president of the bulgarian academy of sciences stefan dodunekov passed away on august 5, 2012, after a short illness at the age of 66. stefan dodunekov was a distinguished expert the field of coding theory. his results in the characterization of the optimal binary linear codes of dimension 7 and 8 established him as one of the leading scientists in the field. his later results on near-mds codes, spherical codes and designs, self-dual codes, were much acknowledged by the coding theory community and fueled the research of many young scientists. stefan dodunekov was also one of the pioneers in using computer algebra in the research. stefan dodunekov was born in 1945 in kilifarevo and graduated from sofia university in 1968 with a degree in “mathematics”. he was doctor of mathematics (phd) since 1975, doctor of sciences (dsci) since 1986, professor since 1990 and academician (full member of the bulgarian academy of sciences) since 2008. he was director of the institute of mathematics and informatics at the bulgarian academy of sciences since 1999 and was elected president of the bulgarian academy of sciences on 11.06.2012. he had over 35 years of lecturing experience in bulgarian and foreign universities. he is author of of three books and 12 textbooks in mathematics for secondary schools. stefan dodunekov was chairman of the union of the bulgarian mathematicians since 2001 and president of the mathematical society of south-eastern europe (massee) for two mandates. he was member of the information theory society, american mathematical society, combinatorial society of china. stefan dodunekov was twice chairman of the international conferences on mathematical methods and models in biosciences (biomath 2011 and biomath 2012). he actively participated in the meetings and projects of the biomathematics departments of the the institute of mathematics and informatics, to mention the scopes project funded by the swiss national science fund. he highly supported the research in the field of mathematical modeling in biology in bulgaria. for the people who had the privilege of knowing stefan dodunekov, he was a friend who was always ready to help. on august 8, 2012, hundreds of colleagues and friends of stefan dodunekov said to him “last farewell” in the big hall of the main building of the bulgarian academy of sciences. academician blagovest sendov’s farewell speech at this meeting is included next in this volume. http://dx.doi.org/10.11145/j.biomath.2012.09.305 page 1 of 1 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.09.305 original article biomath 1 (2012), 1212107, 1–5 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum a linear complementarity numerical approach to the non-convex problem of structures environmentally damaged and strengthened by cable-bracings dedicated to academician blagovest sendov’s 80th anniversary konstantinos liolios∗, stefan radev†, asterios liolios‡, ivan georgiev§, krassimir georgiev¶ ∗department of environmental engineering, democritus university of thrace, xanthi, greece kliolios@env.duth.gr †institute of mechanics, bulgarian academy of sciences, sofia, bulgaria stradev@imbm.bas.bg ‡department of civil engineering, democritus university of thrace, xanthi, greece aliolios@civil.duth.gr §institute of mathematics and informatics, bulgarian academy of sciences, sofia, bulgaria john@parallel.bas.bg ¶institute of information and communication technologies, bulgarian academy of sciences, sofia, bulgaria georgiev@parallel.bas.bg received: 1 august 2012, accepted: 10 december 2012, published: 29 december 2012 abstract—a computational treatment is presented for the mathematically rigorous analysis of civil engineering structures, which have been environmentally damaged and subsequently strengthened by cable-elements. the problem is treated as an inequality one, where the governing conditions are equalities as well as inequalities. the cable behavior is considered as nonconvex and nonmonotone one and is described by generalized subdifferential relations including loosening, elastoplastic fracturing and other effects. using piece-wise linearization for the cable behavior, a linear complementarity problem, with a reduced number of unknowns, is solved by optimization algorithms. finally, an example from civil and environmental engineering praxis is presented. keywords-civil and environmental engineering; nonconvex analysis; computational mechanics; cable-braced structures; optimization algorithms; i. introduction environmental actions can often cause significant damages to civil engineering structures, see e.g. [1], [2]. main such defect is the strength degradation, causing a reduction of the load bearing capacity. to handle such defects, sometimes cable-like members are used as a first strengthening and repairing procedure. these cable-like members can undertake tension but buckle and become slack and structurally ineffective when subjected to a sufficiently large compressive force. thus the governing conditions take an equality as well as an inequality form and the problem becomes nonlinear. so, the problem of structures containing as above cable-like members belongs to the so-called inequality problems of mechanics, as their governing conditions are of both, equality and inequality type [3]–[6] . a realistic numerical treatment of such problems citation: k liolios, s radev, a liolios, i georgiev, k georgiev, a linear complementarity numerical approach to the non-convex problem of structures environmentally damaged and strengthened by cable-bracings, biomath 1 (2012), 1212107, http://dx.doi.org/10.11145/j.biomath.2012.12.107 page 1 of 5 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.12.107 k liolios at al., a linear complementarity numerical approach to the non-convex problem of structures ... can be obtained by mathematical programming methods (optimization algorithms) [3]–[5]. the early numerical realizations of these approaches were based mainly upon the principle of minimum complementary energy. thus, an equivalence principle for the analysis of statically undetermined structures with unilateral constraints has been proposed and proven by g. nitsiotas [3]. further, for the rigorous mathematical investigation of the problem, convex analysis and the variational or hemivariational inequality concept have been introduced and used, see panagiotopoulos [6], [7] and [11]. the aim of this paper is to deal with the development of a simple numerical procedure for the rigorous analysis of civil engineering structures containing cablelike members by using a version of the direct stiffness (displacement) method of structural analysis. the present procedure is based on the finite element method and the equivalence principle, proposed by g. nitsiotas in [3]. using this principle, the analysis of such structures can be reduced to a linear complementarity problem (lcp), which can be solved by various effective quadratic programming algorithms [5]. a numerical example shows the direct applicability on the computer and the effectiveness of the procedure presented herein. ii. method of analysis a. problem formulation a frame structure containing n cable-like members is considered. the structure is discretized according to the natural finite element method [4], [6]. for the cables, pin-jointed bar elements with unilateral behavior are used. following panagiotopoulos [6], [7]. the behavior of the cables, including loosening, elastoplastic or/and elastoplastic-softening-fracturing and unloading reloading effects, can be expressed mathematically by the subdifferential relation: si(di) ∈ ∂̂si(di) (1) here si and di are the (tensile) force and the deformation (elongation), respectively, of the i-th cable element, ∂̂ is the generalized gradient and si is the superpotential function [6], [7], [10], [11]. by definition, relation (1) is equivalent to the following hemivariational inequality, expressing the virtual work principle for inequality problems: s ↑ i (di, ei −di) ≥ si(di).(ei −di) (2) here s↑i denotes the subderivative of si , and ei, di are kinematically admissible (virtual) deformations. for numerical treatments of practical inequality problems, a piece-wise linearization is usually applied to relations (1) and (2), see e.g. [4], [8]. so, the unilateral behavior for the i-th cable-element (i = 1, . . . , n) is expressed by the following relations [3], [4]: ei = f0isi + ei0 −vi (3) si ≥ 0, vi ≥ 0, si.vi = 0. (4) here ei , f0i, si, ei0, and vi denote the strain (elongation), natural flexibility constant, stress (tension), initial strain and slackness, respectively. from (3) it is clear that the slackness vi can be considered as an unknown initial strain which constitutes a reversible negative elongation [3]. further, relations (4) express that either a nonnegative stress-tension or a nonnegative slackness exists on any cable. for the remaining structure (besides the cables), the usual frame finite element models, which exhibit a bilateral behavior, are used. b. numerical solution approach now the equivalence principle, proposed by g. nitsiotas in [3], is applied for the whole structure. according to this principle, the structure under consideration behaves as an equivalent, linearly elastic structure, under the condition that in each cable-element either a nonnegative stress or a fictitious, unknown, nonnegative slackness appears-see relations (3). thus, collecting in (n×1) vectors t and v the stress and slackness behavior of all the n cable-elements, corresponding, the following linear complementarity conditions hold: t ≥ 0, v ≥ 0, tt v = 0. (5) further, following the stiffness (displacement) method of structural analysis, we consider the cableelement as solidified rods and we assume that the so-modified structure is a statically stable one with bilateral rod-elements. so, the tension vector t is decomposed as follows [9]: t = cv + t0 (6) here t0 is the stress vector of the solidified cableelements, now acting as normal bilateral rods, due to external actions and c is the natural influence matrix of v on t. for both it is assumed a linearly elastic, bilateral behavior for the stable structure, where. the cables are considered as already solidified bars. so, the natural biomath 1 (2012), 1212107, http://dx.doi.org/10.11145/j.biomath.2012.12.107 page 2 of 5 http://dx.doi.org/10.11145/j.biomath.2012.12.107 k liolios at al., a linear complementarity numerical approach to the non-convex problem of structures ... stiffness matrix c is symmetric and in general positive semi-definite. thus, if t0 is known, then vectors t and v can be determined by solving the linear complementarity problem (lcp) formed by relations (5) and (6). for the solution of this problem, various effective algorithms are available [3]–[10]. most of these algorithms reduce the above linear complementarity problem to a quadratic programming one of the form: min { 1 2 vt cv + vt t0 : v ≥ 0 } (7) after the previous preparation we can now formulate the following numerical procedure for the analysis of structures containing cable like members: a) considering the cables as having been solidified (normal bilateral bars), the vector t0 due to external actions is determined by the finite element method. b) under the same assumption and by the same method as in a), the influence matrix c is determined. in this matrix, cij is the stress (axial force) in the solidified cable-element i caused by a unit-shortening vj = 1 imposed in the solidified cable-member j, (i, j = 1, . . . , n). c) the linear complementarity problem of relations (5) and (6) is solved to provide the sought vector v. so it is computed which cable-elements are activated (under tension) and which are not (under non-zero slackness). d) the final stress state of the structure is determined by taking into account the external actions and the computed forces t of the active cable-elements. thus, the whole procedure requires the linear elastic analysis of the modified (with solidified cable-elements) structure (n + 2) times, where n is the number of the cables, and the solution of a quadratic programming problem or a lcp. alternatively, after having computed t, the structure is analyzed due to external actions by omitting the slack cables for which the step c) has given zero tension values. iii. numerical example the reinforced concrete plane frame structure of fig. 1 had been initially analyzed, designed and constructed to bear the shown loads, without the shown cableelements. the concrete class is c40/50 and the elasticity modulus eb = 3.5×107kn/m2. the shown dimensions width/height of rectangular sections are in centimeters [cm]. the analysis of the above frame without cableelements is obtained by using any available finite element method code, e.g. sap2000 code [13]. due to environmental actions [1], [2], corrosion and cracking had been taking place. this had caused a reduction for the section inertia moments, which, according to [14], is estimated to be 10% for the columns and 50% for the beams. so it was necessary for the system to be strengthened. as a first repairing and strengthening procedure, ten (n = 10) cable members with crosssectional area fr = 8 cm2 have been added as shown in fig. 1. the steel class is s1400/1600 with an elasticity modulus es = 210gpa. these cables are placed as counter diagonals. as it is not known in advance which of them are activated or not by the given loads, the purpose here is to compute what happens. the application of the presented numerical procedure gives first the values of the slackness of the not activated cable-elements: v1 = 0.848×10−3m, v3 = 10.321×10−3m, v5 = 1.082×10−3m, v8 = 9.564×10−3m, v10 = 1.652×10−3m. further, the elements of vector t, where t = [s1, s2, . . . , s10]t , are computed to have the following values (in kn) for the non-active cables: s1 = s3 = s5 = s8 = s10 = 0.0, whereas for the active cables it holds: s2 = 10.17kn, s4 = 346.04kn, s6 = 18.84kn, s7 = 342.08kn, s9 = 25.81kn. thus, cables 2,4,6,7 and 9 are the only ones active, having zero slackness. the other cables 1,3,5,8 and 10 cannot contribute to the system resistance under the given loads in fig. 1. using the previous results, the final stress state is computed. in fig. 2 is shown indicatively the final bending moments diagram for the strengthened frame containing the active cable-elements only. comparing the diagram in fig. 2 with the corresponding one for the initial frame without cables, the efficiency of the strengthening can be checked. iv. conclusion the inequality problem of the cable-braced civil engineering structures can be treated numerically by the herein presented approach. this approach takes into account the unilateral behavior of cable elements, uses the equivalence principle of g. nitsiotas [3] and so leads to a linear complementarity problem, with a reduced number of problem unknowns. thus, the numerical realization of biomath 1 (2012), 1212107, http://dx.doi.org/10.11145/j.biomath.2012.12.107 page 3 of 5 http://dx.doi.org/10.11145/j.biomath.2012.12.107 k liolios at al., a linear complementarity numerical approach to the non-convex problem of structures ... 4,6 m 3,4 m 3,1 m 4,9 m 18 kn 180 kn 4,1 m 3,0 m 24 kn 32kn/m 52 kn/m 220 kn 8 kn/m 30/40 30/50 30/60 30/60 30/60 30/45 30/50 30/65 30/55 30/70 30/65 30/50 30/30 25/25 25/30 30/35 10 9 1 2 3 4 5 6 7 8 fig. 1. the cable-braced structural system of the numerical example. fig. 2. bending moments diagram (in knm) for the frame with the 5 active cable-elements no 2,4,6,7 and 9. biomath 1 (2012), 1212107, http://dx.doi.org/10.11145/j.biomath.2012.12.107 page 4 of 5 http://dx.doi.org/10.11145/j.biomath.2012.12.107 k liolios at al., a linear complementarity numerical approach to the non-convex problem of structures ... the proposed approach is obtained by available computer codes of the finite element method and of mathematical programming algorithms. moreover, as it has been verified in an example, the herein developed approach can treat in a realistic way the general problem of cablebraced structures in civil and environmental engineering praxis. an extension of the presented approach for the case of the earthquake response of such structures has been recently presented [12]. acknowledgment this work was partially supported by the bulgarian science fund under the project dmu 03-62. references [1] h. s. peavy, d. r. rowe, and g. tchobanoglous, environmental engineering, mcgraw-hill, new york, 1985. [2] a. moncmanova, environmental deterioration of materials, wit press, southampton, 2007. http://dx.doi.org/10.2495/978-1-84564-032-3 [3] g. nitsiotas, die berechnung statisch unbestimmter tragwerke mit einseitigen bindungen, ingenieur-archiv, vol. 41, pp. 46-60, 1971. http://dx.doi.org/10.1007/bf00536162 [4] g. maier, a quadratic programming approach for certain classes of non-linear structural problems, meccanica , vol. 3, pp. 121-130, 1968. http://dx.doi.org/10.1007/bf02129011 [5] r. w. cottle, fundamentals of quadratic programming and linear complementarity, in m.z. cohn, d.e. grierson and g. maier (eds.), ”engineering plasticity by mathematical programming”, pergamon press, oxford, 1979. [6] p. d. panagiotopoulos, inequality problems in mechanics and applications. convex and nonconvex energy functions, birkhuser verlag, basel, boston, 1985. http://dx.doi.org/10.1007/978-1-4612-5152-1 [7] p. d. panagiotopoulos, hemivariational inequalities. applications in mechanics and engineering, springer verlag, berlin, 1993. http://dx.doi.org/10.1007/978-3-642-51677-1 [8] a. a. liolios, a linear complementarity approach for the nonconvex seismic frictional interaction between adjacent structures under instabilizing effects. journal of global optimization, vol. 17, no. 1-4, pp. 259-266, 2000. http://dx.doi.org/10.1023/a:1026789817828 [9] ang. liolios, numerical simulation of structures with unilateral constraints and solution by the finite element method, graduate degree thesis, department of civil engineering, democritus university of thrace, xanthi, greece, 2007. [10] e. s. mistakidis and g. e. stavroulakis, nonconvex optimization in mechanics. smooth and nonsmooth algorithmes, heuristic and engineering applications, kluwer, london, 1998. [11] r. t. rockafellar, convex analysis, princeton univ. press, princeton, 1970. [12] ang. liolios, k. chalioris, ast. liolios, st. radev, and kon. liolios, a computational approach for the earthquake response of cable-braced reinforced concrete structures under environmental actions. in: lirkov, i., margenov, s. and wasniewski, j. (eds): ”large-scale scientific computing”, lncs 7116, pp. 590-597, springer-verlag, berlin heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-29843-1 [13] computers and structures, inc. sap2000, integrated finite element analysis and design of structures, berkeley, 2005. [14] t. pauley and m. j. n. priestley, seismic design of reinforced concrete and masonry buildings, wiley, new york, 1992. http://dx.doi.org/10.1002/9780470172841 biomath 1 (2012), 1212107, http://dx.doi.org/10.11145/j.biomath.2012.12.107 page 5 of 5 http://dx.doi.org/10.2495/978-1-84564-032-3 http://dx.doi.org/10.1007/bf00536162 http://dx.doi.org/10.1007/bf02129011 http://dx.doi.org/10.1007/978-1-4612-5152-1 http://dx.doi.org/10.1007/978-3-642-51677-1 http://dx.doi.org/10.1023/a:1026789817828 http://dx.doi.org/10.1007/978-3-642-29843-1 http://dx.doi.org/10.1002/9780470172841 http://dx.doi.org/10.11145/j.biomath.2012.12.107 introduction method of analysis problem formulation numerical solution approach numerical example conclusion references original article biomath 1 (2012), 1211225, 1–4 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum mathematical problems in the theory of bone poroelasticity merab svanadze∗, antonio scalia† ∗institute for fundamental and interdisciplinary mathematics research ilia state university, tbilisi, georgia svanadze@gmail.com †dipartimento di matematica e informatica università di catania, catania, italy scalia@dmi.unict.it received: 15 july 2012, accepted: 22 november 2012, published: 29 december 2012 abstract—this paper concerns with the quasi-static theory of bone poroelasticity for materials with double porosity. the system of equations of this theory based on the equilibrium equations, conservation of fluid mass, the effective stress concept and darcy’s law for material with double porosity. the internal and external basic boundary value problems (bvps) are formulated and uniqueness of regular (classical) solutions are proved. the single-layer and double-layer potentials are constructed and their basic properties are established. finally, the existence theorems for classical solutions of the bvps are proved by means of the potential method (boundary integral method) and the theory of singular integral equations. keywords-bone poroelasticity; double porosity; boundary value problems. i. introduction the concept of porous media is used in many areas of applied science (e.g., biology, biophysics, biomechanics) and engineering. poroelasticity is a well-developed theory for the interaction of fluid and solid phases of a fluid saturated porous medium. it is an effective and useful model for deformation-driven bone fluid movement in bone tissue [1], [2], [3]. the theory of consolidation for elastic materials with double porosity was presented by aifantis and his coworkers [4], [5]. the aifantis’ theory unifies the earlier proposed models of barenblatt’s for porous media with double porosity [6] and biot’s for porous media with single porosity [7]. however, aifantis’ quasi-static theory ignored the cross-coupling effects between the volume change of the pores and fissures in the system. the cross-coupled terms were included in the equations of conservation of mass for the pore and fissure fluid by several authors [8], [9], [10]. the double porosity concept was extended for multiple porosity media by bai et al. [11]. the theory of multiporous media, as originally developed for the mechanics of naturally fractured reservoirs, has found applications in blood perfusion. the double porosity model would consider the bone fluid pressure in the vascular porosity and the bone fluid pressure in the lacunar-canalicular porosity [1], [2], [3]. an extensive review of the results in the theory of bone poroelasticity can be found in the survey papers [1], [2]. for a history of developments and a review of main results in the theory of porous media, see de boer [12]. the investigation of bvps of mathematical physics by the classical potential method has a hundred year history. the application of this method to the 3d bvps of the theory of elasticity reduces these problems to 2d singular integral equations [13]. owing to the works of mikhlin [14], kupradze and his pupils (see [15], [16]), the theory of multidimensional singular integral equations has presently been worked out with sufficient completeness. citation: m svanadze, a scalia, mathematical problems in the theory of bone poroelasticity, biomath 1 (2012), 1211225, http://dx.doi.org/10.11145/j.biomath.2012.11.225 page 1 of 4 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.11.225 m svanadze at al., mathematical problems in the theory of bone poroelasticity this theory makes it possible to investigate 3d problems not only of the classical theory of elasticity, but also problems of the theory of elasticity with conjugated fields. for an extensive review of works on the potential method, see gegelia and jentsch [17]. this paper concerns with the quasi-static theory of bone poroelasticity for materials with double porosity [8], [9], [10]. the system of equations of this theory based on the equilibrium equations, conservation of fluid mass, the effective stress concept and darcy’s law for a material with double porosity. the internal and external basic bvps are formulated and uniqueness of classical solutions are proved. the single-layer and double-layer potentials are constructed and their basic properties are established. finally, the existence theorems for classical solutions of the bvps are proved by means of the boundary integral method and the theory of singular integral equations. ii. basic equations let x = (x1,x2,x3) be a point of the euclidean threedimensional space r3, let t denote the time variable, t ≥ 0, u(x, t) denote the displacement vector in solid, u = (u1,u2,u3); p1(x, t) and p2(x, t) are the pore and fissure fluid pressures, respectively. we assume that subscripts preceded by a comma denote partial differentiation with respect to the corresponding cartesian coordinate, repeated indices are summed over the range (1,2,3), and the dot denotes differentiation with respect to t. in the absence of body force the governing system of field equations of the linear quasi-static theory of elasticity for solid with double porosity consists of the following equations [8], [9], [10]. 1) the equilibrium equations tlj,j = 0, l = 1, 2, 3, (1) where tjl is the component of the total stress tensor. 2) the equations of fluid mass conservation div v(1) + ζ̇1 + β1ėrr + γ(p1 −p2) = 0, div v(2) + ζ̇2 + β2ėrr −γ(p1 −p2) = 0, (2) where v(1) and v(2) are fluid flux vectors for the pores and fissures, respectively; elj is the component of the strain tensor, elj = 1 2 (ul,j + uj,l) , l,j = 1, 2, 3, (3) β1 and β2 are the effective stress parameters, γ is the internal transport coefficient and corresponds to a fluid transfer rate respecting the intensity of flow between the pores and fissures, γ > 0; ζ1 and ζ2 are the increment of fluid in the pores and fissures, respectively, and defined by ζ1 = α1 p1 + α3 p2, ζ2 = α3 p1 + α2 p2, (4) α1 and α2 measure the compressibilities of the pore and fissure systems, respectively, α3 is the cross-coupling compressibility for fluid flow at the interface between the two pore systems at a microscopic level [8], [9], [10]. however, the coupling effect (α3) is often neglected in the literature [4], [5], [6]. 3) the equations of the effective stress concept tlj = t ′ lj − (β1p1 + β2p2) δlj, l,j = 1, 2, 3, (5) where t′lj = 2µelj +λerrδlj is the component of effective stress tensor, λ and µ are the lamé constants, δlj is the kronecker’s delta. 4) the darcy’s law for material with double porosity v(1) = − κ1 µ′ grad p1, v (2) = − κ2 µ′ grad p2, (6) where µ′ is the fluid viscosity, κ1 and κ2 are the macroscopic intrinsic permeabilities associated with the matrix and fissure porosity, respectively. we note that in the real porous media the fissure permeability κ2 is much greater than the matrix permeability κ1, while the fracture porosity is much smaller than the matrix porosity. substituting equations (3)-(6) into (1) and (2), we obtain the following system of homogeneous equations in the linear quasi-static theory of elasticity for solids with double porosity expressed in terms of the displacement vector u, pressures p1 and p2. µ∆u + (λ + µ)∇div u −β1∇p1 −β2∇p2 = 0, k1∆p1 −α1ṗ1 −α3ṗ2 −γ(p1 −p2) −β1divu̇ = 0, k2∆p2 −α3ṗ1 −α2ṗ2 + γ(p1 −p2) −β2divu̇ = 0, (7) where ∆ and ∇ are the laplacian and gradient operators, respectively, and kj = κj µ′ (j = 1, 2). in the follows we assume that the inertial energy density of solid with double porosity is a positive definite quadratic form. thus, the constitutive coefficients satisfy the conditions: µ > 0, 3λ + 2µ > 0, k1 > 0, k2 > 0, α1 > 0, α1α2 −α23 > 0. biomath 1 (2012), 1211225, http://dx.doi.org/10.11145/j.biomath.2012.11.225 page 2 of 4 http://dx.doi.org/10.11145/j.biomath.2012.11.225 m svanadze at al., mathematical problems in the theory of bone poroelasticity if the displacement vector u, the pressures p1 and p2 are postulated to have a harmonic time variation, that is, {u,p1,p2}(x, t) = re [{ u′,p′1,p ′ 2 } (x)e−iωt ] , then from system (7) we obtain the following system of homogeneous equations of steady vibrations in the linear quasi-static theory of elasticity for solids with double porosity µ∆u′ + (λ + µ)∇divu′ −β1∇p′1 −β2∇p ′ 2 = 0, (k1∆ + a1)p ′ 1 + a3p ′ 2 + iω β1 div u ′ = 0, a3p ′ 1 + (k2 ∆ + a2)p ′ 2 + iω β2 div u ′ = 0, (8) where aj = iω αj −γ, a3 = iω α3 + γ (l,j = 1, 2); ω is the oscillation frequency, ω > 0. iii. boundary value problems let s be the closed surface surrounding the finite domain ω+ in r3, s ∈ c2,λ0, 0 < λ0 ≤ 1, ω̄ = ω ∪s, ω− = r3\ω̄+. definition 1. a vector function u = (u′,p′1,p ′ 2) = (u1,u2, · · · ,u5) is called regular in ω− (or ω+) if 1) ul ∈ c2(ω−)∩c1(ω̄−) (or ul ∈ c2(ω+)∩c1(ω̄+)), 2) ul(x) = o(|x|−1), ul,j(x) = o(|x|−1), where |x|� 1, l = 1, 2, · · · , 5, j = 1, 2, 3. the basic bvps of steady vibrations in the linear quasi-static theory of elasticity for solid with double porosity are formulated as follows. find a regular (classical) solution u = (u′,p′1,p ′ 2) to system (8) satisfying the boundary condition lim ω+3x→ z∈s u(x) ≡{u(z)}+ = f (z) in the problem (i)+f , and lim ω−3x→ z∈s u(x) ≡{u(z)}− = f (z) in the problem (i)−f , where f is the known fivecomponent vector function. iv. uniqueness theorems we have the following results. theorem 1. the internal homogeneous bvp (i)+f admits at most one regular solution. theorem 2. the external bvp (i)−f admits at most one regular solution. theorems 1 and 2 can be proved similarly to the corresponding theorems in the classical theory of thermoelasticity (for details see [13, chapter iii]). v. basic properties of elastopotentials the system (8) may be written as b(dx) u(x) = 0, where b(dx) is the matrix differential operator corresponding left-hand side of (8) and dx = ( ∂∂x1 , ∂ ∂x2 , ∂ ∂x3 ). we introduce the following notations: 1) z(1)(x, g) = ∫ s γ(x − y)g(y)dys is the singlelayer potential, 2) z(2)(x, g) = ∫ s [p̃(dy, n(y))γ >(x − y)]>g(y)dys is the double-layer potential, where γ = (γlj)5×5 is the fundamental matrix of the operator b(dx), p̃ = ( p̃lj ) 5×5 is the matrix differential operator of the first order, g is five-component vector, the superscript > denotes transposition. remark 1. in the aifantis’ quasi-static theory (α3 = 0), the fundamental matrix γ(x) is constructed by svanadze [18]. we have the following basic properties of elastopotentials. theorem 3. if s ∈ c2,λ0, g ∈ c1,λ ′ (s), 0 < λ′ < λ0 ≤ 1, then: (a) z(1)(·, g) ∈ c0,λ ′ (r3) ∩c2,λ ′ (ω̄±) ∩c∞(ω±), (b) b(dx) z (1) (x, g) = 0, x ∈ ω±, (c) p(dz, n(z)) z (1) (z, g) is a singular integral, (d) {p(dz, n(z)) z(1)(z, g)}± = ∓ 1 2 g(z) +p(dz, n(z)) z (1) (z, g), z ∈ s, where p(dz, n(z)) is the stress operator in the linear theory of elasticity for solids with double porosity. theorem 4. if s ∈ c2,λ0, g ∈ c1,λ ′ (s), 0 < λ′ < λ0 ≤ 1, then: (a) z(2)(·, g) ∈ c1,λ ′ (ω̄±) ∩c∞(ω±), (b) b(dx) z (2) (x, g) = 0, x ∈ ω±, (c) z(2)(z, g) is a singular integral, (d) {z(2)(z, g)}± = ± 1 2 g(z) + z(2)(z, g), z ∈ s. theorems 3 and 4 can be proved similarly to the corresponding theorems in the classical theory of thermoelasticity (for details see [13, ch. x]). biomath 1 (2012), 1211225, http://dx.doi.org/10.11145/j.biomath.2012.11.225 page 3 of 4 http://dx.doi.org/10.11145/j.biomath.2012.11.225 m svanadze at al., mathematical problems in the theory of bone poroelasticity vi. existence theorem we introduce the notation k1 g(z) ≡ 1 2 g(z) + z(2)(z, g), k2 g(z) ≡− 1 2 g(z) + z(2)(z, g) for z ∈ s. obviously, on the basis of theorem 4, k1 and k2 are the singular integral operators. lemma 1. the singular integral operators k1 and k2 are of the normal type with an index equal to zero. lemma 1 leads to the following existence theorems. theorem 5. if s ∈ c2,λ0, f ∈ c1,λ ′ (s), 0 < λ′ < λ0 ≤ 1, then a regular (classical) solution of the internal bvp (i)+f exists, is unique and is represented by double-layer potential u(x) = z(2)(x, g) for x ∈ ω+, where g is a solution of the singular integral equation k1 g(z) = f (z) for z ∈ s which is always solvable for an arbitrary vector f . theorem 6. if s ∈ c2,λ0, f ∈ c1,λ ′ (s), 0 < λ′ < λ0 ≤ 1, then a regular (classical) solution of the external bvp (i)−f exists, is unique and is represented by sum u(x) = z(2)(x, g) − i z(1)(x, g) for x ∈ ω−, where g is a solution of the singular integral equation kg(z) − i z(1)(z, g) = f (z) for z ∈ s which is always solvable for an arbitrary vector f . theorem 5 and 6 are proved by means of the potential method and the theory of singular integral equations (for details see [13]). vii. conclusion 1. by the above mentioned method it is possible to prove the existence and uniqueness theorems in the modern linear theories of elasticity and thermoelasticity for materials with microstructure. 2. on the basis of theorems 1 to 6 it is possible to obtain numerical solutions of the bvps of the quasistatic theory of elasticity for solids with double porosity by using of the boundary element method. references [1] s.c. cowin, bone poroelasticity, j. biomech. 32:3 (1999), 217– 238. http://dx.doi.org/10.1016/s0021-9290(98)00161-4 [2] s.c. cowin, g. gailani and m. benalla, hierarchical poroelasticity: movement of interstitial fluid between levels in bones, philos.trans.r. soc. a 367 (2009) 3401-3444. http://dx.doi.org/10.1098/rsta.2009.0099 [3] e. rohan, s. naili, r. cimrman and t. lemaire, multiscale modeling of a fluid saturated medium with double porosity: relevance to the compact bone, j. mech. phys. solids 60:5 (2012), 857-881. http://dx.doi.org/10.1016/j.jmps.2012.01.013 [4] r.k. wilson and e.c. aifantis, on the theory of consolidation with double porosity -i, int. j. eng. sci. 20:9 (1982), 1009–1035. http://dx.doi.org/10.1016/0020-7225(82)90036-2 [5] m.y. khaled, d.e. beskos and e.c. aifantis, on the theory of consolidation with double porosity iii, int. j. numer. anal. meth. geomech. 8:2 (1984), 101-123. http://dx.doi.org/10.1002/nag.1610080202 [6] g.i. barenblatt, i.p. zheltov and i.n. kochina, basic concept in the theory of seepage of homogeneous liquids in fissured rocks (strata), j. appl. math. mech. 24:5 (1960), 1286-1303. http://dx.doi.org/10.1016/0021-8928(60)90107-6 [7] m.a. biot, general theory of three-dimensional consolidation, j. appl. phys. 12:2 (1941), 155–164. http://dx.doi.org/10.1063/1.1712886 [8] n. khalili and s. valliappan, unified theory of flow and deformation in double porous media, europ. j. mech. a/solids 15:2 (1996), 321– 336. [9] n. khalili, m.a. habte and s. zargarbashi, a fully coupled flow deformation model for cyclic analysis of unsaturated soils including hydraulic and mechanical hysteresis, comput. geotech. 35:6 (2008), 872–889. http://dx.doi.org/10.1016/j.compgeo.2008.08.003 [10] n. khalili, coupling effects in double porosity media with deformable matrix, geophys. res. letters 30:22 (2003), 2153. [11] m. bai, d. elsworth and j.c. roegiers, multiporosity/multipermeability approach to the simulation of naturally fractured reservoirs, water resour. res. 29:6 (1993), 1621–1633. http://dx.doi.org/10.1029/92wr02746 [12] r. de boer, theory of porous media: highlights in the historical development and current state, springer-verlag, berlin, heidelberg, new york, 2000. [13] v.d. kupradze, t.g. gegelia, m.o. basheleishvili and t.v. burchuladze, three-dimensional problems of the mathematical theory of elasticity and thermoelasticity, northholland, amsterdam, new york, oxford, 1979. [14] s.g. mikhlin, multidimensional singular integrals and integral equations. pergamon press, oxford, 1965. [15] v.d. kupradze, potential methods in the theory of elasticity. israel program for scientific translations, jerusalem, 1965. [16] t.v. burchuladze and t.g. gegelia, the development of the potential methods in the elasticity theory, metsniereba, tbilisi, 1985. [17] t. gegelia and l. jentsch, potential methods in continuum mechanics. georgian math. j., 1:6 (1994), 599–640. http://dx.doi.org/10.1007/bf02254683 [18] m. svanadze, fundamental solution in the theory of consolidation with double porosity, j. mech. behav. mater. 16:1-2 (2005), 123–130. biomath 1 (2012), 1211225, http://dx.doi.org/10.11145/j.biomath.2012.11.225 page 4 of 4 http://dx.doi.org/10.1016/s0021-9290(98)00161-4 http://dx.doi.org/10.1098/rsta.2009.0099 http://dx.doi.org/10.1016/j.jmps.2012.01.013 http://dx.doi.org/10.1016/0020-7225(82)90036-2 http://dx.doi.org/10.1002/nag.1610080202 http://dx.doi.org/10.1016/0021-8928(60)90107-6 http://dx.doi.org/10.1063/1.1712886 http://dx.doi.org/10.1016/j.compgeo.2008.08.003 http://dx.doi.org/10.1029/92wr02746 http://dx.doi.org/10.1007/bf02254683 http://dx.doi.org/10.11145/j.biomath.2012.11.225 introduction basic equations boundary value problems uniqueness theorems basic properties of elastopotentials existence theorem conclusion references editorial biomath 1 (2012), 1210115 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum the art of mathematical biology: a foreword for the launch of biomath we cannot debate that the advances made over the past 30 years in molecular biology, biochemistry and cell biology have dramatically changed biology and the biomedical sciences. academic institutions and funding agencies increasingly recognize the interdisciplinary nature of biology and medicine, including the important role mathematical, statistical and computational modeling play in the analysis and interpretation of data and information. by and large, biologists and biomedical scientists appear to have more appreciation for mathematical biology research because they are increasingly attracted to discover new biological principles and mechanisms in collaboration with mathematicians. as a biomedical scientist active in the field of mathematical biology, i believe that the scope of mathematical biology includes providing novel biological insights that come from the mathematical formulation and analysis of biological problems. i fully recognized, however, that mathematical biology can also stimulate the development of new mathematics. there are excellent examples in the literature of mathematical biology research that have provided novel and important biological insights. classical examples may be found in the work of alan turing [1] and of alan hodgkin and andrew huxley [2]. much has changed since the publication of these papers. a mathematical biology research project is no longer guided by the independent spirit of one or two applied mathematicians working on a biological problem. modern examples show that mathematical biology research is now a team science effort. how do we presently perform mathematical biology research? an answer can be given by dividing the research process into the three steps of performing applied mathematics research [3]. the first step in performing mathematical biology research requires the formulation of the biological problem in mathematical terms. there is a widespread misunderstanding that this requires a proficiency and encyclopedic knowledge of biology. in reality it requires a good biological intuition and insight into the decision of which biological problems to attack. we need to learn to exercise excellent judgment in the formulation of the problem. this entails deciding what approximations to adopt in order to achieve a minimal model. the derivation of a minimal model, without losing the essential mechanisms of the problem, is an art form rather than a science. the mathematical biologist cannot master this art by working independently and as an isolated scientist. biology is so complex that the knowledge, intuitions and insights now require interdisciplinary team work. the second step in performing mathematical biology research requires the solution of the mathematical model formulated. in mathematical biology, this now requires an extensive knowledge of mathematical, computational and statistical methods. the selection of the appropriate mathematical or computational technique will depend on the biological and physicochemical scales of the problem under consideration. the solution of most of the mathematical models requires the implementation of complex computational algorithms. mathematical biologists need to investigate the model dynamics under biologically realistic parameter bounds. at the same time mathematical biologists need to investigate model sensitivity to parameters and initial conditions using statistical approaches and sensitivity analysis. knowledge of mathematics, scientific computing and statistics is obviously necessary, but there is no person with an encyclopedic understanding of these methods. the isolated mathematician again will have http://dx.doi.org/10.11145/j.biomath.2012.10.115 page 1 of 2 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.10.115 limited chances of succeeding. the third step in performing mathematical biology research requires the interpretation of the mathematical model solutions and their empirical verification in experimental terms. this step serves as the culmination of the research, but it cannot be accomplished without the interdependence between the mathematical biologists and the experimentalists. although it would be incorrect to say that all mathematical biology involves this third step (for example, the development of new mathematics and theory may not include this step), mathematical biology research must give priority of the empirical verification and evidence. novel mathematical biology ideas, methods and techniques will need to show usefulness in the biological and biomedical sciences. in the context of the international conferences on mathematical methods and models in the bioscience (biomath 2011 and biomath 2012), held in sofia, bulgaria, it is a great pleasure to see the steps of mathematical biology research effectively applied to investigate a wide range of problems in the biological and biomedical sciences. the biomath conferences will serve to train and catalyze new research avenues for young scientists in bulgaria, eastern europe and elsewhere. mathematical biologists are now actively seeking collaboration with experimental biologists and the need for real application is being emphasized in mathematical biology research. there are tremendous opportunities for the new generation of mathematical biologists in interdisciplinary research. the bulgarian academy of sciences is undertaking an exciting challenge with the new journal biomath. they are launching biomath to publish research being undertaken in the growing field of mathematical and computational biology. biomath will strive to be a leader in the field, publishing new research articles, reviews and communications. the editorial team composed by roumen anguelov (university of pretoria, south africa), svetoslav markov (bulgarian academy of sciences) and nina pesheva ( bulgarian academy of sciences) intends to appeal to a broad audience of researchers who draw on mathematics, statistics and computing, with the aim of providing insight into the life sciences. the bulgarian academy of sciences and editorial team are launching biomath with the determination to make it a success. i sincerely hope that you and your colleagues will enjoy reading the articles in this new journal and will submit your work for publication in biomath. acknowledgment the author is partially supported by the 21st century science initiative for studying complex systems of the james s. mcdonnell foundation and niddk r25 dk088752. he also thanks michelle wynn for providing critical comments. references [1] a. m. turing, (1952) chemical basis of morphogenesis. philosophical transactions of the royal society of london b 327, 37–72. http://dx.doi.org/10.1098/rstb.1952.0012 [2] a. l. hodgkin and a. f. huxley (1952). a quantitative description of membrane current and its application to conduction and excitation in nerve. journal of physiology 117, 500–544. [3] c.c. lin and l. a. segel (1988). mathematics applied to deterministic problems in the natural sciences. classics in applied mathematics, vol. 1. society for industrial and applied mathematics, philadelphia, p. 5. http://dx.doi.org/10.1137/1.9781611971347 santiago schnell department of molecular and integrative physiology department of computational medicine and bioinformatics, and brehm center for diabetes research university of michigan medical school ann arbor, michigan 48105, usa e-mail: schnells@umich.edu biomath 1 (2012), 1210115, http://dx.doi.org/10.11145/j.biomath.2012.10.115 page 2 of 2 http://dx.doi.org/10.1098/rstb.1952.0012 http://dx.doi.org/10.1137/1.9781611971347 http://dx.doi.org/10.11145/j.biomath.2012.10.115 references original article biomath 2 (2013), 1212127, 1–5 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum rough sets in biomedical informatics antony popov and simeon stoykov faculty of mathematics and informatics, sofia university “st. kliment ohridski” 5 james bourchier blvd., 1164 sofia, bulgaria e-mail:simeon@microdicom.com received: 30 october 2012, accepted: 12 december 2012, published: 21 january 2013 abstract—the intent of this paper is to face the essentials of granular computing and in its major component— the rough sets theory, introduced by pawlak, since any rough set represents an information granule. as a part of modern soft computing paradigm, rough sets have been introduced as an interval-like extension of the usual sets with main applications in the intelligent systems. the proposed rough approach provides efficient algorithms for finding hidden patterns in data, finds minimal sets of data (data reduction), evaluates significance of data. applications in medicine via dicom standard are presented, as well as ideas for applications to microbiology. keywords-rough sets; mathematical morphology; molecular biology classifier; medical diagnosis; medical imaging i. introduction in classical set theory a set is uniquely determined by its elements. in other words, it means that every element must be uniquely classified as belonging to the set or not. that is to say the notion of a set is a precise, or crisp one. for instance, the set of integer numbers is crisp because every number can be uniquely represented by its decimal digits. in mathematics traditionally crisp notions are mainly use to ensure precise reasoning. however philosophers and natural scientists for many years were interested also in imprecise notions like feelings, moral categories, beauty, including also many biological features like the color of the skin or a flower. the rough set approach makes the vagueness of the data possible. it provides efficient algorithms for finding hidden patterns in data, finds minimal sets of data (data reduction), evaluates significance of data. applications in medicine via dicom standard are presented in this paper, as well as applications to microbiology and biometrics. strictly speaking, any rough set represents an information granule. as an example, in gray scale images boundaries between object regions are often ill defined because of grayness or spatial ambiguities. this uncertainty can be effectively handled by describing the different objects as rough sets with upper (or outer) and lower (or inner) approximations as follows: let the universe u be an image consisting of a collection of pixels. then if we partition u into a collection of non-overlapping windows of size m × n, each window can be considered as a granule g. given this granulation, object regions in the image can be approximated by rough sets. a rough image is a collection of pixels and the equivalence relation induced partition of an image into sets of pixels lying within each non-overlapping window over the image. ii. information systems and rough sets let u be a non-empty, finite set called the universe and a is a non-empty, finite set of attributes, that is every a ∈ a is a mapping of the form a : u → va, where va is called a value set of a. the elements of u are called objects and interpreted as, e.g. cases, states, processes, patients, observations. attributes are interpreted as features, variables, characteristic conditions, etc. every information system a = (u,a) and nonempty set b ⊆ a determine a b-information function defined by infb(x) = {(a,a(x)) : a ∈ b}. the set {infa(x) : x ∈ u} is called a-information set and it is denoted by inf(a). with every subset of attributes b ⊆ a, an equivalence relation, denoted by inda(b) citation: a popov, s stoykov, rough sets in biomedical informatics, biomath 2 (2013), 1212127, http://dx.doi.org/10.11145/j.biomath.2012.12.127 page 1 of 5 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.12.127 a popov at al., rough sets in biomedical informatics (or ind(b)) called a b-indiscernibility relation, is associated and defined by ind(b) = {(s,s′) ∈ u2 : for every a ∈ b,a(s) = a(s′)}. any minimal subset b ⊆ a such that ind(a) =ind(b) is called a reduct in the information system. in fact, microcalcification on x-ray mammogram is a significant mark for early detection of breast cancer. texture analysis methods can be applied to detect clustered microcalcification in digitized mammograms [2]. in order to improve the predictive accuracy of the classifier, the original number of feature set is reduced into smaller set using feature reduction techniques. in [5] have been introduced rough set based reduction algorithms based on the extracted features. the rough reduction algorithms are tested on mammograms from mammography image analysis society (mias) database [8]. iii. rough set formal definition and main properties rough set theory can be viewed as a specific implementation of fuzzyness and vagueness, i.e., imprecision in this approach is expressed by a boundary region of a set, and not by a partial membership, like in fuzzy set theory. however a rough set can be expressed by a fuzzy membership function, as demonstrated below, but it many cases the textual and table representation of a rough set makes it easier and more efficient to practical implementations rather than the original fuzzy approach (3). moreover, the attributes may be numeric, or in the most cases non-numeric (categorical) quantities, such as big, small, good, malignant, benign etc. as we said previously, we represent the rough. objects x in the universe by their information vector infa(x). thus we can define an equivalence relation r between two objects x and y if their information representation coincides, i.e. infa(x) = infa(y), so x and y belong to a same information granule. then the lower approximation of a rough set x with respect to r is the set of all objects, which can be for certain classified as x with respect to this relation. the upper approximation of a rough set x with respect to r is the set of all objects which can be possibly classified as x with respect to r. the boundary region of a set x with respect to r is the set of all objects, which can be classified neither as x nor as notx with respect to r. for any crisp set the boundary is empty. therefore we should mainly work with rough sets for which the boundary region of x is nonempty. the equivalence class of r determined by element x will be denoted by r(x). formal definitions of approximations and the boundary region follow below. • r-lower approximation of x: r∗(x) = ⋃ x∈u {r(x) : r(x) ⊆ x}. • r-upper approximation of x: r∗(x) = ⋃ x∈u {r(x) : r(x)∩x 6=}. • r-boundary region of x: rnr(x) = r ∗(x)−r∗(x). it is easy to see that r∗(x) ⊆ x ⊆ r∗(x). thus we can define a fuzzy membership function, i.e. a fuzzy representation of the rough set x in the universe x with respect to the relation r, see [4]: µr(x) = #(r(x)∩x) #(x) . note, that in the definitions above, x is a normal precise subset of the universe u, but we have constructed its rough representation with respect to the relation r the pair (r∗(x),r∗(x)) and its fuzzy analog µr(x). here the sign # means the cardinality (the number of the elements) of a set. the lower approximation is sometimes referred to as positive region, while the space of the universe outside the upper approximation is called also negative region. it has been mentioned by bloch [7] that there is an analogy between rough sets and mathematical morphology. namely, the r-upper approximation is an analog of morphological dilation, while the r-lower approximation is an analog of morphological erosion. this fact is not surprising, since the relations between fuzzy sets and operations on them and morphology are well studied [1], and a relation between classical interval operations and morphological ones have been established. moreover, it is evident that the rough approximation of a set is similar to an interval approximation of a real number. on the other hand, interval operations and mathematical morphology have demonstrated their capabilities in solving problems in biomedicine. as an example, due to a complex nature of biomedical images, it is practically impossible to select or develop automatic segmentation methods of generic nature, that could be applied for any type of images, namely for either microand macroscopic images, cytological and histological biomath 2 (2013), 1212127, http://dx.doi.org/10.11145/j.biomath.2012.12.127 page 2 of 5 http://dx.doi.org/10.11145/j.biomath.2012.12.127 a popov at al., rough sets in biomedical informatics ones, mri and x-ray, and so on. medical image segmentation is an indispensable process in the visualization of human tissues. however, medical images always contain a large amount of noise caused by operator performance, equipment and environment. this leads to inaccuracy with segmentation. so, a robust segmentation technique is required. the basic idea behind introducing rough sets is that while some cases may be clearly distinguished as being in a set x (positive region in rough sets theory), and some cases may be clearly labeled as not being in set x (negative region). since we can obtain limited information we are not able to label all possible cases clearly. the remaining cases cannot be distinguished and lie in the boundary region. iv. rough set specification by decision rules for rough separation of the universe u one can use efficiently fuzzy c-means clustering [4]. if we want to separate m data elements into n clusters, by this algorithm we obtain n cluster centers and m n numbers between 0 and 1 showing the degree of membership of ith data element to the jth cluster. thus if this number is not less than 0.75 then the element belongs to the positive region of the cluster, if it is less than 0.25 it belongs to negative region, otherwise it belongs to the border. then by if then else rules we may specify the regions [6]. the rules can be applied to a set of unseen cases in order to estimate their classification power. several application schemes can be envisioned. let us consider one of the simplest which has shown to be useful in practice: 1. when a rough set classifier is presented with a new case, the rule set is scanned to find applicable rules, i.e. rules whose predecessors match the case. 2. if no rule is found (i.e. no rule is fired), the most frequent outcome in the training data is chosen. 3. if more than one rule fires, these may in turn indicate more than one possible outcome a voting process is then performed among the rules that fire in order to resolve conflicts and to rank the predicted outcomes. here are some rough rules which in fact form a decision table: if gene a is up-regulated and gene d is downregulated then tissue is healthy; if transcription factor f binds and transcription factor v binds then gene is co-regulated with gene h; if protein contains motif j then function is magnesium ion binding or copper ion binding; if protein contains motif d and ligand wateroctanol coeff. > c1 then binding affinity is high; if change in frequency of alpha-helix at position x > c2 then resistant to drug w. v. dicom format and its realization acr (the american college of radiology) and nema (the national electrical manufacturers association) formed a joint committee to develop a standard for digital imaging and communications in medicine [9] . this standard is developed in liaison with other standardization organizations including cen tc251 in europe and jira in japan, with review also by other organizations including ieee, hl7 and ansi in the usa. this standard is now designated for almost ct, pet, mri, ultrasound devices used in practice. it is applicable to a networked environment. the previous versions were applicable in a point-to-point environment only; for operation in a networked environment a network interface unit (niu) was required. dicom version 3.0 supports operation in a networked environment using industry standard networking protocols such as osi and tcp/ip. it specifies how devices claiming conformance to the standard react to commands and data being exchanged. previous versions were confined to the transfer of data, but dicom version 3.0 specifies, through the concept of service classes, the semantics of commands and associated data. dicom version 3.0 explicitly describes how an implementor must structure a conformance statement to select specific options. it is structured as a multi-part document. this facilitates evolution of the standard in a rapidly evolving environment by simplifying the addition of new features. iso directives which define how to structure multi-part documents have been followed in the construction of the dicom standard. a single dicom file contains both a header (which stores information about the patient’s name, the type of scan, image dimensions, etc), as well as all of the image data (which can contain information in three dimensions). the dicom header the size of this header varies depending on how much header information is stored. header represents an instance of a real world, referred to as information object. header is constructed of data elements. data elements contain the encoded values of attributes of that object. the specific content and semantics of these biomath 2 (2013), 1212127, http://dx.doi.org/10.11145/j.biomath.2012.12.127 page 3 of 5 http://dx.doi.org/10.11145/j.biomath.2012.12.127 a popov at al., rough sets in biomedical informatics attributes are specified in information object definitions (see ps 3.3 of the dicom standard [9]). the construction, characteristics, and encoding of a data set and its data elements are discussed in ps 3.5 of the dicom standard. pixel data, overlays, and curves are data elements whose interpretation depends on other related elements. as seen below, the data elements can be interpreted as rough set attributes. the main part of a dicom file is the image collection reffered by the textual part described above. data elements a data element is uniquely identified by a data element tag. the data elements in header shall be ordered by increasing data element tag number and shall occur at most once in a data set. a dicom attribute or data element is composed of: • a tag, in the format of group, element (xxxx,xxxx) that identifies the element. • a value representation (vr) that describes the data type and format of the attribute’s value. • a value length that defines the length of the attribute’s value. • a value field containing the attribute’s data. the basic attribute structure is shown below. tag vr value length value field a simple example for a single tag for a ct image is: (0028, 0004), photometric interpretation: monochrome2 here you can see also a genetic data representation: tag term frequency gene(s) gene(s) (6950,0001) response to stress 16 of 106 15.1% prx1, hsp26, pho5, hsp30, ... (6810,0015) transport 15 of 106 14.2% glk1, hxt7, hxt6, pic2, stf2, ... each data element is described by a pair of numbers (group number, data element number). even numbered groups are elements defined by the dicom standard and are referred to as public tags. odd numbered groups can be defined by users of the file format, but must conform to the same structure as standard elements. these are referred to as private tags. the acr-nema version 1 and 2 standards did not use object-oriented analysis or design. instead, attributes (or elements, as they were called) were grouped according to use. for example, there were groups of elements that carried identifying information about the patient and others consisting of elements that described the methods of image acquisition. because they were developed without an entity relationship (e-r) model, these groups do not conform to conventional object-oriented definitions. note, that the e-r data model views the real world as a set of basic objects (entities) and relationships among these objects. for example, a collection of elements used in the acrnema version 2 standard to identify and describe a computer tomographic (ct) image would also contain the patient name. in an entity relationship (e-r) model, however, the patient name is an attribute of the patient object, not of the image object. in other words, the patient name is not needed to describe the ct image, even though it would be needed to identify the image. one might also view these complex objects as consisting of parts of more than one entity in an e-r model. a novel free dicom viewer called microdicom has been created by the second author (10). it gives good opportunities for finding pathological objects. after clustering by 5 features (pixel intensity, mean and standard deviation in a 7 × 7 window, the two x and y sobel operations) four tissue clusters are specified. then by adding rules for finding the connected components associated with the pathology cluster based on contour tracing techniques, the tumor is located, see figure 1. vi. conclusion we tried to explain the power of rough modeling in biomedicine. the microdicom project is under development and further intelligent capabilities based on soft computing, and especially on rough sets theory will be included. acknowledgments this work was supported by ministry of education, youth and technologies under contract do 02359/2008 “computer simulation and innovative modelbased study of bioprocesses”. references [1] popov a. t. (2007) general definition of fuzzy mathematical morphology operations. image and non-image applications, in: nachtegael, m., van der weken, d., kerre, e.e., philips, w. (eds.), soft computing in image processing recent advances, series: studies in fuzziness and soft computing, vol. 210, springer, 355–384. biomath 2 (2013), 1212127, http://dx.doi.org/10.11145/j.biomath.2012.12.127 page 4 of 5 http://dx.doi.org/10.11145/j.biomath.2012.12.127 a popov at al., rough sets in biomedical informatics figure 1. mri slice of a human brain with a tumor detected by microdicom software [2] baeg s., s. batman, e. r. dougherty, v. g. kamat, n. kehtarnavaz, seunghan kim, a. popov, k.sivakumar, r. shah, (1999) unsupervised morphological granulometric texture segmentation of digital mammograms, journal of electronic imaging 8(1), 65–75. http://dx.doi.org/10.1117/1.482685 [3] pawlak, z. (1991) rough sets: theoretical aspects of reasoning about data. volume 9 of series d: system theory, knowledge engineering and problem solving, kluwer. [4] nguyen h. t., e. a.walker (2000) a first course in fuzzy logic (2nd edition), crc press. [5] thangavel k., karnan m., and pethalakshmi a. (2005) performance analysis of rough reduct algorithms in image mammogram, icgst international journal on graphics, vision and image processing 8, 13–21 [6] hvidsten, t.r., lgreid, a., komorowski, j. (2003) learning rule-based models of biological process from gene expression time profiles using gene ontology. bioinformatics 19, 1116– 1123 http://dx.doi.org/10.1093/bioinformatics/btg047 [7] bloch, i (2000) on links between mathematical morphology and rough sets, pattern recognition 33, 1487–1496 http://dx.doi.org/10.1016/s0031-3203(99)00129-6 [8] http://http://www.mammoimage.org/databases/; http://peipa.essex.ac.uk/info/mias.html [9] http://www.dclunie.com/dicom-status/status.html [10] http://www.microdicom.com biomath 2 (2013), 1212127, http://dx.doi.org/10.11145/j.biomath.2012.12.127 page 5 of 5 http://dx.doi.org/10.1117/1.482685 http://dx.doi.org/10.1093/bioinformatics/btg047 http://dx.doi.org/10.1016/s0031-3203(99)00129-6 http://http://www.mammoimage.org/databases/ http://peipa.essex.ac.uk/info/mias.html http://www.dclunie.com/dicom-status/status.html http://www.microdicom.com http://dx.doi.org/10.11145/j.biomath.2012.12.127 introduction information systems and rough sets rough set formal definition and main properties rough set specification by decision rules dicom format and its realization conclusion references www.biomathforum.org/biomath/index.php/biomath original article numerical solutions of one-dimensional parabolic convection-diffusion problems arising in biology by the laguerre collocation method burcu gürbüz, mehmet sezer department of mathematics, manisa celal bayar university, manisa, turkey burcugrbz@gmail.com, mehmet.sezer@cbu.edu.tr received: 2 october 2016, accepted: 4 june 2017, published: 9 june 2017 abstract—in this work, we present a numerical scheme for the approximate solutions of the onedimensional parabolic convection-diffusion model problems which arise in biological models. the presented method is based on the laguerre collocation method used for ordinary differential equations. the approximate solution of the problem in the truncated laguerre series form is obtained by this method. by substituting truncated laguerre series solution into the problem and by using the matrix operations and the collocation points, the suggested scheme reduces the problem to a linear algebraic equation system. by solving this equation system, the unknown laguerre coefficients can be computed. the accuracy and efficiency of the method is studied by comparing with other numerical methods when used to solve some numerical experiments. keywords-convection-diffusion equation models, parabolic problem, laguerre collocation method. i. introduction diffusion models form a reasonable basis for studying insect and animal dispersal and invasion, which arise from the question of persistence of endangered species, biodiversity, disease dynamics, multi-species competition so on. convection-diffusion problem is also a form of heat and mass transfer in biological models [1-3]. fig. 1. (a) flow between imaginary compartments in a continuous one-dimensional system. (b) discrete grid system used in two-dimensional transport models. (c) a close-up of five grid points showing the similarity to compartment models. compartment models are general framework citation: burcu gürbüz, mehmet sezer, numerical solutions of one-dimensional parabolic convection-diffusion problems arising in biology by the laguerre collocation method, biomath 6 (2017), 1706047, http://dx.doi.org/10.11145/j.biomath.2017.06.047 page 1 of 5 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2017.06.047 burcu gürbüz, mehmet sezer, numerical solutions of one-dimensional parabolic convection-diffusion ... fig. 2. a conceptual rate equation with respect to the convection-diffusion model that has many applications in biology, ecosystems and enzyme kinetics which can be mostly shown by forrester diagrams. the system is decomposed into flows of material as possibly large number of discrete compartments which are very useful. conversely, it is also useful for the quantities nominally not flow, for instance, blood or water pressure in animal and plant physiological systems. furthermore, complex interconnection networks can be addressed by these type of models with respect to link many of them together in many different complicated ways (fig. 1). on the other hand, in transport models, we have a physical quantity, such as energy i.e. heat or a quantity of matter, that flows from spatial point to point. there are many forces that could influence the flow of the matter, but the following simplified view uses two that will illustrate the qualitative model formulation. convection moves the substance with a physical flow of water from point to point (i.e. river flow). diffusion moves a substance in any direction according to the concentration of the substance around each point (fig. 2) [4-5]. in this study, we consider the one-dimensional parabolic convection-diffusion problem ∂u ∂t = ∂2u ∂x2 + a(x) ∂u ∂x + b(x)u + f(x,t), 0 ≤ x ≤ l,0 ≤ t ≤ t, (1) with the initial conditions u(x,0) = g(x), 0 ≤ x ≤ l < ∞, (2) and the boundary conditions u(0, t) = h(t), u(l, t) = k(t), 0 ≤ t ≤ t < ∞ (3) where f(x,t),a(x),b(x),g(x) and h(t) are functions defined in [0, l] × [0,t]; l and t are appropriate constants. in this study, we develop the laguerre collocation method given in [9,10] and use to obtain the approximate solution of eq. (1) in the truncated laguerre series form u(x,t) = n∑ r=0 n∑ s=0 ar,slr,s(x,t); (4) lr,s(x,t) = lr(x)ls(t) where ar,s, r,s = 0, ...,n, are the unknown laguerre coefficients and ln(x), n = 0,1,2, ...,n are the laguerre polynomials defined by [6-8] ln(x) = n∑ k=0 (−1)k k! ( n k ) xk,n ∈ n, 0 ≤ x < ∞. (5) ii. numerical method we first consider the series (4) for n = 2, as follows: u(x,t) = 2∑ r=0 2∑ s=0 ar,slr(x)ls(t) = a00l0(x)l0(t) + a10l1(x)l0(t) +a20l2(x)l0(t) + a01l0(x)l1(t) (6) +a11l1(x)l1(t) + a21l2(x)l1(t) +a02l0(x)l2(t) + a12l1(x)l2(t) +a22l2(x)l2(t) then we can generalize the approximate solution (6) for any truncated limit n and can write the obtained series in the matrix form [u(x,t)] = l(x)l(t)a (7) biomath 6 (2017), 1706047, http://dx.doi.org/10.11145/j.biomath.2017.06.047 page 2 of 5 http://dx.doi.org/10.11145/j.biomath.2017.06.047 burcu gürbüz, mehmet sezer, numerical solutions of one-dimensional parabolic convection-diffusion ... where l(x) = [ l0(x) l1(x) · · · ln(x) ] , l(t) =   l(t) 0 · · · 0 0 l(t) · · · 0 ... ... . . . ... 0 0 · · · l(t)   and a = [a0,0 a0,1 · · · a0,n · · · an,0 an,1 · · · an,n] t also, we can put the matrix l(x) in the matrix form l(x) = x(x)h (8) where x(x) and h are defined as x(x) = [ 1 x1 · · · xn ] and h =   (−1)0 0! ( 0 0 ) 0 · · · 0 (−1)0 0! ( 1 0 ) (−1)1 1! ( 1 1 ) · · · 0 . . . . . . . . . . . . (−1)0 0! ( n 0 ) (−1)1 1! ( n 1 ) · · · (−1) n n! ( n n )   moreover, it is clearly seen that the relations between the matrix x(x) and its derivatives x′(x) and x′′(x) are x′(x) = x(x)b and x′′(x) = x(x)b2 (9) where b =   0 1 0 · · · 0 0 0 2 · · · 0 ... ... ... . . . ... 0 0 0 · · · n 0 0 0 · · · 0   . then, by using the expressions (8) and (9) we easily find the matrix relations l′(x) = x(x)bh and l′′(x) = x(x)b2h (10) l(t) = x(t)h and l ′ (t) = x(t)bh (11) now, by means of the relations (7)-(11) we obtain the following matrix forms: [u(x,t)] = l(x)l(t)a = x(x)hx(t)ha (12) [ux(x,t)] = l ′(x)l(t)a = x(x)bhx(t)ha (13) [uxx(x,t)] = l ′′(x)l(t)a = x(x)b2hx(t)ha (14) [ut(x,t)] = l(x)l(t)a = x(x)hx(t)bha (15) by putting the expressions (8), (12), (13), (14) and (15) into eq. (1), we obtain the matrix equation {x(x)hx(t)b−x(x)b2hx(t) −a(x)x(x)bhx(t) (16) −b(x)x(x)hx(t)}ha = f(x,t) or briefly, w(x,t)a = f(x,t) besides, by substituting the collocation points defined by xi = l n i, tj = t n j, i,j = 0,1,2, ...,n, into the eq.(16), we have the system of the matrix equations w(xi, tj)a = f(xi, tj) or briefly the fundamental matrix equation wa = f =⇒ [w; f] by using the same procedure for the initial and boundary conditions we obtain the matrix relations for i,j = 0,1, ...,n: u(xi,0) = x(xi)hx(0)ha = g(xi) = λi u(0, tj) = x(0)hx(tj)ha = h(tj) = µj u(y,tj) = x(y)hx(tj)ha = k(tj) = γj or briefly, ua = [λ]; [u;λ],va = [µ]; [v;µ],za = [γ]; [z;γ]. to obtain the approximate solution of eq. (1) under conditions (2) and (3), we form the augmented matrix biomath 6 (2017), 1706047, http://dx.doi.org/10.11145/j.biomath.2017.06.047 page 3 of 5 http://dx.doi.org/10.11145/j.biomath.2017.06.047 burcu gürbüz, mehmet sezer, numerical solutions of one-dimensional parabolic convection-diffusion ... table i comparison of the absolute errors with tcm and lcm for n = 25, 50 in example 2. tcm lcm tcm lcm x e25 e25 e50 e50 0.0 0.69500e-17 7.000000e-13 0.10000e-18 0.0000000000 0.1 0.15886e-03 3.681538e-06 0.15886e-03 1.040134e-07 0.2 0.63428e-03 2.207214e-05 0.63428e-03 4.157153e-08 0.3 0.14208e-02 4.966673e-05 0.14208e-02 9.332459e-08 0.4 0.25078e-02 5.150761e-05 0.25078e-02 1.652982e-05 0.5 0.38799e-02 2.543790e-11 0.38799e-02 2.569605e-15 0.6 0.55168e-02 1.324508e-04 0.55168e-02 3.676180e-06 0.7 0.73934e-02 3.734395e-04 0.73934e-02 4.964259e-06 0.8 0.94802e-02 7.505606e-04 0.94802e-02 6.423988e-06 0.9 0.11744e-01 1.291408e-03 0.11744e-01 8.044241e-07 1.0 0.14146e-01 2.023575e-03 0.14146e-01 9.812752e-07 [w̃; f̃] =   w; f u;λ v;µ z;γ   hence, the unknown laguerre coefficients are computed by a = ( ˜̃ w) −1 ˜̃ f where [ ˜̃w; ˜̃f] is obtained by using the gauss elimination method and then removing zero rows of augmented matrix [w̃; f̃] [9-11]. by substituting the determined coefficients into eq. (4), we have the laguerre series solution un(x,t) = n∑ r=0 n∑ s=0 ar,slr,s(x,t), lr,s(x,t) = lr(x)ls(t). iii. numerical results test case[11] ∂u ∂t = ∂2u ∂x2 + (2x + 1) ∂u ∂x + x2u + ex+t � , 0 ≤ x ≤ 1,0 ≤ t ≤ 1, (17) with conditions u(x,0) = ex � , 0 ≤ x ≤ 1 u(0, t) = et � , u(y,t) = e1+t � 0 ≤ t ≤ 1, with � = 2.10−4 and the exact solution of the problem is u(x,t) = e x+t � . from table 1, it is seen that the errors from laguerre collocation method (lcm) are in general less than taylor collocation method (tcm). table i. shows the comparison between absolute errors of lcm solutions and tcm solutions for different n values. iv. conclusion we have presented and illustrated the laguerre collocation method is based on computing the coefficients in the laguerre expansion of solution of a one dimensional parabolic convection-diffusion model problems. a considerable advantage of the method is that the laguerre polynomial coefficients of the solution are found very easily by using computer programs; maple and matlab. illustrative example is included to show the validity and applicability of the technique. shorter computation time and lower operation count results in reduction of cumulative truncation errors and improvement of overall accuracy. as a result, the method can also be extended to the system of reaction-diffusion-advection model problems with their residual error analysis, but some modifications are required. acknowledgment this work was financially supported by society for mathematical biology for during the biomath 2016, the annual international conference biomath 6 (2017), 1706047, http://dx.doi.org/10.11145/j.biomath.2017.06.047 page 4 of 5 http://dx.doi.org/10.11145/j.biomath.2017.06.047 burcu gürbüz, mehmet sezer, numerical solutions of one-dimensional parabolic convection-diffusion ... on mathematical methods and models in biosciences and it is performed within the ”numerical solutions of partial functional integro differential equations with respect to laguerre polynomials and its applications” project, manisa celal bayar university department of scientific research projects, with grant ref. 2014-151. references [1] j. d. murray, mathematical biology 1: an introduction, berlin: springer, 2002. 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http://dx.doi.org/10.11145/j.biomath.2017.06.047 introduction numerical method numerical results conclusion references www.biomathforum.org/biomath/index.php/biomath original article a note on proportionate mixing assumption revisited for a model with vertical transmission yannick kouakep tchaptchie∗, duplex elvis houpa danga†, nguelbe alex‡ and guidzavai k. albert§ ∗dpt of mathematics and computer science, faculty of science university of ngaoundere (p.o. box 454, ngaoundere) and aims-cameroon (608, limbe), cameroon kouakep@aims-senegal.org †dpt of mathematics and computer sciencefaculty of science university of ngaoundere (p.o. box 454, ngaoundere), cameroon e houpa@yahoo.com ‡ aims-cameroon (p.o. box 608, limbe), cameroon alex.nguelbe@aims-cameroon.org §dpt of mathematics and computer science university of ngaoundere (p.o. box 454, ngaoundere), cameroon kouguidzavai@gmail.com received: 6 september 2014, accepted: 8 april 2015 , published: 28 april 2015 abstract—we conditionally extend formulas of (dietz and schenzle, j. math. biol. 22: 117-120, 1995) for the ”transmission potential” of an immunizing infection with pre-infection possible before birth and vertical transmission admitted. we look for minimum proportion to be covered to reduce basic reproduction rate below 1 by acting through vaccination. we present also a new criterion allowing the selection of an immunizing vaccination strategy by bringing the reproduction number below 1. we find that reduce vertical transmission, adds chances to eradicate disease. moreover reduce age of vaccination reduces the minimum vaccination coverage inducing global immunization against disease by bringing down the basic reproduction number. keywords-transmission potential, minimum proportion for vaccination immunization, endemic disease, pre-infection. ams classification: 35k55, 92d30, 49j20, 92d25. i. introduction: motivation and formulation of the model a. motivation we study an immunizing infection in a closed population where, as [5, (dietz and schenzle 1985)], susceptible newborns are added according to the constant positive rate λ, infective newborns at constant positive rate λ′ of total infective at the same index c ≥ 0 (seen e.g. as the infection time of infected population or any other biological structure with dc/dt=1), and susceptible individuals die according to the rate µ(a), where a denotes citation: yannick kouakep tchaptchie, duplex elvis houpa danga, nguelbe alex, guidzavai k. albert, a note on proportionate mixing assumption revisited for a model with vertical transmission, biomath 1 (2015), 1504081, http://dx.doi.org/10.11145/j.biomath.2015.04.081 page 1 of 7 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2015.04.081 y. k. tchaptchie, a note on proportionate mixing assumption revisited ... the chronological age of an individual. we use proportionate mixing that is more accurate for an age strutured model describing a closed population (of size n(t,a) at time t and chronological age a) that is not big. vertical transmission is introduced here (contrary to [5, (dietz and schenzle 1985)] and according to [1, (busenberg and cooke 1993)] due to its importance. as main results, we find that reduce vertical transmission, adds chances to eradicate disease. moreover reduce age v of vaccination reduces the minimum vaccination coverage inducing global immunization against disease by bringing down the basic reproduction number. b. formulation of the models and equilibria 1) the model and its aggregated form with perfect vaccine and vertical transmission: in equilibrium, we assume that the size n of the population is: n = λ ∫ ∞ 0 e−m(a)da = λl where m(a) = ∫a 0 µ(s)ds, n(t,a) t→+∞→ n∞(a) = λe−m(a) and l denotes the life expectancy of newborn. we analyse the basic reproduction rate [4, (dietz 1975)] or infectious contact number [7, (hethcote 1976)] r0 without vaccination (resp. r0(ψ) with vaccination rate ψ) estimated from equilibrium force of infection λ0 (resp. λψ). as [5, (dietz and schenzle 1985)] in most cases, we defined (or assume): • k(ψ,λψ) as the smallest contact rate above wich a positive endemic level is possible for the vaccination rate or strategy ψ [5]; • γ(a) as the age-specific per capita contact or activity rate; it takes also into account the age specific (average) probability of becoming infected through a contact with infectious individual; • lim a→+∞ [ ap∞(a)e −m(a) ] = 0 because we assume also that the function a j→ ap∞(a)e −m(a) belongs to l1 (0, +∞); • x(t,a) as the density of susceptibles at time t and age a; • y(t,a,c) as the density of infectives at time t , chronological age a and level c; • d1(a,c) is the additional death rate due to disease to be added to the rate of healing or immunization (we later simplify it into the form d1(a,c) ≡ d1(a)); • we see then that dynamic of the compartment of retired individuals is decoupled from the model studied for our immunizing infection; • c could be greater than a (notion of ”preinfection” included: infection before birth possible); • a consequence is this modified version of the force of infection (compare to [5, (dietz and schenzle 1985, p. 118)]): λ(t) = f n ∫ ∞ 0 ∫ ∞ 0 p(t,a′)y(t,a′,c)dcda′ • f as the probability of infectiousness (depending on c in [5, (dietz and schenzle 1985)] but constant here as an average since several health public policy ignore probability variations at first approximation); • probability that an individual of age a has contact with an individual of age a′given that it has a contact with a member of the population p(t,a,a′) ≡ p(t,a′) = γ(a′)n(t,a′)∫∞ 0 γ(u)n(t,u)du with p(t,a′) t→+∞→ p∞(a′) = γ(a′)n∞(a′)∫∞ 0 γ(u)n∞(u)du • transmission potential r0(ψ) = k(ψ,λψ) k(ψ, 0) we formulate, with the notation ∂z := ∂∂z , a model with vertical transmission by combining approach of [3, (castillo-chavez and feng 1998)] biomath 1 (2015), 1504081, http://dx.doi.org/10.11145/j.biomath.2015.04.081 page 2 of 7 http://dx.doi.org/10.11145/j.biomath.2015.04.081 y. k. tchaptchie, a note on proportionate mixing assumption revisited ... and [5, (dietz and schenzle 1985)]:   (∂t + ∂a) x = − (λ(t) + ψ(a) + µ(a)) x (∂t + ∂a + ∂c)y(t,a,c) = −(d1(a,c) + µ(a)) y(t,a) x(t, 0) =λ ≥ 0 x(0,a) =x0(a) ≥ 0 y(0,a,c) =y0(a,c) ≥ 0 y(t,a, 0) =λ(t)x(t,a) y(t, 0,c) =λ′ ∫ ∞ 0 p(t,a′)y(t,a′,c)da′ (1) remark 1: in fact there are certain probability for the infected population y(t,a′,c) giving a birth to a health newborn, this indicates that there are some input of newborns in to the formulation of x(t, 0) from y(t,a′,c). however, this could be neglected since we assume that infected population is very small compared to healthy one: it justfies also our constant influx of health newborns λ. we will focus later on λ′ since we want to sketch in priority the impact of vertical transmission on basic reproduction rate. for sake of simplicity, we select the special case: d1(a,c) ≡ d1(a) and use the new variable y(t,a) = ∫ ∞ 0 y(t,a,c)dc then the wellposed system (1) rewrites as   (∂t + ∂a) x = −(λ(t) + ψ(a) + µ(a))x(t,a) (∂t + ∂a) y(t,a) = − ( d1(a) + µ(a) ) y(t,a) + λ(t)x(t,a) x(t, 0) =λ ≥ 0 x(0,a) =x0(a) ≥ 0 y(0,a) =y0(a) ≥ 0 y(t, 0) =λ′ ∫ ∞ 0 p(t,a′)y(t,a′)da′ λ(t) = f n ∫ ∞ 0 p(t,a′)y(t,a′)da′ (2) 2) cauchy problem and integrated solutions in brief: the system (2) can be re-written under the form of a cauchy problem: { dw(t) dt = aw(t) + f(t,w(t)) := g(t,w(t)) w(0) = w0 ∈ d(a) (3) with w(t) ≡   0 0 x(t, .) y(t, .)   and d(a) = {0}×{0}× ( w 11(0; +∞) )2 consider v ≡   α β x̂ ŷ   and the banach space x = r×r× ( l1(0; +∞) )2 endowed with the usual norm ‖v‖x = |α| + |β| + ∫ ∞ 0 [|x̂(a)| + |ŷ(a)|] da positive cone of x is x+ = [0; +∞) × [0; +∞) × ( l1+(0; +∞) )2 we define also x0 = {0}×{0}× ( l1(0; +∞) )2 and its positive cone x0+ = {0}×{0}× ( l1+(0; +∞) )2 . we set u ≡   0 0 x̂ ŷ   and the linear and closed operator defined on d(a) by: a : d(a) → x u 7−→   x̂(0) ŷ(0) −dx̂ da − (ψ(.) + µ(.)) x̂ −dŷ da − ( d1(.) + µ(.) ) ŷ   biomath 1 (2015), 1504081, http://dx.doi.org/10.11145/j.biomath.2015.04.081 page 3 of 7 http://dx.doi.org/10.11145/j.biomath.2015.04.081 y. k. tchaptchie, a note on proportionate mixing assumption revisited ... it always exists µ̄ ∈ [0; +∞) such that: ∀a ≥ 0, µ(a) ≥ µ̄. then we have for each λ > −µ̄ (λ−a)−1x+ ⊂ x0+ (4) and (−µ̄,∞) ⊂ ρ(a) with ‖(λ−a)−1‖l(x) ≤ 1 λ + µ̄ , ∀λ > −µ̄. (5) the part a0 of a defined by a0 : d(a0) → x u 7−→   0 0 −dx̂ da − (ψ(.) + µ(.)) x̂ −dŷ da − ( d1(.) + µ(.) ) ŷ   with d(a0) defined by u ≡   0 0 x̂ ŷ   ∈ d(a) : au ∈ d(a), x̂(0) = ŷ(0) = 0   a0 verifies the hille-yosida property: it exists µ̄ ∈ [0; +∞) such that ∀a ≥ 0, µ(a) ≥ µ̄ and we have for each λ > −µ̄ ‖(λ−a0)−1‖l(x0) ≤ 1 λ + µ̄ , ∀λ > −µ̄. (6) and (by lemma 2.1 of [6, ducrot et al. 2010]): x1 := d(a0). assumption 2.2 of [6, ducrot et al. 2010, p. 267] is satisfied. then its lemma 2.3[6, ducrot et al. 2010, p. 267] applies: a0 is the infinitesimal generator of a c0-semigroup (ta0 (t))t≥0 on x1. we define (with λ(t) = f n ∫∞ 0 p(t,a′)ŷ(a′)da′) also the frechet differentiable in the second variable u ( and then ”locally” lipschitz in u) perturbation (for each t ≥ 0): f(t, .) : x0 → x u 7−→ g(t,u(.)) −au with f(t,u(.)) =   −λ −λ′ ∫∞ 0 p(t,a′)ŷ(a′)da′ −λ(t)x̂ λ(t)x̂   the model (3) is well posed with an integrated solution w globally defined in time through a bounded dissipativity property[2], [6], [8], [10], [12], [13]. w satisfies (in bochner sense for integrals): ∫ t 0 w(s)ds ∈ d(a) and w(t) = w0 + a ∫ t 0 w(s)ds + ∫ t 0 f(s,w(s))ds (t ≥ 0) 3) stationary solution of (2): a stationary solution (xψ; yψ) of (2) (with the force of infection at equilibrium λψ) satisfies:   xψ(a) = λe −(λψ(a)+φ(a)+m(a)) yψ(a) = λ ′ ∫ ∞ 0 p∞(a ′)yψ(a ′)da′e−(d1(a)+m(a)) + ∫ a 0 λψxψ(s)e (φ(s)−φ(a)+m(s)−m(a))ds λψ : = f n ∫ ∞ 0 p∞(a ′)yψ(a ′)da′ (7) with: λψ(a) = ∫ a 0 λψda ′ = λψ.a φ(a) = ∫ a 0 ψ(a′)da′ and d1(a) = ∫ a 0 d1(a ′)da′ then λψ is a fixed point of the function g(z) = ( λ′ ∫ ∞ 0 p∞(a ′)e−(d1+m)(a ′)da′ + fλ n ∫ ∞ 0 p∞(a) ∫ a 0 e−(φ(a)+m(a)+s.z)dsda ) .z biomath 1 (2015), 1504081, http://dx.doi.org/10.11145/j.biomath.2015.04.081 page 4 of 7 http://dx.doi.org/10.11145/j.biomath.2015.04.081 y. k. tchaptchie, a note on proportionate mixing assumption revisited ... we defined then the non-increasing function h(z) = λ′ ∫ ∞ 0 p∞(a ′)e−(d1(a ′)+m(a′))da′ + λf n ∫ ∞ 0 p∞(a) ∫ a 0 e−(φ(a)+m(a)+s.z)dsda and the treshold kψ0 := h(0) solution(s) of the equation g(z) = z are z = λ−ψ = 0 (disease free equilibrium) and (if k ψ 0 ≥ 1) z = λ+ψ where λ + ψ is the only non zero solution of the equation: h(z) = 1 corresponding to the endemic equilibrium. ii. transmission potentials as [5, (dietz and schenzle 1985)], we defined k(ψ,λψ) = (h(λψ)) −1 it is obvious that k(ψ,λψ) = k(0,λ0) then the potential transmission is r0(ψ) = k(ψ,λψ) k(ψ, 0) = k(0,λ0) k(ψ, 0) (8) then if we set these two non-increasing functions: a(φ) :=λ′ ∫ ∞ 0 p∞(a ′)e−(d1(a ′)+m(a′))da′ + f λ n ∫ ∞ 0 ap∞(a)e −(φ(a)+m(a))da and b(λ0) : = λ ′ ∫ ∞ 0 p∞(a ′)e−(d1(a ′)+m(a′))da′ +f λ n ∫ ∞ 0 p∞(a) ∫ a 0 e−(m(a)+s.λ0)dsda r0(ψ) = a(φ) b(λ0) (9) rλ ′=0 0 (ψ) is the basic reproduction rate r0 without vertical transmission and rλ ′ 6=0 0 (ψ) is the basic reproduction rate r0 with vertical transmission. we set u1 := ∫ ∞ 0 p∞(a) ∫ a 0 e−(m(a)+s.λ0)dsda and u2 := ∫ ∞ 0 ap∞(a)e −(φ(a)+m(a))da two cases appear: c1) if u1 ≥ u2 then rλ ′=0 0 (ψ) ≤ r λ′ 6=0 0 (ψ) c2) if u1 ≤ u2 then rλ ′=0 0 (ψ) ≥ r λ′ 6=0 0 (ψ). remark 2: c2) is satisfied if e−m(a) ∫ a 0 e−s.λ0ds ≤ ae−(φ(a)+m(a)) that means 1 −e−λ0a λ0 ≤ ae−φ(a) the approximation (for λ0 very small compared to maximal reacheable human age or life expectancy/lifespan) provides the approximation a . ae−φ(a) or 1 . e−φ(a) in that case, the second case c2) is probably less recurrent than obvious case c1) remark 3: another remark for inequality 1 −e−λ0a λ0 ≤ ae−φ(a) coming from case c2), is the fact that it corresponds to a ”massive and agressive” campaign (φ huge) of vaccination that reverse the effect of vertical transmission (λ0 very small). c2) naturally traduces the supplementary infectious cases brought by vertical transmission reduced by vaccination, but not enough to be similar to the case λ′ = 0. because of the term e−φ(a) at the numerator of r0(ψ), we see that r0(ψ) ≤ r0(0) := r0: vaccination reduces the basic reproduction rates (see also [9, (kouakep and houpa 2014)] for the case without vertical transmission). biomath 1 (2015), 1504081, http://dx.doi.org/10.11145/j.biomath.2015.04.081 page 5 of 7 http://dx.doi.org/10.11145/j.biomath.2015.04.081 y. k. tchaptchie, a note on proportionate mixing assumption revisited ... iii. a criterion for ”sufficient” vaccination strategies and minimal proportion for immunization of an almost closed population we observe that if r0 6= 0 r0(ψ) r0 = ( 1 − fλ n ∫∞ 0 ap∞(a)e −m(a) [ 1 −e−φ(a) ] da a(0) ) (10) theorem 4: we assume that r0 > 1. to reach a basic reproduction rate r0(ψ) below 1, the following inequality schould be satisfied by the choosen vaccination rate ψ: ( f λ n ∫∞ 0 ap∞(a)e −m(a) [ 1 −e−φ(a) ] da a(0) ) > 1− 1 r0 (11) a sufficient condition to reach r0(ψ) below 1 is the condition (for a.e a > 0):( f n [ 1 −e−φ(a) ] λ′ λ [1 + e −d1(a) a ] ) > 1 − 1 r0 (12) remark 5: inequality (12) induces a kind of control on age a > 0 (for a vertically transmited disease as hepatitis b) that could reduce globally the number of infected childs and infectives. we define the non-increasing function: g(b) :=λ′ ∫ ∞ b p∞(a ′)e−(d1(a ′)+m(a′))da′ + f λ n ∫ ∞ b ap∞(a)e −m(a)da following [5, (dietz and schenzle 1985)], we propose in the next result a formula showing minimum proportion p∗ to be covered if vaccination takes place at age v : theorem 6: a formula showing minimum proportion p∗ to be covered if vaccination takes place at age v is given by: p∗ = ( 1 − 1 r0 ) g(0) g(v ) (13) iv. discussion formula (11) suggests that a pressure is done by vertical transmission on the inequality to satisfy if we want to bring the transmission potential rate below 1, compared to the situation without vertical transmission. with (12) we see also that if size at equilibrium n of population increases, then achieve global immunization through vaccination is more difficult. we observe a similar situation for the minimal proportion for vaccination: reduce λ′ (vertical transmission), adds chances to satisfy criterion (11) for disease eradication. a discussion is necessary around the formula (13). [11, (sall et al. 2004)] said truth when they pointed out the fact that neglect vertical transmission in vaccinal strategies for sub-saharan africa is a mistake: we see by formula (13) that reduce age v of vaccination (in biological ranges) reduces the minimum vaccination coverage p∗ inducing immunization. a further work will consider an imperfect vaccine and differential infectivity as [3, (castillochavez and feng 1998)] and [9, (kouakep and houpa 2014)] in a specific case as hepatitis b. migrations could be also considered. acknowledgements the authors would like to thank the three anonymous reviewers for valuable comments and questions, which greatly improved the quality of paper. the authors thank also pr békollè david, pr a. ducrot, pr mama foupouagnigni and pr dimi jean-luc for their helpful suggestions. this work was carried out with financial support from the government of canada’s international development research centre (idrc), and within the framework of the aims research for africa project no snmcm2013014s.the authors are biomath 1 (2015), 1504081, http://dx.doi.org/10.11145/j.biomath.2015.04.081 page 6 of 7 http://dx.doi.org/10.11145/j.biomath.2015.04.081 y. k. tchaptchie, a note on proportionate mixing assumption revisited ... solely responsible for the views and opinions expressed in this research (part of y.k.t.’s phd thesis work); it does not necessarily reflect the ideas and/or opinions of the funding agencies (aimsnei or idrc) and university of ngaoundere. references [1] busenberg s., cooke k., vertically transmitted diseases: models and dynamics, springer verlag berlin heidelberg (1993). [2] c. castillo-chavez, h.w. hethcote, v. andreason, s.a. levin, w. liu, epidemiological models with age structure, proportionate mixing, and cross-immunity, j.math. biol. 27:233 (1989). http://dx.doi.org/10.1007/bf00275810 [3] castillo-chavez c. , feng z., global stability of an age-structure model for tb and its applications to optimal vaccination strategies, math. biosci., 151:135-154 (1998). http://dx.doi.org/10.1016/s0025-5564(98)10016-0 [4] dietz, k., transmission and control of arbovirus diseases. in: epidemiology, ludwig, d., cooke, k. l., (eds.), philadelphia: society for industrial and applied mathematics, 104-121 (1975). [5] dietz k., schenzle d., proportionate mixing models for age-dependent infection transmission, j. math. biol. 22:117-120 (1985). http://dx.doi.org/10.1007/bf00276550 [6] a. ducrot, p. magal and k. prevost, integrated semigroups and parabolic equations. part i: linear perburbation of almost sectorial operators, j. evol. equ. 10, 263– 291 (2010). http://dx.doi.org/10.1007/s00028-009-0049-z [7] hethcote, h. w. qualitative analyses of communicable disease models. math. biosci. 28:335-356 (1976). http://dx.doi.org/10.1016/0025-5564(76)90132-2 [8] k. ezzinbi, lectures notes in functional analysis and evolution equations, graduate course delivered at aust, abudja, nigeria (2010). [9] kouakep t. y., houpa d. d. e. optimal test strategies for hepatitis b vaccination with no vertical transmission, gen. math. notes, vol. 20 no 1:19-26, (2014). [10] a. pazy, semigroups of linear operators and applications to partial differential equations, springer-verlag, berlin, (1983). [11] a. sall diallo, m. sarr, fall y., c. diagne, m. o. kane, hepatitis b infection in infantile population of senegal, dakar med. http://www.ncbi.nlm.nih.gov/pubmed/15786625 (2004) [12] n. tanaka, quasilinear evolution equations with nondensely defined operators, differential and integral equations volume 9, number 5, 1067-1106 (1996). [13] n. tanaka, semilinear equations in the ”hyperbolic” case, nonlinear analysis, theory, methods and applications, vol. 24, no. 5, 773-788 (1995). biomath 1 (2015), 1504081, http://dx.doi.org/10.11145/j.biomath.2015.04.081 page 7 of 7 http://dx.doi.org/10.1007/bf00275810 http://dx.doi.org/10.1016/s0025-5564(98)10016-0 http://dx.doi.org/10.1007/bf00276550 http://dx.doi.org/10.1007/s00028-009-0049-z http://dx.doi.org/10.1016/0025-5564(76)90132-2 http://dx.doi.org/10.11145/j.biomath.2015.04.081 introduction: motivation and formulation of the model motivation formulation of the models and equilibria the model and its aggregated form with perfect vaccine and vertical transmission cauchy problem and integrated solutions in brief stationary solution of (2) transmission potentials a criterion for "sufficient" vaccination strategies and minimal proportion for immunization of an almost closed population discussion references original article biomath 1 (2012), 1211117, 1–6 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum predicting and scoring links in anatomical ontology mapping peter petrov∗, milko krachounov∗, ognyan kulev∗, maria nisheva∗, dimitar vassilev† ∗faculty of mathematics and informatics, sofia university st. kliment ohridski 5 james bourchier blvd., 1164 sofia, bulgaria †bioinformatics group, agrobioinstitute 8 dragan tsankov blvd., 1164 sofia, bulgaria email: jim6329@gmail.com received: 12 july 2012, accepted: 11 november 2012, published: 22 december 2012 abstract—the paper presents a work performed in the area of automatic and semi-automatic ontology mapping. a method for inferring additional cross-ontology links while mapping anatomical ontologies is described and the results of some experiments performed with various external knowledge sources and scoring schemes are discussed as well. keywords-ontology; graph; directed acyclic graph; ontology mediation; ontology mapping; ontology merging; scoring scheme; probability; knowledge sharing; knowledge reuse; interoperability i. introduction the term ontology comes from philosophy and has been applied in information systems, information retrieval etc. to represent the formalization of a body of knowledge describing a given domain. ontologies have become increasingly popular because they help to realize many of the most challenging problems in the it field like interoperability, information/knowledge sharing and knowledge reuse. information sources (and ontologies in particular), even from the same problem domain, are usually heterogeneous. in order to enable interoperation between such information sources (ontologies) and to integrate the information/knowledge from multiple sources, one needs to build mappings between ontologies. these mappings establish the semantic correspondence between concepts and relations in different ontologies. as we have noted in [10] there are some terminological differences pertaining to the integration of ontologies within the ontology mapping/merging/matching (om) community. those terminological differences are mostly between the terminology adopted in [1] on one side, and in [11] on the other. in our works, we adopt the terminology of [1]. in the sense of [1], ontology mapping is the process of taking two input ontologies and generating semantic links between their concepts/terms. the generated links are not part of the two input ontologies; they are stored separately from them. two other terms are related to ontology mapping: ontology aligning and ontology merging. ontology aligning [1] can be viewed as an automatic or semi-automatic ontology mapping; it denotes the process of discovery of cross-ontology links by a computer program. again, these links are stored separately from the two input ontologies. ontology merging [1] is the ultimate goal when integrating/mediating two input ontologies; it comes down to taking two input ontologies and generating an output ontology that unifies the knowledge contained in them. it is usually a process which follows the processes of mapping/aligning and which utilizes the intermediate results produced by them; during this process, some pairs of terms (one from each of the two input ontologies) are merged into single nodes of the output ontology, while other input terms are not paired but are just copied unchanged to the output ontology. this paper discusses some issues in automatic mapcitation: p petrov, m krachounov, o kulev, m nisheva, d vassilev, predicting and scoring links in anatomical ontology mapping, biomath 1 (2012), 1211117, http://dx.doi.org/10.11145/j.biomath.2012.11.117 page 1 of 6 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.11.117 p petrov at al., predicting and scoring links in anatomical ontology mapping ping or aligning of species-specific anatomical ontologies by utilization of various knowledge sources. ii. problem formulation given the anatomical ontologies of two different species (model organisms) e.g. mouse and zebrafish, our goal is to establish semantic links between the terms of the two ontologies such that: (i) these links are of one of the following types: r1 = synonymy, r2 = hypernymy, r3 = hyponymy, r4 = holonymy, r5 = meronymy, and (ii) each of these links has some degree of certainty or degree of confidence or confidence score which is a real number in the interval [0, 1]. the semantic relation types rk that we refer to here are well-known and are widely utilized in the areas of linguistics, knowledge representation and ontology engineering. that is why we don’t provide any formal or informal definitions for them here. the two input ontologies are represented in the form of obo files. obo stands for “open biomedical ontology“ and denotes an ontology language and an ontology file format [2] for defining ontologies. it has been used mostly for defining ontologies in the biomedical domain. nowadays obo is adopted by the go project [2], [3], the obo foundry initiative [4], and other communities. iii. formalization of the problem in mathematical terms, each of the two input anatomical ontologies can be considered as a directed acyclic graph together with a function colouring the graph’s edges. the colours model the relations defined within the input ontologies (like is a and part of ) which we call inner-ontology relations. typically, there are other innerontology relations except those two. these additional relations usually pertain to the development of the particular organism and not just to its adult/gross anatomy. such relations are for example start stage, end stage, develops from but practically we don’t deal with them as we are mainly concerned with the organism’s adult/gross anatomy, not with the organism’s growth and development. we shall use further the following notation: o1 = om : dag1 = (v1,e1), f1 : e1 →c = {c1,c2, ...,cn}; o2 = oz : dag2 = (v2,e2), f2 : e2 →c = {c1,c2, ...,cn}. here o1 and o2 are the two input anatomical ontologies; dag1, dag2 are their corresponding directed acyclic graphs; v1 and v2 are the sets of terms of the two input ontologies (each term has an identifier and a name); e1 and e2 are the relations defined within the two input ontologies; f1 and f2 are the edge-colouring functions. two terms u1 and u2 are connected with an edge e if and only if the pair of terms (u1,u2) belongs to the relation represented by e. the relations is a (specialization/generalization) and part of (membership/aggregation) are the two typical examples of inner-ontology relations defined within the ontologies o1 and o2. in our notation, we map relations to colours (through f1 and f2), and we deal only with two relations (is a, part of). so it can be assumed that n = 2, c1 = is a, c2 = part of . thus, if for example, u1 =“brain”, u2 =“central nervous system”, u1,u2∈v1, then there usually exists an edge e between u1 and u2 such that f1(e) = part of (because the brain is part of the central nervous system and anatomical ontologies of most organisms usually declare this fact explicitly). also given are several (typically large) external knowledge sources which might be either biomedical ones or general-purpose ones. they contain anatomical terms and relations (is a, part of , others) between their own terms. three concrete external knowledge sources have been used for the purposes of this work. these are t1 = umls, t2 = fma, t3 = wordnet. umls [5], [14] and fma [6], [15] are biomedical knowledge sources, and wordnet [7], [8], [16] is a general purpose knowledge source. formally stated, each of these knowledge sources ts, s = 1, 2, 3, contains the following information: • terms. ms = {ts1,ts2, ...,tsms} is the set of terms in the knowledge source ts. here tsk = (idsk;namesk); idsk is the identifier within ts of the term tsk; namesk is the textual name within ts of the term tsk; ms (usually 106 ≤ ms ≤ 107) is the number of terms in the knowledge source ts. • relations. these are the is a and part of relations defined within the external knowledge source ts: r ′ ts = ris ats ⊆ms ×ms, r ′′ ts = r part of ts ⊆ms ×ms. typically other relations are also defined within the external knowledge source ts but only these two are relevant to our work. each knowledge source src = ts, s = 1, 2, 3, is up-front assigned a score f(src) which is based on its preciseness in predicting synonymy and parent-child (is a, part of ) relations between terms of the two input ontologies. details on this evaluation (of the three biomath 1 (2012), 1211117, http://dx.doi.org/10.11145/j.biomath.2012.11.117 page 2 of 6 http://dx.doi.org/10.11145/j.biomath.2012.11.117 p petrov at al., predicting and scoring links in anatomical ontology mapping knowledge sources that we use) can be found in [9]. having the notation introduced above, we now seek to find a set of predictions (a set of 4-tuples): d = {(v1k,v2k,rk,sk) |k = 1,2, ..., |d|}, such that v1k∈v1, v2k∈v2, rk∈{r1,r2,r3,r4,r5} and sk∈(0,1]. here, for each k, v1k is a term from the input ontology o1, v2k is a term from the input ontology o2, rk are automatically (i.e. in silico) predicted crossontology links from one of the five types defined in the previous section, and sk is a real number denoting the confidence score of the prediction that the terms v1k and v2k are related/linked by a cross-ontology link of the type rk. requiring that sk ∈ (0,1], we basically imply that the set d which we seek, is in fact a set of cross-ontology predictions or a set of predicted crossontology links between o1 and o2 where each score is probabilistic-based (modeling, given the information we have in the input ontologies and also in the available knowledge sources, the probability that the corresponding prediction is actually true). iv. algorithmic procedures three algorithmic procedures are applied to the graph structures that were described formally in the previous section. each of them adds more links to the set d that is being sought. these three procedures are detailed in [12], here we mention them only briefly. within the first procedure, the two input ontologies are scanned for identity matches between the names of their terms. if t1 ∈ v1 and t2 ∈ v2 have the same names, they are marked as synonyms predicted by what we call the direct matching (dm) procedure. the cross-ontology links discovered/predicted this way are assigned the highest possible scores of 1.0 as these predictions come from information contained entirely in the two input ontologies. during the second procedure, using the information (the terms and the relations) in the external knowledge sources, and identity matches between term names of the two input ontologies and term names of the three external knowledge sources, we build a graph model/structure which aligns each of the two input ontologies to each of the three external knowledge sources. this model contains a set of semantic links (of the types rk, k = 1, 2, ..., 5, that were defined above) between the two input ontologies on the one side, and the three external knowledge sources on the other side. then a set of logical rules is applied, and conclusions are drawn for the semantic relations that exist between terms t1 ∈v1 and t2 ∈v2 of the two input ontologies. the following rules are applied at this stage: • rule (a). if two terms t1 ∈v1 and t2 ∈v2 have been detected as synonyms of the same term t∈ts, then t1 and t2 are marked as predicted cross-ontology synonyms of each other; • rule (b). if tj∈vj has been detected as a synonym of t∈ts (s=1, 2, 3), and if the term t3−j∈v3−j has been detected as an (is a/part of) child/parent of t, then tj is marked as predicted cross-ontology (is a/part of) parent/child of t3−j (here j =1 or j =2). the application of these rules is what we call the source matching predictions (smp) procedure. rule (a), when applied, finds the synonymy relations (i.e. the relations of type r1) between terms from the two input ontologies. rule (b) is a composite (generalized) version of four separate rules (two options for is a/part of by two options for child/parent makes four options in total). these four rules which originate from rule (b), when applied, find the hypernymy, hyponymy, holonymy and meronymy relations (i.e. the relations of types r2, r3, r4, r5) between terms of the two input ontologies. all links predicted through smp are given the score f(src), where src is the knowledge source confirming/implying these predictions. finally, we run a procedure that we denote as the child matching predictions (cmp) procedure. this one tries to find r1, r2, r3, r4 and r5 links between terms of the two input ontologies, t1∈v1 and t2∈v2, for which no links have been predicted either by dm or by smp. the approach cmp takes is to consider patterns of crossontology connectivity (found by dm and smp) between t1 ∈v1 (parent term 1), t2 ∈v2 (parent term 2), and the child terms of the two parent terms t1 and t2. three separate patterns of connectivity are considered by cmp: (i) t1∈v1←−tch1∈v1←→tch2∈v2−→t2∈v2 (we call this an u pattern); (ii) t1∈v1←−tch2∈v2←→tch1∈v1−→t2∈v2 (we call this an x pattern); (iii) t1 ∈v1 ←− tch1 ∈v1 −→ t2 ∈v2 or t1 ∈v1 ←− tch2 ∈v2 −→ t2 ∈v2 (we call these two patterns v patterns). in this notation, the −→ and ←− arrows denote sets of non-cmp parent-child links (the arrows always point from child to parent). these are asymmetrical links. the ←→ arrows denote sets of non-cmp synonymy links these are symmetrical links. the tch1 and tch2 are child terms from the two input ontologies. each occurrence of biomath 1 (2012), 1211117, http://dx.doi.org/10.11145/j.biomath.2012.11.117 page 3 of 6 http://dx.doi.org/10.11145/j.biomath.2012.11.117 p petrov at al., predicting and scoring links in anatomical ontology mapping any of these patterns between t1 and t2 (the two parent terms) we call a pattern instance. all arrows within one pattern instance represent either is a or part of links (we don’t allow mixing these two within a single pattern instance). based on these patterns of connectivity, new crossontology links (cmp links) are introduced (one cmp link per pattern instance) between t1 and t2. we call these links individual cmp links. to assign scores to the individual cmp links, the concepts score of a set of noncmp links between two terms and score of a pattern instance (or score of an individual cmp link) are defined below. also, we introduce two functions, conj and disj, with n ≥ 2 parameters each, which, provided that the probabilities p1,p2, ...,pn of n events are given, define the probabilities of (i) all these events occurring at the same time (conj), and (ii) at least one of these events occurring (disj). we call the conj and disj functions accumulation functions as they accumulate scores of non-cmp links to produce a score of an individual cmp link. finally, all individual cmp links between t1 and t2 are aggregated through what we call an aggregation function (which can be e.g. the max of n ≥ 1 numbers). next, we define in some more detail the concepts which we just introduced in relation to cmp. definition 1 (conj): conj is a function which takes n arguments (each of them in [0, 1]) and returns a result in [0, 1]. we discuss a possible implementation for it below. definition 2 (disj): disj is a function which takes n arguments (each of them in [0, 1]) and returns a result in [0, 1]. we discuss possible implementations for it below. definition 3 (score of a non-cmp link): the score of a non-cmp link between any two terms (which could be from the same ontology or not) is defined as follows: score(sij)=   i if sij is an io link, d if sij is a dm link, f(src) if sij is an smp link which came from the source src ∈ {umls, fma, wordnet}. here io stands for inner-ontology, dm stands for direct matching and smp stands for source matching predictions; sij is one single non-cmp link (i.e. one single evidence); the i and d are constants (typically having the values of 1.0). definition 4 (score of a set of non-cmp links): the score of a set of non-cmp links (score of an evidence set) is defined as follows: score(si) = disjmk=1(score(sik)), where disj is the function from definition 2, sik are non-cmp (i.e. either io or dm or smp) links, and the disj is taken over all non-cmp links taking part in the evidence set si. definition 5 (score of an individual cmp link): the score of an individual cmp link e is defined as: score(e) = p · conjni=1(score(si)), where p ∈ [0, 1] is a cmp penalty constant, conj is the function from definition 1, and the conj is taken over all evidence sets si that take part in the pattern instance, which the link e originates from (note that n = 2 for the v patterns and n = 3 for the x and u patterns). definition 6 (aggregation function): let k be the number of all individual cmp links drawn between t1 ∈ v1 and t2 ∈ v2. an aggregation function is a known function fagg which takes the scores of all these k individual cmp links and produces a single number pcmp (t1,t2) ∈ [0,1], which we call score of the aggregated (final) cmp link drawn between t1 and t2. as a final result from the cmp procedure, this aggregated cmp link is drawn between any two terms t1 and t2 for which at least one pattern (of any of the three types x, u, v) is found. the score of this link is calculated in the way shown above. v. comparison of alternative scoring schemes we have produced several distinct scoring schemes by varying the functions conj, disj and fagg which were defined above. • scheme #1: (1a) conj(s1,s2) = s1s2; conj(s1,s2,...,sn)=conj(conj(s1,s2,...,sn−1),sn) (1b) disj(s1,s2) = s1 + s2 −s1s2; disj(s1,s2,...,sn)=disj(disj(s1,s2,...,sn−1),sn) (1c)fagg(s1,s2, ...,sn) = max(s1,s2, ...,sn) • scheme #2: (2a) conj(s1,s2) = s1s2; conj(s1,s2,...,sn)=conj(conj(s1,s2,...,sn−1),sn) (2b) disj(s1,s2) = s1 + s2 −s1s2; disj(s1,s2,...,sn)=disj(disj(s1,s2,...,sn−1),sn) (2c)fagg(s1,s2, ...,sn) = disj(s1,s2, ...,sn) biomath 1 (2012), 1211117, http://dx.doi.org/10.11145/j.biomath.2012.11.117 page 4 of 6 http://dx.doi.org/10.11145/j.biomath.2012.11.117 p petrov at al., predicting and scoring links in anatomical ontology mapping • scheme #3: (3a) conj(s1,s2) = s1s2; conj(s1,s2,...,sn)=conj(conj(s1,s2,...,sn−1),sn) (3b)disj(s1,s2)=α(s1+s2−s1s2)+(1−α)max(s1,s2); disj(s1,s2,...,sn)=disj(disj(s1,s2,...,sn−1),sn) (3c)fagg(s1,s2, ...,sn) = disj(s1,s2, ...,sn) the disj functions from schemes #1 and #2 are identical to the formula for calculating the probability of the union of two (and respectively n) independent events. fagg from scoring scheme #1 corresponds to the probability of the union of two events such that one is completely dependent on the other. fagg from scoring scheme #2 coincides with disj from the same scoring scheme, which equals the probability of the union of two independent events. therefore in a probabilistic model the expression s1 +s2−s1s2 is a good choice for combining two independent scores, while max(s1,s2) is a good choice for combining scores when one score is completely dependent on the other. in scoring scheme #3 we design a scoring function whose values are between the values of the first two scoring functions (#3 is a linear combination of #1 and #2). the main objective behind the use of this third scoring function is to account for the dependencies between the knowledge sources (umls, fma, wordnet) without completely ignoring the fact that, if more than one of them confirm certain prediction, that usually improves the odds that this prediction is correct. in scheme #3, α ∈ [0, 1] is a parameter of the linear combination defined in (3b). it varies depending on the knowledge source or the combination of knowledge sources, which confirm the predictions whose scores we accumulate in (3b). the α parameter acts as a buffer to prevent the score from growing too fast when adding up cumulative predictions (i.e. when the predictions being accumulated are confirmed by several knowledge sources): when α equals 0.0, the value is growing the quickest (as it should for independent scores); when α equals 1.0, the value is limited by the maximum score of the scores being accumulated. to experimentally show that the choice of disj from (3b) is a reasonable one, we have generated a set of observations on two dependent random variables x1, x2 with boolean (1/0 i.e. true/false) truth values, and we have confirmed that if we substitute the scores s1 and s2 in (3b) with the probabilities p(xi = true) (i = 1, 2), and α with the modulus of the correlation coefficient between the two random variables, we get a very good approximation for the probability p(z = true) of their boolean disjunction z = (x1 or x2). vi. results and discussion let us consider the following two figures which illustrate how the scores generated by the three scoring schemes are related to each other and demonstrate the advantages of scheme #3. figure 1. scatter plot: scheme #1 vs schemes #2 and #3 figure 2. scatter plot: scheme #3 vs schemes #1 and #2 biomath 1 (2012), 1211117, http://dx.doi.org/10.11145/j.biomath.2012.11.117 page 5 of 6 http://dx.doi.org/10.11145/j.biomath.2012.11.117 p petrov at al., predicting and scoring links in anatomical ontology mapping it can be seen on fig.1 that the data in scheme #1 appear clustered around the configured values for knowledge source scores (and combinations of these), because there isn’t anything to account for the amount of available evidence gathered from each source (e.g. the number of patterns confirming a prediction). compared to scheme #1, both schemes #2 and #3 scatter the clusters because the fagg values are growing when more patterns are confirming a given prediction. as fagg in scheme #3 is limited through the α parameter, it causes a more moderate scattering as seen on fig. 1, while scheme #2 causes a very rapid increase. the main advantage of scheme #3 is that it allows us to control the speed at which additional patterns increase the score, while scheme #2 gives control only over the initial value of that score. within scheme #2, when having one pattern confirming the prediction, the scores start somewhere around the configured cmp score value (defined by the penalty constant), and grow with the same speed up to 1.0. within scheme #3 this growth can be slowed down and controlled through the α parameter. the difference between the schemes #2 and #3 can be seen on fig. 2 in red, and it clearly shows how easily some scores approach the value 1.0 when scheme #2 is used. the disj function from scheme #3 also causes a softening effect on the score when there are multiple knowledge sources and algorithmic procedures (dm, smp, cmp) confirming the prediction, because it allows us to control the speed at which the score grows and even to use the actual correlation coefficient between the distinct knowledge sources. this is not directly visible on the figures at this scale, because it largely produces local shifts in the position of the clusters and has the biggest effect on data predicted by the knowledge sources (smp) which constitute the cluster around score=1.0. vii. conclusion we presented in this paper an original algorithmic approach to inferring (predicting and scoring) crossontology links within automatic mapping of distinct species-specific anatomical ontologies. the full mapping procedure assumes that the auto-generated set of predictions will be carefully checked by a curator (a human, an anatomy expert) and his/her input will be utilized to accurately calculate the correlation coefficients between certain pairs of knowledge sources. these correlation coefficients could be used as values for the α parameters of the scoring scheme. the procedures described briefly here and detailed in [12], and the scoring schemes introduced here, are utilized in the software program anatom [10], [13] developed as part of our work on semi-automatic mapping and merging of anatomical ontologies. references [1] j. de bruijn et al., ontology mediation, merging, and aligning. in: j. davies, r. studer, p. warren (eds.), semantic web technologies. wiley, 2006, pp. 95–113. 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[9] ernest a.a. van ophuizen, jack a.m. leunissen, an evaluation of the performance of three semantic background knowledge sources in comparative anatomy. j. integrative bioinformatics, vol. 7, 2010, pp. 124–130. [10] peter petrov, nikolay natchev, dimitar vassilev, milko krachounov, maria nisheva, ognyan kulev, anatom an intelligent software program for semi-automatic mapping and merging of anatomy ontologies. to appear in: proceedings of the 6th international conference on information systems & grid technologies (isgt, sofia, 1–3 june 2012), sofia, st. kliment ohridski university press, 2012. [11] j. euzenat, p. shvaiko, ontology matching. springer, heidelberg, 2007. [12] peter petrov, milko krachunov, ernest a.a van ophuizen, dimitar vassilev, an algorithmic approach to inferring crossontology links while mapping anatomical ontologies. to appear in serdica journal of computing, issn 1312-6555, vol. 6, 2012. [13] peter petrov, milko krachunov, elena todorovska, dimitar vassilev, an intelligent system approach for integrating anatomical ontologies. biotechnology and biotechnological equipment 26(4):3173–3181, 2012 [14] http://www.nlm.nih.gov/research/umls/ – web site of the unified medical language system (umls) by the u.s. national library of medicine (nlm), last accessed: 20 october 2012. [15] http://sig.biostr.washington.edu/projects/fm/ – web site of the foundational model of anatomy (fma) by the university of washington, last accessed: 20 october 2012. [16] http://wordnet.princeton.edu/ – web site of the wordnet project by the princeton university, last accessed: 20 october 2012. biomath 1 (2012), 1211117, http://dx.doi.org/10.11145/j.biomath.2012.11.117 page 6 of 6 http://dx.doi.org/10.1038/75556 http://dx.doi.org/10.1038/nbt1346 http://dx.doi.org/10.1093/nar/gkh061 http://dx.doi.org/10.1016/j.jbi.2003.11.007 http://dx.doi.org/10.1145/219717.219748 http://dx.doi.org/10.11145/j.biomath.2012.11.117 introduction problem formulation formalization of the problem algorithmic procedures comparison of alternative scoring schemes results and discussion conclusion references original article biomath 2 (2013), 1212101, 1–6 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum impact of perfect drug adherence on immunopathogenic mechanism for dynamical system of psoriasis priti kumar roy∗ and abhirup datta∗ ∗ centre for mathematical biology and ecology department of mathematics jadavpur university, kolkata 700032, india emails: pritiju@gmail.com, abhirupdattajumath@gmail.com received: 15 july 2012, accepted: 10 december 2012, published: 04 january 2013 abstract—psoriasis is a frequent autoimmune chronic skin disease differentiated by t-cells agreeable hyperproliferation of epidermal keratinocytes. the feature of tcells held up psoriatic scratches is the epidermal penetration of basically oligoclonal cd8+ t-cells and also of cd4+ t-cells in the dermis. psoriatic lesions are sharply distinguished, red and enlarged scratches together with whitish silver scales. in this research article, we propose a mathematical depiction for psoriasis, involving a set of differential equations, regarding t-cells, dendritic cells, cd8+ t-cells and epidermal keratinocytes. here, we specially introduce the interaction between dendritic cells and cd8+ t-cells to monitor the impact of this interaction upon the system dynamics. we also analyze the mathematical model both in presence and absence of effectiveness of two drugs. we study the system analytically and numerically to comprehend the significance of effectiveness of the drugs, integrated in the model system. here, we reduce the keratinocyte population to restrict psoriasis by applying the combination of two drugs and able to enlighten the perspective of the disease dynamics for psoriasis. keywords-t-cells; dendritic cells; cd8+ t-cells; keratinocytes; mhc; pmhc; t-cells receptor; dermis; epidermis; lymphocytes; monocytes; neutrophils; cytokines; drug efficacy i. introduction in spite of precise fundamental and experimental studies for more than a few decades, many queries continue relating to psoriasis. inflammatory tissues respond along with enormous influxes of t-cells and dendritic cells (nickoloff, 2000). a “perfect cytokine storm” is produced through this multicellular scheme that synchronizes the cellular attack and links mutually with connection of both soluble intermediaries and cellular ingredients (uyemura et. al., 1993, nickoloff and nestle, 2004) [1]. psoriasis has been measured as a dermatological chaos, in which t-cells and epidermal keratinocytes perform a relevant pathogenic function. dcs play an essential role in pathogenesis of psoriasis by attending antigens throughout principal major histo compatibility (mhc) complex ii molecules [2]. psoriasis is observed as a widespread inflammatory skin chaos with an inherited contact. it is illustrious through epidermal hyperplasia by means of cellular diffusion of lymphocytes, monocytes and neutrophils [3]. local production of t-cells is observed as a significant immunological constituent of psoriatic lesions. the enormous numbers of dendritic cells below the hyperplastic epidermis, are surrounded by t-cells within the psoriatic plaques [4]. roy and bhadra [5] have clarified that, suppression made on dendritic cells will reduce the expansion of keratinocytes and will give better effect than suppression made on t-cells. for suppression made on t-cells, the pathogenesis continues due to auxiliary basis in presence of dcs, as the suppression on dcs presents a superior result. in our very recent work, we have formed a set citation: p. roy, a. datta, impact of perfect drug adherence on immunopathogenic mechanism for dynamical system of psoriasis, biomath 2 (2013), 1212101, http://dx.doi.org/10.11145/j.biomath.2012.12.101 page 1 of 6 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.12.101 p. roy et al., impact of perfect drug adherence on immunopathogenic mechanism... of differential equations to exhibit a course of stable connection to the growth of epidermal keratinocytes through negative feedback control, that is comparable to the favorable drug management. we also integrate a time delay in our model to furnish the time from creation of t-cells and dcs to the enhancement of epidermal keratinocytes [6]. in our present research article, we introduce cd8+ t-cells population which interacts with dcs in the dynamical system. this interaction leads to generate keratinocytes, which in turn supports to expand the keratinocytes growth. cyclosporin and fk506 are applied as drugs, that perform to restrict psoriasis [7]. to confine this growth, we apply drug at the interaction between cd8+ t-cells and dcs. another method to create keratinocytes is the interaction between t-cells and keratinocytes itself. here, we also set the drug in that interaction to control the growth of keratinocytes, whose surplus production directs to create psoriasis. in this article, we study the effectiveness of two drugs on the cell biological scheme to build a comparative analysis for the drugs to restrain the disease. ii. the basic assumptions and formulation of the mathematical model we consider the mathematical model of psoriasis to describe the dynamical cell biological system. let us assume l(t), m(t), c(t) and k(t) to represent the densities of t-cells, dendritic cells, cd8+ t-cells and epidermal keratinocytes correspondingly at a specific time t to attain a set of differential equations. in the region proximity, the accumulation of t-cells is considered at a constant rate a and the accumulation of dendritic cells is taken at a constant rate b at the appropriate regime. it is assumed that, the rate of activation of t-cells by dcs is δ and β is the rate of activation of dcs by t-cells. expansion of keratinocytes density is taken to be proportional to the production of t-cells and dcs densities with a rate η. the rate of activation of keratinocytes by t-cells due to tcells mediated cytokines is referred as γ1 and γ2 is the rate at which growth of keratinocytes takes place. the per capita removal rate of t-cells is denoted by µ and µ′ is the per capita removal rate of dendritic cells throughout normal procedure. the premature dendritic cells turn into mature in the course of some cell biological procedures and move into the lymph node. in that lymph node, the mature dcs interrelate with cd8+ t-cells at a rate qn, where q is the average peptide specific t-cells receptor (tcr) and n is the average number of the related pmhc complexes per dcs and this contact gives a negative effect to dcs as well as positive effect to cd8+ t-cells. the cd8+ t-cell proliferation is stimulated by similar antigen presenting dcs at a rate r. we assume here also that, α is the rate of interaction between dcs and cd8+ t-cells. it gives negative impact to cd8+ t-cell population. in addition, keratinocytes are produced through interaction between dcs and cd8+ t-cells at a rate α1. again, we assume ξ and λ as the per capita removal rate of cd8+ t-cells and epidermal keratinocytes respectively. all the parameters, described above, are always positive. here, we assimilate the combination of two drug efficacy parameters u1 and u2, placed between the interaction of t-cells and epidermal keratinocytes and dendritic cells and cd8+ t-cells respectively to restrain the growth of epidermal keratinocytes, whose excess production is one of the main reasons to form psoriasis. accumulating collectively the above assumptions, we can formulate the mathematical model given below: dl dt = a − δlm − γ1lk(1 − u1) − µl, dm dt = b − βlm − qnmc − µ′m, dc dt = rqnmc − αmc(1 − u2) − ξc, (1) dk dt = ηlm + γ2lk(1 − u1) + α1mc(1 − u2) − λk, where l(0) > 0, m(0) > 0, c(0) > 0 and k(0) > 0 at a specific time period t. the communication is organized as follows: we comprise the general outlook and discuss about the effectiveness of drugs on the cell biological system of psoriasis in section i. in section ii, we represent the mathematical model of psoriasis including basic assumptions. section iii describes theoretical analysis of the model system (1). this section is also integrated with two equilibrium points of the system dynamics. theoretical explanation of the model parameters, centering on its stability and associated features are discussed in the same section. in section iv, we include results from numerical simulation of the system and finally section v ends with the conclusion of the model dynamics. iii. local stability analysis for the system the rhs of the equation (1) is a smooth function of l(t), m(t), c(t) and k(t) and also the parameters, as long as these quantities are non-negative. for that reason, local existence and uniqueness properties hold in the positive octant. biomath 2 (2013), 1212101, http://dx.doi.org/10.11145/j.biomath.2012.12.101 page 2 of 6 http://dx.doi.org/10.11145/j.biomath.2012.12.101 p. roy et al., impact of perfect drug adherence on immunopathogenic mechanism... 0 20 40 60 80 100 0 200 400 600 800 1000 time ( day −1 ) c e l l p o p u l a t i o n ( m m − 3 ) (a) λ=0.4 0 20 40 60 80 100 0 200 400 600 800 1000 time ( day −1 ) l(t) m(t) c(t) k(t) (b) λ=0.6 0 20 40 60 80 100 0 200 400 600 800 time ( day −1 ) (c) λ=0.8 fig. 1. behaviors of different cell biological masses of the system (1) with u1=0.5 and u2=0.7 for λ = 0.4 (panel a), λ = 0.6 (panel b) and λ = 0.8 (panel c), keeping other parameters at their standard values as in table 1. a. equilibria of the model system the model equation (1) has two equilibrium points, i.e., ẽ(l̃, m̃, 0, k̃) and e∗(l∗,m∗,c∗,k∗). now, m̃= b βl̃+µ′ , k̃= a−δl̃m̃−µl̃ γ1l̃(1−u1) and l̃ is the positive root of the equation al̃3 − bl̃2 + cl̃ + d = 0, (2) where a = βδγ1γ2m̃(1 − u1) + βγ1γ2µ(1 − u1) > 0, b = aβγ1γ2(1 − u1) + bηγ21 (1 − u1) + δγ1m̃(βλ− γ2µ ′) + γ1µ(βλ − γ2µ′) + γ1γ2µ′u1(δm̃ + µ) > 0, c = aγ1(βλ − γ2µ′) + γ1µ′(aγ2u1 − δλm̃ − µλ) > 0, d = aγ1µ′λ > 0. this cubic equation (2) has positive real root if the coefficients of l̃3, −l̃2 and l̃ are positive. now, considering descartes’ rule of sign, we may conclude that the equation al̃3 −bl̃2 + cl̃ + d = 0 has two positive real roots (multiplicities of roots are adequate) [8] if and only if the following conditions are hold: (i) βλ > γ2µ ′ and (ii) aγ2u1 > λ(δm̃ + µ). from the second equation of system (1), we include m̃ is always positive by our necessary assumptions. from the first equation of system (1), we state that k̃ is realistic if a > l̃(δm̃ + µ). as a result, if (i) and (ii) are persuaded, then we may bring to an end that, the equation (2) has two positive real roots and henceforth positive equilibrium point ẽ(l̃, m̃, 0, k̃) of the system (1) exists. finally, for the interior equilibrium point e∗(l∗,m∗,c∗,k∗), l∗, m∗, c∗ and k∗ are the nontrivial solutions of the model equation (1). remark 1. the system (1) exists if the two conditions are hold, (a) the product of the rate of activation of dcs by t-cells and the per capita removal rate of keratinocytes should be greater than the product of the rate of growth of keratinocytes due to t-cells mediated cytokines and the per capita removal rate of dcs and (b) the rate of accumulation of t-cells itself and the product of the rate of accumulation of t-cells, the rate of growth of keratinocytes due to t-cells mediated cytokines and the first drug efficacy parameter must be greater than a pre-assigned positive quantity. the characteristic equation of the matrix related to the equilibrium point ẽ(l̃, m̃, 0, k̃) in presence of effectiveness of both drugs (u1 = u2 = 1) is illustrated by, (−λ − φ)(rqnm̃ − ξ − φ)[φ2 − (trace v )φ +det v ] = 0, where trace v = −(βl̃ + δm̃ + µ + µ′) < 0 and det v = βµl̃ + δµ′m̃ + µµ′ > 0. now, φ1 (=−λ) is always negative, φ2=rqnm̃−ξ and the roots of the equation φ2 − (trace v )φ + det v = 0 are negative since trace v < 0 and det v > 0. hence the equilibrium point ẽ(l̃, m̃, 0, k̃) in presence of effectiveness of both drugs is stable only if m̃ < ξ rqn . remark 2. the cd8+ t-cells free equilibrium point in presence of effectiveness of both drugs is stable if dc population is less than some pre-determined positive value. the characteristic equation of the matrix related to the equilibrium point ẽ(l̃, m̃, 0, k̃) in absence of effectiveness of both drugs (u1 = u2 = 0) is furnished by, (rqnm̃ − αm̃ − ξ − ψ)(ψ3 + a1ψ2 + a2ψ + a3) = 0. here, ψ1=rqnm̃ − αm̃ − ξ and from routh-hurwitz criterion, a1 > 0 if β > γ2, a3 > 0 if ηγ1 > δγ2, βλ > γ2µ ′ and k̃ l̃ > γ2µ γ1λ and a1a2 − a3 > 0 if β > γ2. thus the equilibrium point ẽ(l̃, m̃, 0, k̃) in absence of effectiveness of both drugs is stable if m̃ < ξ rqn−α , β >max[γ2, γ2µ ′ λ ] and γ2 γ1 <min[ η δ , λk̃ µl̃ ], provided rqn > α. biomath 2 (2013), 1212101, http://dx.doi.org/10.11145/j.biomath.2012.12.101 page 3 of 6 http://dx.doi.org/10.11145/j.biomath.2012.12.101 p. roy et al., impact of perfect drug adherence on immunopathogenic mechanism... 0 20 40 60 80 100 0 200 400 600 800 1000 time ( day −1 ) c e l l p o p u l a t i o n ( m m − 3 ) u 1 =u 2 =0.5 (a) 0 20 40 60 80 100 0 200 400 600 800 1000 time ( day −1 ) l(t) m(t) c(t) k(t) u 1 =0.9, u 2 =0.5 (b) 0 20 40 60 80 100 0 200 400 600 800 1000 time ( day −1 ) (c) u 1 =0.5, u 2 =0.9 fig. 2. behaviors of different cell biological masses of the system (1) for different values of two drug efficacy parameters u1 and u2, keeping other parameters at their standard values as in table 1. remark 3. the cd8+ t-cells free equilibrium point in absence of effectiveness of both drugs is stable if (1) dc population is less than some pre-assigned positive quantity, (2) the rate of activation of dcs by t-cells should be always greater than the maximum of [γ2, γ2µ ′ λ ] and (3) the ratio of γ2 and γ1 should be always less than the minimum of [ η δ , λk̃ µl̃ ]. also we study another two cases, i.e., first drug (u1) is present and second drug (u2) is absent and vice-versa in the system dynamics. the characteristic equation of the matrix related to the equilibrium point ẽ(l̃, m̃, 0, k̃) in presence of effectiveness of first drug (u1 = 1) and absence of effectiveness of second drug (u2 = 0) is illustrated by, (−λ − ϕ)(rqnm̃ − αm̃ − ξ − ϕ)[ϕ2 − (trace w)ϕ +det w ] = 0, where trace w = −(βl̃ + δm̃ + µ + µ′) < 0 and det w = βµl̃ + δµ′m̃ + µµ′ > 0. now, ϕ1 (=−λ) is always negative, ϕ2=rqnm̃−αm̃−ξ and the roots of the equation ϕ2−(trace w)ϕ+det w = 0 are negative since trace w < 0 and det w > 0. hence the equilibrium point ẽ(l̃, m̃, 0, k̃) in presence of effectiveness of first drug (u1 = 1) and absence of effectiveness of second drug (u2 = 0) is stable only if m̃ < ξ rqn−α , provided rqn > α. remark 4. the cd8+ t-cells free equilibrium point in presence of effectiveness of first drug and absence of effectiveness of second drug is stable if dc population is less than some pre-determined positive value, provided the product of the rate at which cd8+ t-cell proliferation is stimulated by antigen presenting dcs, average peptide specific t-cells receptor (tcr) and average number of the related pmhc complexes per dcs is greater than the rate of interaction between dcs and cd8+ t-cells. the characteristic equation of the matrix related to the equilibrium point ẽ(l̃, m̃, 0, k̃) in absence of effectiveness of first drug (u1 = 0) and presence of effectiveness of second drug (u2 = 1) is demonstrated by, (rqnm̃ − ξ − χ)(χ3 + b1χ2 + b2χ + b3) = 0. here, χ1=rqnm̃ − ξ and from routh-hurwitz criterion, we obtain β > γ2, ηγ1 > δγ2, βλ > γ2µ ′ and k̃ l̃ > γ2µ γ1λ . thus the equilibrium point ẽ(l̃, m̃, 0, k̃) in absence of effectiveness of first drug (u1 = 0) and presence of effectiveness of second drug (u2 = 1) is stable if m̃ < ξ rqn , β >max[γ2, γ2µ ′ λ ] and γ2 γ1 <min[ η δ , λk̃ µl̃ ]. remark 5. if (1) dc population is less than some pre-assigned positive quantity, (2) the rate of activation of dcs by t-cells should be always greater than the maximum of [γ2, γ2µ ′ λ ] and (3) the ratio of γ2 and γ1 should be always less than the minimum of [ η δ , λk̃ µl̃ ], then the cd8+ t-cells free equilibrium point in absence of effectiveness of first drug and presence of effectiveness of second drug is stable. now, we analyze the roots of the characteristic equation of the matrix related to the interior equilibrium point in presence of effectiveness of both drugs and study it’s stability analysis. we discard the cases where (a) effectiveness of both drugs are absent and (b) one is present and other is absent and vice-versa, as they are not realistic for the interior equilibrium in true sense. the characteristic equation is furnished by, (−λ − τ)(τ3 + c1τ2 + c2τ + c3) = 0. here, τ1 (=−λ) is always negative and from routhhurwitz criterion, we have δ > rqn and m∗ < ξ rqn . hence the interior equilibrium point e∗(l∗, m∗, c∗, k∗) in presence of effectiveness of both drugs (u1 = u2 = 1) is stable if rqn <min[δ, ξ m∗ ]. biomath 2 (2013), 1212101, http://dx.doi.org/10.11145/j.biomath.2012.12.101 page 4 of 6 http://dx.doi.org/10.11145/j.biomath.2012.12.101 p. roy et al., impact of perfect drug adherence on immunopathogenic mechanism... 0 20 40 60 80 100 0 200 400 600 800 1000 time ( day −1 ) c e l l p o p u l a t i o n ( m m − 3 ) u 1 =u 2 =0 (a) 0 20 40 60 80 100 0 200 400 600 800 1000 time ( day −1 ) l(t) m(t) c(t) k(t) (b) u 1 =1, u 2 =0 0 20 40 60 80 100 0 200 400 600 800 1000 time ( day −1 ) (c) u 1 =0, u 2 =1 fig. 3. behaviors of different cell biological masses of the system (1) for perfect absence and presence of two drug efficacy parameters u1 and u2, keeping other parameters at their standard values as in table 1. remark 6. the interior equilibrium point in presence of effectiveness of both drugs is stable if the product of the rate at which cd8+ t-cell proliferation is stimulated by antigen presenting dcs, average peptide specific tcells receptor (tcr) and average number of the related pmhc complexes per dcs should be less than the minimum of [δ, ξ m∗ ]. iv. numerical simulation of the system in the previous section, we have initiated analytical techniques for qualitative study of the system with effectiveness of two drugs. in this part, we perform numerical simulation of the model system (1). we approximate the parameters in the course of our analytical outcomes and conditions. numerical values of the model parameters, used in our numerical simulation, have been specified in table 1. table 1. parameters used in the model equation (1) paradefault values assigned meter a 9 mm−3 day−1 [6] b 14 mm−3 day−1 [6] δ 0.01 mm3 day−1 [6] β 0.065 mm3 day−1 [6] η 1.5 mm3 day−1 [6] γ1 0.0002 mm 3 day−1 [6] γ2 0.0001 mm 3 day−1 (estimated) r 0.9 day−1 (estimated) q 0.001 day−1 (estimated) n 7 (estimated) α 0.0007 mm3 day−1 (estimated) α1 0.0005 mm 3 day−1 (estimated) µ 0.07 day−1 (estimated) µ′ 0.002 day−1 [6] ξ 0.08 day−1 (estimated) λ 0.4 day−1 [6] we are trying to monitor the cell behavioral patterns of different cells, involved in the system dynamics for variation in the values of the parameters. firstly, we set u1 = 0.5 and u2 = 0.7 and assume the value of the decay rate of keratinocytes (λ) as 0.4 day−1, then we notice that, keratinocytes initially increase around 900 cells/mm3, decline slightly just below 800 cells/mm3 and finally become stable, displayed in fig. 1(a). when the value of λ is 0.6 day−1, keratinocytes raise up to 800 cells/mm3, after that decrease below 600 cells/mm3, at last turn into stable, shown in fig. 1(b). finally, considering the value of λ as 0.8 day−1, keratinocytes increase below 800 cells/mm3, next decrease at about 400 cells/mm3 and lastly develop into stable in nature, portrayed in fig. 1(c). thus we conclude that, due to increase in the value of decay rate of keratinocytes, its population must be decreased. next, we monitor the performances of four different cells in the system for alteration in the values of two drug efficacy parameters (u1 and u2). at fig. 2(a), we consider u1 and u2 are both as 0.5. when the value of u1 is increased by 0.4 at fig. 2(b) (i.e., u1 = 0.9), then we study that, tcells are increased than the earlier case. again, when u1 is decreased to 0.5, the behavior of t-cells at fig. 2(c) is rolled back like fig. 2(a). besides, we notice that, keratinocytes arrive quicker in the direction of the stable region than for the lower value of effectiveness of first drug. no specific change is occurred for the presence and increase of effectiveness of second drug in the system. at fig. 3(a), we assume perfect absence of effectiveness for both the drugs, effectiveness of first drug is perfectly present and effectiveness of second drug is absent perfectly, illustrated at fig. 3(b) and vice-versa at fig. 3(c). the increasing and decreasing natures of t-cells are repeated as like fig. 2 for the perfect absence and presence of effectiveness of two drugs. there is not at all any significant change in the behavioral pattern, observed in dc and cd8+ t-cell biomath 2 (2013), 1212101, http://dx.doi.org/10.11145/j.biomath.2012.12.101 page 5 of 6 http://dx.doi.org/10.11145/j.biomath.2012.12.101 p. roy et al., impact of perfect drug adherence on immunopathogenic mechanism... population for variation in the values of the system parameters and also for the change (increase or decrease) in the values of two drug efficacy parameters. v. conclusion the effectiveness of first drug has an imperative effect rather than the second drug upon our proposed model. in presence of first drug, the system develops into stable condition very smoothly. there is not at all any significant difference between absence and presence of second drug. thus the second drug has not any significant impact on the interaction between dcs and cd8+ t-cells, which are not influenced for change in the value of the model parameters. they maintain the same pattern, which is independent of any model parameter of the system. it is numerically revealed that, increase in the value of decay rate of keratinocytes leads to the decrease in keratinocytes and hence increase in t-cells is occurred simultaneously. enhancing in the value of decay rate of keratinocytes has no impact on dcs and cd8+ t-cells. with the increasing value of decay rate of keratinocytes, population of keratinocyte is decreased and thus t-cell population is gradually increased. increasing in the value of u1 forwards to the increase in t-cells, because increase in the value of u1 reduces keratinocytes and thus t-cells are increased. keratinocytes reach stable situation more rapidly with the increase in the value of first drug efficacy parameter. but unfortunately, the second drug has not any significant effect on the system, specially on the interaction between dcs and cd8+ t-cells and thus the second drug does not take part to control the keratinocyte population in broad sense. we may also predict another important feature that, decrease in keratinocytes provides increase in t-cells but increase in t-cells does not essentially lead to decrease in keratinocytes or the decrease is not too prominent to observe. we have used the combination of two drugs to get enhanced result but from analytical and numerical points of view, the first drug has the significant effect compared to the second drug on our proposed mathematical model for psoriasis. hence, to put the drug in the interaction between dcs and cd8+ t-cells may not be successful enough to restrict the expansion of keratinocytes growth. thus applying the drug at the place of interaction between t-cells and dcs on psoriatic patients in a systematic way would be able to obtain the improved outcome for better cure of the disease psoriasis. acknowledgment research is supported by the council of scientific and industrial research, government of india, ref. no. 38(1320)/12/emr-ii, dated 3rd april, 2012. references [1] b. j. nickoloff, b. k. bonish, d. j. marble, k. a. schriedel, l. a. dipietro, k. b. gordon and m. w. lingen, “lessons learned from psoriatic plaques concerning mechanisms of tissue repair, remodeling, and inflammation,” journal of investigative dermatology symposium proceedings, vol. 11, pp. 16–29, 2006. http://dx.doi.org/10.1038/sj.jidsymp.5650010 [2] f. prignano, f. ricceri, b. bianchi and t. lotti, “quantity, distribution and immunophenotypical modification of dendritic cells upon biological treatments in psoriasis,” int. j. immunopathol pharmacol, vol. 22, no. 2, pp. 379–387, 2009. [3] k. j. zhu, h. cheng, x. h. mao, l. m. lao, j. p. cen and j. ye, “increased endocytic activity in monocyte-derived dendritic cells in patients with psoriasis vulgaris,” indian j med res, vol. 123, pp. 43–50, 2006. [4] f. o. nestle, l. a. turka and b. j. nickoloff, “characterization of dermal dendritic cells in psoriasis. autostimulation of t lymphocytes and induction of th1 type cytokines,” j. clin. invest., vol. 94, no. 1, pp. 202–209, 1994. http://dx.doi.org/10.1172/jci117308 [5] p. k. roy and j. bhadra, “comperative study of the suppression on t-cell and dendritic cells in a mathematical model of psoriasis,” international journal of evolution equation, vol. 5, no. 3, pp. 309–326, 2010. [6] p. k. roy and a. datta, “negative feedback control may regulate cytokines effect during growth of keratinocytes in the chronic plaque of psoriasis: a mathematical study,” international journal of applied mathematics, vol. 25, no. 2, pp. 233– 254, 2012. [7] p. k. roy, a. datta and a. n. chatterjee, “saturation effects on immunopathogenic mechanism of psoriasis: a theoretical approach,” acta analysis functionalis applicata, vol. 13, no. 3, pp. 310–318, 2011. [8] j. lou, l. chen and t. ruggeri, “an impulsive differential model on post exposure prophylaxis to hiv-1 exposed individual,” journal of biological systems, vol. 17, no. 4, pp. 659–683, 2009. http://dx.doi.org/10.1142/s0218339009002934 biomath 2 (2013), 1212101, http://dx.doi.org/10.11145/j.biomath.2012.12.101 page 6 of 6 http://dx.doi.org/10.1038/sj.jidsymp.5650010 http://dx.doi.org/10.1172/jci117308 http://dx.doi.org/10.1142/s0218339009002934 http://dx.doi.org/10.11145/j.biomath.2012.12.101 introduction the basic assumptions and formulation of the mathematical model local stability analysis for the system equilibria of the model system numerical simulation of the system conclusion references www.biomathforum.org/biomath/index.php/biomath original article on the hausdorff distance between the shifted heaviside step function and the transmuted stannard growth function anton iliev ∗†, nikolay kyurkchiev †, svetoslav markov † ∗faculty of mathematics and informatics paisii hilendarski university of plovdiv, bulgaria aii@uni-plovdiv.bg †institute of mathematics and informatics bulgarian academy of sciences, sofia, bulgaria nkyurk@math.bas.bg, smarkov@bio.bas.bg received: 4 july 2016, accepted: 4 september 2016, published: 14 september 2016 dedicated to professor roumen anguelov on the occasion of his 60th anniversary abstract—in this paper we study the one-sided hausdorff distance between the shifted heaviside step–function and the transmuted stannard growth function. precise upper and lower bounds for the hausdorff distance have been obtained. we present a software module (intellectual property) within the programming environment cas mathematica for the analysis of the growth curves. numerical examples, illustrating our results are given, too. keywords-transmuted stannard growth function; heaviside step function; hausdorff distance; upper and lower bounds. i. introduction and preliminaries the stannard function finds numerous applications in many scientific fields, including population dynamics, bacterial growth, population ecology, plant biology, chemistry, agriculture, demography, financial mathematics, statistics and fuzzy set theory [1]–[5]. definition 1. for γ ∈ r define the shifted heaviside step function as [12]: hγ(t) =   0, if t < 0, [0, 1], if t = γ, 1, if t > γ. (1) definition 2. define the shifted stannard growth function s(t) as [1]–[5]: s(t) = 1( 1 + e −(β+k(t−γ)) m )m , (2) where β, k and m ∈ r are the growth parameters. we note that the slope of (2) at t = γ is equal to: ke− β m( 1+e− β m )m+1 . definition 3. a random variable t is said to have a transmuted distribution if its cumulative distribution function (cdf) is given by [6], [7]: g1(t) = (1 + λ)f1(t) −λf21 (t), |λ| ≤ 1, (3) where f1(t) is the cdf of the base distribution. citation: anton iliev, nikolay kyurkchiev, svetoslav markov, on the hausdorff distance between the shifted heaviside step function and the transmuted stannard growth function, biomath 5 (2016), 1609041, http://dx.doi.org/10.11145/j.biomath.2016.09.041 page 1 of 6 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2016.09.041 a. iliev et al., on the hausdorff distance between the shifted heaviside ... ”λ transmuting” of (cdf) is a familiar technique from the field of probability distributions with application to insurance mathematics. definition 4. the hausdorff distance ρ(f,g) between two interval functions f,g on ω ⊆ r, is the distance between their completed graphs f(f) and f(g) considered as closed subsets of ω × r [8], [9], [12]. more precisely, we have ρ(f,g) = max{ sup a∈f(f) inf b∈f(g) ||a−b||, (4) sup b∈f(g) inf a∈f(f) ||a−b||}, wherein ||.|| is any norm in r2, e. g. the maximum norm ||(t,x)|| = max{|t|, |x|}; hence the distance between the points a = (ta,xa), b = (tb,xb) in r2 is ||a−b|| = max(|ta − tb|, |xa −xb|). sigmoidal growth curves typically have three parts (phases, time intervals): lag, log and stationary parts. it is a challenging question to characterize mathematically these phases. the lag time (interval) is practically important in many medical and biotechnological applications as this time is responsible for the acceleration or inhibition of the process and the possibility of controlling the lag time depends on the understanding of the hidden mechanisms of the corresponding process [10], [11]. usually the lag time is defined by means of the uniform distance between the sigmoidal function and the induced cut function. we propose a new definition for the lag time by means of the hausdorff distance between the sigmoidal function and the induced step function. in this work we prove estimates for the one– sided hausdorff approximation of the shifted heaviside step–function by transmuted stannard growth function. let us point out that the hausdorff distance is a natural measuring criteria for the approximation of bounded discontinuous functions [12], [13]. fig. 1. approximation of the shifted heaviside step function by transmuted stannard growth function for the following data: k = 16, m = 0.52, β = 0.01, tr = 5; hausdorff distance d = 0.0801797. fig. 2. approximation of the shifted heaviside step function by transmuted stannard growth function for the following data: k = 26, m = 2.1, β = 1, tr = 5; hausdorff distance d = 0.112237. ii. main results for γ,β,m ∈ r consider the following transmuted stannard function s∗(t) = 1 + λ( 1 + e −(β+k(t−γ)) m )m− (5) λ( 1 + e −(β+k(t−γ)) m )2m , |λ| ≤ 1. biomath 5 (2016), 1609041, http://dx.doi.org/10.11145/j.biomath.2016.09.041 page 2 of 6 http://dx.doi.org/10.11145/j.biomath.2016.09.041 a. iliev et al., on the hausdorff distance between the shifted heaviside ... function s∗(t) from (5) satisfies: s∗(γ) = 1 + λ( 1 + e− β m )m − λ( 1 + e− β m )2m = 12 (6) hence λ = 0.5(1 + z)2m − (1 + z)m (1 + z)m − 1 ; z = e− β m . (7) we study the hausdorff approximation d of the heaviside step function hγ(t) by the transmuted stannard function (5)–(7) and look for an expression for the error of the best one–sided approximation. let a = (1 + λ) ( 1 + e− β m )−m −λ ( 1 + e− β m )−2m b = 1 − 2e− β m ( 1 + e− β m )−1−2m kλ + e− β m ( 1 + e− β m )−1−m k(1 + λ), k ∈ r. (8) the following theorem gives upper and lower bounds for d. theorem 2.1 for the hausdorff distance d between the function hγ(t) and the transmuted stannard function (5)–(7) the following inequalities hold for |λ| ≤ 1 and b > 4: dl = a 2b < d < a ln(2b) 2b = dr. (9) proof. we need to express d in terms of k, β and m. the hausdorff distance d satisfies the relation f(d) := s∗(γ −d) = 1 + λ( 1 + e− β−kd m )m− (10) λ( 1 + e− β−kd m )2m −d = 0. consider the function g(d) = a−bd. fig. 3. the functions f(d) and g(d) for k = 16, m = 0.52, β = 0.01, tr = 5. by means of taylor expansion we obtain g(d) −f(d) = o(d2). hence g(d) approximates f(d) with d → 0 as o(d2) (see fig. 3). further, for |λ| ≤ 1 and b > 4 we have g(dl) = a 2 > 0, g(dr) = a (1 − 0.5 ln(2b)) < 0. this completes the proof of the theorem. some computational examples using relations (9) are presented in table 1. the last column of table 1 contains the values of d computed by solving the nonlinear equation (10). table i bounds for d computed by equation (9) for various β, k, m. biomath 5 (2016), 1609041, http://dx.doi.org/10.11145/j.biomath.2016.09.041 page 3 of 6 http://dx.doi.org/10.11145/j.biomath.2016.09.041 a. iliev et al., on the hausdorff distance between the shifted heaviside ... fig. 4. a simple module implemented in cas mathematica for the computation and visualization of the hausdorff distance between the heaviside step function and the transmuted stannard growth function. biomath 5 (2016), 1609041, http://dx.doi.org/10.11145/j.biomath.2016.09.041 page 4 of 6 http://dx.doi.org/10.11145/j.biomath.2016.09.041 a. iliev et al., on the hausdorff distance between the shifted heaviside ... iii. conclusion remarks new estimates for the hausdorff distance between an interval heviside step function and its best approximating stannard function are obtained. on fig. 1 and fig. 2 appropriate illustrations of some approximations of the shifted heaviside step function by transmuted stannard growth function are given. we propose a software module within the programming environment cas mathematica for the analysis of the considered growth curves (see fig. 4). the module offers the following possibilities: i) generation of the shifted stannard curve under user-defined values for k,m,β; ii) automatic check of the condition |λ| ≤ 1 that guarantees the existence of sigmoidality of the transmuted stannard curve; iii) software tools for animation and visualization. the hausdorff approximation of the interval step function by the logistic and other sigmoidal functions is discussed from various approximation, computational and modelling aspects in [14]–[27]. acknowledgments the authors would like to thank the anonymous reviewers for their helpful comments that contributed to improving the final version of the presented paper. references [1] c. stannard, a. williams, p. gibbs, temperature/growth relationship for psychotropic food– spoilage bacteria, food microbiol. 2 (1985) 115–122. [2] a. khamis, z. ismail, kh. haron, a. mohammed, nonlinear growth models for modeling oil palm yield growth, journal of mathematics and statistics 1(3) (2005) 225–233, doi:10.3844/jmssp.2005.225.232 [3] m. zwietering, i. jongenburger, f. rombouts, k. riet, modeling of the bacterial growth curve, appl. and environmental microbiology 56(6) (1990) 1875–1881. [4] m. panik, growth curve modeling: theory and applications, wiley (2014), isbn: 978-1-118-76404-6, doi:10.1002/9781118763971 [5] n. kyurkchiev, a. iliev, on some growth curve modeling: approximation theory and applications, int. j. of trends in research and development, 3(3) (2016) 466–471. [6] w. shaw, i. buckley, the alchemy of probability distributions: beyond gram-charlier expansion and a skewkurtotic-normal distribution from rank transmutation map, research report (2007). 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(eds), scientific computing, computer arithmetic, and validated numerics, 16th international symposium, springer scan 2014, lncs 9553 (2016) 3–13, doi:10.1007/978-3-319-31769-4 [13] n. kyurkchiev, a. andreev, approximation and antenna and filter synthesis: some moduli in programming environment mathematica, lap lambert academic publishing, saarbrucken (2014), isbn 978-3659-53322-8. [14] d. costarelli, r. spigler, approximation results for neural network operators activated by sigmoidal functions, neural networks 44 (2013) 101–106, doi:10.1016/j.neunet.2013.03.015 [15] n. kyurkchiev, on the approximation of the step function by some cumulative distribution functions, compt. rend. acad. bulg. sci. 68(12) (2015) 1475–1482. [16] n. kyurkchiev, s. markov, on the hausdorff distance between the heaviside step function and verhulst logistic function, j. math. chem. 54(1) (2016) 109–119, doi:10.1007/s10910-015-0552-0 [17] n. kyurkchiev, s. markov, sigmoidal functions: some computational and modelling aspects, biomath communications 1(2) (2014) 30–48, doi:10.11145/j.bmc.2015.03.081 [18] a. iliev, n. kyurkchiev, s. markov, on the approximation of the cut and step functions by logistic and gompertz functions, biomath 4(2) (2015) 1510101, doi:10.11145/j.biomath.2015.10.101 [19] a. iliev, n. kyurkchiev, s. markov, on the approximation of the step function by some sigmoid functions, biomath 5 (2016), 1609041, http://dx.doi.org/10.11145/j.biomath.2016.09.041 page 5 of 6 http://dx.doi.org/10.11145/j.biomath.2016.09.041 a. iliev et al., on the hausdorff distance between the shifted heaviside ... mathematics and computers in simulation (2015), doi:10.1016/j.matcom.2015.11.005 [20] n. kyurkchiev, s. markov, on the approximation of the generalized cut function of degree p+1 by smooth sigmoid functions, serdica j. computing 9(1) (2015) 101–112. [21] n. kyurkchiev, s. markov, sigmoid functions: some approximation and modelling aspects, lap lambert academic publishing, saarbrucken (2015), isbn 978-3-659-76045-7. [22] v. kyurkchiev, n. kyurkchiev, on the approximation of the step function by raised-cosine and laplace cumulative distribution functions, european international journal of science and technology 4(9) (2015) 75–84. [23] n. kyurkchiev, s. markov, a. iliev, a note on the schnute growth model, int. j. of engineering research and development 12(6) (2016) 47–54. [24] n. kyurkchiev, a. iliev, a note on some growth curves arising from box-cox transformation, int. j. of engineering works 3(6) (2016) 47–51. [25] n. kyurkchiev, a note on the new geometric representation for the parameters in the fibril elongation process, compt. rend. acad. bulg. sci. 69(8) (2016) 963–972. [26] d. costarelli, g. vinti, pointwise and uniform approximation by multivariate neural network operators of the max-product type, neural networks (2016), doi:10.1016/j.neunet.2016.06.002 [27] iliev, a., n. kyurkchiev, s. markov, approximation of the cut function by stannard and richards sigmoid functions, ijpam 109(1) 2016 119–128. biomath 5 (2016), 1609041, http://dx.doi.org/10.11145/j.biomath.2016.09.041 page 6 of 6 http://dx.doi.org/10.11145/j.biomath.2016.09.041 introduction and preliminaries main results conclusion remarks references communication/review biomath 2 (2013), 1312312, 1–5 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum 130th anniversary of prof. dr. vladimir markov prof. ivan mitov, md, phd, dept. medical microbiology, medical faculty medical university of sofia prof. dr. vladimir markov this year marks 130 years since the birth of the founder and leader (untill 1958) of the department of microbiology at the medical university sofia in bulgaria, the eminent and internationally recognized scholar prof. dr. vladimir nestorov markov, full member of the bulgarian academy of sciences (academician). a contemporary with the liberation from ottoman rule, vladimir markov was assigned by fate to become the father of microbiology in bulgaria and creator of the national microbiological school. he elevated national microbiology to world-class level at that time, leaving a profound legacy in the development of medicine in bulgaria. born in the city of veliko tarnovo, vladimir markov was brought up in a family of patriotic bulgarians. his father nestor markov was an active participant in the struggle for spiritual and political liberation. his brother marko markov was a member of the detachment of yane sandanski and was killed in a battle of the liberation of macedonia. vladimir markov completed his secondary education in the state agricultural school “obraztzov chiflik” in rousse. in 1904, he went to germany, where he graduated in veterinary medicine in munich and berlin. upon his return to bulgaria, vladimir markov embarked on a career as a veterinarian doctor in panagiurishte and oriahovo. in lieu of his excellent training in veterinary medicine and by using bacteriological methods of investigation, he discovered the cause of infectious pneumonia in calves in the kabuik horse farm near shumen and provided science-based treatment for successful eradication of that epizootic. this investigative discovery brought to the forefront his special interest and specific talent for bacteriology as well as his ability to illuminate a purely practical question. this led to him devoting his first scientific work in 1909 to this subject. vladimir markov’s qualities were noticed by his colleagues and his superiors and in the autumn of 1909, after successful competitive examination, he was sent to specialize in bacteriology and serology in munich and berlin. he worked in various research institutes including the institute for infectious diseases “robert koch”. vladimir markov received excellent microbiological training by prominent scholars such as t. kit, p. ulenhut, a. wasserman, w. loeffler and others. in 1911 he was promoted to a doctor of veterinary medicine at citation: ivan mitov, 130th anniversary of prof. dr. vladimir markov, biomath 2 (2013), 1312312, http://dx.doi.org/10.11145/j.biomath.2013.12.312 page 1 of 5 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2013.12.312 i mitov, 130th anniversary of prof. dr. vladimir markov the university of berlin. after his return from germany, vladimir markov was much sought after as a specialist in microbiology and in the course of six years he consecutively worked in the veterinary bacteriological station and the hygiene institute of sofia. during the balkan wars and the first world war he held key positions on various fronts as bacteriologist and set-up new laboratories and work stations. he examined the etiology of some vague but typical human and animal diseases and successfully battled cholera, dysentery, typhus and other diseases spread among the troops and the civilian population. he gave scientific significance to his practical experience with a number of publications recognized in the scientific literature both abroad and in bulgaria, which defined him as an established researcher in the late twenties. on these grounds, after the establishment of the medical faculty of sofia, vladimir markov was invited to be a chief assistant at first, and shortly after, in the middle of 1921 was elected private associate professor and head of the department of bacteriology and serology. he remained in this position for over four decades. vladimir markov began his teaching activities with an interesting introductory lecture and discourse on “agglutination as colloidal reaction” in october 1921. among others he lectured and organized practical training in microbiology for medical students, agronomists, chemists, pharmacists and biologists till the end of his life. he developed and delivered the first lecture course in technical microbiology in the new found state polytechnic school. all specialists in microbiology were trained and developed under his leadership. his lectures were full of content and real-life examples and together with his rich experience and originality, they aroused great interest. looking very harsh and inaccessible on the outside, vladimir markov loved his students very much. he engaged with them on many levels and he had discussions with them on various professional, current and political issues. he sympathized with the progressive studentship and cultivated in his students the love for microbiology and science in general, and the selfless and enthusiastic service to the people. vladimir markov’s students remember him encouraging them to participate in the student’s movements in 1932 against the antidemocratic government. vladimir markov’s great love and dedication to the preparation of students consisted also of his constant efforts to provide them with textbooks and training aids. vladimir markov wrote more than ten original textbooks, manuals and other materials on microbiology for students of various disciplines at sofia university. vladimir markov established and maintained an original style, method and traditions in the teaching of microbiology. he actively contributed to the creation of new departments of microbiology at sofia university and other universities that were opened in the years after world war ii, as well as in the sanitary and epidemiological stations and the scientific-research institutes. naturally those institutions became followers of his work, developing it further under the new conditions after the war. as a scientist with an alert public awareness, vladimir markov was closely linked to prominent public figures. his closest friends were prof. as. zlatarov, prof. as. hadzhiolov, academician georgi uzunov, dr. racho angelov, who were brought together by long-term collaborative work in professional and social areas. “microbes in the service of life”—that was the motto of vladimir markov, left to his followers. this is one of the factors that can explain why till the end of his days vladimir markov associated his entire scientific and research work with the needs of society. he and his department were the team that developed and organized the microbiological thought and microbiologists all over the country, and in 1923 vladimir markov initiated the establishment of bulgarian microbiological society, which he chaired till the end of his life in 1962. there is hardly any field in microbiology to which vladimir markov did not contribute scientifically. the scientific heritage left by vladimir markov is outstanding. for 53 years, he worked dedicatedly, strenuously and enthusiastically, with the expertise of a true scientistexperimentalist-bacteriologist. he wrote more than 180 works devoted to general microbiology, infectiology, infection and immunity, serology, special microbiology, industrial and soil microbiology, food microbiology, etc. he was highly appreciated and quoted as early as the 1920s in many german essays, textbooks and magazines and was placed on a par with all most prominent german researchers. vladimir markov’s achievements on the problems of variability of microorganisms should be noted among his most important scientific developments. regarding the discussions that occurred at the beginning of the century on the problems of variability, vladimir markov in 1922, for the first time in science, founded and described the dissociation in the anthrax bacillus later biomath 2 (2013), 1312312, http://dx.doi.org/10.11145/j.biomath.2013.12.312 page 2 of 5 http://dx.doi.org/10.11145/j.biomath.2013.12.312 i mitov, 130th anniversary of prof. dr. vladimir markov called 8 ik forms of bacteria, and proved the possibility of existence of naturally attenuated forms, named by him “vaccine in nature”. this discovery has not lost its theoretical and practical significance to the present day. he explained experimentally the phenomena paraagglutination as a result of mixed infection in which bacteria of the saprophytic intestinal flora acquire some properties of the pathogenic intestinal microbes. the formulations of his experiments on para-agglutination were adopted and then used for decades in science and references. these studies of vladimir markov significantly contributed to the explanation of the origin of the so-called atypical forms of bacteria which is very important for microbiological diagnosis. vladimir markov defended his science-based views on the problems of variability with the dignity of a scholar. even in the politically laden period of 1948-1950 he rejected the theory of some politically supported pseudo-scientists about the possible conversion of microbial species from one into another, which was attributed by the ruling party as his great sin. vladimir markov devoted a lot of effort to the study of etiology and epidemiology of many bacterial, viral and parasitic diseases in humans and domestic animals. he identified the cause agents of typhoid borreha recurrence, on piroplasmosis in domestic animals, found in bulgaria for the first time in 1912 as well as the cause agents of typhus, anaerobic infections, etc. he discovered the non-described, till then anaerobic bacillus, and gave it the name of bacillus anaerobus hemoliticus. we should also recall the incredibly active research and practical work of vladimir markov in the fight against intestinal infections. he studied the large epidemic of typhoid fever in sofia (1932), kyustendil (1938–1939), a number of toxico-infectious outbreaks, most important among which were the poisoning with foods, bonito fish, cheese, the dysentery epidemic in svishtov, diseases of glanders, anthrax, rhinoscleroma and others. vladimir markov was the first in our country back in 1954 to point out the task and manage extensive research on proving the presence of enteropathogenic bacteria coli in our bodies and its importance for pathology. in his studies on the biology of both pathogenic and beneficial microbes to man, vladimir markov created a number of more comprehensive elective nutrient media to facilitate cultivation and microbiological diagnosis. early in his scientific quest vladimir markov was interested in the problems of the process of infection, immunity and serology. in 1911 he was the first to develop cell-free extracts of anti-anthrax precipitating serum and studied the bactericidal action of normal serum on bacteria. his theory concerning the antibodies analyzed in his work “antibodies and sub-antibodies” published in 1928, as well as his works on obtaining specific sera against streptococcus, diphtheria bacteria and others were valued as highly original. in addition, the combined active and passive immunization against rabies was a method developed by vladimir markov and remains of great importance even today for the successful control of this extremely dangerous infection. the biochemical studies of vladimir markov related to the natural cycle of substances and the role of microorganisms contributed theoretically to the studies of oceanography, balneology and toxicology. such are his publications devoted to biochemical microbiological processes in the formation of the therapeutic mud in the cities of pomorie and tuzlata as well. within the context of the industrial microbiology vladimir markov also contributed greatly to development in bulgaria. his scientific research helped support industrial production and was used in the making of valuable manufactured goods, food, beverages, etc. his studies from 1923 on the optimization of the wine producing process by introducing pure cultures of selected yeasts became very popular. he provided those to the interested wine-makers for free till the creation of specialized institutes and laboratories in 1950. based on his studies, vladimir markov gave the breadmaking industry a pure culture of bread yeast and his own original method for detecting tampering with hop yeast bread. this helped improve the quality of bread and facilitated workers in the bread making process. during the hungry years of the second world war, his research into mold bread resulted in the preparation of bread yeast of lactic acid bacteria for the production of potato bread. he conducted extensive research on the problem of nitrogen fixation and on the use of organic fertilizers with his chemical “radiksoya”—a problem that has not lost its importance today. he carried out comprehensive and in-depth studies on “bacterial flora in bulgarian yogurt, plain bulgarian cheese, sheep’s milk and rennet”. asked for assistance by soap manufacturers whose export shipments were returned as unfit, vladimir markov explored the causes, described two new types of bacteria, called serratia saponaria i and ii, studied their biochemical biomath 2 (2013), 1312312, http://dx.doi.org/10.11145/j.biomath.2013.12.312 page 3 of 5 http://dx.doi.org/10.11145/j.biomath.2013.12.312 i mitov, 130th anniversary of prof. dr. vladimir markov characteristics and proposed measures to prevent rancidity of the soap. the issues of antibiotics interested vladimir markov in many aspects and his theoretical and practical work on “volatile antibiotics”, affecting bacteria from a distance, provoked great interest as well. he and prof. sv. bardarov together examined the resistance to penicillin, developed and proposed the neutralization of the enzyme and recovery of the efficiency of the antibiotic during treatment with anti-penicillinase serum. according to his understanding of the infectious process in the fifties, vladimir markov developed the hypothesis for the viral genesis of tumors, whereby the virus and the cell produced organically bound complex which is foreign to the body. moreover, the emergence of this complex was determined by a number of exogenous and endogenous factors, a pattern characterizing the infectious process. in his promotional activities he managed different levels of involvement. vladimir markov pointed out the great importance of the control of infectious diseases, of women and their role in public life, of alcoholism and the fight against it. a great patriot and admirer of bulgaria, he described its beauty and passionately promoted tourism and sports. the ultimate goal of his activity was to help people to “master the art of living” and devoted more than 250 works to these causes. vladimir markov, the great scientist and doctor was also a great fighter for peace. as a scientist, microbiologist and a public figure he was not afraid to protest angrily against the preparation of bacteriological warfare in the mid-thirties. in a brochure issued by him, he condemned the supporters and scientists who were developing those weapons of mass extermination. academician vladimir markov led the battle against the bacteriological weapons till the end of his life. his creed on the problems of life, peace and war he expressed as follows: “progress and culture are created only in an environment of peace”. this is a brief summary of the creative scientific activity of vladimir markov. what he had mastered and created in science, he readily and with dedication shared in the press, in magazines, in the media, in lectures and discussions at a variety of forums. he also contributed to various bulgarian and foreign magazines. he was an editor-in-chief in some of them. in the years after world war ii, vladimir markov’s qualities as a skilled scientist and academic supervisor were well founded and he established the national school of medical microbiology, which even today enjoys recognition. vladimir markov’s scientific achievements were recognized when he was elected a full member of the academy of sciences (academician) and doctor honoris causa of the faculty of medicine at the university of sofia. he was also awarded the prestigious “dimitrov prize”, the title of “distinguished doctor”, the medals “red flag”, “people’s republic of bulgaria grade ii”. his life and work at the department of microbiology for the medical faculty of the university of sofia and as founder of microbiology in bulgaria, academician prof. dr. vladimir nestorov markov remains an epic model and example in the history of medicine in bulgaria. selected references1 markov, wl., studien ueber die variability der bakterien. zugleich ein beitrag zur morphologie und biologie des milzbrand-bazillus, ztschr. f. infektionskrankh., parasil krankh. u. hyg. d. haustiere, bd. 12, 1912, 137– 158. bernhardt, g., w. markoff., ueber modifikationen bei bakterien. beitrag zur frage der sogenannten “mutation” bei bakterien. ztrbl. f. bakteriol. etc., i abt., orig., bd. 65, 1912, 1–4. markov, wl., experimentelle studien ueber das wesen der para-agglutination, ztrbl. f. bakteriol. etc., i abt., orig.bd. 75, 1916, 372–383. gildenmeister, e., markoff, wladimir n. experimentelle studien ueber das wesen der panagglutination. i. mitteilung. (c. f. bakt., i abt., orig., bd. 78, 1916, p. 372), ztrbl. f. bakteriol. etc., i abt., ref., bd. 66, 1918, h. 19, p. 489. markov, wl., spezifische und subspezifische antikorper. wirkung von bakteriophagen auf bakterielle antigene und kompleentbindungsversuche, ztschr. f. immunforsch., bd. 56, 1928, 95–106. markov, wl., z. k. jatschewa. eine todliche episootie unter den forellen im mussalahsee, ztrbl. f. bakteriol. etc., ii abt. orig., bd. 100, 1939, 194–201. markov, wl., zum problem der seefischfaulnis, ztrbl. f. bakteriol. etc., ii abt, orig, bd. 101, 1939, 151–171. gildemeister, e., markoff, wl. n., zum problem der seefischfaulnis. (ztrbl. f. bakter., 11, orig, 101, 151171, 1939), ztrbl. f. bakteriol. etc., i abt., ref., bd. 137, 1940, 1/2, p. 44. 1for a full bibliography see: markov, k., v. valchanov, vladimir markov—bibliography, in: bibliography of bulgarian scientists, bulgarian academy of sciences publ. house, 1960, p. 201. biomath 2 (2013), 1312312, http://dx.doi.org/10.11145/j.biomath.2013.12.312 page 4 of 5 http://dx.doi.org/10.11145/j.biomath.2013.12.312 i mitov, 130th anniversary of prof. dr. vladimir markov gaethgens, w., markoff, wl. n., erfahrungen mit dem behringschen diphtherieheilserum in bulgarien. (z. immunforsch., 99, 27-39, 1940), ztrbl. f. bakteriol. etc., i abt., ref., bd. 140, 1941, 1/2, p. 12. markov, wl., die menge der antitoxischen einheiten (ae) bei der behandlung der toxischen diphtherie, wiener med. wschr., bd. 91, 1941, 14, 279–280. markoff, w. n., die menge der antitoxischen einheiten (ae) bei der behandlung der toxischen diphtherie. (wien. med. wschr., 1941, 279–281), ztrbl. f. bakteriol. etc., i abt., ref., bd. 140, 1941, 21/22, p. 445. markov, wl., die verbeugung der garungsdyspepsie, wiener med. wschr., bd. 92, 1942, 20, p. 1–2. markov, w., the normal intestinal flora in man. annales medicales, revue de l’union des medicins bulgares, sofia, 1948,40/10 (1174–1180). excerpta medica, sect. iv, medical microbiology and hvgiene, vol. ii, 1949, 10, abstr. 4947. markov, wl., serratia saponaria i und ii. schadliche bakterien in der seife, ztrabl. f. bakteriol. etc., i abt., orig. bd. 110, 1956, 26–31. markov, wl., g. mitov, g. saev. die eigenschaften der biogenen substituenten der fluchtigen antibiotika und ihre verbreitung. zentralblatt fur baktertologie, parasitenkunde, infektlonskrankheiten und hygiene, i orig. 173. 129–140 (1958). biomath 2 (2013), 1312312, http://dx.doi.org/10.11145/j.biomath.2013.12.312 page 5 of 5 http://dx.doi.org/10.11145/j.biomath.2013.12.312 original article biomath 1 (2012), 1210043, 1–6 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum on the computation of output bounds for compartmental in-series models under parametric uncertainty diego de pereda∗, sergio romero-vivo†, beatriz ricarte† and jorge bondia∗ ∗institut universitari d’automàtica i informàtica industrial universitat politècnica de valència, spain emails: dpereda@upvnet.upv.es, jbondia@isa.upv.es †institut universitari de matemàtica multidisciplinar universitat politècnica de valència, spain emails: sromero@imm.upv.es, bearibe@imm.upv.es received: 15 july 2012, accepted: 4 october 2012, published: 21 december 2012 abstract—in this work, the problem of obtaining tight output bounds for compartmental in-series models under parametric uncertainty is addressed. it is well-known that current methods used to compute a solution envelope may produce a significant overestimation. however, monotonicity analysis enables us to estimate a tight solution envelope. our main aim is to get an equivalent model to the initial one, which is usually non-monotone, by means of a suitable combination of equations. in this new model the system monotonicity with respect to the uncertain parameters depends on the elimination rate values of the original model. if the equivalent model is monotone, no overestimation occurs in the computation of the output bounds. keywords-uncertainty; parametric uncertainty; compartmental systems; interval simulation; monotonicity i. introduction mathematical models have appeared in many different real situations emerging from biology, economics, engineering, medicine, human sciences and many other research fields. the most common mathematical models used to mimic real processes are compartmental systems, in which each compartment represents a state of the system. however, as a mathematical model is usually a simplified version of a real process, a mismatch between the behaviour of the model and the reality is produced. this mismatch yields non-modelled dynamics. moreover, this kind of processes is also characterized by its variability, leading to parametric uncertainty. therefore, the exact values of the model parameters are unknown, but they can be bounded by intervals. while there is only one possible behaviour for a model with constant parameters, parametric uncertainty produces a large set of different possible solutions. traditionally, monte carlo methods have been used to deal with uncertainty [1], owing to the fact that a large number of solutions can be easily computed. however, independently of the number of simulations executed, the output bounds obtained cannot ensure the inclusion of all the possible solutions [2]. this inclusion guarantee is needed for error-bounded parametric identification and constraint-satisfaction problems. in the former, the range (or a tight enclosure) of the output trajectory must be computed and compared with measurements to estimate intervals for the model parameters guaranteeing data consistency. in the latter, the computed range must be compared with the constraints to be satisfied so as to obtain an inner and outer approximation of the output set for the decision variables. furthermore, the citation: d. de pereda, s. romero-vivo, j. bondia, on the computation of output bounds for compartmental in-series models under parametric uncertainty, biomath 1 (2012), 1210043, http://dx.doi.org/10.11145/j.biomath.2012.10.043 page 1 of 6 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.10.043 d. de pereda et al., on the computation of output bounds for compartmental in-series models... computational cost of monte carlo methods increases proportionally to the number of simulations performed to cover the uncertain input space sufficiently. for these reasons, other methods have been considered to compute output bounds, such as region-based and trajectory-based approaches [3] mainly founded on interval analysis [4] and monotone systems theory [5], [6]. the aim of this work is to compute output bounds for compartmental in-series models with parametric uncertainty. that is, a tight solution envelope must be computed to ensure the inclusion of all the possible solutions for the model as well as to minimise the overestimation. otherwise, if the overestimation is high, it could not be useful from a practical point of view, for instance, in an insulin therapy for diabetes patients [7]. this work has been organised as follows: in section 2, a monotonicity analysis approach is introduced. in section 3, compartmental in-series models are presented. in section 4, a new method is proposed for the analysis of the system monotonicity with respect to the parameters. in section 5, the proposed method is applied to compute the output bounds of a linear glucose model. finally, section 6 outlines the conclusions of this study. ii. uncertain systems continuous-time systems under parametric uncertainty are described by an initial-value problem (ivp): ẋ(t, p) = f (x, p), x(t0) = x0, x ∈ rn, t ∈ r, p ∈ rnp (1) where f is the vector function with components fi, x is the state vector, p is the parameter vector, and np is the number of parameters. the solution of (1) is denoted by x(t; t0, x0, p). we consider that the parameters and the initial conditions are unknown, but they can be bounded by intervals. representing intervals in bold, interval vectors p and x0 include all the possible values for the parameters p and for the initial conditions x0 of the model, respectively. the set of possible solutions considering parametric uncertainty is denoted by x(t; t0, x0, p): x(t; t0, x0, p) = {x(t; t0, x0, p) | x0 ∈ x0, p ∈ p}. the computation of solution envelopes plays a key role in the simulation of systems under parametric uncertainty. such a computation can be performed by using one-step-ahead iteration based on previous approximations of a set of point-wise trajectories generated by the selection of particular values of the parameters p ∈ p and initial conditions x0 ∈ x0 by using heuristics such as a monotonicity analysis of the system [8]. monotone systems have very robust dynamical characteristics, since they respond to perturbations in a predictable way. the interconnection of monotone systems may be studied in an analytical way [9], by considering a flow x(t) = φ(x0, t). a system is monotone if x0 � y0 ⇒ φ(x0, t) � φ(y0, t) for all t ≥ 0, where � is a given order relation. cooperative systems form a class of monotone dynamical systems [5] in which ∂fi ∂xj ≥ 0, for all i 6= j, t ≥ 0. in order to calculate a solution envelope, an upper bounding model and a lower bounding model are computed. in an upper bounding model, the cooperative states with respect to the output take their upper bound value, while the monotone but non-cooperative states, known as competitive states, take the value of their lower bound. on the other hand, a lower bounding model is obtained taking account of the lower bound of the cooperative states, and the upper bound of the competitive states. in both cases, the non-monotone states are still computed as intervals that produce a significant overestimation. the model parameters are considered as invariant states to carry out the monotonicity analysis, where ẋ1(t) = f1(t, x1(t), x2(t), ..., xn(t), p1(t), p2(t), ...) ... ẋn(t) = fn(t, x1(t), x2(t), ..., xn(t), p1(t), p2(t), ...) ṗi(t) = 0 iii. compartmental in-series models it is well known that a compartmental system consists of a finite number of interconnected subsystems called compartments. the interactions among compartments are transfers of material according to the law of conservation of mass. these are natural models useful for many application areas subject to that law, which appear in physiology, chemistry, medicine, epidemiology, ecology, pharmacokinetics and economy [10]. the state variables biomath 1 (2012), 1210043, http://dx.doi.org/10.11145/j.biomath.2012.10.043 page 2 of 6 http://dx.doi.org/10.11145/j.biomath.2012.10.043 d. de pereda et al., on the computation of output bounds for compartmental in-series models... of these systems represent the amount of material contained in each compartment and then they are restricted to be non-negative over time; that is, they belong to the broader class of positive systems. a general in-series model composed of n compartments is represented in figure 1. q 1 k1,2 q 2 e1 q n k2,1( k2,3( k3,2( kn-1,n( kn,n-1( e2 en u un fig. 1. diagram of a compartmental in-series model. an in-series model is named bidirectional if the fluxes between the compartments go forward and backward. however, if the fluxes just go forward, the in-series system is called unidirectional. bidirectional in-series models are given by the equations: q̇1(t) = u(t) − (k1,2(·) + e1)q1(t) +k2,1(·)q2(t) q̇i(t) = ki−1,i(·)qi−1(t) + ki+1,i(·)qi+1(t) −(ki,i−1(·) + ki,i+1(·) + ei)qi(t) q̇n(t) = un(t) + kn−1,n(·)qn−1(t) −(kn,n−1(·) + en)qn(t) q1(0) = q10 , qi(0) = qi0 qn(0) = qn0 (2) for i ∈ {2, ..., n − 1}, where the states of the model qj (t), j ∈ {1, ..., n}, are the in-series compartments, and qn(t) is the output of the model. furthermore, u(t) and un(t) represent the inputs and the parameters ej , j ∈ {1, ..., n}, are the elimination rates for each compartment, while ki,i+1(·) and ki+1,i(·), i ∈ {1, ..., n−1}, are non-negative scalar functions that represent the flux from the compartment i to the compartment j and they may depend on the states of the model, i.e., ki,j (·) = ki,j (q1(t), . . . , qn(t)) ≥ 0. iv. analysis of the system monotonicity in this section, we analyse compartmental in-series models by focusing on the monotonicity of the dynamical system with respect to the states and the parameters. the general system described by equations (2) can be non-monotone with respect to the states since it is not possible to determine the exact sign of the partial derivatives ∂q̇i(t) ∂qj (t) , i, j ∈ {1, ..., n}, i 6= j. notice that, for instance, the sign of the following partial equation cannot be determined: ∂q̇1(t) ∂q2(t) = − ∂k1,2(·) ∂q2(t) q1(t) + k2,1(·) + ∂k2,1(·) ∂q2(t) q2(t) therefore the monotonicity analysis cannot be accomplished with respect to the states and the parameters of the model. nevertheless, as we are focused on the output of the model, this fact can be avoided by the transformation of this system into an equivalent system having the same output, given by: ṡ1(t) = u(t) + un(t) − ∑n−1 j=1 ej (sj (t) − sj+1(t)) −ensn(t) ṡi(t) = un(t) + ki−1,i(·)(si−1(t) − si(t)) −ki,i−1(·)(si(t) − si+1(t)) − ∑n−1 j=i ej (sj (t) − sj+1(t)) − ensn(t) ṡn(t) = un(t) + kn−1,n(·)(sn−1(t) − sn(t)) −(kn,n−1(·) + en)sn(t) (3) for i ∈ {2, ..., n − 1}, where si = ∑n j=i qj (t), ∀i ∈ {1, ..., n}. it is worth mentioning that all the fluxes ki,j in this new system may depend on the new states si, such that ki,j (·) = ki,j (s2(t) − s1(t), . . . , si+1(t) − si(t), . . . , sn(t)) note that this equivalent system has been obtained by the combination of equations (2) and the output compartment is not modified, because sn(t) = qn(t), which enable us to compute the output bounds of original system (2) by means of the conclusions obtained by the monotonicity analysis of new equivalent system (3) for i ∈ {2, ..., n − 1}: ∂ṡ1(t) ∂sj (t) = ej−1 − ej (2 ≤ j ≤ n), ∂ṡi(t) ∂sj (t) = ∂ki−1,i(·) ∂sj (t) (si−1(t) − si(t)) −∂ki,i−1(·) ∂sj (t) (si(t) − si+1(t)) (j < i − 1), biomath 1 (2012), 1210043, http://dx.doi.org/10.11145/j.biomath.2012.10.043 page 3 of 6 http://dx.doi.org/10.11145/j.biomath.2012.10.043 d. de pereda et al., on the computation of output bounds for compartmental in-series models... ∂ṡi(t) ∂si−1(t) = ∂ki−1,i(·) ∂si−1(t) (si−1(t) − si(t)) −∂ki,i−1(·) ∂si−1(t) (si(t) − si+1(t)) + ki−1,i(·), ∂ṡi(t) ∂si+1(t) = ∂ki−1,i(·) ∂si+1(t) (si−1(t) − si(t)) −∂ki,i−1(·) ∂si+1(t) (si(t) − si+1(t)) + ki,i−1(·) + (ei − ei+1), ∂ṡi(t) ∂sj (t) = ∂ki−1,i(·) ∂sj (t) (si−1(t) − si(t)) −∂ki,i−1(·) ∂sj (t) (si(t) − si+1(t)) + (ej−1 − ej ) (j > i + 1), ∂ṡn(t) ∂sj (t) = ∂kn−1,n(·) ∂sj (t) (sn−1(t) − sn(t)) −∂kn,n−1(·) ∂sj (t) sn(t) (j < n − 1), ∂ṡn(t) ∂sn−1(t) = ∂kn−1,n(·) ∂sn−1(t) (sn−1(t) − sn(t)) −∂kn,n−1(·) ∂sn−1(t) sn(t) + kn−1,n(·) under the conditions ∂ki,i+1(·) ∂sj ≥ 0 and ∂ki+1,i(·) ∂sj ≤ 0, ∀i, j : i ∈ {1, ..., n − 1}, j ∈ {1, ..., n}, we remark that the compartments of model (3) are cooperative among each other if ej ≥ ej+1, ∀j ∈ {1, ..., n − 1}, since the partial derivative equations ∂ṡi(t) ∂sj (t) , i, j ∈ {1, ..., n}, i 6= j are always non-negative. furthermore, in this same case, the inputs u(t) and un(t), and the functions kj,j+1(·), j ∈ {1, ..., n−1} are also cooperative, while the elimination rates ej , j ∈ {1, ..., n} and the functions kj+1,j (·), j ∈ {1, ..., n − 1} are competitive. but, note that as qn(t) = sn(t) and qi(t) = si(t) − si+1(t), i ∈ {1, ..., n − 1}, then: ∂k(·) ∂s1 = n∑ s=1 ∂k(·) ∂qs ∂qs ∂s1 = ∂k(·) ∂q1 ∂k(·) ∂sj = n∑ s=1 ∂k(·) ∂qs ∂qs ∂sj = ∂k(·) ∂qj − ∂k(·) ∂qj−1 (j ∈ {2, ..., n}) where k(·) represents both ki,i+1(·) and ki+1,i(·), i ∈ {1, ..., n − 1}. thus, we can sum up these relations between both kinds of systems in the following lemma: lemma iv.1. consider a bidirectional in-series model (2) that satisfies: (a) the elimination rate for each compartment is greater than or equal to the elimination rate for the next compartment, i.e. ej ≥ ej+1, ∀j ∈ {1, ..., n − 1}. (b) the forward fluxes among the compartments satisfy that ∂ki,i+1(·) ∂qj − ∂ki,i+1(·) ∂qj−1 ≥ 0, whereas the backward fluxes satisfy that ∂ki+1,i(·) ∂qj − ∂ki+1,i(·) ∂qj−1 ≤ 0, ∀i, j : i ∈ {1, ..., n − 1}, j ∈ {2, ..., n}, where ∂ki,i+1(·) ∂q0 = ∂ki+1,i(·) ∂q0 = 0. then, there is an equivalent model (3) that satisfies the following properties: (i) the equivalent system is cooperative with respect to the states si, i ∈ {1, ..., n}, the inputs u(t) and un(t), and the fluxes kj,j+1(·), j ∈ {1, ..., n − 1}. (ii) the equivalent system is competitive with respect to the elimination rates ej , j ∈ {1, ..., n}, and the fluxes kj+1,j (·), j ∈ {1, ..., n − 1}. v. linear glucose model in the sequel, we illustrate the result presented in the previous section through a linear glucose example. namely, we turn the non-monotone system describing the cobelli et al. model [11] into an equivalent monotone system, in which output bounds are easily computed without overestimation. plasma glucose concentration in blood is maintained within a narrow range with the help of the insulin hormone. insulin is secreted by the pancreas with the role of reducing glucose concentration in blood. under normal circumstances, a decrease in plasma glucose concentration is followed by a decrease in insulin secretion. on the other hand, insulin secretion increases when plasma glucose concentration increases, for instance after an ingestion. q 2 k2,1 k2,0 k1,2 k3,0 q 3 q 1 k3,1 k1,3 egp fig. 2. diagram of the linear glucose model developed by cobelli et al. the analysis of glucose kinetics is essential to analyse the insulin secretion by the pancreas. cobelli et al. developed a physiological model to study the insulin system and the control exerted by glucose on insulin secretion. this compartmental model describes the nonaccessible portion of the system, and it is composed of three compartments, as seen in figure 2. the output compartment is the concentration of the accessible pool, biomath 1 (2012), 1210043, http://dx.doi.org/10.11145/j.biomath.2012.10.043 page 4 of 6 http://dx.doi.org/10.11145/j.biomath.2012.10.043 d. de pereda et al., on the computation of output bounds for compartmental in-series models... displayed in the central position. the mass balance and measurement equations are given by: q̇1(t) = −(k1,2 + k1,3)q1(t) + k2,1q2(t) +k3,1q3(t) + egp q̇2(t) = k1,2q1(t) − (k2,1 + k2,0)q2(t) q̇3(t) = k1,3q1(t) − (k3,1 + k3,0)q3(t) g(t) = q1(t) vi (4) where q1(t) is the accessible pool of the plasma glucose, q2(t) and q3(t) illustrate peripheral compartments, respectively, in rapid and slow equilibrium with the accessible pool, and the output of the model is given by the plasma glucose concentration g(t), which depends on the central compartment q1(t). the parameter vi is the volume of plasma in the accessible compartment, the parameter egp denotes the input, the constant parameters k1,2, k1,3, k2,1 and k3,1 are the fluxes between the compartments, while the parameters k2,0 and k3,0 stand for the elimination rates of the peripheral compartments. in this model there is no elimination rate in the accessible pool. the parameters values have been obtained from [12]. performing a monotonicity analysis, it is deduced that the system is cooperative with respect to the compartments, as the partial derivatives among the model compartments are all non-negative. furthermore, the input egp is also a cooperative parameter, while vi , and the elimination rates k2,0 and k3,0 are competitive parameters. the monotonicity evaluation with respect to the fluxes k1,2, k1,3, k2,1 or k3,1 is not possible. cobelli et al. model (4) can be analysed as two compartmental in-series models interconnected, where the central compartment is the output of both in-series models. this system is equivalent to ṡ1(t) = −(k1,2 + k1,3)s1(t) + k2,1(s2(t) − s1(t)) +k3,1(s3(t) − s1(t)) + egp ṡ2(t) = −k1,3s1(t) − k2,0(s2(t) − s1(t)) +k3,1(s3(t) − s1(t)) + egp ṡ3(t) = −k1,2s1(t) − k3,0(s3(t) − s1(t)) +k2,1(s2(t) − s1(t)) + egp g(t) = s1(t) vi (5) where s1 = q1, s2 = q1 + q2 and s3 = q1 + q3. as ki,i+1(·) and ki+1,i(·), i ∈ {1, ..., n − 1}, are constant parameters then ∂ki,i+1 ∂sj = 0 and ∂ki+1,i ∂sj = 0. moreover, as there is no loss in the output compartment and k2,0 ≥ 0 and k3,0 ≥ 0, then the lemma iv.1 conditions hold. thus, equivalent system (5) is cooperative with respect to the parameters k2,1 and k3,1, the initial conditions and the input egp . furthermore, the equivalent system is competitive with respect to the parameters k1,2 and k1,3, the elimination rates k2,0 and k3,0, and the volume vi . 0 5 10 15 20 50 100 150 200 250 300 tim e [m inutes] g lu c o s e [ m g /d l ] (a) 0 5 10 15 20 80 100 120 140 160 tim e [m inutes] g lu c o s e [ m g /d l ] (b) fig. 3. output bounds for the linear glucose model developed by cobelli et al. (a) monotonicity approach. (b) using lemma iv.1. the black dashed lines in figure 3 display the computed output bounds, while the light grey lines represent several numerical simulations executed by the variation of the parameters and initial conditions values. first of all, the computation of output bounds is performed following the traditional monotonicity approach, where the parameters k1,2, k1,3, k2,1 and k3,1 have to be considered as intervals. the solution envelope in figure 3a illustrates the overestimation produced in this case. on the other hand, when lemma iv.1 is applied the system is monotone with respect to all the states and parameters, thus none of them have to be considered as intervals. therefore, it is possible to compute the output bounds without overestimation, as shown in figure 3b. biomath 1 (2012), 1210043, http://dx.doi.org/10.11145/j.biomath.2012.10.043 page 5 of 6 http://dx.doi.org/10.11145/j.biomath.2012.10.043 d. de pereda et al., on the computation of output bounds for compartmental in-series models... vi. conclusion different approaches in the literature have tackled the problem of parametric uncertainty for ordinary differential equations. in this work, a new method for the computation of output bounds on the compartmental in-series models is proposed. this method has been compared with previous approaches in a linear glucose model. the most common method in the literature to compute a tight solution envelope is to perform a monotonicity analysis of the system for a trajectory-based approach. after this method is applied, only non-monotone compartments and parameters produce an overestimation in the computation of output bounds. this happens in compartmental in-series models, as they have non-monotone compartments and parameters. our proposal consists in obtaining an equivalent model by the combination of the differential equations of the original model, but without altering the output compartment. then, by this way, a monotonicity analysis of the equivalent model is performed, obtaining that the new model is monotone with respect to all the compartments and parameters (cooperative or competitive) if the lemma iv.1 conditions meet. thus, this approach allows us to compute a solution envelope adjusted to numerical simulations, in which no overestimation is produced, and computing just two simulations, one for the upper bound and another one for the lower bound. our proposed method outperforms previous approaches for the computation of output bounds on compartmental in-series models, as it computes the solution envelope without overestimation. acknowledgment this work was partially supported by the spanish ministerio de ciencia e innovación through grant dpi-201020764-c02, and by the generalitat valenciana through grant gv/2012/085. references [1] j. hammersley and d. handscomb, monte carlo methods. taylor & francis, 1975. [2] d. de pereda, s. romero-vivo, and j. bondia, “on the computation of output bounds on parallel inputs pharmacokinetic models with parametric uncertainty,” mathematical and computer modelling, 2011. http://dx.doi.org/10.1016/j.mcm.2011.11.031 [3] v. puig, a. stancu, and j. quevedo, “simulation of uncertain dynamic systems described by interval models: a survey,” in 16th ifac world congress, p. 207, 2005. [4] n. nedialkov, vnode-lp: a validated solver for initial value problems in ordinary differential equations. technical report cas-06-06-nn, department of computing and software, mcmaster university, hamilton, ontario, canada, l8s 4k1, 2006. [5] h. smith, monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. ams bookstore, 2008. [6] e. sontag, “monotone and near-monotone biochemical networks,” systems and synthetic biology, vol. 1, no. 2, pp. 59–87, 2007. http://dx.doi.org/10.1007/s11693-007-9005-9 [7] d. de pereda, s. romero-vivo, b. ricarte, and j. bondia, “on the prediction of glucose concentration under intra-patient variability in type 1 diabetes: a monotone systems approach,” computer methods and programs in biomedicine, 2012, doi: 10.1016/j.cmpb.2012.05.012. [8] n. nedialkov, “interval tools for odes and daes,” in scan 2006: 12th gamm-imacs international symposium on scientific computing, computer arithmetic and validated numerics, pp. 4–4, ieee, 2006. [9] m. kieffer and e. walter, “guaranteed nonlinear state estimator for cooperative systems,” numerical algorithms, vol. 37, no. 1, pp. 187–198, 2004. http://dx.doi.org/10.1023/b:numa.0000049466.96588.a6 [10] j. jacquez, compartmental analysis in biology and medicine. biomedware, 1996. [11] c. cobelli, g. toffolo, and e. ferrannini, “a model of glucose kinetics and their control by insulin, compartmental and noncompartmental approaches,” mathematical biosciences, vol. 72, no. 2, pp. 291–315, 1984. http://dx.doi.org/10.1016/0025-5564(84)90114-7 [12] e. carson and c. cobelli, modelling methodology for physiology and medicine. academic press, 2001. biomath 1 (2012), 1210043, http://dx.doi.org/10.11145/j.biomath.2012.10.043 page 6 of 6 http://dx.doi.org/10.1016/j.mcm.2011.11.031 http://dx.doi.org/10.1007/s11693-007-9005-9 http://dx.doi.org/10.1023/b:numa.0000049466.96588.a6 http://dx.doi.org/10.1016/0025-5564 (84)90114-7 http://dx.doi.org/10.11145/j.biomath.2012.10.043 introduction uncertain systems compartmental in-series models analysis of the system monotonicity linear glucose model conclusion references www.biomathforum.org/biomath/index.php/biomath original article identification of hiv dynamic system in the case of incomplete experimental data p. mathye∗, i. fedotov∗ and m. shatalov∗† ∗department of mathematics and statistics tshwane university of technology, pretoria, south africa email: mathyeph@gmail.com; fedotovi@tut.ac.za; shatalovm@tut.ac.za †manufacturing and materials council for scientific and industrial research (csir), pretoria, south africa received: 6 october 2015, accepted: 14 december 2015, published: 17 january 2016 abstract—in this paper we apply an inverse method that estimates parameters of deterministic mathematical models to an hiv model. we consider the case where experimental data concerning the values of some variables is incomplete or unknown. the objective is to estimate the parameters and to restore the information concerning the behaviour of the incomplete data. the method is based on integrating both sides of equations of a dynamic system, and applying some minimization methods (for example least square method). such an approach was first suggested in [7] and [8]. analysis of the hiv model and a corresponding numerical example is presented. keywords-inverse problem, least square methods, parameter estimation, hiv model, incomplete data. i. introduction several hiv vaccine models have been developed over the recent years (e.g., [1] and [4]). these models describe and predict the potential epidemiological impact of vaccination. for the models to give insight into the transmission dynamics of hiv, model parameters are of great significance. the parameters for these model are estimated based on hiv seroprevalence data. most often the data on testing and treatment history is incomplete (missing). this barrier can be as a result of the stigma attached to and the discrimination against people living with hiv and aids. in studying the dynamics of this world pandemic, the availability of recorded valuable data is thus a challenge. the proposed method is based on eliminating the unknown (missing) state variables from the original system algebraically. the resultant system is then used to estimate the unknown model parameters. the discussion on the identification of dynamic systems with incomplete data and restoring the missing data is elementary and can be suitable for students in basic courses on dynamic systems identification. a four dimensional deterministic model for transmission dynamics of hiv in the presence of a preventive vaccine is considered as an example. the model is identified in the cases of incomplete data. it is assumed that the population sizes of individuals infected by the wild type strain and by both the wild and the vaccine strains is unknown. for this study, the data is artificially generated from the given parameters from gumel’s paper [4]. we then ’forget’ about the these parameters and citation: p. mathye, i. fedotov, m. shatalov, identification of hiv dynamic system in the case of incomplete experimental data, biomath 4 (2015), 1512141, http://dx.doi.org/10.11145/j.biomath.2015.12.141 page 1 of 7 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2015.12.141 p. mathye et al., identification of hiv dynamic system in the case of incomplete ... part of the generated data. applying the proposed method we estimate the parameters and then restore the ’forgotten’ data. the performance of the proposed method with real life data remains to be investigated in the future. the remainder of the paper is structured as follows. section ii briefly outlines the formulation of the problem. section iii discusses the inverse method used in solving the problem. section iv gives a description of the mathematical model that is used as an example. sections v, vi and vii discusses the method of solution, parameter estimations and a numerical simulation respectively. finally, concluding remarks are made in section viii. ii. problem formulation consider a system of ordinary differential equations of the form dx dt = f(t,x,y,θ), (1) dy dt = g(t,x,y,θ), (2) where (x,y) is a vector-function [0,t] 3 t → (x,y) ∈ rm × rn subject to experimental information concerning the values of x(tj) at points tj ∈ [0,t], (tj = 0, 1, . . . ,n) are known. now, suppose the information concerning the values y(tj) is either incomplete or unknown. it may be, for example, some statistical data of the form: table 1: experimental data. t0 · · · tj · · · tn x0 = x(t0) · · · xj = x(tj) · · · xn = x(tn ) y0 = ? · · · yj = ? · · · yn = ? the parameter θ is l dimensional, that is θ ∈ a ⊂ rl where a can coincide with rl (no constrains between the entries of θ) and a can be subset of rn (there are constraints). the purpose of this study is to identify the model parameters θ and to restore (recover) the information concerning the behaviour of the incomplete or unknown of y(t). iii. inverse method solve (1) with respect to y to get y = h ( t,x, dx dt ,θ ) . (3) substitute y by h ( t,x, dx dt ,θ ) in (2) to obtain d dt h ( t,x, dx dt ,θ ) = g ( t,x,h ( t,x, dx dt ,θ ) ,θ ) . (4) for this study consider the case where (4) is linear with respect to the coefficients ck(θ): d dt h0 ( t,x, dx dt ) = n∑ k=1 ck(θ)hk ( t,x, dx dt , d2x dt2 ) , (5) where ck : a 3 θ 7→ ck(θ) is a scalar function. the parameter identification for the model is based on the direct integration of the dynamic system with posterior application of a quadrature rule (for example, the adaptive trapezoidal rule). some minimization methods (for example, least squares method (see for example [5])) is then applied to find the estimates. integrating (5) twice with respect to t from t0 to ti, (i = 1, 2, . . . ,n) yields: aα−h = 0 (6) where a = ∫ ti t0 (ti − τ)hk ( τ,x(τ), dx(τ) dτ , d2x(τ) dτ2 ) dτ, α = ck(θ), and h = ∫ ti t0 h0 ( τ,x(τ), dx(τ) dτ ) dτ. the values of the unknown parameters α in (6) can be determined using method of least squares. suppose there are some constraints to be satisfied amongst the parameters α. the problem is thus to minimize the lagrangian l(α,λ) = (aα−h)> (aα−h) + 2λ> (cα−b) , (7) where λ is the lagrange multiplier and cα = b (8) biomath 4 (2015), 1512141, http://dx.doi.org/10.11145/j.biomath.2015.12.141 page 2 of 7 http://dx.doi.org/10.11145/j.biomath.2015.12.141 p. mathye et al., identification of hiv dynamic system in the case of incomplete ... the constraints that must be satisfied. the necessary conditions for a constrained minimum at α∗ is the existence of vector λ* such that ∇αl(α∗,λ∗) = (α∗)>a>a−h>a + (λ∗)>c = 0 (9) and ∇λl(α∗,λ∗) = cα∗ −b = 0. (10) that is, we obtain the system:( a>a c> c 0 )( α∗ λ∗ ) = ( a b ) (11) where the matrix:( a>a c> c 0 ) (12) is a square matrix with a non-zero determinant. the solution to the linear system (11) is given by ( α∗ λ∗ ) = ( a>a c> c 0 )−1 ( a b ) . (13) with the the parameters α = ck(θ) known, the information concerning the behaviour of the incomplete or unknown state variable y(t) can now be restored from (3). iv. mathematical model the model monitors four populations namely: hiv susceptible (x), unvaccinated individuals infected by wild type (yw), individuals uninfected by the wild type but infected by the vaccine strain (yv) and individuals dually-infected with the vaccine and wild strain (yvw). the total (sexual activity) population size is n = n(t) = x + yv + yw + yvw. the model is a modified version of those studied by blower and gumel in [1] and [4] respectively. a. hiv susceptible (x) individuals are recruited, by birth or immigration, into this population at the rate p1 . this population is reduced by the natural cessation of sexual activity at a rate µ, infection with the vaccine strain at the rate αv, the wild strain and infected by the dual infected individuals at the rate αw. in this case it is assumed that the dual infected individuals can only transmit the wild strain. thus, dx dt = p1 −µ1x −αv xyv n −αw x(yw + yvw) n (14) b. individuals uninfected by the wild type but infected by the vaccine strain (yv) this population increases through the new susceptible being vaccinated at the rate p2 and by infection by the vaccine strain at the rate αv. it is decreased by infection with the wild strain and infected by the dual infected individuals at the rate γ. the parameter γ is of the form (1−ψ)cβw where ψ is the degree of protection that the vaccine provides against infection with the wild-type strain, c the number of sexual partners and βw the rate of infection by the wild-type strain. the population is further decreased by natural cessation from sexual activity and by death induced by infection with the wild strain at a rate µ2. dyv dt = p2 + αv xyv n −γ yv(yw + yvw) n −µ2yv, (15) c. unvaccinated individuals infected by wild type (yw) this population increases through the susceptible being infected by the wild strain and by the dual infected individuals at the rate αw. the population is decreased by natural cessation from sexual activity and by death induced by infection with the wild strain at a rate µ3. dyw dt = αw x(yw + yvw) n −µ3yw, (16) d. individuals dually-infected with the vaccine and wild strain (yvw) this population increases through infection with the wild strain and infected by the dual infected individuals at the rate γ. the population is decreased by natural cessation from sexual activity biomath 4 (2015), 1512141, http://dx.doi.org/10.11145/j.biomath.2015.12.141 page 3 of 7 http://dx.doi.org/10.11145/j.biomath.2015.12.141 p. mathye et al., identification of hiv dynamic system in the case of incomplete ... and by death induced by infection with the wild strain at a rate µ4. dyvw dt = γ yv(yw + yvw) n −µ4yvw. (17) the complete model is thus: dx dt = p1 −µ1x −αv xyvn −αw x(yw+yvw) n , dyv dt = p2 + αv xyv n −γyv(yw+yvw) n −µ2yv, dyw dt = αw x(yw + yvw) n −µ3yw, dyvw dt = γ yv(yw + yvw) n −µ4yvw,   (18) adding all the equations of the system (18) gives dn dt = p1+p2−µ1x−µ2yv−µ3yw−µ4yvw (19) all model parameters are nonnegative. v. method of solution suppose that n = n(t),x = x(t),yv = yv(t) are known and that yw and yvw are unknown. from n = x + yv + yw + yvw, (20) it is clear that the sum yw + yvw is also known. in fact, yw + yvw = n −x −yv, (21) let z = yw + yvw. (22) the system with equations (14, 15, 16) and (17) then becomes dx dt = p1 −µ1x −αv xyv n −αw xz n , (23) dyv dt = p2 + αv xyv n −γ yvz n −µ2yv, (24) dyw dt = αw xz n −µ3yw, (25) dyvw dt = γ yvz n −µ4yvw. (26) by using the inverse method the parameters in equations (23) and (24) can be determined. that is, parameters p1,p2,µ1,µ2,αv,αw and γ can now be considered known. adding equations (25) and (26) and rearranging yields µ3yw + µ4yvw = αw xz n + γ yvz n − dz dt . (27) let m = αw xz n + γ yvz n − dz dt , (28) equation (27) then becomes µ3yw + µ4yvw = m. (29) from (22) and (29), form the system yw + yvw = z, (30) µ3yw + µ4yvw = m. (31) solving this system we obtain yw = µ4z −m ∆ (32) and yvw = m −µ3z ∆ (33) where ∆ = µ4 −µ3. (34) differentiating equation (32) yields, dyw dt = µ4 ∆ dz dt − 1 ∆ dm dt (35) equating (25) and (35) yields, µ4 ∆ dz dt − 1 ∆ dm dt = αw xz n −µ3yw (36) simplifying µ4 ∆ dz dt − 1 ∆ dm dt = αw xz n − µ3µ4 ∆ z+ µ3 ∆ m (37) rearranging gives, µ4 ∆ dz dt − 1 ∆ dm dt = αw xz n − µ3µ4 ∆ z+ µ3 ∆ m (38) for convenience, a1 dz dt + a2 dm dt + a3m + a4z + f = 0 (39) biomath 4 (2015), 1512141, http://dx.doi.org/10.11145/j.biomath.2015.12.141 page 4 of 7 http://dx.doi.org/10.11145/j.biomath.2015.12.141 p. mathye et al., identification of hiv dynamic system in the case of incomplete ... where a1 = µ4 ∆ ,a2 = − 1 ∆ ,a3 = − µ3 ∆ , a4 = µ3µ4 ∆ and f = −αw xz n . the relationship between the parameters a1,a2,a3 and a4 is given by the following constrains a1 + a3 = 1, (40) and a1a3 −a2a4 = 0. (41) suppose that the parameters a1,a2,a3 and a4 are obtained, then µ4 = − a1 a2 and µ3 = a1 − 1 a2 . finally, with z, m, µ3 and µ4 known, the unknown variables yw and yvw can be restored from (32) and (33). vi. parameter estimation recall (23), dx dt = p1 −µ1x −αv xyv n −αw xz n , (42) integrating both sides with respect to t from t0 to ti, gives ∆xi = p1ti −µ1ji −αvki −αwpi, (43) where the integrals ∆xi = ∫ ti t0 dx(τ) dt dτ, ki = ∫ ti t0 x(τ)yv(τ) n(τ) dτ, ji = ∫ ti t0 x(τ)dτ and pi = ∫ ti t0 x(τ)z(τ) n(τ) dτ, are evaluated using the trapezoidal rules. the system (43) is of the form aa = h, where a =   t1 −j1 −k −p1... ... ... ... tn −jn −k −pn   , a =   p1 µ1 αv αw   and h =   ∆x1... ∆xn   . solving the regression problem minimize‖aa − h‖2 (44) using least squares method we obtain the estimate ã. thus the parameters p1,µ1,αv and αw are found. now recall (24), dyv dt = p2 + αv xyv n −γ yvz n −µ2yv, (45) similarily, integrating both sides of (45) yields ∆yv = p2ti + αvki −γq1 −µ2ii, (46) where ∆yv = ∫ ti t0 dyv(τ) dt dτ, ki = ∫ ti t0 x(τ)yv(τ) n(τ) dτ, qi = ∫ ti t0 x(τ)z(τ) n(τ) dτ and ii = ∫ ti t0 y (τ)dτ. the system (46) is of the form aa = h, where a =   t1 −q −i1... ... ... tn −q −in   , a =   p2γ µ2   and h =   ∆y1 −αvk1... ∆yn −αvkn   . solving the regression problem minimize‖aa − h‖2 (47) using least squares method we obtain the estimate . thus the parameters γ,µ2 and p2 are found. lastly recall (39), a1 dz dt + a2 dm dt + a3m + a4z + f = 0 (48) subject to the constraints a1 + a3 − 1 = 0 and a1a3 −a2a4 = 0. (49) integrating both sides of (48) with respect to t from t0 to ti, gives a1∆zi +a2∆mi +a3ui +a4wi + ∆fi = 0 (50) biomath 4 (2015), 1512141, http://dx.doi.org/10.11145/j.biomath.2015.12.141 page 5 of 7 http://dx.doi.org/10.11145/j.biomath.2015.12.141 p. mathye et al., identification of hiv dynamic system in the case of incomplete ... where ∆zi = ∫ ti t0 dz(τ) dt τ, ∆mi = ∫ ti t0 dm(τ) dt τ, ui = ∫ ti t0 m(τ)dτ, wi = ∫ ti t0 z(τ)dτ and ∆fi = −αw ∫ ti t0 x(τ)yv(τ) n(τ) dτ. to find the unknown parameters a1,a2,a3 and a4, the lagrangian l3 can be written with one constraint as l3(a) = 1 2 n∑ i=0 [ a1(zi −ui) + a2mi + a4wi+ (∆fi + ui) ]2 + λ(a1 −a21 −a2a4) (51) where λ1 is the lagrange multipliers and a = (a1,a2,a4). to minimize the least-square functional l3, set ∂l3 ∂a1 = ∂l3 ∂a2 = ∂l3 ∂a4 = ∂l3 ∂λ = 0. (52) the resultant system of equations is then solved using the method proposed by fedotov et al [3] for finding roots of transcendental algebraic equations. the parameter value a3 is finally obtained from a3 = 1 −a1. vii. numerical simulation in order to illustrate the effectiveness of the method a numerical example is presented. the system (18) is solved numerically by adams method using a mathematical software mathcad. the following parameter values and initial conditions from gumel [4] are used. p1 = 400, p2 = 1.6×103, αv = 2.5, αw = 2.25, γ = 0.9µ1 = 0.031, µ2 = 0.0331, µ3 = 0.0281, µ4 = 0.231, x 0 = 8 × 104, y 0v = 2000, y 0w = 8000, y 0 vw = 8000 the solution vectors obtained are taken as experimental data. we then assume that the experimental data concerning the state variables yw, yvw and the model parameter values to be unknown. the method discussed in the sections above is then applied to estimate the model parameter values and restore the vectors yw and yvw. the results given in the table below, show a comparison of the actual parameters used, the estimated parameters and the percentage error given by ‖α− α̃‖/‖α‖× 100. table 2: the parameter estimates and errors. parameter actual value estimated value %error α α̃ p1 400 400.142 0.035 p2 1.6 ×103 1.6 ×103 0.000 µ1 0.031 0.031 0.000 µ2 0.331 0.326 1.511 µ3 0.281 0.282 0.356 µ4 0.231 0.229 0.866 αv 2.5 2.495 0.200 αw 2.25 2.25 0.000 γ 0.9 0.905 0.556 from table 2 it can be seen that the estimate parameters are close enough to the actual ones. the percentage relative errors for this estimates are mostly low than 1 %. let yw := ( u<3> ) i and ( yvw := u <4> ) i be the solution vectors obtained from solving the system (18) and y ywi and y yvwi the estimated vectors. from figures 1 and 2, it can be seen that the estimated vectors, y ywi and y yvwi are within acceptable limit of error. viii. conclusion in this paper a method to identify dynamic mathematical models with incomplete (missing) data was discussed. the method was applied to a four dimensional hiv vaccination model. the model parameters and the unknown (missing) data were restored. the proposed method gives a direct hint of what is necessary to measure in practice and what data can be analytically restored (found). biomath 4 (2015), 1512141, http://dx.doi.org/10.11145/j.biomath.2015.12.141 page 6 of 7 http://dx.doi.org/10.11145/j.biomath.2015.12.141 p. mathye et al., identification of hiv dynamic system in the case of incomplete ... fig. 1. comparison of the estimated solutions, y y wi, with the “experimental data”, u<3>i , and the percentage relative error. references [1] s. m. blower, k. koelle, d. krischner & j. mills, live attenuated hiv vaccine: predicting the trade-off between efficacy and safety, proc nat acad sci, 98(6) : 3618 − 3623, 2001. [2] f. ding, g. liu & x. p. liu, parameter estimation with scarce measurements, automatica, 47 : 1646 − 1655, 2001. [3] i. fedotov, m. shatalov & j. n. mwambakana, roots of transcedental algebraic equations: a method of bracketing roots and selecting initial estimations, buffelsfontein time 2008 peer-review conference procedings, 22−26 september, 2008. [4] a. b. gumel, s. m. moghadas & r. e. mickens, effect of a preventive vaccine on the dynamics of hiv transmission, communication in nonlinear science and numerical simulations, 9 : 649−659, 2004. fig. 2. comparison of the estimated solutions, y y vwi, with the “experimental data”, u<4>i , and the percentage relative error. [5] c. l. lawson & r. j. hanson, solving least square problems, new jersey, america: prentice–hall, 1974. [6] m. a. nowak& a. r. mclean, a mathematical model of vaccination against hiv to prevent the development of aids, rgmia, victoria university, 10(1, 2) : 106−116, 2007. [7] m. shatalov & i. fedotov, on identification of dynamic systems parameters from experimental data, rgmia, victoria university, 10(1, 2) : 106−116, 2007. [8] m. shatalov,i. fedotov & s. v. joubert, a novel method of interpolation and extrapolation of functions by a linear initial value problem, buffelsfontein time 2008 peerreview conference procedings, 22−26 september, 2008. biomath 4 (2015), 1512141, http://dx.doi.org/10.11145/j.biomath.2015.12.141 page 7 of 7 http://dx.doi.org/10.11145/j.biomath.2015.12.141 introduction problem formulation inverse method mathematical model hiv susceptible (x) individuals uninfected by the wild type but infected by the vaccine strain (yv) unvaccinated individuals infected by wild type (yw) individuals dually-infected with the vaccine and wild strain (yvw) method of solution parameter estimation numerical simulation conclusion references original article biomath 1 (2012), 1211211, 1–7 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum position-induced phase change in a tasep with a double-chain section (a model of biological transport) nina c. pesheva∗ and jordan g. brankov† ∗institute of mechanics-bas, sofia, bulgaria email: nina@imbm.bas.bg † joint institute for nuclear research, dubna, russian federation email: brankov@theor.jinr.ru received: 31 august 2012, accepted: 21 november 2012, published: 21 december 2012 abstract—the totally asymmetric simple exclusion processes (tasep) has been used since 1968 to model different biochemical processes, including kinetics of protein synthesis, molecular motors traffic, collective effects in genetic transcription. here, we consider tasep defined on an open network consisting of simple head and tail chains with a double-chain section in-between. our results of monte carlo simulations show a novel property of the model when the simple chains are in the maximum-current phase: upon moving the double-chain defect from the central position forward or backward along the network, keeping fixed the length of both the defect and the whole network, a position-induced phase change in the parallel defect chains takes place. this phenomenon is explained in terms of finite-size dependence of the effective injection and removal rates at the ends of the double-chain defect. some implications of the results for molecular motors cellular transport along such networks are suggested. however, at present these are just speculations which need further examinations. keywords-tasep; molecular motors traffic; kinetics of protein synthesis; traffic flow models; non-equilibrium phase transitions; non-equilibrium statistical physics i. introduction the world of non-equilibrium phenomena is more diverse and much more interesting as compared to our experience with its equilibrium counterpart. rigorously put, true equilibrium phenomena are an idealization which is seldom met in nature. the development of a fundamental and comprehensive understanding of physics far from equilibrium is currently under way. that is why the study of simple non-equilibrium models like the totally asymmetric simple exclusion process (tasep) (see, e.g., [1, 2, 3, 4, 5, 6, 7, 8, 9]) is very informative and helpful. this approach—the study of simple model systems, has shown to be very effective in the equilibrium statistical mechanics and now it is intensively exploited also in the non-equilibrium case. one can see that recently more methods and concepts from non-equilibrium statistical physics are applied to model processes in living systems and biological phenomena [10, 11, 12, 13, 14, 15, 16, 17, 18]. this is quite natural since the object of non-equilibrium statistical physics are open many-particle systems with macroscopic currents of energy and/or particles. biological systems, on the other hand, are rather complex systems which in order to function properly need energy and matter flows. there are biological transport phenomena which can be considered to be restricted to an effectively one-dimensional track, e.g., stepping of kinesins and dyneins along microtubules, translocation of rna polymerase (rnap) on dna during transcription, ribosomes on messenger rna (mrna) during protein syntheses—a process referred to as translation. kinesins and dyneins are cytoskeletal citation: n. pesheva , j. brankov, position-induced phase change in a tasep with a double-chain section (a model of biological transport), biomath 1 (2012), 1211211, http://dx.doi.org/10.11145/j.biomath.2012.11.211 page 1 of 7 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.11.211 n. pesheva et al., position-induced phase change... motors: kinesin moves cargo inside cells away from the nucleus along microtubules and dynein transports cargo along microtubules towards the cell nucleus. all these stochastic processes are of special interest due to their fundamental importance for the functioning of living cells. hence, they are a challenging object for mathematical modeling and discrete stochastic models seem adequate for that purpose. usually a large number of such agents move unidirectionally along the same track with excluded volume interaction, which makes the simple models of vehicular traffic appropriate for incorporation in more sophisticated ones. for example, stochasticity and traffic jams in the transcription of ribosomal rna have been considered by klumpp and hwa [14]. in the present study, we are concerned with one specific example of application of a simple non-equilibrium model, the tasep, to the protein synthesis. since 1968 this model has been used to model different biological processes [10, 11, 12, 13, 14, 15, 16, 17, 18] including the phenomenon of protein synthesis [10]. in the last twenty years, the non-equilibrium statistical physicists [19, 20, 21, 22, 23, 24] are very much interested in the study of different kind of models which are expected to provide deep understanding of the generic behavior of non-equilibrium systems. another challenging problem, from both biological and mathematical point of view, is the consideration of biochemical transport phenomena on networks with non-trivial topology. our goal here is to present a study of the effects, arising in tasep, defined on a simple example of such a network: a linear chain of attachment sites with a double-chain defect inserted in it [25]. for other studies of tasep on topologies more complex than a single segment see [26, 27, 28, 29]. recently, applications to biological transport have motivated generalizations of the tasep to cases when the entry rate is chosen to depend on the number of particles in the reservoir (tasep with finite resources) [30, 31]. this year, the cases of multiple competing taseps with a shared reservoir of particles [32, 33], and tasep with langmuir kinetics and memory reservoirs [34] were studied too. the next section is devoted to the single chain tasep, then a short overview is given on the tasep with a double-chain section in-between [25]. the last section is devoted to our new monte carlo simulation results displaying a novel property of the model with the double-chain section in the maximum-current phase. ii. model and applications a. single chain tasep one of the simplest driven (non-equilibrium) models of many-particle systems with particle conserving stochastic dynamics is the asymmetric simple exclusion process (asep). it has been extensively studied on simple chains with periodic, closed and open boundary conditions. in the extremely asymmetric case particles are allowed to move with in one direction only this is the totally asymmetric simple exclusion process (tasep). it was first introduced in [10] as a model of protein synthesis; in the context of interacting markov processes, see [1]. its steady states are exactly known for both open and periodic boundary conditions, for continuous-time and several kinds of discrete-time dynamics. here we shall focus our attention on the steady states of the open tasep with continuous-time stochastic dynamics on a simple chain, illustrated in fig. 1. for a review on the exact results for the stationary states of tasep under different kinds of stochastic dynamics, and its numerous applications, we refer the reader to [4, 22]. the continuous-time dynamics is modeled by the so called random-sequential update: in the algorithm one chooses with equal probability any one of the lattice sites (the left reservoir is included as an additional site), and, if the chosen site is occupied by a particle, moves it (with rate p = 1) to the nearest-neighbor site on the right, provided the target site is empty. in the case of open system, particles are injected at the left end with rate α and removed at the right end with rate β when the last site is occupied. when α, β ∈ (0, 1] the boundary conditions correspond to coupling of the system to reservoirs of particles with constant densities α and 1 − β, respectively. as predicted by krug [21], the change of the boundary rates induces non-equilibrium phase transitions between different stationary phases. in the thermodynamic limit, the phase diagram of the stationary states in the plane of the particle injection and removal rates is shown in fig. 2. it exhibits three distinct phases: a low-density freeflow phase (region ai ∪ aii), a high-density congested traffic one (region bi ∪ bii), and a maximum current phase (region m c), characterized by a synchronized flow in which jams and free-flow coexist at intermediate densities. these phases are separated by lines of non-equilibrium first-order and second-order phase transitions. here we need to mention some basic facts obtained in the case of continuous-time dynamics: (a) the correlations in the bulk of an infinite chain vanish biomath 1 (2012), 1211211, http://dx.doi.org/10.11145/j.biomath.2012.11.211 page 2 of 7 http://dx.doi.org/10.11145/j.biomath.2012.11.211 n. pesheva et al., position-induced phase change... fig. 1. schematic representation of the open tasep on a simple chain; for details see text. fig. 2. phase diagram of the open tasep on a simple chain. the regions of the different phases are explained in the text. and the dependence of the stationary current of particles j on the average density ρ is given by j = ρ(1−ρ); (b) in the maximum current phase ρ = 1/2 and j = 1/4; when j < 1/4, there are two densities which support that current: ρ±(j) = [1 ± (1 − 4j)1/2]/2, (1) (ρ−) is the bulk density in the low-density (high-density) phase. fig. 3. schematic representation of the network: a single chain with a two-chain incertion. the segments c2 and c3 have equal length l2(l3 = l2) in the case under consideration. the particles are injected at the left end with a rate α and removed at the right end with a rate β. the particles move from left to right, at the branching point pb = l1 they take with equal probability the upper or the lower branch. b. tasep with a double-chain section the idea of studying networks, composed of chain segments, which exhibit the bulk behavior of an open tasep under boundary conditions given in terms of effective input and output rates, was first advanced in our work [25]. the network considered there is shown schematically in fig. 3. the appearance of correlation effects, close to the ends of the chain segments, as well as of cross-correlations in the double-chain segment was found. the same approach was applied in ref. [26] to an open network consisting of one vertex with two incoming chains, coupled to one reservoir, and one outgoing chain, coupled to another reservoir. different versions of simple networks were studied also in refs. [25] and [26]. in the latter work the notion of particle-hole symmetry in the presence of a junction was carefully analyzed and an appropriate interpretation on the microscopic level was given. tasep with parallel update on single multipleinput—single-output junctions has been investigated too [29]. the main concern in the above works was the construction of the phase diagram under different open boundary conditions. here we continue the investigation of the network considered in [25], see fig. 3. note, that the last site i = l1 of the head chain is a branching point, from which the particles can take the upper or the lower branch of the two-chain section with equal probability. simultaneous and independent traffic of particles on the two equivalent branches was simulated. the parallel branches merge at site i = l1 + l2, where the particles have to wait for the first site of the tail chain i = l1 + l2 + 1 to become empty. we have denoted the phase structure of the model by (x1, x2,3, x4), where xk (k = 1, 2, 3, 4) biomath 1 (2012), 1211211, http://dx.doi.org/10.11145/j.biomath.2012.11.211 page 3 of 7 http://dx.doi.org/10.11145/j.biomath.2012.11.211 n. pesheva et al., position-induced phase change... stands for one of the stationary phases of the chain segment ck: ld—low density, hd—high density, m c— maximum current, and cl—coexistence line. our analytical analysis of the allowed phase structures, based on the properties of single chains in the thermodynamic limit, and the neglect of the pair correlations between the nearest-neighbor occupation numbers at the junctions of different chain segments, yielded 8 possibilities. here we focus our investigation on 3 of the most interesting cases (m c, ld, m c), (m c, cl, m c), and (m c, hd, m c), which appear under the conditions α > 1/2, β > 1/2, corresponding to the maximum current phase of a single chain. the phase state of the chains in the double-chain defect depends on the effective injection rate α∗ of particles at the first site of each of the chain segments c2,3 and on the effective removal rate β∗ of particles from the last site of each of these chains. as in the case of a single infinite chain, the density profiles of c2 and c3 are similar to the ones in the ld, cl, and hd phases when β∗ < α∗ < 1/2, β∗ = α∗ < 1/2, and α∗ < β∗ < 1/2, respectively. the crucial difference now is that the above effective rates depend on the finite size of the head and tail simple chain segments. in the present interpretation, the hard-core particles represent individual molecular motors. iii. results and discussion as a result of monte carlo simulations we have found a novel property of the model in the maximum-current phase, i.e., when α > 1/2 and β > 1/2. then the current j2,3 trough each of the chains c2,3 equals half of the maximum current, i.e., j2,3 = 1/8. therefore, due to the fundamental relationship j = ρ(1 − ρ), in the thermodynamic limit these chains can be found either in a low-density phase with bulk density j(1/8) = [1 − √ 0.5]/2 ≈ 0.14645 , (2) or in the high-density phase with bulk density j(1/8) = [1 + √ 0.5]/2 ≈ 0.85355 , (3) or on the coexistence line of these two phases. upon moving the double-chain defect along the network, keeping fixed the lengths of both the defect and the whole network, a position-induced phase change in the defect chains takes place. this change from the coexistence line to a lowor high-density phase is observed in the density profile of each of the chains forming the defect. in fig. 4 we show our simulation results for the density distributions for a rather small system of fixed fig. 4. simulation results: density profiles as a function of the scaled distance x = i/lk, for the system with the (m c, cl, m c) phase structure, appearing when . the symmetric case with l1 = l2,3 = l4 = 50 is shown with red squares. the change of the density profiles in the double-chain section is clearly seen: when l1 = 25, l4 = 75 its shape is characteristic of the hd phase (blue circles); when l1 = 75 and l4 = 25 its shape is characteristic of the ld phase (green triangles). total length ltot = l1 + l2,3 + l4 = 150 sites and fixed size of the double-chain section, l2 = l3 = 50. the ensemble averaging was performed over 200 independent runs and after 3 000 000 monte carlo steps were omitted in order to ensure that the system had reached a stationary state. one can easily see the sharp change, which the density profiles undergo, when the position of the loop is shifted. as a reference, the results for the density profiles of the system with segments of equal length l1 = l2,3 = l4 = 50 are shown with red squares. grey circles in the latter case the two branches of the defect section are on the coexistence line. however, when the head chain is shorter, e.g., when l1 = 25 and, respectively, l4 = 75, the density distribution in the double-chain section is typical for the hd phase (see the results shown with blue circles). in the opposite case, when the head chain is longer than the tail one, l1 = 75 and l4 = 25, the density distribution of the double chain-section has the typical shape of the ld phase (shown with green triangles). the spatial behavior of the correlations between nearest-neighbor occupation numbers, shown in figure 5, is also typical for the corresponding phases. an explanation of the phenomenon can be given in terms of finite-size dependence of the effective injection and removal rates at the ends of the double-chain defect. in the symmetric case, when l1 = l4, we observe biomath 1 (2012), 1211211, http://dx.doi.org/10.11145/j.biomath.2012.11.211 page 4 of 7 http://dx.doi.org/10.11145/j.biomath.2012.11.211 n. pesheva et al., position-induced phase change... fig. 5. simulation results: nearest-neighbor correlations fcorr , in the (m c, cl, m c) phase state of the system, as a function of the scaled distance x = i/lk, for different positions of the double-chain segment. α∗ ≈ β∗ and, in the thermodynamic limit, the defect chains should be on the coexistence line. this fact is demonstrated by the (almost) linear density profile, changing from ρ−(1/8) at the left end to ρ+(1/8) at the right end. such a linear profile is known to result from a freely moving domain wall separating the low-density and high-density regions. due to the size dependence of the effective rates, on moving the defect to the left (i.e., when l1 decreases and l4 increases, so that l1 + l4 remains constant), α∗ increases and β∗ slightly decreases, thus the condition α∗ > β∗ becomes fulfilled and the chains c2,3 obtain a density profile, characteristic of the high-density phase. in the opposite case, on moving the defect to the right (i.e., when l1 increases and l4 decreases, so that l1 + l4 is constant), α∗ slightly decreases and β∗ increases, so that the condition α∗ < β∗ takes place and the chains c2,3 obtain a density profile, characteristic of the low-density phase. it is interesting to note, that the average velocity of particles v, defined from the relation j = ρv, is higher (lower) in the low-density (high-density) phase than in the head and tail chains, for which vm c = 1/2. indeed, in the ld phase vld = 1/[4(1 − √ 0.5)] ≈ 0.85355 , (4) and in the hd phase vhd = 1/[4(1 + √ 0.5)] ≈ 0.14645 , (5) another notable observation is, that not only the bulk density of a single chain in the double-chain segment in the ld (hd) phase is lower (higher) than the bulk density of the head and tail chains, for which ρm c = 1/2, but the same relation holds for the sum of the bulk densities of both chains in the double-chain segment. indeed, in the ld phase 2ρ−(1/8) = 1 − √ 0.5 ≈ 0.29289 , (6) and in the hd phase 2ρ+(1/8) = 1 + √ 0.5 ≈ 1.7071 , (7) in general, for a multi-chain defect, consisting of n parallel identical chains, in the ld phase we obtain for the total bulk density of particles in the defect nρ±(1/4n) = n[1 ± (1 − n−1)1/2]/2 99k 1/4, n 99k ∞ (8) therefore, the unlimited increase of the number of chains in the defect part, tends to lower the total bulk density of particles in it from 2ρ−(1/8) ≈ 0.29289 down to 0.25. this is a very interesting and useful property. iv. conclusion a possible biochemical interpretation of the model, considered here, can be given in terms of molecular motors moving along linear biopolymers, such as actin filaments, microtubules, dna and rna molecules. our model ignores the possibility of backward steps, as well as the initiation stage, the dissociation from the track and the sequence of intermediate biochemical states, for example, the arrival and binding of a fuel molecule. we have focused on the effect of a non-trivial topology on the transport of hard-core particles. as pointed out by pronina and kolomeisky [26], the realistic description of cellular transport, requires also to include the possibility of motion on lattices with a more complex geometry. for example, there are indications, that the number of proto-filaments, that kinesins walk on, may vary in the microtubules. this indicates the existence of junctions and other lattice defects, which may be responsible for some human diseases. the network with a double-chain defect, considered by us, can be thought of as some sort of genetic malformation or defect, caused by radiation or some other source. our main results concern the bulk density and the average velocity of particles in the defect chains, in the regime of maximum current through the whole network. one can imagine scenarios, when it is needed to minimize or maximize some of the above mechanical parameters, presumably, for engineering novel cellular behavior. then some hints from models of traffic on tracks with parallel sections could biomath 1 (2012), 1211211, http://dx.doi.org/10.11145/j.biomath.2012.11.211 page 5 of 7 http://dx.doi.org/10.11145/j.biomath.2012.11.211 n. pesheva et al., position-induced phase change... be helpful. from the point of view of statistical physics, one is interested in a number of issues. a fundamental question concerns the ”stability” of steady-state properties with respect to model modifications. which changes of the microscopic model details will lead to changes of the macroscopic behavior? also, while for equilibrium systems basic notions of universality and independence from dynamic details are well understood, only initial steps are taken towards extending these notions towards non-equilibrium systems and more specifically towards non-equilibrium steady states [35, 36]. we would like to conclude by pointing out that even though such simple models may not permit immediate comparisons with available experimental data, due to the significant amount of simplification and/or abstraction involved, they can still be quite useful in guiding future experimental work. acknowledgment n.p. acknowledges a financial support by the project bg051po001/3.3-05.001 ”science and business”, financed by the operational programme ”human resources development” of esf, under 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(sofia) 36, 57–76 (2006). biomath 1 (2012), 1211211, http://dx.doi.org/10.11145/j.biomath.2012.11.211 page 7 of 7 http://dx.doi.org/10.1007/bf01018556 http://dx.doi.org/10.1103/physreva.38.4271 http://dx.doi.org/10.1103/physrevlett.67.1882 http://dx.doi.org/10.3367/ufne.0181.201106d.0647 http://dx.doi.org/10.11145/j.biomath.2012.11.211 introduction model and applications single chain tasep tasep with a double-chain section results and discussion conclusion www.biomathforum.org/biomath/index.php/biomath original article permanence and periodic solution for a modified leslie-gower type predator-prey model with diffusion and non constant coefficients a. moussaoui1, m. a. aziz alaoui2, r. yafia3 1department of mathematics, university of tlemcen, tlemcen, algeria moussaoui.ali@gmail.com 2normandie university, le havre, france ulh, lmah, fr cnrs 3335, le havre, france aziz.alaoui@univ-lehavre.fr 3université ibn zohr, faculté polydisciplinaire, ouarzazate, morocco yafia@yahoo.fr received: 5 march 2017, accepted: 10 july 2017, published: 19 july 2017 abstract—in this paper we study a predator-prey system, modeling the interaction of two species with diffusion and t -periodic environmental parameters. it is a leslie-gower type predator-prey model with holling-type-ii functional response. we establish some sufficient conditions for the ultimate boundedness of solutions and permanence of this system. by constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. numerical simulations are presented to illustrate the results. keywords-reaction-diffusion equations, predatorprey model, functional response, permanence. i. introduction and mathematical model the dynamical properties of the predator-prey models can be used to analyze the relations between the prey and predator and to predicate whether they can coexist. as we known, one of the earliest and also the best known predatorprey models is the leslie-gower model [16], [17], which is a modificiation of the lotka-volterra model [22]. the leslie-gower type model can be described by the following autonomous bidimensional system [16], [17]  du dt = u(a− bu) −αuv, dv dt = v ( c− βv u ) , (1) where u is the population of the prey and v is the population of the predator. in (1) we assume the prey grows logistically with carrying capacity k = a b and intrinsic growth rate a in the absence of predation. the predation is assumed to be proportional to the population size of the prey. the predator grows logistically with intrinsic growth rate c and carrying capacity c β u(t) proportional to the population size of prey (or prey abundance). citation: a. moussaoui, m. a. aziz alaoui, r. yafia, permanence and periodic solution for a modified leslie-gower type predator-prey model with diffusion and non constant coefficients, biomath 6 (2017), 1707107, http://dx.doi.org/10.11145/j.biomath.2017.07.107 page 1 of 8 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2017.07.107 a. moussaoui et al., permanence and periodic solution for a modified leslie-gower type ... the parameter β is a measure of the food quality that the prey provides for conversion into predator birth. the stability of the interior equilibrium is studied in [25] by numerical methods. lindstrom [18] investigated the nonexistence, existence and limit cycles. hsu and huang [13] prove that all the solutions are bounded and positive if their initial values are in the first quadrant, and study the globally asymptotical stability of the interior equilibrium using liapunov function and lasalle’s invariance principle. aziz-alaoui and daher okiye [3] argued that a suitable predator-prey model should incorporate some kind of functional response, while the predator species could have other food resource. basing on those assumption, they proposed a predator-prey model with modified leslie-gower and holling-type ii schemes [12] as follow:  du dt = u(a− bu) − αuv u + k1 , dv dt = v ( c− βv u + k2 ) , (2) where k1 is the half-saturation constant in the holling-type ii functional response and k2 is a measure of alternative prey densities in the environment, allowing the predator to persist when the prey population disappears. the authors investigated the boundedness and global stability of the system (2). nindjin et al. [23] further incorporated the time delay to the system considered in [3], and they showed that time delay plays important role on the dynamic behaviors of the system. yafia et al. [27] studied the limit cycle bifurcated from time delay. for more works on leslie-gower predator-prey model, one could refer to [1], [4], [7], [8], [14], [26], [28], [29] and the references cited therein. to achieve further understanding it is now essential to consider more general and hence more ”difficult” models. we will focus here on the case in which the biological or environmental parameters are time-periodic, and will assume that the species are free to move at random throughout some bounded habitat. under these assumptions, we model the species interaction via a system of reaction-diffusion equations of the form  ∂u ∂t −d1∆u=u ( a(t,x)−b(t,x)u ) − α(t,x)uv u+k1(t,x) , ∂v ∂t −d2∆v =v ( c(t,x)− β(t,x)v u+k2(t,x) ) , (3) where the function u(t,x) and v(t,x) determine the densities of prey and predator, respectively, at a point x and time t. here the equations are assumed to be satisfied in a cylinder x ∈ ω̄, 0 < t < ∞, where ω is an open, bounded, smooth domain in rn. these equations are supplemented with homogeneous neumann boundary conditions ∂u ∂n = ∂v ∂n = 0 on ∂ω × (0,∞). where n is the outward unit vector of the boundary ∂ω which we assume is smooth, and the following nonnegative initial values u(0,x) = u0(x) ≥ 0, v(0,x) = v0(x) ≥ 0 in ω. the various coefficients on the right-hand side depend on both t and x modelling the fact that effects vary in both time and space. the periodicity of coefficients models seasonal fluctuations. d1 and d2, are positive diffusion coefficients reflecting the non-homogeneous dispersion of populations. many authors studied the qualitative properties of this system, but for the case for which the parameters are constant, using neumann or dirichlet boundary conditions, see [8], [24], [27], [29]. motivated by the papers mentioned above, we deal here with the permanence and existence of periodic solutions of the diffusive system (3). the content of this paper is as follows. in section 2, we give conditions for the ultimate boundedness of solutions and permanence of the system. in section 3, we establish conditions for the existence of a unique periodic solution of the system. numerical simulations are presented in section 4 to illustrate the feasibility of our results. biomath 6 (2017), 1707107, http://dx.doi.org/10.11145/j.biomath.2017.07.107 page 2 of 8 http://dx.doi.org/10.11145/j.biomath.2017.07.107 a. moussaoui et al., permanence and periodic solution for a modified leslie-gower type ... ii. boundedness and permanence we analyze the permanence (dissipation and persistence) of system (3) with non-negative initial functions, this ensures the long-term survival (i.e., will not vanish in time) of all components of system (3), under some conditions. we first recall a well known result on the logistic equation. lemma 2.1: [30]. assume that u(t,x) is defined by  ∂u ∂t =d1∆u+ru ( 1− u k ) , x∈ω, t>0, ∂u ∂n = 0, x ∈ ∂ω, t > 0, u(x, 0) = u0(x) > 0, x ∈ ω, (4) then, limt→∞u(t,x) = k. throughout the paper we always assume that: (h): functions a(t,x), b(t,x), c(t,x), α(t,x), β(t,x) and ki(t,x), (i = 1, 2) are bounded positive-valued functions on r+ × ω̄ continuously differentiable in t and x, and are periodic in t with a period t > 0. for a bounded function φ(t,x), we denote φm = inf (t,x)∈r+×ω φ(t,x), φm = sup (t,x)∈r+×ω φ(t,x). a. dissipation proposition 2.2: all the solutions of (3) initiated in the positive octant are nonnegative and satisfy lim sup t→+∞ max x∈ω̄ u(t,x) ≤ am bm , lim sup t→+∞ max x∈ω̄ v(t,x) ≤ cm βm (am bm + km2 ) . proof the nonnegativity of the solutions of (3) is obvious since the initial value is nonnegative. we consider now the second part of the theorem. for convenience, we denote a = a(t,x), and similar meaning to b,c,α,β,k1 and k2. from the first equation of system (3), we have ∂u ∂t = d1∆u + u ( a− bu ) − αuv u + k1 , ≤ d1∆u + u ( am − bmu ) . (5) from the comparison principle of the parabolic equations [9], [11], it is easy to verify that u(t,x) ≤ u(t), where u(t) is the spatially homogeneous solutions of  ∂u ∂t =d1∆u+u ( am−bmu ) , x ∈ ω, t > 0 ∂u ∂n = 0, x ∈ ∂ω, t > 0 u(0,x) = u∗ (6) where u∗ = max x∈ω u(0,x). this implies, by using lemma 2.1, that lim sup t→+∞ max x∈ω̄ u(t,x) ≤ am bm . then, for ε > 0 there exists t1 > 0 such that u(t,x) ≤ η1 for t > t1, (7) where η1 = a m bm + ε. therefore, from the second equation of system (3) and (7) and using the same reasoning, we have ∂v ∂t = d2∆v + v ( c− βv u + k2 ) ≤ d2∆v + v ( cm − βmv η1 + k m 2 ) for t > t1. hence there exists t2 > t1 such that for any t > t2 v(t,x) ≤ η2 (8) where η2 = c m βm (a m bm + ε + km2 ) + ε, which implies lim sup t→+∞ max x∈ω̄ v(t,x) ≤ cm βm ( am bm + km2 ) . therefore, any positive solution of system (3) is ultimately bounded, which completes the proof. biomath 6 (2017), 1707107, http://dx.doi.org/10.11145/j.biomath.2017.07.107 page 3 of 8 http://dx.doi.org/10.11145/j.biomath.2017.07.107 a. moussaoui et al., permanence and periodic solution for a modified leslie-gower type ... b. persistence definition 2.3: [5], [6] system (3) is said to be persistent if for any positive initial data (u0(x),v0(x)), there exist positive constants ξ1 = ξ1(u0,v0),ξ2 = ξ2(u0,v0), such that the solution (u(t,x),v(t,x)) of (3) satisfies lim inf t→+∞ min x∈ω̄ u(t,x) ≥ ξ1, lim inf t→+∞ min x∈ω̄ v(t,x) ≥ ξ2 proposition 2.4: assume that amkm1 β m > αmcm (am bm + km2 ) (9) then system (3) is persistent. proof from (3), (7) and (8), it follows that for t ≥ t2, ∂u ∂t = d1∆u + u(a− bu) − αuv u + k1 ≥ d1∆u + u(am − bmu) − αmη2u km1 = d1∆u + u ( am − αmη2 km1 − bmu ) then from the comparison principle of the parabolic equations, it is easy to verify that u(t,x) ≥ u(t), where u(t) is the spatially homogeneous solutions of  ∂u ∂t =d1∆u + u ( am− αmη2 km1 −bmu ) , x ∈ ω, t > 0, ∂u ∂n = 0,x ∈ ∂ω, t > 0, u(x, 0) = u∗ (10) where u∗ = min x∈ω u(0,x). thanks to lemma 2.1, we obtain, lim inf t→+∞ min x∈ω̄ u(t,x) ≥ 1 bm ( am − αmη2 km1 ) . hence, there exists t3 > t2 such that for any t > t3, u(t,x) ≥ ξ1 (11) where, ξ1 = 1 bm ( am − αmη2 km1 −ε ) . from the predator equation, it follows that ∂v ∂t = d2∆v + v ( c− βv u + k2 ) ≥ d2∆v + v ( cm − βmv km2 ) . hence, there exists t4 > 0 such that for any t > t4 v(t,x) > ξ2, (12) where ξ2 = cmkm2 βm −ε. therefore, from (11) and (12), we obtain, lim inf t→+∞ min x∈ω̄ u(t,x) ≥ 1 bm ( am − αmcm βm (a m bm + km2 ) km1 ) , lim inf t→+∞ min x∈ω̄ v(t,x) ≥ cmkm2 βm . (13) thus, system (3) is persistent, which completes the proof of proposition 2.4. a direct application of proposition 2.2 and proposition 2.4 gives the following result. proposition 2.5: (permanence) if condition (9) holds, there exist positive constants 0 < ζ < η, such that, ζ ≤ lim inf t→∞ min x∈ω̄ u(t,x) ≤ lim sup t→∞ max x∈ω̄ u(t,x) ≤ η ζ ≤ lim inf t→∞ min x∈ω̄ v(t,x) ≤ lim sup t→∞ max x∈ω̄ v(t,x) ≤ η that is, model (3) is permanent. biomath 6 (2017), 1707107, http://dx.doi.org/10.11145/j.biomath.2017.07.107 page 4 of 8 http://dx.doi.org/10.11145/j.biomath.2017.07.107 a. moussaoui et al., permanence and periodic solution for a modified leslie-gower type ... iii. periodic solutions a very basic and important problem in the study of a population growth models with periodic environment is the global existence and stability of positive periodic solutions, which plays a similar role as a globally stable equilibrium for autonomous models [5], [11], [15], [19], [21]. in this section, we derive sufficient conditions that guarantee existence, uniqueness and global stability of a t -periodic positive solution of system (3). for this aim, we consider the matrix m, which reads as: m =   2 ( am−2bmζ− α mkm1 ζ (η+km1 ) 2 ) βmη2 (ζ+km2 ) 2 βmη2 (ζ+km2 ) 2 2 ( cm−2βmζ ζ+k m 2 (η+km2 ) 2 )   (14) where ζ and η are the bounds of any non-zero arbitrary solution of system (3), initialing with non-negative function, given by proposition 2.5. proposition 3.1: assume that condition (9) holds, that is system (3) is permanent, if µ(m) < 0, (15) where µ(m) is the maximal eigenvalue of the matrix m. then, system (3) has a unique globally asymptotic stable strictly positive t -periodic solution. proof let (u1(t,x),v1(t,x)) and (u2(t,x),v2(t,x)) be two solutions of system (3), by proposition 2.5, these solutions are bounded by constants ζ and η, where ζ = min{ξ1,ξ2} and η = max{η1,η2}, defined in section 2. consider the function u(t)= ∫ ω ( (u1(t,x)−u2(t,x))2+(v1(t,x)−v2(t,x))2 ) dx (16) one has, du(t) dt = 2 ∫ ω (u1 −u2) ( ∂u1 ∂t − ∂u2 ∂t ) dx +2 ∫ ω (v1 −v2) ( ∂v1 ∂t − ∂v2 ∂t ) dx = 2d1 ∫ ω (u1 −u2)∆(u1 −u2)dx +2d2 ∫ ω (v1 −v2)∆(v1 −v2)dx +2 ∫ ω (u1 −u2)[( u1(a−bu1)−αu1v1u1+k1 ) − ( u2(a−bu2)−αu2v2u2+k1 )] dx +2 ∫ ω (v1−v2) [( v1(c− βv1u1+k2 ) − ( v2(c− βv2u2+k2 )] dx := i1 + i2 + i3 + i4 (17) it follows from the boundary condition in (3) that i1 + i2 = 2d1 ∫ ∂ω (u1 −u2)∇(u1 −u2)dη −2d1 ∫ ω (∇(u1 −u2))2dx +2d2 ∫ ∂ω (v1 −v2)∇(v1 −v2)dη −2d2 ∫ ω (∇(v1 −v2))2dx =−2d1 ∫ ω (∇(u1−u2))2dx−2d2 ∫ ω (∇(v1−v2))2dx. ≤ 0. for the third and fourth term in (17), we have i3 +i4 = 2 ∫ ω (u1−u2) [( (u1−u2) ( a−b(u1 +u2) ) −α u1u2(v1 −v2) + k1(u1v1 −u2v2 (u1 + k1)(u2 + k1) ] dx +2 ∫ ω (v1 −v2) [ c(v1 −v2) −β (v21u2 −v 2 2u1) + k2(v1 −v2)(v1 + v2) (u1 + k2)(u2 + k2) dx ] . note that v21u2−v 2 2u1 = (v1−v2)(v1u2+v2u1)−v1v2(u1−u2) and u1v1 −u2v2 = u1(v1 −v2) + v2(u1 −u2). therefore i3 + i4 = 2 ∫ ω (u1 −u2)2 [ a− b(u1 + u2) − αk1v2(u1+k1)(u2+k1) ] dx +2 ∫ ω (v1−v2)2 [ c−β (v1u2 + v2u1) + k2(v1 + v2) (u1 + k2)(u2 + k2) dx ] biomath 6 (2017), 1707107, http://dx.doi.org/10.11145/j.biomath.2017.07.107 page 5 of 8 http://dx.doi.org/10.11145/j.biomath.2017.07.107 a. moussaoui et al., permanence and periodic solution for a modified leslie-gower type ... +2 ∫ ω (u1 −u2)(v1 −v2)[ βv1v2 (u1+k2)(u2+k2) − αu1 (u1+k1) ] dx ≤ 2 ∫ ω (u1−u2)2 [ am−2bmζ− αmkm1 ζ (η + km1 ) 2 ] dx +2 ∫ ω (v1−v2)2 [ cm−2βmζ ζ + km2 (η + km2 ) 2 ] dx +2 ∫ ω ∣∣∣(u1−u2)(v1−v2)∣∣∣ βmη2 (ζ + km2 ) 2 dx ≤ µ(m) ∫ ω [ (u1−u2)2+(v1−v2)2 ] dx. using (15) yields, u(t) ≤ u(0)eµ(m)t → 0 as t →∞ (18) thus, we have proved that ‖u1(t,x)−u2(t,x)‖→ 0 and ‖v1(t,x) −v2(t,x)‖→ 0 as t →∞, where ‖.‖ denotes the norm of the space l2(ω). let p > n a positive integer and w(t,w0) = (u(t,x,u0,v0),v(t,x,u0,v0)). by applying exactly the same reasoning as in [2], we prove that for some γ ∈ ( 1 2 + n 2p , 1), the solution {w(t,w0)} is relatively compact in the space c1+θ(ω̄,r2), for 0 < θ < 2γ − 1 −n/p. therefore, lim t→∞ sup x∈ω |u1(t,x) −u2(t,x)| = 0, lim t→∞ sup x∈ω |v1(t,x) −v2(t,x)| = 0. (19) now we consider the sequence (u(kt,x,u0,v0),v(kt,x,u0,v0)) = w(kt,w0). then, {w(kt,w0),k ∈ n} is compact in the space c(ω̄) × c(ω̄). let ω̄ be a limit point of this sequence, then w(t,w̄) = w̄. indeed, it follows, from w(t,w(knt,w0)) = w(knt,w(t,w0)) and ω(knt,w(t,ω0))−w(knt,ω0) → 0 as kn →∞, that ‖w(t,w̄)−w̄‖c≤‖w(t,w̄)−w(t,w(knt,w0))‖c +‖w(t,w(knt,w0)) −w(knt,w0)‖c +‖w(knt,w0) − w̄‖c → 0 as n →∞. the sequence {w(kt,w0),k ∈ n} has a unique limit point, otherwise, there are two limit points w̄ = lim t→∞ w(knt,w0) and ŵ = lim t→∞ w(knt,w0). but, thanks to (19) and ŵ = w(knt,ŵ), we get ‖w̄−ŵ‖c ≤ ‖w̄−w(knt,w0)‖c +‖w(knt,w0)−ŵ‖c → 0 as n →∞. (20) thus, w̄ = ŵ. hence, the solution (u(t,x, ū, v̄),v(t,x, ū, v̄)) is the unique periodic solution of system (3). finally, due to (19), we conclude that this periodic solution is globally asymptotically stable. iv. numerical simulations in this section, numerical simulations for a given parameters range of system (3) are done to support our analytical results obtained in sections 3 and 4. we consider system (3) with d1 = 0.5,d2 = 0.8, a = 2 + 0.5 sin(2πt), b = 4 + 0.5 sin(2πt), α = 0.03 + 0.02sin(2πt), k1 = 1 + 0.2 sin(2πt), k2 = 1 + 0.5 sin(2πt), c = 1 + 0.6 sin(2πt) and β = 1 + 0.8 sin(2πt) . obviously, all the parameters have a common period t = 1 in t, by a direct computation, we can prove that all conditions in proposition 4 are satisfied. then, system (3) has a unique positive 1-periodic solution u(t,x),v(t,x) which is globally asymptotically stable. by applying matlab to simulate, we can obtain figures 1-4. from these figures, we see that system (3) is permanent and has positive periodic solution. v. conclusion the interacting species play important roles in real ecosystem. in this paper, we have studied time-periodic leslie-gower type predatorprey model with diffusion and holling-type-ii functional response whose growth rates and interaction rates are periodic functions of time. we have obtained sufficient conditions for the persistence of (3) in proposition 2.4. the conditions are given in term of parameters of the model. biologically speaking, we may expect the coexistence when the predator growth rate is sufficiently small, or if the predation rate α is small enough. biomath 6 (2017), 1707107, http://dx.doi.org/10.11145/j.biomath.2017.07.107 page 6 of 8 http://dx.doi.org/10.11145/j.biomath.2017.07.107 a. moussaoui et al., permanence and periodic solution for a modified leslie-gower type ... 0 0.5 1 1.5 2 0 5 10 15 20 0 0.1 0.2 0.3 0.4 xt u (t ,x ) fig. 1. periodic prey solution with respect to the time and space variables. 0 2 4 6 8 10 12 14 16 18 20 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 t u (t ,x ) fig. 2. the section of fig. 1 with x = 0. next, we have investigated sufficient conditions which ensure the existence of positive t -periodic solutions of (3) in proposition 3.1. the conditions are given in term of the largest eigenvalue of certain matrix. our study demonstrates how parameters of the model which are not constant but vary in response to environmental fluctuations, influence a species prosperity, and gives some valuable suggestions for saving the two species and regulating populations when the ecological and environmental parameters are affected by periodic factors such as the season switching. numerical simulations are carried out to support our theoretical results. 0 0.5 1 1.5 2 0 5 10 15 20 0 0.1 0.2 0.3 0.4 0.5 xt v( t, x) fig. 3. periodic prey solution with respect to the time and space variables. 0 2 4 6 8 10 12 14 16 18 20 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 t v( t, x) fig. 4. the section of fig. 3 with x = 0. acknowledgments the authors would like to thank the editor and anonymous referees for their careful reading of the manuscript and valuable suggestions to improve the quality of this work. references [1] p. aguirre, e. gonzalez-olivares, e. saez, two limit cycles in a leslie-gower predator-prey model with additive allee effect, nonlinear anal. real world appl., 10, 14011416, 2009. 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[31] q. yue, dynamics of a modified leslie-gower predator-prey model with holling-type ii schemes and a prey refuge, springerplus 5: 461, (2016). doi 10.1186/s40064-016-2087-7. biomath 6 (2017), 1707107, http://dx.doi.org/10.11145/j.biomath.2017.07.107 page 8 of 8 http://dx.doi.org/10.11145/j.biomath.2017.07.107 introduction and mathematical model boundedness and permanence dissipation persistence periodic solutions numerical simulations conclusion references original article biomath 1 (2012), 1209262, 1–7 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum modeling and simulations of mosquito dispersal. the case of aedes albopictus c. dufourd ∗, y. dumont† ∗ ird, crvoi, réunion island, france email: claire.dufourd@gmail.com † cirad, umr amap, montpellier, france email: yves.dumont@cirad.fr received: 10 july 2012, accepted: 26 september 2012, published: 31 december 2012 abstract—to prevent epidemics of mosquito-transmitted diseases like chikungunya in réunion island, we develop tools to control its principal vector, aedes albopictus. biological control tools, like the sterile insect technique (sit), are of great interest as an alternative to chemical control tools which are very detrimental to environment. the success of sit is based on a good knowledge of the biology of the insect, but also on an accurate modeling of insects distribution. we model the mosquito dispersal with a system of coupled parabolic pdes. considering vector control scenarii, we show that the environment can have a strong influence on mosquito distribution and in the efficiency of vector control tools. keywords-parabolic equation; existence; mosquito dispersal; modeling; numerical simulation; splitting algorithm; vector control; sterile insect technique. i. introduction chikungunya is an unusual vector-borne disease. first isolated in tanzania in 1953, it is now geographically distributed in africa, india and south-east asia. after a huge epidemic in réunion island and in india in 2006, it appeared for the first time in europe, in italy, in 2007 (see [1], [2] and references therein). the symptoms that characterize chikungunya are high fever, headache, persistant joint pain that can last several weeks. one way to reduce the risk of infection for the population is to control the vector populations. the principal vector for chikungunya, is aedes albopictus mosquito, commonly called the “asian tiger” [3]. standard vector control tools, like adulticide and lavicide, together with mechanical control are useful but cannot always be used for several reasons. firstly, because réunion island is a hot spot of endemicity. secondly, because mosquito can develop resistance to insecticides. thirdly, because only approximately 10% of the island can be treated, due to its chaotic landscape. therefore, it is necessary to consider new sustainable alternatives or additional tools, like the sterile insect technique (sit). sit consists in releasing sterilized male mosquitoes that will mate with wild females which won’t be able to have offspring. consequently, this will lead to the decrease of the vector population [4], [5]. the success of sit is based on a good knowledge of the biology and the behavior of the vector, but also on an accurate modeling of its dispersal to optimize the impact of sterile males. the previous published models were temporal models, and didn’t take into account the spatial component. but, it is necessary to consider a spatiotemporal model to obtain realistic simulations and to simulate several vector control strategies. in a previous paper [6], we have considered a dispersal model with only adult females splitted in two compartments: the blood meal searchers and the breeding site searchers. this led to a system of two partial differential equations. here, we add the aquatic stage, immature females, resting females, wild males, and finally, sterile males. this leads to a system of coupled (nonlinear) partial differential equations. after some theoretical results and the presentation of the compartmental model, citation: c. dufourd, y. dumont, modeling and simulations of mosquito dispersal. the case of aedes albopictus, biomath 1 (2012), 1209262, http://dx.doi.org/10.11145/j.biomath.2012.09.262 page 1 of 7 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.09.262 c. dufourd et al., modeling and simulations of mosquito dispersal. the case of aedes albopictus we present briefly the numerical methods based on the operator splitting technique [7]. finally, we end the paper with some numerical simulations with and without chemical or biological control. ii. the mathematical model aedes albopictus is found in south-east asia, the pacific and the indian ocean islands, and up north through china and japan. in the last twenty years, it has invaded several developed countries in europe, the usa, africa and south america (see [8] and references therein). when mosquitoes are not subject to stimuli, it is possible to assume that they move randomly in any direction [9]. this leads to a diffusion equation which can be extended to take into account the landscape heterogeneity or correlated random walk. therefore, we have to incorporate advection terms or drift terms when mosquitoes, stimulated by attractants, move preferably in certain directions. for simplicity, we present the generic pde that models the spread of a mosquito (sub)population. of course, it is now well recognized that the environment heterogeneity can have an important effect [10], [11]. this is taken into account in the model by assuming spatial and temporal variations in the parameters. so, let u represent the number of insects. a possible modeling is to consider the following general advection-diffusion-reaction-like equation: ∂u ∂t = ∇ · (d∇u) − ∇ · (cu) − ~v · ∇u − g (u) , (1) in a bounded domain ω ⊂ rn (where 1 ≤ n ≤ 3) with a smooth boundary ∂ω. d (x, t) ≥ 0 is the diffusion (dispersion) coefficient or the diffusivity (d can eventually depend on u). entomologists usually admit that there is no passive transportation of aedes albopictus mosquito by the wind. conversely, the blood-seeking mosquitoes will follow odors and carbon dioxide carried by the wind; this is modeled by the term ~v (x, t) .∇u. indeed, it is well known that carbon dioxide (co2), in interaction with other components, acts as an attractant and induces a direct response to guide the mosquito towards the host. the breeding sites or the blood feeding sites attractions are modeled by the term ∇·(c (x, t) u). thus it is necessary to take this important fact into account: the term ∇. · (c(x, t)u) represents a localized attractants (or a repellings) due to the presence or not of (blood or sugar) meals (like animals, humans or fruits...). c(x, t) represents an advection velocity toward favorable “places”. the definition of c takes into account wind’s direction and strength. for the sake of simplicity, the effective attraction areas are represented as ellipses. the attractor is set as one of the foci of the ellipse. the other focus point is calculated as a function of wind’s direction and strength. outside the ellipse, there is no attraction and the related advection term is equal to zero. note that if there is no wind, the attraction area is reduced to a disk of which the center is the attractor. the reaction term g can be nonlinear, and represents the time-evolution of the population (death-birth rates, migration, vector control...), and thus may depend on the mosquito population u, and some environmental parameters (temperature, position in the domain,...). we suppose that u (x, 0) = u0 (x) for x ∈ ω, where u0 can be a continuous (or possibly discontinuous) function. it may be possible to consider generalized boundary conditions, like robin conditions, −→ ∇u·~n + αu = ρ(x, t), for all (x, t) ∈ ∂ω × (0, t ), where ~n stands for the exterior unit normal to the boundary ∂ω. for the sake of simplicity we will consider homogeneous neumann boundary conditions, i.e. α = 0. finally, we deduce the following (quasi)linear parabolic equation:   ∂u (·, t) ∂t = ∇ · (d (·, t) ∇u) − ∇ (c (·, t) u) + ~v (·, t) · ∇u +g (u, ·, t) , x ∈ ω and t > 0, u (x, y, 0) = u0 (x, y) ≥ 0, x ∈ ω−→ ∇u · ~n = 0, for all x ∈ ∂ω, and t > 0, (2) problem (2) is a (quasi)linear parabolic equation, for which it is possible to show the existence of a local solution. moreover, under mild conditions it may be possible to show that the solution is global [12]. a. the compartmental model in order to obtain some biologically interesting simulations, we take into account some biological facts about aedes albopictus [3]. there are two main stages in the development of mosquitoes: an aquatic stage and an adult stage. the aquatic stage gathers eggs, larvae and pupae. the adult stage can be divided into several compartments: immature females, blood feeding females, breeding females, resting females and males. since we assume no sex differences in the aquatic stage, mosquitoes, after emergence, are distributed between the immature female compartment and the male compartment, according to r, the ratio of adult females mosquitoes to the total mosquito population. after mating with males, immature females enter the feeding female compartment and seek biomath 1 (2012), 1209262, http://dx.doi.org/10.11145/j.biomath.2012.09.262 page 2 of 7 http://dx.doi.org/10.11145/j.biomath.2012.09.262 c. dufourd et al., modeling and simulations of mosquito dispersal. the case of aedes albopictus ∂ua ∂t = negg(1 − ua k ) · 1b · ub(x, t)dt − (ηa + ma)ua, ∂uy ∂t = ∇ · (d∇uy ) + ~v · ∇uy − (my + βy )uy + rηaua, ∂uf ∂t = ∇ · (d∇uf ) + ~v · ∇uf − ∇(cf (x)uf ) − (mf + µf r1f )uf + µbf 1bub + βy um um + um s uy when um s 6= 0, = ∇ · (d∇uf ) + ~v · ∇uf − ∇(cf (x)uf ) − (mf + µf r1f )uf + µbf 1b · ub when um s=0, ∂ur ∂t = ∇ · (d∇ur) + ~v · ∇ur − (mf + µrb1b)ur + µf r · uf ,   ∂ub ∂t = ∇.(d∇ub) + ~v · ∇ub − ∇(cb(x)ub) − (mf + µbf 1b)ub + µrb · ur, ∂um ∂t = ∇ · (d∇um ) + ~v · ∇um − ∇(cf (x)um ) − ∇(cb(x)um ) − mm · um + (1 − r)ηa · ua, ∂um s ∂t = ∇ · (d∇um s) + ~v · ∇um s − ∇(cf (x)um s) − ∇(cb(x)um s) − mms · um s + λs · 1s, ∀(x, t) ∈ qt ua(x, y, 0) = ua0 (x, y) x ∈ ω ux (x, y, 0) = 0 x ∈ ω with x ∈ {y, f, b, m, m s} ~∇ux (x, y, t) · ~n = 0 x ∈ δω and 0 < t < t with x ∈ {a, y, f, r, b, m, m s} (3) for blood meals before going into the resting compartment. afterwards, the females pass into the breeding compartment seeking for a breeding site to deposit eggs. once egg deposit is done, breeding females need blood and pass into the feeding compartment again. the eggs laid by the breeding females supply the aquatic stage. in order to simulate sit control, we add a sterile male compartment. sterile males are released in specific places. the transmission rate between the immature females and the feeding females is conditioned by the proportion of non-sterile males to the whole male population, near the immature females. contrary to anopheles mosquito, aedes males are looking for young females and thus, in general, they are located near the breeding sites or near the hosts. we assume that resting females are not subjected to the attraction of blood meals or breeding sites. resting females diffuse slowly, and their direction can be affected by the wind. the rate of transmission from one compartment to another allows us to take into account the average time spent by mosquitoes in each compartment. according to the previous explanations, we derive model (3). after rescaling, we consider ω = [−1, 1]2, qt = ω × (0, t ]. we set three indicator functions, 1b, 1f and 1s. function 1b defines the area where the breeding females found a breeding site to lay eggs and become feeding females. 1f defines the area where feeding females found a blood meal and become resting females. 1s defines the area where sterile males are released. cf represents the attraction due to blood meals, like houses, and cb represents the attraction due to the breeding sites. ma, my , mf , mm and mms are respectively the mortality rates for the aquatic stage, the immature females, the mature females, the wild males and the sterilized males. ηa is the egg hatching rate, βy is the rate at which immature females become bloodfeeding females, µf r is the rate at which blood-feeding females become resting females, µrb is the rate at which resting females become breeding females, µbf is the rate at which breeding females become blood-feeding females, negg is the average number of eggs laid per female, k is the carrying capacity of a breeding site, and λs is the number of sterile males released periodically each τ days. following [4], [5], we assume that the probability that an immature female becomes a feeding female depends on the ratio um um +um s , when sterile males are released. system (3) can be rewritten in the following way, with u = (ua, uy , uf , ur, ub, um , um s) t ,   ∂u ∂t = ∇ · (d∇u) − ∇ (c (x) u) + ~v (x, t) · ∇u+ +mu + γ(x, t) ∈ qt , ∇u · ~n = 0, x ∈ ∂ω and 0 < t < t, u (x, 0) = u0 (x) x ∈ ω, (4) biomath 1 (2012), 1209262, http://dx.doi.org/10.11145/j.biomath.2012.09.262 page 3 of 7 http://dx.doi.org/10.11145/j.biomath.2012.09.262 c. dufourd et al., modeling and simulations of mosquito dispersal. the case of aedes albopictus with γ(x, t) = 0 bbbbbbb@ 0 0 0 0 0 0 λs 1 ccccccca , d (·, t) = d (·, t) 0 bbbbbbb@ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 ccccccca , c (·, t) = 0 bbbbbbb@ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 cf 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 cb 0 0 0 0 0 0 0 cf + cb 0 0 0 0 0 0 0 cf + cb 1 ccccccca , m (·, t) = 0 bbbbbbb@ −(ma + ηa) 0 0 −rηa −(my + βy ) 0 0 βy ( um um +um s ) −(mf + µf r1f ) 0 0 µf r1f 0 0 0 (1 − r)ηa 0 0 0 0 0 0 negg(1 − ua k )1b 0 0 0 0 0 0 0 0 0 0 −(mf + µrb) 0 0 0 µrb −(mf + µbf 1b) 0 0 0 0 −mm 0 0 0 0 −mms 1 ccccccca . problem (4) is a (quasi)linear weakly coupled parabolic system, for which it is possible to show the existence of a local solution. the existence of a unique global solution may be proved using, for instance, [13]. to provide numerical simulations of system (3), we need to construct a reliable algorithm, that preserves most of the properties of the system (positivity of the solution, equilibrium, if any, and its (un)stability properties, ...). b. the numerical methods we consider an operator splitting method [7]. this is an interesting method that enables us to solve separately each term of equation (1). so, we will solve successively the convective term, the diffusive term, and the reaction term, using the most efficient numerical method for each process. the full system can also be rewritten as follows ut = f (x, u) = a (u) + d (u) + r (u) , (5) where a represents the advective (or convective) terms, d the diffusive terms and r the reaction terms. briefly, • the advection process is solved using the corner transport upwind scheme (ctu) [14]. it is a total variation diminishing scheme, that preserves the monotonicity of the solution providing that we verify a cfl condition between the convective parameters, the space-steps and the time-step. moreover, this scheme minimizes the numerical diffusion and is even exact when ∆t is chosen appropriately. • the diffusion process is solved using the method of lines (mol) for which we consider the secondorder finite difference method for the spatial discretization, and the tr-bdf2 (trapezoide rule 2nd order backward difference formula) method for the time discretization. • the reaction process is solved using the nonstandard finite difference method [15]. more details are developed in a forthcoming paper (see also [6]). our scheme permits to provide several simulations with and without constant parameters. in particular, we show that the solution converges, at least numerically, to a steady state. the numerical algorithm is implemented in scilab [16], while the visualization is obtained with “r” [17]. iii. simulations and discussions we present some simulations in a homogeneous landscape, with time independent parameters, where there are 4 breeding sites and 5 blood-feeding sites as in fig. 1. we assume that all these attractors have the same attraction force and domains of attraction. for each attractor, the area of attraction is defined as an ellipse depending on the wind’s direction and strength. note that, if there is no wind, the area of attraction is reduced to a disk. fig. 1. homogeneous landscape with 5 houses (blood meals) and 4 breading sites when the landscape is homogeneous, without wind, the mature females, i.e., the blood feeding females, the biomath 1 (2012), 1209262, http://dx.doi.org/10.11145/j.biomath.2012.09.262 page 4 of 7 http://dx.doi.org/10.11145/j.biomath.2012.09.262 c. dufourd et al., modeling and simulations of mosquito dispersal. the case of aedes albopictus fig. 2. mature females distribution with no wind, no vector control. fig. 3. mature females distribution with north wind, mechanical control and releases of 1000 sterile males per week, from the 20th simulation day. resting females and the breeding females, tend to gather around the houses and the breeding sites (fig. 2). this distribution is quite disturbed when wind and vector control get involved. fig. 3, show the distribution of females when we consider north wind, the removal of the two east breeding sites and the releases of sterile males. we notice that the majority of the mature females tend to migrate north, following odors or co2 carried by the wind. they gather around the western attractors since the two east breeding sites no longer exist. moreover, we notice that the number of mature females has decreased. this decrease can be explained by the fact that sterile males have been released, but also because of the north migration that allows mosquitoes to leave the domain. sit is respectful towards the environment, and is likely to be used as an alternative to the use of chemical products. on a temporal scale, we compare the impact of a 7-days periodic massive spraying of deltamethrin around houses over a one-month treatment with 7-days fig. 4. mature females distribution with north wind, and weekly releases of 1000 sterile males, near the north breading sites, from the 20th simulation day. fig. 5. mature females distribution with north wind, and weekly releases of 1000 sterile males, near the south breading sites, from the 20th simulation day. pulsed sterile males releases near the breeding sites over the same period of time. fig. 6 shows that with an appropriate choice in the periodicity of the releases, and the number of released sterile males, sit could be a promising alternative to massive spraying. however, in order to have the most efficient results using sit, it is important to have a good knowledge on the environment and take it into account. for instance, compare fig. 4 and fig. 5: they show the distribution of mature females controlled with sit near the north breeding sites, and near the south breeding sites, respectively. we know that, with north wind, females tend to migrate towards the north; this is also the case for the released sterile males. that is why northern releases have a lower impact on the number of mature females compared to southern releases that are more efficient. thus, it is more efficient to release sterile males downwind rather then biomath 1 (2012), 1209262, http://dx.doi.org/10.11145/j.biomath.2012.09.262 page 5 of 7 http://dx.doi.org/10.11145/j.biomath.2012.09.262 c. dufourd et al., modeling and simulations of mosquito dispersal. the case of aedes albopictus fig. 6. comparison of the evolution of the number of mature females with a weekly massive spraying control, and sit with weekly releases of 2000, 4000, 10000 and 20000 sterile males over one month. fig. 7. evolution of the density of mature females with northern wind with two spatial strategies of sit ( 1000 sterile males released per week) from the 20th day. upwind (on condition that the wind strength is not to high, otherwise mosquitoes will hide wherever they can rather than fly upwind). this result is even more obvious when we have a look at the temporal evolution of the number of mature females depending on which spatial strategy is chosen for sit releases (fig. 7). iv. conclusion this work presents promising tool for modeling mosquito dispersal. thanks to the numerical simulations we could point out interesting results that can be a great use for optimizing sit. our work can be helpful to propose new field experiments; in particular it may be possible to consider temperature-varying parameters. this will be presented in a forthcoming paper. we could also add other compartments (sugar feeding compartment, sterile female compartment,... ). as we could see, the environment has a non-negligible influence on mosquito dispersal, and some works need to be done about landscape ecology because, so far, little is known about the interactions between landscape, vegetation and aedes albopictus dispersal [11]. acknowledgment the tis project was financially supported by the french ministry of health and the feder convergence réunion 2007–2012 program. this paper is dedicated to axel, for his fifteenth birthday. references [1] y. dumont, f. chiroleu, and c. domerg, “on a temporal model for the chikungunya disease: modeling, theory and numerics,” mathematical biosciences, vol. 213, no. 1, pp. 80–91 (2008). http://dx.doi.org/10.1016/j.mbs.2008.02.008 [2] y. dumont and f. chiroleu, “vector control for the chikungunya disease.” mathematical biosciences and engineering, vol. 7, no. 2, pp. 313–345 (2010). http://dx.doi.org/10.3934/mbe.2010.7.313 [3] h. 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[online]. available: http://www.r-project.org/ biomath 1 (2012), 1209262, http://dx.doi.org/10.11145/j.biomath.2012.09.262 page 7 of 7 http://dx.doi.org/10.1007/s00028-002-8081-2 http://dx.doi.org/10.1016/s0022-0396(03)00165-7 http://dx.doi.org/10.1016/0021-9991(90)90233-q http://www.scilab.org http://www.r-project.org/ http://dx.doi.org/10.11145/j.biomath.2012.09.262 introduction the mathematical model the compartmental model the numerical methods simulations and discussions conclusion references original article biomath 2 (2013), 1312291, 1–7 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum stochastic arithmetic as a tool to study the stability of biological models rene alt laboratoire d’informatique de paris 6 university pierre et marie curie, 4 place jussieu, 75252 paris cedex 05, france e-mail: rene.alt@lip6.fr jean-luc lamotte laboratoire d’informatique de paris 6 university pierre et marie curie, 4 place jussieu, 75252 paris cedex 05, france e-mail: jean-luc.lamotte@lip6.fr received: 27 october 2013, accepted: 29 december 2013, published: 15 january 2014 abstract—the theoretical study of the stability of the numerical solution of a differential system may be complicated or even not feasible when the system is large and nonlinear. here it is shown that such a study can be experimentally done by using stochastic arithmetic and its discrete approach known as the cestac method. the cestac method has been first proposed since more than forty years by m. la porte and j. vignes as an experimental statistical method to estimate the accuracy on the result of numerical program [10], [14]. later an abstract formalization of the theory called stochastic arithmetic has been developed and many of its algebraic properties have been studied [2], [4], [5]. here a brief presentation of stochastic arithmetic, of its main properties and of the different software existing for its implementation are given. then it is demonstrated that the use of stochastic arithmetic in the solver of a differential system can easily reveal whether the computed solution is stable or not. moreover the stability can be studied with respect to the coefficients of the system or with respect to the initial conditions. at the end it is also pointed out that the same method can be used to detect instabilities due to the used solver. some examples taken from the biological literature are given [1], [6], [7]. keywords-stochastic arithmetic, cestac method, stability of differential biological models. i. introduction the detection of instabilities in the numerical solution of differential systems is generally not an easy thing to do. actually instabilities have two main sources. the theoretical differential system can be stiff or inherently unstable and the numerical method used to solve it can also be unstable. a common example of this last situation is the numerical solution of a stiff system using an explicit method and a too large step. the classical approach to know whether a differential system is stable or not is the computation of its jacobian and of its eigenvalues, see for example [12]. this is generally not so easy and requires most of the time manual calculation or the use of a computer algebra system. in the same manner the use of an explicit method to solve a differential system is rather simple but may lead to numerical instabilities if the step happen to be too large even if the method has an automatic calculation of the step. on the contrary an implicit method may not have this step restriction but requires at each step the solution of a system of equations which is non linear when the differential system is non linear. and this is the case for most biological models. here it is shown that a simple method called the cestac method can be easily used to investigate the sensitivity of the computed solution to the coefficients and initial conditions of a differential system and to detect possible instabilities. various numerical examples coming from the modelisation of bacterial growth are presented. the structure of this paper is as follows. first stochastic arithmetic and the cestac method are shortly recalled. then a software called cadna which implecitation: rene alt, jean-luc lamotte, stochastic arithmetic as a tool to study the stability of biological models, biomath 2 (2013), 1312291, http://dx.doi.org/10.11145/j.biomath.2013.12.291 page 1 of 7 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2013.12.291 r alt et al., stochastic arithmetic as a tool to study the stability... ments the cestac method is presented and the use of this software to investigate the stability of the solution of a differential system is detailed. its efficiency is illustrated on several biological models. ii. stochastic arithmetic and the cestac method a. stochastic arithmetic stochastic arithmetic is a model for exact computation on imprecise data. it can be summed-up as follows. let us assume that an imprecise data can be represented as a gaussian distribution with known mean value m and known standard deviation σ, σ ≥ 0. in the following such an imprecise data is called a stochastic number. thus the set of stochastic numbers denoted s is the set of gaussian random variables. one of the main properties of a gaussian distribution and hence of a stochastic number is: for x = (m,σ) the confidence interval of m with a probability p = (1 − η) is defined as [m−λησ,m + λησ] where λη is such that p (x ∈ [m−λησ,m + λησ]) = 1−η, (1) it is well known that for η = 0.05, that is p = 1−η = 0.95, we have λη = 1.96. consequently the number of significant decimal digits on m is the integer part of: cη,x = log10 ( |m| λησ ) , (2) provided that |m|/(λησ) ≥ 10, otherwise we assume cη,x = 0. the ratio λησ/|m| is called the relative error of the stochastic number x. this characteristic is analogous to the relative error of an approximate number. the arithmetic operations on stochastic numbers are defined as the operations on independent gaussian distributions. they are denoted s+, s−, s∗, s/ in [8] and [9] but here the simpler notations +,−, ×, / are preferred. they are: x1 + x2 def = ( m1 + m2, √ σ21 + σ 2 2 ) x1 −x2 def = ( m1 −m2, √ σ12 + σ22 ) x1×x2 def = ( m1m2, √ m22σ12 + m12σ22 + σ 2 1 σ 2 2 ) x1/x2 def = ( m1/m2, √( σ1 m2 )2 + ( m1σ2 m22 )2 + ( σ1σ2 m22 )2) (3) the first three formulae including stochastic multiplication are exact. the formula for the division is only exact to the first order terms in σ/m and must be considered as an approximation. actually it is well known that the distribution of the ratio of two gaussian variables with expected value 0 and variance 1 is not gaussian but follows a cauchy law which has no mean value and no standard deviation but is symmetric and has a mode and a median. more details on stochastic arithmetic can be found in [2], [4], [5], [11], [13]. b. the cestac method when one wants to develop a software to estimate the accuracy of a numerically computed result a first possibility is to use formulae (3) instead of standard floating point operations. this can be easily done as many programming languages such as c++ or fortran 90 allow the overloading (re-definition) of the floating point operations. another approach used in the cestac method, see [3], [14], [9] is to discretize the theoretical gaussian distributions with gaussian random samples and to use their empirical mean values and standard deviations instead of the theoretical ones. this is done in the cestac method in the case of rounding errors coming from the floating point operators. the idea of the cestac method is that each result of a floating point operator (assignment, arithmetic operator) which is not an exact floating point value, is always bounded by two floating point values r− and r+ obtained by rounding up or down the exact result, each of them being representative of the exact result. the random rounding mode consists, at the level of each floating point operation or assignment, in choosing as a result, randomly with an equal probability, either r− or r+. thus when a code is performed n times in a synchronous parallel way with the use of this random rounding mode, n samples rk,k = 1...,n of each computed results are obtained, and then from these samples, the accuracy of the mean value r of these samples, considered as the computed result, may be estimated. hence a probabilistic model of the round-off error on a computed result obtained with the random rounding mode has been developed, see [9]. in this model it is shown that under two simple hypotheses which generally hold in real life problems, each sample obtained by the cestac method may be modelled by a random variable biomath 2 (2013), 1312291, http://dx.doi.org/10.11145/j.biomath.2013.12.291 page 2 of 7 http://dx.doi.org/10.11145/j.biomath.2013.12.291 r alt et al., stochastic arithmetic as a tool to study the stability... z defined by: z ' r + nb∑ i=1 ui(d)2 −pzi, zi ∈]−1,+1[ (4) where r is the unknown exact result, p is the length of the mantissa in the floating point representation of numbers, ui(d) are constants, nb is the number of arithmetic operations and zi are independent centred and equidistributed variables. as a consequence: e(z) ' r, the distribution of z is quasi-gaussian and the estimation of the accuracy on r can be done with the student test which provides a confidence interval for r. hence the number of decimal significant digits cr of r can be estimated by: cr = log10 (√ n ∣∣r∣∣ τησ ) (5) where r = 1 n n∑ k=1 rk, σ 2 = 1 n −1 ( rk −r )2 τη is the value of the student distribution for n − 1 degrees of freedom and a probability level 1−η. from a theoretical point of view the vector of the n empirical values representing a floating point result used in the cestac method is called a discrete stochastic number. in the same manner the cestac method is said to use a discrete stochastic arithmetic (dsa). it must be noted that the cestac method differs from a simple monte carlo method where a gaussian noise would be added to the data and the program would be run several times. in contrast in the cestac method a gaussian noise is actually added to the data but also after each arithmetic operation n results are computed being rounded randomly up or down. moreover the runs are done in such a way that at each test the same branching is performed in all runs. thus after each arithmetic operation or each test the theoretically computed stochastic number appears as a vector of n empirical values really representing the same theoretical value. hence, the number of decimal significant digits of any intermediate result can be computed using formula (5) in the same manner as the one(s) of the final result(s). this would not be the case in a simple monte carlo method. c. the cadna software the cadna software implements the so-called discrete stochastic arithmetic, which is nothing else than the cestac method to which have been added comparison operators, the notion of non-significant result and some more complementary features. it can be freely downloaded from [8]. two versions exist, one in fortran 90 and one in c++. they have been developed as libraries to be added to an already existing code. in this software, new types for single precision and double precision stochastic numbers have been defined and all arithmetic operators and tests have been overloaded so that computing with stochastic numbers is as easy as computing with real numbers. thus, any fortran 90 or c++ code working with real numbers can be almost instantly converted in a code working with stochastic numbers, i.e. numbers with their errors. it has been theoretically and experimentally shown that formula (5) is correct to one digit with n = 3, consequently in the cadna software all stochastic numbers are represented as three samples with gaussian distribution. another feature of the cadna software is that stochastic numbers represent imprecise numbers where the error is due not only to rounding in floating point arithmetic but also to the data which may also be imprecise. thus in this software it is possible to introduce errors in the data so that an imprecise data is represented by a stochastic number with a known mean value and a known standard deviation. this possibility is used in the experimental investigation of the stability of the numerical solution of differential systems especially those coming from the modelisation of biological reactions. iii. application to some biological models many biological models are represented as nonlinear differential systems. studying their stability may be difficult, see for example [6]. but the use of the cadna software leads to an immediate answer to the question: “is the computed solution stable around this special value of this particular parameter?” of course this is not the answer to the more general question “what is the domain of stability of the system?” but as is shown below, it can easily help to analyse the sensitivity of the solution to some parameter or to initial conditions. as illustrations of the efficiency of the cestac method to experimentally investigate the stability of computed solutions, several biological model solutions have been computed using a fourth order runge-kutta method together with the cadna software. biomath 2 (2013), 1312291, http://dx.doi.org/10.11145/j.biomath.2013.12.291 page 3 of 7 http://dx.doi.org/10.11145/j.biomath.2013.12.291 r alt et al., stochastic arithmetic as a tool to study the stability... errors have been introduced in the initial conditions and in some parameters to see the effect of imprecise data on the computed solution. the results are reported in the corresponding figures. in these figures the solutions computed with discrete stochastic numbers are represented as three curves corresponding to the three samples as explained above. a. first model the model is taken from [7] for cyclodextrin-glucanotransferase production by immobilised cells of bacillus circulans atcc21783. the differential system is: dx dt = (µ(s)−γ)x ds dt = −µ(s) x yx/s dp dt = (αµ(s) + β)x (6) with initial conditions s0,x0,p0. in this system x is the biomass concentration, s is the substrate concentration and p is the product concentration. µ(s) is the andrews function: µ(s) = µmax sks+s+kis2 . the experiments have been done with the coefficients: α = 1.11,β = 0.07,γ = 0.06,µmax = 2, yx/s = 26.3, ks = 0.8, ki = 0.12, s0 = 2, x0 = 0.2, p0 = 0 this system has been successively solved with some tolerances introduced on the initial condition for the substrate (s0 = 2 ± 0.2), then for the biomass concentration (x0 = 0.2 ± 0.1), then on the coefficient µmax (µmax = 2 ± 0.3) and on the coefficient yx/s (yx/s = 26 ± 1). at last the system has been solved using a too large step (h = 0.9) so that the numerical integrating method is unstable. the results are reported in figures 1 to 5. a simple observation of these figures shows that errors on the initial conditions or on the coefficients cause two kinds of modifications on the solution. one kind is an increase or decrease of the limit of the product as in figures (1) and (4), the other kind is a modification of the delay after which the biomass and product begin a fast increasing as in figures (2) and (3). another conclusion is that the system is much more sensitive to the initial substrate concentration than to the initial biomass concentration. b. second model the model is taken from alt and markov [1] for e. coli + glucose. in this model the microbial population is subdivided into two subgroups: i) micro-organisms in lag and stationary phase are classified into one subclass with biomass denoted x. it is assumed that micro-organisms in that class experience unfavourable growth conditions and are not able to immediately produce anything; ii) active (viable) micro-organisms in log phase, denoted y, possessing a complete set of active enzymes. bacteria in dying state are modelled by decay terms and need not be assigned to a special subgroup. the system of equations is here: ds/dt = −k1xs−βys, dx/dt = −k1xs + k2y −kdx2, dy/dt = k1xs−k2y + βys, (7) with the initial conditions s(0) = s0, x(0) = x0, y(0) = y0. the terms participating in this system have the following meaning: k1xs models the consumption of s by bacteria x and the transition of (fasting) bacteria x into (viable, active) bacteria y; βys models the consume of s by bacteria y and the increase of bacteria biomass y due to nutrition and reproduction; k2y models the random transitions of bacteria from class y into class x; kdx2 models competition and decay of (starving) bacteria x. this system has been solved with the following coefficients and initial conditions: k1 = 0.23, k2 = 0.85, kd = 0.3, β = 1.0, s0 = 2.0, x0 = 0.25, y0 = 0. to study the stability of the solution with respect to the initial conditions, some relative errors have been successively introduced into them. they were: 5% on s0 and 25% on x0. the results are given in figure (6) and (7). these two figures clearly show that the solution is much more sensitive to an error on the initial amount of substrate than to the initial amount of biomass. in fact a 25% relative error on the biomass perturbs the solution less that a 5% error on the initial substrate. this remark goes in the same direction as the corresponding one for the first model. c. third model the model is taken from [6] for 1,2-dichloroethane (dca) biodegradation by klebsiella oxytoca va 8391 immobilized on granulated carbon. the differential system biomath 2 (2013), 1312291, http://dx.doi.org/10.11145/j.biomath.2013.12.291 page 4 of 7 http://dx.doi.org/10.11145/j.biomath.2013.12.291 r alt et al., stochastic arithmetic as a tool to study the stability... is: dx1/dt = (µ1(s)−d)x1 + kimxim dxim/dt = (µim(s)−kim)xim ds/dt = −( 1 γ µ1(s) + β1)x1 − ( 1γµim(s) + βim)xim +d(sin −s)−kla(1−µ2(s))s dp/dt = ( 1 γ µ1(s) + β1)x1 +( 1 γ µim(s) + βim)xim −dp (8) with: µ1(s) = m1s ks+s+s2/ki µim(s) = mims ks+s+s2/ki µ2(s) = m2s k+s (9) with initial conditions x1(0), xim(0), s(0), p(0). the signification and values of the coefficients of this system can be found in [6]. this model differs from the classical bioreactor models as a phase with immobilised cells has been added to the normal free cells. these cells are attached to carrier particles and can grow and detach from the solid surface to leak into the liquid. in this model x1, xim, s, p respectively represent the free cells, the immobilised cells, the substrate and the product. a detailed theoretical study of the stability of this model has been developed in the cited paper. two experiments are reported here. the solutions have been computed with the coefficients proposed by the authors and initial conditions: x1(0) = 0.02, xim(0) = 9, s(0) = 0.25, p(0) = 0, which satisfy the theoretical stability conditions. in the first experiment relative errors have been simultaneously introduced in the initial conditions: 10% on x1(0), 50% on xim(0) and 40% on s(0), i.e. the initial values have been n times (n = 3) randomly generated in the intervals x1(0)±10%, xim(0)±50%, s(0)±40%. the second experiment has been done with the same coefficients but with 100% simultaneous relative errors in the initial conditions s(0), x1(0) and xim(0). the results are shown in figures (8) and (9). it must be noted that the concentrations of free cells and of immobilized cells are of different order, close to 0.3 kg/m3 for the free cells and close to 200 kg/m3 for the immobilised cells. this is why in the figures the concentrations of immobilised cells have been scaled by a factor 0.001. in the second experiment the cadna software detects several instabilities as some multiplications and divisions have non-significant results. anyhow the corresponding figure (9) shows that the asymptotes of all components do not depend on the initial conditions. indeed this fact which can probably be proved has been numerically experimented by running the program with many different initial conditions and the asymptotes are always identical provided that xim(0) 6= 0 and that s(0) satisfies the stability condition: s0 ≤ dsin −β1x1(0)−βimxim(0) as explained in [6]. iv. conclusion stochastic arithmetic has been proved to be an interesting method for the estimation of the error on a computed result when the data are inaccurate and the arithmetic operators introduce round-off errors. in this paper it has been experimentally shown on several bioreactor models that the cestac method and the corresponding cadna software which are based on stochastic arithmetic can provide an easy alternative to theoretical studies when one wants to know whether the computed solution of a differential system is correct and stable or not. of course the cadna software does not lead to the domain of stability of the system but only to the knowledge of the stability of a particular solution computed with a particular numerical method. but in many cases the interest is not in the whole domain of stability but only in the solution of a particular model of a real experimental bioreactor. as presented on the models taken from the classical literature, the cadna software is particularly efficient in the detection of the coefficients or initial values, a small variation of which introduces a large variation in the solution. in the same idea, the coefficients which have a very little influence on the solution can be detected as well. this gives a precious information to the biologists who make the real experiments on the necessity of knowing some coefficients with a very good accuracy whereas some others can be only roughly known. acknowledgment the authors want to thank the anonymous referees for their extremely useful comments and remarks. biomath 2 (2013), 1312291, http://dx.doi.org/10.11145/j.biomath.2013.12.291 page 5 of 7 http://dx.doi.org/10.11145/j.biomath.2013.12.291 r alt et al., stochastic arithmetic as a tool to study the stability... v. figures 0 20 40 60 80 100 120 0 5 10 15 20 time, (h) cgtase beschkov model, s0=2 +0.2 biomass substrate product fig. 1: sensitivity of system (6) to s0 = 2±0.2 0 20 40 60 80 100 120 0 5 10 15 20 time, (h) cgtase beschkov model, x0=0.2 +0.1 biomass substrate product fig. 2: sensitivity of system (6) to x0 = 0.2±0.1 0 20 40 60 80 100 120 0 5 10 15 20 time, (h) cgtase beschkov model, mumax=2 +0.3 biomass substrate product fig. 3: sensitivity of system (6) to µmax = 2±0.3 0 20 40 60 80 100 120 0 5 10 15 20 time, (h) cgtase beschkov model, yxs=26 +1 biomass substrate product fig. 4: sensitivity of system (6) to yx/s = 26±1 0 20 40 60 80 100 120 0 5 10 15 20 time, (h) cgtase beschkov model, step=0.9 biomass substrate product fig. 5: solution of system (6) with integrating step h = 0.9 0 0.5 1 1.5 2 2.5 0 2 4 6 8 10 12 14 time, (h) s. markov model, relat. err. on initial substrate: 5% substrate biomass log phase biomass lag phase fig. 6: sensitivity of system (7) to s0 = 2±5% biomath 2 (2013), 1312291, http://dx.doi.org/10.11145/j.biomath.2013.12.291 page 6 of 7 http://dx.doi.org/10.11145/j.biomath.2013.12.291 r alt et al., stochastic arithmetic as a tool to study the stability... 0 0.5 1 1.5 2 2.5 0 2 4 6 8 10 12 14 time, (h) s. markov model, relat. err. on initial log phase: 25% substrate biomass log phase biomass lag phase fig. 7: sensitivity of system (7) to x0 = 0.25±25% 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 20 40 60 80 100 120 140 160 time, (h) n. dimitrova model relat err on init cond:s0=0.25+-10%, x0=0.02+-50%, xim0=9+-40% free cells 0.001*immobilised cells substrate product fig. 8: sensitivity of (8) to x1(0), xim(0), s(0) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 20 40 60 80 100 120 140 160 time, (h) n. dimitrova model relat err on init cond:s0=0.25+-100%, x0=0.02+-100%, xim0=9+-100% free cells 0.001*immobilised cells substrate fig. 9: sensitivity of (8) to 100% errors on initial conditions references [1] r. alt, s. markov, theoretical and computational studies of some bioreactor models, computers and mathematics with applications 64 (2012), 350–360. 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[14] j. vignes, a stochastic arithmetic for reliable scientific computation, math. and comp. in sim. 35, 1993, 233–261. biomath 2 (2013), 1312291, http://dx.doi.org/10.11145/j.biomath.2013.12.291 page 7 of 7 http://dx.doi.org/10.1002/9780470724163 http://dx.doi.org/10.1007/978-3-540-78827-0_12 http://dx.doi.org/10.1016/j.camwa.2012.02.067 http://www-pequan.lip6.fr/cadna// http://dx.doi.org/10.1023/b:numa.0000049474.51465.41 http://dx.doi.org/10.11145/j.biomath.2013.12.291 introduction stochastic arithmetic and the cestac method stochastic arithmetic the cestac method the cadna software application to some biological models first model second model third model conclusion figures references original article biomath 1 (2012), 1209022, 1–6 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum modeling of viral dynamics after liver transplantation in patients with chronic hepatitis b and d natalie filmann∗, eva herrmann∗ ∗institute of biostatistics and mathematical modeling department of medicine, goethe university, frankfurt main, germany emails: filmann@med.uni-frankfurt.de, herrmann@med.uni-frankfurt.de received: 15 july 2012, accepted: 2 september 2012, published: 12 october 2012 abstract—viral kinetic models have become an important tool for understanding the main biological processes behind the dynamics of chronic viral diseases and optimizing effectiveness of anti-viral therapy. we analyzed the dynamics of hepatitis b and d co-infection (hbv/hdv) and the pharmacokinetics/pharmacodynamics of the reinfection prophylaxis with polyclonal antibodies after liver transplantation. therefore we developed a mechanistic model consisting of a system of ordinary differential equations. this model was fitted by analyzing the kinetics of hbv/hdv viremia after liver transplantation in patient data and correlated with the reinfection prophylaxis dosing schemes. the results suggest that this modeling approach may help to optimize reinfection prophylaxis. keywords-infectious diseases; hepatitis b and d; viral dynamics; pk/pd i. introduction hepatitis b is an infectious disease of the liver caused by the hepatitis b virus. although vaccination is possible nowadays, hepatitis b is still a major concern in global health. approximately 2 billion people have been infected with the hepatitis b virus (hbv) ( [3], [4]) and it is estimated that 350-400 million people are chronic carriers of hbv [5]. persistent hepatitis b infection comprises a high risk for liver cirrhosis or hepatocellular carcinoma [3]. in these cases liver transplantation often remains the only therapy option. a. hepatitis b virus the hepatitis b virus is a dna virus that belongs to the family hepadnaviridae. it replicates in the liver by utilization of an rna-mediate and reverse transcription. the produced virus is secreted into serum, where it might infect hepatocytes or be detected by the immune system and degraded. the virus itself is non-cytopathic, but apoptosis of infected hepatocytes might be induced by immune response (especially ctl-response). a viral protein of particular clinical significance is the hepatitis b surface antigen (hbsag), the envelope of the hepatitis b virus. hbsag particles (lacking of virus dna) are produced in excess by infected hepatocytes: the ratio of hbsag to complete virus particles in serum is approximately 1000-10000:1. hepatitis b surface antibodies (anti-hbs) are directed to the hepatitis b surface antigen and may prevent the entry of the virus by binding and neutralizing circulating virions [9]. b. delta hepatitis delta hepatitis is considered as the most severe form of chronic viral hepatitis frequently leading to endstage liver disease and hepatocellular carcinoma. it is caused by the hepatitis d virus (hdv), a single-stranded rna genom which depends on the hepatitis b virus surface antigen for complete replication and transmission. therefore, hdv infection only occurs in hbsagpositive individuals either as acute co-infection or as citation: n. filmann , e. herrmann, modeling of viral dynamics after liver transplantation in patients with chronic hepatitis b and d, biomath 1 (2012), 1209022, http://dx.doi.org/10.11145/j.biomath.2012.09.022 page 1 of 6 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.09.022 n. filmann et al., modeling of viral dynamics after liver transplantation in patients with chronic hepatitis b and d superinfection in patients with chronic hepatitis b [7]. c. liver transplantation liver transplantation (ltx) remains the only therapy option for patients with end-stage liver disease due to chronic hepatitis b virus (hbv) infection or hepatitis b and d (hbv/hdv) co-infection. to prevent reinfection of the graft, caused by circulating virions, hepatitis b immune globuline (hbig) and hbv polymerase inhibitors are administered. hepatitis b immune globuline (hbig) is a blood plasma product containing polyclonal antibodies (anti-hbs) against hbsag. this protection by anti-hbs in the liver transplant setting, however, is not sterile: hbv dna is detectable in the new liver even in cases with effective prophylaxis. lamivudine inhibits the production of hepatitis b virions, but neither the production nor release of hbsag particles nor delta virions. with the introduction of hbig and hbv polymerase inhibitors as standard prophylaxis, the risk for a reinfection has decreased from approximately 80% to less than 10%. despite these progresses there does not exist any rational basis for hbig doses schedules up to now, typically hbig is given during the anhepatic phase, followed by daily infusions at a fixed dose until hbsag is negative. there is large interest to optimize/individualize hbig treatment schedules, since high doses of antibodies can be a burden for the patient and hbig is very expensive [6]. ii. modeling of virus dynamics in hepatitis models for hepatitis b virus dynamics are mostly derived from the basic model for hepatitis c, introduced by neumann et al. [10]: dv dt = pi(t) − cv (t) di dt = βt (t)v (t) − δi(t) dt dt = λ − βt (t)v (t) − dt (t) in this model the uninfected cell population is denoted by t , infected cells by i and free virus particles in serum by v . uninfected cells t are assumed to be produced at a constant rate λ and to die at a rate d. free virus particles v are produced at a rate p proportional to i and are removed from the system at a rate c. target cells t are infected at a rate β proportional to t v . infected cells i are killed by the immune system at a rate δ. the effect of antiviral therapy may be modeled by partial blocking of release of virions (hence (1 − �)p, 0 < � < 1) and/or partial blocking of infection of hepatocytes ((1 − η)β, 0 < η < 1). there exist several extensions of this basic model. for example, dahari et al. introduced proliferation of (uninfected and infected) hepatocytes and a curing rate of infected liver cells, which allows modeling of complex decline profiles [11]. de sousa et al. proposed a model for chronic hbv/hdv co-infection [12]: the basic model was extended by including compartments for circulating delta virions, hdv-mono-infected and hbv/hdv co-infected liver cells. forde modeled the dynamics of chronic hbv/hdv co-infection under consideration of the patients immune response (hbvand hdv-specific ctl-response, not published). for the setting of liver transplantation only few models exist for hepatitis c ( [13], [14]). since in hepatitis c an infection of the liver graft is unavoidable with current treatments and extrahepatic compartments might play a significant role, these models may not be transferred to the case of hbv/hdvor hbv-induced liver transplantation. neumann et al. examined the effect of a single dose of monoclonal anti-hbs in patients with chronic hepatitis b [15]. he assumed that anti-hbs not only acts by neutralizing circulating hbsag and virions, but also may enter hepatocytes and reduce the release of virions and hbsag particles. iii. dynamics after liver transplantation we propose that the dynamics after liver transplantation can be described as shown in figure 1: hbig (i.e. anti-hbs particles) is injected intravenously and immediately available. anti-hbs are cleared by metabolism at a constant rate. due to binding to circulating hbsag particles, and hepatitis b virions, we have an accelerated clearing of anti-hbs, hbsag, hbv, and hdv. we assume that formed immune complexes dissociate with a certain probability. at the time of transplantation, we assume that all hepatocytes are uninfected and susceptible. free virions infect hepatocytes of the graft at a constant rate. since the replication cycle for hbv takes 1-2 days [19], we introduce two different kinds of compartments of infected cells, one, that does not secrete virus and hbsag particles yet and a compartment of mature infected cells, that does. our model is based on the basic model by neumann et al. [10], a standard one-compartment pk-model, and on the delay differention equation model for hbv by gourley et al. [18]. the corresponding ode system is biomath 1 (2012), 1209022, http://dx.doi.org/10.11145/j.biomath.2012.09.022 page 2 of 6 http://dx.doi.org/10.11145/j.biomath.2012.09.022 n. filmann et al., modeling of viral dynamics after liver transplantation in patients with chronic hepatitis b and d decribed as below: da(t) dt = d(t) vd(t) − ke1a(t) − ka(t)(h(t) + v1(t)+ v2(t)) + kdah(t) + kdav1(t) + kdav2(t) dh(t) dt = ph1i1(t) + ph12i12(t) − ca(t)h(t) − δh h(t) + cdah(t) dv1(t) dt = p1i1(t) + p12i12(t) − ca(t)v1(t) − δ1v1(t) + cdav1(t) dv2(t) dt = p2i12(t) − ca(t)v2(t) − δ2v2(t) + cdav2(t) dah(t) dt = ca(t)h(t) − cdah(t) − δah ah(t) dav1(t) dt = ca(t)v1(t) − cdav1(t) − δav1 av1(t) dav2(t) dt = ca(t)v2(t) − cdav2(t) − δav2 av2(t) dt (t) dt = λ − δt (t) − β1v1(t)t (t) − β2v2(t)t (t) e1(t) dt = β1v1(t)t (t) − δe1(t) − β2v2(t)e1(t) − e−δτ β1v1(t − τ )t (t − τ ) e2(t) dt = β2v2(t)t (t) − δe2(t) − β1v1(t)e2(t) e12(t) dt = β2v2(t)e1(t) + β1v1(t)e2(t) + β2v2(t)i1(t) − δe12(t) − e−δτ (β2v2(t − τ )e1(t − τ ) + β1v1(t − τ )e2(t − τ ) + β2v2(t − τ )i1(t − τ )) di1(t) dt = e−δτ βv1(t − τ )t (t − τ ) − δi1 i1(t) − β2v2(t)i1(t) di12(t) dt = e−δτ (β2v2(t − τ )e1(t − τ ) + β1v1(t − τ )e2(t − τ ) + β2v2(t − τ )i1(t − τ )) − δi12 where a(t), the level of anti-hbs in serum, h(t), hbsag level in serum, v1(t), hbv dna in serum, v2(t), hdv rna in serum, ah(t), anti-hbs-hbsag immune complexes, av1(t), anti-hbs-hbv immune complexes, av2(t), anti-hbs-hdv immune complexes, t (t), target cells, target cells t hbv e1 hbv & hdv e12 hdv e2 hbv i1 hbd & hdv i12 hdv v2 hbv v1 anti-hbs a immune complex a-v2 generation of target cells λ cell death δ infection of target cells β2 t v 2 β1 t v1 cell death δ β 2 e1 v2 β1 e2 v1 β2 i1 v2 aging aging cell death δ cell death δ12 cell death δ1 ph1 i1 p12 i12 p1 i1 p2 i12 clearing δ2 δ1 clearing δh infusion d(t)/vd clearance ke1 production of virus cell death δ aging hbsag h immune complex a-v1 immune complex a-h kd k cd cd cd c c c binding & dissociation fig. 1. the model of the main mechanism during treatment with anti-hbs after ltx. e1(t), hbv mono-infected cells not replicating yet, e2(t), hdv mono-infected cells (cannot replicate), e12(t), hbv/hdv co-infected cells not replicating yet, i1(t), replicating hbv mono-infected cells, and i12(t), replicating cells co-infected with hbv/hdv. the compartments are described as follows: a. anti-hbs a anti-hbs a is assumed to be administered intravenously with complete and immediate bioavailability. to model the pharmacokinetics of anti-hbs we use a standard one-compartment intravenous infusion model. we assume a zero order infusion rate constant d(t) > 0 during time intervals [t starti , t stop i ], i = 1, . . . , n and d(t) = 0 for t /∈ [t starti , t stop i ], i = 1, . . . , n, and a constant volume of distribution vd. the loss of anti-hbs a due to metabolism is modeled as a first order elimination with a constant rate ke1, corresponding to the half-life of log(2)/ke1 of hbig in immunosuppressed patients [ [17], [16]]. the additional loss of anti-hbs caused by binding biomath 1 (2012), 1209022, http://dx.doi.org/10.11145/j.biomath.2012.09.022 page 3 of 6 http://dx.doi.org/10.11145/j.biomath.2012.09.022 n. filmann et al., modeling of viral dynamics after liver transplantation in patients with chronic hepatitis b and d of anti-hbs to circulating hbsag particles, hepatitis b virions, and delta virions is modeled with a constant rate k proportional to v1, and h. since the formation of immune complexes ah, av1, and av2 is a reversible reaction, we introduce immune complex compartments ah, av1, and av2 and a dissociation rate kd. b. hbsag h, hbv dna v1, and hdv rna v2 hbsag particles h are produced at constant rates ph1 and ph12 proportional to the number of infected cells i1 and i12, eliminated at a constant rate δh and are bound to anti-hbs a at a constant rate c proportional to a. the dissociation rate of hbsag is calculated as cd = ckd k . the hbv and hdv compartments v1 and v2 are described analogously, except that delta virions are exclusively produced in co-infected cells. c. immune complexes ah, and av1, and av2 the immune complex compartment ah is characterized by a constant association rate k proportional to a and h, a constant dissociation rate kd proportional to ah, and a constant clearing rate δah . the immune complex compartments av1 and av2 are described analogously. d. target ccells t target cells are infected by hepatitis b and delta virions v1 and v2 at constant rates β1 and β2 proportional to t , v1, and v2, die at a constant rate δ and are produced at a constant rate λ. e. infected cells e1, e2, e12, i1 and i12 since the replication cycle of hbv takes 1-2 days [19], for the delta virus we assume the same length, we incorporate a delay in our model: we employ the age structured model after mckendrick-forster, as it was introduced for the setting of chronic hepatitis b infection by gourley et al. [18]. target cells t infected with hbv v1 begin after τ units of time to secrete virions. cells mono-infected with hdv e2 are not able to produce delta virions (due to the lack of the helper virus), in case they are superinfected with hbv, they begin after τ units of time to secrete hbsag h, hbv v1, and hdv v2. note, that delta virus may decrease the production rates of hbv and hbsag severely in co-infected cells. infected cells not secreting virus yet e1, e2, and e12 die at the constant rate δ. we use the same death rate δ as for the target cells t , because we assume these cells are not recognized by the immune system before they start to secrete virions. if a mono-infected cell e1 is superinfected with the delta virus, we assume it will start to secrete hbsag h, hbv v1, and hdv v2 after τ units of time and neglect a possible release of hbv and hbsag particles beforehand. the increase in the number of mature infected cells i1 and i12 is proportional to the number of cells that have been infected before τ units of time and the number of free virus at the time t−τ . mature infected cells i1 and i12 die at constant rates δi1 and δi12 . iv. simple variant of the model since a reinfection with hbv or hbv/hdv after liver transplantation can be successfully prevented in most cases nowadays (the risk is less than 10% in hbv mono-infected patients, in hdv/hbv even smaller), we assume that the amount of hepatocytes that will be infected after transplantation is rather small and may be neglected. hence, we propose a simplified variant of our model that focus on the clearance of hbv, hdv and hbsag and the dose-effect relationship of anti-hbs and hbsag/hbv/hdv and does neither include liver cell nor immune complex compartments: da(t) dt = d(t) vd − ke1a(t) − ka(h(t) + v1(t) + v2(t)) dh(t) dt = −ch a(t)h(t) − δh h(t) dv1(t) dt = −c1a(t)v1(t) − δ1v1(t) dv2(t) dt = −c2a(t)v2(t) − δ2v2(t) note, that due to different methods of quantification for hbsag, hbv dna, and hdv rna, we consider different binding rates ch , c1, and c2 here. a. application of the simple model to analyze the dynamics after liver transplantation and to evaluate our model assumptions, we fitted the simplified model to data on co-infected patients that underwent liver transplantation at hannover medical school between 1994-2009. viral load (hbv and hdv), hbsag and hbig (anti-hbs) were measured serially before and after liver transplantation. since in most cases hbv dna was negative or below the limit of detection at the time of liver transplantion we only analyzed the kinetics of hdv rna, hbsag and anti-hbs. note that a previous analysis of this data with a different pharmacokinetics was published in journal of hepatology [2]. biomath 1 (2012), 1209022, http://dx.doi.org/10.11145/j.biomath.2012.09.022 page 4 of 6 http://dx.doi.org/10.11145/j.biomath.2012.09.022 n. filmann et al., modeling of viral dynamics after liver transplantation in patients with chronic hepatitis b and d 0 2 4 6 8 10 12 10 -2 10 0 10 2 10 4 10 6 time after ltx [days] h d v -r n a [ c o p ie s /m l] h b s a g [ c o p ie s /m l] , a n ti -h b s [i u /l ] 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 -2 10 0 10 2 10 4 10 6 time after ltx [days] h d v -r n a [ c o p ie s /m l] h b s a g [ c o p ie s /m l] , a n ti -h b s [i u /l ] hdv rna hdv rna below limits of detection hdv rna hbsag hbsag below limits of detection hbsag hbig administration fig. 2. fitting results of representative patients. 1) parameter fitting: the parameter ke1 was fixed to 0.028, δ2 and δh were fixed to 0.69. the parameters c2, ch , vd and k were estimated individually. the algorithms were implemented in matlab (matlab 7.10.0, mathworks inc, natick, ma, usa) using a stiff differential equation solver (ode23s, based on a modified rosenbrock formula of order 2) and nonlinear optimization routines (fminsearch, based on the nelder-mead simplex method). hereby a maximum likelihood approach was used for non-linear fitting of the model function; values below the limit of detection were considered as random variables following a normal distribution. b. results we observed a strong correlation between hdv and hbsag decline, anti-hbs increase and hbig dose rates. despite the high interpatient variation we observed an overall similar kinetic pattern with a nearly parallel decline of hdv rna and hbsag (figure 2). the decline of hbsag and hdv rna seems to be determined almost exclusively by anti-hbs administration: in cases of intermittent hbig administration, the decline was delayed. this was also reflected in our modeling approach, as there were no systematic deviations from the model fit. v. conclusion we showed that it is possible to model the dynamics of hbv/hdv-infected patients after liver transplantation with the simplified model without taking reinfection into account. the strong correlation between hdv and hbsag decline, anti-hbs increase and hbig dose rates which is also displayed by our model suggest that this approach may help to individualize and optimize hbig dosing schemes in patients undergoing hbv/hdvor hbv-indicated liver transplantation. currently hbig is mostly given at a fixed daily dose until hbsag level becomes negative. the next step is to simulate reinfections after liver transplantation by means of our general model and further variants. for example it might be important to include resistance mutations, because resistance mutations caused by lamivudine therapy might lead to reduced antigenity of hbsag and hence resistence to hbig [8]. by means of these extended models which take reinfection into account, the factors which indicate an upcoming (chronic) reinfection shall be specified by monte carlo filtering and the necessary hbig dose rate to successfully prevent reinfection shall be quantified. acknowledgment the authors would like to thank their medical cooperation partners ingmar mederacke and heiner wedemeyer. references [1] m. a. nowak, and r. m. may, virus dynamics: mathematical principles of immunology and virology, england: oxford university press, 2000. 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reaction-diffusion equation; stationary solutions; weak solutions i. introduction a parabolic reaction-diffusion system is proposed in [14] to model the wnt signaling pathway in primary hair follicle initiation in mice. the authors in [14] use a modified version of the well-known activator-inhibitor (gierer-meinardt) model [3], [4] with saturation and without source terms. an important characteristic of the model is that both species share the same (up to scaling) non-linear production term for both activator and inhibitor. a modified version of this model was studied in [12] as a proxy to reduce the parameter complexity and to capture the dynamics of the original model. global existence of solutions of both the original and the modified systems was demonstrated by estimating time-independent upper bounds for the solutions. a parameter space analysis indicated the range of the existence of turing patterns. it is demonstrated that heterogeneous solutions arise not only because of diffusion-driven instability, but also due to convergence to far-from-equilibrium solution branches. this short note compares stationary solutions in the singularly perturbed problem (letting the inhibitor’s diffusion rate tend to 0) and the reduced problem (setting the inhibitor’s diffusion rate equal to 0) based on the modified equations from [12] for the case of a one-dimensional domain. stability of the stationary solutions is analysed. we show that all strictly positive, spatially heterogeneous, regular solutions of the reduced problem are unstable. furthermore, for some parameter citation: peter rashkov, regular and discontinuous solutions in a reaction-diffusion model for hair follicle spacing, biomath 3 (2014), 1411111, http://dx.doi.org/10.11145/j.biomath.2014.11.111 page 1 of 12 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2014.11.111 p. rashkov, regular and discontinuous solutions in a reaction ... values, the spatially homogeneous solution is also unstable. the only asymptotically stable solutions are weak solutions where the activator exhibits jump discontinuities. this work is organised as follows: first we define the model system and its reduced variant as a coupled ode-reaction-diffusion system. then we restrict our attention to a one dimensional domain and convert the problem to an auxiliary two-point boundary value problem. energy methods are employed to construct the regular and weak stationary solutions. finally we establish the stability properties of the different solutions. ii. the model problem let ω ∈ rn be a bounded domain with sufficiently regular boundary ∂ω. consider the following problem that describes the spatio-temporal dynamics of two interacting species ut = � 2∆u + f(u,v), vt = d∆v + g(u,v), ∂xu(·,x) = ∂xv(·,x) = 0, x ∈ ∂ω, (1) with nonlinearities f,g given by f(u,v) = ρu u2 v(1 + κu2) −µuu, g(u,v) = ρv u2 v(1 + κu2) −µvv. (2) the functions u = u(t,x),v = v(t,x) describe the concentrations of the species at x ∈ ω for time t > 0. the initial conditions u(0, ·),v(0, ·) are sufficiently smooth so that the second derivatives in space are well-defined. the model parameters ρu,ρv,µu,µv,κ have the following physical interpretation. κ is saturation parameter for the production law for u and v, which is scaled respectively by ρu,ρv. µu,µv denote the decay rates of u and v. the diffusion constants �,d describe the diffusion speeds in the domain ω. the model equations are based on the equations proposed in [14] to model hair follicle spacing in mice. of particular interest are the properties of the non-negative stationary solutions of (1), i.e. those pairs (u,v) such that ut = vt = 0. these are those pairs (u,v) solving the problem of two coupled elliptic pdes 0 = �2∆u + f(u,v), 0 = d∆v + g(u,v), ∂xu(·,x) = ∂xv(·,x) = 0, x ∈ ∂ω. (3) a. diffusion-driven instability the mechanism of diffusion-driven (or turing) instability has been used used in mathematical and biological models to motivate the emergence of patterns and forms (spatial heterogeneities) in development processes. the classical form of the mechanism is described by a reaction-diffusion model system with two morphogens that react and diffuse in the domain producing heterogeneous spatial patterns [10]. let us recall the conditions for diffusion-driven instability of a steady state (û, v̂) of (3). the jacobian of the reaction-kinetic system ut = f(u,v),vt = g(u,v) evaluated at this steady state is j = ( fu fv gu gv ) . (4) from the definition of diffusion-driven instability, in the absence of diffusion d = 0, the steady state (û, v̂) must be locally unstable to spatially inhomogeneous perturbations ũ(t,x) = u(t,x) − û, ṽ(t,x) = v(t,x) − v̂, but locally stable to spatially homogeneous perturbations ũ(t) = u(t) − û, ṽ(t) = v(t) − v̂. under the ansatz ũ(t,x) = ueikxeλt, ṽ(t,x) = veikxeλt, where u,v are scalars, ũ(t,x), ṽ(t,x) will be growing in time t if the eigenvalue λ associated to the wave number k > 0 satisfies reλ > 0. for the spatially homogeneous perturbation (k = 0) the eigenvalue λ associated to the wave number k = 0 satisfies reλ < 0. both conditions can be written in terms of the dispersion relation m(λ,k) relating the eigenvalue λ to the wavenumber k. the conditions for biomath 3 (2014), 1411111, http://dx.doi.org/10.11145/j.biomath.2014.11.111 page 2 of 12 http://dx.doi.org/10.11145/j.biomath.2014.11.111 p. rashkov, regular and discontinuous solutions in a reaction ... diffusion-driven instability can be formulated as follows. first, we require the equation m(λ, 0) = 0 to have solutions with reλ < 0. second, there must exist at least one k0 > 0 such that the equation m(λ,k0) = 0 has a solution with reλ > 0. in particular, this implies that the diffusion constants must be different d 6= �2. in particular biological applications the diffusion constant �2 may be so small to be negligible or u may not diffuse at all. the reduced problem with � = 0 may also exhibit diffusion-driven instability, but the above conditions may have different significance. in particular, the properties of the stationary solutions cannot be derived from a linear stability analysis of the spatially homogeneous steady state because these solutions are far-fromequilibrium solutions. we remark that spatial heterogeneities arise also in reaction-diffusion systems where the nonlinearities do not even allow the existence of a spatially homogeneous steady state (û, v̂) [12]. in this note, our attention is restricted to the onedimensional case ω = [0, 1]. for simplicity we set ρu = ρv = 1 in (2). we begin by providing some a priori estimates for the solutions u,v > 0 of the stationary problem (3). b. a priori estimates throughout the rest of the discussion we let ω = [0, 1]. lemma 1. let f,g be given by (2). assume that the pair (u,v) with positive u,v ∈ c2(ω) solves (3). then max ω u ≤ e √ µu/� min ω u, max ω v ≤ e √ µv/d min ω v. proof: for shortness we shall prove this for v. the computations for u > 0 are analogous. since we are looking for a solution (u,v) > 0, we can divide both sides of the equation for v in (3) by v and integrate over ω, µv = d ∫ 1 0 ∆v v dx + ∫ 1 0 u2 v2(1 + κu2) dx. integration by parts using the boundary condition ∂xv(0) = ∂xv(1) = 0 gives µv = d ∫ 1 0 ( ∂xv v )2 dx + ∫ 1 0 u2 v2(1 + κu2) dx, whence ∫ 1 0 ( ∂xv v )2 dx ≤ µv d . choose xmin,xmax ∈ ω such that v(xmin) = min ω v, v(xmax) = max ω v. then log max v − log min v = ∫ xmax xmin ∂xv v dx ≤ |xmin −xmax|1/2 · ∣∣∣∣∣ ∫ xmax xmin ( ∂xv v )2 dx ∣∣∣∣∣ 1/2 ≤ √ µv d , implying max v ≤ e √ µv/d min v. theorem 1. let (u,v) ∈ c2(ω) solve (3). then there exist monotone functions ψ↑,ψ↓,α such that α,ψ↑ are monotone increasing, ψ↓ is monotone decreasing, and ψ↑(�) < u(x) < ψ↓(�), (5) α(min ω u) ≤ v(x) ≤ α(max ω u). (6) the functions ψ↑,ψ↓,α are independent of d. proof: solving the equation g(u,v) = 0 (2), v can be expressed in terms of u as v = α(u) :≡ ( u2 µv(1 + κu2) )1/2 . α(u) is a monotone increasing function of u. observe that for a fixed u g(u,v) < 0 iff v > α(u) and g(u,v) > 0 iff v < α(u). choose xmin,xmax ∈ ω such that v(xmin) = min ω v, v(xmax) = max ω v. biomath 3 (2014), 1411111, http://dx.doi.org/10.11145/j.biomath.2014.11.111 page 3 of 12 http://dx.doi.org/10.11145/j.biomath.2014.11.111 p. rashkov, regular and discontinuous solutions in a reaction ... because ∆v(xmin) ≥ 0, ∆v(xmax) ≤ 0 the equality d∆v + g(u,v) = 0 implies g(u(xmin),v(xmin)) ≤ 0, g(u(xmax),v(xmax)) ≥ 0. therefore, min v = v(xmin) ≥ α(u(xmin)), min v = v(xmax) ≤ α(u(xmax)), and due to monotonicity of α we obtain min ω v ≥ α(min ω u), max ω v ≤ α(max ω u), whence α(min ω u) ≤ v(x) ≤ α(max ω u), x ∈ ω. (7) from the equation f(u,v) = 0 we express v = β(u) :≡ u (1 + κu2)µu , for u > 0. observe that for a fixed u > 0 f(u,v) < 0 iff v > β(u) and f(u,v) > 0 iff v < β(u). choose xmin,xmax ∈ ω such that u(xmin) = min ω u, u(xmax) = max ω u. because ∆u(xmin) ≥ 0, ∆u(xmax) ≤ 0 the equality �2∆u + f(u,v) = 0 implies f(u(xmin),v(xmin)) ≤ 0, f(u(xmax),v(xmax)) ≥ 0. therefore, v(xmin) > β(u(xmin)), v(xmax) < β(u(xmax)). we obtain the following estimates max ω v(x) > β(min ω u), min ω v(x) < β(max ω u). (8) combining estimates (7) and (8) we have β(min ω u) < α(max ω u), β(max ω u) > α(min ω u). note that α is monotone increasing in u, so the estimates in lemma 1 transforms these inequalities to β(min ω u) < α(e √ µu/� min ω u), β(max ω u) > α(e− √ µu/� max ω u). therefore, min u ≥ ζ1, where ζ1 is the solution of β(z) = α(e √ µu/�z), (9) and max u ≤ ζ2 , where ζ2 is the solution of β(z) = α(e− √ µu/�z). (10) hence we obtain the functions ψ↑,ψ↓ by setting ψ↑(�) = ζ1(�), ψ↓(�) = ζ2(�), and applying lemma 2 completes the proof. remark: the functions ψ↑,ψ↓ may have jump discontinuities. to estimate the behaviour of ζi, i = 1, 2, we need lemma 2. the equation (9) has as unique solution ζ1 = 0 if µv < µ2u. otherwise, the equation (9) has a unique non-negative solution ζ1(�) which is an increasing function of �. furthermore, as � → 0, ζ1(�) → 0, and as � →∞, ζ1(�) → 1κ( µv µ2u − 1). the equation (10) has a unique non-negative solution ζ2(�) which is a decreasing function of �. furthermore, as � → 0, ζ2(�) → ∞, and as � → ∞, ζ2(�) → 0 if µv < µ2u and ζ2(�) → 1κ( µv µ2u − 1) else. proof: note that α(0) = β(0) = 0, and α′ > 0, while β has a maximum at z = 1√ k . let s = e± √ µu/�,c = µv µ2u . the problem reduces to solving the equation β(z) = α(sz) or c (1 + κz2)2 = s2 1 + κs2z2 . this is a quadratic in z2, s2 −c + (2 −c)κs2z2 + κ2s2z4 = 0. (11) this quadratic has real solutions if its discriminant d = κ2s2(c2s2 + 4c − 4cs2) ≥ 0. biomath 3 (2014), 1411111, http://dx.doi.org/10.11145/j.biomath.2014.11.111 page 4 of 12 http://dx.doi.org/10.11145/j.biomath.2014.11.111 p. rashkov, regular and discontinuous solutions in a reaction ... first, note that s = e √ µu/� ≥ 1, so by descartes’ rule of signs (11) has no positive solutions for c < 1. for 1 ≤ c < 2 (11) has a positive solution iff c > s2. the positive solution of (9) is the positive square root ζ1 = √ z where z(s) = c − 2 2κ + √ c2s2 + 4c − 4cs2 2κs . for c ≥ 2 (11) has at least one positive solution iff d > 0. observe that as � → ∞,s → 1 so the positive solution ζ1 tends to the square root of lim s→1 z(s) = c − 1 κ . second, note that s = e− √ µu/� ≤ 1, so d > 0 (solutions are always real). for c < 1, descartes’ rule of signs shows (11) has a positive solution iff s2 < c. for c ≥ 1 descartes’ rule of signs shows that (11) has always one non-negative solution. hence we conclude that for µv µ2u < 1, ζ1 = 0, and for µv µ2u ≥ 1, the solution ζ1 of (9) is an increasing function of �, and lim �→0 ζ1(�) = 0, lim �→∞ ζ1(�) = √ 1 κ ( µv µ2u − 1). furthermore, ζ2(�) →∞ as � → 0, while lim �→∞ ζ2(�) = 0 for µv µ2u < 1, but lim �→∞ ζ2(�) = √ 1 κ ( µv µ2u − 1) for µv µ2u ≥ 1. in the singular-perturbation limit � → 0, the problem (3) will exhibit spike solutions with the spikes in u having small support in ω. we refer to [2], [15] for construction and analytic properties of such solutions. fig. 1 shows a typical spike solution. fig. 1. a pattern with spikes. parameter values are � = 0.01,d = 0.1,µu = 1,µv = 1.2. c. reduced problem we show that the reduced problem (� = 0) admits another class of solutions. by setting � = 0 in (3) the resulting reduced problem is an algebraicpde system 0 = f(u,v ), 0 = d∆v + g(u,v ), 0 = ∂xv, x ∈ ∂ω. (12) in the following, we shall characterise solutions of (12) and their stability properties. let us recall some basic definitions. a solution (u,v ) to the problem (12) is called regular if there exists a function h ∈ c1(r) such that the a solution u(x) of f(u,v ) = 0 given by u(x) = h(v (x)) for all x ∈ ω. if the function h is not unique, there is more than one way to choose the solution for u in the first equation in (12), so the problem may have only piecewise continuous solutions u on ω. hence it is convenient to study such solutions in a weak sense. the weak solution (u,v ) of (12) belongs to the class l∞(ω)×h1(ω) and satisfies 0 = f(u,v ), a.e. x ∈ ω, d〈∇v,∇ψ〉 = 〈g(u,v ),ψ〉, ψ ∈ h1(ω), (13) where 〈·, ·〉 denotes the h1-scalar product. proposition 1. suppose the problem (3) has a spatially homogeneous steady state (û, v̂). diffusionbiomath 3 (2014), 1411111, http://dx.doi.org/10.11145/j.biomath.2014.11.111 page 5 of 12 http://dx.doi.org/10.11145/j.biomath.2014.11.111 p. rashkov, regular and discontinuous solutions in a reaction ... driven (turing) instability at (û, v̂) occurs if u is self-activating there, in other words, fu(û, v̂) > 0. proof: the dispersion relation is a quadratic in λ. solving the dispersion relation m(λ, 0) = 0 for λ we obtain the conditions trj = fu + gv < 0, (14) det j = fugv −fvgu > 0, (15) so that reλ1,2 < 0. the derivatives are evaluated at (û, v̂). next we solve the dispersion relation m(λ,k) = 0 for λ. by vieta’s formulae λ1 + λ2 = −dk2 + trj, λ1λ2 = −fudk2 + det j, whence λ1 + λ2 < 0, for all k > 0. if fu ≤ 0, λ1λ2 > 0, so reλ1,2 < 0 for all k > 0, and no diffusion-driven instability would be possible. this proves the claim. iii. auxiliary problem in this section, we consider an auxiliary elliptic problem when at least one of u,v > 0 on some subinterval of ω. suppose that the equation f(u,v ) = 0 can be solved (not necessarily) uniquely on a subset i ⊂ ω. let u(x) = h(v (x)),x ∈ i, with h ∈ c1(r). then every regular solution of (12) on i satisfies the elliptic problem 0 = d∆v + φ(v ), x ∈ i, (16) with φ(v ) = g(h(v ),v ). the solutions of (16) can be constructed using an energy method for two-point boundary value problems. there are two cases for the function φ, depending on whether u = 0 on i or u > 0 on i. let us consider each case separately. if u(x) = 0,x ∈ i, we have h ≡ 0, so φ = −µvv almost everywhere on i. the problem (16) is reduced to the elliptic problem (17), 0 = d∆v −µvv, x ∈ i, (17) which has a non-trivial solution only under dirichlet or robin boundary conditions. next we classify the solutions when h 6≡ 0 on i. we solve formally for u in (12), u = hi(v ) = 1 ± √ 1 − 4κµ2uv 2 2κµuv , i = 1, 2. (18) and use u2 v (1 + κu2) = µuu, x ∈ i. when u > 0, the equation f(u,v ) = 0 may have locally at most two solutions. then (16) becomes 0 = d∆v + µuhi(v ) −µvv, (19) so we must solve a two-point boundary-value problem in two cases i = 1, 2 (for each solution branch for u), 0 = ∆v + 1 d (µuhi(v ) −µvv ), x ∈ i. (20) we set φi(y) = 1 d (µuhi(y) −µvy) = 1 d ( 1 ± √ 1 − 4κµ2uy2 2κy −µvy ) (21) with φ1 denoting the choice of positive square root and φ2 denoting the choice of negative square root in (21). the auxiliary elliptic problem is thus formulated: solve for v = y(x) such that 0 = y′′ + φi(y), x ∈ i, i = 1, 2. (22) recall that only solutions y ∈ (0, 1 2µu √ k ) are considered in order for the square root in (21) to be real-valued. problem (22) can be rewritten as the equivalent system (23) of first-order equations y′ = z, z′ = −φi(y). (23) note that y < 1 2µu √ κ in order for the square root in (21) to be well-defined in r. hence, without loss of generality we may assume that v < 1 2µu √ κ on i. else, we restrict the domain to a subset {x : v (x) < 1 2µu √ κ }⊂ i. biomath 3 (2014), 1411111, http://dx.doi.org/10.11145/j.biomath.2014.11.111 page 6 of 12 http://dx.doi.org/10.11145/j.biomath.2014.11.111 p. rashkov, regular and discontinuous solutions in a reaction ... we employ an energy formulation to describe the solutions y,z of (23). we set e as the total energy, u as the potential energy. the first integral of (23) is z2 2 + u(y) = e, u′ = φi. (24) to see this, differentiate the left-hand side of (24) and apply (22) and (23),( z2 2 + u(y) )′ = zz′ + u′y′ = zz′ + φiz = 0. let u have a local minimum at y0. choose total energy e such that u(0) > e > u(y0). according to [1, satz, p.92], for e > u(y0), the equation (24) defines a closed smooth curve in the (y,z)plane, which is symmetric with respect to the yaxis. then there exists some t > 0 such that z(0) = z(t), corresponding to a solution of problem (22) under homogeneous neumann boundary conditions on (0, t). as in [1] we express the solution y of the ode using (24) as y′ = ± √ 2(e −u(y)). choosing the positive value of y′ (corresponding to a monotone increasing y), rearranging the above as 1 = y′√ 2(e −u(y)) (25) and integrating both sides of (25) over x, we obtain for every l > 0, l = ∫ l 0 y′(x)√ 2(e −u(y(x)) dx = ∫ y(l) y(0) dy√ 2(e −u(y)) . (26) let 0 < y1 < y0 < y2 < 12µu √ κ be such that u(y1) = u(y2) = e, but u′(yi) 6= 0, i = 1, 2. then the integral i(e) = ∫ y2 y1 dy√ 2(e −u(y)) := l 2 . (27) is convergent [1, p.93]. then (26) defines implicitly a continuous solution y of (22) such that y(0) = y1,y( l 2 ) = y2. this solution can be continued periodically on r and the periodic function has a period l 2 . therefore, every such closed curve for suitable l will correspond to a solution of the system (23) under homogeneous neumann boundary conditions on (0, l 2 ). the properties of the solutions of (23) will depend, therefore, on the properties of the integral (27), i(e). the following lemma is a modification of a well-known result. for the idea of proof we refer to [11, lemma 3.1] or [5, lemma 5.3-5.5]. lemma 3. let u have a local minimum at x0 and a local maximum at 0. suppose 0 < x1 < x0 < x2 are such that u(x1) = u(x2) = e,u′(x1),u′(x2) 6= 0. then i(e) is a continuous function in e, and lim e→u(x0) i(e) = πu′′(x0) , lim e→u(0) i(e) = ∞. the extrema of the potential energy ui depend on the zeros of the functions φi. let us examine φi’s zeros for i = 1, 2. note that in order for the square root in (21) to be real-valued, y > 0 is such that 1 − 4κµ2uy2 ≥ 0, so we search for zeros in the interval (0, 1 2µu √ k ). lemma 4. the equation φ1(y) = 0 has no solutions in (0, 1 2µu √ k ). for µv < µ2u or µv > 2µ 2 u, φ2(y) = 0 has no solution in (0, 1 2µu √ k ). for µ2u < µv < 2µ 2 u, the solution of φ2(y) = 0 is y0 = √ µv−µ2u κµ2v . proof: after rearrangement of the terms and squaring both sides, we obtain 1 − 4κµ2uy2 = (1 − 2κµvy2)2, and after cancellation of y2 from both sides, κµ2vy 2 + µ2u − µv = 0. for µv < µ2u, the lefthand side is strictly positive, hence neither φi has a positive root. if µv > µ2u, a direct computation shows that the solution of the above quadratic is y0 = √ µv−µ2u κµ2v . yet, y0 is a root of φ2 only. biomath 3 (2014), 1411111, http://dx.doi.org/10.11145/j.biomath.2014.11.111 page 7 of 12 http://dx.doi.org/10.11145/j.biomath.2014.11.111 p. rashkov, regular and discontinuous solutions in a reaction ... when µv < 2µ2u, 1 − 2κµvy2 > 1 − 4κµ2uy2 ≥ 0, so φ1(y) = 1 d ( 1 − 2κµvy2 + √ 1 − 4κµ2uy2 2κy ) > 0. therefore, for µ2u < µv < 2µ 2 u, φ1(y) = 0 has no solution y ∈ r+. for µv > 2µ2u, the number y0 lies outside the domain of definition of the square root, (0, 1 2µu √ k ). hence none of φi, i = 1, 2 has a zero in (0, 1 2µu √ k ). now we are able to characterise the extrema of the potential energies u1,u2 on (0, 1 2µu √ k ). lemma 5. let µu,µv > 0. the potential energy u1 has a maximum at 1 2µu √ k and no local minima. • for µv < µ2u, u2 has a minimum at 0; • for µv ∈ (µ2u, 2µ2u), u2 has a local maximum at 0 and a local minimum at y0 = √ µv−µ2u κµ2v ; • for µv > 2µ2u, u2 has a maximum at 0. proof: lemma 4 implies that φ1 never changes sign on the interval (0, 1 2µu √ k ). thus, it is clear that u1 has no local extrema in (0, 1 2µu √ k ). in fact, lim y→0 u1(y) = −∞, and u1 has a maximum at 1 2µu √ k . furthermore, u2 has an extremum at y0 =√ µv−µ2u κµ2v , see lemma 4. note that for this y0,√ 1 − 4κµ2uy20 = ∣∣∣∣1 − 2µ2uµv ∣∣∣∣ = 2µ2uµv − 1. next, we compute u′′2 (y0) = φ′2(y0). note that φ′2(y) = 1 d ( 1 2κy2 √ 1 − 4κµ2uy2 − 1 2κy2 −µv ) . if µv ∈ (µ2u, 2µ2u), u′′2 (y0) = φ′2(y0) = 2µv(µv −µ2u) d(2µ2u −µv) > 0. hence, u2 has a local minimum at y0. if µv = 2µ2u, the point y0 = 1 2µu √ κ coincides with the endpoint of the interval, so it is of no interest. we compute by l’hôpital’s rule lim y→0 φ2(y) = 0, showing 0 is an extremum for u2. next we examine the extremum properties of 0 by applying again l’hôpital’s rule lim y→0 φ′2(y) = 1 d (µ2u −µv). therefore, 0 is a maximum for u2 if µ2u < µv, and a minimum if µ2u > µv. this completes the proof. lemma 5 and [1, satz, p.92] allow us to relate the existence of regular solutions of problem (12) on ω to the extrema of the potential energies associated to the auxiliary problem (22). we conclude that no regular solutions (u,v ) can be constructed using the potential energy u1 because it does not have local minima. the only possibility to construct regular solutions (u,v ) is by using the potential energy u2 when µ2u < µv < 2µ2u. the following lemma provides an important property of the integral i(e) which will be employed in the construction of regular solutions of (12). lemma 6. i(e) is monotone in e on (u2(y0),u2(0)). proof: using reasoning as in [5, lemma 5.5] it is enough to show that u′′′2 ≤ 0 on (0, 12µu√κ). then we estimate u′′′2 (y) = φ′′2 (y) = 2 y3 (1 − (1 − 4µ2uκy2)− 1 2 ) − 4κµ 2 u y (1 − 4κµ2uy2)− 3 2 , but (1 − 4µ2uκy2)− 1 2 ≥ 1 so u′′′2 (y) < 0. iv. stationary solutions combining the results on the auxiliary problem, we can classify the regular and the weak solutions of the problem (22). biomath 3 (2014), 1411111, http://dx.doi.org/10.11145/j.biomath.2014.11.111 page 8 of 12 http://dx.doi.org/10.11145/j.biomath.2014.11.111 p. rashkov, regular and discontinuous solutions in a reaction ... a. regular stationary solutions we first remark on the ‘trivial’, constant solution to (12). if µ2u < µv, there is a constant solution of (12), û = √ µv −µ2u κµ2u , v̂ = √ µv −µ2u κµ2v . (28) for µ2u < µv < 2µ 2 u we can construct a regular solution using the auxiliary problem and the potential energy u2 as follows. proposition 2. let lmin := min e i(e) = π√ u′′2 (√ µv−µ2u κµ2v ) and set n = max{n ∈ n : nlmin ≤ 1}. the two-point boundary value problem (20) with homogeneous neumann boundary conditions has the following solutions • the spatially homogeneous solution v = v̂ given in (28). • if lmin ≤ 1, there exists a unique monotone increasing solution v↑(x), and a unique monotone decreasing solution v↓(x) = v↑(1 −x) • for all 2 ≤ n ≤ n there exists a unique n-periodic solution vn,↑ which is monotone increasing on (0, 1 n ), as well as a unique n-periodic solution vn,↓ which is monotone decreasing on (0, 1 n ). proof: the validity of the first claim is obvious. the remaining claims use the properties of the integral i in lemma 3 and 6. these imply that for all n ≤ n, there exists a unique energy level en : i(en) = 1n, corresponding to a unique monotone increasing solution vn,↑(x) on (0, 1 n ), and a monotone decreasing solution vn,↓ = vn,↑( 1 n −x) on (0, 1 n ). if n ≥ 2, either solution can be extended to the entire domain ω by the folding principle [1], [11]. b. weak stationary solutions the previous section showed that for values of µu,µv such that µv < µ2u (12) has only weak solutions. these solutions are constructed piecewise using the auxiliary problem for each branch of u = hi(v ). then u ∈ l∞(ω) and v is continuous on ω. start on the y-axis at x = 0 and begin tracing along any admissible trajectories defined by • (y,z) : z = ± √ 2e −u0(y), • (y,z) : z = ± √ 2e −u1(y), or • (y,z) : z = ± √ 2e −u2(y). here u0(y) = −µvd y 2 is the potential energy associated with the problem (17). the potential energies u1,2 are represented in closed form (up to a constant) by u1(y) = 1 dκ log y + 1 2dκ √ 1 − 4κµ2uy2 − µv 2d y2 − 1 2κ log ( 1 2µu √ k + √ 1 4µ2uκ −y2 ) , u2(y) = − 1 2dκ √ 1 − 4κµ2uy2 − µv 2d y2 + 1 2dκ log ( 1 2µu √ k + √ 1 4µ2uκ −y2 ) . continue tracing until returning to the y-axis at x = 1 (fig. 4). in this way we obtain a partitioning of the domain ω into subintervals ii. on each ii the solution v is given by the y-coordinate of the admissible solution trajectories in the (y,z)-space. of course, under this construction u may be discontinuous in ω as on each subinterval ii u is given by a different branch hi. however, note that v := y belongs to c1(ω) because by construction z = y′ is continuous at the intersection of such trajectories. on fig. 2 and fig. 3 are plotted several solution curves corresponding to an energy level e for the different cases of potential energies ui. note, for example, that a weak solution can be traced by following a trajectory given by u0,u2 (in that order) when µv < µ2u or u2,u1 (in that order) when µv > 2µ2u. biomath 3 (2014), 1411111, http://dx.doi.org/10.11145/j.biomath.2014.11.111 page 9 of 12 http://dx.doi.org/10.11145/j.biomath.2014.11.111 p. rashkov, regular and discontinuous solutions in a reaction ... u1 e z y y z y y u0 e fig. 2. solution curves in (y,z)-space associated to potential energies u1 and u0. fig. 3. solution curves in (y,z)-space associated to potential energies u2 for the different cases (from left to right) µv > 2µ2u,µ 2 u < µv < 2µ 2 u,µv < µ 2 u. v. stability of stationary solutions for a stationary solution (u,v ) we can establish linear stability in the classical sense: for small perturbations ũ(t,x) = u(t,x) − u(x), ṽ(t,x) = v(t,x)−v (x), the behaviour of the solutions u,v of (3) is governed locally by the linear approxiz yv (0) v (1) fig. 4. a weak solution on ω following different energy trajectories. mation. by setting an ansatz ũ(t,x) = u(x)eλt, ṽ(t,x) = v(x)eλt, for local stability we have to consider the sign of reλ. λ is an eigenvalue of the linearised differential operator l = diag(0,d∆) + j, where j = ( 2u v (1+κu2)2 −µu − u 2 v 2(1+κu2) 2u v (1+κu2)2 − u2 v 2(1+κu2) −µv ) . if every eigenvalue λ of l has negative real part, the stationary solution (u,v ) is locally stable. note that the spectrum of l need not be discrete. in the one-dimensional case we formulate results on the stability of spatially nonuniform stationary solutions of the system. easy linear stability analysis leads to proposition 3. let µ2u < µv < 2µ 2 u, the constant solution (û, v̂) is locally asymptotically unstable. when µv > 2µ2u, the constant solution (û, v̂) is locally asymptotically stable. proof: a linearisation of the right-hand side of (12) at (û, v̂) gives the following: l = diag(0,d∆) + ( 2µ3u µv −µu −µv 2µ3u µv −2µv ) . note that the saturation parameter κ does not influence the stability of the constant solution. l has constant coefficients and its spectrum can be computed by matrix eigenvalue analysis. for values µv > 2µ2u, the spectrum of l lies entirely in the left half-plane. hence, the constant solution (û, v̂) is locally asymptotically stable. for values µv < 2µ2u, l has positive eigenvalues. this proves the claim. the following result establishes the local instability of spatially heterogeneous regular solutions. theorem 2. let µ2u < µv < 2µ 2 u. any spatially heterogeneous regular solution (u,v ) of (12) on the interval ω is unstable. proof: for the proof we use a result on the spectrum of l established in [6, corollary 2.7], biomath 3 (2014), 1411111, http://dx.doi.org/10.11145/j.biomath.2014.11.111 page 10 of 12 http://dx.doi.org/10.11145/j.biomath.2014.11.111 p. rashkov, regular and discontinuous solutions in a reaction ... namely the effect of autocatalysis in the nonlinearity f which implies that the spectrum of l comprises eigenvalues with positive real part. this fact implies linear instability of the stationary solution (u,v ). note that f(0,v ) = 0 for all v ∈ r. let (u,v ) be a regular solution of (12). then due to the energy method construction, u = h2(v ) > 0 is continuous on i, and so is fu(u,v ) = 2u v (1 + κu2)2 −µu. furthermore, using the formula (18) we see that u = h2(v ) < κ −1/2. we estimate, first, using the relation f(u,v ) = 0 on i, fu(u,v ) = 2µu 1 + κu2 −µu < µu second, fu(u,v ) = 2µu 1 + κu2 −µu > 2µu 1 + 1 −µu = 0. the result of [6, corollary 2.7] implies that the linearised differential operator l has eigenvalues with positive real part. hence, (u,v ) is an unstable solution. the result of theorem 2 is also a consequence of the results in [7] that establish instability of heterogeneous solutions for semilinear diffusion equations. however, in this particular case instability of the regular solution can be established by direct computation. theorem 3. let µv < 2µ2u. any weak solution (u,v ) of (12) such that u < κ−1/2 on ω is unstable. the proof is identical to that of theorem 2. theorem 4. let n ∈ n, 0 ≤ x1 < x2 < ... < xn ≤ 1. the pair (u,v ) defined by u(x) > 0,x = xj,u(x) = 0,x 6= xj, and v a solution to (17) with robin boundary conditions, is a solution to (13). moreover, any such (u,v ) is locally asymptotically stable. proof: for any (u,v ) of the given type, the linearised operator l looks almost everywhere like l = diag(0,d∆) + ( −µu 0 0 −µv ) . therefore, the spectrum of l is bounded away from 0 in the left halfplane almost everywhere. hence, any stationary solution (u,v ) is locally asymptotically stable. vi. discussion theorem 3 and theorem 4 imply that the locally stable weak solutions must necessarily exhibit large jump discontinuities of amplitude at least κ−1/2. in the literature such solutions are said to exhibit striking patchiness [8], [9] or transition layers [13]. the results show that all spatially heterogeneous, strictly positive, regular solutions of (12) over a one-dimensional domain ω are unstable. this is in contrast to the singularly perturbed system (3) where stable spike solutions exist [15]. the asymptotically stable solutions, which are different from the homogeneous steady state (û, v̂) (28) are discontinuous solutions that exhibit large transition layers. fig. 5 shows such a pattern. in contrast to the results in [5], [6], the problem (12) does not fulfil the autocatalysis condition in [6, corollary 2.7], whose estimates do not hold for all bounded weak solutions of (12). that explains the existence of stable weak solutions with large transition layers. fig. 5. a discontinuous pattern with large amplitude. parameter values are µu = 1,µv = 1.2,κ = 0.1,d = 0.1. biomath 3 (2014), 1411111, http://dx.doi.org/10.11145/j.biomath.2014.11.111 page 11 of 12 http://dx.doi.org/10.11145/j.biomath.2014.11.111 p. rashkov, regular and discontinuous solutions in a reaction ... acknowledgment the author acknowledges the support of the centre for synthetic microbiology in marburg, promoted by the loewe excellence program of the state of hesse, germany. current address: college of life and environmental sciences, university of exeter, stocker road, exeter ex4 4qd, united kingdom. references [1] v. i. arnold, gewöhnliche differentialgleichungen. veb deutscher verlag der wissenschaften, berlin, 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[2] a. doelman, r. a. gardner, and t. j. kaper. large stable pulse solutions in reaction-diffusion equations. indiana univ. math. j. 50 (1) 443–507, 2001. [3] a. gierer and h. meinhardt. a theory of biological pattern formation, kybernetik 12, 30–39, 1972. [4] a. j. koch and h. meinhardt. biological pattern formation: from basic mechanisms to complex structures. rev. mod phys. 66 (4), 1481–1507, 1994. [5] a. marciniak-czochra, g. karch, and k. suzuki. unstable patterns in reaction-diffusion model of early carcinogenesis. j. math. pures appl. 99, 509–543, 2013. doi:10.1016/j.matpur.2012.09.011 [6] a. marciniak-czochra, g. karch, and k. suzuki. unstable patterns in autocatalyctic reaction-diffusion systems. preprint. [7] h. matano. asymptotic behavior and stability of solutions of semilinear diffusion equations. publ. rims, kyoto univ. 15, 401-454, 1979. [8] m. mimura and j.d. murray. on a diffusive prey-predator model which exhibits patchiness. j. theor. biol. 75, 249– 262, 1979. [9] m. mimura, m. tabata, and y. hosono. multiple solutions of two-point boundary value problems of neumann type with a small parameter. siam j. math. anal. 11 (4), 613–631, 1980. [10] j. d. murray, mathematical biology. springer, new york, 1993. [11] y. nishiura. global structure of bifurcating solutions of some reaction-diffusion systems. siam j. math. anal. 13 (4), 555–593, 1982. [12] p. rashkov. remarks on pattern formation in a model for hair follicle spacing. preprint. [13] k. sakamoto. construction and stability analysis of transition layer solutions in reaction-diffusion systems. tôhoku math. j. 42, 17–44, 1990. [14] s. sick, s. reinker, j. timmer, and t. schlake. wnt and dkk determine hair follicle spacing through a reaction-diffusion mechanism. science 314 (5804), 1447– 1450, 2006. doi:10.1126/science.1130088 [15] f. veerman and a. doelman. pulses in a gierermeinhardt equation with a slow nonlinearity. siam j. dyn. sys. 12(1), 28–60, 2013. doi:10.1137/120878574 biomath 3 (2014), 1411111, http://dx.doi.org/10.11145/j.biomath.2014.11.111 page 12 of 12 http://dx.doi.org/10.1016/j.matpur.2012.09.011 http://arxiv.org/abs/1301.2002 http://dx.doi.org/10.1126/science.1130088 http://dx.doi.org/10.1137/120878574 http://dx.doi.org/10.11145/j.biomath.2014.11.111 introduction the model problem diffusion-driven instability a priori estimates reduced problem auxiliary problem stationary solutions regular stationary solutions weak stationary solutions stability of stationary solutions discussion references original article biomath 2 (2013), 1312061, 1–10 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum parameter identification in population models for insects using trap data claire dufourd∗, christopher weldon†, roumen anguelov∗, yves dumont§ ∗department of mathematics and applied mathematics, university of pretoria, pretoria, south africa, {claire.dufourd, roumen.anguelov}@up.ac.za †department of zoology and entomology, university of pretoria, pretoria, south africa, cwweldon@zoology.up.ac.za §cirad, umr amap, montpellier, france, yves.dumont@cirad.fr received: 18 october 2013, accepted: 6 december 2013, published: 23 december 2013 abstract—traps are used commonly to establish the presence and population density of pest insects. deriving estimates of population density from trap data typically requires knowledge of the properties of the trap (e.g. active area, strength of attraction) as well as some properties of the population (e.g. diffusion rate). these parameters are seldom exactly known, and also tend to vary in time, (e.g. as a result of changing weather conditions, insect physiological condition). we propose using a set of traps in such a configuration that they trap insects at different rates. the properties of the traps and the characteristics of the population, including its density, are simultaneously estimated from the insects captured in these traps. the basic model is an advection-diffusion equation where the traps are represented via a suitable advection term defined by the active area of the traps. the values of the unknown parameters of the model are derived by solving an optimization problem. numerical simulations demonstrate the accuracy and the robustness of this method of parameter identification. keywords-partial differential equation; advectiondiffusion equation; parameter identification; inverse problem; trap interference; population density. i. introduction this work is motivated by the need to develop a reliable and efficient method for detecting the presence and estimating population density of the invasive fruit fly, bactrocera invadens drew, tsuruta & white (diptera: tephritidae) in south africa. bactrocera invadens is a fruit fly species introduced from asia to africa where it was first described and recorded in kenya in 2003 [9], [20]. in 2010, b. invadens was detected in the northern part of the limpopo province in south africa [21]. its capacity for rapid population growth, high invasive potential [16], and wide range of fruit hosts [26] represents a major threat for all fruit industries in south africa. fruit flies are a perennial problem in south africa because in addition to b. invadens there are three endemic species that already represent economic pests. fruit flies have historically been controlled in south africa by the application of insecticide cover sprays. current practice, however, involves the use of alternative control strategies due to regulationand consumer-driven requirements for fruit to be free of insecticide residues. the primary techniques used in fruit fly control are the application of bait sprays [21], m3 bait stations [22], or the ‘male annihilation technique’ [21]. all three techniques work on the same principal: a food or sex attractant, which is fed on by adult flies, is mixed with an insecticide such as malathion or gf120. with regard to b. invadens, male annihilation technique has been applied to control incursions in south africa [21]. another control strategy for this pest may include mass-trapping, which uses male attractants to capture and kill males of a population, leading to reduced female mating and possibly causing local extinction of the population [12]. alternatively, the citation: claire dufourd, christopher weldon, roumen anguelov, yves dumont, parameter identification in population models for insects using trap data , biomath 2 (2013), 1312061, http://dx.doi.org/10.11145/j.biomath.2013.12.061 page 1 of 10 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2013.12.061 c dufourd et al., parameter identification in population models for insects using trap data sterile insect technique (sit) may represent a useful approach to control incursions of b. invadens. sit involves the release of large numbers of sterilized males that compete with wild males for female fertilization, which leads to no production of viable offspring [18] and its success can be measured using the ratio of sterile: wild insects captured in and array of surveillance traps [17]. regardless of the alternative control strategy used, their successful application requires a good knowledge of the distribution of the pest, and their dispersal capacity and density. the density of an insect population, however, is a parameter that cannot easily be obtained by direct field observations because traps usually sample only a small proportion of all individuals. to overcome this problem, it is often the case that captures of insects in traps are compared to simulated data [14]. an advection-diffusion equation is considered for modelling the dynamics of fruit flies, where density is the initial value of the model. such a model requires knowledge of the properties of the trap such as the active area [5] and the strength of attraction, as well as some properties of the population, like its diffusion rate [25]. these parameters are seldom exactly known, and also tend to vary with changing weather [23] and landscape heterogeneity [11],[10]. determining the values of these parameters is actually an inverse problem, that is, given the solution of the model, or at least part of it, one or more of the model parameters can be identified. the parameter identification problem consists of finding a unique and robust estimation for the parameter values. this problem leads to solving a global optimization problem in order to find the set of parameters that minimizes an objective function. mathematically, the existence and uniqueness of this global minimum relies on the well-posedness of the inverse problem, while its robustness relies on its well-conditionedness. however, inverse problems are typically ill-posed or conditioned, [19], [7], [24]. in this paper, we show that by using different settings of interfering traps we obtain a parameter identification problem which can be solved numerically in a reliable way. it is essential in this approach that interfering traps generate different incoming streams of insects. thus, more information about the characteristics of the insect population is provided. indeed, as the relationship between the setting and the traps is highly non linear and not well understood, several settings of traps are considered and the robustness of the estimates are compared. we demonstrate empirically that using this approach, the problem of simultaneously identifying a set of unknown parameters is well-posed and well-conditioned. the numerical procedure falls under the well-known trial-anderror method of regularization theory [28]. ii. the insect trapping model: the direct problem the model is formulated on a domain ω ⊂ r2 which is assumed to be isolated, i.e. there is no immigration and no emigration of insects. it is also assumed that when there is no stimulus, the insects individually follow a random walk. because insects are often in large abundance, we can apply a diffusion equation to model the dispersal of insects at population level [29]. the traps set on ω are attractive. thus, the active area of the trap [5] is the area where the concentration of the attractant is above the threshold of concentration at which the fruit flies can detect it. therefore, in this area the insects will be influenced to move in a preferred direction towards the trap. this can be modelled using an advection equation [3]. finally, we assume that our experiments take place over a short period of time, thus we may omit reaction terms. using the above assumptions, the insect dynamics can be modelled via an advection-diffusion equation. ∂u ∂t −∇(s(x)∇u) + ∇(a(x)u) = 0, ∂u ∂n |∂ω = 0, u|t=0 = u0. (1) u(t,x) denotes the population density at time t and at the point x = (x1,x2) ∈ ω. the advection function a(x) is space-dependent and determines the attractiveness of the trap with respect to the distance to the center of the trap. the traps are circular of radius rtrap. assume that the active area of a trap is defined by a disk of radius rmax from the center of the trap. then the insects that are beyond this disk are not subjected to advection and we assume that their dynamics are only governed by the diffusion term. as the insect gets closer to the trap, the force of attraction increases and reaches its maximum at a distance rmin from the center of the trap. if n is the number of traps distributed on the domain, then: a(x) = n∑ t=1 at (x), at (x) = amaxα(||xt −x||) xt−x||xt−x||, (2) where xt is the coordinate of trap t , and the function biomath 2 (2013), 1312061, http://dx.doi.org/10.11145/j.biomath.2013.12.061 page 2 of 10 http://dx.doi.org/10.11145/j.biomath.2013.12.061 c dufourd et al., parameter identification in population models for insects using trap data α(d) is defined for d ∈ [0, +∞), as follows. α(d) =  amax d sin ( πd 2rtrap ) , if d < rtrap amax d , if rtrap ≤ d < rmin amax 2d ( cos ( π d−rmin rmax−rmin ) +1 ) , if rmin ≤ d < rmax 0 if rmax ≤ d (3) the function α(d) is represented in fig. 1. note that the value of the advection inside the trap does not really matter, and we make α(d) decrease to 0 from the distance rtrap to ensure the continuity of a(x). fig. 1. graph of the function α(d) the diffusion coefficient s(x) is also space-dependent. it is assumed to be constant, s(x) = σ, outside the active areas of the traps. since the insects do not escape from the traps there should be no diffusion across the trap boundary. in order to ensure the existence and uniqueness of the (weak) solution of (1) we assume that inside a trap the function s(x) has a positive value ε which is so small that the implied diffusion effect in the time interval of observation can be neglected. in order to further ensure continuity of s we take s(x) = σ − n∑ t=1 st (||xt −x||), st (d) =  σ−ε, if d ≤ rtrap (σ−ε) ( 1− d−rtrap rmin−rtrap ) , if rtrap<d ≤ rmin 0, if rmin < d. (4) iii. the parameter identification problem the trap parameters, rmin, rmax and amax, as well as the diffusion parameter σ are seldom know. therefore, although our main goal is to estimate the initial population density u0, these other parameters are also needed. for simplicity, we assume that the initial population density is a constant, that is, u0(x) = u0 ∈ r,∀x ∈ ω. denote p = (u0,σ,rmin,rmax,amax) the vector of the parameters to identify. let p be a compact subset of r5 to which p belongs. when solving the direct problem (1), we are given p and we find a function u satisfying the differential equation in (1) and the respective boundary and initial conditions. this way, we define a mapping φ from the domain p of parameters to the space of solutions: u = φ(p) (5) it is well-know that the solution operator φ is continuous and injective. therefore, using the compactness of p we obtain that the operator φ−1 : φ(p) −→p is continuous ([28], p.29). thus, the inverse problem to (1) is wellposed. then if a solution u of (1) is given, the value of p = φ−1(u) can be determined by well-known methods, e.g. minimizing a norm of u−φ(p). however, in practice u is commonly not known, at least not on the whole domain ω × [0, +∞) [24]. what is usually available is a function b(u) referred to as observation operator [4]. therefore the parameter identification problem is stated as follows: given an observation ψ, find p such that (b ◦φ) (p) = ψ. (6) in the setting of problem (1) the observation operator consists of the insect count in traps at given times t1, t2, ..., tk. more precisely, we have b(u) =   b1(u,t1) . . . b1(u,tk)... ... bn (u,t1) . . . bn (u,tk)   , where bt (u,tk) is the total number of insects captured in trap t until time tk, t = 1, ...,n, k = 1, ...,k. hence the observation ψ in (6) is an n ×k real matrix ψ =   ψ1,t1 . . . ψ1,tk... ... ψn,t1 . . . ψn,tk   . note that b is an array of numbers representing averaged values of u on particular areas of the domain at finite number of points in time. hence, the injectivity of b◦φ is problematic. furthermore, the observation ψ contains both model and measurement error which further complicates the well-posedness of equation (6). in particular, ψ is not necessarily an element of (b◦φ)(p). as usual for such situations, p is obtained as a solution of (6) in a least square sense, that is, p is a solution of the optimization problem, φ(p) = ‖(b ◦φ) (p) −ψ‖2f −→ min, (7) biomath 2 (2013), 1312061, http://dx.doi.org/10.11145/j.biomath.2013.12.061 page 3 of 10 http://dx.doi.org/10.11145/j.biomath.2013.12.061 c dufourd et al., parameter identification in population models for insects using trap data where ‖a‖f = ( m∑ i=1 n∑ j=1 |ai,j|2 )1/2 . let p∗ be the minimizer of φ, that is p∗ = arg min p∈p φ(p) (8) in the setting of the current model, both the observation operator and the model (1) depend on the distribution of the traps. our aim is to find trap configurations for which the minimizer of φ is unique and can be reliably determined by a numerical procedure. in order to investigate the properties of (7), equivalently (8), we solve the optimization problem iteratively using a random multistart approach [30] over a set of initial values of p ∈ p. for each randomly selected starting value of p, a local minimum of φ is found using the gauss-newton line search algorithm. the solution of (7) is identified as the estimated parameter values of the minimum of the local minima. thus, increasing the number of starting values increases the chances of finding the global minimum of the objective function. furthermore, for each trap setting, we consider how well the global minimum can be discriminated from the other local minima. this gives an important indication on how well the parameter values can be identified in the presence of noise and we refer to it as robustness. here we investigate the influence of the choice of the settings of traps on the accuracy and robustness of the method. in general, one can expect that increasing n and k leads to a more regular problem. however, as shown in [2], an incoming stream can be quite accurately identified by using relatively small number of observations. furthermore, if the traps are far enough from each other, the rows of b(u) are the same. therefore, in the considered setting, just increasing the size of the matrix b(u) would not improve substantially the regularity of the problem. our approach is to have some of the traps close enough so that due to interferences, they produce different streams of trapped insects. this will increase the rank of the matrix b which can be reasonably expected to improve the regularity of the problem. empirical evidence supporting this conjecture is provided in the next section, where the effect of different configuration is examined. iv. description of the experiments we consider four trap settings: • (a) single trap, • (b) nine traps in a square formation, • (c) five traps in a z-formation, • (d) five traps in a kite formation. these four setting are shown on fig. 2. note that in all cases the traps are sufficiently far from the boundary of the domain ω so that in the considered period of observation the boundary condition in (1) remains reasonable. fig. 3 represents the population distribution after 15 fig. 2. distribution of the traps on ω for each setting. the symbols identify the traps that produce identical incoming streams within each setting. time units using setting (d). the interference between the traps can be observed in fig. 4, which is a zoom in of fig. 3. note that in these experiments, because of the symmetry of the trap distribution, identical trap counts are observed in several traps. in fig. 2, the traps identified with the same symbols have identical incoming streams. therefore, by using the nine traps of setting (b) for instance, we multiply the amount of information by three compared to the one-trap setting (a). in settings (c) and (d), five traps are used. due to their configuration, setting (c) produces three distinct incoming streams, whereas setting (d) produces four distinct incoming streams. the distinct incoming streams obtained in each setting are represented in fig. 5. considering the different settings of traps we run numerical simulations in order to estimate, one, two, three, and finally four parameters simultaneously using the cummulative trap counts over 15 time units (k = 15). biomath 2 (2013), 1312061, http://dx.doi.org/10.11145/j.biomath.2013.12.061 page 4 of 10 http://dx.doi.org/10.11145/j.biomath.2013.12.061 c dufourd et al., parameter identification in population models for insects using trap data fig. 3. insect distribution after 15 time units using setting (d). fig. 4. interference between the traps using setting (d). zoom in of fig. 3. since the parameter of main interest to identify here is the initial population density u0, it is always among the parameters to estimate. the diffusivity of the insects σ, is the second parameter of main interest when dealing with insect dispersal. the choice of the remaining parameters to estimate is based on a sensitivity analysis. we selected in priority parameters on which the model has the highest sensitivity. indeed, when solving the inverse problem, the more sensitive the original problem on a parameter is, the more accurate the parameter identification is. the sensitivity of a parameter is a measure of the change in the output of the model caused by a change in this parameter value. however, since the parameters under consideration have different order of magnitude, it is more appropriate to measure their elasticity index [15], i.e. the proportional change in the output of the model caused by a change in the value of the parameter. the fig. 5. cumulative number of captured insects in each trap, using the trap settings (a), (b), (c) and (d) of fig. 2. elasticity indices of each parameter with respect to each setting of traps is given in table ii. note that the output of the model is the most sensitive to a change in the parameter for which the elasticity index is the greatest. therefore, in order the output of the model is more sensitive to changes of values of amax than of rmax for which the model is more sensitive to changes of rmin. therefore, we proceeded to the parameter identification of p1 = (u0), p2 = (u0,σ), p3 = (u0,σ,amax) and p4 = (u0,σ,amax,rmax). our aim is to find p∗κ that satisfies (8), where κ denotes the number of parameters to be estimated (κ = {1; 2; 3; 4}). we limited our numerical simulations to the simultaneous estimation of 4 parameters due to limitations in computational capacity. indeed, when solving the inverse problem, each evaluation of the objective function requires solving problem (1). since we do not have real field data, we simulated the data over a period of 15 time units with the values of parameters given in table i. assuming that the period biomath 2 (2013), 1312061, http://dx.doi.org/10.11145/j.biomath.2013.12.061 page 5 of 10 http://dx.doi.org/10.11145/j.biomath.2013.12.061 c dufourd et al., parameter identification in population models for insects using trap data table i values of the parameters used to simulate the field data parameter unit value u0 insects/unit area 27.5 σ m2/time unit 5.23 amax m/time unit 5.75 rmax m 8 rmin m 2 table ii elasticity of the parameters with respect to the incoming streams of insects. let ν denote a parameter, and z(ν) the incoming stream of insects with respect to ν, then, the elasticity index of parameter ν is eν = ν‖z(ν) −z(ν + ∆ν)‖2/ (‖z(ν)‖2∆ν) setting trap u0 σ amax rmax rmin (a) 1.00 0.92 1.42 1.26 0.60 (b) center 1.00 1.02 1.26 0.83 0.54 corner 1.00 0.96 1.35 1.06 0.57 median 1.00 0.98 1.31 0.95 0.55 (c) center 1.00 0.98 1.33 1.01 0.56 side 1.00 0.95 1.38 1.15 0.59 top/bottom 1.00 0.95 1.36 1.08 0.58 (d) center 1.00 0.98 1.29 0.90 0.55 side 1.00 0.95 1.36 1.10 0.58 top 1.00 0.97 1.33 1.01 0.56 bottom 1.00 0.94 1.39 1.17 0.59 between two consecutive collects of data is equal to one time unit, we extract from the simulated data only those that correspond the collect time. a noise of 5% is added to the simulated data, and this gives ψ = (ψi,t1 . . .ψi,tk )i=1..n , where n is the number of traps. the use of simulated data also allows us to compare the estimation with the real values of the parameters. the numerical solution of the initial value boundary problem (1) is obtained by using the crank-nicolson scheme [8], [27]. the optimisation problem (8) is solved by using the matlab function fminunc which performs the gauss-newton line search algorithm [13]. since this function finds only a local minimum, we run it with many randomly selected initial points in order to have a reasonable certainty that the global minimum is among the local minima found. here we choose u(0)0 ∈ [5; 45], σ(0) ∈ [4; 6], a(0)max ∈ [2; 8], r (0) max ∈ [3; 10] as ranges of the initial parameter values. v. results and discussion the results of the experiments described above are presented in table iii. the values of the local minima of φ, and the respective estimated values of the parameter vectors p̃κ, with κ = {1; 2; 3; 4}, are given according to the trap setting. according to our method, when several local minima of φ are found, the identified set of parameter values corresponds to the global minimizer of φ, that is, the minimizer of the smallest local minimum. the ∗ on the right of the table indicates the values of the parameters identified as global minimizers of φ. in these experiments, we investigate the accuracy and the robustness of the identified parameter values. the accuracy of p∗κ is measured by calculating the relative error to the exact solution p̄κ (table i) i.e. erel=‖p̄κ−p∗κ‖/‖p̄κ‖. in this setting the concept of robustness of the method represents how well the global minimum value of φ is discriminated against the values of φ of other minima. table iii local mimima of the objective function φ, with the four settings of traps with respect to the number of parameters estimated simultaneously. ts stands for the trap setting used (fig. 2). the * indicates the set of parameters identified as the global minimizer of φ. p ts φ ũ0 σ̃ ãmax r̃max erel p 1 = (u 0 ) (a) 240004 27.63 – – – 0.5% * (b) 259813 27.43 – – – 0.3% * (c) 572150 27.45 – – – 0.2% * (d) 620115 27.31 – – – 0.7% * p 2 = (u 0 , σ ) (a) 219149 21.49 3.31 – – 22.5% * 222543 25.43 4.73 – – (b) 256632 27.01 5.15 – – 1.8% * (c) 572066 27.52 5.24 – – 0.1% * 826874 19.69 2.45 – – (d) 619615 27.46 5.26 – – 0.2% * p 3 = (u 0 , σ , a m a x ) (a) 204279 17.79 4.30 7.89 – 34.9% * 214067 21.14 6.44 7.90 – (b) 256473 26.70 5.16 5.81 – 2.8% * 297564 18.37 4.48 7.65 – (c) 572055 27.61 5.23 5.73 – 0.4% * 636640 17.35 4.18 8.07 – (d) 598689 25.17 5.48 6.31 – 8.4% * 647548 17.43 4.45 8.14 – p 4 = (u 0 , σ , a m a x , r m a x ) (a) 210462 33.31 3.25 5.07 5.83 22.1% * 210466 34.38 3.08 4.90 5.71 210471 32.75 3.35 5.17 5.89 210503 32.02 3.47 5.30 5.98 210533 31.53 3.56 5.39 6.03 210679 30.22 3.81 5.65 6.20 (b) 255885 26.93 5.09 5.64 8.15 2.1% * 277462 18.24 4.00 6.80 8.56 (c) 571732 27.51 5.28 5.85 7.88 0.6% * 609828 17.47 3.84 7.05 8.61 (d) 597514 25.40 5.37 6.07 8.18 7.2% * 615604 17.66 4.06 7.08 8.61 as a preliminary experiment, table iv presents the identified values and relative errors of p when a single parameter value is unknown. p = (u0), p = (σ), p = (amax) and p = (rmax) are successively identified using each setting of traps. these results are in agreement with biomath 2 (2013), 1312061, http://dx.doi.org/10.11145/j.biomath.2013.12.061 page 6 of 10 http://dx.doi.org/10.11145/j.biomath.2013.12.061 c dufourd et al., parameter identification in population models for insects using trap data the sensitivity analysis provided in table ii, i.e. the more sensitive the output of the model to a certain parameter, the more accurate its estimation. table iv identified values of the single value parameter p and its relative error with respect to the trap setting, ts. the values of the estimates are rounded to two digits after the decimal point. ts p ũ0 σ̃ ãmax r̃max r̃min erel (a) p = (u0) 27.63 – – – – 0.48% p = (σ) – 5.20 – – – 0.60% p = (amax) – – 5.77 – – 0.37% p = (rmax) – – – 8.03 – 0.41% p = (rmin) – – – – 2.02 0.81% (b) p = (u0) 27.43 – – – – 0.24% p = (σ) – 5.24 – – – 0.21% p = (amax) – – 5.74 – – 0.17% p = (rmax) – – – 7.98 – 0.24% p = (rmin) – – – – 1.99 0.64% (c) p = (u0) 27.45 – – – – 0.19% p = (σ) – 5.24 – – – 0.19% p = (amax) – – 5.74 – – 0.13% p = (rmax) – – – 7.99 – 0.17% p = (rmax) – – – – 1.99 0.05% (d) p = (u0) 27.31 – – – – 0.69% p = (σ) – 5.27 – – – 0.71% p = (amax) – – 5.72 – – 0.49% p = (rmax) – – – 7.95 – 0.60% p = (rmin) – – – – 1.96 1.76% a. do interfering trap-settings provide better results than non-interfering trap settings? from table iii we can see that, using setting (a) i.e. without trap interference, the number local minima found increases, as the number of parameters to identify simultaneously increases. conversely, using the setting with trap interferences, a maximum of two minima were found. moreover, the minima found using one trap are in a maximum range of 5% of the optimal value of φ and within this range, the norms of the different minima can differ by 15%. more precisely, for p = p2 (resp. p = p3), the minima are found withing a range of 2% (resp. 15%) of the optimal value of φ. in particular, when p = p4, 6 minima are found within a range of 0.1% of the optimal value of φ where the norm of the minima can differ by 10.6%. this shows that, without interferences, as the number of parameters to identify increases, the reliability of the estimates decreases. furthermore, we can see from table iii, that the one-trap setting provides a poor accuracy of the estimates when several parameter values are identified simultaneously, compared to the other setting. in fact, using one trap, the relative error of the estimates is above 22% when two or more parameter values are estimated, whereas, using the other settings, the relative error is always below 8.4%. therefore, when two or more parameters are to be identified, we may conclude that interfering trap settings provide estimates with better accuracy than non-interfering trap settings. however, from table iii, note that if a single parameter needs to be identified, setting (a), using only one trap, would provide a sufficiently accurate and reliable estimate. in this case, adding more traps is not really helpful. this suggests that one must choose an appropiate setting depending on the parameters that need to be identified. by investigating further on the results, when p = p3 using one trap, we simulated the incoming streams using the global minimizer of φ (p3 = p∗3), (table iii). the curve obtained with the latter simulation as well as the curve representing the incoming stream simulated using the real value of p , given in table i (p3 = p̄3), are represented in fig. 6. as we can see, the two curves are hardly distinguishable, meaning that two different sets of parameters can lead to very similar incoming streams. this illustrates the ill-conditionedness of the problem where two very similar streams of trapped insects are produced using two very different sets of parameters (erel = 34.9%, table iii). fig. 6. simulated cumulative number of captures when estimating three parameters with setting (a), using the real values, p̄3, and the identified values, p∗3. b. interfering trap-setting strategies we have shown above that using a setting of traps that are interfering provide better results for parameter identibiomath 2 (2013), 1312061, http://dx.doi.org/10.11145/j.biomath.2013.12.061 page 7 of 10 http://dx.doi.org/10.11145/j.biomath.2013.12.061 c dufourd et al., parameter identification in population models for insects using trap data fication in terms of robustness and accuracy when several parameters are identified simultaneously. however, little is known on the actual role of the interferences between the traps and their effect on the regularity of the problem. in order to understand the effect of these interactions, several interfering trap settings are compared. we study the results obtained using the nine-trap setting (b) and the setting (d) of five traps. one could expect better accuracy and robustness of the estimates using setting (d) since in this setting four distinct incoming streams are produced whereas only three are produced using setting (b). however, apart from the case p = p1, the relative error of the estimate using setting (d) is always lower than using setting (b). considering the fact that the noisy data are averaged over the traps producing the same streams, in setting (b), the corner and median noisy data are averaged over three traps, whereas only the noisy data of the two side traps are averaged using setting (d). therefore, the data used to identify the parameters using setting (b) are smoother than those used with setting (d). this reduces the effect of the noise and may explain why more accurate and robust estimations are obtained using nine traps. in setting (c), five traps are used producing three distinct incoming streams as for the nine-trap setting (b). the estimates obtained using setting (c) are the most accurate compared to all the other experiments that were carried out (erel = 6% for p = p4). the robustness, however, is not as good as for the nine trap setting. for instance, when p = p3, two local minima of φ are found within 10% (resp. 14%) of its optimal value for setting (c) (resp. setting (b)). despite this, the global minimum of φ is clearly identified. this counter intuitive result shows that the interference phenomenon is not trivial. in particular, we can see in fig. 5 that the 3 incoming streams produced by the nine trap setting are more distinct than those obtained using the five-trap setting. a possible explanation may relate to the sensitivity of the parameters. indeed, the parameters amax and rmax are more sensitive using setting (c) than when using setting (b) (table ii). such a result suggest that there must be an optimal setting which would allow to obtain robust and accurate parameter estimation using as few traps as possible. these results show that trap interference can be used to make the problem more regular, however, the relationship between the regularity of the problem and the interfence between the traps is highly nonlinear and therefore difficult to analyse. vi. conclusion parameter identification is challenging, in particular when the direct problem is defined by pdes, since it often leads to solving inverse problems that are illposed or ill-conditioned. we demonstrate numerically that the interferences between traps can be used to make the problem well-posed and well-conditioned using a trial-and-error approach. this method enables the identification of parameter values that describe population characteristics, i.e the population density, its diffusion rate, attractiveness of the traps and their maximum radius of attraction. by choosing biologically realistic ranges of parameter values to estimate, the parameters are identified as the global minimizer of function φ using starting values over these ranges. we show numerically that trap interferences can be used not only to increase the accuracy of the estimates, but also to find a global minimum which can be well descriminated from the other local minima. the trial-and-error method over the choice of the trap setting, followed with the multistart approach to solve the optimization method increases the chances of succeeding in solving the parameter identification problem. for each trap setting trial, we found numerically the global minimum of the objective function and we measured how well it is discriminated from the other minima. the setting where the global minimum can be the best discriminated from the other minima provides an inverse problem that is well-conditioned and thus provides the most robust solution to the parameter identification problem. note that, once a “good” setting of trap is identified, other methods, such as random search methods, could be interesting alternatives to finding the global minimum of the objective function [30], [6], [1]. furthermore this work investigates how experiments using traps should be conducted in the field so that sufficient information is recorded, using as few traps as possible. indeed, the optimal setting of traps depends on the parameters that need to be identified. for instance, we showed that a setting using only one trap would be sufficient to identify a single parameter. however when several parameters are unknown, interfering traps provide more accurate and more reliable estimates. in particular, setting (c) stands out from the other settings. due to a highly nonlinear relationship between the trap interferences and the regularity of the problem, it is challenging to find an optimal setting of traps providing highly accurate and robust estimates. a global sensitivity of the model to the parameters will be considered in a future work to give more insight into this relationship. biomath 2 (2013), 1312061, http://dx.doi.org/10.11145/j.biomath.2013.12.061 page 8 of 10 http://dx.doi.org/10.11145/j.biomath.2013.12.061 c dufourd et al., parameter identification in population models for insects using trap data this is a promising method, not only from the numerical and theoretical perspective for parameter identification, but may also prove to be of practical importance for the determination of insect population density in the field with the use of traps. acknowlegment the support of the national research fundation of south africa and citrus research international ltd. 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[29] p. turchin, quantitative analysis of movement: measuring and modeling population redistribution in animals and plants. sinauer associates sunderland, 1998. [30] a.a. zhigljavsky, theory of global random search. kluwer academic publishers, dordrech, 1991. biomath 2 (2013), 1312061, http://dx.doi.org/10.11145/j.biomath.2013.12.061 page 10 of 10 http://dx.doi.org/10.1016/j.ecocom.2011.10.002 http://dx.doi.org/10.1603/0013-8746(2008)101[331:hpahpp]2.0.co;2 http://dx.doi.org/10.1603/0013-8746(2008)101[331:hpahpp]2.0.co;2 http://dx.doi.org/10.1137/1.9780898717938 http://dx.doi.org/10.11145/j.biomath.2013.12.061 introduction the insect trapping model: the direct problem the parameter identification problem description of the experiments results and discussion do interfering trap-settings provide better results than non-interfering trap settings? interfering trap-setting strategies conclusion references original article biomath 3 (2014), 1407231, 1–15 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum modelling and analysis of mirna regulation svetoslav g. nikolov institute of mechanics, bulgarian academy of sciences, bulgaria department of systems biology and bioinformatics, university of rostock university of transport, sofia, bulgaria e-mail: s.nikolov@imbm.bas.bg received: 16 may 2014, accepted: 23 july 2014, published: 15 september 2014 abstract—mirna regulation is involved in many important biological processes such as cell proliferation, apoptosis and metabolism. computational predictions of mirna targets suggest that up 30 % of human proteins coding genes may be regulated by mirnas. this makes mirnas one of the most abundant classes of regulatory genes in humans. in the present paper we develop a time delay model of a feedback system regulated via mirna. the model resulted in three ddes with three discrete time delays. since this system is a classical case study, covering several essential features of mirna and genetic regulatory mechanisms, general conclusions about design principles and role of time delays in the stability of gene circuits can be suggested. the basic view that time delays are a key factor in the dynamic behaviour of the system was confirmed by the analytical calculations and numerical simulations. keywords-mirna regulation; time delay; andronov-hopf bifurcation; numerical analysis i. introduction cells are the structural and functional units of all living organisms as protein synthesis is an essential function of a cell. protein synthesis is achieved in a two-step process. firstly, when a protein is needed in a cell, a messenger rna (mrna) is created as a copy of the information from the appropriate gene (process called transcription). then, the mrna molecule is used as a template for the creation of the protein (process called translation). the processes of transcription and translation in eukaryotic cells are very complicated, since there are more levels of control of gene expression. transcription and translation are processes which require a certain amount of time to complete. thus, these two processes of gene expression induce time delays in the biochemical systems. smolen and coauthors in [28] describe a time delay associated to the translocation of proteins and mrnas between 50 and 100 min. rateitschak and wolkenhauer in [24] describe a time delay for gene transcription between 10 and 40 min. finally in [29] a time delay around 7 min is defined and estimated for nucleocytoplasmic shuttling. recently, an additional level of regulation in protein expression has been introduced following the identification of short non-protein-coding rnas [22]. micrornas (mirnas) are small regulatory rnas with length of 18–25 nucleotides (typically ≈ 22nt size in human), which function citation: svetoslav g. nikolov, modelling and analysis of mirna regulation, biomath 3 (2014), 1407231, http://dx.doi.org/10.11145/j.biomath.2014.07.231 page 1 of 15 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2014.07.231 s. nikolov, modelling and analysis of mirna regulation is to regulate the activity and stability of specific messenger rna targets. a large body of data suggests that mirnas are supposed to account for 1–5% of animal genes [3, 12] (bind through imperfect base paining to the 3’ utr of their target mrnas and interfere with translational output [16]), making them one of the most abundant gene product regulators. mirnas are embedded in complex regulatory networks that involved gene activation, posttranslational regulation and protein-protein interactions. therefore, this makes mirnas as one of the most abundant classes of regulatory genes in animals. a key recurring function of mirna in networks is to reinforce the gene expression of differentiated cellular states. for example, regulatory networks involving mirnas have been described in the asymmetric differentiation of left-right neurons in c.elegans, eye and sensory organprecursor development in drosophila, and granulocytic differentiation in human [8]. thus, mirna provide an ideal way to regulate rapid and localized protein synthesis. a recent study [31] classified gene regulatory networks involving mirnas in two types of circuits: in type a circuits the transcription of the mirnas and their targets are positively coregulated, while in type b circuits the transcription of the mirnas and their targets is oppositely regulated by common upstream factors or processessee fig.1. type a circuits may be more prevalent in networks operating in homeostasis, to maintain in protein steady state and in general to reduce the noise in transcriptional/translational fluctuations. type b circuits instead can serve as surveillance mechanisms to suppress ‘leaky’ transcription of target gene, and in general as positive feedback loops where a transient signal can be converted in a long-lasting cellular response such as irreversible cellular differentiation [8]. time lags in continuous systems can produce complex dynamics and instabilities. in the recent years ordinary differential equations (odes) with retarded argument(s) have been widely used in modelling and analyzing regulatory systems involve many genes, factors and complex interactions (gene regulatory networks). in these models, the gene and mrna concentrations are timedependent variables, interactions are represented by functional and differential between variables, and retarded arguments are usually the time of transcription and translation, or time delay feedback loop. for the concentration xi (t) of the factor i at time t, the evolution in time is described by the system: dxi (t) dt = axi (t) + bxi (t− τ) = fi (xj (t) , xj (t− τ) , j ∈ ni) (i = 1, 2, ..., n) , (1) where fi is a nonlinear real-valued function of the states (xj (t) , xj (t− τ)) of the vertices j ∈ ni, that interact with the factor i. an interesting and important problem appears due to the nonlinearity of fi, since an analytical solution of the system is usually not possible. in this sense, the recent molecular biological discoveries (like the mirnas and their complex regulatory effects) clearly show the need to develop mathematical models that take into consideration the post-transcriptional regulation. also, the models should take into account that the interactions can be complex, consisting of cyclic dependences, cooperative regulations (including transcription factors (tfs) that act simultaneously or that assemble in complex structures), dna looping regulation, etc. these criteria seem difficult to satisfy with a standard modeling formalism, and answers may lie in hybrid approaches [11]. as an intuitive example of nonlinearity by characterizing the behavior of a system in terms of stimulus and response is the following: if we give the system a “kick” or “input signal” and observe a certain response to that kick, then we can ask what happens if we kick the system twice as hard. if the response is not twice as large (it might be larger or smaller), then we say the system’s behavior is nonlinear [7]. what is the fuss over nonlinearity? the basic idea is that, for nonlinear systems, a small change in a parameter can lead to sudden and dramatic changes in both the qualitative and biomath 3 (2014), 1407231, http://dx.doi.org/10.11145/j.biomath.2014.07.231 page 2 of 15 http://dx.doi.org/10.11145/j.biomath.2014.07.231 s. nikolov, modelling and analysis of mirna regulation mir-x pol iienhancer promoter mirna type a mirna circuit type b mirna circuit gene z enhancer promoter z mrna no z protein steady state levels of z protein mirna degradation/inhibition pol ii gene y pol iienhancer promoter transcription off y mrna promoter no y protein fig. 1. integrating mirnas into regulatory networks. type a circuits might be involved in maintaining protein steady-state levels and homeostasis. type b circuits can serve as a surveillance mechanism to suppress leaky transcription of target genes. quantitative behavior of the system. for one value, the behavior might be stable, for another value only slightly different from the first, the behavior might be periodic or completely aperiodic. when time-delay, feedback-loops and nonlinearity appear combined in systems like mirna networks, a suitable approach for investigating its dynamics is the systems bifurcation analysis. bifurcation theory studies persistence and exchange of qualitative properties of dynamical systems under continuous perturbations. from the point of view of the dynamic systems theory, the hopf bifurcation theorem [22] together with other elements of the bifurcation theory are basic analytical tools to investigate pathological conditions in biological systems. a typical case is that of systems depending continuously on a single parameter. under small variations of the parameter, the systems may lose stability, and re-stabilize near another equilibrium or a closed orbit, or a larger attractor. the type of transition can be continuous, gentle and smooth, with the new equilibrium being far from the “trivial” one [2, 27]. the first case corresponds to a local bifurcation, the second one is a global bifurcation. local bifurcations can be of two types: i) the system leaves its equilibrium state and reaches a new equilibrium state; ii) the system goes from an equilibrium state to an invariant subset generally composed of several equilibriums and curves connecting them, closed orbits, or tori, etc. for example, two component mechanisms with autocatalysis easily generate oscillations and bistability. they also exhibit a rich structure of bifurcations to more complicated behavior, for example pitchfork bifurcation, saddle-node bifurcation or takens-bogdanov bifurcation [26]. the most popular and elementary situation is the andronovhopf bifurcation, characterized by the onset of a closed orbit, starting near the trivial equilibrium (from focus type), which is the phase portrait of a periodic solution with a period close to some fixed biomath 3 (2014), 1407231, http://dx.doi.org/10.11145/j.biomath.2014.07.231 page 3 of 15 http://dx.doi.org/10.11145/j.biomath.2014.07.231 s. nikolov, modelling and analysis of mirna regulation number [1, 27]. in this work, we will restrict our investigation to the andronov-hopf bifurcation. the aim of this article is to elucidate how the dynamics of the gene expression (associated with the time delays) is regulated by the mirna. thus, the plan of the paper is as follows: in section 2 we formulate the mathematical model. it results in three delayed differential equations (ddes). a qualitative analysis of the model equations is performed in section 3. in section 4 we explore numerically the model. finally, in section 5, we discuss and unify results from previous sections. ii. model there are two possible ways in which mirnas can play a role in gene expression [33]. one is that they can accelerate the degradation of mrnas, while the other is that they repress translation by forming silencing complexes (see fig. 2). in the version of the model discussed here we suppose that the system acts as a feedback loop where the protein, y2, controls its own synthesis through the repression of mrna, y1. here we construct one equation, describing the rate of change in the concentration of mirna, y3, taking into account only the first possible mechanism. we assume also (similarly to [33]) that the production of mirna occurs at a constant rate, l, and we define k6 as its degradation rate. according to the first mechanism (in which mirnas are regarded as an enhancer of mrna degradation) an extra degradation term, k4y1y3, to the rate equation for mrna and mirna (where k4 is the degradation rate) is added. based on these hypotheses, the system can be represented by the following mathematical model in time delayed differential equations dy1 dt = k1 k2 + k3y n1 2 (t− τ1) −γ1y1 −k4y1y3, dy2 dt = k5y1 (t− τ2) −γ2y2, dy3 dt = l−k6y3 (t− τ3) −k4y1y3, (2) mrna protein delayed negative feedback mirna mrna degradation risc τ1 τ2 gene τ3 fig. 2. scheme of transcription ( τ1) and translation ( τ2) time delay negative feedback control of protein synthesis where τ1 is the time delay for translation, τ2 is the time delay for transcription, τ3 is the average time delay for degradation of mirna, n1 is often referred as a hill coefficient or a cooperativity coefficient, ki (i = 1, ..., 6) and γ1, γ2 are the kinetic rate constants. in the present work, according to [20, 34], we consider one representative value for hill coefficient, n1 = 2. iii. qualitative analysis in this section, we consider the system (2) when the hill coefficient, n1, is equal to two and all constants of the model are real positive numbers. furthermore, we investigate the bifurcation structureparticularly the andronov-hopf bifurcationfor system (2), using time delays τ1, τ2 or τ3 as bifurcation parameters. here, we note that system (2) is a private case of the general system considered in [23], and for our analysis we use a specific theoretical approach obtained there. the fixed points of the system, e = (y1, y2, y3), represented by eq. (2) can be analytically estimated and are defined by the following set of algebraic equations, including the rate constants of the model: biomath 3 (2014), 1407231, http://dx.doi.org/10.11145/j.biomath.2014.07.231 page 4 of 15 http://dx.doi.org/10.11145/j.biomath.2014.07.231 s. nikolov, modelling and analysis of mirna regulation y42 + k5 (k4l + k6γ1) k4γ1γ2 y32 + k2 k3 y22 + k5 k3k4γ1γ2 [k2 (k4l + k6γ1) −k1k4] y2 − k1k 2 5k6 k3k4γ1γ 2 2 = 0,y1 = γ2 k5 y2, y3 = k5l k5k6 + γ2k4y2 . (3) according to descarte’s rule [10, 14], the first equation in (3) has only one real positive root, which ensures that system has only one physiologically feasible fixed point. let us consider a small perturbation around the fixed point e of the system (2) defined as: yk = yk + xk (k = 1, 2, 3) (4) in the case when n1 = 2, the function k1 k2+k3y 2 2 (t−τ1) can be written as a maclaurin series: k1 k2 + k3y 2 2 (t− τ1) = k1 δ + k3η = k1 δ ( k3 δ η + 1 ) = k1 δ ( 1 − k3 δ η + ( k3 δ )2 η2 − ( k3 δ )3 η3 + ... ) (5) where δ = k2 + k3y22 and η = 2y2x2 (t− τ1) + x22 (t− τ1). if we take only linear term from (5) and after substitution of (4) into differential equation (2) we have dx1 dt = −c1x1 − c2`−χτ1x2 − c3x3 − c4`−2χτ1x22 −k4x1x3, dx2 dt = k5` −χτ2x1 −γ2x2, dx3 dt = −c5x1 − ( k6` −χτ3 + c3 ) x3 −k4x1x3, (6) where c1 = γ1 + k4y3, c2 = 2k1k3y2 δ2 , c3 = k4y1, c4 = k1k3 δ2 , c5 = k4y3. (7) the characteristic equation of (6) has the form det ( −c1 −χ −c2`−χτ1 −c3 k5` −χτ2 −γ2 −χ 0 −c5 0 −(c3 + k6`−χτ3 ) −χ ) = 0 (8) that is χ3 + k1χ 2 + k2χ + k3 = ` −χτ3 (t1χ 2 + t2χ +t3) + ` −χ(τ1+τ2) (t4χ + t5) + t6` −χ(τ1+τ2+τ3) (9) where k1 = γ2 + c1 + c3, k2 = γ2c1 + c3 (γ2 + c1 − c5) ; k3 = γ2c3 (c1 − c5) , t1 = −k6, t2 = −k6(γ2 + c1), t3 = −γ2k6c1, t4 = −k5c2, t5 = −k5c2c3, t6 = −k5k6c2. (10) remark 1. we note that χ = 0 is a root of (9) if and only if k3 = t3 = t5 = t6 = 0. in the absence of delays ( τ1 = τ2 = τ3 = 0), e is locally asymptotically stable if p = γ2 + k6 + c1 + c3 > 0, q = γ2 (k6 + c1 + c3) + k5c2 + c3 (c1 − c5) > 0, r = γ2c3 (c1 − c5) + γ2c1k6 + k5c2 (k6 + c3) > 0, r = pq −r > 0. (11) biomath 3 (2014), 1407231, http://dx.doi.org/10.11145/j.biomath.2014.07.231 page 5 of 15 http://dx.doi.org/10.11145/j.biomath.2014.07.231 s. nikolov, modelling and analysis of mirna regulation here we note that (9) is a third-degree exponential polynomial in χ with three discrete time delays. because of the presence of three different discrete delays into (9), the analysis of the sign of the real parts of the eigenvalues is very complicated, and a direct approach cannot be considered. thus, in our analysis we will use a method consisting of determining the stability of the steady state when firstly two delays are equal to zero, and secondly when one delay is equal to zero. the case τ1 = τ3 = 0, τ2 > 0. generally, in eukariotes translation takes longer than transcription and one of the reasons is intron splicing. it has been suggested that the length of the introns is fundamental to cell timing [30]. hence, we assume that the finite time delay τ2 of transcription is longer than the time delay τ1 of translation and finite time delay τ3 of degradation of mirna. setting τ1 = τ3 = 0 into (9), the characteristic equation becomes χ3 + k11χ 2 + k21χ + k31 = ` −χτ2 (t4χ + t51) , (12) where k11 = k1 −t1, k21 = k2 −t2, k31 = k3 −t3, t51 = t5 + t6. (13) it is well-known that that the stability of the equilibrium state e depends on the sign of the real parts of the roots of (12). we recall that a steady state is locally asymptotically stable if and only if all roots of (12) have negative real parts, and its stability can only be lost if these roots cross the vertical axis, that is if purely imaginary roots appear. generally speaking, the transcendental equation (12) (for nonzero delay) cannot be solved analytically and has an indefinite number of roots. in essence, we have two main tools besides direct numerical integration; firstly, the linear stability analysis in the case of a small time delay, and secondly, the hopf bifurcation theorem. because from biological point of view it is known that time delay of transcription, τ2, in many cases is bigger than one [15, 20, 21, 35] here we use hopf bifurcation theorem. thus, we let χ = m + in (m, n ∈ r), and rewrite (12) in terms of its real and imaginary parts as ∣∣∣∣∣∣∣∣∣∣ m3 − 3mn2 + k11 ( m2 −n2 ) + k21m + k31 = `−mτ2 [(t4m + t51) cosnτ2 + t4nsinnτ2] , −n3 + 3m2n + 2k11mn + k21n = `−mτ2 [t4ncosnτ2 − (t4m + t51) sinnτ2] . (14) to find the first bifurcation point we look for purely imaginary roots χ = ±in, n ∈ r of (12), i.e. we set m = 0. then the above two equations are reduced to ∣∣∣∣ −k11n2 + k31 = t51cosnτ2 + t4nsinnτ2,−n3 + k21n = t4ncosnτ2 −t51sinnτ2, (15) or another one cos nτ2 = t51(k31−k11n2)+t4n2(k21−n2) t251+t 2 4 n 2 , sin nτ2 = n[t4(k31−k11n2)−t51(k21−n2)] t251+t 2 4 n 2 . (16) it is clear that if the first bifurcation point is( n0b, τ 0 b ) , then the other bifurcation points (nb, τb) satisfy nbτb = n0bτ 0 b + 2νπ, ν = 1, 2, ..., ∞. one can notice that if n is a solution of (15) (or (16)), then so is −n. hence, in the following we only investigate for positive solutions n of (15), or (16) respectively. by squaring the two equations into system (15) and then adding them, it follows that: biomath 3 (2014), 1407231, http://dx.doi.org/10.11145/j.biomath.2014.07.231 page 6 of 15 http://dx.doi.org/10.11145/j.biomath.2014.07.231 s. nikolov, modelling and analysis of mirna regulation n6 + ( k211 − 2k21 ) n4 + (k221 − 2k11k31 −t 24 )n 2 + k231 −t 2 51 = 0. (17) as e is locally asymptotically stable at τ2 = 0, satisfies the routh-hurwitz conditions for stability for a cubic polynomial [9, 19]. equation (17) is a cubic in n2 and the left-hand side is positive for very large values of n2 and negative for n = 0 if and only if t 251 > k 2 31, i.e. when eq. (17) has at least one positive real root. moreover, to apply the hopf bifurcation theorem, according to [9], the following theorem in this situation applies: theorem 1. suppose that nb is the last positive simple root of (17). then, in (τb) = inb is a simple root of (12) and m (τ2) + in (τ2) is differentiable with respect to τ2 in a neighbourhood of τ2 = τb. to establish an andronov-hopf bifurcation at τ2 = τb, we need to show that a pair of complex eigenvalues crosses the imaginary axis with nonzero speed, i.e. the following transversality condition d(reχ)dt |τ=τb 6= 0 is satisfied. from (15) we know that τbk corresponding to nb is τbk = 1 nb arccos [ (−t4n4b + (t4k21 −t51k11)n 2 b +t51k31)/(t 2 51 + t 2 4 n 2 b) ] + 2kπ nb , k = 0, 1, 2, ... (18) because for τ2 = 0, equilibrium e is stable, by butler’s lemma [5], equilibrium remains stable for τ2 < τbk , where τb = τbk as k = 0. we have now to show that d(reχ)dt |τ=τb 6= 0. hence, if denote h (χ, τ2) = χ 3 + k11χ 2 + k21χ + k31 − `−χτ2 (t4χ + t51) , (19) then dχ dτ2 = − ∂h ∂τ2 /∂h ∂χ = (−χ`−χτ2 (t4χ + t51)) /(3χ2 + 2k11χ + k21 + τ2` −χτ2 (t4χ + t51) − `−χτ2t4). (20) evaluating the real part of this equation at τ2 = τb and setting χ = inb yield dm dτ2 |τ2=τb = d (reχ) dt |τ2=τb = (n2b[3n 4 b + 2 ( k211 − 2k21 ) n2b + k 2 21 − 2k11k31 −t 24 ]) / (l2 + i2) (21) where l = −3n2b + k21 + τ2 ( −k11n2b + k31 ) −t4cosnbτ2, i = 2k11nb + τ2 ( −n3b + k21nb ) + t4sinnbτ2. (22) let θ = n2b, then, (17) reduces to g (θ) = θ3 + ( k211 − 2k21 ) θ2 + ( k221 − 2k11k31 −t 2 4 ) θ + k231 −t 2 51. (23) then, for g′ (θ) we have g′(θ)|τ2=τb = dg dθ |τ2=τb = 3θ2 + 2 ( k211 − 2k21 ) θ + k221 − 2k11k31 −t 2 4 . (24) if nb is the least positive simple root of (17), then biomath 3 (2014), 1407231, http://dx.doi.org/10.11145/j.biomath.2014.07.231 page 7 of 15 http://dx.doi.org/10.11145/j.biomath.2014.07.231 s. nikolov, modelling and analysis of mirna regulation dg dτ2 |θ=n2b > 0. (25) hence, dm dτ2 |τ2=τb = d (reχ) dτ2 |τ2=τb = n2bg ′ ( n2b ) l2 + i2 > 0. (26) according to the hopf bifurcation theorem [17], we define the following theorem 2: theorem 2. if nb is the least positive root of (17), then an andronov-hopf bifurcation occurs as τ2 passes through τb. corollary 2.1. when τ2 < τb, then the steady state e of system (2) is locally asymptotically stable. the case τ1 = 0; τ2, τ3 > 0. we return to the study of (9) which with τ2, τ3 > 0 has the form χ3 + k1χ 2 + k2χ + k3 = `−χτ3 ( t1χ 2 + t2χ + t3 ) + `−χτ2 (t4χ + t5) + t6` −χτ23, (27) where τ = [τ2, τ3, τ23 = τ2 + τ3] t denotes a point in the time delay space, i.e. τ ∈ ω ⊂ r3+. ω is the time delay space and r3+ denotes the set of nonnegative real numbers. in order to assess the stability of e with respect to any delay τ, one should know where all χ roots of (27) lie on the complex plane. eq. (27) has infinitely many roots on the complex plane due to the transcendental term `−χτ . this makes the analytical stability assessment intractable. previously, we obtain that in the absence of delays, e is locally asymptotically stable if the conditions (11) are valid. by remark 1, this implies that χ = 0 is not root of (27). further, we introduce the following simple result (which is proved in [25]) using rouche’s theorem lemma 1. consider the exponential polynomial p (χ, `−χτ1, ..., `−χτm ) = χn + p (0) 1 χ n−1 + ... + p (0) n−1χ + p (0) n + [p (1) 1 χ n−1 + ... + p (1) n−1χ + p(1)n ]` −χτ1 + ... + [p (m) 1 χ n−1 + ... + p (m) n−1χ + p(m)n ]` −χτm, where τi ≥ 0 (i = 1, 2, ..., m) and p (i) j (i = 0, 1, ..., m; j = 1, 2, ..., n) are constants. as (τ1, τ2, ..., τm) vary, the sum of the order of the zeros of p (χ, `−χτ1, ..., `−χτm )on the open right half plane can change only if a zero appears on or crosses the imaginary axis. obviously, in (n > 0) is a root of (27) if and only if n satisfies −n3i−k1n2 + k2ni + k3 = (cosnτ3 − isinnτ3) ( −t1n2 + t2ni + t3 ) + (cosnτ2 − isinnτ2) (t4ni + t5) + t6 (cosnτ23 − isinnτ23) . (28) separating the real and imaginary parts into (28), we obtain ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ −k1n2 + k3 = t4nsinnτ2 + t5cosnτ2 + ( −t1n2 + t3 ) cosnτ3 + t2nsinnτ3 +t6cosnτ23, −n3 + k2n = t4ncosnτ2 −t5sinnτ2 − ( −t1n2 + t3 ) sinnτ3 + t2ncosnτ3 −t6sinnτ23 . (29) we square and add the equations (29), and after simplifying, we get that τ and n must be biomath 3 (2014), 1407231, http://dx.doi.org/10.11145/j.biomath.2014.07.231 page 8 of 15 http://dx.doi.org/10.11145/j.biomath.2014.07.231 s. nikolov, modelling and analysis of mirna regulation among the real solutions of n6 + ( k21 −t 2 1 − 2k2 ) n4 + (k22 −t 2 2 −t 2 4 − 2k1k3 + 2t1t3)n2 + t 26 + k 2 3 −t 2 3 −t 2 5 = 2{[t2t4n2 + t5(t1n2 + t3)]cosn(τ2 − τ3) + n[t4(−t1n2 + t3) −t2t5]sinn(τ2 − τ3) −t6[(k1n2 −k3)cosnτ23 + (−n3 + k2n)sinnτ23]} (30) we note that the right-hand side of (30) is always less than 2[|t2t4|n2 + |t5|| − t1n2 + t3| + (|t4|| − t4n2 + t3| − |t2t5|)n − |t6| ( |k1n2 −k3| + |−n3 + k2n| ) ]. hence if the inequality ω6 + ( k21 −t 2 1 − 2k2 ) ω4 + ( k22 −t 2 2 −t 2 4 − 2k1k3 + 2t1t3 ) ω2 + t 26 + k 2 3 −t 2 3 −t 2 5 > 2[|t2t4|ω 2 + |t5||−t1ω2 + t3| + |t4||−t4ω2 + t3|ω −|t6| ( |k1ω2 −k3|− |−ω3 + k2ω| ) ] (31) has no real solution on 0 < ω < n+, then (30) cannot be satisfied. note that n+ is the positive solution of first equation in (29), which we write as k1n 2 = ψ (n) = [k + (t1cosnτ3) n 2 − (t4sinnτ2 + t2sinnτ3) n−t5cosnτ2 −t3cosnτ3 −t6cosnτ23] ≤ k3 + |t1|n2 − (|t2| + |t4|)n−|t3|− |t5|− |t6|, i.e. (k1 −|t1|) n2 + (|t2| + |t4|) − (k3 −|t3|− |t5|− |t6|) = 0. (32) thus, for n+ we have n+ = 1 2a ( −b + √ b2 + 4ac ) , (33) where a = k1 −|t1| 6= 0, b = |t2| + |t4|, c = k3 −|t3|− |t5|− |t6|. it is clear that n ≤ n+. rearranging terms, we write (31) as ( ω|−ω2 + k2|− |t6| )2 + ( |k1ω2 −k3| + |t6| )2 + ( −t4ω2 + t3 )2 + (|t2|ω + |t5|) + t 24 ω 2 >( |−t1ω2 + t3| + |t5| )2 + (|−t4ω2 + t3| + t4ω) 2 + (t2 + t4) 2 ω2 + t 22 ω 2 + t 25 . (34) hence, the following theorem can be formulated theorem 3. let k3 −|t3|− |t5|− |t6| 6= 0 and (34) hold. then there is no change in stability of e. remark 2. in the special case that τ2 = τ3, the characteristic equation (27) becomes χ3 + k1χ 2 + k2χ + k3 = = `−χτ2 ( t1χ 2 + t24χ + t35 ) + t6` −2χτ2, (35) where t35 = t3 + t5 and t24 = t2 + t4. therefore, this case is a private one of theorem 3. corollary 3.1. if conditions of theorem 3 are not valid and τbif2́ defined as in theorem 2, then according to lemma 1 for any τ2 ∈ [0, τb), there exists a τbif23 (τ2) > 0 ( τ bif 1 (τ2) > 0 respectively) such that the steady state e of system (2) is unstable when τ3 ∈ [0, τbif3 (τ2)) ( τ1 ∈ [0, τbif1 (τ2)) respectively), and an andronovhopf bifurcation takes place. the general case τ1, τ2, τ3 > 0. similar to previous section, we set that χ = in (n > 0) is a root of (9) if and only if n satisfies biomath 3 (2014), 1407231, http://dx.doi.org/10.11145/j.biomath.2014.07.231 page 9 of 15 http://dx.doi.org/10.11145/j.biomath.2014.07.231 s. nikolov, modelling and analysis of mirna regulation n3i−k1n2 + k2in + k3 = (cosnτ3 − isinnτ3) ( −t1n2 + t2in + t3 ) + (cosnτ4 − isinnτ4) (t4in + t5) + (cosnτ5 − isinnτ5) t6, (36) where in this case τ = [τ3, τ4 = τ1 + τ2, τ5 = τ1 + τ2 + τ3] t denotes a point in the time delay space, i.e. τ ∈ ω ⊂ r3+. here, ω is the time delay space and r3+ denotes the set of nonnegative real numbers. separating the real and imaginary parts into (36), we have ∣∣∣∣ −k1n2 + k3 −a1 = b1 + t6cosnτ5,−n3 + k2n + a2 = b2 −t6sinnτ5 (37) where a1 = ( −t1n2 + t3 ) cosnτ3 + t2nsinnτ3, a2 = ( −t1n2 + t3 ) sinnτ3 −t2ncosnτ3, b1 = t4nsinnτ4 + t5cosnτ4, b2 = t4ncosnτ4 −t5sinnτ4. (38) adding up the squares of both equations into (37), we have n6 + ( k21 + t 2 1 − 2k2 ) n4 + t 23 −t 2 5 −t 2 6 + ( k22 + t 2 2 −t 2 4 − 2k1k3 − 2t1t3 ) n2 = 2{t6 (−t4nsinnτ3 + t5cosnτ3) − ( −n3 + k2n ) [ ( −t1n2 + t3 ) sinnτ3 −t2ncosnτ3] − ( k1n 2 −k3 ) [ ( −t1n2 + t3 ) cosnτ3 + t2nsinnτ3]}, (39) where τ5−τ4 = τ3. clearly, the right-hand side of (38) is always less than 2[t6|−t4n + t5|− (−n3 −k1n2 + k2n + k3)|−t1n2 + t2n + t3|]. (40) hence if the inequality λ6 + ( k21 + t 2 1 − 2k2 ) λ4 + ( k22 + t 2 2 −t 2 4 − 2k1k3 − 2t1t3 ) λ2 + t 23 −t 2 5 −t 2 6 > 2[t6|−t4λ + t5| − ( −λ3 −k1λ2 + k2λ + k3 ) |−t1λ2 + t2λ + t3|], (41) has no real solution on 0 < λ < n+, then (39) cannot be satisfied. similar to previous section, we note that n+ is the positive solution of first equation into (37), which is written as k1n 2 = ψ (n) = k3 − [ ( −t1n2 + t3 ) cosnτ3 + t2nsinnτ3 + t4nsinτ4 + t5cosnτ4 +t6cosnτ5] ≤ k3 + |t1|n2 − (|t2| + |t4|) n −|t3|− |t5|− |t6|, (42) i.e. we obtain the same formula as (32) (and (33) respectively). rearranging terms, we write (41) as ( |−t1λ2 + t2λ + t3|−λ3 −k1λ2 + k2λ + k3 )2 + ( |t2λ + t3| + t1λ2 )2 + ( λ2 + k1λ )2 + (t2 −t3λ)2 + ( t 25 + 2k 2 2 ) λ2 + k23 > (|−t4λ + t5| + t6)2 + (t5λ + t4)2 + (k2λ + k3) 2 + (t2λ + t3) 2 + ( λ3 −k3 )2 + ( k1λ 2 −k2λ )2 + t 21 λ 4 + t 23 λ 2. (43) therefore, the following theorem can be formulated biomath 3 (2014), 1407231, http://dx.doi.org/10.11145/j.biomath.2014.07.231 page 10 of 15 http://dx.doi.org/10.11145/j.biomath.2014.07.231 s. nikolov, modelling and analysis of mirna regulation theorem 4. let k3 −|t3|− |t5|− |t6| 6= 0 and (43) hold. then there is no change in stability of e. remark 3. in the special case that τ1 = τ2 = τ3, the characteristic equation (9) becomes χ3 + k1χ 2 + k2χ + k3 = ` −χτ3 (t1χ 2 + t2χ + t3) + ` −2χτ3 (t4χ + t5) + t6` −3χτ3. (44) hence, this case is a private one of theorem 4. corollary 4.1. if conditions of theorem 4 are not valid and τbif3 is defined as in theorem 2, then according to lemma 1 for any τ3 ∈ [0, τb), there exists a τbif1 ((τ3)) > 0 ( τ bif 2 (τ3) > 0; τbif4 (τ3) > 0; τ bif 5 (τ3) > 0 respectively) such that the steady state e of system (2) is unstable when τ1 ∈ [0, τbif1 (τ3)) ( τ2 ∈ [0, τ bif 2 (τ3)); τ4 ∈ [0, τbif4 (τ3)); τ5 ∈ [0, τ bif 5 (τ3)) respectively), and an andronov-hopf bifurcation takes place. in next section, we illustrate numerically the existence of the behavior predicted for some values of the rate constants ki (i = 1, ..., 6) , γ1, γ2, l and the time delays τ1, τ2 and τ3. iv. numerical analysis in the previous section, we proposed the analytical tools and used them for a qualitative analysis of the system, obtaining predictions about dynamics of the system, i.e. the stability and existence of periodic solutions via andronov-hopf bifurcation in time delay model (2). in this section, we perform a numerical analysis of model (2), based on the results previously obtained. some of the parameter values used in the numerical analysis were selected according to [4, 20, 24, 29, 33] in the form k1 = 0.3 [ min−1 ] , k2 = k3 = 1 [ min−1 ] , k4 = 0.3 [ min−1 ] , k5 = 0.5 [ min−1 ] , γ1 = 0.1 [ min−1 ] , γ2 = 0.2 [ min−1 ] , τ1 ∈ [1, 8] , τ2 ∈ [12, 35] . (45) according to [6, 32] proteins (or rnas) degraded with a probability that depends on their structure because some of the degradation mechanisms involve multiple steps. therefore, individual protein (or rna) senesces through time. mirnas which are incorporated into the risc complex, do not degrade with their targets but return to the cytosol a new round of target mrna repression. it is plausible, however, that mirna may be degraded after a few cycles of target mrna binging [13]. since we do not know the exact values of the average time delay for degradation of mirna, we set τ3 in large boundariesfrom few minutes to few hours and its degradation rate k6 ∈ [0.05, 0.3]( min−1 ) . our model include also one additional parameter for which no values are available and his estimations require further experimental studies. thus, we assume here that l = 0.1 in minutes. in order to compare the predictions with numerical results, the governing equations of the model (2), were solved numerically using matlab [18]. in figure 3, the stable solutions for the concentration of mrna ( y1), the concentration of protein ( y2) and the concentration of mirna ( y3) are shown for absence of time delay (i.e. τ1 = τ2 = τ3 = 0) see fig. 3a, and for τ1 = τ3 = 0, τ2 = 12 see fig. 3b. it is evident that after several physiological acceptance fluctuations, the solution of system (2) approaches a constant value (stable equilibrium state). in other words the system possesses a stable equilibrium state which corresponds to a normal mirna regulation process. this conclusion is in accordance with the theorem 2 (corollary 2.1) proofed in previous section 3. biomath 3 (2014), 1407231, http://dx.doi.org/10.11145/j.biomath.2014.07.231 page 11 of 15 http://dx.doi.org/10.11145/j.biomath.2014.07.231 s. nikolov, modelling and analysis of mirna regulation (a) (b) fig. 3. stable regime of system (2) for a) τ1 = τ2 = τ3 = 0 and b) for τ1 = τ3 = 0, τ2 = 12. model parameters: k1 = k4 = 0.3; k2 = k3 = 1; γ1 = 0.1; γ2 = 0.2; k5 = 0.5; k6 = 0.05; n1 = 2; l = 0.1. figure 4 depicts the cases when τ2 and τ3 are varied. it is seen that for larger values of τ2 (see fig. 4a) and τ3 (see fig.4b) than bifurcation one the stable limit cycle (self oscillations) occur and sustained oscillations take place. in other words, in these cases the conditions of theorem 3 and theorem 4 are not satisfied and the steady state of system (2) is unstable. from biological point of view, the occurrence of oscillation implies that if the average time delay for degradation of mirna, τ3, can increase, then the effect of mirna on gene expression is initially destabilizing. if the average time delay for degradation of mirna increases further, then the effect of mirna on gene expression can promote stabilitysee fig. 4c, and again instabilitysee fig. 4d. thus gene expression follows a cyclic pattern (from stable to unstable behaviour and vice versa) as function of average time delay for degradation of mirna. this cyclic regime is shown in figure 5. (a) (b) biomath 3 (2014), 1407231, http://dx.doi.org/10.11145/j.biomath.2014.07.231 page 12 of 15 http://dx.doi.org/10.11145/j.biomath.2014.07.231 s. nikolov, modelling and analysis of mirna regulation (c) (d) fig. 4. dynamic behaviour of system (2). unstable regime (periodic oscillations) for τ1 = τ3 = 0, τ2 = 20 (a); and τ1 = 0, τ2 = 13, τ3 = 10 (b). stable regime for τ1 = 1, τ2 = 13, τ3 = 37 (c). unstable regime (sustained oscillations) for τ1 = 1, τ2 = 13, τ3 = 60 (d). in figure 5, the stable and unstable zones in the (τ3, τ1 + τ2) parameter space are shown. it is seen that these zones are with different size. it is also interesting to note that in unstable zones sustained oscillations with period one and quasiperiodic motion take place. 21 ττ + 10 3τ25 49 78 99 135 147 unstable 300 fig. 5. stable and unstable zones of system (2) at τ1 = 1, τ2 = 13 and τ3 ∈ [10, 300]. v. conclusions gene expression in the human organism is posttranscriptionally regulated by mirnas. mirnas are embedded in complex regulatory networks that involved gene activation, post-translational regulation and protein-protein interactions. therefore, this makes mirnas as one of the most abundant classes of regulatory genes in animals. in the present paper we develop a time delay model of a feedback system regulated via mirna. our hypothesis (according [33] is that mirna can participate in the regulation of gene expression by accelerating the degradation of mrna or by repressing the translation process. the model resulted in three ddes with three discrete time delays. since this system is a classical case study, covering several essential features of mirna and genetic regulatory mechanisms, general conclusions about design principles and role of time delays in the stability of gene circuits can be suggested. the basic view that time delays τ1, τ2 and τ3 are a key factor in the dynamic behaviour of the system was confirmed by the analytical calculations and numerical simulations. in more details, under the assumption that an equilibrium exists, we have estimated the length of delays for which local asymptotic stability will be preserved. we have also derived criteria for which no change biomath 3 (2014), 1407231, http://dx.doi.org/10.11145/j.biomath.2014.07.231 page 13 of 15 http://dx.doi.org/10.11145/j.biomath.2014.07.231 s. nikolov, modelling and analysis of mirna regulation in stability will occur. if system (2) starts with a stable equilibrium, which for some delay(s) becomes unstable, it will likely destabilize by means of an andronov-hopf bifurcation leading to small amplitude periodic solutions. our investigation of such a behaviour is devoted to the use of bifurcation analysis. particularly, a hopf bifurcation theorem was employed. from the viewpoint of the qualitative theory of ddes, time delay(s) appears as a bifurcation parameter on whose values the altered (stable or unstable) behaviour of the model depends. for time delays longer than τb, the gene expression system regulated by means of mirna would present sustained oscillations with coupled periodic variations on the concentration of the mrna, protein and the mirna. in contrast, a time less than τb would provoke damped oscillations around a stable steady state. we can say that in this case time delays have a destabilizing role. from a physiological point of view, the loss of stability might be related to emergence of new configurations in the regulatory gene circuit that could lead the system to a pathological state. to conclude, mathematical modelling and analysis can enable to understand the mechanism underlying an observed biological process, and at the same time, provide a testable hypothesis for future studies. in this paper we investigate the main processes of the formation of a proteintranscription time (starts with splicing and polyadenylation of the initial transcript), translation time (the timespan from the emergence of mrna) and average time delay for degradation of mirna. thus, our dynamical predictions can be tested in future experiments. references [1] a. andronov, a. witt and s. chaikin theory of oscillations, addison-wesley, 1966. 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[35] s. zeiser, o. rivera, c. kuttler and et al., oscillations of hes7 caused by negative autoregulation and ubiquiti nation, comput. biol. chem. 32 (2008) 48–52. biomath 3 (2014), 1407231, http://dx.doi.org/10.11145/j.biomath.2014.07.231 page 15 of 15 http://dx.doi.org/10.1016/j.molcel.2007.05.018 http://dx.doi.org/10.11145/j.biomath.2014.07.231 introduction model qualitative analysis numerical analysis conclusions references www.biomathforum.org/biomath/index.php/biomath original article biological control of sugarcane caterpillar (diatraea saccharalis) using interval mathematical models josé renato campos ∗, edvaldo assunção †, geraldo nunes silva ‡ and weldon alexander lodwick § ∗ area of sciences federal institute of education, science and technology of são paulo, votuporanga, sp, brazil jrcifsp@ifsp.edu.br, jrcifsp@gmail.com † department of electrical engineering unesp univ estadual paulista, ilha solteira, sp, brazil edvaldo@dee.feis.unesp.br ‡ department of applied mathematics unesp univ estadual paulista, são josé do rio preto, sp, brazil gsilva@ibilce.unesp.br § department of mathematical and statistical sciences university of colorado, denver, colorado, usa weldon.lodwick@ucdenver.edu received: 20 august 2015, accepted: 23 april 2016, published: 2 may 2016 abstract—biological control is a sustainable agricultural practice that was introduced to improve crop yields and has been highlighted among the various pest control techniques. however, real mathematical models that describe biological control models can have error measurements or even incorporate lack of information. in these cases, intervals may be feasible for indicating the lack of information or even measurement errors. therefore, we consider interval mathematical models to represent the biological control problem. specifically, in the present paper, we illustrate the solution of a discrete-time interval optimal control problem for a practical application in biological control. to solve the problem, we use single-level constrained interval arithmetic [9] and the dynamic programming technique [3] along with the idea proposed in [23] for the solution of the interval problem. keywords-interval optimal control problem; interval mathematical models; single-level constrained interval arithmetic; dynamic programming; biological control. i. introduction sugarcane culture plays an important role in the brazilian economy. it is estimated that the country has more than 8 million hectares of cultivated area [1] and that sugarcane is responsible for over 4.5 million jobs [38]. in addition to the production of sugar, ethanol and various other byproducts, it is also used to produce electricity with the use citation: josé renato campos, edvaldo assunção, geraldo nunes silva, weldon alexander lodwick, biological control of sugarcane caterpillar (diatraea saccharalis) using interval mathematical models, biomath 5 (2016), 1604232, http://dx.doi.org/10.11145/j.biomath.2016.04.232 page 1 of 11 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2016.04.232 josé renato campos et al., biological control of sugarcane caterpillar ... of biomass (bagasse and straw). thus, sustainable management of this culture is fundamental. among the various types of management that can be implemented (control of pests and weeds, soil handling, etc.) and the various methods of manufacture (biological control, use of insecticide and herbicide, manual and mechanical control, etc.), pest control through biological control stands out. biological control is sustainable because it does not affect the environment. for the culture of sugarcane, the control involves the caterpillar and wasp. the caterpillar (diatraea saccharalis) is an insect that causes damage to the crop, and its natural predator, cotesia flavipes, is a wasp that deposits its eggs on the caterpillar and inhibits the development of the caterpillar. hence, the caterpillar dies without completing its life cycle and without causing economic loss to the crop. the spread of the caterpillar can cause damage to the crop such as weight loss and reduction in germination, leading to the death of germinating plants, which directly reflects on the costs of production. thus, the biological control of pests is a good alternative to the feasibility of such crops for the country. in addition, the biological control process is part of the integrated crop protection [11] that is a benchmark for sustainable farming practices. control theory study began in the usa in the 1930s with studies of problems in electrical engineering and mechanical engineering [8]. in the 1950s, with optimization methods developed by bellmann in 1957 (see [2]) and pontryagin in 1958 (see [29], [30]), modern control theory or optimal control theory was born. such theory brought advances in several areas such as agriculture, biology, economics, engineering and medicine. in agriculture or biology, deterministic optimal control problems are widely studied, and some biomathematical models illustrating deterministic models can be found in [7], [15], [16], [19], [37]. in these studies, conventional models were assumed with fixed coefficients. for problems with uncertain parameters, the optimal control problem usually utilizes stochasticity [4], [16] or, more recently, fuzzy set theory [12], [10], [28]. in the two cases, the coefficients are viewed as random variables or as fuzzy sets, and it is assumed that their probability distributions or membership functions, respectively, are known. in biological problems, uncertainty arises frequently because it is inherent to the determination of biological data; for example, uncertainty arises due to measurement errors, inaccuracies in the equipment, climatic factors, and lack of specification, among many others. thus, we propose interval uncertainty to describe the uncertainty in obtaining data in biological problems. we can represent a parameter of the model, such as the mortality rate of predators, as an interval. this is relevant because we can model an environment with several variations in the mortality rates of predators and not have to consider a unique rate for all the predators, especially if this information has been obtained imprecisely. optimal control problems involving uncertain systems are described in [6], [13], [14], [39]. however, in these approaches, the functional is a real number and thus differs from the approach proposed in this paper. additionally, the problem discussed here does not include state feedback. references on control problems that present interval uncertainty but still differ from that proposed in this paper can be found in [20], [17], [32]. thus, in this work, we consider a new kind of problem called the interval optimal control problem. the interval arithmetic used in this approach is described in [9], [21], [22] and is different from the standard interval arithmetic proposed in [24]. to solve the interval optimal control problem, we choose single-level constrained interval arithmetic [9] because it eliminates certain problems related to other types of interval arithmetic, such as the existence of the additive inverse or the distributive law property. single-level constrained interval arithmetic also has properties closer to the space of real numbers. therefore, we study the discrete time interval optimal control problem with the interval initial condition or interval parameters in the dynamic equation. biomath 5 (2016), 1604232, http://dx.doi.org/10.11145/j.biomath.2016.04.232 page 2 of 11 http://dx.doi.org/10.11145/j.biomath.2016.04.232 josé renato campos et al., biological control of sugarcane caterpillar ... the paper is arranged as follows. section ii presents the application in biology, and some biological aspects will be demonstrated. we also present the deterministic and interval optimal control problem. in section iii, we present the solutions of the discrete time interval optimal control problems previously proposed. the discussion of the results is provided in section iv. ii. the biomathematical model the biological situation studied is a problem encountered in sugarcane culture. according to silva and bergamasco [35], the environmental management of sugarcane culture requires performance prediction in production and environmental risk at various levels of control in sugarcane production because manipulation of the soil, planting depth and density, pest and diseases, among other factors, and biological control have proven to be effective in operational management of the culture. thus, the problem studied corresponds to a model of competition between the wasp (cotesia flavipes) and the caterpillar (diatraea saccharalis) in terms of sugarcane, represented using the lotkavolterra two-species model. tusset and rafikov in [37] ran a simulation of the dynamics of the system without application control and showed that the system begins to stabilize at 350 days and that during this period, economic losses are experienced. thus, we need to apply control in previous periods, and the application of control corresponds to the introduction of wasps in sugarcane culture. tusset and rafikov [37] solve the continuous deterministic optimal control problem using the riccati equation. campos [7] also solved the deterministic and discrete problem using dynamic programming, and the results are similar for the two approaches. the goal here is to present the interval optimal control problem and solve the biological control problem encountered in sugarcane culture. we analyze the biological situation and describe the biomathematical model. according to tusset and rafikov [37], the lotka-volterra two-species model used in the problem of sugarcane culture is given by { ẋ = x (a−γ x− cy) ẏ = y (−d + rx) + u∗ + u , (1) where x(t) is the number of preys and y(t) is the number of predators for t ≥ 0. here, u∗ is the control that carries the system to the desired equilibrium point, and u is the control that stabilizes the system at this point. the dynamic model (1) is a lotka-volterra model for the case of the caterpillar that is the sugarcane parasitoid, where the coefficient a represents the interspecific growth of the preys, the coefficient d represents the mortality of the predators, c represents the capture rate, r is the maximum rate of growth of the predator population, and γ is the self-inhibition coefficient of growth of the preys due to restriction of food. according to [37], the parameter a is calculated assuming the absence of predators in (1). then, we obtain ẋ = x (a−γ x), (2) where we suppose that γ = a/k. solving the differential equation (2) and isolating the value of the parameter a, we obtain a = − 1 t [ ln ( k−x x k−x0 x0 )] . assuming k = 25000 and considering that the caterpillar lives on average 70 days and after mating lays on average 300 eggs (see [27]), we find that t = 70 days with x(70) = 300 caterpillars per hectare. assuming an initial number of preys equal to x0 = 2 caterpillars per hectare, it follows that the interspecific growth of the caterpillar is a = 0.0716 caterpillars per hectare per day. the calculation of the other parameters of the dynamic equation of problem (1) can be found in [37], following a similar analysis. thus, in this work, we obtain the numerical coefficients a, γ, c, d and r in [37] as well as the expression for the functional of the optimal control biomath 5 (2016), 1604232, http://dx.doi.org/10.11145/j.biomath.2016.04.232 page 3 of 11 http://dx.doi.org/10.11145/j.biomath.2016.04.232 josé renato campos et al., biological control of sugarcane caterpillar ... problem. the problem proposed in [37] with a quadratic objective function subject to nonlinear restrictions is given by minc = 1 2 ∫ tf 0 8(x−x∗)2+0.2841(y−y∗)2+u2dt subject to { ẋ = x (0.0716 − 0.0000029 x− 0.0000464 y) ẏ = y (−1 + 0.000520235 x) + u∗ + u , (3) where tf is the final time, the initial conditions are x0 = 5000 and y0 = 1500, and the final conditions are the desired equilibrium point (x∗, y∗). from a biological point of view, segato et al. [33] show that when the number of preys (diatraea saccharalis) reaches 5000 caterpillars per hectare, application of control u corresponds to the release of predators (cotesia flavipes). the calculations for the numerical coefficients of the states x and y in the functional of problem (3) are extensive and can be found in [37]; such calculations are based on [34], [31]. furthermore, tusset and rafikov consider in [37] a positive semidefinite and symmetric quadratic functional in order to take the system to the desired equilibrium point the fastest way possible when considering only small oscillations in the path of the system. this is important for the biological control problem studied. to solve problem (3), tusset and rafikov in [37] considered a problem with a linear dynamic equation. the linearization of the model is feasible because we suppose that the linear and nonlinear dynamic system behaviors are qualitatively equivalent in the vicinity of the equilibrium point (see [25], grobman-hartman theorem). thus, the dynamic equation of problem (3) is linearized (see [25]) assuming that the initial conditions are near the equilibrium point (2000, 1418.10). in a real system, this is possible when we apply a value several times that of the control. according to botelho and macedo [5] for the sugarcane crop, greater than or equal to 2500 caterpillars per hectare causes damage to the culture. we fix x∗ = 2000 (a value that does not cause damage) and hence obtain the value y∗ using the equation f(x∗, y∗) = 0, where f(x, y) = 0.0716 − 0.0000029 x − 0.0000464 y. therefore, the desired equilibrium point for the prey and the predator is represented by (x∗, y∗) = (2000, 1418.10) and used in the final condition of the problem. finally, the optimal control problem with a quadratic objective function subject to linear restrictions proposed in [37] is given by minc = 1 2 ∫ tf 0 8 z21 + 0.2841 z 2 2 + u 2 dt subject to ż = [ −0.0058 −0.0928 0.7386 0.0405 ] z + [ 0 1 ] u, (4) with initial conditions z1 0 = 3000 and z2 0 = 80.17 due to translation to the equilibrium point. note that z = (z1, z2)t = (x−x∗, y−y∗)t , where z is the translation of the point of equilibrium (x∗, y∗) to the origin and t denotes the transposed vector. in particular, the change in coordinates to problem (4) is performed assuming that we are close to the fixed point; furthermore, the change in coordinates facilitates the computational implementation. to find the solution of the problem of biological control of the sugarcane caterpillar (4) with a discrete dynamic programming method, campos [7] discretized problem (4). the discrete model (and match) proposed in [7] is minc = h 2 n∑ k=0 8 z21k + 0.284 z 2 2k + u 2 k subject to zk+1 = [ 0.960 −0.093 0.743 1.006 ] zk + [ −0.047 1.009 ] uk, (5) biomath 5 (2016), 1604232, http://dx.doi.org/10.11145/j.biomath.2016.04.232 page 4 of 11 http://dx.doi.org/10.11145/j.biomath.2016.04.232 josé renato campos et al., biological control of sugarcane caterpillar ... where zk = (z1k, z2k)t and the initial conditions are z1 0 = 3000 and z2 0 = 80.17. here, k denotes the discrete iterations in days for the problem. furthermore, for problem (5), the simulation period equals n = 18 days, and hence, tf = hn = 18 days. campos in [7] used the zero-order hold method (function c2d in matlab 7.4) to discretize the dynamic equation of problem (4). thus, the zeroorder hold method provides an exact match between the continuous dynamic system of problem (4) and the discrete dynamic system of problem (5). for the biological analysis of the optimal control problem, we are assuming that the control decision uk, introduction of predators, occurs only once a day. the discretization of the functional of problem (4) introduces an error because it is approximated using a numerical quadrature. however, the error in the discretization of the functional does not change the behavior of the dynamic equations of problems (4) and (5). furthermore, the weight assigned to the coefficient of control uk in the functional of problem (5) can be modified and adapted according to the costs involved in the operations. next, we illustrate the formulation of interval control problems for two distinct situations. the first involves the problem with the interval initial condition. the second formulation considers an interval coefficient in the dynamic equation. a. uncertainty in the initial condition suppose that the model (5) uses the interval initial condition because we consider there to be inaccurate information in the data. we use an interval initial condition of z1 0 = [2970, 3030], which represents an error of 2%. the second initial condition used is z2 0 = 80.17 and represents a degenerate interval. therefore, the problem with the interval initial condition is described below. it is given by min c = h 2 ⊗ n∑ k=0 8 ⊗z21k ⊕0.284 ⊗z 2 2k ⊕u 2 k subject to { z1k+1 = 0.960⊗z1k 0.093⊗z2k 0.047⊗uk z2k+1 = 0.743⊗z1k⊕1.006⊗z2k ⊕ 1.009⊗uk (6) where z1k, z2k, uk and c are intervals and the initial conditions are z1 0 = [2970, 3030] and z2 0 = 80.17. for the interval problem, the symbols ⊕, ,⊗ and � represent the sum, subtraction, multiplication and division of intervals, respectively, according to single-level constrained interval arithmetic. this model is presented in [7]; however, here it is presented as an interval problem. in particular, the initial condition is also an interval. problem (6) is called the interval optimal control problem. furthermore, we emphasize that the functional is an interval and that its optimality is given by the order relation of single-level constrained interval arithmetic (see [18]). according to leal [18], given two intervals a = [a, ā] and b = [b, b̄], the order relation between them is given by a ≤sl b iff a(λ) ≤ b(λ) for all λ ∈ [0, 1], where ≤sl denotes the inequality between intervals according to single-level constrained interval arithmetic and a(λ) and b(λ) are the convex constraint functions associated with a and b, respectively. note that a(λ) = (1 −λ) a + λā, 0 ≤ λ ≤ 1. initially, the interval optimal control problem (6) can be transformed into a real classic problem using single-level constrained interval arithmetic. thus, the interval optimal control problem (6), rewritten as the single-level constrained interval arithmetic [9], is given by minc = h 2 n∑ k=0 8z21k(λ) + 0.284z 2 2k(λ) + u 2 k (λ) subject to zk+1(λ)= [ 0.960 −0.093 0.743 1.006 ] zk(λ)+ [ −0.047 1.009 ] uk(λ), (7) biomath 5 (2016), 1604232, http://dx.doi.org/10.11145/j.biomath.2016.04.232 page 5 of 11 http://dx.doi.org/10.11145/j.biomath.2016.04.232 josé renato campos et al., biological control of sugarcane caterpillar ... where zk(λ) = (z1k(λ), z2k(λ))t and the initial conditions arez1 0(λ) = 2970+60 λ and z2 0(λ) = 80.17, 0 ≤ λ ≤ 1. here, zk(λ) and uk(λ) are the convex constraint functions associated with intervals zk and uk, respectively. furthermore, we also suppose zk(λ) and uk(λ) to have the appropriate dimensions. now, problem (7) is a classic optimal control problem for all fixed λ ∈ [0, 1]. therefore, we use dynamic programming as our solution technique for the discrete time optimal control problem. the advantage of dynamic programming is that it determines the optimal solution of a multistage problem by breaking it into stages, where each stage is a subproblem. solving a subproblem is a simpler task in terms of calculation than dealing with all the stages simultaneously. moreover, a dynamic programming model is a recursive equation that links the different stages of the problem, ensuring that the optimal solution at each stage is also optimal for the entire problem (see [36]). details on dynamic programming can be found in [3]. finally, we solve problem (7) for all fixed λ ∈ [0, 1], and we present the solution in the interval space in accordance with the ideas proposed in [9] and [23], i.e., we return the solution to the interval space using the minimum and maximum of the values obtained for each stage of the problem, provided that the minimum and maximum exist. b. uncertainty in the dynamic equation for the interval problem with uncertainty in the dynamic equation, we consider again the biomathematical model (5) described previously. suppose that, due to some biological factors, the first parameter of the first dynamic equation is an interval. specifically, consider that due to some inaccuracy in obtaining the data for the model, the interval optimal control problem represents the first parameter of the dynamic equation as an interval, that is, the value 0.960 is substituted by the interval [0.760, 1.160]. this interval represents 41.67% of the error in relation to the deterministic value. therefore, the interval optimal control problem is min c = h 2 ⊗ n∑ k=0 8 ⊗z21k ⊕ 0.284 ⊗z 2 2k ⊕u 2 k subject to { z1k+1 =[0.760, 1.160]⊗z1k 0.093⊗z2k 0.047⊗uk z2k+1 =0.743⊗z1 k⊕1.006⊗z2 k⊕1.009⊗uk (8) where z1k, z2k, uk and c are intervals and the initial conditions are z1 0 = 3000 and z2 0 = 80.17 (degenerate intervals) due to translation of the equilibrium point. similar to subsection ii-a, we rewrite the interval problem according to single-level constrained interval arithmetic [9]. we then solve the corresponding problem using dynamic programming [3]. according to the methodology proposed in [23] and [9], we find the solution interval. the numerical solution to the problems (6) and (8) will be presented in the next section. iii. numerical analysis and simulations the implementation and adaptation of the dynamic programming algorithm to solve problems (6) and (8) were performed using matlab 7.4. furthermore, problems (6) and (8) were solved using a microcomputer with a dual-core amd e 300 processor and 3 gb of memory. for the interval problems, we chose n = 18 days. the computational time to solve problem (6) was approximately 4.5 minutes, and the computational time required to solve problem (8) was approximately 26 minutes. the interval cost found in the solution is called the optimal interval cost. the interval state obtained is called the optimal interval state, and the interval control obtained for each iteration in the interval optimal control problem is called the optimal interval control. the following figures represent the numerical results of problems (6) and (8). the deterministic and discrete solutions are also introduced in the figures. in the solutions presented, the translation of the solution has been reversed. in addition, the points representing the deterministic and interval biomath 5 (2016), 1604232, http://dx.doi.org/10.11145/j.biomath.2016.04.232 page 6 of 11 http://dx.doi.org/10.11145/j.biomath.2016.04.232 josé renato campos et al., biological control of sugarcane caterpillar ... fig. 1. preys for problem (6). solutions to the problem are connected by line segments for facilitating the visualization of the temporal evolution. the solutions given by the minimum and maximum values correspond to the optimal interval solutions. graphical solutions are provided for the situations described in problem (6). figure 1 illustrates the number of preys for the problem with uncertainty in the initial condition. figure 2 illustrates the number of predators for the same problem. the predators are introduced in figure 3, and the negative values that appear in the figure correspond to the number of predators that should be removed using some sustainable agricultural practice. the optimal cost of the deterministic problem is 1.3716× 108. the optimal interval cost of problem (6) is [1.3443 × 108, 1.3992 × 108]. thus, the interval uncertainty inserted in the initial condition of the problem results in a variation in the cost of approximately 4.00% compared with the deterministic solution. the graphical solution to problem (8) is presented below. figure 4 illustrates the number of preys for this problem. figure 5 illustrates the number of predators for (8). the values of the control variable are presented in figure 6. the optimal interval cost of problem (8) is fig. 2. predators for problem (6). fig. 3. introduction of predators for problem (6). [7.8523 × 107, 2.7181 × 108]. the uncertainty introduced into the dynamic equation generated a variation of approximately 140.92% in the functional in relation to the deterministic solution. remark 3.1: the solutions of the interval problems (6) and (8) converge to the desired equilibrium point. the interval solutions converge to the desired equilibrium point if the distance between them tends to zero according to the definition of the distance between intervals given by [9]. thus, the approximate interval x to x∗ means that the biomath 5 (2016), 1604232, http://dx.doi.org/10.11145/j.biomath.2016.04.232 page 7 of 11 http://dx.doi.org/10.11145/j.biomath.2016.04.232 josé renato campos et al., biological control of sugarcane caterpillar ... fig. 4. preys for problem (8). fig. 5. predators for problem (8). distance between them, given by max 0≤λ≤1 |x(λ)−x∗| where x(λ) is a convex constraint function associated with x, tends to zero. further, analyzing the interval problems (6) and (8) according to the associated convex constraint functions (see, for example, problem (7)), we have that the corresponding optimal control problems are classical optimal control problems for all fixed λ ∈ [0, 1] and satisfy the stability criterion (see [26], [3]) for optimal control problems with quadratic functional and linear constraints. fig. 6. introduction of predators for problem (8). remark 3.2: other interval optimal control problems can be investigated, such as the problem with interval initial conditions and interval parameters in the interval dynamic equation. thus, considering the interval optimal control problem given by min c = h 2 ⊗ n∑ k=0 8 ⊗z21k ⊕ 0.284 ⊗z 2 2k ⊕u 2 k subject to { z1k+1 =[0.760, 1.160]⊗z1k 0.093⊗z2k 0.047⊗uk z2k+1 = 0.743 ⊗z1 k ⊕ 1.006 ⊗z2 k ⊕ 1.009 ⊗uk (9) where z1k, z2k, uk and c are intervals and the interval initial conditions are z1 0 = [2970, 3030] and z2 0 = 80.17, we have that the optimal interval cost is given by [7.6959 × 107, 2.7729 × 108]. furthermore, the solution of the interval problem (9) shows basically the same qualitative behavior as that of the solution of the interval problem (8). iv. discussion of the results in the problems studied, the initial condition or the dynamic equation has intervals because the data are generally inaccurate and may be represented by interval uncertainty. consequently, biomath 5 (2016), 1604232, http://dx.doi.org/10.11145/j.biomath.2016.04.232 page 8 of 11 http://dx.doi.org/10.11145/j.biomath.2016.04.232 josé renato campos et al., biological control of sugarcane caterpillar ... this implies a variation in the functional, state and control at each iteration (cost, state and control represented by intervals). the decision maker should consider whether it is feasible to run the model for the values obtained in these intervals. therefore, to analyze if the number of preys or predators achieves the minimum or maximum values is an important question in the decisionmaking process of a manager because it can lead to financial loss and environmental damage. furthermore, the analysis of interval costs is also very important for the company. we now emphasize some points from the solutions obtained previously. a. analysis of the interval problem (6) in the solution presented for the interval problem (6), we found consistency with the deterministic results as can be seen from figures 1, 2 and 3. the behaviors of the interval state variable and interval control variable are also quite regular and in accordance with the variation of the deterministic solution. the extremes of the intervals of the state interval solutions x and y approached the desired value, as was observed with the interval control u. therefore, the decision maker obtains values close to those found for the deterministic solution; associated with this, we observe only a small variation in the functional. thus, an error caused by lack of information in obtaining the initial condition generated small variations in cost and did not result in drastic changes for the decision maker. b. analysis of the interval problem (8) for the interval problem (8), the behaviors of the interval state variable x and interval control variable u followed the same trajectory as that of the deterministic solution after the thirteenth day. thus, for the state variable x (preys) and with the introduction of predators u, there was no large variation in comparison with the deterministic solution after the thirteenth day. however, in the initial periods, the introduction of predators u presented a large variation, with direct implications for agricultural practice of pest control. we emphasize the large variation of the interval state variable y , which represents the variation of the predators (figure 5). for the third period, we obtained a variation of 5.1270×103 up to 1.4565× 104 corresponding to the y optimal interval state given by the interval [5.1270× 103, 1.4565× 104]. for this variable, we obtained an approximation of the extremes of the interval, which represents the interval solution, to the deterministic solution after the fifteenth day. furthermore, the problem presents a large variation in the optimal interval cost. finally, we can conclude that the facts described above will certainly influence the company’s decision making. c. conclusion in section iii, we perceive that the optimal interval state x was approximately 2000 in problems (6) and (8). the optimal interval state y (predators) also approximated the desired value. the optimal interval control tends to the value of 16 wasps per day for the two situations. these values approximated the results presented in [5]. botelho and macedo in [5] show that the application of control in the population of caterpillars in the state of são paulo brazil utilizing the parasitoid cotesia flavipes stabilized the number of caterpillars to x = 1900 per hectare. the number of wasps per hectare stabilized to y = 1423 with the average rate of introduction of 16.4 wasps per day. thus, considering the deterministic or interval problem, the values that represent the solution to the problem are near the desired values and in accordance with the actual situation practiced in the state of são paulo. for the implementation of biological control in practice, the simulation results show us that we should introduce a daily number of predators (cotesia flavipes) in the tillage, and this number should be contained in the interval solution. we remark that inserting large numbers of predators does not necessarily guarantee a higher cost compared to the costs that are contained in the optimal interval cost and does not necessarily guarantee biomath 5 (2016), 1604232, http://dx.doi.org/10.11145/j.biomath.2016.04.232 page 9 of 11 http://dx.doi.org/10.11145/j.biomath.2016.04.232 josé renato campos et al., biological control of sugarcane caterpillar ... a control of the infestation in a shorter time, although this is a likely outcome for both interval problems studied. we only know that independent of the number of predators inserted in tillage, and because this number of predators is contained in the interval solution, we can control the infestation with a cost contained in the optimal interval cost state and control contained in the optimal interval state and optimal interval control, respectively. furthermore, the daily number of predators inserted in tillage corresponds to the difference, in absolute value, between the number of predators inserted the previous day and the number that will be inserted the day after. acknowledgment the authors wish to express their sincere thanks to the referees for valuable suggestions that improved the manuscript. the authors also thank the capes – coordination for the improvement of higher education personnel. the author edvaldo assunção was partially supported by the cnpq – brazilian national council for scientific and technological development – under grant number 300703/20139. the author geraldo nunes silva was partially supported by the são paulo state research foundation (fapesp – cepid) under grant number 2013/07375-0. references [1] agrianual 2012: anuário da agricultura brasileira, são paulo: fnp, 2012. 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[39] l. yu, j. m. xu and q. l. han, optimal guaranteed cost control of linear uncertain systems with input constraints, proceedings of the fifth world congress on intelligent control and automation, 553–557, 2004. http://dx.doi.org/10.1109/wcica.2004.1340636 biomath 5 (2016), 1604232, http://dx.doi.org/10.11145/j.biomath.2016.04.232 page 11 of 11 http://dx.doi.org/10.1007/s00500-013-1006-x http://dx.doi.org/10.1109/scan.2006.27 http://dx.doi.org/10.1080/00207177408932688 http://dx.doi.org/10.1109/wcica.2004.1340636 http://dx.doi.org/10.11145/j.biomath.2016.04.232 introduction the biomathematical model uncertainty in the initial condition uncertainty in the dynamic equation numerical analysis and simulations discussion of the results analysis of the interval problem (6) analysis of the interval problem (8) conclusion references www.biomathforum.org/biomath/index.php/biomath original article a particular solution for a two-phase model with a sharp interface david a. ekrut, nicholas g. cogan department of biological mathematics florida state university tallahassee, florida, united states ekrut@math.fsu.edu, cogan@math.fsu.edu received: 23 october 2014, accepted: 8 march 2015, published: 28 april 2015 abstract—two-phase models can be used to describe the dynamics of mixed materials and can be applied to many physical and biological phenomena. for example, these types of models have been used to describe the dynamics of cancer, biofilms, cytoplasm, and hydrogels. frequently the physical domain separates into a region of mixed material immersed in a region of pure fluid solvent. previous works have found a perturbation solution to capture the front velocity at the initial time of contact between the polymer network and pure solvent, then approximated the solution to the sharp-interface at other points in time. the primary purpose of this work is to use a symmetry transformation to capture an exact solution to this two-phase problem with a sharp-interface. this solution is useful for a variety of reasons. first, the exact solution replicates the numeric results, but it also captures the dynamics of the volume profile at the boundary between phases for arbitrary time scales. also, the solution accounts for dispersion of the network further away from the boundary. further, our findings suggest that an infinite number of exact solutions of various classes exist for the two-phase system, which may give further insights into the behaviors of the general two-phase model. keywords-multi-phase modeling; two-phase modelling; free boundary problems; gel dynamics; analytic solutions; exact solutions. i. introduction two-phase models are useful for capturing the interactions between fluids and/or viscoelastic material. each phase is averaged over a control volume, where the volume-averaged phases are incompressible. there is no inertial component to the system, and the phases are immiscible. each phase is governed by conservation equations. these models have been successful at describing how emergent structures develop though the interactions of the two phases. there are several known applications. breward et. al. [1] developed a two-phase model to understand the role of viscosity and dragfriction in avascular tumor growth. an asymptotic solution solved explicitly for the volume fraction revealed that in the absence of viscosity and friction, tumor growth was regulated by oxygen tension. numerical simulations showed increases in either the drag coefficient or viscosity parameter reduces the speed tumor growth. this leads credence to the notion that the invasiveness of tumor cells is related to the viscosity of the cells. well-differentiated cells are known to grow more slowly and considered more viscous due to overlapping filopodia. whereas, poorly differentiated citation: david a. ekrutl, nicholas g. cogan, a particular solution for a two-phase model with a sharp interface, biomath 1 (2015), 1503081, http://dx.doi.org/10.11145/j.biomath.2015.03.081 page 1 of 14 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2015.03.081 d. a. ekrutl, a particular solution for a two-phase model ... (less viscous) cells repel one another, contributing to the spread of tumors. an extension of this model with an additional phase [2] contrasts the role of the expansive growth (passive response) and foreign body hypotheses (active response) in tumorigenesis. numerical simulations showed capsule formation could not result from an active response. another model [3] was used to describe avascular tumor as a two-phase system where tumor spheroids exist in two states, one solid and one liquid. time independent solutions reveal tumor size increases at an optimal rate of cell proliferation under nutrient-rich stress-free conditions. simulations also provided a critical region for which a necrotic core forms at the tumor’s center. several forces are required to balance conservation of momentum. for two-phase models, the viscosity of the phases and interstitial friction must be accounted, but for biofilm morphology, in addition to hydrostatic pressure, osmotic pressure is also needed. one such model [4] describes the role of a network comprised of an extra-cellular polymeric substance (eps) in structural development in biofilm. numerical simulations indicate as eps is produced by bacteria, a rise in osmotic pressure contributes to the expansion of the biofilm region. two-phase dynamics have also been used to simulate biofilm growth and cell motility [5], [6]. a mobile cell contains polymer network phase comprised of actin filaments, intermediate filaments, and microtubules. this phase is the exoskeleton to a cytoplasmic phase. the network contracts to propel the cell forward. numerical simulations of these models have shown to contain traveling wave solutions. another biological model describes to formation of channels in biofilm [7]. steady-state analysis suggests that there is an optimal range for the pressure gradient to drive the formation of a channel between two flat plates. when regions occupied by differing materials have free boundaries, numerical methods are useful to track the sharp interface. the location of the interface can be followed explicitly by interface tracking methods [8]. alternatively, interface capturing can be used to implicitly solve the same equations throughout the domain by capturing the appropriate interface conditions [9]. one such interface capturing method given by du et. al. [10] has analyzed the behavior of a free boundary problem of a two-phase viscous fluid mixture with a prevalent viscosity in a single phase. the solution found by du et. al. is perturbation solution of the front velocity at time t = 0 for a vanishing solvent phase. this solution was built to explore how the velocity of the interface moves in a consistent manner to develop numerical methods to handle the free boundary problem. the velocity is then tracked numerically for various initial profiles with the interface capturing method developed by the group. in each instance, the numerical solution is compared to the asymptotic solution and found to be accurate. in part, the purpose of this paper is to explore the accuracy of the perturbation solution given by [10] in comparison to an exact solution, which was found using symmetry analysis, also called lie’s classical method. in each model previously discussed, numerical, perturbation, and semi-analytic methods were used to provide insights into the behaviors of interest. and though these methods have had some successes in assessing two-phase models, few attempts have been made to attain generalized behavior of these systems with exact solutions. lie’s method produces symmetry transformations which can reduce a system of partial differential equations (pdes) in one spacial dimension to a system of odes. these symmetries are generated by introducing infinitesimal transformations, which leave the original system invariant. for classical symmetry analysis, expansion of this infinitesimal transformation, produces a linear system of pdes, called determining equations, whose solutions provide the forms for the symmetry transformations. non-classical methods have also been developed which, in some cases, lead to additional symmetries. the infinitesimal transformations give rise to highly non-linear determining equations and can be difficult or impossible to biomath 1 (2015), 1503081, http://dx.doi.org/10.11145/j.biomath.2015.03.081 page 2 of 14 http://dx.doi.org/10.11145/j.biomath.2015.03.081 d. a. ekrutl, a particular solution for a two-phase model ... solve. for this reason, the analysis in the paper only includes the classical method, as it recovers the solution given by [10] that we are seeking. lie’s classical method for producing symmetries has been successful in generating exact solutions for a system of pdes describing viscous flow through expanding channels [11]. in this work conservation laws and point symmetries provide reductions, some of which lead to exact solutions of the flow in deformable channels. for elliptic, hyperbolic and mixed-type pdes for ricci flow, wang [12] found several solutions, including traveling wave solutions, to hyperbolic geometric flow of riemann surfaces. the work by cimpoiasu et. al [13] used lie symmetries to produce classes of solutions for the 2d nonlinear heat equation. it has also been shown that lie symmetries generate the similarity solution for a class of (2+1) nonlinear wave equation [14]. in this paper, we generate an exact solution for the two-phase model using a point symmetry. in the first section, we outline a derivation of a two-phase system that represents the simplest version of the model and can be adapted for a variety of physical situations. next, we briefly discuss how to develop symmetries and find a time translation, scaling symmetry, and a general galilei time group. in the third section, we use a symmetry transformation to reduce the system of pdes to an invariant system of odes. we make parameter assumptions similar to du et. al. [10] to recover the exact velocity for their asymptotic solution and compare the exact to the perturbation solution. it is shown that the approximated free boundary solution is a close approximation to the general solution for t = 0. in the fourth section, we vary which physical driving forces dominate the two-phase model and generate additional exact solutions to the system. in the final section of this work, we discuss potential uses of exact solutions for the two-phase model and future directions of this work. ii. the two-phase model in this section, we derive the equations to describe a two-phase model as seen in the kinetics of biological gels as described in [5], [10]. gels swell and deswell due to ionic fluctuations and chemical triggers. an example of this occurs in crawling cells. myosin converts chemical energy in the form of atp into mechanical energy by causing actin filament to contract, propelling cells into motion. neutrophils and macrophages, cells integral to the immune system of humans, respond in this manner. chemical gradients are left by cells foreign to the immune system, leaving a chemotactic trail for the immunological cells to follow [15]. like in [10], we assume the viscous terms are prominent forces and inertial terms are negligible. gels are composed of a polymeric network given by φ1 and a fluid solvent φ2. both phases are treated as newtonian fluids that are immiscible. when considering the redistribution of mass within a control volume, the flux of the network is given by ∇· (φ1u1), where the network moves with a velocity u1. a similar argument is made for the solvent to give the following equations to conserve mass. ∂ ∂t (φ1) + ∂ ∂x (uφ1) = 0, (1) ∂ ∂t (φ2) + ∂ ∂x (vφ2) = 0, (2) where the sum of the volumes saturate to a fixed control volume, φ1 + φ2 = 1. several forces act upon the network. the first is the force due to the network stress tensor σ1, which includes the viscous stress tensor and mass production. σ1 = µ̂1(∇u1 + ∇ut1 ) + λ1∇·u1, where µ̂1 is the shear viscosity and λ1 is the bulk viscosity. in 1-d, this becomes σ1 = µ1 d dx u1, (3) where µ1 = 2µ̂1 + λ1. another force that we include is the frictional force created by interstitial interactions between biomath 1 (2015), 1503081, http://dx.doi.org/10.11145/j.biomath.2015.03.081 page 3 of 14 http://dx.doi.org/10.11145/j.biomath.2015.03.081 d. a. ekrutl, a particular solution for a two-phase model ... phases. if both fluids move in unison or if either volume fraction becomes negligible, drag will vanish. with a frictional coefficient given by ξ, this drag force is given by ξφ1φ2(φ1 −φ2). next, we need to account for both the hydrostatic pressure and osmotic pressure caused by swelling. if p is the total hydrostatic pressure, then the total pressure p acting on the network is given by φ1∇p . ionizing chemicals in the solvent can cause the gel to absorb or release the fluid solvent, causing an osmotic pressure gradient ∇ψ(φ1) acting on the network. for this reason, φ1 is considered the active phase. for the form of the osmotic pressure term, we follow cogan et. al. [5] and the references therein, and assume that ψ(φ1) = k2φ21(φ1 −φ0). the constant k2 accounts for the effects of the ionic environment, polymeric structure, and solvent concentration that contribute to swelling and deswelling. the value of φ0 is a reference volume fraction. this structure allows for osmotic pressure to vanish in the event of φ1 = 0 or at some reference fraction φ0 that can be determined experimentally for various physical applications. assuming constant shear and bulk viscosity, the momentum of these moving fluids can be given by balancing the forces described above. µ1 ∂ ∂x ( φ1 ∂ ∂x u ) + φ1 ∂ ∂x p(φ1,φ2) (4) − ∂ ∂x ψ(φ1) − ξφ1φ2(u−v) = 0 similar arguments can be made to derive the forces of momentum within the solvent. the solvent is a newtonian fluid with only viscous stresses acting on it. fluid pressure acts on the solvent, but osmosis does not create pressure on the fluid itself. the fluid is actively absorbed and released by the gel. the final force is the drag or frictional force created by interstitial interactions. combining these gives the momentum for the solvent. µ2 ∂ ∂x ( φ2 ∂ ∂x v ) + φ2 ∂ ∂x p(φ1,φ2) (5) + ξφ1φ2(u−v) = 0, where µ2 is the viscosity of the solvent. summing (4) and (5) gives the following equation. µ1 ∂ ∂x ( φ1 ∂ ∂x u ) + µ2 ∂ ∂x ( φ2 ∂ ∂x v ) + (φ1 + φ2) ∂ ∂x p(φ1,φ2) − ∂ ∂x ψ(φ1) = 0. since φ1 + φ2 = 1, this becomes µ1 ∂ ∂x ( φ1 ∂ ∂x u ) + µ2 ∂ ∂x ( φ2 ∂ ∂x v ) (6) + px − ∂ ∂x ψ(φ1) = 0, where px = ∂∂xp(φ1,φ2). solving for px gives px = ∂ ∂x ψ(φ1) −µ1 ∂ ∂x ( φ1 ∂ ∂x u ) (7) − µ2 ∂ ∂x ( φ2 ∂ ∂x v ) . next, we substitute φ2 = 1−φ1 in the equations of mass (1) and (2), and the momentum equation (4) to find the following system for analysis. ∂ ∂t (φ1) + ∂ ∂x (uφ1) = 0, (8) − ∂ ∂t (φ1) + ∂ ∂x (v(1 −φ1)) = 0, (9) µ1 ∂ ∂x ( φ1 ∂ ∂x u ) − ∂ ∂x ψ(φ1) (10) +φ1px − ξφ1(1 −φ1)(u−v) = 0. together equations (8-10) can be reduced to a system of odes using the following transformation. u = f(t−αx), v = g(t−αx), (11) φ1 = m(t−αx), biomath 1 (2015), 1503081, http://dx.doi.org/10.11145/j.biomath.2015.03.081 page 4 of 14 http://dx.doi.org/10.11145/j.biomath.2015.03.081 d. a. ekrutl, a particular solution for a two-phase model ... where f, g, and m are to be determined and α is an arbitrary constant describing wave speed. traveling wave solutions have been shown to exist for the two phase system [6]. for this reason, if one were to guess an invariant transformation to reduce this system, the general traveling wave solution (11) may seem like an obvious first choice. but, this specific transformation came from a more general transformation found using symmetry analysis. before producing the general transformation, a brief explanation of symmetry analysis is given in the following section. iii. symmetry analysis in this section, we give a brief explanation of the method for generating the invariant transformations that will be used to generate exact solutions in later sections. for systems of pdes in 1-d, symmetry transformations reduce the pdes to a system of odes. derived by sophus lie [16], symmetry analysis is the mathematical method for finding transformations to a system of pdes that leaves the set of equations invariant, or unchanged. more recently, there has been substantial literature regarding symmetry methods. for further details, we refer the reader to books by hydon [17], bluman and kumei [18], and olver [19]. the following coordinate change is called the infinitesimal transformations. these can be thought of as a local perturbation on the original coordinate system. φ̄1 = φ1 + φ1(t,x,u,v)� + o(� 2), t̄ = t + t(t,x,u,v)� + o(�2), x̄ = x + x(t,x,u,v)� + o(�2), ū = u + u(t,x,u,v)� + o(�2), (12) v̄ = v + v (t,x,u,v)� + o(�2), where φ1, t , x, u, and v are called the infinitesimals. in general, one seeks to find invariance of a system of differential equations of the form fi(t,x,u,v,φ1,ut,vt,φ1t,ux,vx,φ1x, ...) = 0, (13) with i = 1, 2, . . . ,n, where u, v, φ1 are functions of t, x. in the specific case of our two-phase model, the system fi is given by the equations (810). under (12), a set of differential equations is produced for the infinitesimals t , x, u, and v . these differential equations are called the determining equations because they determine the form for the infinitesimals. solving these determining equations produced by (12) provides invariant transformations for the differential equations given by (13). the following is called the invariant surface condition, so called because it leaves the solution surface invariant under the change of coordinates. tut + xux = u, (14) tvt + xvx = v, (15) tφ1t + xφ1x = φ1. (16) when the infinitesimals are solved in conjunction with the invariant surface condition given by (14-16), the solutions u, v, and φ1 provide a transformation which reduces the original pde (13) to an ode. in other words, by using lie’s method to find an infinitesimal change of coordinates, a two variable pde can be reduced to an equation of a single variable to become an ordinary differential equation (ode). taking the physical nature of the problem into account, these reductions can lead to exact solutions to the pde. applying the transformation given by (12) on (8-10) yields a large system of linear pdes. the determining equations are solved interactively to give the forms of the infinitesimals. φ1 = 0, t = α, x = δx + γ(t), (17) u = δu + d dt γ(t), v = δv + d dt γ(t). due to the size of the equations, details of the determining equations are omitted. for more biomath 1 (2015), 1503081, http://dx.doi.org/10.11145/j.biomath.2015.03.081 page 5 of 14 http://dx.doi.org/10.11145/j.biomath.2015.03.081 d. a. ekrutl, a particular solution for a two-phase model ... details on an example, see the details given in the appendix (a). in order for the pdes given by the determining equations to be satisfied, two cases arise. either δ = 0 or δ 6= 0. if δ 6= 0, then the friction coefficient ξ given in the momentum equations vanishes. the transformation given by α is a time translation, δ is a scaling symmetry, and γ(t) is a general time dependant galilei group, as used in fluid mechanics [20]. these symmetries can be used to find invariant reductions in the original system. notice that for δ = 0 and γ(t) = 1 in (17) and solving for u, v, and φ1 in (14-16) gives the transformation u = 1 α + f̂(t−αx), v = 1 α + ĝ(t−αx), φ1 = m(t−αx), letting f̂ = f(t−αx)− 1 α and ĝ = g(t−αx)− 1 α gives the transformation (11). it should be noted that for our purposes, we are only interested in pursuing a classical symmetry analysis to recover the solution presented by du et. al. [10]. it is possible that more solutions will arise from other methods as well. non-classical symmetries arise in many cases. in the work performed by arrigo et. al. [21], a nonclassical symmetry is emitted by a class of burgers’ system. the steinbergs symmetry method has provided exact solutions and reductions to the calogerobogoyavlenskii-schiff equation [22]. the gardner method can generate an infinite hierarchy of symmetries, as was shown with the kdv equations, camassa-holm, and sine-gordon equations [23]. non-classical symmetries have also been generated for the fourth-order thin film equation using non-classical methods [24]. further analysis could include any of these methods, as well as a classification of parameters which has the potential to produce more symmetries. the purpose of this work is not an exhaustive search for symmetries, but an introduction to using symmetry methods to recover a more general solution to the two-phase problem described above and partially recovered by du et. al. [10]. iv. recovering the exact solution for a free boundary problem as discussed in [10], since the viscosity of the solvent is of a much higher magnitude than that of the fluid, we assume the solvent viscosity µ2 is zero. since, φ1 + φ2 = 1, we have φ2 = 1 −φ1. now, we replace φ2 in the equations of momentum (4-5) and find µ1 ∂ ∂x ( φ1 ∂ ∂x u ) + φ1px − ∂ ∂x ψ(φ1) (18) −ξφ1(1 −φ1)(u−v) = 0, (1 −φ1)px + ξφ1(1 −φ1)(u−v) = 0, (19) where u, v, and φ1 are all functions of t, x as previously discussed and px is the pressure gradient. next, we solve (19) for px to find px = −ξφ1(u−v). (20) we see the mass equations (1-2) have now become ∂ ∂t (φ1) + ∂ ∂x (uφ1) = 0, − ∂ ∂t (φ1) + ∂ ∂x (v(1 −φ1)) = 0, summing these two equations of mass gives ∂ ∂x (uφ1 + v(1 −φ1)) = 0. imposing the average velocity is zero, we have uφ1 + v(1 −φ1) = 0, which gives v = − φ1 1 −φ1 u. (21) to match the form of equations given by du et. al. [10], we let the osmotic swelling term take the form ψ(φ1) = φ1ψ(φ1). this, together with biomath 1 (2015), 1503081, http://dx.doi.org/10.11145/j.biomath.2015.03.081 page 6 of 14 http://dx.doi.org/10.11145/j.biomath.2015.03.081 d. a. ekrutl, a particular solution for a two-phase model ... multiphase φ1,φ2>0 φ1=0 γ ω1 ω2 fluid fig. 1. this shows the region ω2 of pure solvent (φ1 = 0) separated at the boundary γ from the region ω1 containing the mixture of both phases. (20-21), reduces the equation (18) to the following equation. (µ1φ1(u)x)x−(φ1ψ(φ1))x− ξφ1 1 −φ1 u = 0. (22) as in figure (1), we assume the mixture occupies the interior region (ω1), while pure solvent occupies the external region (ω2). as the gelmixture swells/deswells, the interface between the regions (γ) moves. to specify the motion du et. al. impose standard jump conditions: [µ1φ1(u)x −φ1ψ(φ1)] = 0 [u] = 0. the solution found by du et. al. [10] approximates the front velocity for the free boundary problem at time t = 0. the solution for a piecewise constant profile is given by, φ1 = { φ− if x < 0 φ+ if x > 0 , and the following can be derived u = { ceβ−x if x < 0 ce−β+x if x > 0 , (23) where β± = √ ξ µ1(1 −φ1±) , and c = −φ+ψ(φ+) + φ−ψ(φ−) µ1(φ+β+ + φ−β−) . the solution (23) was derived by assuming φ1+ → 0 at t = 0. in biological gels, regions of gel separate from regions of pure solvent. so, it is reasonable to assume that the network phase vanishes in this region of pure solvent. to make a graph of the solution given by (23), we assign the following initial profile. φ1 = { φ− = 1 6 if x < 0 φ+ = 0 if x > 0 . (24) the parameters used to generate the graphs are taken from [10], but are repeated in (i) for convenience. the graph figure (2) represents the velocity front for a swelling gel in contact with a fluid solvent. this perturbation solution is an approximation for the velocity front at t = 0. however, there exists an exact solution to this system that captures this behavior for all values of φ1 at any point in time. for the infinitesimals given by (17), let δ = 0 and γ(t) = 1. solving the invariant surface condition for u, v, and φ1 will lead to (11) in terms of the variable r = t − αx. as with the case found with solving for (23) , we assume the viscosity of the second phase is negligible in comparison to that of the first phase, letting µ2 = 0. to make the analysis easier, we allow only for swelling in the active phase, making φ0 = 0. applying (11) reduces (8-10) to a single ode. µ1α 4(αf − 1)2f ′2 −µ1α3(αf − 1)3f ′′ (25) +3k2γ 2α3f ′ + ξ(αf − 1)4 = 0, where g = 1 α , (26) m = γ αf − 1 , (27) biomath 1 (2015), 1503081, http://dx.doi.org/10.11145/j.biomath.2015.03.081 page 7 of 14 http://dx.doi.org/10.11145/j.biomath.2015.03.081 d. a. ekrutl, a particular solution for a two-phase model ... fig. 2. this is the perturbation solution given by (23) at time t = 0. this shows the velocity for a region of vanishing network (top) in the region x > 0 for an initial profile (24) (bottom). this would represent expectations of a velocity front for a swelling gel to contact a fluid solvent. it should be noted that we could have just as easily solved (8-10) for f and left m to be determined. we are choosing to leave f general to assess the behavior of the velocity, because we wish to show the exact solution approximated by du et. al. [10] can be recovered. multiple solutions exist for (25). first, we attempt to recover an exponential solution similar in form to (23). if we assume the viscosity of the network phase φ1 has a greater impact on the system than viscosity and interstitial friction, as was assumed by du et al [10]. we can divide by µ1. this gives the following equation from (25). α4(αf − 1)2f ′2 −α3(αf − 1)3f ′′ (28) +3 k2 µ1 γ2α3f ′ + ξ µ1 (αf − 1)4 = 0. for µ1 of a much larger magnitude than k2 and ξ, this becomes the following: α(αf − 1)2f ′2 − (αf − 1)3f ′′ = 0, (29) whose solution is f(r) = eα(κr+λ) α + 1 α . (30) this makes the analytic solution for the original system (1-2) and (4-5) to be φ1 = γ̂ αf − 1 = γe−α(κ(t−αx)+λ), φ2 = 1 −γe−α(κ(t−αx)+λ), u = eα(κ(t−αx)+λ) α + 1 α , v = 1 α , with µ1 = 0, k2 = 0, ξ = 0. the parameters of this solution can be matched to the parameters of the solution given by (23). we can see that if k = − β α2 and λ = 1 α ln(αc − 1), then the solution found above becomes: φ1 = γ c e β α t−βx, φ2 = 1 − γ c e β α t−βx, u = ce− β α t+βx + 1 α , (31) v = 1 α , the parameter α remaining in the velocity of (31) gives flexibility on scaling time and adjusting the orientation of the velocity. notice, as α →∞, this solution is the same as (23). the velocity becomes identical, and the volume fraction becomes constant, as in the perturbation solution provided by (23). so, in essence, we have recovered the time function that was missed by the perturbation biomath 1 (2015), 1503081, http://dx.doi.org/10.11145/j.biomath.2015.03.081 page 8 of 14 http://dx.doi.org/10.11145/j.biomath.2015.03.081 d. a. ekrutl, a particular solution for a two-phase model ... method used to find (23). next, we match the numerical results of (23) for the parameters given by (i). to do this, we set t = 0 and separate the solution for the velocity as follows. u = { ceβx + 1 α , if x ≤ 0 ce−βx + 1 α if x > 0 , (32) in figure (3), we can see the solutions given by (23) and (32) super-imposed on the same graph for parameter values given by (i). it is clear that the perturbation solution is a close approximation for the exact solution for large values of α. as we would expect from inspection of the solution (32), smaller values of α will adjust the exact solution away from the perturbation solution. the largest impact α has on the system is in regards to the time scale and solvent velocity. large values of α require larger time steps for movement in the system, while decreasing the solvent velocity. β µ1 ξ α 1 0.0108 0.018 1000 10 0.0037 0.616 10000 100 0.000338 5.64 100000 table i the parameters given in the row beginning with β = 1 generates the results in (3) top. the next row for β = 10 gives (3) middle with the final row generating (3) bottom with β = 100. there are several benefits of finding the exact solution, instead of using numerical methods. first, numerical results have a difficult time capturing the behavior at the region of contact between the phases, while the analytical solution easily gives interface behaviour without computationally expensive coding, as can be seen in 4. here we can see the region of network at t = 0 moving uniformly away from the initial contact region x = 0. smaller values of β fail to capture the sharp interface. but as β increases to β = 100, we see the interface remains sharp as time increases. this is expected, as these results coincide with the numerical simulations found in [10] by a moving mesh. fig. 3. these are the perturbation solutions given by (23) at time t = 0 graphed with the solution given by 32 with β = 1 and the corresponding values for µ1 and ξ described in (i) given by the top, β = 10 middle, and β = 100 on bottom. the perturbation solutions are a close approximations for the exact solutions near the region of separation. we can see that the shape of each solution is preserved for each set of parameters, though the scale is modified. biomath 1 (2015), 1503081, http://dx.doi.org/10.11145/j.biomath.2015.03.081 page 9 of 14 http://dx.doi.org/10.11145/j.biomath.2015.03.081 d. a. ekrutl, a particular solution for a two-phase model ... fig. 4. for network (φ1) profiles for t = 0 (top) and t = 1000 (bottom). as β increases, the exact solution becomes more accurate at capturing the expected behavior at the sharp interface. as we have seen, the analytic solution recovers the perturbation solution as well as the numerical results given by du et. al. [10]. however, this is just a single solution to the nonlinear equation given by (25). it is possible that the other solutions are extraneous, but more likely, additional solutions describe other physical or biological phenomenon yet to be determined. further exploration will be required to fit these solutions, but we look at the others here. a. other solutions to (25) being a non-linear system, the solution to (25) is not unique. even though the transformation given by (11) will clearly give traveling wave solutions, the structure of the traveling wave for each solution can vary widely as can be seen with the next two examples. if the viscosity of the network is negligible µ1 = 0, the following solution to (25) is given by f(r) = − k 1 3 2 γ 2 3 ξ 1 3 α 1 3 (r + δ) 1 3 + 1 α , (33) where r = t−αx. the structure of this solution is different from (30) in several ways. when plotting at a single moment in time t = 0, it looks like a pulse as seen by the first curve in figure (5). when animated (30) can be seen as a traveling wave solution, given by the black curves which moves in the positive t direction. fig. 5. the solution given by (33) plotted at t = (0, 10). the first curve is at t = 0. as seen by the black curves, the velocity front travels like a wave as time increases. alternatively, if the osmotic pressure has less of an impact than viscosity and friction, then with k2 = 0 as seen in the absence of ionizing agents for gels, we find the following solution f(r) = e α(−κr+λ)+ ξ 2µ1 r2 α + 1 α . (34) like (30) this solution is exponential, but as seen in (6) the quadratic term gives an unbounded traveling wave. the velocity at t = 0 is given by the first curve. as time increases, the front velocity biomath 1 (2015), 1503081, http://dx.doi.org/10.11145/j.biomath.2015.03.081 page 10 of 14 http://dx.doi.org/10.11145/j.biomath.2015.03.081 d. a. ekrutl, a particular solution for a two-phase model ... travels as a wave moving to the right. this does not seem to have any physical analogue since the velocities are unbounded. fig. 6. the solution given by (34) plotted at t = [0, 10]. the first curve is at t = 0. the velocity front shifts to the right as time increases, moving as a traveling wave. v. additional solutions to the two-phase model this section also provides theoretical solutions, which may or may not have physical relevance. we explore them here to account for the multitude of solutions that are emitted by the system (8-10). there are other two-phase models from physics that might have solutions contained here. for example, one such model describes granular flows where air is considered a non-viscous (nondense) phase with the rocks, debris, and other materials considered as a second highly viscous (dense) phase [25], [26]. within these works, numerical simulations describe the flow behaviors air has on granular flow. the results suggest that drag has more than a negligible effect on the flow of granular materials of finite mass. the traveling wave solutions provided by (11) are given by a simple choice for γ(t) in (17). here, we explore different choices for the transformation and follow the reduction of the pdes to odes. then, we derive solutions to the odes by considering various changes in the physical nature of the problem. by adjusting which physical parameters are the dominating driving force in the problem, we can generate different solutions, which may prove useful in exploring the nature of physical and biological phenomenon. first, we let δ = 0 and γ = 1 t in (17). solved with (14-16) will give the following transformation. u = 1 αt + f ( x− ln(t) α ) , v = 1 αt + g ( x− ln(t) α ) , φ1 = m ( x− ln(t) α ) . neglecting solvent viscosity, µ2 = 0, reduces the original system (8-10) to the following ode, µ1f 2(α−f)f ′2 −µ1f3f ′′ (35) −3k2α(2φ0f − 3α)(α−f)f ′ − ξf5 = 0. with m = α f , g = αf α−f . again, if we assume the dominating force is the viscosity and set k2 and ξ2 to zero, this can be solved to give f = κeλ(x− ln(t) α ) = κ α √ tλ eλx. (36) the complete solution to (1-2) and (4-5) becomes u = 1 αt + κ α √ tλ eλx, v = 1 αt + α α− κα√ tλ eλx κ α √ tλ eλx, φ1 = κ α √ tλ eλx, φ2 = 1 − κ α √ tλ eλx. if we assume friction and pressure dominate and let µ1 = 0, the ode yields no real solution without further assumptions on the constants of integration. biomath 1 (2015), 1503081, http://dx.doi.org/10.11145/j.biomath.2015.03.081 page 11 of 14 http://dx.doi.org/10.11145/j.biomath.2015.03.081 d. a. ekrutl, a particular solution for a two-phase model ... this may imply that viscosity is required for nonconstant solutions. another choice for (17) is to let δ remain arbitrary and to consider ξ = 0, a requirement for invariance to be satisfied. again we consider cases where the viscosity of the solvent is negligible, µ2 = 0. choosing γ = 0 we find the following transformation u = f(xe− δ α t)e δ α t, v = g(xe− δ α t)e δ α t, φ1 = m(xe − δ α t), which reduces (8-10) to the following three odes. (αf −rδ)m′ + αmf ′ = 0, −(αg −rδ)m′ + α(1 −m)g′ = 0, (37) (1 −m)(µ1f ′ −k2m(2φ0 − 3m))m′ +µ1m(1 −m)f ′′ = 0, with r = xe− δ α t. if both osmotic pressure and viscosity are negligible such that k2 = 0 and µ1 = 0, then the following solution satisfies (37). f = γe−λr + δ λr − 1 λα , g = (λr − 1)δeλr λα(δeλr − 1) + κ λα(δeλr − 1) , m = δeλr. the complete solution to (1-2) and (4-5), is u = γe−λxe − δ α t + δ λxe− δ α t − 1 λα e δ α t, v = (λxe− δ α t − 1)δeλxe − δ α t λα(δeλxe − δ α t − 1) + κ λα(δeλxe − δ α t − 1) (38) φ1 = δe λxe− δ α t , (39) φ2 = 1 −φ1. it should be noted that if either viscosity is the dominating force with k2 = 0, or if osmotic pressure is the dominating force with µ2 = 0, then m, f, and g are constant. this implies that friction is required for non-constant solutions. this is different from before, where we found viscosity to be the driving force for the model. in summary, we have found that each solution requires a dominating force to generate nonconstant solutions. this gives flexibility in assessing the two-phase model and suggests that exact solutions may exist for many differing physical phenomenon of interest. for example, it is possible that the solution given by (39) can be matched to results consistent to granular flow, since friction as a necessary component for granular flow [25], [26]. vi. discussion in this work, we found an exact solution which accurately replicates the results from a previously found numerical results. it has been shown that for α → ∞, the analytic solution found here is exactly the perturbation solution found by du et. al. [10]. the exact solution has the benefit of time dependence, which is useful for assessing behavior of the two-phase system without the implementation of numerical methods. additionally, we showed that many traveling wave solutions arise from the two-phase problem. due to the time dependent general galilei group, we have an unlimited number of choices to adjust the speed of the wave through time. these solutions also require specific dominating forces to attain. it is possible that such solutions only arise in specific physical circumstances. though some of these solutions may be extraneous, further investigation is warranted to determine their uses. although asymptotic and numerical methods yield useful information concerning the behavior of multi-phase systems, these methods require substantial efforts. exact solutions have the benefit of being computationally inexpensive to simulate, and with lie symmetries, are relatively simple to generate. there are several directions for future analysis that arise from this work. first, exploring the behavior of the additional solutions may give further biomath 1 (2015), 1503081, http://dx.doi.org/10.11145/j.biomath.2015.03.081 page 12 of 14 http://dx.doi.org/10.11145/j.biomath.2015.03.081 d. a. ekrutl, a particular solution for a two-phase model ... insights into the nature of dominating forces in the two phase system. this may give insight into specific physical phenomenon in which these additional solutions may be esoterically relevant. also, additional symmetries may exist, which could be found using non-classical methods. solutions arising from non-classical methods would then need to be assessed to determine relevant matching physical or biological behavior. additionally, biofilms typically include growth terms to account for the production of new network. it is possible that symmetry solutions can capture this behavior as well. appendix deriving the infinitesimals for the two-phase model generates a large system of linear pdes. for this reason, we have provided details of the process by way of an example in this appendix. for further details, see [27]. consider the following nonlinear first order pde ut = u 2 x (40) under the transformation t̄ = t + �t(t,x,u) + o(�2), x̄ = x + �x(t,x,u) + o(�2), ū = u + �u(t,x,u) + o(�2), to order �2 (40) becomes ut + utuu −ut (tt + uttu) −ux (xt + utxu) − 2ux(ux + uxuu −ut (tx + uxtu) − ux (xx + uxxu)) = 0. using the original equation (40) to eliminate ut and grouping coefficients of ux, we have ut + u 2 xuu −u 2 x ( tt + u 2 xtu ) − ux ( xt + u 2 xxu ) − 2ux(ux + uxuu − u2x (tx + uxtu) −ux (xx + uxxu)) = ut − (xx + 2ux) ux + (2xx −tt −uu) u2x = (xu + 2tx) u 3 x + tuu 4 x = 0. invariance requires the coefficients of ux to be zero, providing us with the following system. u(t,x,u)t = 0, x(t,x,u)x + 2u(t,x,u)x = 0, 2x(t,x,u)x −t(t,x,u)t −u(t,x,u)u = 0, xu + 2tx = 0, tu = 0. these are called the determining equations, because they determine the forms of the infinitesimals. these are linear pdes, which are easily solved with standard techniques of integration. so, we have the following form for the infinitesimals. t(t,x,u) = c1 + c2t + c3x + c4t 2 + c5tx + c6x 2, x(t,x,u) = c7 + c8 + c9x + c4tx + 1 2 k5x 2 − (2k3 + 2k5t− 4k6x) u, u(t,x,u) = k10 − 1 2 k8x− 1 4 k4x 2 + (2k9 −k2) u + k5xu− 4k6u2, where ci and ki are arbitrary constants of integration. together with the invariant surface condition given by tut + xux = u we can find a transformation to reduce (40) to an ode. the form for the transformation will vary depending on choices for the constants ci and ki. references [1] c. j. breward, h. m. byrne, c. e. lewis the role of cell-cell interactions in a two-phase model for avascular tumor growth j. math. biol, 45 (2002) 125-152 [2] s. r. lubkin, t jackson multiphase mechanics of capsule formation in tumors j. biomech. eng.124 (2002) 237-243 http://dx.doi.org/10.1115/1.1427925 [3] h. m. byrne, l. preziosi modelling solid tumor growth using the theory of mixtures math. med. biol. 20 (2003) 341-366 http://dx.doi.org/10.1093/imammb/20.4.341 [4] n. g. cogan, j. p. keener the role of the biofilm matrix in structural development math. med. biol. 21 (2005) 147166 [5] n. g. cogan, r. d. guy multiphase flow models of biogels from crawling cells to bacterial biofilms hsfp journal4 (2009) 11-25 x biomath 1 (2015), 1503081, http://dx.doi.org/10.11145/j.biomath.2015.03.081 page 13 of 14 http://dx.doi.org/10.1115/1.1427925 http://dx.doi.org/10.1093/imammb/20.4.341 http://dx.doi.org/10.11145/j.biomath.2015.03.081 d. a. ekrutl, a particular solution for a two-phase model ... 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(e.g. appl math comput. 215 (2010) 3645–3654; math biosci. 233 (2011) 90– 97) we present a detailed analytical and numerical study of the evolution in time of the pressure gradient across one wavelength. an analytical expression for the pressure gradient is obtained in terms of mittag-leffler functions and its behavior is analyzed. for numerical computation the fractional adams method is used. the influence of the different material parameters is discussed, as well as constraints on the parameters under which the model is physically meaningful. keywords-riemann-liouville fractional derivative; mittag-leffler function; viscoelastic flow; fractional oldroyd-b constitutive model; peristalsis. i. introduction recently, fractional calculus has gained considerable popularity mainly due to its numerous applications in diverse fields of science and engineering. fractional calculus allows integration and differentiation of arbitrary order, not necessarily integer. more precisely, it deals with integrodifferential operators, where the integrals are of convolution type with weakly singular power-law kernels. extensive applications of fractional calculus can be found in the constitutive modeling of viscoelasticity, see [3], [4], [10], [18], [20] and the references cited there. the fractional order constitutive models (proposed in the beginning in an implicit way, see for a historical overview [19], [22], [29]) appear to be a valuable tool for describing viscoelastic properties. unlike the classical models which exhibit exponential relaxation, the models of fractional order show power-law behavior which is widely observed in a variety of experiments. they provide a higher level of adequacy preserving linearity and give the possibility for relatively simple description of the complex behavior of non-newtonian viscoelastic fluids. the generalized fractional oldroyd-b constitutive law belongs to the class of linear fractional models of viscoelastic fluids. it is obtained by replacing the first order derivatives in the classical oldroyd-b model by derivatives of fractional order. the corresponding constitutive equation in the one-dimensional case is given by (1 + λα1 d α t ) τ(t) = η ( 1 + λ β 2d β t ) ε̇(t), (1) where τ(t) is the shear stress, ε(t) shear strain, citation: emilia bazhlekova, ivan bazhlekov, peristaltic transport of viscoelastic bio-fluids with fractional derivative models, biomath 5 (2016), 1605151, http://dx.doi.org/10.11145/j.biomath.2016.05.161 page 1 of 12 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2016.05.161 e. bazhlekova et al., peristaltic transport of viscoelastic bio-fluids ... the over-dot denotes the first time derivative, η > 0 is the dynamic viscosity of the fluid, λ1,λ2 ≥ 0 are parameters related to the relaxation and retardation times, respectively, and dαt and d β t are fractional riemann-liouville derivatives in time of orders α and β, where 0 < α ≤ 1, 0 < β ≤ 1. the generalized oldroyd-b model (1) encompasses a large class of fluids: newtonian fluid (λ1 = λ2 = 0), fractional second grade fluid (λ1 = 0, λ2 > 0), fractional maxwell fluid (λ2 = 0, λ1 > 0). in [23] a very good fit with experimental data is achieved for the fractional oldroyd-b model. unidirectional flows of viscoelastic fluids with the fractional oldroyd-b constitutive law are studied in [5], [11], [13], [17], [21], to mention only few of many recent publications. the transportation of many biophysical fluids is controlled by a special mechanism called peristalsis. the mechanism includes involuntary periodic contraction followed by relaxation and expansion of the ducts through which the fluids pass. it is inducted by the propagation of electrochemically generated waves along the vessels containing fluids. examples from physiology where the peristaltic transport is prevalent are the movement of chyme in the small intestine, transport of bile in bile ducts, transport of lymph in the lymphatic vessels, etc. the complex physical nature of peristaltic flows of non-newtonian fluids stimulated significant attention in the applied mathematics and engineering sciences research communities. for recent research on this topic we refer to [1], [2], [8], [9], [15], [28]. fractional derivative models for peristaltic transport of viscoelastic fluids are derived in a series of papers by tripathi et al. (e.g. [24], [25], [26], [27]) as noted in [26] such models are appropriate for describing the chyme movement in the small intestine, by considering the gastric chyme as a viscoelastic fluid. in [26] and [27] peristaltic transport through a cylindrical tube of fractional oldroyd-b fluid is studied, with 0 < α ≤ β ≤ 1. in [26] inclined tube is considered and in [27] wall slip conditions are assumed. in [24], [25] the particular case of a fractional maxwell model (λ2 = 0) is considered. for the numerical computations the adomian decomposition and homotopy analysis methods are used in the aforementioned articles. in the present work, peristaltic flow of viscoelastic fluid through a uniform channel is considered under the assumptions of long wavelength and low reynolds number. the viscoelastic properties of the fluid are governed by the fractional oldroyd-b constitutive equation. we employ the model proposed in [24] for the particular case of fractional maxwell fluid (λ2 = 0) and generalize it in a straightforward way to cover also the case λ2 6= 0. since the considered model is non-stationary in nature and contains parameters which are timerelated such as the orders of the fractional time derivatives α and β, the relaxation and retardation times λ1, λ2, it is natural to study the time evolution of the physical quantities described by the model and the influence of the different parameters on this evolution. our main contribution is a detailed analytical and numerical analysis of the time evolution of the pressure gradient across one wavelength in the peristaltic flow. to the best of our knowledge, this issue has not been discussed before in the general case λ2 6= 0. an explicit expression for the pressure gradient in terms of the mittag-leffler functions is derived and its behavior is studied. for the numerical computations a technique based on the fractional adams method [6], [7] is used. results of several numerical examples are given and the influence of the different material parameters is discussed as well as constraints on the parameters under which the model is physically meaningful. the rest of the paper is organized as follows. in section ii the equation for the pressure gradient is derived. in section iii an analytical representation for the pressure gradient is obtained in terms of mittag-leffler functions and its behavior is analyzed. in section iv the numerical method used for the computations is described. the obtained numerical results are given in section v and the influence of the different material parameters is discussed. section vi contains conclusions. some biomath 5 (2016), 1605151, http://dx.doi.org/10.11145/j.biomath.2016.05.161 page 2 of 12 http://dx.doi.org/10.11145/j.biomath.2016.05.161 e. bazhlekova et al., peristaltic transport of viscoelastic bio-fluids ... basic definitions and results from fractional calculus which are used in this work are summarized in an appendix. ii. mathematical model in this section we derive the equation for the pressure gradient in a peristaltic flow of a viscoelastic fluid with fractional oldroyd-b constitutive model. this equation was originally proposed in [24] for the case λ2 = 0. the necessary generalizations to the case λ2 6= 0 are straightforward. here we give for completeness the main steps in the derivation. for further details we refer to [24] and the related works [17], [21], [25], [26], [27]. the fundamental equations governing the unsteady motion of an incompressible viscoelastic fluid are the continuity equation: ∇·v = 0 and the general cauchy momentum equation: ρ ( ∂v ∂t + (v ·∇)v ) = ∇·σ (2) where v is the velocity vector, ρ fluid density, σ cauchy stress tensor: σ = −pi + τ, (3) where p is the pressure and τ is the shear stress tensor, which for a viscoelastic fluid with the generalized fractional oldroyd-b model satisfies the equation( 1 + λα1 dα dtα ) τ = η ( 1 + λ β 2 dβ dtβ ) a1. (4) here η > 0 is the dynamic viscosity of the fluid, λ1 and λ2 are parameters related to relaxation and retardation times, respectively, satisfying (see [23]) λ1 ≥ λ2 ≥ 0, (5) α and β are fractional parameters, 0 < α ≤ 1, 0 < β ≤ 1, (6) a1 is the first rivlin-ericksen tensor given by a1 = ∇v + (∇v)t (7) and dγ dtγ denotes the upper convected time derivative dγτ dtγ = d γ t τ +(v ·∇)τ − (∇v) ·τ −τ · (∇v) t , where dγt is the riemann-liouville fractional derivative, see (51) in the appendix for the definition. it is assumed that the relevant reynolds number is small enough for inertial effects to be negligible and the wavelength to diameter ratio is large enough for the pressure to be considered uniform over the cross-section of the channel. as in [24] we consider a uniform horizontal two-dimensional channel with h being the transverse displacement of the walls. denote by x the axis along the channel and by y the transverse coordinate. let u be the velocity of the flow in the direction of the channel. first, the above equations are rewritten in dimensionless form (for details see [24]). here, for simplicity we keep the same notations. according to the assumption of low reynolds number, one obtains inserting (3) in the momentum equation (2): ∂p ∂x = ∂τxy ∂y , (8) ∂p ∂y = 0. (9) on the other hand, the constitutive equation (4) gives: (1 + λα1 d α t ) τxy = ( 1 + λ β 2d β t ) ∂u ∂y . (10) applying the operator (1 + λα1 d α t ) to both sides of (8) the following identity is deduced: (1 + λα1 d α t ) ∂p ∂x = (1 + λα1 d α t ) ∂τxy ∂y (11) differentiating with respect to y both sides of equation (10) one gets (1 + λα1 d α t ) ∂τxy ∂y = ( 1 + λ β 2d β t ) ∂2u ∂y2 . (12) biomath 5 (2016), 1605151, http://dx.doi.org/10.11145/j.biomath.2016.05.161 page 3 of 12 http://dx.doi.org/10.11145/j.biomath.2016.05.161 e. bazhlekova et al., peristaltic transport of viscoelastic bio-fluids ... from (11) and (12) one deduces the following equation for the pressure gradient ∂p ∂x (1 + λα1 d α t ) ∂p ∂x = ( 1 + λ β 2d β t ) ∂2u ∂y2 . (13) boundary conditions for the velocity are given by ∂u ∂y ∣∣∣∣ y=0 = 0, u|y=h = 0. (14) in the following integrations we essentially use the fact that the pressure does not depend on the transverse coordinate y, see (9), i.e. p = p(x,t). integrating eq. (13) with respect to y twice and using boundary conditions (14) one obtains successively (1 + λα1 d α t ) ∂p ∂x y = ( 1 + λ β 2d β t ) ∂u ∂y . (15) y2 −h2 2 (1 + λα1 d α t ) ∂p ∂x = ( 1 + λ β 2d β t ) u. (16) denote by q the volumetric flow rate q =∫h 0 u dy. then after one more integration (16) implies − h3 3 (1 + λα1 d α t ) ∂p ∂x = ( 1 + λ β 2d β t ) q. (17) following [24] it is assumed that the wall of the channel undergoes contraction and relaxation given by h = 1 −φ cos2(πx), (18) where φ is the amplitude of the wave. the transformations between the wave and the laboratory frames (an established procedure in peristaltic fluid dynamics, see [14]) are given in dimensionless form by x = x−t, y = y, u = u−1, θ = q−h. (19) let q denotes the averaged volumetric flow rate q = ∫ 1 0 q dt. (20) using the following relation from [14] q = θ + 1 − φ 2 = q−h + 1 − φ 2 , (21) eq. (17) gives (1 + λα1 d α t ) ∂p ∂x = ( 1 + λ β 2d β t ) a, (22) where a = − 3 h3 ( q + h− 1 + φ/2 ) . (23) for further details on this derivation we refer to [24]. let us emphasize that the function a defined by (23) does not depend on time, but it depends on the spatial variable x via the peristaltic wave parameters h,φ and q. therefore we write a = a(x). iii. analytical properties of the pressure gradient function since a(x) is independent of t equation (22) can be rewritten in the form (1 + λα1 d α t ) ∂p ∂x = ( 1 + λ β 2 t−β γ(1 −β) ) a(x). (24) here we have used the identity d β t 1 = t−β γ(1 −β) , 0 < β < 1, (25) obtained by applying the definition (51) of the riemann-liouville fractional derivative and identity (50). eq. (24) implies that the pressure gradient can be expressed as follows ∂p ∂x = a(x)y(t), (26) where the function y(t) is a solution of the equation (1 + λα1 d α t ) y(t) = 1 + λ β 2 t−β γ(1 −β) . (27) therefore, the time evolution of the pressure gradient is determined by the behavior of the function y(t). in the present work our study is limited to the properties of this function. in what follows we assume λ1 6= 0. let us rewrite (27) in the form of the following fractional order equation dαt y(t) = − 1 λα1 y(t) + f(t), (28) biomath 5 (2016), 1605151, http://dx.doi.org/10.11145/j.biomath.2016.05.161 page 4 of 12 http://dx.doi.org/10.11145/j.biomath.2016.05.161 e. bazhlekova et al., peristaltic transport of viscoelastic bio-fluids ... where f(t) = 1 λα1 ( 1 + λ β 2 t−β γ(1 −β) ) . (29) suppose the physically reasonable initial condition ∂p ∂x ∣∣∣∣ t=0 < ∞. (30) therefore, from (26), y(0) < ∞ and thus lim t→0 (j1−αt y)(t) = 0, (31) where j1−αt is a riemann-liouville fractional integral, see (47) in the appendix. according to (63) the solution of equation (28) is given by y(t) = ∫ t 0 τα−1eα,α ( − τα λα1 ) f(t− τ) dτ, (32) where eα,α(·) denotes the mittag-leffler function (see (55) for the definition). from (32), (29) and the definition (47) of the fractional integration operator we deduce the representation y(t) = 1 λα1 ( j1t + λ β 2j 1−β t )( tα−1eα,α ( − tα λα1 )) , which by the use of identities (61) and (59) implies that the function y(t) can be expressed in terms of the mittag-leffler functions as follows y(t) = 1 −eα,1 ( − tα λα1 ) (33) + λ β 2 λα1 tα−βeα,α+1−β ( − tα λα1 ) . inserting (55) into (33) one obtains the following series expansion y(t) = 1 − ∞∑ k=0 (−1)ktαk λαk1 γ(αk + 1) (34) + λ β 2 ∞∑ k=0 (−1)ktαk+α−β λ α(k+1) 1 γ(αk + α−β + 1) . in the particular case α = β (33) reduces to y(t) = 1 − ( 1 − ( λ2 λ1 )α) eα,1 ( − tα λα1 ) , (35) while for λ2 = 0 it gives y(t) = 1 −eα,1 ( − tα λα1 ) . (36) based on the obtained expressions we study the behavior of the function y(t). recall the restrictions on the parameters λ1 ≥ λ2 ≥ 0, 0 < α ≤ 1 and 0 < β ≤ 1. in the simplest case λ2 = 0, based on representation (36) and the properties of mittagleffler function, we easily infer that y(t) is a monotonically increasing function with y(0) = 0 and y(+∞) = 1. moreover, for small times t the function y(t) grows faster when α is smaller, while for large t it grows faster (and approaches the value 1) when α is larger. this behavior can be seen also on fig. 1. qualitatively similar behavior can be deduced from the representation (35) for the case α = β, taking into account that λ1 ≥ λ2 (see also fig. 5 and fig. 6). however, there is one essential difference: in this case y(t) does not vanish at t = 0, more precisely, (35) implies y(0) = ( λ2 λ1 )α , α = β. (37) to find the asymptotic behavior of y(t) for t → 0 we take the first terms in the series representation (34) and obtain for λ2 6= 0: y(t) = λ β 2 λα1 tα−β γ(1 + α−β) + o ( tmin{α,2α−β} ) , (38) and for λ2 = 0: y(t) = 1 λα1 tα γ(1 + α) + o ( t2α ) . (39) therefore, for λ2 = 0 as well as for λ2 6= 0 and α > β the function y(t) vanishes for t → 0. if λ2 6= 0 and α = β then the initial value y(0) is as in (37). however, if λ2 6= 0 and α < β, then the asymptotic expansion (38) implies that the function y(t) has a weak singularity as t → 0 (see also fig. 3). this contradicts initial condition (30) and raises the question whether the model is physically correct in this case. biomath 5 (2016), 1605151, http://dx.doi.org/10.11145/j.biomath.2016.05.161 page 5 of 12 http://dx.doi.org/10.11145/j.biomath.2016.05.161 e. bazhlekova et al., peristaltic transport of viscoelastic bio-fluids ... concerning large time behavior, asymptotic expansion (56) in the appendix implies y(t) = 1 −λα1 t−α γ(1 −α) + λα2 t−β γ(1 −β) + o ( t−min{α+β,2α} ) , t → +∞. (40) this asymptotic expansion is valid in all of the considered cases. therefore, in all cases lim t→+∞ y(t) = 1. (41) this can also be observed on the figures. iv. numerical method let us note first that the explicit representation (34) derived from the series expansions of the mittag-leffler functions is appropriate for numerical computation of the function y(t) only for sufficiently small times. in [24], [25], [26], [27] two semi-numerical techniques are used for the solution of eq. (27): adomian decomposition method (adm) and homotopy analysis method (ham). these two methods give a series of functions, which first terms are used for the numerical computation of y(t). it appears that the obtained approximations of y(t) by these two methods (for the chosen parameters in ham ~ = −1 and p0 = 0) are the same as if we take the first terms of the series in (34). therefore, it can be expected that the numerical techniques proposed in these studies retain the aforementioned disadvantage of using the series expansion (34) for numerical computation: they work only for sufficiently small times. in contrast, the numerical technique used in the present work is appropriate for all times. for numerical computation of the function y(t) we use an algorithm based on its representation as a solution of an integral equation. applying the operator jαt to both sides of equation (28) and using (31), (52), and the semi-group property (49), we obtain that y(t) satisfies the following integral equation y(t) = − 1 λα1 ∫ t 0 (t− τ)α−1 γ(α) y(τ) dτ +h(t), (42) where h(t) = 1 λα1 ( tα γ(α + 1) + λ β 2 tα−β γ(α−β + 1) ) . (43) equation (42) is used here for the numerical computation of the function y(t), applying the socalled fractional adams method, originally proposed and analyzed by diethelm et al. [6], [7]. this is a predictor-corrector method in which as predictor the fractional adams-bashforth method is used and as corrector the fractional adamsmoulton method. for completeness, here we give the numerical scheme. to find a numerical solution of eq. (42) in the time interval t ∈ [0,t] consider a uniform grid {tj = jh,j = 0, 1, ...,n} with some integer n and h = t/n. denote by yj the approximation for y(tj). for the sake of brevity the notation λ = −λ−α1 is used. the predictor ypk+1 is determined by the formula ypk+1 = λ γ(α) k∑ j=0 bj,k+1yj + h(tk), (44) where bj,k+1 = hα α ((k + 1 − j)α − (k − j)α). (45) the corrector formula is given by yk+1 = λ γ(α)   k∑ j=0 aj,k+1yj + ak+1,k+1y p k+1   + h(tk+1), where aj,k+1 = hα α(α + 1) aj,k+1 (46) and aj,k+1 are defined by aj,k+1=   kα+1−(k−α)(k+ 1)α if j = 0, (k − j + 2)α+1 + (k − j)α+1 −2(k − j + 1)α+1 if 1 ≤j ≤k, 1 if j = k + 1. using this numerical algorithm, the function y(t) is computed for several values of the parameters. the performed numerical experiments indicate biomath 5 (2016), 1605151, http://dx.doi.org/10.11145/j.biomath.2016.05.161 page 6 of 12 http://dx.doi.org/10.11145/j.biomath.2016.05.161 e. bazhlekova et al., peristaltic transport of viscoelastic bio-fluids ... that this method is fast and stable. although, due to the calculation of the integral, it is more time consuming for larger t , the numerical experiments with t = 10 and t = 100 indicate that the method works sufficiently well also for large time intervals. v. numerical results and discussion in this section we discuss some results for the function y(t), obtained by the numerical technique described above. recall that the graphs of y(t) represent the time evolution of the pressure gradient. on fig. 1 and fig. 2 plots of the function y(t) are presented for λ2 = 0, which corresponds to the case of fractional maxwell model, considered in [24]. comparing these two figures to [24], fig. 1 and fig. 2, we observe the same behavior (for better comparison we have chosen the same values for the parameter α as in [24]). the time profiles on fig. 1 exhibit increasing pressure gradient with time as for smaller α it increases faster for small t and slower for large t, whereas for larger α the situation is opposite: it increases slower for small t and faster for large t. this is in agreement with the theoretical observations in section iii based on the analytical representation (36). unlike the figures in [24], where only the time interval t ∈ [0, 1] is considered, on fig. 1 we also give plots for t > 1, which reveal that the pressure gradient does not grow infinitely with time and approaches a certain value (a(x)). this confirms the theoretical observations in section iii. on fig. 2, where the influence of the relaxation time λ1 is illustrated, we see that the pressure gradient is smaller for larger values of λ1. therefore this parameter resists the movement of the flow. figures 3–6 correspond to the general case λ2 6= 0. on fig. 3 the behavior of the pressure gradient function for α < β is illustrated. it is seen that the pressure gradient has a singularity at t = 0 (y(0) = +∞). this was also observed in section iii. to the best of our knowledge, this feature of the model has not been discussed before and raises the question of its physical adequacy. in the works 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 t y λ 1 =1, λ 2 =0 α=1/3 α=1/2 α=2/3 α=1 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 t y λ 1 =1, λ 2 =0 α=1/3 α=1/2 α=2/3 α=1 fig. 1. time profile of the pressure gradient for a = 1, λ1 = 1, λ2 = 0, and various values of α. time interval [0,1] (above) and [0,10] (below). [26] and [27], where the case α ≤ β is considered, this issue has not been addressed. on fig. 4 the case α > β is illustrated. comparing fig. 3 and fig. 4 it is seen that the behavior for α > β is qualitatively different from those for α < β. for small times the pressure gradient is monotonically decreasing for α < β (fig. 3) and monotonically increasing for α > β (fig. 4). however, this monotonic behavior is not retained for all t. the influence of the fractional parameters α and β observed on both figures is as follows. the effect of the fractional parameter α for small times is opposite to that for large times. the same holds for the fractional parameter β. in addition, the effects of the parameters α and β are found to be opposite to each other. plots for the case α = β are given on fig. 5 and fig. 6. the influence of the relaxation time biomath 5 (2016), 1605151, http://dx.doi.org/10.11145/j.biomath.2016.05.161 page 7 of 12 http://dx.doi.org/10.11145/j.biomath.2016.05.161 e. bazhlekova et al., peristaltic transport of viscoelastic bio-fluids ... 0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t y λ 2 =0, α=0.5 λ 1 =0.4 λ 1 =1.0 λ 1 =3.0 λ 1 =10. fig. 2. time profile of the pressure gradient for a = 1, λ2 = 0, α = 0.5 and various values of λ1. 0 1 2 3 4 5 6 7 8 9 10 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 t y λ 1 =λ 2 =1, α=0.2 β=0.4 β=0.6 β=0.8 β=1.0 0 1 2 3 4 5 6 7 8 9 10 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 t y λ 1 =λ 2 =1, β=0.8 α=0.1 α=0.3 α=0.5 α=0.7 fig. 3. time profiles of the pressure gradient for α < β, a = 1 and λ1 = λ2 = 1. λ1 and retardation time λ2 observed on fig. 5 is as follows. the pressure gradient increases with the retardation time λ2 whereas it decreases with the relaxation time λ1. on fig. 6 the effect of 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 t y λ 1 =λ 2 =1, α=0.8 β=0.1 β=0.3 β=0.5 β=0.7 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 t y λ 1 =λ 2 =1, β=0.2 α=0.4 α=0.6 α=0.8 α=1.0 fig. 4. time profiles of the pressure gradient for α > β, a = 1 and λ1 = λ2 = 1. the fractional parameter α is examined. in order to capture the peculiarities of the function y(t) a larger time interval is considered t ∈ [0, 100]. the influence of the fractional parameter resemble those observed on fig. 1 in the case of fractional maxwell model. this is in agreement with the similarity in the explicit expressions (35) and (36). vi. conclusions employing the mathematical tools of fractional calculus we study in this work the time evolution of the pressure gradient in a viscoelastic peristaltic flow with fractional oldroyd-b constitutive model. the analysis of the effect of different parameters shows that for α < β there is an unphysical singularity. this means that from the previously considered in [26] and [27] range 0 < α ≤ β ≤ 1 only for α = β the model is physically meaningful. this is also in agreement with the biomath 5 (2016), 1605151, http://dx.doi.org/10.11145/j.biomath.2016.05.161 page 8 of 12 http://dx.doi.org/10.11145/j.biomath.2016.05.161 e. bazhlekova et al., peristaltic transport of viscoelastic bio-fluids ... 0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t y α=β=0.5, λ 2 =0.2 λ 1 =0.4 λ 1 =1.0 λ 1 =3.0 λ 1 =10 0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t y α=β=0.5, λ 1 =20 λ 2 =0.4 λ 2 =1.0 λ 2 =3.0 λ 2 =10 fig. 5. time profiles of the pressure gradient for α = β = 0.5, a = 1 and various values of λ1 and λ2, λ1 > λ2. statement in [30], that the oldroyd-b constitutive law is thermodynamically compatible only if the fractional orders α and β coincide and λ1 ≥ λ2. in both cases of physical interest: λ2 = 0, 0 < α ≤ 1 (fractional maxwell model) and λ1 ≥ λ2 > 0, 0 < α = β ≤ 1 (thermodynamically compatible oldroyd-b model) the pressure gradient across one wavelength is monotonically increasing with time and approaches a certain stationary value. the same qualitative behavior will hold for the pressure rise and friction force. the technique used in this work can be applied to more general fractional derivative viscoelastic models of peristaltic transport, such as models with more complicated geometry (non-uniform, cylindrical, inclined channels), flows with slip effects, flows in porous media, etc. 0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t y λ 1 =10, λ 2 =1.0 α=β=0.2 α=β=0.4 α=β=0.6 α=β=0.8 0 10 20 30 40 50 60 70 80 90 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t y λ 1 =10, λ 2 =1.0 α=β=0.2 α=β=0.4 α=β=0.6 α=β=0.8 fig. 6. time profiles of the pressure gradient for α = β, a = 1, λ1 = 10, λ2 = 1. time interval [0,10] (above) and [0,100] (below). acknowledgments the work is partially supported by grant dfni-i02/9 from the bulgarian national science fund; and the bilateral research project between bulgarian and serbian academies of sciences (2014-2016) ”mathematical modeling via integraltransform methods, partial differential equations, special and generalized functions, numerical analysis”. appendix here we summarize some facts from fractional calculus, for details see [12], [16]. the fractional order riemann-liouville integral jαt is defined by jαt f(t) = ∫ t 0 ωα(t− τ)f(τ) dτ, α > 0, (47) biomath 5 (2016), 1605151, http://dx.doi.org/10.11145/j.biomath.2016.05.161 page 9 of 12 http://dx.doi.org/10.11145/j.biomath.2016.05.161 e. bazhlekova et al., peristaltic transport of viscoelastic bio-fluids ... where ωα(t) = tα−1 γ(α) , α > 0, t > 0. (48) here γ(·) is the gamma function. basic properties of this function are γ(1) = 1, γ(α + 1) = αγ(α). therefore ω1(t) ≡ 1. the operators of fractional integration satisfy the semi-group property: jαt j β t = j α+β t , α,β > 0, (49) or, equivalently, jαt ωβ = ωα+β, α,β > 0. (50) the riemann-liouville fractional derivative dαt of order α ∈ (0, 1] is defined by d1t = d/dt and dαt = d 1 t j 1−α t . (51) the riemann-liouville fractional derivatives and integrals are related via the identities: dαt j α t f = f, α > 0, and jαt d α t f = f − (j 1−α t f)(0)ωα(t), α ∈ (0, 1). (52) application of the laplace transform l{f(t)}(s) = f̂(s) = ∫ ∞ 0 e−stf(t) dt to the operator of fractional integration gives l{jαt f}(s) = s −αf̂(s), α > 0, (53) which implies the following identity for the riemann-liouville fractional derivative of order α ∈ (0, 1): l{dαt f}(s) = s αf̂(s) − (j1−αt f)(0). (54) denote as usual by eα,β(·) the two-parameter mittag-leffler function eα,β(z) = ∞∑ k=0 zk γ(αk + β) . (55) for α ∈ (0, 2),β > 0, the mittag-leffler function has the following asymptotic expansion as t → +∞ eα,β(−t) = − n−1∑ k=1 (−t)−k γ(β −αk) + o(t−n ). (56) an important particular case is e1,1(−t) = exp(−t) and some properties of the function eα,1(−t) for 0 < α < 1 resemble the behavior of the exponential function: eα,1(−t) is monotonically decreasing with eα,1(0) = 1, eα,1(−∞) = 0. however, unlike the fast exponential decay of exp(−t) for large t, the mittag-leffler function admits a slow algebraic decay, which is slower for smaller α. at t = 0 the opposite picture is observed: the mittag-leffler function admits a fast decay ( ddteα,1(−t) →∞ for t → 0, see (60)), and this decay is faster for smaller α. recall the laplace transform pairs l{ωα(t)}(s) = s−α, (57) l { tβ−1eα,β(λt α) } (s) = sα−β sα −λ . (58) the following identity is often useful j γ t ( tβ−1eα,β(λt α) ) = tβ+γ−1eα,β+γ(λt α), (59) where α,β,γ,t > 0. it can be proven by applying laplace transform and using (53) and (58). another useful property is the following d dt eα,1(λt α) = λtα−1eα,α(λt α). (60) it can be deduced again by applying laplace transform and using (58) or directly from the series representation (55) of the mittag-leffler function. integrating (60) and using that eα,1(0) = 1 we obtain for α > 0 and t > 0 j1t ( tα−1eα,α(λt α) ) = − 1 λ (1 −eα,1(λtα)) (61) let f(t) be an integrable on (0,t) function and let α ∈ (0, 1). then the differential equation of fractional order dαt y(t) = λy(t) + f(t), t > 0, (62) biomath 5 (2016), 1605151, http://dx.doi.org/10.11145/j.biomath.2016.05.161 page 10 of 12 http://dx.doi.org/10.11145/j.biomath.2016.05.161 e. bazhlekova et al., peristaltic transport of viscoelastic bio-fluids ... has a unique solution given by y(t) = y0t α−1eα,α(λt α) (63) + ∫ t 0 τα−1eα,α(λτ α)f(t− τ) dτ, where y0 = limt→0 j 1−α t y. this result can be found in [16], p. 137. the easiest way to prove it is by applying laplace transform. references [1] n.s. akbar, m. raza, r. ellahi, peristaltic flow with thermal conductivity of h2o + cu nanofluid and entropy generation, results phys. 5 (2015) 115–124. http://dx.doi.org/10.1016/j.rinp.2015.04.003 [2] n.s. akbar, m. raza, r. ellahi, influence of induced magnetic field and heat flux with the suspension of carbon nanotubes for the peristaltic flow in a permeable channel, j magn magn mater. 38 (2015) 405–415. http://dx.doi.org/10.1016/j.jmmm.2014.12.087 [3] t.m. atanacković, s. pilipović, b. stanković, d. zorica, fractional calculus with applications in mechanics: vibrations and diffusion processes, john wiley & sons, 2014. 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fedotovi@tut.ac.za; shatalovm@tut.ac.za †manufacturing and materials council for scientific and industrial research (csir), pretoria, south africa received: 13 march 2015, accepted: 23 april 2016, published: 9 may 2016 abstract—an inverse numerical method that estimates parameters of dynamic mathematical models given some information about unknown trajectories at some time is applied to examples taken from biology and ecology. the method consists of determining an overdetermined system of algebraic equations using the experimental data. the solution of the overdetermined system is then obtained using, for example the least-square method. to illustrate the effectiveness of the method an analysis of examples and a numerical example for the model that monitors the dynamics of hiv is presented. keywords-inverse problem; least squares methods; parameter estimation; dymamic systems; predator-prey system; i. introduction function approximation on a fixed interval by means of an initial value problem of an ordinary differential equation with unknown coefficients and unknown initial values as presented by m shatalov, i. fedotov and s.v. joubert in [7], is central to this study. in their paper al method to determine both the unknown coefficients and initial values of a dynamic system by minimizing a certain goal function is presented. in earlier collaborations shatalov and fedotov suggested the use of such an approach in identifying dynamic systems’ parameters from experimental data, see [6]. several other parameter estimation methods are presented in literature, for instance the stochastic models; the bayesion approach, the monte carlo technique, the numerical method with combined adomain/alienor approach, the differential evolution (de) and the hybrid taguchi-differential evolution algorithm. the method used is based on integrating both sides of equations of a dynamic system, and applying regression methods to the overdetermined system of linear algebraic equations with possible constraints. the unknown parameters and initial values can then be obtained using the method of least squares. in this paper, the proposed method gives parameter estimates that have a percentage relative error that is mostly less than 0.4% for artificially generated data and parameters that are in the expected range for real data. as an illustration of the method of identificacitation: a.n. pete, p. mathye, i. fedotov, m. shatalov, determination of parameters for cauchy’s problem for systems of odes with application to biological modelling, biomath 5 (2016), 1604231, http://dx.doi.org/10.11145/j.biomath.2016.04.231 page 1 of 7 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2016.04.231 a.n. pete et al., determination of parameters for cauchy’s problem ... tion, we present the analysis for problems from biology and ecology. the following mathematical models are taken as examples: model of a free population (with negligible mortality and with mortality different from zero), population with negligible mortality and unknown initial conditions, interspecies modifications of the lotka-voltera model and a model that monitors the dynamics of hiv. numerical results with artificially generated data and real data is then given for the model that monitors the dynamics of hiv. a. inverse methods the general approach to the inverse system identification of a n –dimensional parameter a ∈ a ⊂ rn, where a can coincide with rn (no constrains between entries of a) and a can be subset of rn (there are constrains) in a system of ordinary differential equations of the following form ẋ = f(t,x,a), (1) where x is vector-function [0,t] 3 t −→ x (t) ∈ rm subject to experimental information concerning the values x (tj) at the point tj ∈ [0,t] is known: t0 · · · tj · · · tn x0 = x(t0) · · · xj = x(tj) · · · xn = x(tn ) (2) the general approach to find a solution of the formulated problem (1) consists of determining an overdetermined system of algebraic equations using the experimental data (2) aa = h, (3) with respect to unknown vector a. the solution of the system (3) is then obtained using any method of solution of the overdetermined system, for example the least-square method which minimize the difference aa − h using euclidean metric. it is known that in this case (see, for example [5]), the solution of (3) can be obtained by solving the following system a>aa = a>h, (4) to obtain a. ii. mathematical models 1) problem 1: free population: consider a single species that grows by sexual reproduction. assume that the individuals move in the population like brownian motion particles (or that the population is colonial), then the frequency of contact between the individuals is proportional to the squared population density [1]. further assuming that mortality is different from zero and is independent of the population size, the scalar function f(t,x,a) is given by f(t,x,a) = a1x 2 x + a2 −a3x, (5) with the unknown vector a = (a1,a2,a3)> ∈ r3, n = 3, m = 1 and a1 > a3. the parameters a1 and a2 represent per capita birth rate (fecundity) and the population at which half of the females are able to reproduce, respectively. the mortality rate of the population is represented by a3. the parameters a1, a2 and a3 are positive and x(t) = x(t) ∈ r is a scalar function. we also assume that x0 is specified. a special case arises if we assume that the population is of negligible mortality. that is, if a3 = 0 the scalar function f(t,x,a) is then given as f(t,x,a) = a1x 2 x + a2 . (6) 2) problem 2: population with negligible mortality and x0 unknown: here we consider the special case of problem 1 (that is, if a3 = 0) but with the value x0 considered as an incorrect value which must be corrected. the statement of such a problem naturally appears since the initial value x0 = x(t0) plays an important role, namely; it defines the cauchy’s problem for equation (1) and, secondly x0 is included widely in computations below. therefore if x0 given from an experiment has low accuracy, it would be desirable to define this value with more accuracy. 3) problem 3:nonlinear predation at small prey population: in this problem we consider an interspecies modification of the lotka-voltera model biomath 5 (2016), 1604231, http://dx.doi.org/10.11145/j.biomath.2016.04.231 page 2 of 7 http://dx.doi.org/10.11145/j.biomath.2016.04.231 a.n. pete et al., determination of parameters for cauchy’s problem ... where there is a nonlinear predation at small prey population. consider the system ẋ = −a1 x2y xp + a2 + a3x, ẏ = a4 x2y xp + a2 −a5y,   (7) where p = 1, 2, a1,a2,a4 > 0, a3,a5 ≥ 0 and n = 5, m = 2. the prey and predator densities are represented by x(t) and y(t), respectively. in this model, the parameters a1 represent the rate of the consumption of prey by the predator population, a2 the prey population density at which the predator’s consumption is half the maximum value [1], a3 the prey’s growth rate, a4 is rate at which the prey contributes to the predator’s growth rate and a5 is the predator’s death rate. 4) problem 4: a model that monitors the dynamics of hiv: consider the following twodimensional model that monitors the dynamics of hiv. the model considers two sub-populations: hiv susceptible (x), the hiv infected population (y). the total population size is given by n = x + y. the model is described by ẋ = −a1 xy n −a2x + a3, ẏ = a1 xy n −a4y,   (8) where a1, a2, a3 and a4 are all positive constants model parameters. the parameter a1 and a2 denotes the average rate of infection by hiv, a2 the natural cessation of sexual activity, a3 the recruitment rate of susceptible and a4 denotes the death rates of the infected population due to hiv. this model is a modified version of the three dimensional one by gumel (see, [3]). the vector a = (a1,a2,a3,a4) t ∈ r4 is unknown. note that in this case n = 4 and m = 2. iii. construction of overdetermined systems consider equation (1) where f is defined by f(t,x,a) = a1x 2 x + a2 , (9) the resultant equation can be rewritten as ẋx + a2ẋ = a1x 2, (10) integration of (10) with respect to t from t0 to tj (j = 1, 2, . . . ,n) gives −a1pj + a2∆xj = −∆hj, (11) where ∆xj = x (tj) −x (t0) , pj = ∫ tj t0 x2dt, ∆hj = [ 1 2 x2(tj) − 1 2 x2(t0) ] .   (12) the integral pj can be calculated by using a quadrature rule, for example trapezoidal rule. thus, system (3) is solved with a =   −p1 ∆x1... ... −pn ∆xn   , a = ( a1 a2 ) and h =   ∆h1... ∆hn   . now, suppose that (1) is defined as in (10) and the initial condition is unknown. let us replace in (11) ∆xj and ∆hj by those given in (12): −a1pj + a2xj − ( a2x0 + 1 2 x20 ) = −hj, (13) where hj = 12x 2 j under the assumption that for pj we use the old values of x0 defined by (2) for j = 1 and for j ≥ 2 the values of pj are evaluated by an open quadrature rule. formally speaking system (13) is nonlinear but setting − ( a2x0 + 1 2 x20 ) = a3, (14) we obtain the linear system of the form (3) with a =   −p1 x1 1... ... ... −pn xn 1   , a =   a1a2 a3   ,h =   −h1... −hn   . biomath 5 (2016), 1604231, http://dx.doi.org/10.11145/j.biomath.2016.04.231 page 3 of 7 http://dx.doi.org/10.11145/j.biomath.2016.04.231 a.n. pete et al., determination of parameters for cauchy’s problem ... the value of x0 can then be obtained from equation (14). equation (1) with f defined by (5) can be written in the form xẋ + a2ẋ = a1x 2 −a3x2 −a2a3x, (15) in contrast to (10) two supplementary terms −a3x2 and −a2a3x arise. integrating (15) we obtain (a1 −a3)pj −a2xj + ( 1 2 x20 + a2x0 ) −a2a3xj = hj, (16) where xj = ∫ tj t0 xdt. introducing the new vector b = (b1,b2,b3,b4) where b1 = a1 −a2, b2 = a2, b3 = 1 2 x20 + a2x0 and b4 = a2a3 the system can be written in the form (3) where a =   p1 −x1 1 −x1... ... ... ... pn −xn 1 −xn   , b =   b1 b2 b3 b4   and h =   h1... hn   in other words, we obtain the following system b1pj − b2xj + b3 − b4xj = hj, (j = 1, . . .n) (17) since the jacobian is ∂b ∂a = a2 + x0 6= 0, we can find the vector b and hence the unknown vector a. the system (7) of an interspecies modification of the lotka–voltera model for nonlinear predation at small prey population can be written as ẋxp + a2ẋ = −a1x2y + a3xp+1 + a2a3x ẏxp + a2ẏ = a4x 2y −a5yxp −a2a5y } (18) where p = 1, 2, a1,a2,a4 > 0 and a3,a5 ≥ 0. integrating (18) we obtain ∆hj = −a1zj −a2∆xj + a3pj + a2a3xj sj = −a2∆yj + a4zj −a5qj −a2a5yj } (19) where,using the same notation as in (12): ∆xj = x (tj) −x (t0) , pj = ∫ tj t0 x2dt, ∆hj = [ 1 2 x2(tj) − 1 2 x2(t0) ] .   (20) and xj = ∫ tj t0 x(t)dt,yj = ∫ tj t0 y(t)dt, (21) zj = ∫ tj t0 y(t)x2(t)dt,qj = ∫ tj t0 xp(t)y(t)dt, sj = ∫ tj t0 xp(t)dt, (22) with j = 1 . . .n. introducing the new vector b = (b1,b2,b3,b4,b5,b6,b7) > where b1 = a1, b2 = a2, b3 = a3, b4 = a4, b5 = a5, b6 = a2a3 and b7 = a2a5, we write the system (19) in the form (3) where a=   −z1 −∆x1 p1 0 0 x1 0 −z2 −∆x2 p2 0 0 x2 0 ... ... ... ... ... ... ... −zn −∆xn pn 0 0 xn 0 0 −∆y1 0 z1 −q1 0 −y1 0 −∆y2 0 z2 −q2 0 −y2 ... ... ... ... ... ... ... 0 −∆yn 0 zn −qn 0 −yn   and h = (∆h1, . . . , ∆hn,s1, . . . ,sn ). in other words, we obtain the following system of 2n equations with seven unknowns: ∆hj = −b1zj − b2∆xj + b3pj + b6xj, sj = −b2∆yj + b4zj − b5qj − b7yj, } (23) subject to −b6 + b2b3 = 0, −b7 + b2b5 = 0. } (24) to evaluate numerically the right hand side of sj can be considered as the riemann–stietjes integral sj = ∫ tj 0 xp(t)dy(t), (see, for example dragomir and fedotov [3]). otherwise this integral can be computed using numerical differentiation. biomath 5 (2016), 1604231, http://dx.doi.org/10.11145/j.biomath.2016.04.231 page 4 of 7 http://dx.doi.org/10.11145/j.biomath.2016.04.231 a.n. pete et al., determination of parameters for cauchy’s problem ... another approach to construct the over determined system is generally appealing since the system obtained is simpler than (19). this method is possible due to the special form of the system (7); namely the system can be written as ẋ = −a1g(x,y,a2) + a3x, ẏ = a4g(x,y,a2) −a5y, } (25) where g(x,y,a2) = x2y xp+a2 . eliminating g(x,y,a2) we get ẏ + c1y + c2ẋ− c3x = 0, (26) where a5 = c1, a4 a1 = c2 and a4a3 a1 = c3. (27) using the integration techniques that we discussed earlier, we determine the vector c = (c1,c2,c3) > and hence c3 c2 = a3, which will be considered as unknown in the next step. multiplying the first equation of (7) by (xp + a2), we obtain xpẋ + a2ẋ = −a1x2y + a3xp+1 + a2a3x. (28) after integration of (28) the overdetermined system can be written as a1zj + a2 (∆xj −a3xj) = −∆hj + a3pj, (29) where j = 1, . . . ,n, and zj, pj, xj, ∆hj, ∆xj were defined in (20) and (22). the unknown parameters a1 and a2 may then be determined from equation (29) and a4 and a5 from (27). a. a model that monitors the dynamics of hiv the system (8) can be written as ẋ = −a1pj −a2xj + a3, ẏ = a1pj −a4yj. } (30) where ∆xj = x(tj) −x(t0), ∆yj = y(tj) −y(t0), pj = tj∫ 0 x(t)y(t) x(t) + y(t) dt, xj = tj∫ 0 x(t)dt and yj = tj∫ 0 y(t)dt. the system (30) can be written in the form (3) with a =   −p1 −x1 1 0 −p1 −x2 1 0 ... ... ... ... p1 0 0 −y1 p1 0 0 −y2 ... ... ... ...   and h = (∆x1, ∆x2, . . . , ∆y1, ∆y2, . . .) > iv. numerical extraction of model parameters in order to illustrate the effectiveness of the method the parameter identification of the system (8) is presented. using parameter values from gumel [4], we generate points of solutions of the system (8) by the adaptive runge-kutta method. a mathematical software mathcad was used for the adaptive runge-kutta method and for the integration by quadratue rules. these solutions are then perturbed by a normal distribution with mean x̄ and standard deviation δ = 0.5 and further taken as “experimental data”. furthermore, system (8) is studied in the context of the gauteng province, south africa. data used in the numerical simulation is obtained from the acturial society of south africa (assa) (see [2]) hiv prevalence estimates. the data is compiled from the population sensus, antenatal survey and registered deaths [2]. the assa 2003 tables gives the population n and the number infected with hiv y. from the relationship n = x + y, (31) we obtain the susceptible population x as x = n −y. (32) the results given in table 2 below, show a comparison of the parameters used (from gumel [4]) , the estimated parameters and the percentage error given by ||α− α̃||/||α||× 100. biomath 5 (2016), 1604231, http://dx.doi.org/10.11145/j.biomath.2016.04.231 page 5 of 7 http://dx.doi.org/10.11145/j.biomath.2016.04.231 a.n. pete et al., determination of parameters for cauchy’s problem ... table 2: problem 4: the parameter estimates and errors. parameter actual value estimated value % error α α̃ a1 0.24 0.2404 0.375 a2 0.03125 0.0313 0.16 a3 2000 2000 0.000 a4 0.531 0.532 0.16 applying the parameter estimation method in the case of the transmission dynamics of hiv in the gauteng province, south africa the following estimates are obtained:  a1 a2 a3 a4   =   0.393 0.049 565143 0.149   . figures 1 and 2 below show the comparison of the estimated solutions with the “experimental data” and the absolute error in the decimal logarithmic scale. the initial value problem was solved with the new coefficients and old initial conditions. 0 10 20 2 .10 4 4 .10 4 x i x1i ti 0 10 20 2 1 0 log xi x1i−( ) ti fig. 1. comparison of the estimated solutions, xi, with the “experimental data” and the absolute error in the decimal logarithmic scale. 0 10 20 100 200 yi y1i ti 0 10 20 4 2 log yi y1i−( ) fig. 2. comparison of the estimated solutions, yi, with the “experimental data” and the absolute error in the decimal logarithmic scale. from the comparison of the parameters in table 2, figures 1 and 2, it can be seen that the estimated parameters are close enough to the actual values. the estimated parameters of dynamic hiv mathematical models for the gauteng province are all nonnegative and also in the expected ranges. v. conclusion and suggestions for further research a method for estimating parameters of dynamic mathematical models given some information about unknown trajectories at some time, by shatalov, fedotov and joubert [7], was applied to problems from biology and ecology. in problem 2 where the initial value was assumed to be unknown with accuracy, the method was applied to find this value. to illustrate the efficiency of the method, a numerical example for the model that monitors the dynamics of hiv was presented. the estimated parameters for the artificially generated data are close enough to the actual parameter values and for the gauteng province the estimated parameters biomath 5 (2016), 1604231, http://dx.doi.org/10.11145/j.biomath.2016.04.231 page 6 of 7 http://dx.doi.org/10.11145/j.biomath.2016.04.231 a.n. pete et al., determination of parameters for cauchy’s problem ... are in the expected ranges. the method for estimating parameters values could be improved by incorporating a suitable penalty term that minimizes the error caused by numerical quadrature and high observational noise levels in the real data. an improvement of the method will be investigated in future studies. references [1] a. d. bazykin, nonlinear dynamics of interacting populations, singapore, world scientific, 1998. [2] r. e. dorrington, l. f. johnson, d. brandshaw & t. daniel, the demographic impact of hiv/aids in south africa: national and provincial indicators, cape town: centre for acturial research, south african medical research council and actuarial society of south africa, 2006. [3] s. s. dragomir & i. fedotov, a gruss type inequality for mapping abounded variation and applications for numerical analysis, nonlinear functions. anal., appl., 6(3) : 425−433, 2001. [4] a. b. gumel, a competitive numerical method for a chemotherapy model of two hiv subtypes, applied mathematics and computations, 131 : 329−337, 2002. [5] c. l. lawson & r. j. hanson, solving least square problems, new jersey, america: prentice–hall, 1974. [6] m. shatalov & i. fedotov, on identification of dynamic systems parameters from experimental data, rgmia, victoria university, 10(1, 2) : 106−116, 2007. [7] m. shatalov,i. fedotov & s. v. joubert, a novel method of interpolation and extrapolation of functions by a linear initial value problem, buffelsfontein time 2008 peerreview conference procedings, 22−26 september, 2008. biomath 5 (2016), 1604231, http://dx.doi.org/10.11145/j.biomath.2016.04.231 page 7 of 7 http://dx.doi.org/10.11145/j.biomath.2016.04.231 introduction inverse methods mathematical models problem 1: free population problem 2: population with negligible mortality and x0 unknown problem 3:nonlinear predation at small prey population problem 4: a model that monitors the dynamics of hiv construction of overdetermined systems a model that monitors the dynamics of hiv numerical extraction of model parameters conclusion and suggestions for further research references www.biomathforum.org/biomath/index.php/biomath original article which matrices show perfect nestedness or the absence of nestedness? an analytical study on the performance of nodf and wnodf n. f. britton∗, m. almeida neto †, gilberto corso ‡ ∗department of mathematical sciences and centre for mathematical biology university of bath, bath ba2 7ay, uk. email: n.f.britton@bath.ac.uk †departmento de ecologia, universidade federal de goiás, 74001-970 goiânia-go, brazil, email: marioeco@gmail.com ‡departamento de biofı́sica e farmacologia, centro de biociências, universidade federal do rio grande do norte, 59072-970 natal-rn, brazil, email: corso@cb.ufrn.br received: 25 april 2015, accepted: 17 december 2015, published: 16 january 2016 abstract—nestedness is a concept employed to describe a particular pattern of organization in species interaction networks and in site-by-species incidence matrices. currently the most widely used nestedness index is the nodf (nestedness metric based on overlap and decreasing fill), initially presented for binary data and later extended to quantitative data, wnodf. in this manuscript we present a rigorous formulation of this index for both cases, nodf and wnodf. in addition, we characterize the matrices corresponding to the two extreme cases, (w)nodf=1 and (w)nodf=0, representing a perfectly nested pattern and the absence of nestedness respectively. after permutations of rows and columns if necessary, the perfectly nested pattern is a full triangular matrix, which must of course be square, with additional inequalities between the elements for wnodf. on the other hand there are many patterns characterized by the total absence of nestedness. indeed, any binary matrix (whether square or rectangular) with uniform row and column sums (or marginals) satisfies this condition: the chessboard and a pattern reflecting an underlying annular ecological gradient, which we shall call gradient-like, are symmetrical or nearly symmetrical examples from this class. keywords-biogeography, interaction networks, nestedness, bipartite networks i. introduction observing nature is one of the most fascinating experiences in life. a honeybee visits a daisy, a rosemary, and other ten different species. another bee of the same family is specialized in just one flower that by its turn is visited by twenty diverse pollinators. once we put together the community of pollinators and flowers an intricate mutualist network arises [5]. in the opposite side of life a caterpillar feed on two asteraceae species which are eaten by another couple of insects, the full set of herbivorous and plants forms a complex antagonist network. an central quest in ecology citation: n. f. britton, m. almeida neto, gilberto corso, which matrices show perfect nestedness or the absence of nestedness? an analytical study on the performance of nodf and wnodf, biomath 4 (2015), 1512171, http://dx.doi.org/10.11145/j.biomath.2015.12.171 page 1 of 9 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2015.12.171 n. f. britton et al., which matrices show perfect nestedness or the absence of nestedness ... of communities today is the search for patterns in networks that can distinguish between mutualist and antagonist webs [13, 21]. one network pattern that is part of this answer is nestedness, the subject of this manuscript. nestedness is a concept used in ecology to study a specific formation pattern in species interaction networks and in site-by-species incidence matrices. in general terms, nestedness is a specific kind of topological organization in adjacency matrices of bipartite networks where any vertex s, with m links, tend to be connected to a subset of the vertices connected to any other vertex with n links, where n > m. the nestedness concept was first introduced by [8] to characterize species distribution pattern in a spatial set of isolated habitats such as islands. in a perfectly nested pattern site-by-site incidence matrix there is a hierarchy of sites such that the set of species inhabiting any site is a subset of the set inhabiting any site further up the hierarchy. when applied to describe the topological organization in ecological interaction networks this new nestedness concept was first used to networks formed by pollinators and flowering plants and by seed dispersers and flesh-fruited plants [4, 12]. in cases a network is perfectly nested if (i) there is a hierarchy of plant species such that the set of animal (pollinator or seed disperser) species interacting with any plant is a subset of the set of animals interacting with any plant further up the hierarchy, and (ii) there is a similar hierarchy of animals. it is clear that in such a network generalist species interact with specialists and generalists, but specialists do not interact with each other. the proper mathematical framework for introducing nestedness is in the context of bipartite networks. from a general perspective we consider a bipartite network formed by two sets s1 and s2. nestedness is characterized by several indices [22, 18] and it is not the objective of this work to compare them. here we focus on the nodf index, which has a clear mathematical definition that allows further analytic developments. the nodf index, an acronym for nestedness metric based on overlap and decreasing fill, is an index that was introduced in [2] and that has been widely used in the literature. an extension of this index to quantitative networks, wnodf , was recently proposed [3], and we include it in our analysis because of the importance of quantitative networks, specially for networks of interacting species [9, 13]. null models are an important methodological tool widely used in ecology to test model fitting, perform statistical tests or test the validity of indices and measures [10]. in order to assess an index a large set of empirical or artificial data is used as a data bank to explore its limitations and fragility. this process has already been used to test a set of nestedness indices [22]. null models are necessary because statistical tests are otherwise always questionable by limitation in the range of tested parameters, interpretation bias of the results, or equivocal choice of random models. these studies emphasis the necessity of analytic results to strength confidence about nestedness indices and their applications. the original definition of the nodf index depends on how the rows and columns are ordered, and a frequently used software for calculating nodf explicitly asks the user if they would like to order the matrix according to row and column sums (or marginals) [11]. in this paper we employ a definition of (w)nodf in which the matrix is previously sorted before the computation of the index. in this paper we give rigorous definitions of nodf and wnodf and prove two mathematical theorems in each case. for the sake of clarity, and for historical reasons, we explore separately qualitative (binary) and quantitative (weighted) networks. the treatment of the qualitative case is more intuitive and helps the reader to follow the analytic developments. in section 2 we start with a formal definition of nodf and wnodf and present two theorems that characterize the extreme cases, nodf = 0 and wnodf = 0 corresponding to absence of nestedness, and nodf = 1 and wnodf = 1 corresponding biomath 4 (2015), 1512171, http://dx.doi.org/10.11145/j.biomath.2015.12.171 page 2 of 9 http://dx.doi.org/10.11145/j.biomath.2015.12.171 n. f. britton et al., which matrices show perfect nestedness or the absence of nestedness ... to the perfectly nested arrangement. in section 3 we summarize the main ideas of the work and put the results in a broader context. ii. analytic treatment we shall consider a bipartite network of set s1, containing m elements, and set s2, containing n elements, with quantitative data for the frequency wij of the interactions between element i of set s1 and element j of set s2. in the simplest case wi,j is equal to 1 or 0, a situation corresponding to the binary network, qualitative network or presence/absence matrix. the adjacency matrix for the network is the m×n matrix a = (aij), where aij is defined by: aij =   1 if wij 6= 0, so that element i of s1 and element j of s2 are linked 0 if wij = 0, so that they are not linked. (1) we define the row and column marginal totals mt ri and mt c l by mt ri = n∑ j=1 aij and mt c l = m∑ k=1 akl, (2) so that mt ri is the number of elements of s2 interacting with element i of s1, and mt cl is the number of elements of s1 interacting with element l of s2. define the row and column decreasing-fill indicators df rij and df c kl by df rij = { 1 if mt ri > mt r j , 0 if mt ri ≤ mt r j , (3) df ckl = { 1 if mt ck > mt c l , 0 if mt ck ≤ mt c l . (4) note that, if i < j, so that row i is above row j, then df rij = 1 if and only if element i of set s1 is linked with more elements of set s2 than element j of s1; similarly, if k < l, so that column k is to the left of column l, then df ckl = 1 if and only if element k of s2 is linked with more elements of set s1 than element l of s2. it is always possible to permute the rows and columns of the matrix so that mt ri ≥ mt r j whenever i < j, and mt c k ≥ mt r l whenever k < l, but the definition does not require this to be done. a. qualitative matrices, the case nodf in order to define nodf we start with the row paired-overlap quantifier porij as the fraction of unit elements in row j that are matched by unit elements in row i, and the column paired-overlap quantifier pockl as the fraction of unit elements in column l that are matched by unit elements in row k, so that porij = ∑n p=1 aipajp∑n p=1 ajp , pockl = ∑n q=1 akqalq∑n q=1 alq . (5) note that porij is the fraction of elements of s2 linked to element j of s1 that are also linked to element i of s1, and similarly for pockl. define the row paired nestedness np rij between rows i and j, and the column paired nestedness np ckl between columns k and l, by np rij = df r ijpo r ij + df r jipo r ji, (6) np ckl = df c klpo c kl + df c lkpo c lk. (7) note that these definitions are valid whatever the signs of mt ri −mt r j and mt c k −mt c l . finally, define the row and column nestedness metrics nodf r and nodf c by nodf r = ∑m i=1 ∑m j=i+1 np r ij 1 2 m(m−1) , (8) nodf c = ∑n k=1 ∑n l=k+1 np c kl 1 2 n(n−1) , (9) and the overall nestedness metric nodf as a weighted average of these, by nodf = m∑ i=1 m∑ j=i+1 np rij + n∑ k=1 n∑ l=k+1 np ckl 1 2 m(m−1) + 1 2 n(n−1) . (10) biomath 4 (2015), 1512171, http://dx.doi.org/10.11145/j.biomath.2015.12.171 page 3 of 9 http://dx.doi.org/10.11145/j.biomath.2015.12.171 n. f. britton et al., which matrices show perfect nestedness or the absence of nestedness ... 1) conditions for nodf = 0: our objective is to characterize all matrices for which nodf = 0. it is clear that nodf = 0 if and only if both nodf r = 0 and nodf c = 0, so let us first consider the conditions for which nodf r = 0. this is true if and only if np rij = 0 for all pairs (i, j) of rows. from equation (6), np rij = 0 if and only if either mt ri = mt r j , so that df rij = df r ji = 0, or ∑n p=1 aipajp = 0, so that porij = po r ji = 0. in other words, either rows i and j have the same number of unit elements, so that elements i and j of s1 interact with the same number of elements of s2, or there is no p in s2 that interacts with both i and j. if our bipartite network is connected, then it is possible to move from any i in s1 to any other j in s1 by following a path composed of edges of the network from s1 to s2 to s1 and so on. hence, in this connected case, nodf r = 0 if and only if all elements of s1 are linked to the same number of elements of s2. similarly, for a connected network, nodf c = 0 if and only if all elements of s2 are linked to the same number of elements of s1, and nodf = 0 if and only if both these conditions hold. if our network is disconnected, then nodf = 0 if and only if all elements of s1 are linked to the same number of elements of s2, and all elements of s2 are linked to the same number of elements of s1 within each connected component, or compartment. this is a necessary and sufficient condition for nodf = 0. there are many networks that satisfy this condition. for example in figure 1 we show a 9×6 network where each of the nine elements of s1 interact with a different pair of elements of s2, so that each element of s2 interacts with three elements of s1. figure 1(c) does not resemble any of the nodf = 0 configurations exhibited in the literature [4, 15], which are all (including the chessboard after row and column permutation) compartmented with full connectivity within the compartments. case 1(d) seems to reflect an underlying cyclic ecological gradient [15], and we call it gradient-like. the requirement that the gradient be cyclic is manifest in the occupied cell at the bottom left of the matrix, and it is occupied to fulfil the rule that there should be two nonzero elements in each row and three in each column. it is interesting that the dimensions (m, n) of the adjacency matrix obey a constraint in the nodf = 0 case. the total number of matrix elements that is distributed along columns and rows should follow the relation: n∑ i=1 mt ci = m∑ j=1 mt rj . (11) as mt ci and mt r j are constants we can rewrite 11 in the form nmt c = mmt r. 2) conditions for nodf = 1: we now wish to characterize all matrices for which nodf = 1, see figure 2. it is clear that nodf = 1 if and only if both nodf r = 1 and nodf c = 1, so let us first consider the conditions under which nodf r = 1. this is true if and only if np rij = 1 for all pairs (i, j) of rows. from equation (6), np rij = 1 implies that mt r i 6= mt r j , so that either df rij = 1 or df r ji = 1. if there are more elements of s2 interacting with element i in s1 than with j in s1, then mt ri > mt r j , so that df rij = 1, df r ji = 0. then we also require that∑n p=1 aipajp = ∑n p=1 ajp, so that po r ij = 1, in other words that aip = 1 whenever ajp = 1. thus all elements of s2 interacting with element j in s1 also interact with element i in s1, or the set of elements of s2 interacting with j in s1 is nested within (or a proper subset of) the set of elements of s2 interacting with i in s1 . similarly, if there are more elements of s2 interacting with j in s1 than with i in s1, then the set of elements of s2 interacting with i in s1 must be nested within the set of elements of s2 interacting with j in s1. similar results hold for nodf c = 1, so that the set of elements of s1 interacting with any k in s2 must be a proper subset or superset of the set of s1 elements interacting with any other l in s2. for nodf = 1, all (s1 and s2) interaction sets must be proper subor supersets, so that by the pigeonhole principle we must have m = n, and it must be possible to permute the rows and columns of the matrix a so that aij = 1 if i ≥ j, aij = 0 otherwise. the matrix with nodf = 1 is the biomath 4 (2015), 1512171, http://dx.doi.org/10.11145/j.biomath.2015.12.171 page 4 of 9 http://dx.doi.org/10.11145/j.biomath.2015.12.171 n. f. britton et al., which matrices show perfect nestedness or the absence of nestedness ... fig. 1: some nodf = 0 patterns. panels (a) and (b) represent the same matrix after permutation of lines and columns; this non-chessboard tiling is a composition of three disconnected networks. panels (c) and (d) show two connected networks that have nodf = 0, since mt ci = 3 and mt r j = 2 for all i and j respectively. case (d) represents a gradient-like structure. biomath 4 (2015), 1512171, http://dx.doi.org/10.11145/j.biomath.2015.12.171 page 5 of 9 http://dx.doi.org/10.11145/j.biomath.2015.12.171 n. f. britton et al., which matrices show perfect nestedness or the absence of nestedness ... full triangular matrix, unique up to permutation of rows and columns. b. quantitative matrix, the case wnodf to construct the wnodf index we define the row-pair dominance quantifier drij as the fraction of non-zero weights in row j that are dominated by (less than) the corresponding weight in row i, and the column-pair dominance quantifier dckl as the fraction of non-zero weights in column l that are dominated by the corresponding weight in column k, so that drij = ∑n p=1 h(wip −wjp)h(wjp) mt rj , (12) dckl = ∑m q=1 h(wqk −wql)h(wql) mt cl , (13) where h is the heaviside step function with h(0) = 0. note that drij is the fraction of elements of s2 interacting with j in s1 that interact more strongly with i in s1, and similarly for dckl. note that, when calculating nodf for qualitative networks, the quantity corresponding to drij is the row-pair overlap quantifier porij which is the fraction of elements of s2 interacting with j in s1 that also interact with i in s1, and similarly for dckl; the requirement that the interaction be stronger is not (and cannot be) applied. this is the essential difference between the index wnodf for quantitative networks and the index nodf for qualitative ones. now define the row-pair dominance nestedness between rows i and j, and the column-pair dominance nestedness between columns k and l, by dnrij = df r ijd r ij + df r jid r ji, (14) dnckl = df c kld c kl + df c lkd c lk. (15) note that these definitions are valid whatever the signs of mt ri −mt r j and mt c k −mt c l . for example, (i) if mt ri > mt r j then df r ij = 1 and df rji = 0, so dn r ij = d r ij, (ii) if mt r i < mt r j then df rij = 0 and df r ji = 1, so dn r ij = d r ji, and (iii) if mt ri = mt r j then df r ij = df r ji = 0, and dnrij = 0. finally, define the row and column weighted nestedness metrics wnodf r and wnodf c by wnodf r = ∑m i=1 ∑m j=1 dn r ij m(m−1) , (16) wnodf c = ∑n k=1 ∑n l=1 dn c kl n(n−1) , (17) and the overall weighted nestedness metric wnodf as a weighted average of these, by wnodf = m∑ i=1 m∑ j=1 dnrij + n∑ k=1 n∑ l=1 dnckl m(m−1) + n(n−1) . 1) conditions for wnodf = 0: the treatment of wnodf = 0 shares some similarities with the previous analysis of nodf = 0. to characterize all matrices for which wnodf = 0 we proceed as follows. it is clear that wnodf = 0 if and only if both wnodf r = 0 and wnodf c = 0, so let us first consider the conditions for which wnodf r = 0. this is true if and only if dnrij = 0 for all pairs (i, j) of rows. from equation (14), dnrij = 0 if and only if either (i) mt ri = mt r j , so that df rij = df r ji = 0, or (ii) mt r i > mt r j and ∑n p=1 h(wip − wjp)h(wjp) = 0, so that drij = df r ji = 0, or (iii) mt r i < mt r j and∑n p=1 h(wjp − wip)h(wip) = 0, so that d r ji = df rij = 0. in case (i), the elements i and j of s1 interact with the same number of s2 elements. in case (ii), i in s1 interacts with more elements of s2 than does j in s1, but any interaction between j and any element p of s2 is at least as strong as the corresponding interaction between i and p. although i in s1 strictly dominates j in s1 in terms of the number of its interactions, j in s1 (not necessarily strictly) dominates i in s1 in terms of the strength of the interactions it does have. case biomath 4 (2015), 1512171, http://dx.doi.org/10.11145/j.biomath.2015.12.171 page 6 of 9 http://dx.doi.org/10.11145/j.biomath.2015.12.171 n. f. britton et al., which matrices show perfect nestedness or the absence of nestedness ... fig. 2: the maximal nestedness pattern exemplified for qualitative (a) and quantitative (b) cases. in the second situation the weight of the link between species is indicated by grey tones. (iii) is analogous, with i and j interchanged. there are many possible ways to obtain wnodf r = 0, and similarly wnodf c = 0 and wnodf = 0. in particular any connected bipartite network in which all elements of s1 interact with the same number of elements of s2, and all elements of s2 interact with the same number of elements of s1, has wnodf = 0, as does any network in which each element of w is either 0 or 1. note that wnodf is not a continuous function of the elements of w ; for example, if w is a 2 × 2 matrix with w11 = 1 + ε, w12 = w21 = 1, w22 = 0, then wnodf(w) = 0 if ε = 0 but wnodf(w) = 1 if ε is positive, however small it is. c. conditions for wnodf = 1 we now wish to characterize all matrices for which wnodf = 1, see figure 2. this demonstration has some points in common with the case nodf = 1. it is clear that wnodf = 1 if and only if both wnodf r = 1 and wnodf c = 1, so let us first consider the conditions under which wnodf r = 1. this is true if and only if dnrij = 1 for all pairs (i, j) of rows. from equation (15), dnrij = 1 implies that mt r i 6= mt r j , so that either df rij = 1 or df r ji = 1. if there are more elements of s2 interacting with i in s1 than with j in s1, then mt ri > mt r j , and df rij = 1, df r ji = 0. then we also require that ∑n p=1 h(wip − wjp)h(wjp) = mt r j , so that drij = 1, in other words that wip ≥ wjp whenever wjp 6= 0. thus all elements of s2 interacting with j in s1 not only interact with i in s1, but interact more strongly with i than with j. the set of elements of s2 interacting with j in s1 not only has to be nested within (or a proper subset of) the set of s2 elements interacting with i in s1, but all the interactions with i in s1 must be stronger than the corresponding interaction with j in s1. similarly, if there are more s2 elements interacting with j in s1 than with i in s1, then the set of s2 elements interacting with i in s1 must be nested within the set of s2 elements interacting with j in s1, and each interaction with j in s1 must be stronger than the corresponding interaction with i in s1. similar results hold for wnodf c = 1, so that the set of elements of s1 interacting with any k in s2 must be a proper subset or superset of the set of s1 elements interacting with any other l in s2, corresponding interactions in subsets must be weaker, and corresponding interactions in supersets stronger. for wnodf = 1, all (s1 and s2) interaction sets must be proper subor supersets, so that by the pigeon-hole principle we must have m = n, and it must be possible to biomath 4 (2015), 1512171, http://dx.doi.org/10.11145/j.biomath.2015.12.171 page 7 of 9 http://dx.doi.org/10.11145/j.biomath.2015.12.171 n. f. britton et al., which matrices show perfect nestedness or the absence of nestedness ... permute the rows and columns of the matrix w so that wij > 0 if i + j ≤ n + 1, wij = 0 otherwise. any matrix with wnodf = 1 has the same adjacency matrix, up to permutation of rows and columns, and also satisfies the row and column strict dominance properties wik > wjk for all i < j whenever wjk > 0, wki > wkj for all i < j whenever wkj > 0. iii. final remarks this work focuses on probably the most commonly used nestedness index: the nestedness metric based on overlap and decreasing fill. initially we introduce a rigorous formulation for nodf and wnodf . we then elucidate the patterns of maximal and minimal nestedness, (w)nodf = 1 and (w)nodf = 0. the maximal nestedness pattern is already known in the literature [15, 2], but an understanding of the minimum nestedness pattern is substantially extended in this work. the literature usually presents the chessboard pattern as the prototype of the zero nestedness arrangement; but this work shows that there is in fact a large class of matrices that fulfil this condition. we cite the completely compartmented networks with equal modules (of which the chessboard is a special case) and gradient-like matrices. but there is another class of non-symmetrical matrices that also have zero nestedness as long as the row and column sums of the adjacency matrix are uniform. the theoretical discussion about nestedness today resembles the debate around diversity and its measurements [14, 16, 17]. in both cases the community of ecologists is aware of the importance of the concept in understanding and quantifying patterns in ecological processes. in both contexts, also, there is a dynamic debate about the true meaning of the concepts, and the most adequate way to transform them into an index [1, 18, 20]. intriguingly, the comparison between diversity and nestedness is not just a curiosity in the story of theoretical ecology, but also a challenging aspect of theory itself, because beta diversity and nestedness show common similarities and dissimilarities [6, 19]. we hope that this rigorous work that highlight the nestedeness of (w)nodf will contribute to the discussion about the general meaning of nestedness by clarifying the extreme cases: zero and maximal nestedness. the basics of the mathematical framework presented here is flexible enough to encourage further developments using alternative pairwise nestedness indices. despite the large number of nestedness indices, there are few analytic results relating the properties of a nestedness index and the characteristics of the corresponding adjacent matrix; an exception is [7]. with the exact results shown in this manuscript we add new elements to the debate about the real meaning of nestedness and the best way to measure it. acknowledgements financial support to gilberto corso from cnpq (conselho nacional de desenvolvimento cientı́fico e tecnológico) is acknowledged. references [1] m. almeida-neto, d. m. b. frensel, and w. ulrich. rethinking the relatioship between nestedness and beta diversity: a comment on baselga(2010). global ecology and biogeography, 21:772–777, 2012. [2] m. almeida-neto, p.r. guimarães, p. r. guimarães jr, r. d. loyola, and w. ulrich. a consistent metric for nestedness analysis in ecological systems: reconciling concept and measurement. oikos, 117:1227, 2008. [3] m. almeida-neto and w. ulrich. a straightforward computational approach for measuring nestedness using quantitative matrices. enviromental modeling & software, 26:1713, 2011. [4] j. bascompte, p. jordano, c. j. melián, and j. m. olesen. the nested assembly of plantanimal mutualistic networks. proc. natl acad. sci usa, 100:9383, 2003. [5] jordi bascompte and pedro jordano. mutualistic networks. princeton university press, 2013. [6] a. baselga. partitioning the turnover and nestedness components of beta diversity. biomath 4 (2015), 1512171, http://dx.doi.org/10.11145/j.biomath.2015.12.171 page 8 of 9 http://dx.doi.org/10.11145/j.biomath.2015.12.171 n. f. britton et al., which matrices show perfect nestedness or the absence of nestedness ... global ecology and biogeography, 19:134– 143, 2010. [7] g. corso, a. l. de araujo, and a. m. de almeida. connectivity and nestedness in bipartite networks from community ecology. journal of physics: conference series, 285:012009, 2011. [8] j. p. darlington. zoogeography: the geographical distribution of animals. wiley, 1957. [9] luis j. gilarranz, juan m. pastor, and javier galeano. the architecture of weighted mutualistic networks. oikos, 121:1154, 2011. [10] n. j. gotelli and g.r. graves. null models in ecology. smithsonian institution press, washington, d.c., 1996. [11] p. r. guimarães jr. and p. guimarães. improving the analyses of nestedness for large sets of matrices. environmental modeling & software, 21:1512, 2007. [12] p. r. guimarães jr., v. rico-gray, s. f. dos reis, and j. n. thompson. asymmetries in specialisation in ant–plant mutualistic networks. proc. r. soc b, 273:2041, 2006. [13] t. c. ings, j. m. montoya, j. bascompte, n. bluthgen, l. brown, c. f. dormann, f. edwards, d. figueroa, u. jacob, j. i. jones, r. b. lauridsen, m. e. ledger, h. m. lewis, j. m. olesen, f.j. frank van veen, p. h. warren, and g. woodward. ecological networks beyond food webs. journal of animal ecology, 78:253, 2009. [14] l. jost. entropy and diversity. oikos, 113:363–375, 2006. [15] t. m. lewinsohn, p. i. prado, p. jordano, j. bascompte, and j. m. olesen. structure in plant-animal interaction assemblages. oikos, 113:174, 2006. [16] a. e. magurran. measuring biological diversity. blackwell publishing company, 2004. [17] o. parkash and a. k. thukral. statistical measures as measures of diversity. international journal of biomathematics, 3:173, 2010. [18] j. podani and d. schmera. a comparative evaluation of pairwise nestedness measures. ecography, 35:1, 2012. [19] d. schmera and j. podani. comments on separating components of beta diversity. community ecology, 12:153–160, 2011. [20] p. p. a. staniczenko, j. c. kopp, and s. allesina. the ghost of nestedness in ecological networks. nature communications, 4:1391, 2013. [21] e. thébault and c. fontaine. stability of ecological communities and the architecture of mutualistic and trophic networks. science, 329:853, 2010. [22] w. ulrich, m. almeida-neto, and n. j. gotelli. a consumer’s guide to nestedness analysis. oikos, 118:3, 2009. biomath 4 (2015), 1512171, http://dx.doi.org/10.11145/j.biomath.2015.12.171 page 9 of 9 http://dx.doi.org/10.11145/j.biomath.2015.12.171 introduction analytic treatment qualitative matrices, the case nodf conditions for nodf = 0 conditions for nodf = 1 quantitative matrix, the case wnodf conditions for wnodf = 0 conditions for wnodf = 1 final remarks www.biomathforum.org/biomath/index.php/biomath review article can mathematics be biology’s next microscope in disease research at the interface? keynote presentation at the 50th anniversary conference of the mammal research institute of university of pretoria roumen anguelov ∗, armanda bastos† ∗ department of mathematics and applied mathematics † mammal research institute, department of zoology and entomology university of pretoria, south africa roumen.anguelov@up.ac.za, adbastos@zoology.up.ac.za abstract—in this paper we discuss how mathematics can be integrated into biological research, as well as the benefits and challenges related to this process. the focus is on research of diseases at the interface between wildlife, humans and livestock with some illustrative examples of the applications of mathematical models to disease research that we have personally been involved with. keywords-mathematical modelling; thresholding; sensitivity analysis; endemicity; extinction i. introduction mathematical models and methods have become important tools for research in the biosciences in general, and for population dynamics and disease epidemiology in particular [4]. in this paper we focus primarily on research at the disease interface, however many of the issues dealt with are of more general relevance. disease research at the interface (henceforth denoted “the interface”) refers to the study of infectious disease transmission at the interface between free-ranging wildlife on the one side and livestock and / or humans on the other. demographic and socio-economic pressures force wildlife, humans and their domestic animals to co-exist in close proximity, thus intensifying disease transmission potential at the interface. the interface could be linear (e.g. along a fence), or patchy, reflecting habitat preference of disease hosts. it could be focal (at a point), or diffuse, when a range of resources are shared over wide area [2]. wherever the interface occurs, disease transmission is potentially bi-directional. indigenous diseases that are typically maintained in wildlife may cross the interface to livestock or humans (eg. foot and mouth disease, african swine fever, ebola), whereas diseases that are exotic to an area are often introduced by human activity and may cross the interface to wildlife (eg. bovine tuberculosis, canine distemper, rinderpest). a specific challenge for research at the interface arises from the two highly divergent approaches to managing humantransformed landscapes versus natural areas within which wildlife are generally confined. whilst strict veterinary and health policies are applied in the citation: roumen anguelov, armanda bastos, can mathematics be biology’s next microscope in disease research at the interface?, biomath 5 (2016), 1612237, http://dx.doi.org/10.11145/j.biomath.2016.12.237 page 1 of 6 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2016.12.237 r anguelov, a bastos, can mathematics be biology’s next microscope in disease research at the interface? former, a policy of least human intervention is applied to the later. this increased complexity as well as specific factors such as limited opportunities for observation, difficulties and restrictions in collecting data, cost of conducting experiments (if at all possible), etc., make the use of mathematical models and methods a relevant and appropriate research approach. mathematical modelling can be considered as the process of embedding existing knowledge into a mathematical construct. the embedded knowledge is of (i) a qualitative nature, e.g. the law of mass action for the interaction of two species, or of (ii) a quantitative nature, e.g. the space density of each species at given times. further, and possibly equally important, one can embed in the model hypotheses in order to test their feasibility and validity. the need for mathematical modelling is motivated by the substantially high complexity of the biological phenomena considered by contemporary biosciences. indeed, due to technological advances, the possibilities of studying an entire system rather than individual elements or aspects as well as collecting measurements of many observable variables at the same time has increased tremendously. further, solutions to problems of conservation, agriculture, health are sought in the context of a larger system and the long term sustainability analysis is founded on a good understanding of the dynamics of this system. mathematics provides a suitable medium, where the existing qualitative and quantitative knowledge of the system can be adequately represented as a model. the theoretical analysis and practical simulation may produce results which are beyond the reach of normal observation, experimentation or intuition, thus, giving substance to the statement that “mathematics is biology’s next microscope” [4]. ii. integrating mathematics into biological research the classical modus operandi of using mathematical/statistical methods in biological research follows the sequence: research question biological observations and/or experiments — collection of data — mathematical/statistical analysis. this typical modus operandi, reflects the limited use of mathematical methods, which are often only restricted to the analysis of the data, e.g. establishing correlations1. such approaches seldom result in a practically relevant and useful model. we would like to suggest that for mathematical modelling to be efficient and useful it needs to be considered an integral part of all stages of the respective research project. for example, the research question can be considered in the setting of a mathematical model which represents the existing knowledge, allowing for identification of gaps, the relative importance of the involved parameters, the dependence of the parameters of interests on the observable variable as well as any pre-existing bifurcation states. the experimental and field work can be informed by these finding so that when data are collected, hypotheses within the model can be accepted or rejected, aspects of the model can be improved, and the values of parameters of interest reliably identified. the mathematical analysis can also reveal the need for further research leading to a more complete and realistic model, which is also useful for providing answers to the original research questions, possibly generating additional questions along the way. a forum for bringing mathematical expertise and/or collaboration at the early stages of a project is essential for the proposed integration of mathematics into biological research and should form a central tenet upon which future research initiatives are planned. iii. output of mathematical models and their analysis in order to avoid being too general, and so that practically meaningful conclusions can be reached, we will mainly consider models representing dynamical systems, typically formulated in terms of ordinary and/or partial differential equations. possible outputs of the analysis of these models are listed below. 1correlation should not be confused with causality as the former does not imply the latter biomath 5 (2016), 1612237, http://dx.doi.org/10.11145/j.biomath.2016.12.237 page 2 of 6 http://dx.doi.org/10.11145/j.biomath.2016.12.237 r anguelov, a bastos, can mathematics be biology’s next microscope in disease research at the interface? • internal consistency of the set of assumption and/or established knowledge represented via the equations of the model. • bifurcation (thresholding) analysis providing the parameter domains of qualitatively different behaviour of the system and characterising this behaviour, e.g. species invading or not, pathogen persisting or becoming extinct. • testing the validity of assumptions (hypotheses) by comparing model properties with existing observations of the system. • identify the values of parameters of interest by using the values of known parameters and data on the values of observable variables (microscope function). • identify knowledge gaps. • project scenarios into the future (timetelescope function). • quantify uncertainty (guaranteed range, confidence intervals, sensitivity). to illustrate these benefits we provide examples from our own work on mathematical modelling of two infectious vector-borne disease agents: bartonella and african swine fever virus (asfv). the barnonella model [3], considers two vertebrate hosts, rattus rattus and rattus norvegicus and two vectors, ticks (ixodidae family) and fleas (genus xenopsylla). the model takes into account the infection dynamics in each of the four species interlinked via host to vector and vector to host forces of infection (figure 1). the model itself is written in terms of ordinary differential equations. it provides interesting insights on the infection dynamics and the relative importance of its parameters. in particular, the high sensitivity of the model on the rate of vertical transmission motivated further research, conducted as part of an msc study which aimed to accurately estimate its value. further, the mathematical analysis proved that differences in the ectoparasite load between the two rattus species cannot alone explain the difference in their bartonella infection rates. the main aim of the development of the asfv model was to investigate a possible natural mechanism for a locally occurring extinction and was fig. 1. bartonella in r. rattus, r. norvegicus , ticks and fleas (s-susceptible, e-exposed, i-infective, r-recovered). motivated by the apparent disappearance of the virus from a previously positive tick population in the mkuze game reserve, south africa [1]. the model focuses on the infection dynamics within a burrow infested with ornithodoros soft ticks. the movements of warthog sounders in or out of the burrow are relatively fast processes and thus modeled via impulses. the presence of naı̈ve warthogs in the burrow is an important seasonal virus amplification event for the tick population. the flow chart represented in figure 2 is implemented in terms of a system of impulsive differential equations (ordinary and partial)[8]. fig. 2. asvf in warthogs (vertebrate host) and ticks (vector) in a burrow, impulsive recruitment = warthog sounder moving in (s-susceptible, e-exposed, i-infective, r-recovered). the model reveals that the prevalence in a burrow is maintained via the interplay between the mentioned virus amplification event and vertical transmission in ticks. disrupting this mechanism biomath 5 (2016), 1612237, http://dx.doi.org/10.11145/j.biomath.2016.12.237 page 3 of 6 http://dx.doi.org/10.11145/j.biomath.2016.12.237 r anguelov, a bastos, can mathematics be biology’s next microscope in disease research at the interface? (burrow not inhabited by neonatal warthogs for sufficiently long time) may lead to virus extinction. iv. challenges to mathematics despite the abstract nature of mathematics, its development is primarily driven by human practicalities. physics was for a long time the main inspiration and driver behind many mathematical theories. in contemporary research, biology is increasingly assuming its place [4] and poses new challenges for mathematics. here we will list two that are particularly relevant to dynamical systems. • analysis of nonlinear dynamical systems. biological systems are highly nonlinear. despite all of the achievements, the theories of analysis and the analysis of differential equations are largely linear theories. examples that can be included: (i) linear diffusion vs nonlinear diffusion (ii) nonlinear death rate and extinction in finite time. • dealing with uncertainty. parameters of models of biological systems are typically only known to within a particular range rather than to have a precise value. mathematical theories dealing with set-valued data need further development. one approach is via interval analysis, [7]. constructing mathematical models also poses challenges for biology: • quantitative analysis of interactions. mathematical models are built on knowledge of the causal relationships between the model variables, e.g. populations, subpopulations, factors, etc., resulting in adequate functional representation of these relationships, e.g. the number of new infections is proportional to the number of existing infectives. such knowledge is seldom available even on topics that have been well researched. if biologists and epidemiologists are serious about using mathematical modelling, future research needs to give attention to the quantitative analysis of functional links between the intended model variables. • relative completeness of knowledge. a mathematical model is a complete construct in the sense that it functions autonomously with prescribed data and parameters. therefore a model can be useful only if based on adequate knowledge of all the main contributing factors to the studied phenomenon. when constructing a model we necessarily make simplifying assumptions. we take into account some factors, e.g. the life cycle of a disease vector and disregard others (e.g. the phase of the moon when a vector bites the host mammal), based on their (possibly perceived) impact. the model needs knowledge on all relevant aspects of all factors considered of importance. v. integrated approach to disease studies at the interface the processes at the interface arise from interactions between transformed habitats (inhabited by humans and domestic/ated animals) on the one side and natural ecosystems (with wildlife) on the other, in turn result from processes inherent to each of these two sides as well as to global abiotic factors such as climate. integrating knowledge on these three components can potentially provide better insight into disease dynamics and opportunities for control. the usefulness of some links have been recognised, e.g. seasonal calving of wildebeest and the risk of malignant catarrhal fever for cattle [9], or earth satellite images used in identifying vegetation as a proxy for predicting the distribution of dung beetles and herbivores [6]. the opportunities for such integrated approaches have increased with the availability of computer technology for storing vast amounts of data as well as algorithms to search and link data from different sources and in different formats the so called big data science. in this setting an integrated approach has the potential to be an important factor driving future mammal research and research on diseases at the interface. in particular, it facilitates mathematical modelling since a mathematical model can embed relevant knowledge from all disciplines and spheres of research. biomath 5 (2016), 1612237, http://dx.doi.org/10.11145/j.biomath.2016.12.237 page 4 of 6 http://dx.doi.org/10.11145/j.biomath.2016.12.237 r anguelov, a bastos, can mathematics be biology’s next microscope in disease research at the interface? vi. mammal research at university of pretoria the university of pretoria’s mammal research institute (mri) celebrated its 50th anniversary by convening a four-day conference from 12th to 16th september 2016 at mopani camp in kruger park. the conference was aimed at setting the stage for the next 20 years of african mammal research activities in the face of ongoing socio-political, socio-economic and environmental changes and drafting a blueprint to guide ongoing and new initiatives that will prepare graduates, scientists, policy makers and ngos to cope with these anticipated challenges. these initiatives were centered around three main themes: (i) people and wildlife; (ii) environmental stressors; (iii) diseases at the interface between wildlife, livestock and humans. the keynote talk presented in this paper relates to the theme “diseases at the interface”. the field of biomathematics has been actively promoted over the last decade and was given a formal status in south africa by the establishment in 2013 of the dst/nrf sarchi chair on mathematical methods and models in biosciences and bioengineering. the department of mathematics and applied mathematics is a partner to the mri and to other institutes and centers of biological research. the integrative approach promoted in this paper has resulted in a number of initiatives such as an informal weekly meeting under the name of “biomath coffee” which facilitates the early involvement of mathematics in biological research. within this forum presentation of student projects still at the stage of formulation of research question are encouraged, with presenters benefitting from the comments and advice of biologists and mathematicians present and, at least in some cases, fruitful collaboration was established [3], [5], [6]. as previously mentioned, an integrated approach to mammal research and specifically to disease studies at the interface may benefit substantially by using big data science. in this regard, the recent establishment of an institute for big data at the university of pretoria is a timely initiative. vii. conclusion this paper considers some essential principles in relating mathematics to biological research, in general, and to epidemiological research on diseases at the interface between wildlife, humans and livestock, in particular. the impact of the interaction between mathematics and biosciences on each of these two disciplines as well as the benefits of this interaction for each discipline are discussed with comment on recent developments and initiatives that will facilitate this further. the exposition is not meant to be a complete text on the topic. it has the much more modest aim of raising issues of importance and giving some pointers to guide integrative, inter-disciplinary research. references [1] l.f. arnot, j.t. du toit,a.d.s. bastos, molecular monitoring of african swine fever virus using surveys targeted at adult ornithodoros ticks: a re-evaluation of mkuze game reserve, south africa, onderstepoort journal of veterinary research 76(2009), 385-392. [2] r.g. bengis, r.a. kock, j. fisher, infectious animal diseases: the wildlife/livestock interface, scientific and technical review of the office international des epizooties 21(1)(2002), 53-65 [3] h. brettschneider, r. anguelov, chimimba, a.d.s. bastos, a mathematical epidemiological model of gramnegative bartonella bacteria: does differential ectoparasite load fully explain the differences in infection prevalence of rattus rattus and rattus norvegicus?, journal of biological dynamics 6(2)(2012), 763-781 [4] j.e. cohen, mathematics is biology’s next microscope, only better; biology is mathematics’ next physics, only better. plos biol 2(12): e439 (2004), doi:10.1371/journal.pbio.0020439 [5] c. dufourd, c. weldon, r. anguelov, y. dumont, parameter identification in population models for insects using trap data , biomath 2 (2013), 1312061, http://dx.doi.org/10.11145/j.biomath.2013.12.061 [6] i. engelbrecht, m. robertson, m. stoltz, j.w. joubert, reconsidering environmental diversity (ed) as a biodiversity surrogacy strategy. biological conservation 197(2016), 171179 [7] s. markov, biomathematics and interval analysis: a prosperous marriage, aip conf. proc. 1301(26) (2010), http://dx.doi.org/10.1063/1.3526621 biomath 5 (2016), 1612237, http://dx.doi.org/10.11145/j.biomath.2016.12.237 page 5 of 6 http://dx.doi.org/10.11145/j.biomath.2013.12.061 http://dx.doi.org/10.1063/1.3526621 http://dx.doi.org/10.11145/j.biomath.2016.12.237 r anguelov, a bastos, can mathematics be biology’s next microscope in disease research at the interface? [8] p. sivakumaran, mathematical epidemiological models with finite time extinction: the case of african swine fever virus, msc thesis, university of pretoria, 2016. [9] l. wambua, p.n. wambua, a.m. ramogo, d. mijele, m.y. otiende, wildebeest-associated malignant catarrhal fever: perspectives for integrated control of a lymphoproliferative disease of cattle in sub-saharan africa, archives of virology 161(2016), 1-10 biomath 5 (2016), 1612237, http://dx.doi.org/10.11145/j.biomath.2016.12.237 page 6 of 6 http://dx.doi.org/10.11145/j.biomath.2016.12.237 introduction integrating mathematics into biological research output of mathematical models and their analysis challenges to mathematics integrated approach to disease studies at the interface mammal research at university of pretoria conclusion references original article biomath 1 (2012), 1209021, 1–5 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum multiscale analysis of composite structures claudia timofte faculty of physics university of bucharest, bucharest, romania email: claudiatimofte@yahoo.com received: 15 july 2012, accepted: 2 september 2012, published: 9 october 2012 abstract—the goal of this paper is to present some homogenization results for diffusion problems in composite structures, formed by two media with different features. our setting is relevant for modeling heat diffusion in composite materials with imperfect interfaces or electrical conduction in biological tissues. the approach we follow is based on the periodic unfolding method, which allows us to deal with general media. keywords-homogenization; the periodic unfolding method; dynamical boundary condition i. introduction and setting of the problem the analysis of diffusion phenomena in highly heterogeneous materials has been a subject of huge interest in the last decades. the purpose of this paper is to analyze the effective behavior of the solution of some nonlinear problems arising in the modeling of diffusion in a periodic structure formed by two media with different properties, separated by an active interface. our setting is relevant for modeling heat conduction in composite materials with imperfect interfaces or electrical conduction in biological tissues. we assume first that both components are connected. using the periodic unfolding method, which allows us to deal with general heterogeneous media, we can describe the evolution in time of the homogenized solution. the model we obtain at the macroscale is a bidomain model, which conceives the composite material, despite of its discrete structure, as the coupling of two continuous superimposed domains. our model is a generalization of the so-called barenblatt model, arising in the context of diffusion in partially fissured media. a similar model appears also in the study of the bioelectrical activity of the heart at a macroscopic level. in this case, at the microscopic scale, we deal with a medium composed of two different conductive phases (the intracellular and extracellular spaces), separated by a dielectric interface (the cellular membranes), which has a capacitive and a nonlinear conductive behavior. the electric potential verifies elliptic equations in the two conductive regions, coupled by a suitable evolutive boundary condition involving the potential jump at the interfaces between the two phases. the evolution in time of the homogenized potential is governed exactly by a bidomain model. we shall also briefly discuss a different geometric situation, in which only one phase is connected, while the other one is disconnected. in this case, we are led at a different macroscopic model. let ω be a bounded domain in rn (n ≥ 3), with a lipschitz boundary ∂ω consisting of a finite number of connected components. we consider the case in which ω is a periodic structure formed by two components, ωε and πε, representing two materials with different features, separated by an interface sε. we assume that both ωε and πε = ω \ ωε are connected, but only ωε reaches the external fixed boundary of the domain ω. here, ε represents a small parameter related to the characteristic size of the our two regions. more precisely, let y1 be a lipschitz open connected subset of the unit cell y = (0, 1)n. let y2 = y \ y1. we suppose that y2 has a locally lipschitz boundary γ and we assume that the intersections of the boundary of y2 with the boundary of y are identically reproduced on opposite faces of the cell, which are denoted, for any 1 ≤ i ≤ n, by σi = {y ∈ ∂y | yi = 1}, σ−i = {y ∈ ∂y | yi = 0}. citation: c. timofte, multiscale analysis of composite structures, biomath 1 (2012), 1209021, http://dx.doi.org/10.11145/j.biomath.2012.09.021 page 1 of 5 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.09.021 c. timofte, multiscale analysis of composite structures we suppose that repeating y by periodicity, the union of all the sets y1 is connected and has a locally c 2 boundary. also, we assume that the origin of the coordinate system is set in a ball contained in this union (see [8]). let zε = {k ∈ zn | εk + εy ⊆ ω}, kε = {k ∈ zε | εk ± εei + εy ⊆ ω, ∀i = 1, n}, where ei are the elements of the canonical basis of rn. we define πε = int( ⋃ k∈kε (εk + εy2)) and ωε = ω \ πε and we set θ = ∣∣y \ y2∣∣. let α1, β1 ∈ r such that 0 < α1 < β1. we denote by m(α1, β1, y ) the set of all the square matrices a ∈ (l∞(y ))n×n such that, for any ξ ∈ rn, we have (a(y)ξ, ξ) ≥ α1 | ξ |2, | a(y)ξ |≤ β1 | ξ |, almost everywhere in y . we consider a family of matrices aε(x) = a( x ε ) defined on ω, where a ∈ m(α1, β1, y ) is a symmetric smooth y -periodic matrix. we shall denote the matrix a by a1 in y1 and by a2, respectively, in y2. if (0, t ) is the time interval, we shall analyze the macroscopic behavior of the solutions of the following system:  −div (aε1∇u ε) + β(uε) = f in ωε × (0, t ), −div (aε2∇v ε) = f in πε × (0, t ), aε1∇u ε · ν = aε2∇v ε · ν on sε × (0, t ), aε1∇u ε · ν + αε ∂ ∂t (uε − vε) = aεg(vε − uε) on sε × (0, t ), uε = 0 on ∂ω × (0, t ), uε(0, x) − vε(0, x) = c0(x) on sε. (1) here, ν is the unit outward normal to ωε, a > 0, f ∈ l2(0, t ; l2(ω)), c0 ∈ h10 (ω), α > 0 and β and g are continuous functions, monotonously non-decreasing and such that β(0) = 0 and g(0) = 0. we shall suppose that there exist a positive constant c and an exponent q, with 0 ≤ q ≤ n/(n − 2), such that |β(v)| ≤ c(1 + |v|q), |g(v)| ≤ c(1 + |v|q). (2) as examples of such functions, we mention the case of langmuir or freundlich kinetics. for the case of electrical conduction in biological tissues, we may consider that f = 0, β = 0 and g is, in r3, a cubic function, like in the fitzhugh-nagumo model (see, for instance, [9]). results concerning the well posedness of problem (1) in suitable function spaces and proper energy estimates were obtained in [1], [3] and [9]. using the periodic unfolding method recently introduced by d. cioranescu, a. damlamian, g. griso, p. donato and r. zaki (see [4] and [5]), we can prove that the asymptotic behavior of the solution of our problem is governed by a new nonlinear system (see (3)). at a macroscopic level, the composite material can be represented by a continuous model, which describes it as the superimposition of two interpenetrating continuous media, coexisting at every point of the domain. our macroscopic model is a degenerate parabolic system, as the time derivatives involve the unknown v − u. if we deal with a different geometry, i.e. we consider that only one phase is connected, while the other one is disconnected, we are led to a different macroscopic model (see remark 2.). similar problems have been considered, using different techniques, in [1] and [9], for studying electrical conduction in biological tissues. the results presented in this paper constitute a generalization of those obtained in [2], [8] and [10]. as already mentioned, our approach is based on the periodic unfolding method, which allows us to deal with general media (see remark 3.). for dealing with such two-component domains, we use unfolding operators, which map functions defined on oscillating domains into functions defined on fixed domains. in such a way, we can avoid the use of extension operators. therefore, using this general method, we can deal with media with less regularity than those usually considered in the literature (composite materials and biological tissues are highly heterogeneous and their interfaces are not very smooth, in general). the plan of the paper is as follows: in the second section, we give the main convergence result of this paper. the last section is devoted to the proof of our result. ii. the main result using the periodic unfolding method, we can pass to the limit in the variational formulation of problem (1) and we obtain the effective behavior of the solution of our microscopic model. theorem 1. the solution (uε, vε) of system (1) converges, as ε → 0, to the unique solution (u, v), with u, v ∈ l2(0, t ; h10 (ω)), ∂ ∂t (u − v) ∈ l2(0, t ; l2(ω)) biomath 1 (2012), 1209021, http://dx.doi.org/10.11145/j.biomath.2012.09.021 page 2 of 5 http://dx.doi.org/10.11145/j.biomath.2012.09.021 c. timofte, multiscale analysis of composite structures and u, v ∈ c0([0, t ]; h10 (ω)), of the following macroscopic problem:  α | γ | ∂ ∂t (u − v) − div (a1∇u) + θβ(u)− a | γ | g(v − u) = θf in ω × (0, t ), α | γ | ∂ ∂t (v − u) − div (a2∇v)+ a | γ | g(v − u) = (1 − θ)f in ω × (0, t ), u(0, x) − v(0, x) = c0(x) on ω. (3) in (3), a 1 and a 2 are the homogenized matrices, defined by: a 1 ij = ∫ y1 ( aij + aik ∂χ1j ∂yk ) dy, a 2 ij = ∫ y2 ( aij + aik ∂χ2j ∂yk ) dy and χ 1k ∈ h1per(y1)/r, χ2k ∈ h 1 per(y2)/r, k = 1, ..., n, are the weak solutions of the cell problems  −∇y · ((a1(y)∇yχ 1k ) = ∇ya1(y)ek, y ∈ y1, (a1(y)∇yχ 1k ) · ν = −a1(y)ek · ν, y ∈ γ,  −∇y · ((a2(y)∇yχ 2k ) = ∇ya2(y)ek, y ∈ y2, (a2(y)∇yχ 2k ) · ν = −a2(y)ek · ν, y ∈ γ. so, at a macroscopic scale, we obtain a new system, which is similar to the bidomain model, appearing in the context of diffusion in partially fissured media or in the context of electrical activity of the heart (for this case, f = 0, β = 0). remark 2. if we consider the case of a different geometry, i.e. if we assume that ωε is still connected, but πε is disconnected, then the homogenized matrix a 2 = 0 and system (3) consists in the coupling of a partial differential equation and an ordinary differential one. iii. proof of the main result we shall only sketch the proof of our main convergence result. for details, we refer to [11]. let us consider the variational formulation of problem (1):∫ t 0 ∫ ωε aε1∇u ε ·∇ϕdxdt + ∫ t 0 ∫ πε aε2∇v ε ·∇ϕdxdt+ ∫ t 0 ∫ ωε β(uε)ϕdxdt + αε ∫ t 0 ∫ sε (uε − vε) ∂ ∂t [ϕ]dσdt+ αε ∫ sε (uε − vε)(0)[ϕ](0)dσ+ aε ∫ t 0 ∫ sε g(vε − uε)[ϕ]dσdt = ∫ t 0 ∫ ω f ϕdxdt, (4) for any ϕ ∈ l2(ω × (0, t )) such that ϕ|ωε ∈ l 2(0, t ; h1(ωε)), ϕ|πε ∈ l 2(0, t ; h1(πε)), [ϕ] ∈ h1(0, t ; l2(sε)), ϕ vanishes on ∂ω×(0, t ) and ϕ vanishes at t = t . here, we have denoted by [ϕ] the difference of the traces of ϕ|ωε and ϕ|πε on s ε. there exists a unique weak solution (uε, vε) of (4), with uε ∈ l2(0, t ; h1∂ω(ω ε)), vε ∈ l2(0, t ; h1(πε)), where h1∂ω(ω ε) = { u ∈ h1(ωε) | u = 0 on ∂ω ∩ ∂ωε}. under the above hypotheses on the data, we can obtain suitable a priori estimates, independent of ε, for our solution (see [6], [9] and [10]):∫ t 0 ∫ ωε aε1∇u ε · ∇ϕdxdt + ∫ t 0 ∫ πε aε2∇v ε · ∇ϕdxdt+ ε ∫ sε (uε − vε)2dσ ≤ c, where 0 < t < t and c is independent of ε. for dealing with such two-component domains, we use two unfolding operators, t ε1 and t ε 2 , which map functions defined on the oscillating domains into functions defined on fixed domains. in such a way, we can avoid the use of extension operators (see [4] and [7]). also, we shall make use of the boundary unfolding operator, t εb , introduced in [5]. therefore, using the above mentioned unfolding operators, we can prove that there exist u, v ∈ l2(0, t ; h10 (ω)), û ∈ l 2((0, t ) × ω; h1per(y1)), v̂ ∈ l2((0, t ) × ω; h1per(y2)) such that, up to a subsequence, for ε → 0, we have: t ε1 (u ε) → u strongly in l2((0, t ) × ω, h1(y1)), t ε1 (∇ u ε) ⇀ ∇ u+∇yû weakly in l2((0, t )×ω×y1), t ε2 (v ε) → v strongly in l2((0, t ) × ω, h1(y2)), t ε2 (∇ v ε) ⇀ ∇ v+∇yv̂ weakly in l2((0, t )×ω×y2). moreover, ∂ ∂t (u − v) ∈ l2(0, t ; l2(ω)) and u, v ∈ c0([0, t ]; h10 (ω)). biomath 1 (2012), 1209021, http://dx.doi.org/10.11145/j.biomath.2012.09.021 page 3 of 5 http://dx.doi.org/10.11145/j.biomath.2012.09.021 c. timofte, multiscale analysis of composite structures in order to obtain the limit problem (3), we take, in a first step, φ1, φ2 ∈ c∞0 (ω) = d(ω) and ψ ∈ c∞0 ((0, t )) = d(0, t ). we have:∫ t 0 ∫ ωε aε1∇u ε · ∇φ1ψdxdt+ ∫ t 0 ∫ πε aε2∇v ε · ∇φ2ψdxdt + ∫ t 0 ∫ ωε β(uε)φ1ψdxdt +αε ∫ t 0 ∫ sε (uε − vε)(φ2 − φ1) dψ dt dσdt+ aε ∫ t 0 ∫ sε g(uε − vε)(φ2 − φ1)ψdσdt = ∫ t 0 ∫ ωε f φ1ψdxdt + ∫ t 0 ∫ πε f φ2ψdxdt. (5) applying the corresponding unfolding operators in (5) and passing to the limit, with ε → 0, we get (see, for details, [4], [6], [10] and [11]):∫ t 0 ∫ ω×y1 a1(∇u + ∇yû) · ∇φ1ψdxdydt+ ∫ t 0 ∫ ω×y2 a2(∇v + ∇yv̂) · ∇φ2ψdxdydt+ ∫ t 0 ∫ ω×y1 β(u)φ1ψdxdydt+ α ∫ t 0 ∫ ω×γ (u − v)(φ2 − φ1) dψ dt dxdσdt+ a ∫ t 0 ∫ ω×γ g(u − v)(φ2 − φ1)ψdxdσdt = ∫ t 0 ∫ ω×y1 f φ1ψdxdydt+ ∫ t 0 ∫ ω×y2 f φ2ψdxdydt. (6) in a second step, we take the test functions wεi = εφi(x)ϕi( x ε )ψ(t), with i = 1, 2, where φ ∈ d(ω), ϕi ∈ h1per(yi), ψ ∈ d((0, t )). observing that t εi (w ε i ) → 0 strongly in l2((0, t )×ω×yi) and t εi (∇w ε i ) → φi∇yϕi, strongly in l2((0, t )×ω×yi), we can pass to the limit and we get:∫ t 0 ∫ ω×y1 a1(∇u + ∇yû) · ∇yϕ1φ1ψdxdydt+ ∫ t 0 ∫ ω×y2 a2(∇v + ∇yv̂) ·∇yϕ2φ2ψdxdydt = 0. (7) putting together (6) and (7) and using standard density arguments, we obtain exactly the variational formulation of the limit problem (3). we can easily pass to the limit, with ε → 0, in the initial condition and we obtain u(0, x) − v(0, x) = c0(x), ∀x ∈ ω. as u and v are uniquely determined (see [9]), the whole sequences of microscopic solutions converge to a solution of the unfolded limit problem and this completes the proof of theorem 1. remark. 3 the above results can be extended to the case in which aε is a sequence of matrices in m(α1, β1, ω) such that t εi (a ε) → a a.e. in ω × y, with i = 1, 2 and a = a(x, y) ∈ m(α1, β1, ω × y ). the only difference is that in this case the homogenized matrices are no longer constant and depend on x. iv. conclusion using the periodic unfolding method, the effective behavior of the solution of some problems arising in the modeling of diffusion processes in a periodic structure formed by two media with different properties, separated by an active interface, was analyzed. two interesting geometric situations were discussed, leading to different macroscopic models. our setting is relevant for studying the heat conduction in composite materials with imperfect interfaces or the electrical conduction in biological tissues. references [1] m. amar, d. andreucci, p. bisegna and r. gianni,“on a hierarchy of models for electrical conduction in biological tissues”, math. meth. appl. sci., vol. 29 no. 7, pp. 767–787, 2006. http://dx.doi.org/10.1002/mma.709 [2] g. i. barenblatt, y. p. zheltov and i. n. kochina, “basic concepts in the theory of seepage of homogeneous liquids in fissured rocks (strata)”, prikl. mat. mekh., vol. 24, pp. 852–864, 1960. [3] h. brézis, ”problèmes unilatéraux”, j. math. pures et appl., vol. 51 no. 1, pp. 1–168, 1972. [4] d. cioranescu, a. damlamian and g. griso, “the periodic unfolding method in homogenization”, siam j. math. anal., vol. 40 no. 4, pp. 1585–1620, 2008. http://dx.doi.org/10.1137/080713148 [5] d. cioranescu, p. donato and r. zaki, “asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions”, asymptotic anal., vol. 53 no. 4, pp. 209–235, 2007. [6] c. conca, j. i. dı́az and c. timofte, “effective chemical processes in porous media”, math. models methods appl. sci. (m3as), vol. 13 (10), pp. 1437–1462 (2003). biomath 1 (2012), 1209021, http://dx.doi.org/10.11145/j.biomath.2012.09.021 page 4 of 5 http://dx.doi.org/10.1002/mma.709 http://dx.doi.org/10.1137/080713148 http://dx.doi.org/10.11145/j.biomath.2012.09.021 c. timofte, multiscale analysis of composite structures [7] p. donato, k. h. le nguyen, and r. tardieu, “the periodic unfolding method for a class of imperfect transmission problems”, journal of mathematical sciences, vol. 176 no. 6, pp. 891-927, 2011. http://dx.doi.org/10.1007/s10958-011-0443-2 [8] h. i. ene and d. polisevski, “model of diffusion in partially fissured media”, z. angew. math. phys., vol. 53 no. 6, pp. 1052– 1059, 2002. http://dx.doi.org/10.1007/pl00013849 [9] m. pennacchio, g. savaré and p. c. franzone, “multiscale modeling for the bioelectric activity of the heart”, siam j. math. anal., vol. 37 no. 4, pp. 1333–1370, 2005. http://dx.doi.org/10.1137/040615249 [10] c. timofte, “multiscale analysis in nonlinear thermal diffusion problems in composite structures”, central eur. j. physics, vol. 8 no. 4, pp. 555–561, 2010. http://dx.doi.org/10.2478/s11534-009-0141-6 [11] c. timofte, “multiscale analysis of diffusion processes in composite media”, in preparation, 2012. [12] m. veneroni, “reaction-diffusion systems for the microscopic cellular model of the cardiac electric field”, math. meth. appl. sci., vol. 29 no. 14, pp. 1631–1661, 2006. http://dx.doi.org/10.1002/mma.740 biomath 1 (2012), 1209021, http://dx.doi.org/10.11145/j.biomath.2012.09.021 page 5 of 5 http://dx.doi.org/10.1007/s10958-011-0443-2 http://dx.doi.org/10.1007/pl00013849 http://dx.doi.org/10.1137/040615249 http://dx.doi.org/10.2478/s11534-009-0141-6 http://dx.doi.org/10.1002/mma.740 http://dx.doi.org/10.11145/j.biomath.2012.09.021 introduction and setting of the problem the main result proof of the main result conclusion references original article biomath 1 (2012), 1211114, 1–6 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum modelling of a fed-batch culture applying simulated annealing olympia roeva∗ and tanya trenkova† ∗ department of bioinformatics and mathematical modelling institute of biophysics and biomedical engineering, sofia, bulgaria email: olympia@biomed.bas.bg † department of geography national institute of geophysics, geodesy and geography, sofia, bulgaria email: ttrenkova@gmail.com received: 15 july 2012, accepted: 11 november 2012, published: 22 december 2012 abstract—in this paper the metaheuristic simulated annealing (sa) is applied for parameter identification of non-linear model of cultivation process. sa algorithm is a stochastic relaxation technique, using the metropolis algorithm based on the boltzmann distribution in statistical mechanics, for solving nonconvex optimization problems. a real e. coli mc4110 fed-batch cultivation process is considered. the mathematical model is presented by a system of five ordinary differential equations, describing the regarded cultivation process variables biomass, substrate, acetate, dissolved oxygen and bioreactor volume increasing. the obtained criteria values show that the developed model is adequate and has a high degree of accuracy. the presented results are a confirmation of successful application of the sa algorithm and of the choice of sa algorithm parameters. keywords-metaheuristics; simulated annealing; optimization; e. coli; cultivation process i. introduction classical biotechnology is the science of production of human-useful products under controlled conditions, applying biological agents micro-organisms, plant or animal cells, their exoand endo-products, e.g. enzymes, etc. [7]. cultivation of recombinant micro-organisms e.g. e. coli, in many cases is the only economical way to produce pharmaceutical biochemicals such as interleukins, insulin, interferons, enzymes and growth factors. to maximize the volumetric productivity of bacterial cultures it is important to grow e. coli to high cell concentration. in order to optimize a real biotechnological production process, the model must describe those aspects of the process that significantly affect the process performance. the cost of development of mathematical models for bioprocess improvements is often too high and the benefits are too low. the main reason for this is related to the intrinsic complexity and non-linearity of biological systems. mathematical forms and their parameters used to describe cell behavior constitute the key problem of bioprocess modelling. the model building leads to information deficiency and to non unique parameter identification. while searching for new, more adequate modeling metaphors and concepts, methods which draw their initial inspiration from nature have received the early attention. in this paper a simulated annealing (sa) algorithm is proposed to identify the unknown parameters in a non-linear mathematical model of a fed-batch cultivation process. sa as an optimization technique first was introduced to solve problems in discrete optimization, mainly combinatorial optimization. subsequently, this technique has been successfully applied to solve optimization problems over the space of continuous decision variables. sa is a local search method where the search mechanism is modelled on the metropolis et al. [4] algocitation: o. roeva , t. trenkova, modelling of a fed-batch culture applying simulated annealing, biomath 1 (2012), 1211114, http://dx.doi.org/10.11145/j.biomath.2012.11.114 page 1 of 6 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.11.114 o. roeva et al., modelling of a fed-batch culture applying simulated annealing rithm and the principles of thermodynamic annealing. kirkpatrick et al. [3] and cerny [2] were the first to follow such a technique to solve optimization problems. sa can deal with arbitrary systems and cost functions; statistically guarantees finding an optimal solution (sa has the ability to avoid getting stuck at local minima); guarantees a convergence upon running sufficiently large (infinite) number of iterations; is relatively easy to code, even for complex problems. this makes annealing an attractive option for optimization problems this paper is organized as follows. outline of the introduced sa algorithm is described in section 2. in section 3 the problem formulation is presented. in section 4 a discussion of the obtained numerical results of e. coli cultivation process model parameter identification is presented. concluding remarks are made in section 5. ii. outline of the simulated annealing algorithm sa is a stochastic relaxation technique [3]. sa is named by analogy to the annealing of solids, in which a crystalline solid is heated to its melting point and then allowed to cool gradually until it is again in the solid phase at some nominal temperature. at the absolute zero final temperature, the resulting solid achieves its most regular crystal configuration corresponding to a (global) minimal value of the system’s energy. the sa algorithm performs the following steps: the algorithm generates a random trial point. initial temperature is 100. the sa chooses the distance of the trial point from the current point by a probability distribution with a scale depending on the current temperature. the trial point distance distribution could be set as a function that generates a point based on the current point and the current temperature using different distributions. here boltzman distribution is used: p r {e} = exp {e (i) /kt} /z (t ) (1) where e is the system energy; t – the system temperature; k – boltzman’s constant (k = 1.380650 × 10−23m2kgs−2k−1); z(t ) – a normalization function of the form: z (t ) = ∑ h exp {e (h) /kt} (2) the algorithm determines whether the new point is better or worse than the current point. if the new point is better than the current point, it becomes the next point. if the new point is worse than the current point, the algorithm can still make it the next point. the algorithm accepts a worse point based on an acceptance function. the probability of acceptance is: 1 1 + exp (∆/max (t )) (3) where ∆ = new objective — old objective, t — current temperature. since both ∆ and t are positive, the probability of acceptance is between 0 and 1/2. smaller temperature leads to smaller acceptance probability. also, larger ∆ leads to smaller acceptance probability. the algorithm systematically lowers the temperature, storing the best point found so far. the function that the algorithm uses to update the temperature is: t = t00.95 r (4) where r denotes the annealing parameter. reannealing sets the annealing parameters to lower values than the iteration number, thus raising the temperature in each dimension. the annealing parameters depend on the values of estimated gradients of the objective function in each dimension: ri = log   t0 ti max j (sj ) si   (5) where ri is the annealing parameter for component i, t0 — the initial temperature of component i, ti — current temperature of the component i, si — gradient of the objective in direction i times difference of bounds in direction i. the algorithm stops when the average change in the objective function is sufficiently small with respect to the predefined tolerance. the sa algorithm can be described by the scheme: find initial solution (by generating it randomly) set initial value for the parameter t = t0 set a value for r, the rate of cooling parameter j = 0 generate a new solution s′ calculate the difference in cost: ∆ = cost(s′) − cost(s) examine the new solution and decide: accept or reject if accepted, it becomes the current solution; otherwise, keep the old one; j = j + 1 reduce the t and generate a new solution until some stopping criterion is applied biomath 1 (2012), 1211114, http://dx.doi.org/10.11145/j.biomath.2012.11.114 page 2 of 6 http://dx.doi.org/10.11145/j.biomath.2012.11.114 o. roeva et al., modelling of a fed-batch culture applying simulated annealing iii. problem formulation the important part of model building is the choice of a certain optimization procedure for parameter estimation, so that to calibrate the model with a given set of experimental data in order to reproduce the experimental results in the best possible way. the estimation of model parameters with high parameter accuracy is essential for successful model development. a. e. coli mc4110 fed-batch cultivation model the mathematical model of the considered process can be represented by [5]: dx dt = µmax s ks + s x − fin v x (6) ds dt = − 1 ys/x µmax s ks + s x + fin v (sin − s) (7) da dt = 1 ya/x µmax a ka + a x − fin v a (8) dpo2 dt = − 1 ypo2/x µmax po2 kpo2 + po2 x+ kla(po ∗ 2 − po2) − fin v po2 (9) dv dt = fin (10) where: x is biomass concentration, [g/l]; s substrate concentration, [g/l]; a acetate concentration, [g/l]; po2 dissolved oxygen concentration, [%]; po∗2 saturation concentration of dissolved oxygen, [%]; fin feeding rate, [l/h]; v bioreactor volume, [l]; sin substrate concentration in the feeding solution, [g/l]; µmax maximum value of the specific growth rate, [h−1]; ki saturation constants; kla volumetric oxygen transfer coefficient, [h−1]; yi/x yield coefficients, [-]. for the parameter estimation problem real experimental data of the e. coli mc4110 fed-batch cultivation process are used. the cultivation experiments are performed in the institute of technical chemistry, university of hannover, germany during the collaboration work with the institute of biophysics and biomedical engineering, bas, bulgaria, granted by dfg. the cultivation conditions and the experimental data are presented in details in [1], [6]. b. optimization criterion the optimization criterion is a certain factor, which value defines the quality of an estimated set of parameters. the objective function is presented as a minimization of a distance measure j between experimental and model predicted values, represented by the vector y: j = n∑ i=1 m∑ j=1 {[yexp(i) − ymod(i)]j} 2 → min (11) where n is the number of data for each state variable m; yexp the experimental data; ymod model predictions with a given set of the parameters. iv. numerical results and discussion a series of parameter identification procedures for the considered model eq. (6) (10), using sa is performed. the computer specifications to run all optimization procedures are intel core i5-2320 cpu @ 3.00ghz, 8 gb memory (ram), windows 7 (64bit) operating system, matlab 7.5 environment. a. tuning of sa algorithm parameters each algorithm has its own influential parameters that affect its performance in terms of solution quality and computational time. in order to increase the performance of the sa it is necessary to provide the adjustments of the parameters depending on the problem domain. parameters of the fa are tuned on the basis of a large number of pre-tests according to the parameter identification problem, considered here. the algorithm performance is tested varying the following function and parameters: 1) annealing function af a) generates a point based on the current point and the current temperature using student’s distribution (af ast); b) generates a point based on the current point and the current temperature using multivariate normal distribution (aboltz ). 2) temperature function t f a) uses fast annealing by updating the current temperature based on the initial temperature and the current annealing parameter k (tf ast); b) “temperatureexp” uses exponential annealing schedule by updating the current temperature based on the initial temperature and the current annealing parameter k (texp); biomath 1 (2012), 1211114, http://dx.doi.org/10.11145/j.biomath.2012.11.114 page 3 of 6 http://dx.doi.org/10.11145/j.biomath.2012.11.114 o. roeva et al., modelling of a fed-batch culture applying simulated annealing table i objective values obtained for different af and t f at it = 100 and ri = 100 af af ast aboltz tf tf ast 6.6348 7.6207 texp 6.3744 2.0137 tboltz 12.3012 9.3365 c) “temperatureboltz” implements boltzman annealing by updating the current temperature based on the initial temperature and the current annealing parameter k (tboltz ). 3) reannealing interval: ri ∈ [1, 100]. 4) initial temperature: it ∈ [0.001, 100]. it is started with it = 100 and ri = 100. the two afs and the three tfs are tested for considered it and ri. the obtained main objective functions j are presented in table i. on the basis of the obtained results functions af ast for af and tf ast for t f are chosen as more appropriate. using these functions the it and ri are varied in the considered ranges. first, the it is decreased to 50. the resulting main j is 6.4575. then the ri is decreased to 50. at ri = 50 j is 6.4262 and at both it and ri set to 50 j = 6.3801. further decreasing of ri do not improve considerably the objective function for ri = 40 mean j is 6.3798 and for ri = 10 mean j is 6.3857. after that further decreasing of it is considered. for the values t i of 10, 1, 0.5, 0.1 and 0.001 the resulting main j are as follows: 7.4391, 6.1989, 6.2442, 6.4040 and 6.8768. the lowest mean value of j is obtained at it = 1 and ri = 50. after tuning procedures the main sa parameters are set to the following optimal settings: af – af ast; t f – tf ast; it = 1 and ri = 50. b. results from parameter optimization procedure because of the stochastic characteristics of the applied algorithm a series of 30 runs is performed. the best, the worst and the average results of the 30 runs, for the j value and execution time are observed. the obtained results from parameter identification procedures are presented in table ii. the resulting model parameters values are in admissible range according to [8], [9], [10]. in fig. 1 some graphical results about algorithms performance (parameters estimation through generations) are presented. a quantitative measure of the differences between modelled and measured values is important criterion table ii numerical results from parameter identification using simulated annealing parameters average best worst µmax 0.47 0.46 0.45 ks 0.01 0.012 0.02 ka 0.56 0.59 0.51 kpo2 0.02 0.023 0.021 ys/x 0.5 0.497 0.52 ya/x 0.14 0.13 0.16 ypo2/x 0.2 0.201 0.22 kla 57.71 55.87 50.23 j value 6.2661 6.1989 6.4022 execution time, s 88.723 80.954 91.298 0 500 1000 1500 2000 2500 3000 0 0.5 1 1.5 2 2.5 3 3.5 4 function evaluation m m a x k s 1 /y x /s mmax ks 1/y x/s fig. 1. parameters estimation through generations for the adequacy of a model. the model predictions of the state variables are compared to the experimental data points of the real e. coli mc4110 cultivation. the graphical results of the comparison between the model predictions of state variables, based on sa estimations, and the experimental data points of the real e. coli cultivation are presented in the figs. 2–4. model predicted data are presented with solid line. the presented graphics show a very good correlation between the experimental and predicted data. the model predicts successfully the process variables dynamics during the fed-batch cultivation of e. coli mc4110. however, graphical comparisons can clearly show only the existence or absence of systematic deviations between model predictions and measurements. it is evident that a quantitative measure of the differences between calculated and measured values is an important criterion for the adequacy of a model. the most important criterion for the valuation of models is that the deviations between measurements and model calculations (j ) should be as small as possible. the obtained criteria values show that the developed model is adequate and has a high degree of accuracy. as biomath 1 (2012), 1211114, http://dx.doi.org/10.11145/j.biomath.2012.11.114 page 4 of 6 http://dx.doi.org/10.11145/j.biomath.2012.11.114 o. roeva et al., modelling of a fed-batch culture applying simulated annealing a result of the identification procedure accurate model parameters estimations are obtained. the presented results are a confirmation of successful application of the sa algorithm and of the appropriate choice of sa algorithm parameters. 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 0 1 2 3 4 5 6 7 8 9 10 time, [h] b io m a ss , [g /l] ; s u b st ra te *1 0 , [g /l] biomass concentration substrate concentration fig. 2. biomass and substrate concentration modelled and real experimental data fig. 3. acetate concentration modelled and real experimental data v. conclusion in this paper the metaheuristic sa is applied for parameter identification of non-lineal model of cultivation process. real e. coli mc4110 fed-batch cultivation process is considered. the mathematical model is presented by a system of ordinary differential equations, describing the regarded cultivation process variables. particular procedure for model parameter identification is performed using sa. numerical and simulation results reveal that correct and consistent results are obtained. 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 20 20.2 20.4 20.6 20.8 21 21.2 time, [h] d is so lv e d o xy g e n , [% ] dissolved oxygen fig. 4. dissolved oxygen modelled and real experimental data resulting non-linear model predicts adequately and to a high degree of approximation the variation of the considered state variables. simulation results reveal that accurate and consistent estimates can be obtained using sa algorithm. acknowledgment this work is partially supported by the national science fund grants dmu 02/4 “high quality control of biotechnological processes with application of modified conventional and metaheuristics methods” and did 0229 “modeling processes with fixed development rules (modprofix)”. references [1] m. arndt and b. hitzmann, “feed forward/feedback control of glucose concentration during cultivation of escherichia coli”, 8th ifac int. conf. on comp. appl. in biotechn., canada, 2001, pp. 425–429. [2] v. cerny, “thermodynamical approach to the travelling salesman problem: an efficient simulation algorithm”, journal of optimization theory and applications, vol. 45, 1985, pp. 41– 51. http://dx.doi.org/10.1007/bf00940812 [3] s. kirkpatrick, c. d. gelatt and m. p. vecchi, optimization by simulated annealing, science, new series, vol. 220(4598), 1983, pp. 671–680. [4] n. metropolis, a. rosenbluth, m. rosenbluth, a. teller and m. teller, “equation of state calculations by fast computing machines”, journal of chemical physics, vol. 21, 1953, pp. 1087–1092. http://dx.doi.org/10.1063/1.1699114 [5] o. roeva, “parameter estimation of a monod-type model based on genetic algorithms and sensitivity analysis”, lncs, springer, vol. 4818, 2008, pp. 601–608. [6] o. roeva, t. pencheva, b. hitzmann and st. tzonkov, “a genetic algorithms based approach for identification of escherichia coli fed-batch fermentation”, int j bioautomation, vol. 1, 2004, pp. 30–41. biomath 1 (2012), 1211114, http://dx.doi.org/10.11145/j.biomath.2012.11.114 page 5 of 6 http://dx.doi.org/10.1007/bf00940812 http://dx.doi.org/10.1063/1.1699114 http://dx.doi.org/10.11145/j.biomath.2012.11.114 o. roeva et al., modelling of a fed-batch culture applying simulated annealing [7] u. viesturs, d. karklina , i. ciprovica, “bioprocess and bioengineering”, jeglava, 2004. [8] o. georgieva, m. arndt and b. hitzmann, “modelling of escherichia coli fed-batch fermentation”, in international symposium ”bioprocess systems’2001 — biops’2001”, 1–3.x, sofia, bulgaria, 2001, pp. i.61–i.64. [9] d. levisauskas, v. galvanauskas, s. henrich, k. wilhelm, n. volk and a. lubbert, “model-based optimization of viral capsid protein production in fed-batch culture of recombinant escherichia coli”, bioprocess and biosystems engineering, vol. 25, 2003, pp. 255–262. [10] b. zelic, d. vasic-racki, c. wandrey and r. takors, “modeling of the pyruvate production with escherichia coli in a fed-batch bioreactor”, bioprocess and biosystems engineering, vol. 26, 2004, pp. 249–258. http://dx.doi.org/10.1007/s00449-004-0358-0 biomath 1 (2012), 1211114, http://dx.doi.org/10.11145/j.biomath.2012.11.114 page 6 of 6 http://dx.doi.org/10.1007/s00449-004-0358-0 http://dx.doi.org/10.11145/j.biomath.2012.11.114 introduction outline of the simulated annealing algorithm problem formulation e. coli mc4110 fed-batch cultivation model optimization criterion numerical results and discussion tuning of sa algorithm parameters results from parameter optimization procedure conclusion references www.biomathforum.org/biomath/index.php/biomath original article efficient implicit runge-kutta methods for fast-responding ligand-gated neuroreceptor kinetic models edward t. dougherty department of mathematics rowan university glassboro, nj, usa email: doughertye@rowan.edu received: 11 may 2015, accepted: 31 december 2015, published: 19 january 2016 abstract—neurophysiological models of the brain typically utilize systems of ordinary differential equations to simulate single-cell electrodynamics. to accurately emulate neurological treatments and their physiological effects on neurodegenerative disease, models that incorporate biologically-inspired mechanisms, such as neurotransmitter signalling, are necessary. additionally, applications that examine populations of neurons, such as multiscale models, can demand solving hundreds of millions of these systems at each simulation time step. therefore, robust numerical solvers for biologically-inspired neuron models are vital. to address this requirement, we evaluate the numerical accuracy and computational efficiency of three l-stable implicit runge-kutta methods when solving kinetic models of the ligandgated glutamate and γ-aminobutyric acid (gaba) neurotransmitter receptors. efficient implementations of each numerical method are discussed, and numerous performance metrics including accuracy, simulation time steps, execution speeds, jacobian calculations, and lu factorizations are evaluated to identify appropriate strategies for solving these models. comparisons to popular explicit methods are presented and highlight the advantages of the implicit methods. in addition, we show a machinecode compiled implicit runge-kutta method implementation that possesses exceptional accuracy and superior computational efficiency. keywords-implicit runge-kutta; neuroreceptor model; numerical stiffness; ode simulation i. introduction mathematical modeling and computational simulation provide an in silico environment for investigating cerebral electrophysiology and neurological therapies including neurostimulation. traditionally, volume-conduction models have been used to emulate electrical potentials and currents within the head cavity. in particular, these models can reproduced electroencephalograph (eeg) surface potentials [1]–[3], and have been successful in predicting cerebral current density distributions from neurostimulation administrations [1], [4]– [7]. as these models become more refined, their utility in diagnosing, treating, and comprehending neurological disorders greatly increases. progress in field of computational neurology has motivated a migration towards models that incorporate cellular-level bioelectromagnetics. for example, bidomain based models have been used to simulate the effects of extracellular electrical citation: edward t. dougherty, efficient implicit runge-kutta methods for fast-responding ligand-gated neuroreceptor kinetic models, biomath 4 (2015), 1512311, http://dx.doi.org/10.11145/j.biomath.2015.12.311 page 1 of 16 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2015.12.311 e. t. dougherty, efficient implicit runge-kutta methods for fast-responding ... current on cellular transmembrane voltage(s) [8]– [13]. in addition, multiscale models have reproduced eeg measurements originating from action potentials [14], [15], and have also demonstrated an ability to simulate the influence of transcranial electrical stimulation on neuronal depolarization [16]. these models typically utilize a system of ordinary differential equations (odes) to emulate cellular-level electrophysiology. while the computational expense of simulating a single cell is essentially negligible, this is not the case with large-scale applications that may include hundreds of millions of cells; in multiscale applications, solving this set of odes is the computational bottleneck [17]. in these applications, choosing an appropriate numerical solver and using efficient implementation approaches become paramount. alterations in neurotransmitter signalling is a hallmark of many neurodegenerative conditions and treatments. parkinson’s disease (pd), for example, which affects approximately one million individuals in the united states alone [18], culminates with pathological glutamate and γ-aminobutyric acid (gaba) binding activity throughout the basal ganglia-thalamocortical network [19], [20]. as a treatment for pd, deep brain stimulation (dbs) electrically stimulates areas of the basal ganglia, such as the subthalamatic nucleus (stn) [21], to restore normal glutamate and gaba synaptic concentrations [22]– [24]. therefore, models that incorporate fundamental neurotransmitter-based signalling provide utility to the neurological research community. models of metabotropic and slow-responding ligand-gated receptors, such as the gabab and n-methyl-d-aspartate (nmda) glutamate receptors, can be efficiently solved with explicit runge-kutta (erk) methods [25]. on the contrary, fast-responding ionotropic receptors, such as the α-amino-3-hydroxy-5-methyl-4isoxazolepropionic acid receptor (ampar) and the gabaa receptor (gabaar) result in models that are classified as stiff [26], which is an attribute of an ode system that demands relatively small step sizes in portions of the numerical solution [27]. for these ode systems, l-stable implicit runge-kutta (irk) solvers with adaptive timestepping are ideal given their exceptional stability properties [28]. in this paper, we examine l-stable irk methods when solving models that represent the ampa and gabaa neuroreceptors. three l-stable irk methods that are highly effective at solving stiff ode systems were selected and implemented with custom matlab [29] programming. features including adaptive step-sizing, embedded error estimation, error-based step size selection, and simplified newton iterations are incorporated [30]. numerical experiments were then used to identify the optimal maximum number of inner newton iterations for each method. then, for both the ampar and gabaar models, simulation time step results of each irk method are compared to commonly used erk methods. in addition, the numerical accuracy and computational efficiency of each irk method is compared to one other, as well as the highly-popular fifth order, variable step size dormand-prince method. finally, a c++ based irk implementation demonstrates exceptionally accurate and expedient performances, showcasing its potential to support large-scale multi-cellular brain simulations. ii. materials and methods a. neuroreceptor models 1) ampa: glutamate is the single most abundant neurotransmitter in the human brain [31]. it is produced by glutamatergic neurons, and is classified as excitatory in the sense that it predominately depolarizes post-synaptic neurons towards generating action potentials [32]. given the large concentration of glutamate in the nervous system, alterations in its production are associated with many neurodegenerative diseases and treatments. in pd patients, for example, stimulating the stn with dbs causes a cascade of cellular effects within the basal-ganglia thalamocortical pathway through its afferent and efferent projections, including increased glutamate secretion to the globus biomath 4 (2015), 1512311, http://dx.doi.org/10.11145/j.biomath.2015.12.311 page 2 of 16 http://dx.doi.org/10.11145/j.biomath.2015.12.311 e. t. dougherty, efficient implicit runge-kutta methods for fast-responding ... pallidus external (gpe), globus pallidus internal (gpi), and substantia nigra pars reticulata (snr) [23]. ligand-gated ampa receptors for glutamate are permeable to sodium and potassium, have a reversal potential of 0 mv, and possess fast channel opening rates. therefore, these receptors produce fast excitatory post-synaptic currents [33]. figure 1a displays the markov kinetic binding model for the ligand-gated ampar that was utilized in this paper [34]. in this network, there is the unbound ampar form c0, singly and doubly bound receptor forms c1 and c2, which can lead to desensitized states d1 and d2, respectively, and the open receptor form o [35]. in addition, variable t represents neurotransmitter concentration. mass action kinetics gives the following system of odes for the ampa neuroreceptor model: dc0 dt = −kbc0t + c1ku1, (1a) dc1 dt = kbc0t + ku2c2 + kudd1 −ku1c1 −kbc1t −kdc1, (1b) dc2 dt = kbc1t + kudd2 + kco −ku2c2 −kdc2 −koc2, (1c) dd1 dt = kdc1 −kudd1, (1d) dd2 dt = kdc2 −kudd2, (1e) do dt = c2ko −kco, (1f) dt dt = −kbc0t + ku1c1 −kbc1t + ku2c2. (1g) state transition rates were assigned as follows: kb = 1.3 x 107, ko = 2.7 x 103, kc = 200, ku1 = 5.9, ku2 = 8.6 x 104, kd = 900, and kud = 64, each with units [1/sec]. initial concentrations of c1, c2, d1, d2, and o were set to 0 m [33], and initial values for c0 and t were computed from a nonlinear least squares fit of the model to the whole cell recording data in destexhe et al. [35]. 2) gaba: gaba is the most abundant inhibitory neurotransmitter in the human brain [36]. like glutamate, gaba concentrations are altered by neurological disease and treatment. in stn dbs, for example, increased glutamate to the gpe increases gaba secretion to the gpi and snr, resulting in greater gaba neuroreceptor binding in these regions [24]. there are two main categories of gaba neuroreceptors. metabotropic gabab receptors are slow-responding due to the secondary messenger biochemical network cascade necessary for ion channel activation. on the contrary, ligand-gated gabaa receptors are fast-responding due to their expedient ion channel opening rates. gabaa receptors are selective to chlorine with a reversal potential of approximately -70 mv. in addition, this receptor has two bound forms that can both trigger channel activation [35]. figure 1b displays the kinetic binding model for the gabaa receptor that was utilized in this paper [26]. in this model, there is the unbound receptor form c0, singly and doubly bound receptor forms c1 and c2, slow and fast desensitized states ds and df , and singly open and doubly open receptor forms o1 and o2. this model incorporates the minimal forms needed to accurately reproduce gabaar kinetics [37]. mass action kinetics gives the following ode system for the gabaa neuroreceptor model: dc0 dt = −2kbc0t + kuc1, (2a) dc1 dt = 2kbc0t −kuc1 + kudsds −kdsc1 + 2kuc2 −kbc1t + kc1o1 −ko1c1, (2b) dc2 dt = kbc1t − 2kuc2 + kc2o2 −ko2c2 + kudfdf −kdfc2, (2c) dds dt = kfsdf −ksfdst + kdsc1 −kudsds, (2d) ddf dt = ksfdst −kfsdf + kdfc2 −kudfdf, (2e) biomath 4 (2015), 1512311, http://dx.doi.org/10.11145/j.biomath.2015.12.311 page 3 of 16 http://dx.doi.org/10.11145/j.biomath.2015.12.311 e. t. dougherty, efficient implicit runge-kutta methods for fast-responding ... c 0 c 1 c 2 o d 1 d 2 k b t k b t k u1 k u2 k d k ud k d k ud k o k c (a) ampa receptor c 0 c 1 c 2 d s d f o 1 o 2 2k b t k b t k u 2k u k o1 k c1 k o2 k c2 k ds k uds k df k udf k sf t k fs (b) gaba receptor fig. 1: kinetic models for ligand-gated neuroreceptors. do1 dt = ko1c1 −kc1o1, (2f) do2 dt = ko2c2 −kc2o2, (2g) dt dt = kuc1 − 2kbc0t + 2kuc2 −kbc1t + kfsdf −ksfdst. (2h) transition rates for the gabaar ode system were assigned as follows: kb = 5 x 106, ku = 131, kuds = 0.2, kds = 13, kc1 = 1100, ko1 = 200, kc2 = 142, ko2 = 2500, kudf = 25, kdf = 1250, kfs = 0.01, and ksf = 2, each with units [1/sec]. initial values of c1, c2, ds, df , o1, and o2 were set 0 m, and c0 and t were assigned the values 1 x 10−6 m and 4096 x 10−6 m, respectively [26]. b. stiff ordinary differential equations the stiffness ratio is defined as l = max |re(λi)| min |re(λi)| , where λi is the ith eigenvalue of the local jacobian matrix [38], given by jij = ∂fi(t, ȳ) ∂yj . a general non-linear ode system is stiff when l � 1. for each neuroreceptor model, we estimated the eigenvalues numerically; a local jacobian matrix is computed at each simulation time step using finite differences, and then its eigenvalues are computed using matlab’s eig function [39]. for the ampar model l = 1.6 x 1011, and for the gabaar model l = 3.5 x 1011. thus, both of these systems are classified as stiff. c. implicit runge-kutta methods runge-kutta methods are a family of numerical integrators that solve ode systems with trial steps within the time step. these methods can be expressed with the following formulas: z̄i = h s∑ j=1 aijf̄(tn + cjh,ȳn + z̄j), i = 1, ...,s (3a) ȳn+1 = ȳn + h s∑ j=1 bjf̄(tn + cjh,ȳn + z̄j), (3b) where ȳn is the current solution at time tn, h is the current time step, [aij] is the runge-kutta matrix, f̄ is the ode system, [cj] represents intertime trial step nodes, [bj] is the trial step solution weights, s is the number of stages, and ȳn+1 is the biomath 4 (2015), 1512311, http://dx.doi.org/10.11145/j.biomath.2015.12.311 page 4 of 16 http://dx.doi.org/10.11145/j.biomath.2015.12.311 e. t. dougherty, efficient implicit runge-kutta methods for fast-responding ... numerical solution at time tn+1 [28]. a rungekutta method can be fully defined with a butcher table, i.e. a specific [aij], [bj], and [cj] [40]. l-stable irk methods are highly effective at solving stiff ode systems [30]; these methods have no step size constraint to maintain numerical stability and quickly converge [41]. methods with second and third order accuracy were considered as these orders best match the numerical accuracy of fractional step algorithms typically employed with partial differential equation based multiscale models [16], [39]. the following l-stable irk methods were selected for examination: sdirk(2/1) [42], esdirk23a [17], and radauiia(3/2) [30], [43]. each has demonstrated accuracy and computational efficiency when solving extremely stiff ode systems. in addition, each provide an efficient local error estimator that enables error-based adaptive timestepping. for simplicity, these solvers will be referred to as sdirk, esdirk, and radau for the remainder of this paper. butcher tables for these methods are displayed in fig. 2. γ γ 1 1 −γ γ b 1 −γ γ b̂ 1 − γ̂ γ̂ (a) sdirk(2/1) 1 3 5 12 1 12 1 3 4 1 4 b 3 4 1 4 b̂ 3 4 − √ 6 4 1 4 + √ 6 12 (b) radauiia(3/2) 0 0 2γ γ γ 1 b̂1 b̂2 γ 1 b1 b2 b3 γ b 6γ−1 12γ −1 (24γ−12)γ −6γ2+6γ−1 6γ−3 γ b̂ −4γ2+6γ−1 4γ −2γ+1 4γ γ 0 (c) esdirk23a fig. 2: butcher tables for the three implicit rungekutta methods evaluated in this paper. in fig. 2a, γ = 1− √ 2 2 and γ̂ = 2− 5 4 √ 2, and in fig. 2c, γ = 0.4358665215. in each butcher table, b̂ specifies the lower-order trial step solution weights. the sdirk method is second order with an embedded first order formula for local error estimation. each trial step, z̄i, of the sdirk solver can be solved for sequentially. specifically, since a12 = 0 (see fig. 2a), the first stage of this method can be written as z̄1 = h ( a11f̄(tn + c1h,ȳn + z̄1) ) , and z̄1 can be solved for first and used directly in the solution of z̄2 = h ( a21f̄(tn + c1h,ȳn + z̄1) + a22f̄(tn + c2h,ȳn + z̄2) ) . the radau method has two stages like the sdirk method (see fig. 2b), but has third order accuracy with a second order error formula. this method’s runge-kutta matrix is full, therefore the trial stages are solved as a coupled implicit system: z̄1 = h[a11f̄(tn + c1h,ȳn + z̄1)+ a12f̄(tn + c2h,ȳn + z̄2)], z̄2 = h[a21f̄(tn + c1h,ȳn + z̄1)+ a22f̄(tn + c2h,ȳn + z̄2)]. trial steps in the esdirk method are solved sequentially like the sdirk method, after the initial explicit first stage (see fig. 2c). this method is third order with an embedded second order formula for local error estimation, similar to the radau solver. d. implementation the three irk methods were programmed in matlab using principles specified in [30] and [44]; we refer these resources for a detailed explanation of runge-kutta method implementation and in this section provide just a brief overview of key aspects utilized in our implementations. for each irk method, newton’s method is used in solving system (3a). typically, each inner newton iteration involves computing the local jacobian matrix and performing an lu factorization. to greatly decrease run-time, at each time step the jacobian computation and lu factorization are performed just once on the first newton iteration and retained for all remaining iterations. execution time is further decreased by retaining the jacobian in the subsequent time step if the irk method converges with just one newton iteration, or ‖z̄ k+1−z̄k‖ ‖z̄k−z̄k−1‖ ≤ 10 −3, where k is the number of biomath 4 (2015), 1512311, http://dx.doi.org/10.11145/j.biomath.2015.12.311 page 5 of 16 http://dx.doi.org/10.11145/j.biomath.2015.12.311 e. t. dougherty, efficient implicit runge-kutta methods for fast-responding ... inner iterations for convergence and ‖ · ‖ is an error-normalized 2-norm [30], [45]. efficient starting values for each newton iteration are produced via a lagrange interpolation polynomial of degree s [30], [42]. for the radau method, for example, we use the data points: q(0) = 0,q( 1 3 ) = z̄1, and q(1) = z̄2, and obtain the following lagrange polynomial: q(w) =q(0) (w − 1 3 )(w − 1) (0 − 1 3 )(0 − 1) + q ( 1 3 ) (w − 0)(w − 1) ( 1 3 − 0)( 1 3 − 1) + q(1) (w − 0)(w − 1 3 ) (1 − 0)(1 − 1 3 ) = w(w − 1) −2 9 z̄1 + w(w − 1 3 ) 2 3 z̄2. newton iteration starting values are then given by: z̄1 = q(1 + wc1) + ȳn − ȳn+1, z̄2 = q(1 + wc2) + ȳn − ȳn+1, where w = hnew hold . for each time step, local error is calculated and used for (i) step acceptance and (ii) subsequent step size prediction. the error at time step tn+1 can be computed by err = ŷn+1 − ȳn+1, where ŷn+1 = ȳn+b̂0hf̄(tn, ȳn)+ h s∑ j=1 b̂jf̄(tn + cjh,z̄j + ȳn). (4) the error calculations in the sdirk and esdirk methods are suitable for stiff systems [39], [41]. for the radau method, however, ŷn+1−ȳn+1 will become unbounded and is therefore not appropriate for stiff systems [46]. instead, we use the formula err = (i −hb̂0j)−1(ŷn+1 − ȳn+1) which is equivalent to err =(i −hb̂0j)−1[b̂0hf̄(tn, ȳn) + (b̂1 − b1)hf̄(tn + c1h,z̄1 + ȳn) + (b̂2 − b2)hf̄(tn + c2h,z̄2 + ȳn)], (5) where i is the identity matrix, j is the jacobian, and b̂0 = √ 6 6 [46]. we can write ŷn+1 − ȳn+1 as follows [47]: ŷn+1 − ȳn+1 = b̂0hf̄(tn, ȳn) + e1z̄1 + e2z̄2. (6) to identify the coefficients e1 and e2, we substitute z̄1 and z̄2 (3a) into (6): ŷn+1 − ȳn+1 = b̂0hf̄(tn, ȳn) +e1[ha11f̄(tn + c1h,z̄1 + ȳn)+ ha12f̄(tn + c2h,z̄2 + ȳn)] +e2[ha21f̄(tn + c1h,z̄1 + ȳn)+ ha22f̄(tn + c2h,z̄2 + ȳn)]. collecting terms gives: ŷn+1−ȳn+1 = b̂0hf̄(tn, ȳn)+ (e1a11 + e2a21)hf̄(tn + c1h,z̄1 + ȳn)+ (e1a12 + e2a22)hf̄(tn + c2h,z̄2 + ȳn). (7) from (5) and (7), we end up with the following system of equations: b̂1 − b1 = e1a11 + e2a21, b̂2 − b2 = e1a12 + e2a22. using the radau butcher table (fig. 2b) gives (e1,e2) = b̂0 (−9 2 , 1 2 ) . the error estimation is used to predict step size via the strategy proposed by gustafsson [45]. further, step size following a rejected step due to excessive local error, namely ‖err‖ > 1, is 1 3 h. for large-scale simulations, e.g. multiscale applications, hundreds of millions of ode systems may be solved at each time step. for these computationally intensive simulations, scripting languages such as matlab are not ideal, and machinecompiled programs are generally necessary to achieve simulation results within reasonable computing time [48]. due to its superior accuracy in solving both the gabaar and ampar models (see sec. iii), we selected the radau method and configured a c++ implementation of it. execution results of this version provide a measure of optimally expected computational performance. we validated the implementation of each irk method by comparing their gabaar simulation results to those presented in qazi et al. [37], biomath 4 (2015), 1512311, http://dx.doi.org/10.11145/j.biomath.2015.12.311 page 6 of 16 http://dx.doi.org/10.11145/j.biomath.2015.12.311 e. t. dougherty, efficient implicit runge-kutta methods for fast-responding ... and their ampar simulation results to whole cell recording data in destexhe et al. [35]. e. simulations numerical simulations were performed to assess the robustness of the irk methods when solving the ampar and gabaar models. simulations were one second in duration, with rates and initial conditions as specified in section ii-a. absolute and relative error tolerances were both set to 10−8, and initial step size, h, was set to 10−4. for each irk method, the optimal number of maximum newton iterations, kmax, was identified by solving the ampar and gabaar models with kmax = 5, 6, ..., 20. for each value of kmax, the mean execution time of five simulations was computed, and the value of kmax that produced the lowest mean execution time was selected. figure 3a displays the kmax values selected for each model and method. for each method, it was observed that a threshold value of kmax exists, such that higher values do not result in faster simulations. therefore, we selected the minimum kmax value associated with the fastest execution speed. for example, for the radau method solving the gabaar model, simulation times begin to plateau for kmax ≥ 10, and simulation times with kmax ≥ 15 were the same (see fig. 3b). therefore, for this model and irk method, kmax = 15 was selected. figure 3b also shows that faster run times correlate with fewer solution time steps and lu factorizations, until a floor is reached; in the case of the radau method solving the gabaar model, this floor is 29 time steps and 30 lu factorizations. to a point, higher values of kmax increase the probability of newton method convergence, resulting in fewer time steps and fewer computationally expensive lu factorizations [30]. for the radau method solving the gabaar model, values of kmax ≥ 15 yield the fewest number of simulation times steps in addition to no steps where the newton iteration fails to converge. thus, when kmax = 15, time steps and associated lu factorizations are minimized, yielding the fastest execution speeds. model sdirk esdirk radau gabaar 7 10 15 ampar 14 12 17 (a) values of kmax selected for each model and method (b) radau method solving the gabaar model: run time, time steps, and lu factorizations, for kmax = 5, 6, ..., 20 fig. 3: maximum newton iteration metrics and results. to evaluate the advantages that irk methods have when solving fast-responding neuroreceptor models, we first compare the total number of simulation time steps and simulation step sizes of each irk method to the following commonly used erk methods: forward euler (fe), midpoint method (mid), and 4th order runge-kutta (rk4). next, to compare each irk method to one another and to the adaptive 5th order dormand-prince method (dp5) [49], metrics including local and global error, total simulation time steps, step sizes, execution times, and numbers of jacobian computations and lu factorizations were evaluated. absolute and relative error tolerances of the dp5 method were set to 10−8, matching the tolerances of the three implicit methods. to more comprehensively assess performance differences amoung the irk methods, workprecision diagrams using solution run times and scd values, where scd = -log10(‖relative error at t = 1.0 sec ‖∞), were then generated [50]. for the work-precision diagrams, relative error tolerances biomath 4 (2015), 1512311, http://dx.doi.org/10.11145/j.biomath.2015.12.311 page 7 of 16 http://dx.doi.org/10.11145/j.biomath.2015.12.311 e. t. dougherty, efficient implicit runge-kutta methods for fast-responding ... 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 x 10 −7 time (s) o p e n s ta te c o n ce n tr a tio n ( m ) (a) open state concentration solution, o1 + o2 0 0.25 0.5 0.75 1 1.25 1.5 0 0.2 0.4 0.6 0.8 1 x 10 −6 time (ms) c o n ce n tr a tio n ( m ) closed unbound closed bound desensitized open (b) solution of all receptor forms: closed unbound = c0; closed bound = c1 +c2; desensitized = ds +df ; open = o1 + o2 fig. 4: sdirk method solution of gabaar model. were set to rtol = 10−(4+ m 5 ), m = 0, 1, ..., 25, absolute error tolerance was set to 10−4 ·rtol, and initial step size was 10−4. in addition, solution run times presented in these diagrams are the mean of five runs. for all accuracy calculations, solutions with a 5th order adaptive time-stepping l-stable implicit runge-kutta method with a maximum step size of 10−6 and both absolute and relative tolerances set to 10−14 were used as true solutions. finally, the execution time of the radau c++ implementation when solving both neuroreceptor models was assessed. all simulations were run on a linux machine with an intel i7 processor with a clock speed of 2.40 ghz. iii. results and discussion a. gabaar model figure 4 presents the solution of the gabaar model with the sdirk method; esdirk and radau solutions look identical. the sharp transition in the total open state concentration, o1(t) + o2(t), at the onset of neurotransmitter stimulus at t = 0 displays the necessity for smaller time steps in this region of the solution (fig. 4a). upon examining all receptor forms during the first 1.5 ms of the simulation, it is observed that both the unbound closed form, c0(t), and total bound closed form, c1(t) + c2(t), possess concentration transitions even greater than the open receptor form (fig. 4b). these results show the stiffness possessed by the gabaar system. table i displays simulation time step metrics for the three irk methods and the fe, mid, and rk4 erk methods. the maximum step size of each explicit method was calculated with the gabaar model stiffness index and the method’s stability region [28], giving the largest step that can be taken while maintaining numerical stability. then, the number of time steps required for each erk method was computed by dividing the simulation duration by the maximum step size. the fe and mid methods both require 2.1 x 104 time steps, and the rk4 method requires 1.5 x 104, which is lower than the fe and mid methods due to its larger stability region [30]. on the contrary, each implicit method requires less than 30 simulation time steps. as displayed in figure 4a, the majority of these time steps for the sdirk method occur at the beginning of the simulation, within the region of rapid solution transition. similarly, the esdirk and radau solvers demand noticeably more time steps at the onset of neurotransmitter stimulation (fig. 5). rejected steps, totalling three for the esdirk method (fig. biomath 4 (2015), 1512311, http://dx.doi.org/10.11145/j.biomath.2015.12.311 page 8 of 16 http://dx.doi.org/10.11145/j.biomath.2015.12.311 e. t. dougherty, efficient implicit runge-kutta methods for fast-responding ... 0 0.2 0.4 0.6 0.8 1 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 time (s) s te p s iz e ( s) min step size: 1.4e−05 max step size: 0.61 rejected steps:3 step size, h:26 (a) esdirk method 0 0.2 0.4 0.6 0.8 1 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 time (s) s te p s iz e ( s) min step size: 1.2e−05 max step size: 0.49 rejected steps:2 step size, h:29 (b) radau method fig. 5: simulation step sizes for the gabaar model. table i: simulation time steps results for the erk and irk methods when solving the gabaar model. method (order) max step time size (s) steps fe (1) 4.8 x 10−5 2.1 x 104 mid (2) 4.8 x 10−5 2.1 x 104 rk4 (4) 6.8 x 10−5 1.5 x 104 sdirk (2/1) adaptive 28 esdirk (3/2) adaptive 26 radau (3/2) adaptive 29 5a) and two for the radau method (fig. 5b) all occur at time t = 0; once the solution in this region has been accurately resolved, no further rejected steps occur. in addition, for all three irk methods, all newton iterations converged, which was facilitated by identifying optimal kmax values (see sec. ii-e). further, the smallest step sizes of the irk methods, namely 1.4 x 10−5 for the sdirk and esdirk methods and 1.2 x 10−5 for the radau method, have the same order of magnitude as the largest stable step sizes of the erk methods. next the accuracy and computational efficiency of the irk methods were compared to one another 0 0.2 0.4 0.6 0.8 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 x 10 −9 time (s) e rr o r dp5 sdirk esdirk radau fig. 6: gabaar model open state concentration solution error. and with the dp5 method (table ii). while the dp5 method possesses the lowest maximal local true solution deviation (3.2 x 10−10), the 2-norm of its global error is one to two orders of magnitude higher than all three irk methods. these results are explained by the fact that the solution of the dp5 solver oscillates around the true solution (fig. 6). in addition, the dp5 method requires approximately 50,000 simulation time steps and takes 49.0 seconds to run. in comparison, the biomath 4 (2015), 1512311, http://dx.doi.org/10.11145/j.biomath.2015.12.311 page 9 of 16 http://dx.doi.org/10.11145/j.biomath.2015.12.311 e. t. dougherty, efficient implicit runge-kutta methods for fast-responding ... table ii: accuracy and simulation run-time metrics of the dp5 and irk methods when solving the gabaar model. boldface font denotes best results of each column. method (order) ‖ error ‖2 max |error| time steps run time (s) dp5 (5/4) 252.0 x 10−10 3.2 x 10–10 5.0 x 104 49.0 sdirk (2/1) 45.6 x 10−10 19.6 x 10−10 28 0.21 esdirk (3/2) 18.3 x 10−10 8.8 x 10−10 26 0.27 radau (3/2) 8.7 x 10–10 3.7 x 10−10 29 0.69 esdirk method requires 26 time steps and the sdirk method executes in 0.21 seconds. dp5 solution accuracy can be improved with either stricter error tolerances or a decreased time step [51], however, these approaches will result in even greater run times. the radau method has the greatest execution time of the three irk methods, at 0.69 seconds. while the number of simulation time steps amoung the irk methods are comparable, two factors contribute to the longer run time of the radau method. first, this solver generally requires a greater number of iterations for newton’s method to converge (fig. 3a). second, the radau method requires 30 jacobian computations, versus just four for the sdirk and esdirk methods. 1 2 3 4 5 6 10 −1 10 0 10 1 precision (scd) c p u t im e (s e c s ) sdirk esdirk radau sdirk esdirk radau fig. 7: gabaar work-precision diagram with solver run time vs. scd for each irk method. integer exponential tolerances, i.e. 10−4, 10−5, ..., are presented with enlarged symbols. the symbol for rtol = 10−6 is distinguished by the yellow circle. despite its run time disadvantages amongst the irk methods, the accuracy of the radau method stands out as superior. it has the lowest global error 2-norm (8.7 x 10−10), and its maximal deviation from the true solution (−3.7 x 10−10) is comparable to that of the 5th order dp5 method, the only irk method examined where this is the case. further, the radau method has greater accuracy at every time step than both the sdirk and esdirk methods. these findings are reinforced by the workprecision diagram for the three irk methods when solving the gabaar model (fig. 7). this diagram highlights the higher precisions attained by the third order methods, and in addition, also confirms the slower execution speeds achieved by the radau method. however, when comparing graph points of similar relative tolerances, such as the symbols marked in yellow that represent rtol = 10−6, the radau method is consistently more accurate. b. ampar model figure 8 presents solution results of the ampar model solved with the radau method. like the gabaar model, the rapid transition in the open state concentration upon neurotransmitter stimulation demands a greater number of time steps (fig. 8a). specifically, the first 10% of the simulation (0.1 sec) encompasses approximately 96% of the simulation time steps. once beyond this initial region, step size eventually increases by seven orders of magnitude (fig. 8b). similar to the gabaar model, both unbound closed and bound closed forms contribute to the system’s stiffness. a noticeable difference, compared to the gabaar simulation results, is the number of time biomath 4 (2015), 1512311, http://dx.doi.org/10.11145/j.biomath.2015.12.311 page 10 of 16 http://dx.doi.org/10.11145/j.biomath.2015.12.311 e. t. dougherty, efficient implicit runge-kutta methods for fast-responding ... 0 0.02 0.04 0.06 0.08 0.1 0 5 10 15 20 x 10 −6 time (s) o p e n s ta te c o n ce n tr a tio n ( m ) (a) open state concentration solution, o 0 0.2 0.4 0.6 0.8 1 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 time (s) s te p s iz e ( s) max step size: 1.0 min step size: 4.4e−07 step size, h:199 rejected steps:2 nonconvergent steps:2 (b) simulation step sizes fig. 8: radau method solution of the ampar model. steps needed by the implicit methods to solve the ampar model. the radau method, for example, requires 199 time steps (fig. 8b), a 586% increase from the 29 steps needed to solve the gabaar model. similar increases are observed with the sdirk and esdirk solvers, most notably the 531 steps required by the sdirk method (table iii). in addition, the smallest step sizes of the irk methods are two orders of magnitude lower with the ampar model (fig. 8b), due to the stiffness index of the ampar system [27]. despite the elevated simulation time step counts, each irk method still outperforms the explicit methods (table iii); maximum stable step sizes and simulation time steps for the explicit methods were again computed with their stiffness indices and stability regions [28]. while greater kmax values eliminated nonconvergent newton iterations in the gabaar model, this is not the case with the ampar model. each irk method has two instances where newton’s method did not converge. in addition, the sdirk method has four rejected steps, and the esdirk and radau methods each have two, all occurring at time t = 0. table iv displays accuracy and execution efficiency results for the irk methods. an interesting table iii: simulation time steps results for the erk and irk methods when solving the ampar model. method (order) max step time size (s) steps fe (1) 1.7 x 10−5 5.9 x 104 mid (2) 1.7 x 10−5 5.9 x 104 rk4 (4) 2.4 x 10−5 4.2 x 104 sdirk (2/1) adaptive 531 esdirk (3/2) adaptive 211 radau (3/2) adaptive 199 result is the seemingly uncorrelated relationship between simulation time steps and run time. for example, despite having the lowest number of simulation time steps, the radau method has the longest run time. along these same lines, the radau method has less than 50% of the simulation time steps of the sdirk method, yet no noticeable computational advantage. moreover, the esdirk method has approximately 40% of the sdirk method’s time steps, yet it requires 72% of its runtime. with a comparable number of rejected and non-convergent steps (table v), a culprit for this behavior is the number of jacobian computations biomath 4 (2015), 1512311, http://dx.doi.org/10.11145/j.biomath.2015.12.311 page 11 of 16 http://dx.doi.org/10.11145/j.biomath.2015.12.311 e. t. dougherty, efficient implicit runge-kutta methods for fast-responding ... table iv: accuracy and simulation run-time metrics of the dp5 and irk methods when solving the ampar model. boldface font denotes best results of each column. method (order) ‖ error ‖2 max |error| time steps run time (s) dp5 (5/4) 3.3 x 10−8 2.7 x 10−9 1.1 x 105 32.4 sdirk (2/1) 3.0 x 10−8 2.7 x 10−9 531 1.34 esdirk (3/2) 1.7 x 10−8 2.7 x 10−9 211 0.97 radau (3/2) 1.6 x 10–8 2.7 x 10−9 199 1.38 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 0 10 20 30 40 50 60 70 80 90 time (s) c o u n t sdirk lu factorizations: 85 sdirk jac computations: 24 esdirk lu factorizations: 86 esdirk jac computations: 51 (a) sdirk and esdirk jacobian computations and lu factorizations 0 0.2 0.4 0.6 0.8 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 x 10 −9 time (s) e rr o r dp5 sdirk esdirk radau (b) open state concentration solution error fig. 9: method comparison when solving the ampar model. performed by these solvers. figure 9a displays the jacobian computations and lu factorizations of the sdirk and esdirk methods. each method has a near identical number of lu factorizations, however, the esdirk method requires 51 jacobian computations, which is more than double the 24 performed by the sdirk method. in addition, the radau method requires 162 jacobian computations. therefore, despite having a lower number of simulation time steps, the computational advantages of the esdirk and radau methods are diminished due to this elevated number of jacobian computations. once again, the accuracy and computational performances of the irk methods were compared to the dp5 method (table iv). as observed with the gabaar model, the dp5 method has inferior execution performance, requiring 1.1×105 simulatable v: number of rejected and nonconvergent steps for each irk method when solving the ampar model. model rejected non-convergent sdirk 2 2 esdirk 4 2 radau 2 2 tion time steps and 32.4 seconds for a numerically stable solution, both of which are significantly greater than results attained with the irk methods. all four methods generate the same maximum local error (2.7×10−9), which occurs at t = 0 for all methods. also, differences among the global errors are relatively smaller with the ampar model. the oscillatory nature of the dp5 solution around the true solution (fig. 9b) contributes to its biomath 4 (2015), 1512311, http://dx.doi.org/10.11145/j.biomath.2015.12.311 page 12 of 16 http://dx.doi.org/10.11145/j.biomath.2015.12.311 e. t. dougherty, efficient implicit runge-kutta methods for fast-responding ... global error 2-norm (3.3 × 10−8), which is again larger than those of the three irk methods. the radau method once again has the lowest global error 2-norm (1.6×10−8) of all methods inspected. 0 1 2 3 4 5 6 7 8 9 10 −1 10 0 10 1 10 2 precision (scd) c p u t im e (s e c s ) sdirk esdirk radau fig. 10: ampar work-precision diagram with solver run time vs. scd for each irk method. integer exponential tolerances, i.e. 10−4, 10−5, ..., are presented with enlarged symbols. the symbol for rtol = 10−6 is distinguished by the yellow circle. the work-precision diagram for the ampar model (fig. 10) again confirms the higher precision achieved by the third order esdirk and radau solvers. more noticeable in this graph are the differences in the “slopes” of the curves, where “flatter” curves, i.e. esdirk and radau, have more precision per unit cpu time [30]. for the ampar model, the radau method is slower than the esdirk method at all work-precision tolerances examined, yet at relative tolerances greater than 10−6, the radau method becomes faster than the sdirk method. further, the radau method is generally the most accurate of all three irk methods. c. c++ radau implementation the radau method consistently demonstrates the greatest accuracy of the methods examined, however, its main disadvantage is execution speed. for this reason, we selected the radau method and configured a c++ implementation of it. table vi displays execution times for the previous radau matlab implementation, as well as the new c++ version. as expected, the c++ version is significantly faster. specifically, the gabaar model has a 99.6% decrease in execution time, and the ampar model has a 99.7% decrease in execution time. because the implementation algorithms between the two versions are the same, the c++ version maintains the accuracy of the matlab prototype. table vi: run times (seconds) for the matlab and c++ radau method when solving the gabaar and ampar models. implementation gabaar ampar matlab 0.69 1.38 c++ 2.7 x 10-3 3.5 x 10-3 iv. conclusions computational neurology is a valuable contributor in the diagnosis, treatment, and comprehension of neurological disease. to provide maximal utility to the scientific community, computational simulations should incorporate highlydetailed, neurotransmitter-based neuron models. therefore, large-scale simulations involving populations of neurons will inevitably produce computational challenges. in this paper, we have shown that appropriate numerical solvers with efficient implementation strategies can alleviate computational difficulties. commonly used explicit methods are capable in solving a limited number of fast-responding ligand-gated neuroreceptor models. however, we have shown that poor stability properties make them non-ideal for large-scale applications. rather, by addressing the stiffness possessed by these models, we show that implicit methods are highly advantageous. in particular, we demonstrate that l-stable implicit runge-kutta methods offer superior accuracy and run-time efficiency compared biomath 4 (2015), 1512311, http://dx.doi.org/10.11145/j.biomath.2015.12.311 page 13 of 16 http://dx.doi.org/10.11145/j.biomath.2015.12.311 e. t. dougherty, efficient implicit runge-kutta methods for fast-responding ... to their explicit siblings when solving biologicallybased ampa and gabaa neuroreceptor models. to accelerate solutions, we utilize a range of strategies including embedded error estimators and simplified newton iterations. in addition, we show that optimal execution times are achieved when costly jacobian computations and lu factorizations are minimized. the third order radau irk method demonstrates exceptional local and global accuracy compared to all other explicit and implicit methods examined. in addition, its numerical stability properties yield a relatively low number of simulation time steps and efficient step sizes when solving the ampa and gabaa neuroreceptor models. further, a c++ implementation of the radau solver displays the computational faculty to enable largescale multi-cellular simulations. in future work, we plan to continue our investigation of numerical solvers for neurotransmitter-based neuron models by comparing the irk methods to multi-step methods and exponential integrators. acknowledgment the author is grateful to professor jeff borggaard and professor james turner for useful discussions related to this manuscript, and frank vogel for assistance with the radau c++ code. references [1] a. datta, x. zhou, y. su, l. c. parra, and m. bikson, 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[51] m. caberlin, stiff ordinary and delay differential equations in biological systems, ser. mcgill theses. mcgill university, 2002. biomath 4 (2015), 1512311, http://dx.doi.org/10.11145/j.biomath.2015.12.311 page 16 of 16 http://dx.doi.org/10.1007/3-540-33437-8 http://dx.doi.org/10.1007/3-540-33437-8 http://dx.doi.org/10.1017/cbo9780511995569 http://dx.doi.org/10.1017/cbo9780511995569 http://dx.doi.org/10.1023/b:bitn.0000046811.70614.38 http://dx.doi.org/10.1023/b:bitn.0000046811.70614.38 http://dx.doi.org/10.1134/s0965542508110092 http://dx.doi.org/10.1016/j.apnum.2014.09.003 http://dx.doi.org/10.1016/j.apnum.2014.09.003 http://dx.doi.org/10.1016/s0377-0427(97)00141-6 http://dx.doi.org/10.1016/s0377-0427(97)00141-6 http://dx.doi.org/10.1145/198429.198437 http://dx.doi.org/10.1115/1.4001907 http://dx.doi.org/10.1115/1.4001907 http://dx.doi.org/10.1007/s006070170013 http://dx.doi.org/10.1016/0771-050x(80)90013-3 http://dx.doi.org/10.1016/0771-050x(80)90013-3 http://www.dm.unipi.it/∼testset/testsetivpsolvers/ http://www.dm.unipi.it/∼testset/testsetivpsolvers/ http://dx.doi.org/10.11145/j.biomath.2015.12.311 introduction materials and methods neuroreceptor models ampa gaba stiff ordinary differential equations implicit runge-kutta methods implementation simulations results and discussion gabaar model ampar model c++ radau implementation conclusions references original article biomath 3 (2014), 1407191, 1–18 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum a generic modeling of fire impact in a tree-grass savanna model a. tchuinté tamen∗, j. j. tewa∗, p. couteron†, s. bowong∗, y. dumont‡ ∗umi 209 ummisco, grimcape, yaounde, cameroon email: tamenalexis@yahoo.fr; tewajules@gmail.com; sbowong@gmail.com †ird, umr amap, montpellier, france email: pierre.couteron@ird.fr ‡cirad, umr amap, montpellier, france email: yves.dumont@cirad.fr received: 12 november 2013, accepted: 19 july 2014, published: 28 august 2014 abstract—we propose and study a model for treegrass interactions in the context of savannas which are subjected to fire pressure. several theoretical models in the literature which have highlighted the impact of fire on tree-grass interactions did not explicitly deal with the indirect feedback of dry grass biomass onto tree dynamics through fire intensity and frequency. the novelty in our work is to consider a fairly generic modeling of fire impact on woody biomass by means of a family of increasing and bounded functions of grass biomass. the characteristic feature of this family of functions is that, it could include several forms: linear as well as non-linear ones (sigmoidal or not). since the nonlinear shape brings more diverse results than the previous attempts using a linear function, it could be used to show that several vegetation equilibria exist with some of them showing tree-grass coexistence features. we show that the number of equilibria with both grass and trees depends on the choice of the fire impact function. we also established thresholds defining the stability domains of the equilibria and highlighted some bifurcation parameters to provide numerical simulations complying with the theoretical properties of the model. keywords-savanna modeling; tree-grass interactions; stability; nonstandard finite difference method; bifurcation. i. introduction savannas are complex ecosystems mixing trees and grasses to create physiognomies that are neither grassland nor forest [44]. savannas occur in areas where the mean annual temperature is higher than 170c and where mean annual rainfall is between 250 and 2100 mm [60]. for instance, the mean annual rainfall is between 1350 and 1400 mm [48], [61], for the regions between mbam and sanaga in cameroon, where the so-called ”soudano-guinean” savannas dominate [28]. africa contains by far the largest area of savannas, with as much as 15.1 million km2, or 50% of the continent surface [15], [31]. in central africa, savannas spread across northern cameroon, southern chad and the central african republic. furthermore, a large share of these central african savannas fringes extensive areas of moist tropical forests, as do littoral savannas which citation: a. tchuinté tamen, j. j. tewa, p. couteron, s. bowong, ,y. dumont, a generic modeling of fire impact in a tree-grass savanna model, biomath 3 (2014), 1407191, http://dx.doi.org/10.11145/j.biomath.2014.07.191 page 1 of 18 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2014.07.191 a. tchuinté tamen et al., a generic modeling of fire impact in a tree-grass savanna model... spread from the plains of gabon to dcr and angola [19], or inside the continent, the nairi and bateke savannas which extend soudano-zambezian vegetation from angola up to gabon [27]. all these savannas observed under humid climates are rather prone to fires which tend to counteract natural reforestation. most often at times, because savannas are frequently burned and occupy wide areas, their management may influence the regional and possibly global energy, water and carbon balances [40], [41]. during the last decades, various explanations of the long-lasting coexistence of trees and grasses mixtures in savannas have been proposed. some of them invoke limiting resources (soil moisture or nutrients) and possible niche separation of rooting zones that could result in differential access to limiting resources between tree and grass [26], [55], [57], [58]. other explanations emphasized the role of disturbance regimes in preventing trees to reach canopy closure [2], [23], [44]. [38] showed that limiting water resource is probably pervasive in the driest part of the rainfall gradient while disturbances (fire, herbivory) are probably central to tree-grass coexistence under wetter climates. among these disturbances, fire is recognized as quintessential since it is able to suppress young trees and shrubs that lay within the flame zone thereby preventing them to reach maturity [11], [37], [38] and depress grass biomass by shading [29], [43]. in savanna environment, fire intensity is tightly linked to the dried grass biomass that remains during the dry season. large trees having crowns above the flame zone (say 2 m high) are rarely affected by fires, but recurrent fires prevent a great number of these trees from escaping the flame zone, and tree biomass becomes sufficiently low to have no depressure effect on grass biomass production. this feedback loop between grass production and fire intensity is a key pattern of savanna ecosystems that are observed under sufficiently wet climates [47]. since tree-grass coexistence involves complex retroactions which are moreover contextdependent, modeling has to play an important role to understand dynamical processes that shape savanna vegetation. several recent modeling efforts have built on systems of ordinary differential equations expressing the asymmetric competition between tree and grass. in this line of research initiated by [52], the superior competitor (mature tree) is the one that always displaces the inferior competitor (grass) when they both occur in a site, and the inferior competitor can neither invade nor displace the superior competitor from a site. however, the coexistence of trees and grass and to a lesser extent the dominance of grass over trees can result from the suppression of tree seedlings by grass competition in a way either direct or fire-mediated. indeed, a mathematical analysis of tilman’s and derived models shows that it is possible to have a globally asymptotic stable (gas) tree-grass equilibrium. consequently, we found it desirable to consider modeling options of fire as follows: (i) split fire frequency from fire intensity. since fire intensity increases principally with grass abundance [47], while fire frequency also depends on management choices. (ii) use distinct functions of fire impact on grass vs. woody biomass, since it is observed that, grasses and trees are affected differently. (iii) allow for various conditions of bistability in order to render the diversity of physiognomies that are observed in the field. we will show that this last condition may be ensured by incorporating a fairly generic modeling of the impact of fire on woody biomass, by means of a family of functions which could take linear or non-linear shapes. we will also establish that the number of equilibria with tree-grass coexistence depends on the characteristics of the function. our aim in the present paper is therefore to propose a new tree-grass model following the above objectives and achieve a complete theoretical analysis of this model. we shall highlight three thresholds that summarize the dynamics of the system. we shall equally illustrate the theoretical results through numerical simulations, obtained with an appropriate nonstandard finite difference scheme biomath 3 (2014), 1407191, http://dx.doi.org/10.11145/j.biomath.2014.07.191 page 2 of 18 http://dx.doi.org/10.11145/j.biomath.2014.07.191 a. tchuinté tamen et al., a generic modeling of fire impact in a tree-grass savanna model... complying with the theoretical properties of the model. we will also highlight some bifurcation parameters. ii. the mathematical model consider the following simple model of treegrass dynamics taking into account fire as continuous events:  dg dt = (γg −δg0)g−µgg2 −γtgtg−λfgfg, dt dt = (γt − δt )t −µtt2 −λftfω(g)t, (1) with, t(0) = t0 and g(0) = g0 positive initial conditions, t and g are tree and grass biomasses (t.ha−1) respectively, γt and γg are the tree and grass biomass productivity (yr−1). in our model, nutrients and water are modelled implicitly via competition and production terms of vegetation. δt and δg0 are the biomass loss by respiration and natural death, µt and µg are the additional death due to the intra-specific competition, γtg is the mortality due to tree-grass competition, f = 1 τ is the fire frequency with τ the period between two consecutive fires, λft and λfg represent the specific loss of tree and grass biomasses due to fire, and ω(g) is the function of grass biomass which expresses the causality between grass biomass and fire intensity and models the impact of fire on the woody biomass. for convenience, and with limited loss of generality, we make assumptions about ω(g). other than smoothness it satisfies these following three conditions: • ω(0) = 0, • ω(g) > 0, and ω ′ (g) > 0, • lim g→∞ ω(g) < ∞. a. some qualitative results of the system (1) to introduce this section, we first state, with proof, the existence and uniqueness lemma. lemma ii.1. with the initial conditions t(0) = t0 and g(0) = g0, system (1) has a unique maximal solution. proof: let us set x = (g,t)t ∈ r2+, f(x) = (f1(x),f2(x)) t ∈ r2, and   f1(x) = (γg −δg0)g−µgg2 −γtgtg−λfgfg, f2(x) = (γt −δt )t −µtt2 −λftfω(g)t. (2) system (1) becomes dx dt = f(x), (3) with x(0) = (g0,t0)t . f is a continuously differentiable map (c1), because f1 and f2 are c1. then, by the cauchylipschitz theorem, system (1) with the initial condition x(0) = x0 admits a unique maximal solution. models require that trajectories remain positive and that trajectories do not tend to infinity with increasing time. if the set s is such that all trajectories that start in s remain in s for all positive time, then s is said to be ”positively invariant”. (if trajectories remain in s for both positive and negative time, s is said to be invariant). hence, the basic condition for positivity (of dependent variables) can be stated as ”the positive cone is positively invariant for the dynamical system generated by system (1)”. lemma ii.2. the positive cone r2+ is positively invariant for the system (1). proof: g = 0, and t = 0 are vertical and horizontal null clines respectively. then, no trajectory can cut these axes. thus, r2+ is positively invariant for (1) because, all trajectories that start in r2+ remain in r 2 + for all positive time. the dynamical system is said to be ”dissipative” if all positive trajectories eventually lie in a bounded set. this is sufficient to ensure that all solutions of system (1) exist for all positive times. lemma ii.3. the compact γ, given by{ (g,t)t ∈ r2+/t ≤ γt −δt µt ,g ≤ γg −δg0 −λfgf µg } is attracting for system (1) (e.g., all trajectories of system (1) that reach the neighbourhood of γ biomath 3 (2014), 1407191, http://dx.doi.org/10.11145/j.biomath.2014.07.191 page 3 of 18 http://dx.doi.org/10.11145/j.biomath.2014.07.191 a. tchuinté tamen et al., a generic modeling of fire impact in a tree-grass savanna model... converge inside for all positive time). γ is call the ω-limit set. proof: from system (1), with initial conditions t(0) = t0 > 0 and g(0) = g0 > 0, we have  dg dt ≤ (γg −δg0 −λfgf)g−µgg2, dt dt ≤ (γt −δt )t −µtt2, g(0) = g0, t(0) = t0. (4) using the maximum principle, we deduce that   g(t) ≤ g0 g0 ge + (1 − g0 ge )exp{−geµgt} , t(t) ≤ t0 t0 te + (1 − t0 te )exp{−teµtt} , (5) where, ge = γg −δg0 −λfgf µg and te = γt −δt µt . we obtain,   lim t→+∞ g(t) ≤ ge = γg −δg0 −λfgf µg , lim t→+∞ t(t) ≤ te = γt −δt µt . (6) then, all trajectories of system (1) that reach the neighbourhood of γ converge inside as t tends to infinity. thus γ is attracting for the system (1). let ψ : r2 × r → r2 be a function of two variables, such as ψ(y,t) = x(t), where x(t) is the solution of system (3) satisfying the initial condition x(0)=y. definition ii.1. in the terminology of dynamical systems, a steady state or an equilibrium point of system (3) is an element p ∈ r2+ such that ψ(p,t) = p for all t ∈ r. similarly, a periodic orbit is one that satisfies ψ(p,t + t) = p for all t and for some fixed number t. the corresponding solution of system (3) will be a periodic function. it is observed that, system (1) admits these following nonnegative equilibria, • (0; 0) corresponding to bare soil (always unstable), • ete = (0; te) = ( 0; γt −δt µt ) , i.e., a wooded savanna equilibrium, • ege = (ge; 0) = ( γg −δg0 −λfgf µg ; 0 ) , i.e., a grassland savanna, • e∗ = (g∗; t∗), i.e., the tree-grass coexistence equilibrium. in biological systems, eventual behaviours and asymptotic properties of trajectories need to be determined. it is also important to know when limit cycle (periodic solution) occurs or not. lemma ii.4. there is no limit cycle for system (1) in the positive cone r2+. proof: let (g,t)t ∈ r2+, and β(t,g) = 1 tg . we have, βf1 = (γg −δg0 −λfgf) −µgg−γtgt t βf2 = (γt −δt ) −µtt −λftfω(g) g . then, ∂βf1 ∂g + ∂βf2 ∂t = − µg t − µt g < 0. using dulac criterion, we conclude that, system (1) has no periodic solution in r2+. concerning the stability thresholds of equilibria, the following proposition holds. proposition ii.1. the stability of the equilibria depends on the following thresholds: biomath 3 (2014), 1407191, http://dx.doi.org/10.11145/j.biomath.2014.07.191 page 4 of 18 http://dx.doi.org/10.11145/j.biomath.2014.07.191 a. tchuinté tamen et al., a generic modeling of fire impact in a tree-grass savanna model... • r10 = γt −δt λftf 1 ω(ge) , related to the savanna vs. grassland equilibrium. it is an increasing function of tree biomass. it represents the net production of tree biomass relative to the fire-induced biomass loss at the grassland equilibrium. • r01 = γg −δg0 −λfgf γtg µt γt −δt , related to the savanna vs. forest equilibrium. it represents the net primary production of grasses after fire, relative to the grass production loss due to the tree biomass at the wooded savanna equilibrium. • r∗11 = r10r01 ω(ge) ω ′ (g∗)ge , related to the mixed tree-grass equilibrium. proof: see appendix a. remark ii.1. there is an obvious relation between the last threshold and the two previous ones. these thresholds are positive considering the reasonable ecological parameters. 1) model without fire : when fire frequency f = 0 (tilman’s model), all equilibria and their related stability properties are summarized in table i. table i: stability/instability results for tilman’s model threshold conditions ecological stable unstable equilibria equilibria equilibria < 1 (0; 0), ete , ete (0; 0), and ege (gas) ege r01 > 1 (0; 0), ege e ∗ (0; 0), ete , and e ∗ (gas) ege , ete table i means that, without fire, when the threshold r01 < 1, trees grow toward their carrying capacity ete , while it is not the case for grass. this can be explained by the fact that, tree canopy reduces light availability which is necessary for grass growth. then, when the shading effect is higher, only trees can persist. this is supported by studies demonstrating that competition effects of grass are not strong enough to prevent rapid recruitment of trees into savannas when fire is excluded [23], [44]. on the other hand, when, r01 > 1, the tree-grass node exists and is gas, while ete becomes unstable. then, the tilman’s two-species model exhibits the transcritical bifurcation. these results join those in [17]. thus, if trees and grass in savanna are not inflammable, there is only one possible bifurcation namely a transcritical bifurcation. 2) model with continuous fire forcing : we now investigate system (1) with f 6= 0 in order to determine whether multiple stable states exist and analyze how the system can veer off from a stable equilibrium to another depending on some thresholds parameters. ? let us consider ω(g) = g as assumed in [55]. we summarize all the results in the table ii bellow table ii: stability/instability results when ω(g) = g. ecological stable unstable conditions equilibria equilibria equilibria (0; 0) (0; 0), r01 > 1 r∗11 > 1 ete , ege e ∗ (gas) ege , and e∗ and ete r10 > 1 (0; 0) (0; 0), r01 < 1 ege , ete ete (gas) and ege (0; 0) (0; 0), r01 > 1 ege , ete ege (gas) and ete r10 < 1 (0; 0) bi-stability (0; 0) r01 < 1 r∗11 < 1 ete , ege ege (las) and and e∗ ete (las) e ∗ table ii exhibits two different bifurcations. the first one is the same as in table i, say the transcritical bifurcation. more explicitly, when r01 < 1, the forest equilibrium ete is gas, and becomes unstable when r01 > 1 while, the coexistence tree-grass equilibrium which is gas exists. the second bifurcation is the pitchfork bifurcation. this previous bifurcation occurs because a unique coexisting equilibrium which is gas exists when r∗11 > 1, and becomes unstable when r ∗ 11 < 1 while, woodland and grassland equilibria are stable. the bistability between woodland and grassland equilibria corroborates with the theoretical results in [18]. however, the model without fire do not present this bistability. then, fires, turn in favour of grass by damaging young trees and shrubs. these negative feedbacks of grass on trees through fires were considered by [44] which modelled the impact of fires on trees by differentiating tree compartment in two ways: sensitive trees (like biomath 3 (2014), 1407191, http://dx.doi.org/10.11145/j.biomath.2014.07.191 page 5 of 18 http://dx.doi.org/10.11145/j.biomath.2014.07.191 a. tchuinté tamen et al., a generic modeling of fire impact in a tree-grass savanna model... fig. 1: different equilibria and their stability/instability properties when r01 > 1 and ω(ge)r10 > 1. solid circles indicate a stable equilibrium, and open circles indicate an unstable equilibrium. there are four equilibria. only coexistence equilibrium is gas (see tables i and ii). shrubs) and non sensitive trees (mature trees). as a result, fire promotes tree-grass coexistence and the occurrence of bistability of woodland and grassland. let us recall that in our study ω(g) is a generic function. our interest here is to explore to what extent the shape of ω(g) may favour multiple stable states and thereby explain the tendency of savannalike ecosystems to shift among different stable states. according to the choice of the response function ω(g), we obtain several configurations. for instance, as assumed in [2], ω(g) = gβ gβ + αβ , where α controls the location of the point where ω is half of its maximum value and β controls the rate of increase of ω. the particular cases of this function are the holling functions type ii and iii which are also referred to the michaelis-menten function and the sigmoidal response function respectively. here, to show various implications and configurations due to the choice of ω, we have considered these previous two particular cases. the ecological models are developed and their mathematical properties are analyzed. to simplify the calculations, we set g0 = αβ. ?? first, let us consider the holling type ii, and then six different configurations can arise depending on certain thresholds. (a) (b) fig. 2: equilibria and their stability/instability properties when r01 < 1 and ω(ge)r10 > 1. in (a), there is no coexistence and only wooded savanna equilibrium is gas as in table i and table ii. in (b), there are two internal equilibria: e∗1 (unstable) and e ∗ 2 stable. a separatrix divides the plane into two basins of attraction: one to the stable woodland equilibrium; one to e∗2 (bistability). fig. 3: different equilibria and their stability/instability properties when r01 > 1 and ω(ge)r10 < 1. same as in fig 1, there are four equilibria and only coexistence equilibrium is gas. our analysis suggests that, using the holling type ii in tilman’s two-species model provides richer qualitative behaviour than the linear form. for instance two different coexistence equilibria can exist and bistability can occur to one of them with stable forest equilibrium. this result joins those of [1], [10], [22], [23] which modelled biomath 3 (2014), 1407191, http://dx.doi.org/10.11145/j.biomath.2014.07.191 page 6 of 18 http://dx.doi.org/10.11145/j.biomath.2014.07.191 a. tchuinté tamen et al., a generic modeling of fire impact in a tree-grass savanna model... (a) (b) fig. 4: different equilibria and their stability/instability properties when r01 < 1 and ω(ge)r10 < 1. (a) presents two basins of attraction divide by a separatrix: one to the stable grassland equilibrium; one to the stable woodland equilibrium (bistability as in table ii). (b) is the same as (b) in fig 2 except with ω(ge)r10 < 1. savanna and forest as alternative stable states. thus, trajectories of the model can evolve either to the woodland equilibrium or the coexistence equilibrium, depending on initial conditions. ? ? ? now, we consider the holling type iii. here we have seven configurations depending on certain thresholds. fig 7 is obtained when thresholds r01 and ω(ge)r10 are greater than 1. the first one r01 > 1 indicates that the net primary production of grasses after fire, relative to the grass production loss due to the tree biomass at the wooded savanna equilibrium. this competition turns in favour of trees (see fig 6-(a) above). it leads to the existence of the upper coexistence equilibrium e∗1 with higher tree biomass and lower grass biomass. this equilibrium represents the longlasting coexistence of trees and grass due to the inter-specific competition between grass and trees. (a) (b) fig. 5: equilibria and their properties when r01 < 1 and ω(ge)r10 > 1. same as in fig 2, in (a), there is no coexistence and only woodland equilibrium is gas. in (b), there are two feasible internal equilibria. a separatrix divides the plane into two basins of attraction; one to the stable woodland equilibrium; and one to the lower internal equilibrium (bistability). it is obvious that the threshold ω(ge)r10 is linked to r10 which represents the net production of tree biomass relative to the fire-induced biomass loss at the grassland equilibrium. when ω(ge)r10 > 1, the lower coexistence equilibrium e∗2 exists with lower tree biomass and higher grass biomass (see fig 5-(b) above). e∗2 represents the tree-grass coexistence due to the indirect feedback of grass on tree biomass through fire. our interest in this section, was to show various configurations and implications due to the choice of ω. in this case, we have considered two particular functions for w(g): holling type ii and type iii. following [47], we showed that these sigmoidal forms of ω(g) make multiple stable equilibria possible. there are between 0 and 3 internal equilibria and between 0 and 2 stable internal equilibria. in addition, we highlighted two specific thresholds r01 and ω(ge)r10 which regulate different treebiomath 3 (2014), 1407191, http://dx.doi.org/10.11145/j.biomath.2014.07.191 page 7 of 18 http://dx.doi.org/10.11145/j.biomath.2014.07.191 a. tchuinté tamen et al., a generic modeling of fire impact in a tree-grass savanna model... (a) (b) fig. 6: different equilibria and their stability/instability properties when r01 > 1 and ω(ge)r10 < 1. opposite to fig 3, we have two situations. (a) shows two feasible internal equilibria, the lower equilibrium is unstable and the upper is stable. it presents the bistability between the upper coexistence equilibrium and the grassland equilibrium. we could not observe this situation previously, when we used the holling type ii or a linear form of ω. other point is that in (b), grassland is gas. note that this is not the case with holling type ii. grass patterns. however, recall that in this work our function ω(g) is a generic one and therefore could take a linear or non-linear form. concerning the above sigmoidal functions, the results indicate that holling response type iii qualitatively allows richer behaviours for the tilman’s model. to perform our simulations, some parameter values were based on the literature and others from the ecological plausible domains such that they obey the reality. the parameter values are summarized in table iii. iii. a nonstandard algorithm in order to keep all qualitative properties of our model, we design a nonstandard finite difference (nsfd) schemes ([3], [7], [33], [35]). fig. 7: stability/instability properties of equilibria when r01 > 1 and ω(ge)r10 > 1. stable equilibria are shown with solid circles; unstable equilibria are shown with open circles. opposite to fig 1, there are three internal equilibria where the null clines meet. the intermediate equilibrium point is unstable and the lower and upper equilibria are stable. a separatix which divides the plane into two basins of attraction passes through the unstable equilibrium. depending on the initial condition, trajectories will evolve either to the left or right (bistability). table iii: parameter values units values references f yr−1 0 − 1 [55] γg yr −1 0.4(1) − 4.6(2) (1) [35] (2) [32] γt yr −1 0.456 − 7.2 [14] µg ha.t −1.yr−1 0.1 assumed µt ha.t −1.yr−1 0.3 assumed δt yr −1 0.03 − 0.3 [1] δg0 yr −1 0.1 [55] λfg yr −1 0.1(∗) − 1(∗∗) (∗) [54] (∗∗) [1] λft yr −1 0.005(1∗) − 1(2∗) (1∗) [23] (2∗) assumed γt g ha.t −1.yr−1 0.19 assumed α t.ha−1 0.54 − 1.73 assumed recent works have shown that nsfd schemes are appropriate to simulate various compartmental models in epidemiology ([5], [6], [21]) and in ecology ([4], [20], [59]). these schemes are able to preserve important properties, like global asymptotic stability of equilibria, backward bifurcation, dissipativity properties, invariant sets, etc. for the numerical approximation of the model (1), we replace the continuous time variable t ∈ [0,∞) by discrete nodes tn = nh, n ∈ n where h = ∆t > 0 is the step size. we wish to find approximate solutions gn and tn at the time t = biomath 3 (2014), 1407191, http://dx.doi.org/10.11145/j.biomath.2014.07.191 page 8 of 18 http://dx.doi.org/10.11145/j.biomath.2014.07.191 a. tchuinté tamen et al., a generic modeling of fire impact in a tree-grass savanna model... (a) (b) fig. 8: stability/instability of equilibria when r01 < 1 and ω(ge)r10 < 1. like in fig 4: (a) shows two stable equilibria: the stable grassland equilibrium and the stable woodland equilibrium (bistability). fig 6-(b) is different from fig 4-(b). here, tree and grass coexistence does not occur; the woodland equilibrium is gas. then, the stable internal equilibrium disappears when ω(ge)r10 < 1. thus trees dominate the vegetation. this situation does not occur when we use holling type ii. tn. the standard denominator h in each discrete derivative is replaced by a time-step function 0 < φ(h) < 1, such that φ(h) = h + o(h2). a possible simple nsfd scheme of model (1) reads as,   gn+1 −gn φ(h) = γgg n −δg0gn+1 −µggngn+1 −γtgtngn+1 −λfgfgn+1, tn+1 −tn φ(h) = γtt n −δttn+1 −µttntn+1 −λftfω(gn)tn+1. (7) let xn = (gn,tn)t , be an approximation of x(tn) = (g(tn),t(tn)) t . a nonstandard matrix form of (7) is given by xn+1 = a(xn)xn, (8) where a(xn) is a diagonal matrice with diagonal terms defined as follows: a11 = 1 + φ(h)γg 1 + φ(h) (δg0 + λfgf + µgg n + γtgtn) , and, a22 = 1 + φ(h)γt 1 + φ(h) (δt + λftfω(g n) + µttn) . it is obvious that a(xn) is nonnegative. thus, xn ≥ 0 ⇐⇒ xn+1 ≥ 0, ∀n ∈ n. (9) definition iii.1. [7] a numerical scheme is called elementary stable whenever it has no other fixed points than those of the continuous system it approximates, the local stability of these fixed points is the same for both the discrete and the continuous dynamical systems for each value of h. let us set x∗ = (g∗; t∗)t , an equilibrium of the continuous model (1) and assume that x∗ is hyperbolic. lemma iii.1. the numerical scheme (7) and the continuous system (1) have the same equilibria. the proof of lemma iii.1 is provided in appendix b. let a function ϕ : r −→ r, satisfy  ϕ(z) = z + o(z2), 0 < ϕ(z) < 1, (10) for z > 0. the denominator function that is needed in (7) can be taken to be φ(h) = ϕ(qh) q , (11) where q is any number which can capture the dynamics of the model (1). q satisfies q ≥ max { |λ|2 2|reλ| } , (12) biomath 3 (2014), 1407191, http://dx.doi.org/10.11145/j.biomath.2014.07.191 page 9 of 18 http://dx.doi.org/10.11145/j.biomath.2014.07.191 a. tchuinté tamen et al., a generic modeling of fire impact in a tree-grass savanna model... where λ denotes an eigenvalue of j∗ ≡ j(x∗) which is the jacobian matrix of the right hand side of system (1) at x∗. we have following results. lemma iii.2. the jacobian matrix j∗ is diagonalisable. see appendix c for the proof of lemma iii.2. theorem iii.1. the nsfd scheme (7) is elementary stable whenever φ(h) is chosen according to (11) and (12). the proof of theorem iii.1 is done in appendix d. according to theorem iii.1, the continuous and discrete systems (1) and (7) have the same dynamics, at least locally. we now provide some numerical simulations in order to highlight some bifurcation parameters. iv. numerical simulations and discussion using the previous scheme, we will show that some parameters are bifurcation parameters. 3) bifurcation due to fire period τ: in arid and semi-arid savannas, frequent fire pressure influences significantly the balances between tree and grass [45]. moreover, fire is considered as a major determinant of the ecology and distribution of africa’s savanna and grassland vegetation types [12], [23], [61]. to understand the effects due to fire period τ = 1 f , it is helpful to plot some curves. by contrast to the explanations of [9], which show that nonlinear ecosystem dynamics lead to bistable ecological communities that can exist in either a grassland or forest state under the same disturbance frequency, our results suggest that adding a nonlinear fire impact on trees leads to the bistability of two tree-grass coexistence equilibria under certain ecological thresholds. fig. 9 above illustrates this situation. we can observe how, the system can rapidly move between two coexistence equilibria depending upon the starting conditions. however, in response to gradual changes in fire regimes, the ecosystem globally changes. further, at ecological thresholds, small shifts in fire regime can lead to disproportionate 0 0.2 0.98 1.88 2 2.6 2.75 3 3.64 3.22 4 λ ft =0.8, λ fg =0.6, τ=3, γ tg =0.19 g−nullcline t−nullcline e 2 * (unstable) e 3 * (las) e 1 * (las) (a) 0 20 40 60 80 100 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 time(year) t. h a − 1 λ ft =0.8, λ fg =0.6, τ=3, γ tg =0.19 grass biomass tree biomass (b) 0 20 40 60 80 100 0 0.5 1 1.5 2 2.5 3 3.5 4 time(year) t. h a − 1 λ ft =0.8, λ fg =0.6, τ=3, γ tg =0.19 grass biomass tree biomass (c) fig. 9: (a) phase portrait with λft = 0.8, λfg = 0.6, γt g = 0.19, f = 1 τ , and τ = 3. three internal equilibria: e∗1 = (0.2; 3.64), e ∗ 2 = (0.98; 3.22), and e ∗ 3 = (1.88; 2.75). the intermediate point is unstable and the lower and upper points are stable. the trajectory of the system will eventually bifurcate to e∗1 or e∗3 depending on its initial condition. (b) and (c) show respectively the local stability of e∗1 and e ∗ 3 . changes in ecosystems and ecological surprises or sudden changes in state [42]. for example, for one fire every two years, eventually the configuration will change. see figures 10 and 11 below. figures 11-(b) and 11-(c) show that standard methods are not suitably designed for some complex problems. in figure 11-(b), we have used the ode45 routine in matlab: the result is not nice because the grass biomass becomes negative. the positivity of the solutions and their boundedness are not preserved. conversely, figure 11-(c) obtained with the nonstandard scheme preserves all the previous qualitative properties; particularly the global asymptotical stability of the woodland vegetation ete . biomath 3 (2014), 1407191, http://dx.doi.org/10.11145/j.biomath.2014.07.191 page 10 of 18 http://dx.doi.org/10.11145/j.biomath.2014.07.191 a. tchuinté tamen et al., a generic modeling of fire impact in a tree-grass savanna model... 0 1.17 1,695 2 2 2.32 2,6 3.64 λ ft =0.8, λ fg =0.6, τ=2, γ tg =0.19 g−nullcline t−nullcline e 2 * (unstable) e te (las) e 3 * (las) (a) 0 20 40 60 80 100 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time(year) t. h a − 1 λ ft =0.8, λ fg =0.6, τ=2, γ tg =0.19 grass biomass tree biomass (b) 0 20 40 60 80 100 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 time(year) t. h a − 1 λ ft =0.8, λ fg =0.6, τ=2, γ tg =0.19 grass biomass tree biomass (c) fig. 10: the phase plane (a) shows two domains of attraction. the system may move from one domain to another. then, it can be flipped from a stable state ete = (0; 3.64) (figure (b)) to e ∗ 3 = (1.695; 2.32) (figure (c)) by crossing the unstable intermediate point e∗2 . here τ = 2. the effect of an increase in f = 1 τ is to shift the g-null cline down and to the left. hence, the point e∗1 will approach ete along the t-null cline. because the separatrix passes through the point e∗2 , the domain of attraction of e ∗ 1 must shrink, whereas the domain of attraction of e∗3 will expand. then, for higher value of f, the points e∗1 and ete coincide and change their stability. ete becomes stable and e ∗ 1 goes out of domain (fig 9 → 10). when we still increase f, there is only one stable equilibrium for the system ete (fig 10 → 11). all trajectories approach that point. 4) bifurcation due to λfg: the specific loss of grass λfg has an important impact in tree-grass interaction (figure 12 → 13). in the region of bistability, the system converged either to a completely herbaceous state (grassland) or to a woody 0 2 0.5 3.64 4 phase plane g−nullcline t−nullcline e te (gas) (a) 0 20 40 60 80 100 −1 0 1 2 3 4 5 time(year) t. h a − 1 (matlab routine ode45) grass biomass tree biomass (b) 0 20 40 60 80 100 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time(year) t. h a − 1 (nonstandard scheme) grass biomass tree biomass (c) fig. 11: (a): when τ = 1, there is no tree-grass equilibria. the equilibrium ete = (0; 3.64) is globally asymptotically stable. these figures are done for λft = 0.8, λfg = 0.6, τ = 1 and γt g = 0.19. simulations are done with (b) the standard ode45 algorithm; (c) the nonstandard algorithm equilibrium (forest), depending on the initial values of vegetation. in the region of bistability, a coexistence equilibrium which is unstable exists. our results joint results of [8]. increase λfg must shrink the domain of attraction of ege , whereas the domain of attraction of ete will expand. for a higher value of λfg, the points ege and e ∗ 2 coincide and change their stability. ege becomes unstable and there is no coexistence equilibria (figure 12 → 13). then, for larger values of the grass extinction rate, trees become favoured [8]. 5) bifurcation due to γtg: trees exert a competitive pressure on grass via water [57]. figure 14 → 15 illustrate the bifurcation due to γtg. in figure 14, there are two internal equilibria : e∗1 is stable and e ∗ 2 is unstable. the effect biomath 3 (2014), 1407191, http://dx.doi.org/10.11145/j.biomath.2014.07.191 page 11 of 18 http://dx.doi.org/10.11145/j.biomath.2014.07.191 a. tchuinté tamen et al., a generic modeling of fire impact in a tree-grass savanna model... 0 0.976 3.1 4 −0,5 0 0,5 1.12 2,2 2,5 λ ft =0.7, λ fg =0.6, τ=1, γ tg =0.19 g−nullcline t−nullcline e 2 * (unstable) e te (las) e ge (las) (a) 0 20 40 60 80 100 0 0.5 1 1.5 2 2.5 3 3.5 time(year) t. h a − 1 λ ft =0.7, λ fg =0.6, τ=1, γ tg =0.19 grass biomass tree biomass (b) 0 20 40 60 80 100 0 0.5 1 1.5 2 2.5 3 3.5 time(year) t. h a − 1 λ ft =0.7, λ fg =0.6, τ=1, γ tg =0.19 grass biomass tree biomass (c) fig. 12: (a) shows two basins of attraction: the separatrix which passes through the unstable internal equilibrium e∗2 = (0.976; 1.12) separates the two domains. trajectories to the right of the separatrix eventually reach ege = (3.1; 0) and those to the left eventually reach ete = (0; 2.2). (b) stable woody vegetation; (c) stable grassland. 0 4 −2 −1 0 2.2 λ ft =0.7, λ fg =0.8, τ=1, γ tg =0.19 g−nullcline t−nullcline e te (gas) (a) 0 20 40 60 80 100 0 0.5 1 1.5 2 2.5 3 time(year) t. h a − 1 λ ft =0.7, λ fg =0.8, τ=1, γ tg =0.19 grass biomass tree biomass (b) fig. 13: global stability of ete = (0; 2.2). of a decrease in γtg is to shift de g-null cline up and to the right. the points e∗1 and e ∗ 2 will approach each other along the t-null cline and coincide. because the separatrix passes through the point e∗2 , the domain of attraction of e ∗ 1 must shrink, whereas the domain of attraction of ege will expand. for a still lower value of γtg, there is 0 0.25 0.84 3.1 4 −1 −0.5 0 0.5 1.73 2.2 2.5 λ ft =0.7, λ fg =0.6, τ=1, γ tg =0.13 g−nullcline t−nullcline e 2 * (unstable) e ge (las) e * 1 (las) (a) 0 20 40 60 80 100 0 0.5 1 1.5 2 2.5 3 time(year) t. h a − 1 λ ft =0.7, λ fg =0.6, τ=1, γ tg =0.13 grass biomass tree biomass (b) 0 20 40 60 80 100 0 0.5 1 1.5 2 2.5 3 3.5 time(year) t. h a − 1 λ ft =0.7, λ fg =0.6, τ=1, γ tg =0.13 grass biomass tree biomass (c) fig. 14: (a) bistability between e∗1 = (0.84; 2.2) and ege = (3.1; 0). up to the initial conditions, the system may reach: (b) e∗1 , or (c) ege 0 3.1 4 −1 0.0 3,5 λ ft =0.7, λ fg =0.6, τ=1, γ tg =0.1 g−nullcline t−nullcline e ge (gas) (a) 0 20 40 60 80 100 0 0.5 1 1.5 2 2.5 3 3.5 time(year) t. h a − 1 λ ft =0.7, λ fg =0.6, τ=1, γ tg =0.1 grass biomass tree biomass (b) fig. 15: (a) and (b) show that equilibrium ege = (3.1; 0) is gas. no internal equilibrium and ege is globally stable (see fig. 15). v. conclusion it is well-known that fires shape the tree-grass mixture in savanna-like vegetations as soon as rainfall allows sufficient grass-biomass production biomath 3 (2014), 1407191, http://dx.doi.org/10.11145/j.biomath.2014.07.191 page 12 of 18 http://dx.doi.org/10.11145/j.biomath.2014.07.191 a. tchuinté tamen et al., a generic modeling of fire impact in a tree-grass savanna model... (for mean annual rainfall above 600-700 mm). in those ecosystems where rainfall is sufficient to let woodlands and even closed canopy forests develop and perpetuate, fire is indeed the main factor enabling a long-lasting coexistence of tree and grass components (and even the existence of grassland) in spite of climate conditions favourable to forest. simple tree-grass competition models have proven their ability to render the main qualitative behaviour of wet savanna systems with multiple equilibria (i.e. grassland, forest and savanna; [1], [36], [55]). but previous models can be questioned with respect to the way in which the crucial fire factor is modeled. here we have proposed a new tree-grass competition model which explicitly consider fire impact on woody biomass as a generic monotonously increasing function of the grass biomass, which is seen as an indirect proxy of the ignitable dry grass biomass available at the middle of the dry season. this model deals with tree-grass patterns in arid and semi-arid ecosystems and is able to predict, several equilibria, among which pure cover types i.e. bare soil, grassland, forest along with several levels of tree-grass mixtures. notably, the number of equilibria featuring tree-grass coexistence depends on the characteristics of the function ω(g) used to model the fire impact on trees. moreover, our results featured various bistability situations: between forest and grassland; between forest and one of the tree-grass equilibria with low tree biomass; between grassland and another one coexistence equilibrium with low-grass biomass; and between two tree-grass coexistence equilibria (a stable high-grass equilibrium and a stable lowgrass equilibrium). thus the system can occupy multiple stable states, and we have identified three thresholds that summarize the long term dynamics of our system: the threshold r10 which represents the net production of tree biomass relative to the fire-induced biomass loss at the grassland equilibrium, the threshold r01 which represents the net primary production of grasses after fire relative to the grass production loss due to the tree biomass at the wooded savanna equilibrium, and the threshold r∗11 related to the mixed treegrass equilibrium. certainly, in the one hand, our continuous tree-grass competition model shows a wealth of possibilities some of which are still to be explored in the light of more detailed assessment of parameters values relating to specific locations within the savanna biome. on the other hand, and in spite of the potential of the present form of the model, one may discuss the modeling options. first, the tree compartment may also be split into two sub-compartments to distinguish trees sensitive to fires and trees that are not (for instance small trees and tall trees). this has been done in [59], where authors consider also a direct negative impact of grass biomass on sensitive trees. this assumption complexifies the continuous model and allows to treating more diverse ecological situations. second, the modeling of fire as a forcing factor continuous in time may also be questioned. preliminary investigations suggest that discreteevent models (with impulsive differential equations (ide)) can be a way to handle more realistically the influence of fire on tree-grass dynamics. for instance, preliminary results based on the translation of our model into the ide framework show that periodic equilibria may be observed as well as local and global equilibria. although it is beyond the scope of the present paper, the next stage will be a thorough analysis of an impulsional version of our model. it is potentially very important to know to what extent fire management may influence vegetation dynamics in fire-prone savanna-like ecosystems. references [1] accatino, f., de michele, c., vezzoli, r., donzelli, d., and scholes, r. j., (2010) . tree-grass co-existence in savanna: interactions of rain and fire. 13. j. theor. biol. vol. 267, pp. 235-242. http:// dx.doi.org/ 10.1016/ j.jtbi.2010.08.012 [2] andries, j., janssen, m., walker, b., (2002) . grazing management, resilience, and the dynamics of fire-driven rangeland system. ecosystems 5: 23-44. http:// dx.doi.org/ 10.1007/ s10021-001-0053-9 [3] anguelov, r., dumont, y., and lubuma, j., (2012) . ”on nonstandard finite difference method in biosciences”, amitans 2012, aip conf. proc. 1487, pp. 212-223. http:// dx.doi.org/ 10.1063/ 1.4758961 biomath 3 (2014), 1407191, http://dx.doi.org/10.11145/j.biomath.2014.07.191 page 13 of 18 http://dx.doi.org/10.1016/j.jtbi.2010.08.012 http://dx.doi.org/10.1007/s10021-001-0053-9 http://dx.doi.org/10.1063/1.4758961 http://dx.doi.org/10.11145/j.biomath.2014.07.191 a. tchuinté tamen et al., a generic modeling of fire impact in a tree-grass savanna model... 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[57] walker, b. h., ludwig, d., holling, c. s., & peterman, biomath 3 (2014), 1407191, http://dx.doi.org/10.11145/j.biomath.2014.07.191 page 15 of 18 http://dx.doi.org/10.2307/1936967. http://dx.doi.org/10.1038/nature04070. http://dx.doi.org/10.1111/j.1466-8238.2007.00360.x. http://dx.doi.org/10.1007/s00442-004-1683-3 http://dx.doi.org/10.1890/10-1684.1 http://dx.doi.org/10.11145/j.biomath.2014.07.191 a. tchuinté tamen et al., a generic modeling of fire impact in a tree-grass savanna model... r. m., (1981) . stability of semi-arid savanna grazing systems, journal of ecology, vol. 69, n0. 2. pp. 473-498 [58] walter, h., (1971) . ecology of tropical and subtropical vegetation. oliver and boyd, edinburgh, uk. [59] yatat v., dumont y., tewa j.j., couteron p., and bowong s., (2014), mathematical analysis of a size structured tree-grass model in savanna ecosystems, biomath 3 (2014), 1404212 http://dx.doi.org/10.11145/j.biomath.2014.04.212 [60] youta happi, j. (1998). arbres contre graminées: la lente invasion de la savane par la forêt au centrecameroun. thèse, université de paris iv. [61] zogning, a., (1979) . le ”golfe de bafia”, études climatiques. mém. de maı̂trise, univ. de yaoundé, 185 p. appendix a: proof of the proposition ii.1 the jacobian matrix of the system (1) is given by j(g,t) =   j11(g,t) j12(g,t) j21(g,t) j22(g,t)   , where, j11(g,t) = (γg −δg0 −λfgf) − 2µgg−γtgt, j12(g,t) = −γtgg, j21(g,t) = −λftfω ′ (g)t, and j22(g,t) = (γt −δt ) − 2µtt −λftfω(g). now, we analyze the jacobian matrix near the equilibria of system (1). 1) stability of the bare soil (0; 0). the jacobian matrix of system (1) at the equilibrium point (0; 0) is j(0,0) =   (γg −δg0 −λfgf) 0 0 (γt −δt )   . the eigenvalues of j(0; 0) are η1 = γg − δg0 −λfgf, and η2 = γt − δt . in the domain γ, we have η1 > 0, and η2 > 0. since, (0; 0) has two unstable maniflolds, thus (0; 0) is always unstable. 2) stability of the savanna grassland equilibrium ege = (ge; 0) =( γg −δg0 −λfgf µg ; 0 ) . the jacobian matrix of system (1) at the equilibrium point ege is given by j(ge,0) =   j11((ge, 0)) j12((ge, 0)) 0 j22((ge, 0))   , where, j11((ge, 0)) = −(γg −δg0 −λfgf), j12((ge, 0)) = −γtgge, and j22((ge, 0)) = λftfω(ge)(r10 − 1). r10 = γt −δt λftf 1 ω(ge) . the corresponding eigenvalues of j(ge,0) are ν1 = −(γg − δg0 − λfgf) < 0, and ν2 = λftfω(ge)(r10 − 1). therefore, • if r10 < 1, then ege is asymptotically stable. • if r10 > 1, then ege is a saddle point (unstable). 3) stability of the wooded savanna equilibrium ete = (0; te) = ( 0; γt −δt µt ) . according to the system (1), the jacobian matrix at the equilibrium point ete can be written as j(0,te) =   γtgte(r01 − 1) 0 −λftfω ′ (0)te −(γt −δt )   , where, r01 = γg − δg0 −λfgf γtg µt γt −δt . the two eigenvalues of j(0,te) are σ1 = γtgte(r01−1), and σ2 = −(γt −δt ) < 0. we obtain these following conclusions, • if r01 < 1, then ete is asymptotically stable. • if r01 > 1, then ete is a saddle point (unstable). 4) stabilility of the tree-grass coexistence equilibrium e∗ = (g∗; t∗). biomath 3 (2014), 1407191, http://dx.doi.org/10.11145/j.biomath.2014.07.191 page 16 of 18 http://dx.doi.org/10.11145/j.biomath.2014.04.212 http://dx.doi.org/10.11145/j.biomath.2014.07.191 a. tchuinté tamen et al., a generic modeling of fire impact in a tree-grass savanna model... the fixed equilibrium point (g∗; t∗) is determined as the positive solution of system   (γg −δg0 −λfgf) −µgg∗ −γtgt∗ = 0, (γt −δt ) −µtt∗ −λftfω(g∗) = 0. (13) using (13), the jacobian matrix of the system (1) at e∗ = (g∗; t∗) is given by j(g∗;t∗) =   −µgg∗ −γtgg∗ −λftfω ′ (g∗)t∗ −µtt∗   . the eigenvalues of the jacobian matrix at the internal equilibrium e∗ satisfy the following relations • θ1 + θ2 = −µgg∗ −µtt∗ < 0, and • θ1θ2 = γtgλftfω ′ (g∗)g∗t∗(r∗11 − 1), where, r∗11 = µgµt γtgλftf 1 ω ′ (g∗) = r10r01 ω(ge) ω ′ (g∗)ge . clearly, according to θ1θ2 which is the product of the two eigenvalues of j(g∗;t∗), we have, • if r∗11 < 1, then e ∗ is unstable. • if r∗11 > 1, then e ∗is asymptotically stable. this end the proof of the proposition ii.1. appendix b: proof of lemma iii.1 if x∗ is an equilibrium of the continuous system (1), then we have   (γg −δg0 −λfgf)g∗ −µgg2∗ −γtgt∗g∗ = 0, (γt −δt )t∗ −µtt2∗ −λftfω(g∗)t∗ = 0. (14) multiplying both sides of the two equations of (14) by φ(h), leads to the following system:  φ(h)γgg∗ = φ(h)(δg0 + λfgf + µgg∗ + γtgt∗)g∗, φ(h)γtt∗ = φ(h)(δt + λftfω(g∗) + µtt∗)t∗. (15) in (15), adding g∗ and t∗ in both sides of the first equation and the second equation respectively, gives   (1 + φ(h)γg)g∗ = [1 + φ(h)(δg0 + λfgf +µgg∗ + γtgt∗)]g∗, (1 + φ(h)γt )t∗ = [1 + φ(h)(δt + λftfω(g∗) +µtt∗)]t∗. (16) system (16) is equivalent to the following one  a11g∗ = g∗, a22t∗ = t∗. (17) thus, we have a(x∗)x∗ = x∗. (18) then x∗ is an equilibrium of the discrete system (7). appendix c: proof of lemma iii.2 the jacobian matrix of right hand side of the system (1) at x∗ is given by j∗ =   j11(x∗) j12(x∗) j21(x∗) j22(x∗)   , where, j11(x∗) = (γg −δg0 −λfgf) − 2µgg∗ −γtgt∗, j12(x∗) = −γtgg∗, j21(x∗) = −λftfω ′ (g∗)t∗, and j22(x∗) = (γt −δt ) − 2µtt∗ −λftfω(g∗). the eigenvalues of j∗ are solutions of the following equation λ2 − tr(j∗)λ + det(j∗) = 0, (19) where tr(j∗) = j11(x∗) + j22(x∗), and det(j∗) = j11(x∗)j22(x∗) −j12(x∗)j21(x∗). biomath 3 (2014), 1407191, http://dx.doi.org/10.11145/j.biomath.2014.07.191 page 17 of 18 http://dx.doi.org/10.11145/j.biomath.2014.07.191 a. tchuinté tamen et al., a generic modeling of fire impact in a tree-grass savanna model... the sign of each eigenvalue depends of the discriminant ∆ of equation (19): ∆ = (tr(j∗)) 2 − 4 det(j∗) = (j11(x∗) −j22(x∗))2 + 4j12(x∗)j21(x∗). if ∆ > 0, then j∗ is diagonalisable. we have j12(x∗)j21(x∗) = γtgλftfω ′ (g∗)t∗g∗ ≥ 0. then ∆ > 0. therefore j∗ is diagonalisable. appendix d: proof of theorem iii.1 here, we can easily adapted the proof of theorem 8 in [6]. we have shown in lemma iii.1 that, the nsfd scheme (7) has no extra fixed points than those of (1). we have also shown in lemma iii.2 that, the jacobian matrix j∗ is diagonalisable. thus, λ1 and λ2 being the eigenvalues of j∗, there exists a transition matrix p such that p−1j∗p = diag(λ1,λ2). (20) the linearization of system (1) at x∗ reads as dx̃ dt = j∗x̃, (21) where x̃ = x−x∗. system (21) is equivalent to dỹ dt = diag(λ1,λ2)ỹ. (22) thus, applying the nsfd scheme (7) to system (21) or (22), we obtain the linearized scheme x̃n+1 = (i −φ(h)j∗)x̃n, (23) or ỹn+1 = diag( 1 1 −φ(h)λ1 , 1 1 −φ(h)λ2 )ỹn. (24) set ϕ∗ = ϕ((i−φ(h)j∗)−1). it follows from (24) that, ϕ∗ = max { 1 |1 −φ(h)λ1| , 1 |1 −φ(h)λ2| } . (25) recall that, if x∗ is asymptotically stable for (1), then for all i ∈ {1, 2}, we have |re(λi)| = −re(λi). thus ϕ∗ = max 1≤i≤2 { 1√ 1 + 2φ(h)|re(λi)| + φ2(h)|λi|2 } < 1, (26) which shows that x∗ is asymptotically stable for the scheme (7). if x∗ is unstable for (1), then there exists at least one eigenvalue of j∗, λ with positive real part. we then have, 1 |1 −φ(h)λ| = 1√ 1 − 2φ(h)re(λ) + φ2(h)|λ|2 > 1, (27) whenever condition (12) holds. therefore, x∗ is unstable for scheme (7). thus, the discrete scheme (7) preserves stability/instability properties of the continuous model (1). biomath 3 (2014), 1407191, http://dx.doi.org/10.11145/j.biomath.2014.07.191 page 18 of 18 http://dx.doi.org/10.11145/j.biomath.2014.07.191 introduction the mathematical model some qualitative results of the system (1) model without fire model with continuous fire forcing a nonstandard algorithm numerical simulations and discussion bifurcation due to fire period bifurcation due to fg bifurcation due to tg conclusion references www.biomathforum.org/biomath/index.php/biomath original article optimal control of the treatment frequency in a stochastic model of tuberculosis bongor danhree∗, emvudu yves∗ and koı̈na rodoumta ‡ ∗department of mathematics, faculty of science, university of yaounde 1, cameroon sbongordanhree@yahoo.com, yemvudu@yahoo.fr ‡department of mathematics, faculty of exact and applied sciences university of djamena, chad koinarodoumta@yahoo.fr received: 10 march 2016, accepted: 7 may 2017, published: 6 june 2017 abstract—this paper presents a stochastic model of the tuberculosis(tb) infection with treatment in a population composed of four individuals compartments: susceptible individuals, latent infected individuals, active infected individuals and recovered individuals after the therapy. a preliminary survey of the model is performed on the stability before approaching the crucial left of the topic. the aim in this paper is to control the treatment frequency in a stochastic model of the tb infection while minimizing the cost of the measures. then, we formulate an optimal control problem that consists in minimizing the relative cost of the dynamics of tb-model in order to reduce the prevalence and the mortality due to this infection. the optimal problem is solved by applying the projection stochastic gradient method in order to find the optimal numerical solution. finally, we provide some numerical simulations of the controlled model. keywords-stochastic model of tb; local and global stability; optimal control; functional cost; projection stochastic gradient. i. introduction the tuberculosis (tb) continues to make a lot of victims in our societies despite of the existing treatment: the bacillus calmetteguerin. the vaccine anti tubercular is used for preventive treatment for children. nevertheless, other medicines exist as rifampicin, isoniazid, pyrazinamide... for the curative treatment of the patients [23]. the expenses are enormous when the treatment is long. the tuberculosis is one of the causes of elevated mortality in humane communities irrespective of the enormous financial resources made by worldwide governments for the treatment of this disease in the purpose of its eradication. so there is the necessity to integrate to a set of the available control an optimal measure that consists on respecting the dose of the treatment to short length in order to reduce this infection. in this paper, we consider a stochastic model of the tuberculosis (tb) infection in presence of treatment in a population composed of four compartments of individuals: susceptible individuals, latent infected individual, active infected individuals and recovered individuals after the therapy. the mathematical model of tb infection include in addition to the deterministic term, a stochastic term that translates the random noise. the random nature of this model is due to the fact that the citation: bongor danhree, emvudu yves, koı̈na rodoumta, optimal control of the treatment frequency in a stochastic model of tuberculosis, biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 page 1 of 17 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2017.05.077 bongor danhree et al., optimal control of the treatment frequency in a stochastic model of tuberculosis contraction of the mycobacterium tuberculosis, the vector agent of the tb infection and his transmission within the population are done in an random manner according to the variable efficiency of control of the immune system of the individuals. the infection of tb contracts itself mainly by the inhalation of the bacteria distributed by the cough or the sneeze of a sick individual. the vector agent of this infection accommodates itself to the level of the lungs of an individual exposition susceptible of contamination and the immune system of this one controls and maintains the infection in the latent state; otherwise there is the risk that this infection develops itself toward the active state. while supposing that only the infected individuals of active tb transmit the infection, they must observe some hygienic rules, they must adopt a positive behavior with respect to the susceptible individuals (who must also take precaution), to follow the treatment up to finish as early as possible (in less than one year), constitute measures of adequate control. a preliminary survey of the model is performed before introducing a function of control representing the necessary dose of medicines in order to control the frequency of the treatment and to reduce considerably and quickly the prevalence of the disease. the main objective is the control of the treatment frequency in the stochastic model of the tb infection. so we formulate an optimal control problem that consists in minimizing the relative cost of the dynamics of the model in order to reduce the prevalence and mortality due to this infection. to solve this optimal control problem, we are going to apply the stochastic gradient method with projection in order to find the optimal numeric solution. finally, thanks to the numerical simulation tool, we simulate this model without or with control as well as the optimal solution and the associated cost function in order to characterize an optimal decision. in epidemiology and others domain as biology, demography, economy..., many stochastic models deriving from their deterministic formulation. the reference of the literature for a variety of wellknown stochastic models deriving from their deterministic counterparts include the books [1], [5], [6], [7], and [22]. our contribution is first in sub section ii.a, the formulation of a stochastic model of tb with treatment from a deterministic model of tb-only (sharomi [18]) which is formulated along the lines of the model in feng and al. [26]. secondly in sub section ii.b, we change this stochastic model by perturbations or by an affine change of variables affine to lead the survey of the stability of the random equilibrium because the used transformation keeps the law of probability of an random variable [12]. finally in sub section iii.a, we control the treatment frequency in this stochastic model in order to reduce mortality due to the infection. the continuation of the paper is like follows: we recall the results that concern the projection method in sub section iii.b. the gradient projection method is applied to the model in sub section iii.c, and the numerical simulations are plotted in sub section iii.d. ii. stochastic model of tb without control we start this section by the description of the variables and parameters of the model (see table i) then follows it by the presentation of the model. a. diagram and mathematical stochastic model of tb fig. 1. diagram of the stochastic model of tb with treatment biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 page 2 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 bongor danhree et al., optimal control of the treatment frequency in a stochastic model of tuberculosis table i random variables and parameters description variable description st = s(t) susceptible individuals number lt = l(t) number of the tbinfected individuals in the latent state tt = t(t) number of the tb-infected individuals in the active state rt = r(t) number of recovered individuals λi force of tb infection in presence of the treatment λr force of exogenous infection again parameters description λ recruitment rate of susceptible individuals µ naturel mortality rate σ progression rate of tb-infected individuals from latent state to active state ρ infection rate of recovered individuals n proportion of susceptible individuals that enter in (t) by infection βt number of effective contact of susceptible with tb vector δt mortality rate caused by tb ηr proportion of infected individuals by exogenous infection again ηt proportion of recovered individuals by par the active tb-infected τ treatment rate of tb the diagram of the stochastic model of the tb infection is given by fig 1. the mathematical stochastic model of tb infection in the presence of treatment is written under the compact form by the following equation (1) (its formulation uses [1], [5], [6], [7], and [22]) dxt = f(t,xt)dt + g(t,xt)dwt, (1) where xt = (st,lt,tt,rt)t is a 4-dimensional random vector of the states st,lt,tt,rt; wt = (w j t ) t j=1,...,m=10, is a 10-dimensional brownian motion process and is defined on a space of (ω,f,{ft}t≥0,p); f(t,xt) = (fi(t,xt)) t i=1,...,d=4 is a vectorial function of evolution with components fi = fi(t,xt) defined by  f1 = λ − (µ + λi)st, f2 =nλist−(µ+σ+λr)lt+ρrt, f3 = (1−n)λist+(σ+λr) lt−(µ + δt +τ)tt, f4 = τtt − (µ + ρ)rt, (2) g = g(t,xt) = (gij)i=1,...,d=4;j=1,...,m=10 below is a (4 × 10)−dimensional matrix such that g = ( m1 o2×3 o2×3 m2 ) , (3) where o2×3 = ( 0 0 0 0 0 0 ) , m1 = ( g11 g12 g13 g14 0 0 0 0 0 g23 0 g25 g26 g27 ) , m2 = ( g34 0 g36 0 g38 g39 0 0 0 0 g47 0 g49 g410 ) , with g11 = √ λ, g12 = − √ µst, g13 = −g23 = − √ nλist, g14 = −g34 = − √ (1 −n)λist, g25 = − √ µlt, g26 = −g36 = − √ (σ + λr)lt, g27 = −g47 = √ ρrt, g38 =− √ (µ+δt )tt, g39 =−g49 =− √ τtt, g410 = − √ µrt. (4) the tb force of infection λi is defined by: λi = βt tt + ηtrt n (5) with n = st + lt + tt + rt the force of exogenous infection λr is defined by: λr = βt ηrtt n . (6) biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 page 3 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 bongor danhree et al., optimal control of the treatment frequency in a stochastic model of tuberculosis b. analysis of the solution of model and stability in this part, we are going to show the existence and the uniqueness of the global solution positive of the model (1). we also address the existence and stability characterization of the disease free-equilibrium (dfe) and of the endemic equilibrium point. 1) existence and uniqueness of solution: consider a region ω ⊂ r4+ defined by ω ={(st,lt,tt,rt)∈r4+; st+lt+tt+ rt≤ λ µ }. then, we has the following result: theorem 1. let (s0,l0,t0,r0) ∈ ω an initial condition. then there is a unique solution of the stochastic model (1) denoted xt = (st,lt,tt,rt) t such that p{xt = (st,lt,tt,rt)t ∈ ω} = 1 ∀ t ≥ 0. proof: see appendix a. 2) stochastic stability of the random dfe: let us recall the following that will a very helpful in the sequel lemma 1. let p ≥ 2, x,y ∈ r+ and ε > 0 sufficiently small xyp−1 ≤ ε1−p p xp + (p− 1)ε p yp x2yp−2 ≤ 2ε 2−p 2 p xp + (p− 2)ε p yp proof: the inequalities above can be demonstrated with the help of the inequalities of young: for p,q > 0 and 1 p + 1 q = 1, xy ≤ xp p + yp q . proposition 1. the stochastic model (1) admits a random equilibrium point without tb (diseasefree random equilibrium) [ x0 = ( λ µ , 0, 0, 0 )] that is exponentially p-stable if p ≥ 2 and globally asymptotically stable. proof: by translation, we can always bring back a random equilibrium point xe to xe = 0 like in [25]. the existence of x0, disease-free random equilibrium point is proved by the following change variable for the stochastic model (1) s̃t = λ µ −st. (7) as a consequence, the stochastic model (1) reads as dx̃t = f̃(t,x̃t)dt + g̃(t,x̃t)dw̃t, (8) wherein x̃t = (s̃t,lt,tt,rt) t , w̃ = (wi) t , i = 2, ..., 10., f̃(t,xt) = (f̃i(t,xt)) t i=1,...,4 = (f̃i) t i=1,...,4 such that  f̃1 = λ̃i ( λ µ − s̃t ) −µs̃t, f̃2 =nλ̃i ( λ µ −s̃t ) −(µ+σ+λ̃r)lt+ρrt, f̃3 = (1 −n)λ̃i ( λ µ − s̃t ) + (σ + λ̃r)lt − (µ + δt + τ)tt, f̃4 = τtt − (µ + ρ)rt. (9) the noise g̃ = g̃(t,x̃t) is a matrix (4 × 9) given by g̃ = ( m̃1 o2×3 o2×2 m̃2 ) , (10) where o2×2 = ( 0 0 0 0 ) , m̃1 = ( g̃12 g̃13 0 0 0 0 0 g̃23 0 g25 g26 g27 ) , m̃2 = ( g̃34 0 g36 0 g38 g39 0 0 0 0 g47 0 g49 g410 ) , biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 page 4 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 bongor danhree et al., optimal control of the treatment frequency in a stochastic model of tuberculosis with g̃12 = − √ µs̃ t , g̃34 = (1 −n) √ λ̃i ( λ µ − s̃t ) , g̃23 = n √ λ̃i ( λ µ − s̃t ) , g̃13 = √ λ̃i ( λ µ − s̃t ) , λ̃i = βt tt + ηt rt( λ µ − s̃t ) + lt + tt + rt , λ̃r = βt ηrtt( λ µ − s̃t ) + lt + tt + rt . the existence of a disease-free random equilibrium of the model (8) gives the existence of disease-free random equilibrium of (1). in fact, denote by x̃(0) ≡ 0 ∈ r4. the equalities f̃(t, 0) = 0 and g̃(t, 0) = 0 are verified for t ≥ 0. so x̃(0) a disease-free random equilibrium of the model (8). therefore, we have s̃t = 0, lt = 0, tt = 0, rt = 0, that gives st = λ µ , lt = 0, tt = 0, rt = 0, i.e., x0 = ( λ µ , 0, 0, 0 ) is a disease-free random equilibrium of the model (1). now, consider a lyapunov function: v = 1 2p ( k ( λ µ −s̃t )p +k1l p t +k2t p t +k3r p t ) (11) with k > 0,k1 > 0,k2 > 0,k3 > 0,p ≥ 2. let us note by a a differential operator associated to the stochastic model (1), operating on a function v = v (t,x) ∈c1,2(r×rd) by av = ∂v ∂t +f(t,x) ∂v ∂x + 1 2 tr[gt (t,x) ∂2v ∂x2 g(t,x)]. then av =−[k1(µ+σ+λr)l p t +k2(µ+δt +τ)t p t +k3(µ+ρ)r p t ] +k1nλistl p−1 t +k1ρrtl p−1 t + k2(1 −n)λistt p−1 t + k2(σ + λr)ltt p−1 t +k3τttr p−1 t + 1 4 (p−1)[kg211 ( λ µ − s̃t )p−2 +kg212 ( λ µ − s̃t )p−2 +k 1 n g223 ( λ µ − s̃t )p−2 + k1g 2 23l p−2 t + k1g 2 47l p−2 t + k2g 2 34t p−2 t + k2g 2 36t p−2 t + k3g 2 49r p−2 t + k1g 2 25l p−2 t + k1g 2 36l p−2 t + k2g 2 38t p−2 t + k1g 2 49t p−2 t + k3g 2 47r p−2 t + k3g 2 410r p−2 t ] the application of the lemma 1 and the theorem given by afanas’ev in [24], allows us to obtain finally av ≤−[k1(µ + σ + λr)l p t + k2(µ + δt + τ)t p t +k3(µ + ρ)r p t ] av ≤ 0 (necessary to demonstrate). therefore, x0 = ( λ µ , 0, 0, 0 ) is exponentially p−stable (p ≥ 2). for, p = 2, we say that x0 is exponentially 2stable or stable in mean square [24]. in the sense of lyapunov, x0 is globally asymptotically stable. it marks the end of the proof. 3) stability of the endemic random equilibrium: preliminary: suppose that the infection of tb evolves of manner linear i.e. without random noise g(xt, t) ≡ 0. then the model (1) become dxt = f(xt, t)dt which admits a basic reproduction number rτ0 given by: rτ0 = βt (µ + ρ + τηt )[(1 −n)µ + σ] (µ + σ)[(µ + ρ)(µ + δt ) + µτ] + µρτ . (12) if rτ0 > 1, then the model dxt = f(xt, t)dt admits a unique endemic equilibrium point biologically meaningful, x∗ that is locally asymptotically stable [18]. the existence of a random endemic equilibrium [x∗ = (s∗,l∗,t∗,r∗)] is guaranteed by the condition rτ0 > 1 almost surely (see [8]). at this random endemic equilibrium biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 page 5 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 bongor danhree et al., optimal control of the treatment frequency in a stochastic model of tuberculosis [x∗ = (s∗,l∗,t∗,r∗)t ] we have [λi = λ∗i] and [λr = λ ∗ r] such that λ∗i = µ + ρ + τηt ηr(µ + ρ) λ∗r. (13) and a2(λ ∗ i) 2 + a1λ ∗ i + a0 (14) where a0 = (1−rτ0 )(µ+ρ+τηt ){(µ+σ)[(µ+ρ)(µ+δt ) +µτ] +µρτ}, a1 = (µ + ρ + τηt ){(µ + ρ)(µ + σ) + n[δt (µ + ρ) +µτ]+µτ}+ηr(µ+ρ)[(µ+ρ)(µ+δt )+µτ], a2 = ηr(µ+ρ). if rτ0 = 1 ie. a0 = 0, the equation (14) admits a hopeless solution corresponding to x0 the unique equilibrium without tb and it admits another solution to the real negative part corresponding to the endemic equilibrium which is biologically not pertinent. if rτ0 < 1 ie. a0 > 0, then a2a0 < 0 and if the discriminant of (14) is positive i.e. a21−4a2a0 > 0. it follows itself that the equation (14) admits two solutions to part real negatives that corresponding to two equilibriums no pertinent. if rτ0 > 1 ie. a0 < 0 then according to the descartes rule of sign, the equation (14) admits one positive solution λ∗i = −a1 + √ a21 − 2a2a0 2a2 corresponding to an endemic equilibrium x∗. now, suppose that the random noise of the dynamic system of tb has a nature to perturb the states variables st, lt, tt, and rt of the stochastic term g(t,xt) around of s∗, l∗, t∗, and r∗ respectively (see also [25]). then the model (1) becomes dxt = f(t,xt)dt + g(t,xt −x∗)dwt, (15) that can be centered to x∗ by the change variables y1 =st−s∗, y2 =lt−l∗, y3 =tt−t∗, y4 =rt−r∗ (16) the linearized system of (15) around x∗ = (s∗,l∗,t∗,r∗)t as in [4] takes the form dyt = f y(yt)dt + g y(yt)dξt, (17) where fy = fy(yt) = jf (x∗).yt with jf (x∗) the jacobian matrix of f at x∗; yt = y = (y1,y2,y3,y4) t ; ξt = (w it )i=2,...10; fy =   −∂11 ∂12 ∂13 ∂14 ∂21 −∂22 ∂23 ∂24 ∂31 ∂32 −∂33 ∂44 0 0 τ −(µ+ρ)     y1 y2 y3 y4   wherein −∂11 = µ+λ∗i ( 1 − s∗ n∗ ) , ∂12 = −λ∗i s∗ n∗ , ∂13 = (λ ∗ i −βt ) s∗ n∗ ∂14 = (λ ∗ i −βtηt ) s∗ n∗ , ∂21 = nλ ∗ i ( 1 − s∗ n∗ ) + λ∗r l∗ n∗ , −∂22 = nλ∗i s∗ n∗ +λ∗r l∗ n∗ +µ+ρ, ∂23 = −n(λ∗i −βt ) s∗ n∗ + (λ∗r −βtηr) l∗ n∗ , ∂24 = −n(λ∗i −βtηt ) s∗ n∗ + λ∗r l∗ n∗ + ρ, ∂31 = (1 −n)λ∗i ( 1 − s∗ n∗ ) −λ∗r l∗ n∗ , ∂32 = −(1−n)λ∗i s∗ n∗ +λ∗r ( 1 − l∗ n∗ ) +σ, ∂33 = (n−1)(λ∗i−βt ) s∗ n∗ +(λ∗r−βtηr) l∗ n∗ +µ+δt+τ, ∂34 = −(1−n)(λ∗i −βtηt ) s∗ n∗ −λ∗r l∗ n∗ ; and gy(yt) =  g y 12 g y 13 g y 14 0 0 0 0 0 0 0 g y 23 0 g y 25 g y 26 g y 27 0 0 0 0 0 g y 34 0 g y 36 0 g y 38 g y 39 0 0 0 0 0 0 g y 47 0 g y 49 g y 410  , biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 page 6 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 bongor danhree et al., optimal control of the treatment frequency in a stochastic model of tuberculosis with g y 12 = − √ µy1, g y 13 = −g y 23 = − √ nλiy1, g y 14 = −g y 34 = − √ (1 −n)λiy1, g y 25 = − √ µy2, g y 26 = −g y 36 = − √ (σ + λr)y3, g y 27 = −g y 47 = √ ρy4, g y 38 = − √ (µ + δt )y3, g y 39 = −g y 49 = − √ τy3, g y 410 = − √ µy4. (18) . theorem 2. the stochastic model (1) admits a random endemic equilibrium [x∗ = (s∗,l∗,t∗,r∗)] exponentially 2-stable and globally stable if the following conditions (i.), (ii.) are satisfied: (i.) : rτ0 > 1 (ii.) :   ∂11 > 1 2 ( ω1 + κ1βt (1 + ηt ) λ µ ) ∂22 > 1 2 (ω2 + µ) ∂33 > 1 2 ( ω3 + βt λ µ ( c2 c3 + 1 ) + κ2 ) µ + ρ > 1 2 (ω4 + κ3). where, for all real constants ci > 0, i = 1, ..., 4, we have κ1 = 1+n c2 c1 +(1−n) c3 c1 ; κ3 =ρ ( c2 c4 +1 ) +µ; κ2 = σ ( c2 c3 + 1 ) +τ ( c4 c3 + 1 ) +µ+δt ; and ωi > 0, i = 1, ..., 4 such that ω1 = 2λ ∗ i s∗ n∗ + c2 c1 ( nλ∗i ( 1− s∗ n∗ ) +λ∗r l∗ n∗ ) + c3 c1 (1 −n) λ∗i ( 1 − s∗ n∗ ) , ω2 = nλ ∗ i ( 1 − s∗ n∗ ) + nβt (1 + ηt ) s∗ n∗ + 3λ∗r l∗ n∗ + ρ + c3 c2 (λ∗r + σ) ω3 = c1 c3 λ∗i s∗ n∗ + c2 c3 + ( nβt s∗ n∗ + λ∗r ) + λ∗r + (1−n) ( λ∗i ( 1− s∗ n∗ ) +βtηt s∗ n∗ ) +σ ω4 = c1 c4 λ∗i s∗ n∗ + c2 c4 ( nβtηt s∗ n∗ +λ∗r l∗ n∗ +ρ ) + c3 c4 (1 −n)βtηt s∗ n∗ . proof: the trivial solution yt = 0 of the linearized system (17) corresponds to the equilibrium x∗ that the existence is guaranteed by the condition (i). consider now the lyapunov function defined by: v y =v y(y ) = 1 2 4∑ i=1 ciy 2 i , ci>0, i= 1, ..., 4. (19) then av y =−c1∂11y 21 −c2∂22y 2 2 −c3∂33y 2 3 −c4(µ+ρ)y 2 4 + 3∑ i,j=1 4∑ i 6=j ci∂ijyiyj+ 1 2 4∑ i,j=1 tr(gyg yt ij ∂2v y(y ) ∂yi∂yj ) av y = −c1 ( µ + λ∗i(1 − s∗ n∗ ) ) y 21 −c2(nλ ∗ i s∗ n∗ +λ∗r l∗ n∗ +µ+ρ)y 22 −c3[(1−n) (λ ∗ i −βt ) s∗ n∗ +(λ∗r−βtηr) l∗ n∗ +µ+δt +τ]y 2 3 −c4(µ+ρ)y 2 4 + 3∑ i,j=1 4∑ i 6=j ci∂ijyiyj+ 1 2 4∑ i,j=1 tr(gygyt )ij ∂2v y(y ) ∂yi∂yj . to increase the last two terms of av y(y ) that we pose: sum1 = 3∑ i,j=1 4∑ i 6=j ci∂ijyiyj, sum2 = 1 2 4∑ i,j=1 tr(gygyt )ij ∂2v y(y ) ∂yi∂yj . sum1 = 3∑ i,j=1 4∑ i 6=j,∂ij>0 ci∂ijyiyj+ 3∑ i,j=1 4∑ i 6=j,∂ij<0 ci∂ijyiyj biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 page 7 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 bongor danhree et al., optimal control of the treatment frequency in a stochastic model of tuberculosis ≤ 1 2 3∑ i,j=1 4∑ i 6=j,∂ij>0 ci∂ij(y 2 i + y 2 j ) + 3∑ i,j=1 4∑ ,i6=j,∂ij<0 ci∂ijyiyj sum1 ≤ 1 2 3∑ i,j=1 4∑ ,i6=j,∂ij>0 ci∂ij(y 2 i +y 2 j ) sum1 ≤ 1 2 {[2λ∗i s∗ n∗ + c2 c1 ( nλ∗i ( 1 − s∗ n∗ ) + λ∗r l∗ n∗ ) + c3 c1 (1 −n)λ∗i ( 1 − s∗ n∗ ) ]c1y 2 1 + [nλ ∗ i ( 1 − s∗ n∗ ) +nβt (1 + ηt ) s∗ n∗ + 3λ∗r l∗ n∗ + ρ + c3 c2 (λ∗r + σ)]c2y 2 2 + [ c1 c3 λ∗i s∗ n∗ + c2 c3 ( nβt s∗ n∗ + λ∗r ) +(1−n)(λ∗i ( 1 − s∗ n∗ ) +βt ηt s∗ n∗ )+λ∗r +σ]c3y 2 3 +[ c1 c4 λ∗i s∗ n∗ + c2 c4 (nβt ηt s∗ n∗ +λ∗r l∗ n∗ +ρ) + c3 c4 (1−n)×βt ηt s∗ n∗ ]c4y 2 4 }. from where sum1 ≤ 1 2 {ω1c1y 21 + ω2c2y 2 2 + ω3c3y 2 3 + ω4c4y 2 4 } and sum2 = 1 2 {c1(g212 + g 2 13 + g 2 14) + c2(g 2 23 + g 2 25 + g226 + g 2 27) + c3(g 2 34 + g 2 36 + g 2 38 + g 2 39) + c4(g 2 47 + g 2 49 + g 2 410)} + 1 2 {(κ1λi + c1ρ)y1 + c2µy2 + (λr(c2 + c3) + κ2)y3 + κ3y4} sum2 ≤ 1 2 { ( κ1βt λ µ (1 + ηt ) + ρ ) c1y 2 1 + c2µy 2 2 + ( βt λ µ ( c2 c3 + 1 ) + κ2 ) c3y 2 3 + κ3y 2 4 }. hence av y ≤− ( ∂11 − 1 2 ( ω1 + κ1βt (1 + ηt ) λ µ )) c1y 2 1 − ( ∂22 − 1 2 (ω2 + µ) ) c2y 2 2 −[∂33− 1 2 (βt λ µ ( c2 c3 + 1 ) +ω3 + κ2)]c3y 2 3 − ( µ + ρ− 1 2 (ω4 + κ3) ) c4y 2 4 . according to the condition (ii.), we has av y ≤ 0 marking the end of this proof. the random endemic equilibrium [x∗ = (s∗,l∗,t∗,r∗)t ] of the model (1) exists whenever rτ0 > 1 and condition (i.) is fulfilled. it is exponentially 2-stable and globally asymptotically stable in sense of lyapunov if the supplementary condition (ii.) is satisfied. we study in the following section, the optimal control of the treatment frequency in a stochastic model of tb. the condition rτ0 < 1 is needed for the effective stability of tb in a population because the biological pertinence of the endemic equilibrium exists whenever rτ0 > 1 almost surely. the control permits then to adjust this endemic situation unstable. iii. optimal control of the treatment frequency in the tb model a. optimal control problem let (ω,f,{ft}t≥0,p) a complete filtered probability space {ft}t≥0 produced by a standard 10dimensional brownian motion {wt}t≥0. let t > 0 a fixed real number named the horizon of the finite time. let’s note by l2(ω,ft ,r) the space of random variables. ft -measurable to real values and integrable square and by l2f(0;t ,r) a space of process ftadapted to real values and integrable square such that e[ ∫ t 0 |xt|2dt] < +∞. let k ∈ uad a compact convex sub set of l2(0,t ). consider an optimal control problem that consists in minimizing the cost j(., .), the objective function defined for the time t ∈ [0,t ], the state x ∈ r4 and function of control u ∈uad by: j(x,u) = ∫ t 0 e[ϕ(xt,ut)]dt + ∫ t 0 h(ut)dt, (20) relative to the state xt ∈ r4 of the tb model governed in general by:{ dxt =f(t,xt,ut)dt+g(t,xt,ut)dwt, t∈[0,t ] x0 = x(0) ∈ r4 (21) biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 page 8 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 bongor danhree et al., optimal control of the treatment frequency in a stochastic model of tuberculosis and in particular by:{ dxt =f(t,xt,ut)dt+g(t,xt)dwt, t∈[0,t ] x0 = x(0) ∈ r4 (22) where u = ut : τ 7−→ u(t)τ, for all rate τ of (2). this part deals with the study of the particular case where the control doesn’t appear in the stochastic term. the control is said optimal when this dose reached its value optimal positive i.e. u = uop > 0. if this optimal value is not reached, i.e. u ∈ [−1; 0[∪]0; uop[, then the control is said less efficient; it is said without effect when u = 0 and finally the control is said efficient when the optimal value is passed) i.e. u ∈]uop, 1]. the aim is therefore to control the frequencies of the tb treatment in order to reduce number of new cases. the problem of the optimal control is translated to: find an admissible control optimal u = u∗ such that j(x,u∗) = min u∈k⊂uad j(x,u) (23) ie. j(x,u∗) ≤j(x,u) ∀ u ∈ k ⊂uad set f(u) = j(x,u), then the optimal control problem (23) becomes a optimization problem f(u∗) = min u∈k⊂uad f(u), (24) wherein f(u) is a functional convex. b. gradient projection method we want to solve (24) by the projection stochastic gradient method. for this purpose, let us recall the results that concern the projection method on a convex closed k an the stochastic algorithm: proposition 2. let, h a hilbert space, provided with a norm ‖.‖ induced by the scalar product (·|·) and let k ⊂ h a nonempty convex closed set. then for all u ∈ h, 1) an unique ũ ∈ k exists such that ‖u− ũ‖ = min v∈k ‖u−v‖ for all v ∈ k, where ũ = pk(u) is the orthogonal projection of u on k. 2) ũ is charcterized by ũ = pk(u) ⇐⇒ (ũ−u | v − ũ) ≥ 0 proof: 1) the existence of ũ ∈ k holds true because k is closed. let’s suppose that the dimension of h is finite. let us consider k∩b(u;‖u− v‖) the intersection of k with a ball b. on this compact, the function v 7−→‖u−v‖ is continuous. of all minimizing sequence we can extract a convergent sequence, its limit is ũ. the uniqueness comes from the convexity of k and pythagoras’ theorem. 2) for the characterization of ũ; suppose ũ = pk(u) then we has for all v ∈ k ‖u−ũ‖ = min v∈k ‖u−v‖ =⇒‖u−ũ‖≤‖u−v‖ let v ∈ k, pose vε = ũ + ε(v − ũ) ε ∈ ]0; 1[ vε ∈ k which implies that ‖u−ũ‖2 ≤‖u−vε‖2 = ‖u−ũ‖2+ε2‖v−ũ‖2 +2ε(ũ−u | v − ũ) ‖u− ũ‖2 ≤‖u− ũ‖2 + ε2‖v − ũ‖2 +2ε(ũ−u | v − ũ) dividing by ε then we obtain 0 ≤ ε‖v − ũ‖2 + 2(ũ−u | v − ũ) =⇒ (ũ−u | v − ũ) ≥ 0 reciprocally, let’s suppose that (ũ−u | v − ũ) ≥ 0 0 ≥ (u−ũ | v−ũ) = (u−ũ | v−u+u−ũ) 0 ≥‖u− ũ‖2 + (u− ũ | v −u) applying the inequality of cauchy-schwarz, we have 0 ≥‖u− ũ‖2 −‖u− ũ‖‖v −u‖ =⇒‖v −u‖≥‖ũ−u‖ biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 page 9 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 bongor danhree et al., optimal control of the treatment frequency in a stochastic model of tuberculosis proposition 3. the algorithm of the stochastic gradient consists in making evolve the variable u of the optimization problem (24) according to the formula of the following recurrence convergent sequence (un)n≥0 of limit u: un+1 = un + qn(−∇f(un)) where qn > 0 with lim n−→+∞ qn = q and ∇ denotes the gradient.  1. initialization : u0 ∈ h 2. u = un is for n ≥ 0 a) calculate ωn = −∇f(un) choisir qn ≥ 0 such that f(un −qnωn) ≤ f(un −qωn) ∀q > 0 un+1 = un + qnωn b) calculate vn+1 = pk(vn+1) c) test the covergence of the iteration εn = ‖un+1 −un‖ : − if εn < ε stop − otherwise : u = un+1 and repeat iteration. (25) c. projection gradient method applied to the stochastic model of tb with control proposition 4. consider h = u is a hilbert space and uad ⊂ u a closed convex subset. let pk the projection operator on k defined in u by pk(ω) = pkω ∈ k; ∀ ω ∈u, then problem (24) admits an unique solution u or an optimal control such that u = u(·) = pk[u−q(· |f ′(u))] proof: h = u is a hilbert space and uad ⊂u a closed convex subset. the necessary and sufficient condition of the optimality problem (24) is given by (f ′(u) | v −u) ≥ 0 ∀ v ∈ k. let pk the projection operator on k defined in u by pk(ω) = pkω ∈ k; ∀ ω ∈u, such that we have (pkω−ω |pkω−ω) = min u∈k⊂uad (u−ω |u−ω) ∀ω∈u. it is equivalent to (pkω−ω | v−pkω) ≥ 0 ∀v∈k ⇐⇒ ω =pkω. it follows that the solution u of (24) is given by u = u(·) = pk[u−q(· |f ′(u))]. indeed, the optimality condition gives (f ′(u) | v −u) ≥ 0 ∀ v ∈ k, then for q > 0 we have q(· | f ′(h)) | v−u)≥0 =⇒ (q(· | f ′(h)) | v−u)≥0 =⇒ (u−u + q(· | f ′(h)) | v −u) ≥ 0. with ω = u − q(· | f ′(h)), the last implication gives (u−ω |v−u) ≥ 0 ⇐⇒ u = pkω ⇐⇒ u = pk[u−q(· | f ′(h))] for the optimal control problem of the treatment frequency of tb, we are going to define the following iteration scheme for n = 0, 1, ...{ (v | un+1 2 ) = (v | un)−qn(v | f ′n(un)), ∀v∈u un+1 = pk(un+ 1 2 ), (26) where f ′n is the functional approached to the n th iteration of f ′n. the convergence of this scheme, and the calculation of f ′n. are given in [17],[13]. for u(·) an optimal control and x(·), the optimal stat corresponding to x(·) and for v(·) ∈ u ⊂ l2(0,t) such that vp = u(·) + qv(·), 0 < q < 1, then we have for all v ∈ l2(0,t), f ′n(u)(v) = lim q−→0 fn(u + qv) −fn(u) q = e[ ∫ t 0 ϕ′(x)d(x)(v)dt] + ∫ t 0 h′(u)dt, (27) where d(x)(v) = ∫ t 0 [ f ′x(s,x,u)d(x)(v)+f ′ u(s,x,u)v ] ds biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 page 10 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 bongor danhree et al., optimal control of the treatment frequency in a stochastic model of tuberculosis + ∫ t 0 g′x(s,x)d(x)(v)dws, d(d(x)(v)) = [f ′x(t,x,u)d(x)(v)+f ′ u(t,x,u)v]dt +g′x(t,x)d(x)(v)dwt. we define a adjoint functional p, ft-adapted and defined by  −dp = [ϕ′(x) + pf ′x(t,x,u) −pg′x(t,x) .(g′x(t,x)) tr]dt + pg′x(t,x)dwt, p(t ) = 0 (28) such that e[ ∫ t 0 |pt|2dt] < +∞. the right hand side of the equation (27) permits to get finally f ′n(u)(v) from (28), that reads as f ′n(u)(v) = ∫ t 0 e[p(f ′(t,x,u) + h′(u)]vdt, (29) the projection gradient method applied to the stochastic model of tb with control, consist therefore in considering the system (30) of two equations (22) and (28) in order to solve it numerically,   dxt =f(t,xt,ut)dt+g(t,xt,ut)dwt, t∈[0,t ], −dp= [ϕ′(xt)+pf ′x(t,xt,ut)−pg ′ x(t,xt) .(g′x(t,x)) tr]dt + pg′x(t,xt)dwt, x0 = x(0) ∈ r4 p(t ) = 0. (30) the numerical resolution of (30) uses the iteration scheme (31) below for n = 0, 1... and then the euler scheme for the two equations of (30) (see [17]) ,  (v|un+ 1 2 ) = (v|un)−qn(v|e[pn(f ′u(t,xn,un))] + h′(un)), ∀v ∈u un+1 = pk(un+ 1 2 ), (31) where xn, un and pn are the present steps of the functions constructed. d. numerical simulations algorithm[17]: stage 1 to choose the arbitrary initial control for n = 0, 1, · · ·, let u = un, to make the buckle iteration of stage 1 to stage 5; stage 2 to use the implicit euler scheme for the discretization in time of the sde (22) stage 3 to use the implicit euler scheme for the discretization in time of the adjoint equation; (28) stage 4 to use the iteration scheme (31) of the gradient method to update the controls;  um n+ 1 2 =um−qn(e[pm(f ′u(tm,xm,um))] + h′(um)), m = 0, 1, · · ·,mmax umn+1 = pk(un+ 1 2 ); stage 5 calculate en = ‖un − un+1‖. if en is small enough, then exit. otherwise; let u = un+1 repeat the buckle iteration from stage 2 to stage 5. table ii parameter values and references parameters values references λ variable estimate µ 0.02 [18] σ 1/33 [18] ρ 0.04 [18] δt 0.2 [18] n variables estimates ηr, ηt 0.4, 0.06 [18] βt , τ variables estimates for the following figures, we take ϕ(x,u) = (x2 + u2)exp( −t x2 + u2 ), h(x) = x2, n = 0.05, en < 10 −7, p0 = 0.01 and the rest λ, βt , τ, x0 are variable. fig.2 give a schematic plot of the model (1) not depending of u. the aim is to show, for a initial condition given, the asymptotic behavior of the solution around a random endemic equilibrium when the hard epidemic a long time rτ0 > 1. while, fig.3(resp. fig.4) shows a numerical illustration of optimal control u, see (a) and (b) (resp. of cost f(u), see (c) and (d)). the orthogonal projection of the minimum point of f(u) on the closed subset [−1; 1], gives a numerical value of optimal control u∗; e.g. the minimum point • of f(u) represented in (c), is valued as f(u∗) = 2.7066 giving u∗ = 0 if u0 = 1. thanks to matlab, we can value the biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 page 11 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 bongor danhree et al., optimal control of the treatment frequency in a stochastic model of tuberculosis cost f(u∗) and the optimal control u∗ for a control initially chosen u0 as fig.6. in fig.5, the trajectory without control, see (t.1), (resp. with control, see (t.2)) of the active infected individuals t is creasing between 0 and 2 years (resp. decreasing between 0 and 1 year and is annulling constantly thereafter). (t.3) and (t.4) show that a control of treatment intervened 0.5 years equal to 6 months after the infection, permits to reduce to nothing numbers of the active infected individuals of tb. 0 5 10 15 20 0 10 20 30 40 50 60 70 80 90 100 time (years) s , l , t a n d r in d iv id u a ls s without control l without control t without control r without control (i) 0 10 20 30 40 50 60 0 10 20 30 40 50 60 70 80 90 100 time (years) s , l , t a n d r in d iv id u a ls s without control l without control t without control r without control (ii) fig. 2. numerical simulations of the model (1) without control (i.e. not depending of u), showing the asymptotic behavior of the solution when rτ0 > 1 at different initial condition: (i) : x0 = (s0,l0,t0,r0) = (50, 12, 5, 10), λ = 10, βt = 0.8, τ = 0.08, r τ 0 = 2.3710 > 1 and, (ii) x0 = (s0,l0,t0,r0) = (50, 1, 1, 1), λ = 8, βt = 0.9, τ = 0.08. rτ0 = 3.5565 > 1. 0 0.5 1 1.5 2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 p tim a l c o n tr o l u time (years) optimal control u for u 0 =0 (a) 0 0.5 1 1.5 2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 time (years) 0 p tim a l c o n tr o l u w ith d iff e re n te in iti a l v a lu e optimal control u for u 0 =0 optimal control u for u 0 =0.5 optimal control u for u 0 =0.8 optimal control u for u 0 =1 (b) fig. 3. numerical simulations of a control u: (a) and (b) −1 −0.5 0 0.5 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 optimal ontrol u c o st f u n ct io n f (u ) f(u) for u 0 =1 with minimum point • u*=0 (c) −1 −0.5 0 0.5 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 optimal ontrol u c o st f u n ct io n f (u ) f(u) for u 0 =0.5 f(u) for u 0 =0.8 f(u) for u 0 =1 (d) fig. 4. numerical simulation of a cost functionf(u) (c) and (d) biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 page 12 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 bongor danhree et al., optimal control of the treatment frequency in a stochastic model of tuberculosis 0 0.5 1 1.5 2 0 1 2 3 4 5 6 7 8 9 10 time (years) s , l , t a n d r in d iv id u a ls s withoit control l withoit control t withoit control r withoit control (t.1) 0 0.5 1 1.5 2 0 1 2 3 4 5 6 7 8 9 10 time (years) s , l , t ,a n d r s stochastic l stochastic t stochastic r stochastic (t.2) fig. 5. trajectories without and with control of the model (1). for λ = 5, βt = 0.08, τ = 0.08, u0 = 0.08, x0 = (1, 1, 1, 1). let’s note that initial value of sequence qn is chosen as q0 = 0.1 for fig.6 and (d); q0 = 0.6 for fig.4 (c). iv. conclusion the stochastic model (1) of tb without control admits for an initial state x(0), a positive and unique solution xt ∈ ω of probability one. it exist for this model an unique disease equilibrium free (def) exponentially 2-stable and globally asymptotically stable (in lyapunov sense). under a given condition, the model (1) admits a random endemic equilibrium exponentially p-stable (p ≥ 2) and globally stable. the introduction of a treatment control function in model (1) gives an optimal control problem governed by model (22). the projection gradient method permits to determine numerically the optimal control as well as the cost function corresponding to this problem. −1 −0.5 0 0.5 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 optimal ontrol u c o st f u n ct io n f (u ) f(u) for u 0 =0.2 f(u) for u 0 =0.5 f(u) for u 0 =0.8 f(u) for u 0 =1 fig. 6. numerical simulations of the control u and the function cost f(u) for the different initials values of u0. we obtain: if u0 = 0.2 then f(u∗) = 2.7073 and u∗ = uop = 0.03606; if u0 = 0.5 then f(u∗) = 2.9166 and u∗ = uop = 0.01607; if u0 = 0.8 then f(u∗) = 3.3071 and u∗ = uop = 0.0000; and if u0 = 1 then f(u∗) = 3.6673 and u∗ = uop = 0; for example, with a treatment rate equal to τ = 8% and with an initial value equal to u0 = 0.2 of the function control, we obtain u∗ = 0.3606, the admissible optimal control and f(u∗) = 2.7073, the cost. also with τ = 8% and u0 = 1, we obtain u∗ = 0. we therefore deduce that the optimal control is without effect when u0, the initial dose of the medicines taken by a patient ranges from 80% to 100 %. on the other hand the optimal control is efficient admissible when this initial dose is lower to 50 %. thanks to the presence of the optimal control in the stochastic model (1) of tb, we can reduce considerably and quickly (less than one year) the number of the active infected individuals. as in fig.5(t.2) and fig.7 (t.3)-(t.4), the trajectory with control of the active infected individuals t is decreasing between 0 and 1 year and becomes null constantly thereafter. this work is therefore a contribution that enters well in the same line of struggle against mortality due to the infections that several governments as well as biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 page 13 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 bongor danhree et al., optimal control of the treatment frequency in a stochastic model of tuberculosis 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time (years) t t b in fe ct e d a ct iv e t without control t with control (t.3) 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time (years) t t b in fe ct e d a ct iv e t without control t with control (t.4) fig. 7. trajectory without and with control of tb active infected individuals t. with βt = 0.08 for t.3 and βt = 0.8 for t.4. humanitarian associations advocated so much. appendix a: proof of theorem 1 let nt = st +lt +tt +rt, the random variable giving the total number of the population at the time t. we have dnt = (λ −µnt −δttt)dt− ξ<g,dw(t)>. where ξ<g,dw(t)> = ( √ µst + √ µlt + √ (µ + δt )tt+√ µrt − √ λ)d$(t), with $ = wi i = 1, ..., 10. because wi follow the same law of probability, namely the normal law. we need to show that if xt = (st,lt,tt,rt)t ∈ r4+ for all t ∈ [0; t�[ where t� is the explosion time, then we have for p -almost surely (p−as) nt < λ µ . in fact, if xt ∈ r4+ for all t ∈ [0; t�[, then nt is given such that for p−as.: dnt = (λ −µnt −δttt − ξ<g,dw(t)>)dt ≤ (λ −µnt)dt according to the lemma of gronwall, we obtain: nt ≤ λ µ + (n0 − λ µ )e−µt p −as. and as by hypothesis (s0,l0,t0,r0) ∈ ω i.e. n0 − λ µ ≤ 0, we have then nt < λ µ p −as. the terms f(t,xt) and g(t,xt) of the stochastic model (1) being locally lipschitz, there is an unique local solution xt = (st,lt,tt,rt)t for all t ∈ [0; t�[ fixed. therefore, the unique local solution xt = (st,lt,tt,rt)t ∈ r4+. in the sequel we show that xt is global solution p−almost surely i.e. t� = ∞. let n0 > 0, an integer sufficiently large such that (s0,l0,t0,r0) ∈ [ 1 n0 ; n0 ]4 . set et = {st,lt,tt,rt} and for all integer n ≥ n0, we define the stop-times tn = inf {hn} with hn ={ t∈[0,t�] : min et∈ [ 0; 1 n ] or max et∈[n; +∞[ } . (tn)n>0 is an increasing sequence and convergent; denote by t∞ = lim n−→∞ tn then t∞ ≤ t�. let us show that t∞ = ∞ so that we has t� = ∞. for it, let us suppose by absurd that t∞ < ∞, there is θ > 0 such that for all p ∈]0; 1[ we have p{t∞ ≤ θ} > p. consequently, there is an integer n1 ≥ n0 such that for all set an = {tn ≤ θ}, we have p{an} > p n ≥ n1. (32) biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 page 14 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 bongor danhree et al., optimal control of the treatment frequency in a stochastic model of tuberculosis let us consider the function v defined on r4+ and to values in r+ such that v =−ln ( µst λ ) −ln ( µlt λ ) −ln ( µtt λ ) −ln ( µrt λ ) . using the multidimensional itô formula on the interval [0; min(τn; θ)], we have for all t ≥ 0 dv = dv (xt) = [ ∂v (xt) ∂t + 4∑ i=1 fi(t,xt) ∂v (xt) ∂xit + 1 2 4∑ i,j=1 (ggt )ij ∂2v (xt) ∂xt∂x j t ]dt + 4∑ i=1 10∑ j=1 gijdw j t ∂v (xt) ∂xit , where for i = 1, 2, ..., 4; j = 1, 2, ..., 10, g = (gij); and (gg t )ij = 10∑ k=1 gik.gkj. therefore dv = 2[4µ + σ + δt + τ]dt + 5 4 (λi + λr)dt + 1 2 [µ 1 st + (µ + σ) 1 lt + (µ + δt + τ) 1 tt + (µ + ρ) 1 rt ]dt− 1 2 {λ (4st − 1) s2t + (nλist+ρrt) (4lt−1) l2t +[(1−n)λist + (σ+λr)lt] (4tt−1) l2t +τtt (4rt−1) l2t }dt − 1 st (g11dw 1 t + g12dw 2 t + g13dw 3 t + g14dw 4 t ) − 1 lt (g23dw 3 t + g25dw 5 t + g26dw 6 t + g27dw 7 t ) − 1 tt (g34dw 4 t + g36dw 6 t + g38dw 8 t + g39dw 9 t ) − 1 rt (g47dw 7 t + g49dw 9 t + g410dw 10 t ). we further obtain the following inequations: dv (xt) ≤ mdt− 1 st (g11dw 1 t + g12dw 2 t +g13dw 3 t + g14dw 4 t ) − 1 lt (g23dw 3 t +g25dw 5 t + g26dw 6 t + g27dw 7 t ) − 1 tt (g34dw 4 t + g36dw 6 t + g38dw 8 t +g39dw 9 t ) − 1 rt (g47dw 7 t + g49dw 9 t +g410dw 10 t ) p −as. with m = 5 2 [4µ+σ +δt +τ + 1 2 βt (1 +ηt +ηr)] > 0. which implies by integration that∫ ∧tnθ 0 dv ≤ m ∫ ∧tnθ 0 dt−[ 4∑ k=1 (∫ ∧tnθ 0 g1k st dwkt ) + 7∑ k=3,6=4 (∫ ∧tnθ 0 g2k lt dwkt ) ] −[ 9∑ k=4,6=7 (∫ ∧tnθ 0 g3k tt dwkt ) + 10∑ k=7,k 6=8 (∫ ∧tnθ 0 g4k rt dwkt ) ], (33) where ∧tnθ = tn ∧ θ = min(tn; θ). taking the mathematical expectations for all terms of inequations (33), we obtain e[v (xtn∧ θ)] ≤ e[v (x0)] + mθ (34) let a set an = {tn ≤ θ}. denote by ian (resp. i{an) the indicator function of an (resp. of the complementary {an). thus e[v (xtn∧θ)] = e[v (xtn∧θ)ian]+e[v (xtn∧θ)i{an] according to the definition of function v , we have v (xtn∧ θ) ≥ 0. hence e[v (x0)]+mθ≥e[v (x∧tnθ )ian]+e[v (x∧tnθ )i{an] e[v (x0)] + mθ ≥ e[v (xtn)ian] biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 page 15 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 bongor danhree et al., optimal control of the treatment frequency in a stochastic model of tuberculosis thanks to the continuity, it exists at least one of the components xtn equals to n or to 1 n . so v (xtn) ≥ min { −ln (µn λ ) ;−ln ( µ λn )} v (xtn) ≥ min { ln ( λ µn ) ; ln ( λn µ )} , and consequently e[v (x0)] + mθ ≥ e[v (xtn∧ θ)ian] ≥ p{an}× min { ln ( λ µn ) ; ln ( λn µ )} . hence p{an}=p{tn≤θ}≤ e[v (x0)]+mθ min { ln ( λ µn ) ; ln ( λn µ )}. (35) taking the limit when n −→ +∞ in (35), we found that 0<p< lim n−→+∞ p{an}=p{t∞ ≤ θ}= 0 ∀θ≥0, which is a contraction, then t∞ = ∞ i.e. that we have p{t∞ = ∞} = 1. otherwise, t∞ ≤ t�, we therefore conclude that t∞ = t� = ∞ p −as. acknowledgment the authors would like to thank the anonymous referee and the african center of excellence in technologies, information and communication of the university of yaounde 1 (cameroon). references [1] allen l.j.s. and van den driessche, stochastic epidemic models with a bifurcation, mathematical biosciences and engineering, 3(3): 445-457, 2006. [2] a. lahrouz, l. omari, and d. kiouach, global analysis of deterministic and stochastic nonlinear sirs epidemic model, volume 16 , n1, 59-76 .(1992). 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[26] z. feng, c. castillo-chavez and f. capurro, a model for tuberculosis with exogenous reinfection, theor. pop. biol., 57, 235-247, 2000. biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 page 17 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 introduction stochastic model of tb without control diagram and mathematical stochastic model of tb analysis of the solution of model and stability existence and uniqueness of solution stochastic stability of the random dfe stability of the endemic random equilibrium optimal control of the treatment frequency in the tb model optimal control problem gradient projection method projection gradient method applied to the stochastic model of tb with control numerical simulations conclusion references original article biomath 3 (2014), 1411041, 1–8 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum a parameter uniform almost first order convergent numerical method for a semi-linear system of singularly perturbed delay differential equations n. shivaranjani ∗, j.j.h. miller† and s.valarmathi∗ ∗department of mathematics, bishop heber college, tiruchirappalli, tamilnadu, india. email: shivaranjaninagarajan@gmail.com, valarmathi07@gmail.com †institute for numerical computation and analysis, dublin, ireland. email: jm@incaireland.org received: 11 august 2014, accepted: 4 november 2014, published: 24 november 2014 abstract—in this paper an initial value problem for a semi-linear system of two singularly perturbed first order delay differential equations is considered on the interval (0,2]. the components of the solution of this system exhibit initial layers at 0 and interior layers at 1. a numerical method composed of a classical finite difference scheme on a piecewise uniform shishkin mesh is suggested. this method is proved to be almost first order convergent in the maximum norm uniformly in the perturbation parameters. keywords-singular perturbation problems, boundary layers, semi-linear delay differential equations, finite difference schemes, shishkin mesh, parameter uniform convergence. i. introduction singularly perturbed delay differential equations play an important role in the modelling of several physical and biological phenomena like first exit time problems in modelling of activation of neuronal variability [3], bistable devices [8] and evolutionary biology [6] and in a variety of models for physiological processes or diseases [9],[10] and [11]. these systems also find applications in belousovzhabotinskii reaction (bz reaction) models and the modelling of biological oscillators [6]. a model of tumor growth that includes the immune system response and a cycle-phase-specific drug presented in [13] is cited here. the model considers three populations: immune system, population of tumor cells during interphase and population of tumor during mitosis. the governing equations of the system are citation: n. shivaranjani, j.j.h. miller, s.valarmathi, a parameter uniform almost first order convergent numerical method for a semi-linear system of singularly perturbed delay differential equations, biomath 3 (2014), 1411041, http://dx.doi.org/10.11145/j.biomath.2014.11.041 page 1 of 8 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2014.11.041 n. shivaranjani et al., a parameter uniform almost first order ... dti dt =2a4tm − (c1i + d2)ti −a1ti(t− τ) dtm dt =a1ti(t− τ) −d3tm −a4tm − c3tmi −k1(1 −e−k2u)tm di dt =k + ρi(ti +tm ) n α+(ti +tm )n − c2iti − c4tmi −d1i − k3(1 −ek4u)i du dt = −γu with ti(t) =φ1(t) for t ∈ [−τ, 0] tm (t) =φ2(t) for t ∈ [−τ, 0] i(t) =φ3(t) for t ∈ [−τ, 0] u(0) =u0. here, ti(t) population of tumor cells during interphase at time t tm (t) -population to tumor cells during mitosis at time t i(t) -population of immune system at time t u(t) -amount of drug present at time t τ -the resident time of cells in interphase d2ti,d3tm,d1i proportions of natural cell death or apoptosis a1,a4 the rate at which cells cycle are reproduce ci -losses from encounters of tumor cells with immune cells ρi(ti +tm ) n α+(ti +tm )n non-linear growth of the immune population due to stimulus by tumor cells k -constant rate at which the immune cells grow, in the absence of tumor cells ρ,α,n -parameters depending on the type of tumor being considered and the health of the immune system. thus, an initial value problem for a system of semilinear delay differential equations is used to model tumor growth. here, the parameters may take large values, for instance the value of k is 1.3 × 104 in the paper cited. in these cases, the system becomes singularly perturbed. motivated by this, in this paper, the following semilinear system of singularly perturbed delay differential equations is considered: ~t~u = e~u′(x) + ~f(x,u1,u2) + b(x)~u(x− 1) = ~0 on (0, 2], ~u = ~φ on [−1, 0]. (1) for all x ∈ [0, 2], ~u(x) = (u1(x),u2(x))t and ~f(x,u1,u2) = (f1(x,u1,u2),f2(x,u1,u2))t . e, b(x) are 2 × 2 matrices. e = diag(~ε), ~ε = (ε1,ε2) with 0 < ε1 ≤ ε2 ≤ 1, b(x) = diag(~b), ~b = (b1(x),b2(x)). it is assumed that the nonlinear terms satisfy ∂fk(x) ∂uk ≥ β > 0, ∂fk(x) ∂uj ≤ 0, k,j = 1, 2, k 6= j (2) min 1≤i≤2   2∑ j=1 ∂fi(x) ∂uj + bi(x)   ≥ α > 0, (3) bi(x) ≤ 0, i = 1, 2 (4) for x in [0, 2] × c2 where c = c0([−1, 2]) ∩ c1((0, 2]) ∩c2((0, 1) ∪ (1, 2)). these conditions and the implicit function theorem ensure that a unique solution ~u ∈ c2 exists for the problem (1). the solution ~u(x) has initial layers at x = 0 and interior layers at x = 1. both the components u1 and u2 have layers of width o(ε2) and the component u1 has an additional sublayer of width o(ε1). for any vector-valued function ~y on [0, 2] the following norms are introduced: ‖ ~y(x) ‖= maxi |yi(x)|, i = 1, 2 and ‖ ~y ‖= sup{‖ ~y(x) ‖: x ∈ [0, 2]}. a mesh ω̄n = {xi}ni=0 is a set of points satisfying 0 = x0 < x1 < ... < xn = 2. a mesh function v = {v (xi)}ni=0 is a real valued function defined on ω̄n . the discrete maximum norm for the above function is defined by ‖ v ‖ω̄n = maxi=0,1,...,n |v (xi)| and ‖ ~v ‖ω̄n = max{‖ v1 ‖ω̄n ,‖ v2 ‖ω̄n} where the vector mesh functions ~v = (v1,v2) t = {v1(xi),v2(xi)}, i = 0, 1, ..,n. biomath 3 (2014), 1411041, http://dx.doi.org/10.11145/j.biomath.2014.11.041 page 2 of 8 http://dx.doi.org/10.11145/j.biomath.2014.11.041 n. shivaranjani et al., a parameter uniform almost first order ... throughout the paper c denotes a generic positive constant, which is independent of x and of all singular perturbation and discretization parameters. furthermore, inequalities between vectors are understood in the componentwise sense. ii. analytical results the problem (1) can be rewritten in the form ε1u ′ 1(x) + f1(x,u1,u2) + b1(x)φ1(x− 1) = 0 ε2u ′ 2(x) + f2(x,u1,u2) + b2(x)φ2(x− 1) = 0, x ∈ (0, 1] ~u(0) = ~φ(0) (5) and ε1u ′ 1(x) + f1(x,u1,u2) + b1(x)u1(x− 1) = 0 ε2u ′ 2(x) + f2(x,u1,u2) + b2(x)u2(x− 1) = 0, x ∈ (1, 2] ~u(1) known from (5). (6) ~t1~u := e~u ′(x) + ~g(x,u1,u2) = ~0, x ∈ (0, 1] ~t2~u := e~u ′(x) + ~f(x,u1,u2) +b(x)~u(x− 1) = ~0, x ∈ (1, 2] where ~g(x,u1,u2) = ~f(x,u1,u2) + b(x)~φ(x− 1). (7) the reduced problem corresponding to (7) is given by ~g(x,r1,r2) = ~0, x ∈ (0, 1] (8) ~f(x,r1,r2) + b(x)~r(x− 1) = ~0, x ∈ (1, 2]. (9) the implicit function theorem and conditions (2),(3) and (4) ensure the existence of a unique solution for (8) and (9). this solution ~r has derivatives which are bounded independently of ε1 and ε2. hence, |r(k)1 (x)| ≤ c; |r (k) 2 (x)| ≤ c; k = 0, 1, 2, 3; x ∈ [0, 2]. the following shishkin decomposition [1], [2] of the solution ~u is considered: ~u = ~v + ~w, where the smooth component ~v(x) is the solution of the problem e~v′(x) + ~g(x,v1,v2) = ~0, x ∈ (0, 1] e~v′(x) + ~f(x,v1,v2) + b(x)~v(x− 1) = ~0, x ∈ (1, 2] ~v(0) = ~r(0) (10) and the singular component ~w(x) satisfies e~w′(x) + ~g(x,v1 + w1,v2 + w2) −~g(x,v1,v2) = ~0,x ∈ (0, 1] e~w′(x) + ~f(x,v1 + w1,v2 + w2) − ~f(x,v1,v2) +b(x)~w(x− 1) = ~0, x ∈ (1, 2] ~w(0) = ~u(0) −~v(0). (11) the bounds of the derivatives of the smooth component are contained in lemma 1: the smooth component ~v(x) satisfies |v(i)k (x)| ≤ c, k = 1, 2; i = 0, 1 and |v′′k(x)| ≤ cε −1 k , k = 1, 2. proof: the smooth component ~v is further decomposed as follows: ~v = ~̃q + ~̂q where ~̂q is the solution of g1(x, q̂1, q̂2) = 0 (12) ε2 dq̂2 dx + g2(x, q̂1, q̂2) = 0, x ∈ (0, 1] (13) q̂2(0) = v2(0); q̂1(0) = v1(0) (14) and f1(x, q̂1, q̂2) + b1(x)q̂1(x− 1) = 0 (15) ε2 dq̂2 dx + f2(x, q̂1, q̂2) + b2(x)q̂2(x− 1) = 0, x ∈ (1, 2] (16) q̂2(1) and q̂1(1) are known from (12) and (13). ~̃q is the solution of ε1 dq̃1 dx + g1(x, q̃1 + q̂1, q̃2 + q̂2) −g1(x, q̂1, q̂2) = −ε1 dq̂1 dx (17) biomath 3 (2014), 1411041, http://dx.doi.org/10.11145/j.biomath.2014.11.041 page 3 of 8 http://dx.doi.org/10.11145/j.biomath.2014.11.041 n. shivaranjani et al., a parameter uniform almost first order ... ε2 dq̃2 dx + g2(x, q̃1 + q̂1, q̃2 + q̂2) −g2(x, q̂1, q̂2) = 0, x ∈ (0, 1] q̃1(0) = q̃2(0) = 0 (18) and ε1 dq̃1 dx + f1(x, q̃1 + q̂1, q̃2 + q̂2) −f1(x, q̂1, q̂2) +b1(x)q̃1(x− 1) = −ε1 dq̂1 dx (19) ε2 dq̃2 dx + f2(x, q̃1 + q̂1, q̃2 + q̂2) −f2(x, q̂1, q̂2) +b2(x)q̃2(x− 1) = 0, x ∈ (1, 2] (20) q̃1(1) and q̃2(1) are known from (17) and (18). let x ∈ [0, 1]. using (8), (12) and (13), a11(x)(q̂1 −r1) + a12(x)(q̂2 −r2) = 0 (21) ε2 d dx (q̂2 −r2) + a21(x)(q̂1 −r1) +a22(x)(q̂2 −r2) = −ε2 dr2 dx , (22) where, aij(x) = ∂gi ∂uj (x,ξi(x),ηi(x)), i,j = 1, 2; ξi(x),ηi(x) are intermediate values. using (21) in (22), ε2 d dx (q̂2 −r2) + ( a22(x) − a12(x)a21(x) a11(x) ) ×(q̂2 −r2) = −ε2 dr2 dx consider the linear operator, l1(z) := ε2z ′ + ( a22(x) − a12(x)a21(x) a11(x) ) z = −ε2 dr2 dx , (23) where, z = q̂2 −r2. this operator satisfies the maximum principle [1]. thus, ‖ q̂2 −r2 ‖≤ cε2 and ‖ d(q̂2 −r2) dx ‖≤ c. using this in (21), ‖ q̂1 −r1 ‖≤ cε2. hence, ‖ q̂2 ‖≤ c, ‖ dq̂2 dx ‖≤ c and ‖ q̂1 ‖≤ c. differentiating (22), ε2 d2 dx2 (q̂2 −r2) + a′21(x)(q̂2 −r2) + a21(x) d dx (q̂2 −r2) + a′22(x)(q̂1 −r1) + a22(x) d dx (q̂1 −r1) = −ε2 d2r2 dx2 . (24) hence,‖ d2q̂2 dx2 ‖≤ cε−12 . differentiating (21) twice and using the above estimates of d2q̂2 dx2 , ‖ d2q̂1 dx2 ‖≤ cε−12 . (25) from (17) and (18), ε1 dq̃1 dx + a∗11(x)q̃1 + a ∗ 12(x)q̃2 = −ε1 dq̂1 dx (26) ε2 dq̃2 dx + a∗21(x)q̃1 + a ∗ 22(x)q̃2 = 0 (27) q̃1(0) = q̃2(0) = 0 (28) where, a∗ij(x) = ∂gi ∂uj (x,ζi(x),χi(x)), i,j = 1, 2; ζi(x),χi(x) are intermediate values. from equations (26) and (27), ‖ q̃i ‖≤ c, i = 1, 2 (29) ‖ dq̃i dx ‖≤ c, i = 1, 2 (30) ‖ d2q̃i dx2 ‖≤ cε−1i , i = 1, 2. (31) hence from the bounds for ~̃q and ~̂q, the required bounds of ~v follow. biomath 3 (2014), 1411041, http://dx.doi.org/10.11145/j.biomath.2014.11.041 page 4 of 8 http://dx.doi.org/10.11145/j.biomath.2014.11.041 n. shivaranjani et al., a parameter uniform almost first order ... let x ∈ [1, 2]. using (9), (15) and (16), p11(x)(q̂1 −r1) + p12(x)(q̂2 −r2)+ b1(x)(q̂1(x− 1) + r1(x− 1)) = 0 (32) ε2 d dx (q̂2 −r2) + p21(x)(q̂1 −r1) +p22(x)(q̂2 −r2) + b2(x)(q̂2(x− 1) −r2(x− 1)) = −ε2 dr2 dx (33) where, pij(x) = ∂fi ∂uj (x,κi(x),λi(x)), i,j = 1, 2; κi(x),λi(x) are intermediate values. using (32) in (33), ε2 d dx (q̂2 −r2) + ( p22(x) − p12(x)p21(x) p11(x) ) ×(q̂2 −r2) − p21p11 (x)b1(x)(q̂1(x− 1) −r1(x− 1)) + b2(x)(q̂2(x− 1) −r2(x− 1)) = −ε2 dr2 dx consider the linear operator, l2(z) := ε2z ′ + ( p22(x) − p12(x)p21(x) p11(x) ) z +b2(x)z(x− 1) = −ε2 dr2 dx − p21 p11 (x)b1(x)(q̂1(x− 1) −r1(x− 1)), (34) where, z = q̂2 −r2. this operator satisfies the maximum principle [12]. hence using similar arguments as in the interval [0, 1] and the bounds of ~̂q and ~̃q in the interval [0, 1], the required bounds in the interval [1, 2] are derived. lemma 2: the singular component ~w(x) satisfies, for any x ∈ [0, 1], |wi(x)| ≤ ce −αx ε2 ; i = 1, 2 |w′1(x)| ≤ c(ε −1 1 e −αx ε1 + ε−12 e −αx ε2 ) |w′2(x)| ≤ cε −1 2 e −αx ε2 |w′′i (x)| ≤ cε −1 i (ε −1 1 e −αx ε1 + ε−12 e −αx ε2 ), i = 1, 2 for x ∈ [1, 2], |wi(x)| ≤ ce −α(x−1) ε2 ; i = 1, 2 |w′1(x)| ≤ c(ε −1 1 e −α(x−1) ε1 + ε−12 e −α(x−1) ε2 ) |w′2(x)| ≤ cε −1 2 e −α(x−1) ε2 |w′′i (x)| ≤ cε −1 i (ε −1 1 e −α(x−1) ε1 +ε−12 e −α(x−1) ε2 ), i = 1, 2 proof: from equations (11), ε1w ′ 1(x) + s11(x)w1(x) + s12(x)w2(x) = 0 (35) ε2w ′ 2(x) + s21(x)w1(x) + s22(x)w2(x) = 0, x ∈ (0, 1] (36) w1(0) = u1(0) −v1(0); w2(0) = u2(0) −v2(0) and ε1w ′ 1(x) + s ∗ 11(x)w1(x) + s ∗ 12(x)w2(x) +b1(x)w1(x− 1) = 0 (37) ε2w ′ 2(x) + s ∗ 21(x)w1(x) + s ∗ 22(x)w2(x)+ b2(x)w2(x− 1) = 0, x ∈ (1, 2] (38) w1(1) = u1(1) −v1(1); w2(1) = u2(1) −v2(1) here, sij(x) = ∂gi ∂uj (x,νi(x),υi(x)) and s∗ij(x) = ∂fi ∂uj (x,φi(x),φ ∗ i (x)); νi(x),υi(x),φi(x), φ∗i (x) are intermediate values. biomath 3 (2014), 1411041, http://dx.doi.org/10.11145/j.biomath.2014.11.041 page 5 of 8 http://dx.doi.org/10.11145/j.biomath.2014.11.041 n. shivaranjani et al., a parameter uniform almost first order ... from equations (35),(36),(37) and (38), the bounds of the singular component ~w can be derived as in [5] in the domains [0, 1] and [1, 2]. iii. shishkin mesh a piecewise uniform shishkin mesh ω n = ω− n ∪ ω+n where ω− n = {xj} n 2 0 and ω +n = {xj}nn 2 +1 with n mesh-intervals is now constructed on ω = [0, 2],as follows, for the case ε1 < ε2. in the case ε1 = ε2 a simpler construction requiring just one parameter τ suffices. the interval [0, 1] is subdivided into 3 subintervals [0,τ1] ∪ (τ1,τ2] ∪ (τ2, 1]. the parameters τr, r = 1, 2, which determine the points separating the uniform meshes, are defined by τ0 = 0, τ3 = 1 2 , τ2 = min { 1 2 , ε2 α ln n } and τ1 = min {τ2 2 , ε1 α ln n } . (39) clearly 0 < τ1 < τ2 ≤ 12. then, on the subinterval (τ2, 1] a uniform mesh with n4 mesh points is placed and on each of the sub-intervals (0,τ1] and (τ1,τ2], a uniform mesh of n8 mesh points is placed. similarly, the interval [1, 2] is also divided into 3 sub-intervals [1, 1 + τ1], (1 + τ1, 1 + τ2], (1 + τ2, 2] having the same number of mesh intervals as in [0, 1]. note that, when both the parameters τr, r = 1, 2, take on their lefthand value, the shishkin mesh becomes a classical uniform mesh on [0, 2]. iv. discrete problem the initial value problems (5) and (6) are discretised using the backward euler scheme on the piecewise uniform fitted mesh ω̄n. the discrete problem is tn ~u(xj) := ed −~u(xj) + ~g(xj,u1(xj),u2(xj)) = 0, j = 1(1) n 2 (40) t̃n ~u(xj) := ed −~u(xj) + ~f(xj,u1(xj),u2(xj)) = −b(xj)~u(xj − 1), j = n 2 + 1(1)n (41) ~u(0) = ~u(0) and d−~u(xj) = ~u(xj) − ~u(xj−1) xj −xj−1 , j = 1(1)n. lemma 3: for any mesh functions ~y and ~z with ~y (0) = ~z(0), ‖ ~y − ~z ‖≤ c ‖ tn ~y −tn ~z ‖ proof: tn ~y −tn ~z = ed−~y (xj) + ~g(xj,y1(xj),y2(xj)) −ed−~z(xj) −~g(xj,z1(xj),z2(xj)) = ed−(~y − ~z)(xj) + ∂~g ∂u1 (xj,~ξ(xj),~η(xj))(y1 −z1) + ∂~g ∂u2 (xj,~ξ(xj),~η(xj))(y2 −z2) = (t ′n )( ~y − ~z) where t ′n is the frechet derivative of tn and the notation ∂~g ∂ui (xj,~ξ(xj),~η(xj)), i = 1, 2 is used to express the difference between the mid-values for the components g1 and g2. since t ′n is linear, it satisfies the discrete maximum principle and discrete stability result [5].hence ‖ ~y−~z ‖≤ c ‖ t ′n (~y−~z) ‖= c ‖ tn ~y−tn ~z ‖ and the lemma is proved. parameter uniform bounds for the error are given in the following theorem, which is the main result of this paper. theorem 1: let ~u be the solution of the problem (1) and ~u be the solution of the discrete problem (40),(41). then ‖ ~u −~u ‖≤ cn−1 ln n (42) biomath 3 (2014), 1411041, http://dx.doi.org/10.11145/j.biomath.2014.11.041 page 6 of 8 http://dx.doi.org/10.11145/j.biomath.2014.11.041 n. shivaranjani et al., a parameter uniform almost first order ... table i values of dnε , d n, pn, p∗ and cnp∗ for ε1 = η 16 , ε2 = η 4 and α = 0.9. η number of mesh points n 128 256 · · · 8192 16384 20 0.150e-01 0.806e-02 · · · 0.271e-03 0.136e-03 2−3 0.211e-01 0.121e-01 · · · 0.619e-03 0.336e-03 2−6 0.218e-01 0.125e-01 · · · 0.619e-03 0.336e-03 2−9 0.218e-01 0.125e-01 · · · 0.619e-03 0.336e-03 2−12 0.218e-01 0.125e-01 · · · 0.619e-03 0.336e-03 ... ... ... · · · ... ... 2−27 0.218e-01 0.125e-01 · · · 0.619e-03 0.336e-03 dn 0.218e-01 0.125e-01 · · · 0.619e-03 0.336e-03 pn 0.800e+00 0.854e+00 · · · 0.880e+00 cnp 0.249e+01 0.249e+01 · · · 0.196e+01 0.186e+01 computed order of ~ε -uniform convergence, p∗ = 0.8 computed ~ε -uniform error constant, cnp∗ = 2.48 proof: let x ∈ [0, 1]. from the above lemma, ‖ ~u −~u ‖≤ c ‖ tn ~u −tn~u ‖ consider ‖ tn~u ‖=‖ tn~u−tn ~u ‖ hence, ‖ tn~u−tn ~u ‖=‖ tn~u ‖ =‖ tn~u− ~t1~u ‖ = e|(d−~u−~u′)(x)| ≤ e|(d−~v −~v′)(x)| +e|(d−~w − ~w′)(x)| since the bounds for ~v and ~w are the same as in [5] , the required result follows. let x ∈ [1, 2]. from the above lemma, ‖ ~u −~u ‖ ≤ c ‖ t̃n ~u − t̃n~u ‖ ≤ c ‖ b(xj)(~u −~u)(xj − 1) ‖ ≤ c ‖ ~u −~u ‖ ≤ cn−1 ln n v. numerical results the numerical method proposed in this paper is illustrated through an example presented in this section. example consider the initial value problem ε1u ′ 1(x) + 3u1(x) − 1 4 exp(−u21)(x) −u2(x) −x2 + 1 −u1(x− 1) = 0 ε2u ′ 2(x) + 4u2(x) − cos(u2(x)) −u1(x)− ex −u2(x− 1) = 0; x ∈ (0, 1] ~u(x) = ~0; x ∈ [−1, 0]. the above quasi linear problem is solved using the numerical method suggested in this paper utilising the continuation method found in [2]. the maximum pointwise errors and the rate of convergence for this ivp are calculated using the two mesh algorithm in [2] and are presented in table 1. the notations dn,pn,cnp ,c n p∗ and p ∗ bear the same meaning as in [2] but the methods to arrive at them are modified for the vector solution. a graph of the numerical solution is presented in figure 1 for n = 2048 and η = 2−15. the sharper initial layers at x = 0 and interior layers at x = 1 are evident. biomath 3 (2014), 1411041, http://dx.doi.org/10.11145/j.biomath.2014.11.041 page 7 of 8 http://dx.doi.org/10.11145/j.biomath.2014.11.041 n. shivaranjani et al., a parameter uniform almost first order ... fig. 1. numerical solution -2 0 2 4 6 8 10 12 14 0 0.5 1 1.5 2 u1 u2 acknowledgment the first author wishes to acknowledge the financial assistance extended through inspire fellowship by the department of science and technology, government of india. references [1] j. j. h. miller, e. o’riordan, g.i. shishkin, fitted numerical methods for singular perturbation problems, world scientific, revised edition (2012). [2] p.a. farrell, a. hegarty, j. j. h. miller, e. o’riordan, g. i. shishkin, robust computational techniques for boundary layers, applied mathematics & mathematical computation (eds. r. j. knops & k. w. morton), chapman & hall/crc press (2000). [3] c.g.lange, r.m.miura, singular perturbation analysis of boundary-value problems for differential-difference equations. siam j.appl.math.42(3),502-530(1982). http://dx.doi.org/10.1137/0142036 [4] zhongdi cen, a second-order hybrid finite difference scheme for system of singularly perturbed initial value problems, journal of computational and applied mathematics 234, 3445-3457 (2010). http://dx.doi.org/10.1016/j.cam.2010.05.006 [5] s.valarmathi, j.j.h.miller, a parameter uniform finite difference method for a singularly perturbed linear dynamical systems,international journal of numerical analysis and modelling, 7(3), 535-548 (2010). [6] j.d.murray mathematical biology: an introduction (third edition),springer (2002) . 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[13] minaya villasana, ami radunskaya a delay differential equation model for tumor growth. j.math. biol. 47,270294(2003)http://dx.doi.org/10.1007/s00285-003-0211-0 biomath 3 (2014), 1411041, http://dx.doi.org/10.11145/j.biomath.2014.11.041 page 8 of 8 http://dx.doi.org/10.1137/0142036 http://dx.doi.org/10.1016/j.cam.2010.05.006 http://dx.doi.org/10.1103/physreva.26.3720 http://dx.doi.org/10.1016/s0025-5564(00)00006-7 http://dx.doi.org/10.1016/0025-5564(88)90064-8 http://dx.doi.org/10.1016/s0025-5564(02)00099-8 http://dx.doi.org/10.1007/s00285-003-0211-0 http://dx.doi.org/10.11145/j.biomath.2014.11.041 introduction analytical results shishkin mesh discrete problem numerical results references www.biomathforum.org/biomath/index.php/biomath review article building reaction kinetic models for amiloid fibril growth svetoslav markov institute of mathematics and informatics bulgarian academy of sciences sofia, bulgaria smarkov@bio.bas.bg dedicated to the 180th anniversary of the birth of nestor markov, a pioneer of lexicography and phys-math education in bulgaria received: 30 june 2016, accepted: 31 july 2016, published: 8 august 2016 abstract—in this work we discuss some methodological aspects of the creation and formulation of mathematical models describing the growth of species from the point of view of reaction kinetics. our discussion is based on familiar examples of growth models such as logistic growth and enzyme kinetics. we propose several reaction network models for the amiloid fibrillation processes in the citoplasm. the solutions of the models are sigmoidal functions graphically visualized using the computer algebra system mathematica. keywords-reaction kinetics; reaction network; growth models; sigmoidal functions; dynamical systems; ode’s i. introduction a field of considerable interest is the study of various biological growth processes and the application of mathematical models that facilitates the understanding of these processes. growth processes usually evolve in time as sigmoidal functions. there exists a vast literature on sigmoidal functions. the field is characterized by a huge number of studies on real world growth phenomena and attempts to explain the intrinsic mechanisms of these phenomena using various mathematical methods [7], [8], [21], [33]. sigmoidal growth functions are usually introduced in three main ways. often growth functions are defined by an explicite arithmetic expression. another way is to define them as a solution of a problem formulated in terms of a differential equation or a system of (integro-)differential equations. a third way is to formulate a chemical reaction network that induces (via the mass action law) a dynamical system, that in turn imply sigmoidal solution(s). this approach makes use of the reaction network theory—a well established field of applied mathematics (mathematical chemistry) that studies the behavior of real world chemical systems. in many situations the reaction network-approach can be applied to biological phenomena and has the advantage of suggesting possible (bio)-chemical mechanisms of the processes and phenomena under investigation. in this work we are going to illustrate the above mentioned approaches for the formulation and study of growth functions on several familiar citation: svetoslav markov, building reaction kinetic models for amiloid fibril growth, biomath 5 (2016), 1607311, http://dx.doi.org/10.11145/j.biomath.2016.07.311 page 1 of 11 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2016.07.311 s. markov, building reaction kinetic models for amiloid fibril growth examples such as the verhulst logistic function and henri enzyme kinetic reaction network. we shall then formulate some models that can be possibly used to explain the growth of amiloid fibrils in the cell citoplasm. sigmoidal growth curves typically have three parts (phases): lag, log and stationary parts. it is a challenging question to characterize mathematically these phases. the lag phase is practically important in many medical and biotechnological applications as this phase is responsible for the acceleration or inhibition of the process and the possibility of controlling the lag phase depends on the understanding of the hidden mechanisms of the corresponding process. the reaction networkapproach to growth process modelling provides namely certain knowledge about the specific intrinsic mechanisms of the process. ii. biological growth/transition models: three case studies in this section we present three case studies of familiar growth models in order to illustrate possible mechanisms of growth formulated in terms of reaction networks. growth and transition are related processes, as a species (reactant, population) a grows for the expenses of another species b. in some situations this can be also expressed as “species b transits (transforms) into species a”. the transition can be in both directions (reversible). the process can be catalyzed by a third species c, which in particular can coincide with some of the species a,b (autocatalysis). three familiar forms for presentation of the models will be illustrated by means of case studies. explicitly formulated growth curves (models), also known as “empirical” models, are briefly denoted as e-type models. models formulated in terms of systems of differential equations (dynamical systems) are denoted as d-type models. finally, models formulated in terms of a (chemical) reaction network of certain reactants (species, populations) are classified as r-type models. note that a model can have several types of formulation. an r-type model can be reformulated into a d-type model for the concentrations of the reactants by means of the mass action law [6], [32]. for some growth models all three formulation types are available, as shown in case studies 1 and 2 below. a. case study 1: saturated growth the saturated growth is not sigmoidal one, as no lag phase is present. nevertheless this growth model is practically important and is a good illustration of the three model types. a (chemical) reaction network comprises a set of reactants, a set of products (often intersecting the set of reactants), and a set of reactions [6], [30], [32]. consider the simple reaction network consisting of the two species (reactants) s,p and a transition reaction of s transforming into p with rate k, symbolically: s k−→ p. (1) in a situation when it is meaningful to speak of concentrations of the reagents (reacrants), e. g. when reaction (1) takes place in a liquid medium, then the concentrations s,p of the species s,p , resp., obey the mass action law and we obtain the dynamical system ds/dt = −ks, dp/dt = ks. (2) let us formulate an initial value problem (ivp) assuming initial conditions s(0) = s0 = 1, p(0) = p0 = 0. then the first equation of (2) has as solution the exponential decay function: s(t) = exp(−kt). (3) to find an expression for the concentration p of the product species p , we note that the conservation law for system (2): s′ + p′ = 0, induces s(t) + p(t) = s0. substituting s = s0 −p = 1 −p in the second equation of (2) we have dp/dt = k(1 −p). (4) equation (4) with p(0) = 0 has as solution p(t) = 1 − exp(−kt). (5) biomath 5 (2016), 1607311, http://dx.doi.org/10.11145/j.biomath.2016.07.311 page 2 of 11 http://dx.doi.org/10.11145/j.biomath.2016.07.311 s. markov, building reaction kinetic models for amiloid fibril growth in this case study all three formulation types for the growth function p are present: the r-type (1), the d-type (2) and the e-type formulae for the saturation growth of p (5) as well for the decay of s (3). consider now reaction (1) slightly modified by adding a reaction in the reverse direction. the rtype formulation is then: x k−→←− k−1 y, which is a shortcut for the following reaction network: x k−→ y, y k−1−→ x. (6) applying the mass action law to (6) we obtain the following d-type model for the evolution of the concentrations x,y of the reactants x,y , resp.: dx/dt = −kx + k−1y, dy/dt = kx−k−1y. again an ivp can be considered by specifying fig. 1. solutions to the saturation-decay model (6) the initial conditions x(0) = x0, y(0) = y0. the solutions x,y are graphically visualized on fig. 1 for x0 = 1.0, y0 = 0.0. assuming k ≥ k−1 means that x decays and y is growing. the saturation growth process has no lag phase. remark. finding an r-type formulation for a certain d-model is called realization of the dmodel [6]. it is instructive to look for a realization of the malthusian d-type model x′ = kx. a possible realization is: x k−→ x + x. according to this r-type model species x reproduces without using any resources, which may be a rather rough approximation of a real process (in the long term). a generalization of the decay-saturation mechanism for many (more than two) species is discussed in [24]. there we study an extension of the reaction network (6). let us rewrite the latter in the form: x1 k12−→ x2, x2 k21−→ x1. (7) reaction network (7) can be extended for n species x1,x2, ...,xn, so that each pair of species interacts, that is: xi kij−→ xj, i = 1, ...,n, j = 1, ...,n, i 6= j. (8) for example, for n = 3 we have six reactions x1 k12−→ x2, x2 k21−→ x1, x2 k23−→ x3, x3 k32−→ x2, x3 k31−→ x1, x1 k13−→ x3. (9) from the r-type formulations (8), (9) one can straightforward obtain the corresponding d-type models applying the mass action law, cf. [24], [40]. b. case study 2: verhulst logistic growth the logistic function is a smooth sigmoidal function finding numerous applications in biochemical, population and cell growth phenomena [26], [41]–[43]. consider the following autocatalytic reaction network: u + x k−→ x + x. (10) a possible “biological” interpretation of reaction network (10) in the context of population dynamics can be the following: the nutrient substrate (or species) u is utilized (consumed) by species (population) x leading to a reproduction of species x, thereby k is a specific growth rate of the process. assuming that u and x are uniformly spaced in a certain volume (or area), then we may denote the biomass of population x by x and the mass (or concentration) of the (nutrient) substrate u by biomath 5 (2016), 1607311, http://dx.doi.org/10.11145/j.biomath.2016.07.311 page 3 of 11 http://dx.doi.org/10.11145/j.biomath.2016.07.311 s. markov, building reaction kinetic models for amiloid fibril growth u and apply the mass action law, to obtain the dynamical system du/dt = −kxu, dx/dt = kxu. (11) adding initial conditions u(0) = u∗ > 0, x(0) = x∗ > 0 to (11), one obtains a corresponding ivp. the solution of (11) for u∗ = 1.0, x∗ = 0.1 is visualized on fig. 2. the conservation relation du/dt + dx/dt = 0 implies u + x = x∗ + u∗ = const = a. let us assume a = 1 meaning that the initial conditions are normalized by 1/a, setting u∗ := u∗/a < 1, x∗ := x∗/a < 1. we then substitute u = 1 − x in the differential equation for x to obtain the verhulst differential equation: dx dt = kx (1 −x) , x(0) = x∗ < 1. (12) the solution x to ivp (12) passing through the fig. 2. solutions to (11) with u(0) = u∗ = 1.0, x(0) = x∗ = 0.1.. point (0,x(0) = x∗ = 1/2) is the (basic) logistic sigmoid function: x0(t) = 1 1 + e−kt . (13) we see that the logistic model admits all three formulation types: the e-type (13), the d-type (11) or (12) and the r-type (10). in analogy to the generalisation of the decaysaturation r-type mechanism for many (more than two) species (8) we can generalize the logistic rtype mechanism i) introducing a reversible reaction, and ii) introducing many (more than two) species. for the case of a reversible reaction we have: x1 + x2 k12−→ 2x2, 2x2 k21−→ x1. (14) for the case of a (irreversible, closed) food chain of three species we have x1 + x2 k2−→ 2x2, x2 + x3 k3−→ 2x3, x3 + x1 k1−→ 2x1. (15) reaction networks (14), (15) can be easily generalized for n species x1,x2, ...,xn. this demonstrates again the notational power of the r-type model formulations. the basic logistic function (13) is a sigmoidal function with asymptotes limt→−∞x0(t) = 0, limt→∞x0(t) = 1. due to x′′0(0) = 0, x0 has an inflection at 0, inflection point is (0, 1/2), cf. fig. 3. assume that the tangent to the graph of function (13) through the inflection point (0, 1/2) intersects the abscissa at the point (−δ, 0),δ > 0. the slope κ of the tangent through the inflection point (0, 1/2) is equal to κ = x′0(0) = k/4. thus, we have (1/2)/δ = k/4, hence δ = 2/k. the value of δ may be called log time (or high rate time period). c. lag time the so-called lag time is a mathematical characteristic of the concept of lag phase which is the low rate time period of a sigmoidal process. we are going to define and calculate the lag time of the logistic model. to this end consider the shifted logistic function on r: xγ(t) = xγ(k; t) = 1 1 + e−k(t−γ) . (16) function (16) has an inflection point (γ, 1/2) and its slope κ at t = γ is κ = k/4. let (γ − biomath 5 (2016), 1607311, http://dx.doi.org/10.11145/j.biomath.2016.07.311 page 4 of 11 http://dx.doi.org/10.11145/j.biomath.2016.07.311 s. markov, building reaction kinetic models for amiloid fibril growth fig. 3. a logistic function (13) with γ = 0 and reaction rate k = 40. δ, 0), δ = 2/k be the point where the tangent through the inflection point intersects the abscissa. assume that γ ≥ δ. the value tlag = γ −δ = γ − 2/k (17) is called the lag time of the logistic model (16), cf. [39]. expression (17) has been obtained from the e-type logistic model (16), however (17) can be derived from the d-type model as well. indeed, from (12) we obtain x′′ = (kx(1 −x))′ = kx′(1 − 2x), showing that, if the initial condition x(0) is less than 1/2, then (since x′ is positive) at some point γ > 0 with x(γ) = 1/2 there exists an inflection (satisfying 1 − 2x(γ) = 0). thus, at γ we have x(γ) = 1/2,x′(γ) = κ,x′′(γ) = 0. substituting the time t = γ in (12) we obtain x′(γ) = kx(γ)(1−x(γ)) implying κ = k(1/2)(1− 1/2) = k(1/4), hence κ = k/4. thus, again (1/2)/δ = k/4, hence δ = 2/k and (17) holds true. let us examine the solution x0 of the ivp problem (12) with initial condition x0(0) = 1/2. proposition ii.1. the solution x0 of (12) with initial condition x0(0) = 1/2 has an inflection point (0, 1/2). any shifted solution xγ(t) = x0(t − γ) has an inflection point at (γ, 1/2). proof. let x be an arbitrary solution to (12), then we have (1/k)x′ = x(1 − x), hence (1/k)x′′ = x′(1 − x) − xx′ = x′(1 − 2x). due to x′ > 0, x′′ can be zero only for values of t satisfying 1 − 2x = 0, that is x(t) = 1/2. this is true for all solutions of (12), in particular for solution x0, satisfying initial condition x0(0) = 1/2. hence, due to x′′0(0) = 0, we have that x0 has an inflection at 0. � similarly, any shifted solution xγ(t) = x0(t−γ) has an inflection point at t = γ, as xγ(γ) = x0(t−γ)|t=γ = x0(0) = 1/2. problem. find the value of γ so that the shifted function xγ(t) = x0(t − γ) satisfies the initial condition xγ(0) = x0(−γ) = x∗ for a given x∗, 0 ≤ x∗ ≤ 1/2. to solve the above problem we make use of the e-type formulation of the logistic model. let x0(−γ) = x∗ < 1/2, γ > 0, then x∗ is the initial value for the d-type model (12) having as solution the shifted function xγ, that is xγ(0) = x∗. using the e-type presentation (13), the latter gives (1 + ekγ)−1 = x∗, hence γ = 1 k ln 1 −x∗ x∗ . therefore tlag = γ − 2 k = 1 k ( ln 1 −x∗ x∗ − 2 ) . in order to have tlag ≥ 0, the restriction x∗ ≤ (e2 + 1)−1 should be satisfied. we summarize the above calculations in the following proposition ii.2. if the initial value x∗ of the ivp (12) is such that x∗ ≤ (e2 + 1)−1, then the sigmoidal solution x = x(t) has a positive lag time. proposition ii.2 shows that the d-type model (12) should have a sufficiently small initial condition in order to possess a positive lag time. growth processes with positive lag time are typical for bio-chemical reactions, that is reactions involving biomath 5 (2016), 1607311, http://dx.doi.org/10.11145/j.biomath.2016.07.311 page 5 of 11 http://dx.doi.org/10.11145/j.biomath.2016.07.311 s. markov, building reaction kinetic models for amiloid fibril growth functional proteins, such as enzymes, receptors, ligands, as well as population models. finally we shall point out one more characteristic property of the lag time of the logistic growth function. to this end let us first note that any sigmoidal function induces two simple (non-smooth) functions—a so-called cut (or ramp) function and a step function. more specifically consider the shifted logistic function (16) with inflection point (γ, 1/2) and slope κ = k/4 at t = γ. the tangent through the inflection point hits the abscissa t the point a = (γ − δ, 0), δ = 2/k and hits the horizontal line with ordinate 1 in the point b = (γ + δ, 1). the segment of the tangent between the two points a,b together with the parts of the abscissa before a and the part of the horizontal line with ordinate 1 after point b is the graph of the cut function cγ induced by the logistic function xγ. similarly one defines the step function through the inflection point of the logistic function [2]. we can now formulate the following proposition ii.3. [12] the uniform distance between any logistic function and its induced cut function is (1 −e−2)−1. noticing that the uniform distance mentioned in the above proposition is the value of the logistic function at the point γ−δ, we see that this distance does not depend on the slope κ of the logistic function (and the induced cut function as well). for a situation when κ is small it seems not natural to consider the point γ − δ as a definition of the lag time. that is why in a series of papers we propose another definition of lag time, namely γ− δ′ wherein δ′ is the hausdorff distance between the sigmoidal function and the induced step function [17]–[25]. d. case study 3: growth models using henri’s reaction scheme biochemical processes involve functional proteins. the simplest reaction network involving a protein has been first formulated by victor henri [9], [14], [15], [16]. the henri reaction network involves two fractions of the enzyme (free and bound) denoted e and c, resp.: s + e k1−→←− k−1 c k2−→ p + e. (18) it is assumed that the rate parameters k1,k−1,k2 are positive constants such that k1 > k−1. reaction scheme (18) describes the reaction mechanism between an enzyme e with a single active site and a substrate s, forming reversibly an enzyme-substrate complex c, which then yields irreversibly a product p . reaction scheme (18) says that during the transition of the substrate s into product p the enzyme e bounds the substrate into a complex c having specific properties different than the properties of the free enzyme and thus being necessarily considered as a separate substance. denoting the concentrations s = [s], e = [e], c = [c], p = [p ] and applying the mass action law to henri’s reaction scheme (18) we obtain the following system of odes: ds/dt = −k1es + k−1c, de/dt = −k1es + (k−1 + k2)c, dc/dt = k1es− (k−1 + k2)c, dp/dt = k2c, (19) if the three rate constants k1,k−1,k2 are known, system (19) can be treated as an ivp with initial conditions s(0) = s0, e(0) = e0, c(0) = c0, p(0) = p0, (20) usually s0 > 0, e0 > 0, c0 = 0, p0 = 0. system (19) has two conservation laws: e′+c′ = 0 and s′ + c′ + p′ = 0. these two relations can be used to reduce the system to two equations: ds/dt = −k1(e0 − c)s + k−1c, dc/dt = k1(e0 − c)s− (k−1 + k2)c. (21) proposition ii.4. the solution p of the ivp (19)– (20) is a sigmoidal function. proof. the solutions of (19)–(20) are smooth functions. the first equation implies that s is monotonically decreasing function tending to zero with t −→∞ (note that k1 > k−1 and e ≤ e0,c ≤ biomath 5 (2016), 1607311, http://dx.doi.org/10.11145/j.biomath.2016.07.311 page 6 of 11 http://dx.doi.org/10.11145/j.biomath.2016.07.311 s. markov, building reaction kinetic models for amiloid fibril growth fig. 4. solutions to (19) with initial conditions s(0) = 0.5, e(0) = 1.0, c(0) = 0.0 p(0) = 0.2. c0 are bounded). equation p′ = k2c ≥ 0 implies that p is a monotone increasing function. function c increases from c(0) = 0 up to a maximum value c(t∗) achieved at some time t∗ (not necessarily unique) and then decreases tending to 0 with the exhaustion of s, see the second equation in (21), so c′(t∗) = 0. hence p′′(t∗) = k2c′(t∗) = 0. meaning that p has an inflection at t∗. from s′+c′+p′ = 0 we have s+c+p = s0 showing that p(t)t−→∞ = s0. therefore function p is sigmoidal. � the solutions to (19) with initial conditions s(0) = 0.5, e(0) = 1.0, c(0) = 0.0 p(0) = 0.2 are visualized on fig. 4. in practice the rate constants k1,k−1,k2 are often not known and have to be determined for every specific enzyme-substrate pair. the contemporary approach to this task is to consider the rate constants as parameters in the dynamic system (19) and to compute them by fitting the solutions of the system to time course experimentally measured data [11]. the verhulst and henri reaction network models (10), (18) present two useful mechanisms for studying biochemical and biological growth. various combinations of these two models have been proposed for the study of the eps production [27], [28], [34]. iii. r-type models for amiloid fibrillation recent intensive research into the physicochemical properties of amyloid and its formation into fibrils in the citoplasm points attention to growth models [3], [29], [38], [31]. in [39] the authors consider the growth of aminoid fibrils and look for a mechanistic explanation of the process in terms of a biochemical reaction network. fibril is an olygomer composed by monomers, thus shoffner–schnell model [39] involves two reactants: fibril f and monomer m, and additionally the intermediate reactant c. reacrant c is the fibril “in action”, that is the fibril that at the given time moment is in the process of storing the monomer molecule (adding it to self in a compact form). the shoffner-schnell model describes the mechanism of the fibril growth in details and leads to interesting results. for educational purposes we present below several simple r-type models that may be helpful in illuminating certain particular steps of the fibril growth process and certain issues of interest (such as the lag phase). all presented models make use of the verhulst and henri reaction networks. a. a basic verhulst-henri model let m be the total amount (concentration) of monomer, f be the fibril and c = m-f be the monomer-fibril complex at the time t of aggregation [3], [29], [38], [39]. after aggregation the complex c turns into fibril f , that is, the added monomer molecule m converts into (part of) the fibril f . a simple reaction network model of these processes is m + f k+−→←− k− c kc−→ f + f. (22) reaction scheme (22) is almost same as (18) with product p substituted by fibril f (obtained as result of the aggregation of the monomer m). on the other side reaction network (22) can be seen as an extension of verhulst reaction network (10) by involving an intermediate catalyst c. biomath 5 (2016), 1607311, http://dx.doi.org/10.11145/j.biomath.2016.07.311 page 7 of 11 http://dx.doi.org/10.11145/j.biomath.2016.07.311 s. markov, building reaction kinetic models for amiloid fibril growth denoting the concentrations m = [m], f = [f], c = [c] and applying the mass action law to reaction scheme (22) we obtain the following system of three ode’s: dm/dt = −k+fm + k−c, df/dt = −k+fm + (k− + 2kc)c, dc/dt = k+fm− (k− + kc)c. (23) if the three rate constants k+,k−,kc are known, system (23) can be treated as a cauchy ode ivp with initial conditions m(0) = m0, f(0) = f0, c(0) = c0, p(0) = p0, (24) thereby m0 > 0, f0 > 0, c0 = 0, p0 = 0. a conservation law for system (23) is m′ + f ′ + 2c′ = 0 implying the relation m + f + 2c = const = m0 + f0 = a. we have m = a−f − 2c in system (23) reducing it to f ′ = −k+f(a−f − 2c) + (k− + 2kc)c, c′ = k+f(a−f − 2c) − (k− + kc)c. (25) from (25) we have f ′ + c′ = kcc. assuming that the monomer m is abundant in a time interval ∆ we can apply the qssa principle and assume that c′ is approximately equal to 0 in ∆, and accordingly, in ∆ we have approximately f ′ = kcc [1], [4], [5], [13], [32], [35], [36], [37]. hence, for some t∗ ∈ ∆ we have f ′′(t∗) = kcc′(t∗) = 0, hence function f has inflection and has a sigmoidal form. in addition, from f ′ + c′ = kcc and c′(t∗) = 0, we obtain the slope at the inflection point: f ′(t∗) = kcc(t∗) > 0. we thus proved the following proposition iii.1. the solution f of the ivp (23)– (24) is a sigmoidal function. the slope at the inflection point is f ′(t∗) = kcc(t∗) > 0. the solutions m(t),f(t) of model (23) are visualized on figure 5 for the following input data k+ = 0.1,k− = 0.05; kc = 0.2; m0 = 10; f0 = 0.1; c0 = 0 (in the time interval [0, 200]). it should be noted that the fibril growth sigmoid function possesses a clearly expressed lag phase. remark. the inverse problem—so-called parameter identification problem—is to find out valfig. 5. graphs of the solutions m,f of model (23) ues for the rate parameters and the initial conditions by fitting (some of) the solutions to available time course experimental measurements. computational tools for the solution of this problem are proposed in [10], [11]. b. three variants of the basic model our basic model describes the aggregation of the monomer while the monomer is compressed and becomes part of the fibril. below we propose three more reaction networks. all they involve an intermediate product p representing the aggregated monomer before turning into fibril. the three models present possible mechanisms for the transition of the aggregated monomer into fibril particles. reaction network 1. an intermediate product p is added as follows: m + f k+−→←− k− c kc−→ f + p, p kp−→ f. (26) here the transition of the aggregated monomer p into fibril follows the decay-saturation mechanism (1). applying the mass action law we obtain the ode system dm/dt = −k+fm + k−c, df/dt = −k+fm + (k− + kc)c + kpp, dc/dt = k+fm− (k− + kc)c, dp/dt = kcc−kpp. (27) biomath 5 (2016), 1607311, http://dx.doi.org/10.11145/j.biomath.2016.07.311 page 8 of 11 http://dx.doi.org/10.11145/j.biomath.2016.07.311 s. markov, building reaction kinetic models for amiloid fibril growth fig. 6. graph of the solutions m,f,c,p of model (27) a conservation law for system (27) is m′ + f ′ + p′ + 2c′ = 0. the solutions of the system are visualized on fig. 6. for the following data: k1 = 2.62; k2 = 0.62; k3 = 2.62; k4 = 0.63; m0 = 1; f0 = 0.05; c0 = 0; p0 = 0. reaction network 2. an intermediate product p is involved as follows: m + f k+−→←− k− c kc−→ f + p kp−→ f + f. (28) the transition of the aggregated monomer p into fibril according to reaction network (28 follows the verhulst autocatalitic mechanism (10). applying the mass action law we obtain the ode system: dm/dt = −k+fm + k−c, df/dt = −k+fm + (k− + kc)c + kpfp, dc/dt = k+fm− (k− + kc)c, dp/dt = kcc−kpfp. (29) a conservation law for system (29) is m′+f ′+ p′ + 2c′ = 0. reaction network 3. here an intermediate product p is added as follows: m + f k+−→←− k− c kc−→ f + p k′−→←− k′′ cp kp−→ f + f. (30) in the above reaction network (30 the transition of the aggregated monomer p into fibril follows the henri enzyme kinetic mechanism (18). applying the mass action law we obtain: dm/dt = −k+fm + k−c, df/dt = −k+fm + (k− + kc)c + k′′cp −k′fp + 2kpcp, dc/dt = k+fm− (k− + kc)c, dp/dt = kcc−k′fp + k′′cp, dcp/dt = k ′fp− (k′′ + kp)cp. (31) a conservation law for system (31) is m′+f ′+ p′ + 2c′ + 2c′p = 0. iv. conclusion the proposed reaction networks (r-models) (22), (27), (29), (31) together with the implied reaction dynamical systems of equations (d-models) describe possible biochemical mechanisms of the fibril elongation in the citoplasm. the proposed models can be used to fit time course data for real measurements of fibril growth using fitting simulation procedures for the identification of the parameters. the variants producing a good fit will be considered as candidate for a probable mechanism of the amiloid fibrillation process on the base of its generating reaction network. the application of mathematical modelling in the field of amiloid fibrillation necessarily focuses attention to the lag phases of the growth curves that appear as model solutions. therefore it is an open problem to study the above discussed models with respect to this practically important issue. in particular the various reaction networks can be compared with respect to the lag phases of their solutions under various sets of parameters and initial conditions of the related differential ivps. acknowledgment the author is grateful to dr. n. kyurkchiev for his careful reading, useful suggestions, stimulating discussions and technical help. references [1] alt, r., s. markov, theoretical and computational studies of some bioreactor models, computers and mathematics with applications 64 (2012), 350–360. biomath 5 (2016), 1607311, http://dx.doi.org/10.11145/j.biomath.2016.07.311 page 9 of 11 http://dx.doi.org/10.11145/j.biomath.2016.07.311 s. markov, building reaction kinetic models for amiloid fibril growth [2] anguelov, r., markov, s.: hausdorff continuous interval functions and approximations, in: nehmeier, m. et al. 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[42] verhulst, p.-f., recherches mathematiques sur la loi d’accroissement de la population (mathematical researches into the law of population growth increase), nouveaux memoires de l’academie royale des sciences et belles-lettres de bruxelles 18: 1–42 (1845). [43] verhulst, p.-f., deuxieme memoire sur la loi d’accroissement de la population, memoires de l’academie royale des sciences, des lettres et des beaux-arts de belgique 20: 1–32 (1847). biomath 5 (2016), 1607311, http://dx.doi.org/10.11145/j.biomath.2016.07.311 page 11 of 11 http://dx.doi.org/10.11145/j.biomath.2016.07.311 introduction biological growth/transition models: three case studies case study 1: saturated growth case study 2: verhulst logistic growth lag time case study 3: growth models using henri's reaction scheme r-type models for amiloid fibrillation a basic verhulst-henri model three variants of the basic model conclusion references original article biomath 1 (2012), 1210231, 1–7 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum predator-prey model with prey harvesting, holling response function of type iii and sis disease jean jules tewa∗, ramses djidjou demasse† and samuel bowong‡ ∗ national advanced school of engineering, university of yaounde i , umi 209 ummisco, grimcape, yaounde, cameroon email: tewajules@gmail.com † faculty of science, university of yaounde i umi 209 ummisco, grimcape, yaounde, cameroon email: dramsess@yahoo.fr ‡ faculty of science, university of douala umi 209 ummisco, grimcape, yaounde, cameroon email: sbowong@gmail.fr received: 28 june 2012, accepted: 23 october 2012, published: 28 december 2012 abstract—the populations of prey and predator interact with prey harvesting. when there is no predator, the logistic equation models the behavior of the preys. for interactions between preys and predators, we use the generalized holling response function of type iii. this function which models the consumption of preys by predators is such that the predation rate of predators increases when the preys are few and decreases when they reach their satiety. our main goal is to analyze the influence of a sis infectious disease in the community. the epidemiological sis model with simple mass incidence is chosen, where only susceptibles and infectious are counted. we assume firstly that the disease spreads only among the prey population and secondly that it spreads only among the predator population. there are many bifurcations as: hopf bifurcation, transcritical bifurcation and saddle-node bifurcation. the results indicate that either the disease dies out or persists and then, at least one population can disappear because of infection. for some particular choices of the parameters however, there exists endemic equilibria in which both populations survive. numerical simulations on matlab and scilab are used to illustrate our results. keywords-predator; prey; infectious disease; response function; bifurcation; global stability i. introduction there are many epidemiological or ecological models [6], [7], [8], [9], [10], [11], [5] in the literature and also many models which encompass the two fields [3], [4], [8], [9], [10], [11], [12]. dynamic models for infectious diseases are mostly based on compartment structures that were initially proposed by kermack and mckendrick (1927,1932) and developed later by many other researchers. the main questions regarding population dynamics concern the effects of infectious diseases in regulating natural populations, decreasing their population sizes, reducing their natural fluctuations, or causing destabilizations of equilibria into oscillations of the population states. with the holling function response of type iii, it is well known that the predators increase their searching activity when the prey density increases. generally, if x denotes the density of prey population, the holling function of type i is φ1(x) = r x where r is the intrinsic growth rate of preys. the holling function of type ii is φ2(x) = b ω0 x 1 + b ω1 x , where ω0 and ω1 denote respectively the time taking by a predator to search and citation: j. tewa, r. d. demasse, s. bowong, predator-prey model with prey harvesting, holling response function of type iii and sis disease, biomath 1 (2012), 1210231, http://dx.doi.org/10.11145/j.biomath.2012.10.231 page 1 of 7 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.10.231 j. tewa et al., predator-prey model with prey harvesting, holling response function of type iii... capture preys, b is the predation rate per unit of time. in the models considered in this work, the holling function of type iii is used for interactions between predators and preys : φ3(x) = m x2 a x2 + b x + 1 [2], where m and a are positive constants, b is an arbitrary constant. this function models the consumption of preys by predators. it is well known that with this function, the predation rate of predators increases when the preys are few and decreases when they reach their satiety (a predator increases his searching activity when the prey density increases). the functions φ1, φ2 and φ3 are respectively also referred to as lotka-volterra, michaelis-menten and sigmoidal response functions. generally, there are more macroparasitic infections which can affect only preys, only predators or both preys and predators. our goal in this paper is to analyze the influence of a sis infectious disease which spreads only in one of the two populations. the models considered and analyzed here are different from all the models in the literature. moreover, we use numerical simulations on matlab and scilab to illustrate our results. ii. the model formulation the model (s1) is obtained from the classic lotkavolterra model with simple mass action when the disease spreads only inside the prey population. in this model, the infected preys do not reproduce and there is no disease related mortality. the model (s2) is obtained when the disease spreads only inside the predator population. these models are respectively  ẋ = r̃ ( 1 − x k̃ ) x − m̃x2y ãx2 + b̃x + 1 − λ̃x z +γ̃z − h̃1, ż = λ̃x z − γ̃z − m̃1z 2y ãz2 + b̃z + 1 , (s1) ẏ = c̃m̃x2y ãx2 + b̃x + 1 − m̃2z 2y ãz2 + b̃z + 1 − d̃y, x ≥ 0, z ≥ 0, y ≥ 0,  ẋ = r̃ ( 1 − x k̃ ) x − m̃x2y ãx2 + b̃x + 1 − η̃1x 2ω ãx2 + b̃x + 1 −h̃1, ẏ = c̃m̃x2y ãx2 + b̃x + 1 − d̃ y − δ̃ y ω + µ̃ ω, (s2) ω̇ = ẽm̃x2ω ãx2 + b̃x + 1 + δ̃yω − (µ̃ + d̃)ω, x ≥ 0, y ≥ 0, ω ≥ 0. where the variables z and ω denotes respectively the infected preys and infected predators, r̃ denotes the intrinsic growth rate of preys, d̃ is the natural death rate of predators, k̃ is the capacity of environment to support the growth of preys, h̃1 is the rate of preys’s harvesting, γ̃ and µ̃ are the recover rates of infected preys and infected predators respectively, λ̃ is the adequate contact rate between susceptible preys and infected preys while δ̃ is the adequate contact rate between susceptible predators and infected predators. we also assume that infected predators still can catch preys at a different rate η̃1 than sound ones. the parameter η̃1 can be thought to be less than m̃, if the disease affects the ability in hunting of the predators or larger than m̃, if we want to emphasize that the interactions with infected predators cause the preys to die for the disease even if they are not caught. ã and b̃ are positive constants. m̃ > 0 and m̃1 > 0 denote the adequate predation rate between predators and preys. c̃ and ẽ denote the conversion coefficients. m̃2 can be negative (conversion of prey’s biomass into predator’s biomass) or positive (bad effect of the infected preys for the predator population due to disease). trough the linear transformation and time scaling (x, z, y, w, t ) = ( x k̃ , z k̃ , y c̃k̃ , ω ẽk̃ , c̃m̃k̃2t ) , the following simplified systems are obtained from (s1) and (s2),   ẋ = ρx(1 − x) − p(x) y − λ x z + γ z − h1, ż = λ x z − γ z − m1 p(z) y, ẏ = p(x) y − m2 p(z) y − d y, x ≥ 0; y ≥ 0; z ≥ 0, (1)   ẋ = ρx(1 − x) − p(x) y − η1 p(x) ω − h1, ẏ = p(x)y − dy − δ y ω + µ ω, ω̇ = e p(x)ω + δ1 y ω − µ1 ω, x ≥ 0; y ≥ 0; ω ≥ 0, (2) where the parameters are defined as follow ρ = r̃ c̃m̃k̃2 , η1 = η̃1ẽ c̃m̃ , η2 = η̃2ẽ c̃m̃ , h1 = h̃1 c̃m̃k̃3 , λ = λ̃ c̃m̃k̃ , γ = γ̃ c̃m̃k̃2 , m1 = m̃1 m̃ , m2 = m̃2 c̃m̃ , m3 = m̃3 c̃m̃ , d = d̃ c̃m̃k̃2 , δ = δ̃ẽ c̃m̃k̃ , µ = µ̃ẽ c̃2m̃k̃2 , e = ẽ c̃ , δ1 = δ̃ m̃k̃ , µ1 = µ̃ + d̃ c̃m̃k̃2 , a = ãk̃2, b = b̃k̃, p(x) = x2 a x2 + b x + 1 . (3) systems (1) and system (2) are new and different from all the models in the literature. these models without disease give us the same system which has been analyzed without disease in [1]. biomath 1 (2012), 1210231, http://dx.doi.org/10.11145/j.biomath.2012.10.231 page 2 of 7 http://dx.doi.org/10.11145/j.biomath.2012.10.231 j. tewa et al., predator-prey model with prey harvesting, holling response function of type iii... iii. results a. results for the model (1) with disease only in prey population let us set u1(x) = ρx(1 − x) − h1 (1 + m)p(x) − md , r0 = m2(p(η) − d)u21(η) a(p(η) − d)2u21(η) + b(λη − γ)(p(η) − d)u1(η) + (λη − γ)2 the basic reproduction number, and x1 = 1 − √ 1 − 4 h1 ρ 2 , x2 = 1 + √ 1 − 4 h1 ρ 2 , xz = γ λ , x0 = 1 2 , z0 ∈ r∗+, (4) the expressions of the positive real values x0, x1, x2, xz . theorem 1: the equilibrium points of system (1), according to the values of the parameters, are given as follow : • when h1 > ρ 4 , then there is no equilibrium point. • when h1 = ρ 4 , then the unique equilibrium is b0(x0, 0, 0) which is a double point if d 6= 1 a + 2b + 4 and triple point if d = 1 a + 2b + 4 . • when h1 < ρ 4 and a d ≥ 1, then the equilibria are b1(x1, 0, 0) and b2(x2; 0; 0). • when h1 < ρ 4 ; a d < 1 and x3 = x1, then b1(x1, 0, 0) is a double point and b2(x2, 0, 0) exists. • when h1 < ρ 4 ; a d < 1 and x3 = x2, then b1(x1, 0, 0) is simple and b2(x2, 0, 0) is a double point. • when h1 < ρ 4 ; a d < 1 and x3 ∈]x1; x2[, then b1(x1, 0, 0); b2(x2, 0, 0) and b3(x3, 0, y3) exist, where y3 = ρx3(1 − x3) − h1 d > 0. • when h1 < ρ 4 ; a d < 1 and x3 ∈ [0; x1[∪]x2; +∞[, then b1(x1, 0, 0) and b2(x2, 0, 0) exist. • when h1 < ρ 4 ; ad < 1; x4 ∈]η; x2[, x2 > max ( x3; γ λ ) and r0 > 1, then b1(x1, 0, 0); b2(x2, 0, 0) and b4(x4, z4, y4) exist, where x4 > 0, z4 > 0 and y4 > 0. proof : these equilibria are obtained by setting the right hand side of (1) equals to zero. for y = 0 one has equation ρx2 − ρx + h1 = 0. then we have b0, b1 and b2. for z = 0, one has p(x) = d ⇐⇒ (1 − a d)x2 − b dx−d = 0. we deduce x3 and then b3. the condition for existence of b4 is p(z) = 1 m2 (p(x) − d) > 0 ie p(x) − d > 0 ⇐⇒ a d < 1 and x ∈]x3, +∞[. concerning the stability of these equilibria, the following theorem hold. theorem 2: let’s consider system (1). • the equilibria b0 and b1 are always unstable. • the equilibrium b2 is stable if one of the following conditions is satisfied : h1 < ρ 4 , γ λ ≥ x2 and p(x2) ≤ d, or h1 < ρ 4 , γ λ < x2, p(x2) = d and p′′(x2) ≤ 0. • the equilibrium b3 is stable if one of the following conditions is satisfied. h1 < ρ 4 , ad < 1, x3 ∈]x1; x2[ and x3 = γ λ , or h1 < ρ 4 , ad < 1, x3 ∈]x1; x2[, x3 < γ λ and d > 1 a + 2b + 4 , or h1 < ρ 4 , ad < 1, x3 ∈ ]x1; x2[, x3 < γ λ , d < 1 a + 2b + 4 and χ0(x3) < 0, where χ0(x3) is the eigenvalue of x3. • the equilibrium point b4(x4, z4, y4) is asymptotically stable if and only if the following conditions hold : a2 < 0; a2a1 + a0 > 0 and a1a0 > 0, where  a2 = ρ(1 − 2x4) − p′(x4)y4 − λz4 +λx4 − γ − m1p′(z4)y4, a1 = − [ρ(1 − 2x4) − p′(x4)y4 − λz4] × [λx4 − γ − m1p′(z4)y4] −λm1p(z4)y4 − p(x4)p′(x4)y4, a0 = − [ρ(1 − 2x4) − p′(x4)y4 − λz4] × [λx4 − γ − m1p′(z4)y4] + λm2p(x4)p′(z4)y4z4 +p′(x4)y4m1p(z4)(λx4 − γ) +p(x4)p′(x4)y4(λx4 − γ − m1p′(z4)y4). (5) proof : the eigenvalues of the jacobian matrix j(b0) are χ1 = 0; χ2 = λx0 − γ and χ3 = p(x0) − d. a) if γ λ < 1 2 or d < 1 a + 2b + 4 = p(x0), then χ2 > 0 or χ3 > 0 and b0 is unstable. b) if γ λ > 1 2 and d = 1 a + 2b + 4 = p(x0), then χ2 < 0 and χ3 = 0. hence, the stability of b0 is given by the center manifold theorem. the translation (u1, u2, u3) = (x − x0, z, y) brings the singular point b0 to the origin. in the neighborhood of the origin and, since h1 = ρ 4 , system (1) has a new form. the jacobian matrix j(b0) is not diagonalizable and the passage matrix to the jordan’s basis is biomath 1 (2012), 1210231, http://dx.doi.org/10.11145/j.biomath.2012.10.231 page 3 of 7 http://dx.doi.org/10.11145/j.biomath.2012.10.231 j. tewa et al., predator-prey model with prey harvesting, holling response function of type iii... p =   −1 0 10 0 −1 0 1 0  . by the transformation (v1, v2, v3)t = p −1(u1, u2, u3)t , the system becomes:  v̇1 = v2 + p′(x0)(v1v3 − v23) + p′′(x0) 2 (v21 + v 2 3 − 2v1v3)v2 + m1p ′′(0) 2 v2v 2 3 + o(|(v1, v2, v3)| 4), v̇2 = v3 + p′(x0)(v1v3 − v23) + p′′(x0) 2 (v21 + v 2 3 − 2v1v3)v2 − m2p ′′(0) 2 v2v 2 3 + o(|(v1, v2, v3)| 4), v̇3 = χ2v3 − λ(v1v3 − v23) + m1p ′′(0) 2 v2v 2 3 +o(|(v1, v2, v3)|4). (6) we can now find that the center manifold is given by w c = {v3 = 0}. therefore, the system (6) is topologically equivalent, around the origin, to the following system:  v̇1 = v2 + p′′(x0) 2 v21v2 + o(|(v1, v2)| 4), v̇2 = o(|(v1, v2)|4), v̇3 = o(|(v1, v2)|4). then, the singular point b0 is unstable. c) if γ λ = 1 2 and d = 1 a + 2b + 4 = p(x0), then χ2 = 0 and χ3 = 0. applying the center manifold theory as previously, b0 is unstable. d) if γ λ = 1 2 and d > 1 a + 2b + 4 = p(x0), we have χ2 = 0 and χ3 < 0. applying the center manifold theory as previously, b0 is unstable. the stability of b1 is obtained with jacobian matrix. the stability of b2 is obtained using the center manifold theorem. taking into account the fact that p(x3) = d, one find that the characteristic polynomial of the linearized system around the singular point b3 is q(χ) = (χ − λx3 + γ) [ −χ2 + (ρ(1 − 2x3) − p′(x3)y3)χ ] −d(χ − λx3 + γ)p′(x3)y3. the discriminant of q(χ) is ∆3(h1) = ( ρ(1 − 2x3) − p′(x3)y3 )2 − 4dp′(x3)y3. (7) a) if x3 > γ λ , then the eigenvalue χ1 = λx3 −γ > 0. hence, b3 is unstable. b) if x3 < γ λ , then χ1 < 0. b1) when ∆3(h1) = 0 the jacobian matrix at b3 has a double eigenvalue χ0(x3) := ρ(1 − 2x3) − p′(x3)y3 2 . (8) • if d ≥ 1 a + 2b + 4 , then x3 > 1 2 . from where χ0(h1) < 0. therefore, the singular point b3 is stable. • if d < 1 a + 2b + 4 , then: when χ0(h1) < 0 (resp. χ0(h1) > 0) the singular point b3 is stable (resp. unstable). b2) when ∆3 > 0 the eigenvalues of the jacobian matrix at b3 are χ1 < 0, χ2 = χ0(h1) − √ ∆3 2 and χ3 = χ0(h1) + √ ∆3 2 . we have, χ2χ3 = dp′(h1)y3 > 0 and χ2 + χ3 = χ0(h1), where χ0(h1) is defined by (8). • if d ≥ 1 a + 2b + 4 , then the singular point b3 is stable. • if d < 1 a + 2b + 4 , then: when χ0(h1) < 0 (resp. χ0(h1) > 0) the singular point b3 is stable (resp. unstable). b3) if ∆3 < 0, then the eigenvalues of the jacobian matrix at b3 are χ1 < 0, χ2 = χ0(h1) − i √ −∆3 2 and χ3 = χ0(h1) + i √ −∆3 2 , where χ0(h1) is defined by (8). if d ≥ 1 a + 2b + 4 , then the singular point b3 is stable. if d < 1 a + 2b + 4 and χ0(h1) < 0 then, the singular point b3 is stable. if d < 1 a + 2b + 4 and χ0(h1) > 0 then, the singular point is unstable. if d < 1 a + 2b + 4 and χ0(h1) = 0 then, the real central and stable spaces are respectively defined by ec = 〈(1, 0, 0); (0, 0, 1)〉 and es = 〈 (1, −1 − dp′(x3)y3 χ21 , p′(x3)y3 χ1 ) 〉 . then applying the center manifold theorem it comes that the singular point b3 is unstable. the stability of b4 is obtained using the routh-hurwitz conditions. biomath 1 (2012), 1210231, http://dx.doi.org/10.11145/j.biomath.2012.10.231 page 4 of 7 http://dx.doi.org/10.11145/j.biomath.2012.10.231 j. tewa et al., predator-prey model with prey harvesting, holling response function of type iii... b. results for the model (2) with disease only inside predator population let us set u2(x) = e δ [ µ1 e − p(x) ] and v2(x) = (p(x) − d)u2(x) e [ d e − p(x) ] . let x5 the eventual positive root of equation p(x5) = d e and the function g2(x) = ρx(1 − x) − h1 − p(x)u2(x) − η1p(x)v2(x). hypothesis 1 : the attack of non-infected predators is more important than the one of the infected predators i.e. e = ẽ c̃ ≤ 1. theorem 3: the equilibria of system (2), where x0; x1 and x2 are given by (4), according to the values of the parameters, are given as follow. • when h1 > ρ 4 , then there is no equilibrium point. • when h1 = ρ 4 , then c0(x0; 0; 0) is a double point if d 6= 1 a + 2b + 4 and triple point if d = 1 a + 2b + 4 . • when h1 < ρ 4 and a d ≥ 1, then c1(x1; 0; 0) and c2(x2; 0; 0) exist. • when h1 < ρ 4 ; a d < 1 and x3 = x1, then c1(x1; 0; 0) is a double point and c2(x2; 0; 0) exists. • when h1 < ρ 4 ; a d < 1 and x3 = x2, then c1(x1; 0; 0) exists and c2(x2; 0; 0) is a double point. • when h1 < ρ 4 ; a d < 1 and x3 ∈]x1; x2[, then the equilibria are c1(x1; 0; 0); c2(x2; 0; 0) and c3(x3; y3; 0), where y3 = ρx3(1 − x3) − h1 d > 0. • when h1 < ρ 4 ; a d < 1 and x3 ∈ [0; x1[∪]x2; +∞[, then the equilibria are c1(x1; 0; 0) and c2(x2; 0; 0). • when h1 < ρ 4 ; a d < 1; a d e ≥ 1, x6 ∈ ]x1; x2[∩]x3; +∞[; x2 > x3 or h1 < ρ 4 ; a d e < 1, x6 ∈]x1; x2[∩]x3; x5[; x2 > x3; x1 < x5, then the equilibria are c1(x1; 0; 0); c2(x2; 0; 0) and c4(x6; y6; ω6), y6 = u2(x6) and ω6 = v2(x6). proof : the equilibria c0, c1, c2 and c3 are obtained in the same way as in theorem 1, setting the right hand side of the system equals to zero. equilibrium c4 exists when the previous conditions are satisfied. concerning the stability analysis of these equilibria, the following theorem holds. theorem 4: let’s consider the system (2) and suppose that hypothesis 1 holds. • the equilibria c0 and c1 are always unstable. • the equilibrium c2 is stable if h1 < ρ 4 and p(x2) < d. • the equilibrium c3 is stable if and only if one of these conditions is satisfied : h1 < ρ 4 , ad < 1, x3 ∈ ]x1; x2[ and y3 = e δ1 ( µ1 e − d ) or h1 < ρ 4 , ad < 1, x3 ∈]x1; x2[, y3 < e δ1 ( µ1 e − d ) , d > p(x0), or h1 < ρ 4 , ad < 1, x3 ∈]x1; x2[, y3 < e δ1 ( µ1 e − d ) , d < p(x0), ξ0(x3) < 0. • the singular point c4(x6, y6, ω6) is asymptotically stable if and only if the following conditions are satisfied : b2 < 0; b2b1 + b0 > 0 and b1b0 > 0, where  b2 = ρ(1 − 2x6) − p′(x6)(y6 + η1ω6) +p(x6) − d − δω6; b1 = − (ρ(1 − 2x6) − p′(x6)(y6 + η1ω6)) × (p(x6) − d − δω6) + δ1ω6(µ − δy6) −p(x6)p′(x6)y6 − eη1p(x6)p′(x6)ω6; b0 = ep(x6)p′(x6)ω6 [δy6 − µ + η1(p(x6) − d − δω6)] −δ1η1p(x6)p′(x6)y6ω6 −δ1ω6(µ − δy6) (ρ(1 − 2x6) − p′(x6)(y6 + η1ω6)) . (9) proof : the stability of c0 is deduce as for b0 in theorem 2. the jacobian matrix always has a positive eigenvalue. then, c1 is unstable. we obtain the stability of c2 and c3 applying the same arguments as for b2 and b3 in theorem 2. the stability of c4 is obtained using the routh-hurwitz conditions. iv. hopf bifurcation let us introduce the following parameters h10 = ρx3 bx3 + 2 [ 2ax33 + (b − a)x 2 3 + 1 ] , (10) and π = 1 16 [ p(2)(x3) + p (3)(x3) ] − (p′(x3))2 4 √ −∆3(h10) . (11) recalling (4), the flow of system (1) and system (2) respectively undergo a supercritical hopf bifurcation around h10 given by the following result theorem 5: (hopf bifurcation) let h1 < ρ 4 ; ad < 1; x3 ∈]x1, min ( 1 2 , γ λ ) [. thanks to hypothesis 1. then, a unique stable curve of periodic solution bifurcates from the singular points b3 and c3 into the regions h1 > h10 if π < 0 or h1 < h10 if π > 0. the singular points b3 and c3 are stable for h1 < h10 and unstable for biomath 1 (2012), 1210231, http://dx.doi.org/10.11145/j.biomath.2012.10.231 page 5 of 7 http://dx.doi.org/10.11145/j.biomath.2012.10.231 j. tewa et al., predator-prey model with prey harvesting, holling response function of type iii... h1 ≥ h10. this correspond to supercritical stable hopf bifurcation. proof : the proof can be obtained as in [13]. v. numerical simulations 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 3 time (t) x( t), z (t) , y (t) non infected preys infected preys non infected predators (a) 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 0 1 2 infected preys (z) non infected preys (x) n on in fe ct ed p re da to rs (y ) initial condition trajectory initial condition trajectory initial condition trajectory equlibrium b1 equilibrium b2 invariant axis ( 1) (b) fig. 1. phase portraits of system (1) for h1 < ρ 4 ; γ λ = x2 and p(x2) < d. b1 and b2 are unstable. the axis x = γ λ is stable. 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 0 0.5 1 1.5 2 infected preys (z) non infected preys (x) n on in fe ct ed p re da to rs (y ) initial condition trajectory equilibrium b1 equilibrium b2 (a) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.50 0.5 1 1.5 2 2.5 3 time (t) x( t), z (t) , y (t) non infected preys infected preys non infected predators (b) fig. 2. phase portraits of system (1). the case (a) corresponds to h1 < ρ 4 ; γ λ < x2 and p(x2) < d. the case (b) corresponds to h1 = ρ 4 ; γ λ > 1/2 and d = 1 a + 2b + 4 . unstability of b1 and b2. 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 3 time (t) x( t), y (t) , (t ) non infected preys non infected predators infected predators (a) 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 non infected predators (y)non infected preys (x) in fe ct ed p re da to rs ( ) initial condition trajectory equilibrium c1 equilibrium c2 (b) fig. 3. phase portraits of system (2) for h1 < ρ 4 and d > p(x2). stability of c2. 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 time (t) x( t), y (t) , (t ) non infected preys non infected predators infected predators (a) 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 0 10 20 30 40 non infected predators (y)non infected preys (x) in fe ct ed p re da to rs ( ) initial condition trajectory equilibrium c1 equilibrium c2 equilibrium c3 (b) fig. 4. phase portraits of system (2) for h1 < ρ 4 and d < p(x2). unstability of c1, c2 and c3. (a) (b) fig. 5. phase portraits of system (1). the case (a) corresponds to h1 < ρ γ λ ( 1 − γ λ ) . the case (b) corresponds to h1 > ρ γ λ ( 1 − γ λ ) and γ λ < 1 2 . illustration of saddle-node bifurcation phenomenon. vi. conclusion our goal was to analyze the modifications on a predator prey model (generalized gause model) with prey harvesting and holling response type iii : m x2 a x2 + b x + 1 , to account for a disease spreading among one of the two species. the simple epidemiological model sis has been chosen, where only susceptibles and infectives are counted. the results indicate that either the disease dies out, leaving only neutral cycles of generalized gause model, or one species disappears and all individuals in the other one eventually become infected. for some particular choices of the parameters however, endemic equilibria in which both populations survive seem to arise. references [1] r.m. etoua and c. rousseau, bifurcation analysis of a generalized gause model with prey harvesting and a generalized holling response function of type iii , j. differ. equations 249, no. 9, 2316–2356 (2010), issn 0022–0396. [2] a.d. bazykin, a. iosifovich khibnik and b. krauskopf, nonlinear dynamics of interacting populations, world scientific, 1998, 193 pages. [3] k.p. hadeler and h.i. freedman, predator-prey populations with parasitic infection, j. math. biol. 27, (1989) 609–631. http://dx.doi.org/10.1007/bf00276947 [4] e. venturino, the influence of diseases on lotka-volterra systems, rocky mt. j. math. 24, (1994) 381402. 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[6] j.j. tewa, r. fokouop, b. mewoli and s. bowong, mathematical analysis of a general class of ordinary differential equations coming from within-hosts models of malaria with immune effectors, applied mathematics and computation 218, (2012) 7347–7361. http://dx.doi.org/10.1016/j.amc.2011.10.085 [7] j. j. tewa, s. bowong, c. s. oukouomi noutchie; mathematical analysis of two-patch model of tuberculosis disease with staged progression, applied mathematical modelling 36, (2012) 5792– 5807. [8] m. haque, d. greenhalgh, when predator avoids infected prey: a model based theoretical studies. ima j. math. med. biol. 27 (2009) 75–94. http://dx.doi.org/10.1093/imammb/dqp007 [9] n. bairagi, p. k. roy, j. chattopadhyay; role of infection on the stability of a predator-prey system with several response functions—a comparative study, journal of theoretical biology, 248 (1), (2007) 10–25, issn 0022–5193. [10] krishna padas das, kusumika kundu, j. chattopadhyay, a predator-prey mathematical model with both populations affected by diseases, ecological complexity 8, (2011) 68–80. http://dx.doi.org/10.1016/j.ecocom.2010.04.001 [11] m. haque, j. zhen, e. venturino; rich dynamics of lotkavolterra type predator-prey model system with viral disease in prey species; mathematical methods in the applied science 32, (2009) 875–898. [12] j. j. tewa, v. yatat djeumen, s. bowong, predator-prey model with holling response function of type ii and sis infectious disease, applied mathematical modelling, (2012) to appear. [13] y. a. kuznetsov, elements of applied bifurcation theory: third edition, appl. math. sci. 112, springer vergal, new york, 2004. biomath 1 (2012), 1210231, http://dx.doi.org/10.11145/j.biomath.2012.10.231 page 7 of 7 http://dx.doi.org/10.1016/j.amc.2011.10.085 http://dx.doi.org/10.1093/imammb/dqp007 http://dx.doi.org/10.1016/j.ecocom.2010.04.001 http://dx.doi.org/10.11145/j.biomath.2012.10.231 introduction the model formulation results results for the model (??) with disease only in prey population results for the model (??) with disease only inside predator population hopf bifurcation numerical simulations conclusion references www.biomathforum.org/biomath/index.php/biomath original article a parameter uniform almost first order convergent numerical method for a non-linear system of singularly perturbed differential equations r.ishwariya∗, j.princy merlin†, j.j.h.miller‡, s.valarmathi§ ∗department of mathematics, bishop heber college, tiruchirappalli, tamilnadu, india. ishrosey@gmail.com †department of chemistry, bishop heber college, tiruchirappalli, tamilnadu, india. pmej 68@yahoo.co.in ‡institute of numerical computation and analysis, dublin, ireland. jm@incaireland.org §department of mathematics, bishop heber college, tiruchirappalli, tamilnadu, india. valarmathi07@gmail.com received: 30 september 2015, accepted: 11 august 2016, published: 11 september 2016 abstract—in this paper, a biochemical reaction namely michaelis-menten kinetics has been modeled as an initial value problem (ivp) for a system of singularly perturbed first order nonlinear differential equations with prescribed initial values. a new numerical method has been suggested to solve the problem of michaelis-menten kinetics. the novelty is that, a variant of the continuation technique suggested in [2] with the classical finite difference scheme is used on an appropriate shishkin mesh instead of the usual adaptive mesh approach normally used for such problems. in addition to that, a new theoretical result is included which establishes the parameter uniform convergence of the method. from the computational results inferences are derived. keywords-singular perturbation problems, boundary layers, nonlinear differential equations, finite difference schemes, shishkin mesh, parameter uniform convergence. i. introduction a differential equation in which small parameters multiply the highest order derivative and some or none of the lower order derivatives is known as a singularly perturbed differential equation. in this paper, an initial value problem for a sytem of singularly perturbed nonlinear differential equations is considered. in [9], a parameter uniform numerical method for a class of singularly perturbed nonlinear scalar initial value problems is constructed. also results for a problem where two reduced solutions intersect are discussed. in [11], the asymptotic behaviour of the solution for a nonlinear singular singularly perturbed initial value problem is studied. in [3], a numerical method, for a system of singularly perturbed semilinear reaction-diffusion equations, involving an appropriate layer-adapted piecewise citation: r.ishwariya, j.princy merlin, j.j.h.miller, s.valarmathi, a parameter uniform almost first order convergent numerical method for a non-linear system of singularly perturbed differential equations, biomath 5 (2016), 1608111, http://dx.doi.org/10.11145/j.biomath.2016.08.111 page 1 of 8 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2016.08.111 r.ishwariya et al., a parameter uniform almost first order convergent numerical method ... uniform mesh, is constructed and it is proved to be parameter uniform convergent. in [10], a numerical method for a singular singularly perturbed initial value problem for a system of nonlinear differential equations is suggested. singularly perturbed differential equations have wide applications in biology. in [5], a parameter-uniform numerical method for a model of tumour growth, which is a first order semilinear system of singularly perturbed delay differential equations, is suggested and is proved to be almost first order convergent. one of the applications of singularly perturbed differential equations in the field of biochemistry is the modelling of enzyme kinetics. in [12], analytical approximations to the equation for modelling michaelis-menten kinetics are provided. in [13], the enzyme kinetic reaction scheme originally proposed by v. henri is considered and a method of computing the rate constants, whenever time course experimental data are available is also demonstrated. in [14], a new approach is developed for the study of the michaelis-menten kinetic equations at all times t, based on their solution in the limit of large t. in the present paper, an enzyme kinetic reaction is considered in the form of an initial value problem for a system of singularly perturbed nonlinear differential equations of first order. the changes in the solution components are studied and illustrated numerically. it is also proved that the numerical method considered is parameter uniform and is almost first order convergent. a biochemical reaction is a process of interaction of two or more substances to produce another substance. biochemical reactions are continually taking place in all living organisms and in all body processes. such reactions involve proteins called enzymes, which act as remarkably efficient catalysts. enzyme kinetics is the study of the chemical reactions that are catalysed by enzymes. in enzyme kinetics, the reaction rate is measured and the effects of varying the conditions of the reaction are investigated. enzymes react selectively on definite compounds called substrates. one of the most basic enzymatic reactions, that occurs in henrimichaelis-menten kinetics (1900) and michaelismenten kinetics (1913), involves a substrate s reacting with an enzyme e to form a complex se which in turn is converted into a product p and the enzyme. we represent this schematically by s + e k1 k−1 se k2→ p + e, (1) where k1,k−1 and k2 are constant parameters associated with the rates of reaction. as in [6], by using the law of mass action in (1), a system of nonlinear differential equations is obtained ds dt = −k1es + k−1c, de dt = −k1es + (k−1 + k2)c, (2) dc dt = k1es− (k−1 + k2)c, dp dt = k2c, with the initial conditions s(0) = s0, e(0) = e0, c(0) = 0, p(0) = 0, where s,e,c,p denote the concentration of s, e, se, p respectively. it is not hard to get p(t) from the last expression of (2). observing that de dt + dc dt = 0 and introducing the dimensionless quantities as in [6], (2) reduces to du dτ = −u + (u + k −λ)v, ε dv dτ = u− (u + k)v, (3) u(0) = 1, v(0) = 0. biomath 5 (2016), 1608111, http://dx.doi.org/10.11145/j.biomath.2016.08.111 page 2 of 8 http://dx.doi.org/10.11145/j.biomath.2016.08.111 r.ishwariya et al., a parameter uniform almost first order convergent numerical method ... often the enzyme concentration e0 is very small compared with that of substrate concentration s0; in such situations it follows that e0 s0 = ε � 1. this system is said to be partially singularly perturbed, since only the second equation of the system is singularly perturbed. it is to be noted that the right hand sides of (3) are nonlinear. motivated by this, the following more general nonlinear system of singularly perturbed differential equations is now considered. ~t~u(x) = e~u ′(x) + ~f(x,u1,u2) = ~0 on (0, 1], ~u(0) = ~u0, (4) where x ∈ [0, 1], ~u(x) = (u1(x),u2(x))t and ~f(x,u1,u2) = (f1(x,u1,u2),f2(x,u1,u2)) t . e is the 2 × 2 matrix, e = diag(~ε ), ~ε = (ε1,ε2) with 0 < ε1 ≤ ε2 ≤ 1. it is assumed that the nonlinear terms satisfy ∂fk ∂uk ≥ β > 0, ∂fk ∂uj ≤ 0, k,j = 1, 2, k 6= j, (5) min 1≤i≤2   2∑ j=1 ∂fi ∂uj   ≥ α > 0 (6) for all ~f defined on [0, 1] ×r2. these conditions (5), (6) and the implicit function theorem [8], ensure that a unique solution ~u ∈ c2 = c × c, where c = c0([0, 1]) ∩ c2((0, 1]), exists for problem (4). for any vector-valued function ~y on [0, 1] the following norms are introduced: ‖ ~y ‖= max{‖ y1 ‖, ‖ y2 ‖} and ‖ yi ‖= sup x∈[0,1] |yi(x)|. the mesh ω n = {xi}ni=0 is the set of points satisfying 0 = x0 < x1 < ... < xn = 1. a mesh function v = {v (xi)}ni=0 is a real valued function defined on ω n . the discrete maximum norm for such mesh functions is defined by ‖ v ‖ ω n = maxi=0,1,...,n|v (xi)| and ‖ ~v ‖ ω n = max{‖ v1 ‖ωn ,‖ v2 ‖ ωn} where the vector-valued mesh function is defined by ~v = (v1,v2) t = {v1(xi),v2(xi)}t , i = 0, 1, ...,n. throughout the paper c denotes a generic positive constant, which is independent of x and of all singular perturbation and discretization parameters. furthermore, inequalities between vectors are understood in the componentwise sense. ii. analytical results the problem (4) can be rewritten in the form ε1u ′ 1(x) + f1(x,u1,u2) = 0, ε2u ′ 2(x) + f2(x,u1,u2) = 0, x ∈ (0, 1], ~u(0) = ~u0. (7) the reduced problem corresponding to (7) is given by f1(x,r1,r2) = 0, f2(x,r1,r2) = 0, x ∈ (0, 1]. (8) since ~r ∈ c2, ~f(x,r1,r2) is sufficiently smooth. also from (8) and the conditions (5) and (6), the implicit function theorem [8] ensures the existence of a unique solution for (8). this solution ~r has derivatives which are bounded independently of ε1 and ε2. hence, |r(k)1 (x)| ≤ c; |r (k) 2 (x)| ≤ c; k = 0, 1, 2, 3; x ∈ [0, 1]. a shishkin decomposition [1], [2] of the solution ~u is considered: ~u = ~v + ~w, where the smooth component ~v(x) is the solution of the problem e~v ′(x) + ~f(x,v1,v2) = ~0, x ∈ (0, 1], ~v(0) = ~r(0), and the singular component ~w(x) satistfies e~w ′(x) + ~f(x,v1 + w1,v2 + w2) −~f(x,v1,v2) = ~0, x ∈ (0, 1], ~w(0) = ~u(0) −~v(0). (9) the bounds of the derivatives of smooth component are given in biomath 5 (2016), 1608111, http://dx.doi.org/10.11145/j.biomath.2016.08.111 page 3 of 8 http://dx.doi.org/10.11145/j.biomath.2016.08.111 r.ishwariya et al., a parameter uniform almost first order convergent numerical method ... lemma 1: the smooth component ~v(x) satisfies |v(k)i (x)| ≤ c, i = 1, 2; k = 0, 1 and |v′′i (x)| ≤ cε −1 i , i = 1, 2. proof: as in [3] and [5] the smooth component ~v is further decomposed as follows: ~v = ~̃q + ~̂q. here ~̂q is the solution of f1(x, q̂1, q̂2) = 0, (10) ε2 dq̂2 dx + f2(x, q̂1, q̂2) = 0, x ∈ (0, 1], (11) with q̂2(0) = v2(0), and ~̃q is the solution of ε1 dq̃1 dx + f1(x, q̃1 + q̂1, q̃2 + q̂2) −f1(x, q̂1, q̂2) = −ε1 dq̂1 dx , (12) ε2 dq̃2 dx + f2(x, q̃1 + q̂1, q̃2 + q̂2) −f2(x, q̂1, q̂2) = 0, (13) x ∈ (0, 1], with q̃1(0) = q̃2(0) = 0. using (8), (10) and (11), a11(x)(q̂1 −r1) + a12(x)(q̂2 −r2) = 0, (14) ε2 d dx (q̂2 −r2) + a21(x)(q̂1 −r1) + a22(x)(q̂2 −r2) = −ε2 dr2 dx , (15) where aij(x) = ∂fi ∂uj (x,ξi(x),ηi(x)), i,j = 1, 2, and ξi(x), ηi(x) are intermediate values. using (14) in (15) then gives ε2 d dx (q̂2 −r2) + ( a22(x) − a12(x)a21(x) a11(x) ) (q̂2 −r2) = −ε2 dr2 dx . consider now the linear operator lz(x) := ε2z ′(x) + ( a22(x) − a12(x)a21(x) a11(x) ) z(x) = −ε2 dr2 dx , where z = q̂2 −r2. this operator satisfies a maximum principle, see [1]. thus, ‖ q̂2 −r2 ‖≤ cε2 and ‖ d(q̂2 −r2) dx ‖≤ c. using this in (14), ‖ q̂1 −r1 ‖≤ cε2. differentiating (14), we get ‖ d(q̂1 −r1) dx ‖≤ c. hence, ‖ q̂2 ‖≤ c, ‖ dq̂2 dx ‖≤ c and ‖ q̂1 ‖≤ c. differentiating (15), ε2 d2 dx2 (q̂2 −r2) + a′21(x)(q̂1 −r1) + a21(x) d dx (q̂1 −r1) + a′22(x)(q̂2 −r2) + a22(x) d dx (q̂2 −r2) = −ε2 d2r2 dx2 . hence, ‖ d2q̂2 dx2 ‖≤ cε−12 . differentiating (14) twice and using the above estimates, it follows that ‖ d2q̂1 dx2 ‖≤ cε−12 . by using the assumption that ε1 ≤ ε2, we find that ‖ d2q̂1 dx2 ‖≤ cε−11 . expressions (12), (13) and the mean-value theorem for ~f, lead to ε1 dq̃1 dx + a∗11(x)q̃1 + a ∗ 12(x)q̃2 = −ε1 dq̂1 dx , (16) ε2 dq̃2 dx + a∗21(x)q̃1 + a ∗ 22(x)q̃2 = 0, (17) where q̃1(0) = q̃2(0) = 0. (18) here, a∗ij(x) = ∂fi ∂uj (x,ζi(x),χi(x)), i,j = 1, 2; ζi(x), χi(x) are intermediate values. biomath 5 (2016), 1608111, http://dx.doi.org/10.11145/j.biomath.2016.08.111 page 4 of 8 http://dx.doi.org/10.11145/j.biomath.2016.08.111 r.ishwariya et al., a parameter uniform almost first order convergent numerical method ... from equations (16)-(18), by using maximum principle [4], we obtain ‖ q̃i ‖≤ c, i = 1, 2, ‖ dq̃i dx ‖≤ c, i = 1, 2, ‖ d2q̃i dx2 ‖≤ cε−1i , i = 1, 2. hence the bounds for ~v can be obtained using the bounds of ~̃q and ~̂q. lemma 2: the singular component ~w(x) satisfies, for any x ∈ [0, 1], |wi(x)| ≤ cb2(x), i = 1, 2, |w′1(x)| ≤ c(ε −1 1 b1(x) + ε −1 2 b2(x)), |w′2(x)| ≤ cε −1 2 b2(x), |w′′i (x)| ≤ cε −1 i (ε −1 1 b1(x) + ε −1 2 b2(x)), i = 1, 2, where bi(x) = e (−αx εi ) . proof: from equations (9), ε1w ′ 1(x) + s11(x)w1(x) + s12(x)w2(x) = 0, (19) ε2w ′ 2(x) + s21(x)w1(x) + s22(x)w2(x) = 0, (20) for x ∈ (0, 1], and w1(0) = u1(0) −v1(0), w2(0) = u2(0) −v2(0). here, sij(x) = ∂fi ∂uj (x,λi(x),θi(x)), i = 1, 2; λi(x), θi(x) are intermediate values. from (19) and (20), the bounds of the singular component ~w can be derived as in [4]. iii. shishkin mesh for the case ε1 < ε2, a piecewise uniform shishkin mesh ω n with n mesh-intervals is constructed on ω = [0, 1] as follows. the interval [0, 1] is subdivided into 3 sub-intervals [0,τ1] ∪ (τ1,τ2] ∪ (τ2, 1], where τ2 = min { 1 2 , ε2 α lnn } and τ1 = min {τ2 2 , ε1 α lnn } . clearly 0 < τ1 < τ2 ≤ 12. then, on the subinterval (τ2, 1] a uniform mesh with n2 meshpoints is placed and on each of the sub-intervals (0,τ1] and (τ1,τ2], a uniform mesh of n4 points is placed. note that, when both the parameters τr, r = 1, 2, take on their lefthand value, the shishkin mesh becomes a classical uniform mesh on [0, 1]. in the case, ε1 = ε2 a simple construction with just one parameter τ is sufficient. iv. discrete problem the initial value problem (7) is discretised using the backward euler scheme on the piecewise uniform mesh ω n . the discrete problem is ~tn ~u(xj) : = ed −~u(xj) + ~f(xj,u1(xj),u2(xj)) = ~0, j = 1, ...,n, ~u(0) = ~u(0), (21) where d−~u(xj) = ~u(xj) − ~u(xj−1) xj −xj−1 . this discrete problem (21) is to be solved on the shishkin mesh defined above, using the continuation algorithm [2]. lemma 3: for any mesh functions ~y and ~z with ~y (0) = ~z(0), ‖ ~y − ~z ‖≤ c ‖ ~tn ~y − ~tn ~z ‖ . proof: ~tn ~y (xj) − ~tn ~z(xj) = ed−~y (xj) + ~f(xj,y1(xj),y2(xj)) −ed−~z(xj) − ~f(xj,z1(xj),z2(xj)) = ed−(~y − ~z)(xj) + j(~f, ~u)(~y − ~z)(xj) = (~t ′n )( ~y − ~z)(xj), biomath 5 (2016), 1608111, http://dx.doi.org/10.11145/j.biomath.2016.08.111 page 5 of 8 http://dx.doi.org/10.11145/j.biomath.2016.08.111 r.ishwariya et al., a parameter uniform almost first order convergent numerical method ... where ~t ′n is the frechet derivative of ~tn, j(~f, ~u) =   ∂f1 ∂u1 (xj,ξ11,η11) ∂f1 ∂u2 (xj,ξ12,η12) ∂f2 ∂u1 (xj,ξ21,η21) ∂f2 ∂u2 (xj,ξ22,η22)   and ξik(xj),ηik(xj), i,k = 1, 2 are points occuring in the mean value theorem. since ~t ′n is linear, it satisfies the discrete maximum principle and the discrete stability result [4]. hence, ‖ ~y − ~z ‖≤ c ‖ ~t ′n (~y − ~z) ‖= c ‖ ~tn ~y − ~tn ~z ‖ and the lemma is proved. parameter uniform bounds for the error are given in the following theorem, which is the main result of this paper. theorem 1: let ~u be the solution of the problem (4) and ~u be the solution of the discrete problem (21). then, there exists an n0 > 0 such that, for all n ≥ n0, ‖ ~u −~u ‖≤ cn−1lnn. here, c and n0 are independent of ε1,ε2 and n. proof: let x ∈ ωn. from the above lemma, ‖ (~u −~u)(xj) ‖≤ c ‖ (~tn ~u − ~tn~u)(xj) ‖. since ‖ ~tn~u (xj) ‖=‖ (~tn~u− ~tn ~u)(xj) ‖, it follows that ‖ (~tn~u− ~tn ~u)(xj) ‖ = ‖ ~tn~u(xj) ‖ = ‖ (~tn~u− ~t~u)(xj) ‖ = e ‖ (d−~u−~u ′)(xj) ‖ ≤ e ‖ (d−~v −~v ′)(xj) ‖ + e ‖ (d−~w − ~w ′)(xj) ‖ . since the bounds for ~v and ~w are the same as in [4], the required result follows. v. numerical results the numerical method proposed above is illustrated through the example presented in this section. experimental data for the hydrolysis of starch by amylase is used in the numerical illustration. in this experiment starch (c6h12o6)n is the substrate, amylase is the enzyme and maltose n(c12h22o11) is the product, which is represented as follows: (c6h12o6)n starch + nh2o amylase −→ n(c12h22o11) maltose by the procedure explained in the introduction, the enzyme kinetics of the above reaction leads to the formulation of the following example: consider the initial value problem ε du1 dx = u2 − (u2 + k)u1, du2 dx = −u2 + (u2 + k −λ)u1, for x ∈ [0, 1], u1(0) = 0, u2(0) = 1. here x,u1,u2 are the dimensionless quantities as in [6] and are given by, x = k1e0t, u1(x) = c(t) e0 , u2 = s(t) s0 . based on the experiments and analysis carried out in the departments of zoology and chemistry of bishop heber college, tiruchirappalli, tamilnadu, india, and [7], the values of k and λ are taken to be 0.0477mg−1 and 0.0454mg−1. observations: the solution profile of the above problem exhibits interesting facts. the solution component u1 of ~u, representing the ratio of concentration of the complex at time t to initial concentration of the substrate, exhibits initial layer in the neighbourhood of t = 0. the second component u2 exhibits no layer in the domain of the definition of the problem. the ratio of the concentration of the complex to that of enzyme has a rapid increase initially but biomath 5 (2016), 1608111, http://dx.doi.org/10.11145/j.biomath.2016.08.111 page 6 of 8 http://dx.doi.org/10.11145/j.biomath.2016.08.111 r.ishwariya et al., a parameter uniform almost first order convergent numerical method ... as time increases, this ratio remains smooth and reaches equilibrium position. on the other hand, the ratio of the concentration of the substrate at time t to the initial concentration of the substrate decreases smoothly. the maximum pointwise errors and the rate of convergence for this ivp are calculated using a two mesh algorithm for a vector problem which is a variant of the one found in [2] for a scalar problem. the results are presented in table 1 for the domain [0,1]. table 1 values of dn~ε ,d n,pn,p∗ and cnp∗ for α = 0.9. ε number of mesh points n 512 1024 2048 4096 2−1 0.361e-03 0.181e-03 0.905e-04 0.453e-04 2−3 0.143e-02 0.717e-03 0.359e-03 0.180e-03 2−5 0.217e-02 0.123e-02 0.688e-03 0.380e-03 2−7 0.217e-02 0.123e-02 0.688e-03 0.379e-03 2−9 0.217e-02 0.123e-02 0.688e-03 0.379e-03 dn 0.217e-02 0.123e-02 0.688e-03 0.380e-03 pn 0.819e+00 0.840e+00 0.859e+00 cnp∗ 0.833e+00 0.833e+00 0.821e+00 0.799e+00 computed order of ~ε−uniform convergence, p∗ = 0.8194415 computed ~ε−uniform error constant, c∗p∗ = 0.8329233 the notations dn, pn and cnp∗ denote the ~ε-uniform maximum pointwise two-mesh differences, the ~ε-uniform order of convergence and the ~ε-uniform error constant respectively and given by dn = max ~ε dn~ε where dn~ε = ‖ ~u n ~ε − ~u 2n ~ε ‖ωn , p n = log2 dn d2n and cnp∗ = dnnp ∗ 1 − 2−p∗ . then the parameter uniform order of convergence and error constant are given by p∗ = min n pn and c∗p∗ = max n cnp∗. for ε = 2−5 and n = 512, the solution of the example is presented in the form of a figure in the domain [0,1](figure 1). and from figure 2, figure 1. numerical solution for ε = 2−5 and n = 512 in the domain [0,1]. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 u1 u2 figure 2. numerical solution for ε = 2−5 and n = 512 in the domain [0,30]. 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 u1 u2 it could be noted that as the domain increases u1 and u2 approach zero since, as the time increases the concentrations s and c become zero. in both the graphs, the component u1, exhibits an initial layer whereas u2 exhibits no layer. in table 1, it is seen that the parameter uniform order of convergence pn is monotonically increasing towards the value 1 as n increases. this is in agreement with theorem 1. acknowledgment the authors are grateful to prof.svetoslav markov for suggesting improvements to this paper. biomath 5 (2016), 1608111, http://dx.doi.org/10.11145/j.biomath.2016.08.111 page 7 of 8 http://dx.doi.org/10.11145/j.biomath.2016.08.111 r.ishwariya et al., a parameter uniform almost first order convergent numerical method ... further, the authors thank the unknown referees for their suggestions and comments which have improved the paper to its present form. the first author sincerely thanks the university grants commission, new delhi, india, for the financial support through the rajiv gandhi national fellowship to carryout this work. references [1] j.j.h.miller, e.o’riordan, g.i.shishkin, fitted numerical methods for singular perturbation problems, world scientific revised edition(2012). [2] p.a.farrell, a.hegarty, j.j.h.miller, e.o’riordan, g.i.shishkin, robust computational techniques for boundary layers, applied mathematics & mathematical computation (eds. r.j.knops & k.w.morton), champman & hall/crc press (2000). [3] j.l.gracia, f.j.lisbona, m.madaune-tort, e.o’riordan, a coupled system of singularly perturbed semilinear reaction-diffusion equations, a.f. hegarty (ed.), et al., bail 2008-boundary and interior layers, lecture notes in computational science and engineer, vol. 69 (2009), pp. 163172 [4] s.valarmathi, j.j.h.miller, a parameter uniform finite difference method for a singularly perturbed linear dynamical systems, international journal of numerical analysis and modelling, 7(3), 535-548 (2010). [5] n.shivaranjani, j.j.h.miller, s.valarmathi, a parameter uniform almost first order convergent numerical method for a semi-linear system of singularly perturbed delay differential equations, biomath 3 (2014), 1411041. [6] j.d.murray, mathematical biology: an introduction (third edition), springer (2002.) [7] nikola sakac̆, milan sak-bosnar, potentiometric study of α−amylase kinetics using a platinum redox sensor, international journal of electrochemical science, 7(2012) 3008 3017. [8] gerald b.folland, advanced calculus,prentice hall, 2002. [9] eugene oriordan, jason quinn, parameter-uniform numerical methods for some singularly perturbed nonlinear initial value problems, springer (2012) 61:579611. [10] r. e. omalley, jr.andj. e.flaherty, analytical and numerical methods for singular singularly-perturbed initial value problems, society for industrial and applied mathematics, vol. 38, no. 2, april 1980. [11] mo jiaqi and lin wantao, asymptotic solution of singularly perturbed problem for a nonlinear singular equation, appl.math.j.chinese univ.ser.b, 2004,19(2) : 187-190. [12] marko golicnik, exact and approximate solutions for the decades-old michaelis-menten equation: progresscurve analysis through integrated rate equations, biochemistry and molecular biology education, vol. 39, no. 2, pp. 117125, 2011. [13] stanko dimitrov, gergana velikova, venko beschkov, svetoslav markov, on the numerical computation of enzyme kinetic parameters, biomath communications 1 (2014). [14] m. kosmas, e. m. papamichael, e. o. bakalis, an iterative solution of the michaelismenten equations, match commun. math. comput. chem. 70 (2013) 971986,issn 0340 6253. biomath 5 (2016), 1608111, http://dx.doi.org/10.11145/j.biomath.2016.08.111 page 8 of 8 http://dx.doi.org/10.11145/j.biomath.2016.08.111 introduction analytical results shishkin mesh discrete problem numerical results references www.biomathforum.org/biomath/index.php/biomath original article on the approximation of the cut and step functions by logistic and gompertz functions anton iliev∗†, nikolay kyurkchiev†, svetoslav markov† ∗faculty of mathematics and informatics paisii hilendarski university of plovdiv, plovdiv, bulgaria email: aii@uni-plovdiv.bg †institute of mathematics and informatics bulgarian academy of sciences, sofia, bulgaria emails: nkyurk@math.bas.bg, smarkov@bio.bas.bg received: , accepted: , published: will be added later abstract—we study the uniform approximation of the sigmoid cut function by smooth sigmoid functions such as the logistic and the gompertz functions. the limiting case of the interval-valued step function is discussed using hausdorff metric. various expressions for the error estimates of the corresponding uniform and hausdorff approximations are obtained. numerical examples are presented using cas mathematica. keywords-cut function; step function; sigmoid function; logistic function; gompertz function; squashing function; hausdorff approximation. i. introduction in this paper we discuss some computational, modelling and approximation issues related to several classes of sigmoid functions. sigmoid functions find numerous applications in various fields related to life sciences, chemistry, physics, artificial intelligence, etc. in fields such as signal processing, pattern recognition, machine learning, artificial neural networks, sigmoid functions are also known as “activation” and “squashing” functions. in this work we concentrate on several practically important classes of sigmoid functions. two of them are the cut (or ramp) functions and the step functions. cut functions are continuous but they are not smooth (differentiable) at the two endpoints of the interval where they increase. step functions can be viewed as limiting case of cut functions; they are not continuous but they are hausdorff continuous (h-continuous) [4], [43]. in some applications smooth sigmoid functions are preferred, some authors even require smoothness in the definition of sigmoid functions. two familiar classes of smooth sigmoid functions are the logistic and the gompertz functions. there are situations when one needs to pass from nonsmooth sigmoid functions (e. g. cut functions) to smooth sigmoid functions, and vice versa. such a necessity rises the issue of approximating nonsmooth sigmoid functions by smooth sigmoid functions. one can encounter similar approximation problems when looking for appropriate models for fitting time course measurement data coming e. g. from cellular growth experiments. depending on the general view of the data one can decide to use citation: anton iliev, nikolay kyurkchiev, svetoslav markov, on the approximation of the cut and step functions by logistic and gompertz functions, biomath 4 (2015), 15, http://dx.doi.org/10.11145/j.biomath.2015.. page 1 of 12 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2015.. a. iliev et al., on the approximation of the cut and step functions by logistic ... initially a cut function in order to obtain rough initial values for certain parameters, such as the maximum growth rate. then one can use a more sophisticate model (logistic or gompertz) to obtain a better fit to the measurement data. the presented results may be used to indicate to what extend and in what sense a model can be improved by another one and how the two models can be compared. section 2 contains preliminary definitions and motivations. in section 3 we study the uniform and hausdorff approximation of the cut functions by logistic functions. curiously, the uniform distance between a cut function and the logistic function of best uniform approximation is an absolute constant not depending on the slope of the functions, a result observed in [18]. by contrast, it turns out that the hausdorff distance (h-distance) depends on the slope and tends to zero when increasing the slope. showing that the family of logistic functions cannot approximate the cut function arbitrary well, we then consider the limiting case when the cut function tends to the step function (in hausdorff sense). in this way we obtain an extension of a previous result on the hausdorff approximation of the step function by logistic functions [4]. in section 4 we discuss the approximation of the cut function by a family of squashing functions induced by the logistic function. it has been shown in [18] that the latter family approximates uniformly the cut function arbitrary well. we propose a new estimate for the h-distance between the cut function and its best approximating squashing function. our estimate is then extended to cover the limiting case of the step function. in section 5 the approximation of the cut function by gompertz functions is considered using similar techniques as in the previous sections. the application of the logistic and gompertz functions in life sciences is briefly discussed. numerical examples are presented throughout the paper using the computer algebra system mathematica. ii. preliminaries sigmoid functions. in this work we consider sigmoid functions of a single variable defined on the real line, that is functions s of the form s : r −→ r. sigmoid functions can be defined as bounded monotone non-decreasing functions on r. one usually makes use of normalized sigmoid functions defined as monotone non-decreasing functions s(t), t ∈ r, such that lim s(t)t→−∞ = 0 and lim s(t)t→∞ = 1. in the fields of neural networks and machine learning sigmoid-like functions of many variables are used, familiar under the name activation functions. (in some applications the sigmoid functions are normalised so that the lower asymptote is assumed −1: lim s(t)t→−∞ = −1.) cut (ramp) functions. let ∆ = [γ −δ,γ + δ] be an interval on the real line r with centre γ ∈ r and radius δ ∈ r. a cut function (on ∆) is defined as follows: definition 1. the cut function cγ,δ on ∆ is defined for t ∈ r by cγ,δ(t) =   0, if t < ∆, t−γ + δ 2δ , if t ∈ ∆, 1, if ∆ < t. (1) note that the slope of function cγ,δ(t) on the interval ∆ is 1/(2δ) (the slope is constant in the whole interval ∆). two special cases are of interest for our discussion in the sequel. special case 1. for γ = 0 we obtain a cut function on the interval ∆ = [−δ,δ]: c0,δ(t) =   0, if t < −δ, t + δ 2δ , if −δ ≤ t ≤ δ, 1, if δ < t. (2) special case 2. for γ = δ we obtain the cut function on ∆ = [0, 2δ]: cδ,δ(t) =   0, if t < 0, t 2δ , if 0 ≤ t ≤ 2δ, 1, if 2δ < t. (3) biomath 4 (2015), 15, http://dx.doi.org/10.11145/j.biomath.2015.. page 2 of 12 http://dx.doi.org/10.11145/j.biomath.2015.. a. iliev et al., on the approximation of the cut and step functions by logistic ... step functions. the step function (with “jump” at γ ∈ r) can be defined by hγ(t) = cγ,0(t) =   0, if t < γ, [0, 1], if t = γ, 1, if t > γ, (4) which is an interval-valued function (or just interval function) [4], [43]. in the literature various point values, such as 0, 1/2 or 1, are prescribed to the step function (4) at the point γ; we prefer the interval value [0, 1]. when the jump is at the origin, that is γ = 0, then the step function is known as the heaviside step function; its “interval” formulation is: h0(t) = c0,0(t) =   0, if t < 0, [0, 1], if t = 0, 1, if t > 0. (5) h-distance. the step function can be perceived as a limiting case of the cut function. namely, for δ → 0, the cut function cδ,δ tends in “hausdorff sense” to the step function. here “hausdorff sense” means hausdorff distance, briefly h-distance. the h-distance ρ(f,g) between two interval functions f,g on ω ⊆ r, is the distance between their completed graphs f(f) and f(g) considered as closed subsets of ω ×r [24], [41]. more precisely, ρ(f,g) = max{ sup a∈f(f) inf b∈f(g) ||a−b||, (6) sup b∈f(g) inf a∈f(f) ||a−b||}, wherein ||.|| is any norm in r2, e. g. the maximum norm ||(t,x)|| = max |t|, |x|. to prove that (3) tends to (5) let h be the hdistance between the step function (5) and the cut function (3) using the maximum norm, that is a square (box) unit ball. by definition (6) h is the side of the smallest unit square, centered at the point (0, 1) touching the graph of the cut function. hence we have 1 − cδ,δ(h) = h, that is 1 −h/(2δ) = h, implying h = 2δ 1 + 2δ = 2δ + o(δ2). for the sake of simplicity throughout the paper we shall work with some of the special cut functions (2), (3), instead of the more general (arbitrary shifted) cut function (1); these special choices will not lead to any loss of generality concerning the results obtained. moreover, for all sigmoid functions considered in the sequel we shall define a “basic” sigmoid function such that any member of the corresponding class is obtained by replacing the argument t by t − γ, that is by shifting the basic function by some γ ∈ r. logistic and gompertz functions: applications to life-sciences. in this work we focus on two familiar smooth sigmoid functions, namely the gompertz function and the verhulst logistic function. both their inventors, b. gompertz and p.f. verhulst, have been motivated by the famous demographic studies of thomas malthus. the gompertz function was introduced by benjamin gompertz [22] for the study of demographic phenomena, more specifically human aging [38], [39], [47]. gompertz functions find numerous applications in biology, ecology and medicine. a. k. laird successfully used the gompertz curve to fit data of growth of tumors [32]; tumors are cellular populations growing in a confined space where the availability of nutrients is limited [1], [2], [15], [19]. a number of experimental scientists apply gompertz models in bacterial cell growth, more specifically in food control [10], [31], [42], [48], [49], [50]. gompertz models prove to be useful in animal and agro-sciences as well [8], [21], [27], [48]. the gompertz model has been applied in modelling aggregation processes [25], [26]; it is a subject of numerous theoretical modelling studies as well [6], [7], [9], [20], [37], [40]. the logistic function was introduced by pierre françois verhulst [44]–[46], who applied it to human population dynamics. verhulst derived his logistic equation to describe the mechanism of the self-limiting growth of a biological population. the equation was rediscovered in 1911 by a. g. mckendrick [35] for the bacterial growth in biomath 4 (2015), 15, http://dx.doi.org/10.11145/j.biomath.2015.. page 3 of 12 http://dx.doi.org/10.11145/j.biomath.2015.. a. iliev et al., on the approximation of the cut and step functions by logistic ... broth and was tested using nonlinear parameter estimation. the logistic function finds applications in an wide range of fields, including biology, ecology, population dynamics, chemistry, demography, economics, geoscience, mathematical psychology, probability, sociology, political science, financial mathematics, statistics, fuzzy set theory, to name a few [12], [13], [11], [14], [18]. logistic functions are often used in artificial neural networks [5], [16], [17], [23]. any neural net element computes a linear combination of its input signals, and applies a logistic function to the result; often called “activation” function. another application of logistic curve is in medicine, where the logistic differential equation is used to model the growth of tumors. this application can be considered an extension of the above-mentioned use in the framework of ecology. in (bio)chemistry the concentration of reactants and products in autocatalytic reactions follow the logistic function. other smooth sigmoid functions. the integral (antiderivative) of any smooth, positive, “bumpshaped” or “bell-shaped” function will be sigmoidal. a famous example is the error function, which is the integral (also called the cumulative distribution function) of the gaussian normal distribution. the logistic function is also used as a base for the derivation of other sigmoid functions, a notable example is the generalized logistic function, also known as richards curve [37]. another example is the dombi-gera-squashing function introduced and studied in [18] obtained as an antiderivative (indefinite integral) of the difference of two shifted logistic functions. in what follows we shall be interested in the approximation of the cut function by smooth sigmoid functions, more specifically the gompertz, the logistic and the dombi-gera-squashing function. we shall focus first on the verhulst logistic function. iii. approximation of the cut function by logistic functions definition 2. define the logistic (verhulst) function v on r as [44]–[46] vγ,k(t) = 1 1 + e−4k(t−γ) . (7) note that the logistic function (7) has an inflection at its “centre” (γ, 1/2) and its slope at γ is equal to k. proposition 1. [18] the function vγ,k(t) defined by (7) with k = 1/(2δ): i) is the logistic function of best uniform one-sided approximation to function cγ,δ(t) in the interval [γ,∞) (as well as in the interval (−∞,γ]); ii) approximates the cut function cγ,δ(t) in uniform metric with an error ρ = ρ(c,v) = 1 1 + e2 = 0.11920292.... (8) proof. consider functions (1) and (7) with same centres γ = δ, that is functions cδ,δ and vδ,k. in addition chose c and v to have same slopes at their coinciding centres, that is assume k = 1/(2δ), cf. figure 1. then, noticing that the largest uniform distance between the cut and logistic functions is achieved at the endpoints of the underlying interval [0, 2δ], we have: ρ = vδ,k(0) − cδ,δ = 1 1 + e4kδ = 1 1 + e2 . (9) this completes the proof of the proposition. we note that the uniform distance (9) is an absolute constant that does not depend on the width of the underlying interval ∆, resp. on the slope k. the next proposition shows that this is not the case whenever h-distance is used. proposition 2. the function v(t) = v0,k(t) with k = 1/(2δ) is the logistic function of best hausdorff one-sided approximation to function c(t) = c0,k(t) in the interval [0,∞) (resp. in the interval (−∞, 0]). the function v(t), approximates function c(t) in h-distance with an error h = h(c,v) that satisfies the relation: ln 1 −h h = 2 + 4kh. (10) biomath 4 (2015), 15, http://dx.doi.org/10.11145/j.biomath.2015.. page 4 of 12 http://dx.doi.org/10.11145/j.biomath.2015.. a. iliev et al., on the approximation of the cut and step functions by logistic ... fig. 1. the cut and logistic functions for γ = δ = 1, k = 1/2. proof. using δ = 1/(2k) we can write δ + h = (1 + 2hk)/(2k), resp.: v(−δ −h) = 1 1 + e2(1+2hk) . the h-distance h using square unit ball (with a side h) satisfies the relation v(−δ − h) = h, which implies (10). this completes the proof of the proposition. relation (10) shows that the h-distance h depends on the slope k, h = h(k). the next result gives additional information on this dependence. proposition 3. for the h-distance h(k) the following holds for k > 5: 1 4k + 1 < h(k) < ln(4k + 1) 4k + 1 . (11) proof. we need to express h in terms of k, using (10). let us examine the function f(h) = 2 + 4hk − ln(1 −h) − ln 1 h . from f ′(h) = 4k + 1 1 −h + 1 h > 0 we conclude that function f is strictly monotone increasing. consider the function g(h) = 2 + h(1 + 4k) − ln 1 h . then g(h)−f(h) = h + ln(1−h) = o(h2) using the taylor expansion ln(1 − h) = −h + o(h2). hence g(h) approximates f(h) with h → 0 as o(h2). in addition g′(h) = 1 + 4k + 1/h > 0, hence function g is monotone increasing. further, for k ≥ 5 g ( 1 1 + 4k ) = 3 − ln(1 + 4k) < 0, g ( ln(4k + 1) 4k + 1 ) = 2 + ln ln(1 + 4k) > 0. this completes the proof of the proposition. relation (11) implies that when the slope k of functions c and v tends to infinity, the h-distance h(c,v) between the two functions tends to zero (differently to the uniform distance ρ(c,v) which remains constant). the following proposition gives more precise upper and lower bounds for h(k). for brevity denote k = 4k + 1. proposition 4. for the h-distance h the following inequalities hold for k ≥ 5: ln k k − 2 + ln ln k k ( 1 + 1 ln k ) < h(k) < (12) ln k k + 2 + ln ln k k ( ln ln k 1−ln k − 1 ),k = 4k + 1. proof. evidently, the second derivative of g(h) = 2 + h(1 + 4k) − ln(1/h), namely g′′(h) = − 1 h2 < 0, has a constant sign on [ 1 k , ln k k ]. the straight line, defined by the points ( 1 k ,g( 1 k ) ) and( ln k k ,g( ln k k ) ) , and the tangent to g at the point( ln k k ,g( ln k k ) ) cross the abscissa at the points ln k k + 2 + ln ln k k ( ln ln k 1−ln k − 1 ), ln k k − 2 + ln ln k k ( 1 + 1 ln k ), respectively. this completes the proof of the proposition. propositions 2, 3 and 4 extend similar results from [4] stating that the heaviside interval-valued step function is approximated arbitrary well by biomath 4 (2015), 15, http://dx.doi.org/10.11145/j.biomath.2015.. page 5 of 12 http://dx.doi.org/10.11145/j.biomath.2015.. a. iliev et al., on the approximation of the cut and step functions by logistic ... logistic functions in hausdorff metric. the hausdorff approximation of the heaviside step function by sigmoid functions is discussed from various computational and modelling aspects in [28], [29], [30]. iv. approximation of the cut function by a squashing function the results obtained in section 3 state that the cut function cannot be approximated arbitrary well by the family of logistic functions. this result justifies the discussion of other families of smooth sigmoid functions having better approximating properties. such are the squashing functions proposed in [18] further denoted dgsquashing functions. definition 3. the dg-squashing function s∆ on the interval ∆ = [γ −δ,γ + δ] is defined by s (β) ∆ (t) = s (β) γ,δ (t) = 1 2δ ln ( 1 + eβ(t−γ+δ) 1 + eβ(t−γ−δ) )1 β . (13) note that the squashing function (13) has an inflection at its “centre” γ and its slope at γ is equal to (2δ)−1. the squashing function (13) with centre γ = δ: s (β) δ,δ (t) = 1 2δ ln ( 1 + eβt 1 + eβ(t−2δ) )1 β , (14) is the function of best uniform approximation to the cut function (3). indeed, functions cδ,δ and s (β) γ,δ have same centre γ = δ and equal slopes 1/(2δ) at their coinciding centres. as in the case with the logistic function, one observes that the uniform distance ρ = ρ(c,s) between the cut and squashing function is achieved at the endpoints of the interval ∆, more specifically at the origin. denoting the width of the interval ∆ by w = 2δ we obtain ρ = s (β) δ,δ (0) = 1 w ln( 2 1 + eβ(−w) )1/β < (15) ln 2 w 1 β = const 1 β . the estimate (15) has been found by dombi and gera [18]. this result shows that any cut fig. 2. the functions f(d) and g(d). function c∆ can be approximated arbitrary well by squashing functions s(β)∆ from the class (13). the approximation becomes better with the increase of the value of the parameter β. thus β affects the quality of the approximation; as we shall see below the practically interesting values of β are integers greater than 4. in what follows we aim at an analogous result using hausdorff distance. let us fix again the centres of the cut and squashing functions to be γ = δ so that the form of the cut function is cδ,δ, namely (3), whereas the form of the squashing function is s(β)δ,δ as given by (14). both functions cδ,δ and s (β) δ,δ have equal slopes 1/w, w = 2δ, at their centres δ. denoting the square-based h-distance between cδ,δ and s (β) δ,δ by d = d(w; β), w = 2δ, we have the relation s (β) δ,δ (w + d) = 1 w ln ( 1 + eβ(w+d) 1 + eβd )1 β = 1 −d or ln 1 + eβ(w+d) 1 + eβd = βw(1 −d). (16) the following proposition gives an upper bound for d = d(w; β) as implicitly defined by (16): proposition 5. for the distance d the following holds for β ≥ 5: d < ln 2 ln(4βw + 1) 4wβ + 1 . (17) biomath 4 (2015), 15, http://dx.doi.org/10.11145/j.biomath.2015.. page 6 of 12 http://dx.doi.org/10.11145/j.biomath.2015.. a. iliev et al., on the approximation of the cut and step functions by logistic ... proof. we examine the function: f(d) = −βw(1−d)+ln(1+eβ(w+d))+ln 1 1 + eβd . from f ′(d) > 0 we conclude that function f(d) is strictly monotone increasing. we define the function g(d) = −βw + ln(1 + eβw)+ dβ ( w + eβw 1 + eβw ) + ln 1 1 + eβd . we examine g(d) −f(d): g(d) −f(d) = ln(1 + eβw) + eβwβd 1 + eβw − ln(1 + eβ(w+d)). from taylor expansion ln(1 +eβ(w+d)) = ln(1 +eβw) + eβwβd 1 + eβw +o(d2) we see that function g(d) approximates f(d) with d → 0 as o(d2) (cf. fig. 2). in addition g(0) < 0 and g ( ln 2 ln(4βw+1) 4wβ+1 ) > 0 for β ≥ 5. this completes the proof of the proposition. some computational examples using relation (16) and (17) for various β and w are presented in table 1. w β d(w;β) from(16) d(w;β) from(17) 1 30 0.016040 0.027472 5 10 0.012639 0.018288 6 100 0.001068 0.002247 14 5 0.009564 0.013908 50 100 0.000137 0.000343 500 1000 1.38×10−6 5.02×10−6 1000 5000 1.3×10−7 5.8×10−7 table i bounds for d(w;β) computed by (16) and (17), respectively the numerical results are plotted in fig. 3 (for the case β = 5, w = 3; d = 0.0398921) and fig. 4 (for the case β = 10, w = 4; d = 0.0154697). fig. 3. functions cδ,δ and s (β) δ,δ for β = 5, w = 3; d ≤ 0.4. fig. 4. functions cδ,δ and s (β) δ,δ for β = 10, w = 4; d ≤ 0.016. v. approximation of the step function by the gompertz function in this section we study the hausdorff approximation of the step function by the gompertz function and obtain precise upper and lower bounds for the hausdorff distance. numerical examples, illustrating our results are given. definition 4. the gompertz function σα,β(t) is defined for α, β > 0 by [22]: σα,β(t) = e −αe−βt. (18) special case 3. for α∗ = ln 2 = 0.69314718... we obtain the special gompertz function: σα∗,β(t) = e −α∗e−βt, (19) such that σα∗,β(0) = 1/2. biomath 4 (2015), 15, http://dx.doi.org/10.11145/j.biomath.2015.. page 7 of 12 http://dx.doi.org/10.11145/j.biomath.2015.. a. iliev et al., on the approximation of the cut and step functions by logistic ... fig. 5. the gompertz function with α = ln 2 and β = 5; h-distance d = 0.212765. we study the hausdorff approximation of the heaviside step function c0 = h0(t) by gompertz functions of the form (18) and find an expression for the error of the best approximation. the h-distance d = d(α∗,β) between the heaviside step function h0(t) and the gompertz function (19) satisfies the relation σα∗,β(d) = e −α∗e−βd = 1 −d, or ln(1 −d) + α∗e−βd = 0. (20) the following theorem gives upper and lower bounds for d(α∗,β). for brevity we denote α = α∗ in theorem 1 and its proof. theorem 1. the hausdorff distance d = d(α,β) between the step function h0 and the gompertz function (19) can be expressed in terms of the parameter β for any real β ≥ 2 as follows: 2α− 1 1 + αβ < d < ln(1 + αβ) 1 + αβ . (21) proof. we need to express d in terms of α and β, using (20). let us examine the function f(d) = ln(1 −d) + αe−βd. from f ′(d) = − 1 1 −d −αβe−βd < 0 we conclude that the function f is strictly monotone decreasing. consider function g(d) = α − (1 + αβ)d. from taylor expansion α− (1 + αβ)d− ln(1 −d) −αe−βd = o(d2) we obtain g(d)−f(d) = α−(1 + αβ)d− ln(1− d) − αe−βd = o(d2). hence g(d) approximates f(d) with d → 0 as o(d2). in addition g′(d) = −(1 + αβ) < 0. further, for β ≥ 2, g ( 2α− 1 1 + αβ ) = 1 −α > 0, g ( ln(1 + αβ) 1 + αβ ) = α− ln(1 + αβ) < 0. this completes the proof of the theorem. some computational examples using relation (20) are presented in table 2. β d(α∗,β) 2 0.310825 5 0.212765 10 0.147136 50 0.0514763 100 0.0309364 500 0.00873829 1000 0.00494117 table ii bounds for d(α∗,β) computed by (20) for various β. the calculation of the value of the h-distance between the gompertz sigmoid function and the heaviside step function is given in appendix 1. the numerical results are plotted in fig. 5 (for the case α∗ = ln 2, β = 5, h-distance d = 0.212765) and fig. 6 (for the case α∗ = ln 2, β = 20, h-distance d = 0.0962215). remark 1. for some comparisons of the gompertz and logistic equation from both practical and theoretical perspective, see [6], [8], [40]. as can be seen from figure 6 the graph of the gompertz function is “skewed”, it is not symmetric with respect to the inflection point. in biology, the gompertz function is commonly used to model growth process where the period of increasing growth is shorter than the period in which growth decreases [8], [33]. biomath 4 (2015), 15, http://dx.doi.org/10.11145/j.biomath.2015.. page 8 of 12 http://dx.doi.org/10.11145/j.biomath.2015.. a. iliev et al., on the approximation of the cut and step functions by logistic ... fig. 6. the logistic (dotted line) and the gompertz function (dense line) with same point and same rate (at that point). remark 2. for k > 0,β > 0 consider the differential equation y′ = ke−βty, k β = α. (22) we have dy dt = ke−βty; dy y = ke−βtdt ln y = − k β e−βt = −αe−βt; y = e−αe −βt . we see that the solution of differential equation (22) is the gompertz function σα,β(t) (18) [6]). as shown in [28], equation (22) can be interpreted as y′ = ksy, wherein s = s(t) is the nutrient substrate used for the growth of the population; one see that s is a decay exponential function in the gompertz model (a similar interpretation can be found in [21]), [40]). for other interpretations see [6]), [8], [20]. vi. conclusion in this paper we discuss several computational, modelling and approximation issues related to two familiar classes of sigmoid functions—the logistic (verhulst) and the gompertz functions. both classes find numerous applications in various fields of life sciences, ecology, medicine, artificial neural networks, fuzzy set theory, etc. bigskip we study the uniform and hausdorff approximation of the cut functions by logistic functions. we demonstrate that the best uniform approximation between a cut function and the respective logistic function is an absolute constant not depending on the (largest) slope k. on the other side we show that the hausdorff distance (h-distance) depends on the slope k and tends to zero with k →∞. we also discuss the limiting case when the cut function tends to the heaviside step function in hausdorff sense, thereby extending a related previous result [4]. the approximation of the cut function by a family of squashing functions induced by the logistic function is also discussed. we propose a new estimate for the h-distance between a cut function and its best approximating squashing function. our estimate extends a known result stating that the cut function can be approximated arbitrary well by squashing functions [18]. our estimate is also extended to cover the limiting case of the heaviside step function. finally we study the approximation of the cut and step functions by the family of gompertz functions. new estimates for the h-distance between a cut function and its best approximating gompertz function are obtained. references [1] a. akanuma, parameter analysis of gompertz function growth model in clinical tumors, european j. of cancer 14 (1978) 681–688. 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[50] m. h. zwietering, h. g. cuppers, j. c. de wit and k. van′t riet, evaluation of data transformations and validation of a model for the effect of temperature on bacterial growth, appl. environ. microbiol. 60(1) (1994) 195–203. biomath 4 (2015), 15, http://dx.doi.org/10.11145/j.biomath.2015.. page 11 of 12 http://dx.doi.org/10.1017/s0370164600025426 http://dx.doi.org/10.11145/j.biomath.2014.07.121 http://dx.doi.org/10.1093/jxb/10.2.290 http://dx.doi.org/10.1093/gerona/57.2.b69 http://dx.doi.org/10.1016/j.mehy.2012.02.004 http://dx.doi.org/10.1007/978-94-009-0673-0 http://dx.doi.org/10.1016/s0740-0020(85)80004-6 http://dx.doi.org/10.11145/j.biomath.2013.11.261 http://dx.doi.org/10.1016/0047-6374(94)90095-7 http://dx.doi.org/10.11145/j.biomath.2015.. a. iliev et al., on the approximation of the cut and step functions by logistic ... appendix the module “computation of the distance d and visualization of the cut function c∆ and squashing function s(β)∆ ” in cas mathematica. biomath 4 (2015), 15, http://dx.doi.org/10.11145/j.biomath.2015.. page 12 of 12 http://dx.doi.org/10.11145/j.biomath.2015.. a. iliev et al., on the approximation of the cut and step functions by logistic ... fig. 7. module in programming environment mathematica. fig. 8. the test provided on our control example. biomath 4 (2015), 15, http://dx.doi.org/10.11145/j.biomath.2015.. page 13 of 12 http://dx.doi.org/10.11145/j.biomath.2015.. introduction preliminaries approximation of the cut function by logistic functions approximation of the cut function by a squashing function approximation of the step function by the gompertz function conclusion references www.biomathforum.org/biomath/index.php/biomath original article effects of discrete time delays and parameters variation on dynamical systems ibrahim diakite∗, benito m. chen-charpentier† ∗department of global health and social medicine harvard medical school, boston, usa ibrahim diakite@hms.harvard.edu †department of mathematics university of texas arlington, arlington, usa bmchen@uta.edu received: 29 august 2014, accepted: 20 may 2015, published: 8 june 2015 abstract—delay differential equations (dde’s) have received considerable attention in recent years. while most of these articles focused on the effects of the time delays on the stability of the equilibrium points and on the bifurcation that they may raised, very few papers address the key roles that system parameters play on if and how the discrete delays induce stability changes of the equilibria and produce bifurcations near such equilibria. in this article we focus on that question in a general setting, that is, if one has a system of dde’s with one or multiple discrete time delays, what are the results of changing the system parameters values on the effects of the discrete time delays on the dynamic of the system. we present general results for one equation with one and two delays and study a specific example of one equation with one delay. we then establish the procedure for n equations with multiple delays and do a specific example for two equations with two delays. we compute the steady states and analyze their stability as both chosen bifurcation parameters, the discrete time delay τ and a local equation parameter µ, cross critical values. our analysis shows that while changes in both parameters can destabilize the steady state, the discrete time delay can only cause stability switches of the steady state for certain values of µ, while the effects of the local equation parameter on the steady state do not necessarily depend on the value of τ. while µ may cause the system to go through different type of bifurcations, the discrete time delay can only introduce a hopf bifurcation for certain values of µ. keywords-delay differential equations; bifurcation; predator-prey. i. introduction it is well known, that the values of the parameters play a crucial role in the behavior of dynamical systems and that changes in the values can change the behavior significantly. it has also been shown by many researchers (perelson[1],allen[2],bellen[3]) that there is a need to incorporate discrete time delays in dynamical systems (biological systems, physical systems,...) as studied. models that incorporate such delays are referred to as delay differential equations (dde’s). dde’s have been extensively studied by many researchers including pioneers bellman[4], driver [5], and in more recent years by culshaw[6], gakkhar[7], bellen[3], and a superb monograph on the subject citation: ibrahim diakite, benito m. chen-charpentier, effects of discrete time delays and parameters variation on dynamical systems, biomath 1 (2015), 1505201, http://dx.doi.org/10.11145/j.biomath.2015.05.201 page 1 of 17 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2015.05.201 i. diakite et al., effects of discrete time delays and parameters ... by gopalsamy [8]. while most of these research papers focus on issue of the stability changes caused by the delay(s), the main motivation of this paper is to study how a local bifurcation parameter of the system may affect the changes in stability caused by the delay (s). published papers have shown that the incorporation of discrete time delays can highly impact the dynamics of the system, since they can switch the stability of a steady state point, and can also cause the system to go through a hopf bifurcation near that steady state point (culshaw[6], gakkhar[7], bellen[3]). in this paper we consider a system of n delay differential equations (dde’s) with one parameter µ as the bifurcation parameter and also with one or more discrete time delays, τ, which can also behave as bifurcation parameters. we are interested in investigating how the parameters µ and τ affect the stability of the steady state points of the system, and, more important, how their effects on the system are correlated to each other. we present general results in the one dimensional case (propositions 1 to 3) for necessary and sufficient conditions for a stability switch and present a specific example to illustrate these conditions. for the n dimensional case (n ≥ 2) we establish the main ideas, but since there are multiple possible cases, we consider only a specific example. we present a non-kolmogorov type of predator-prey model similar to the model presented by ruan [9]. in this model we introduce two delays, τ1 > 0 and τ2 > 0, to represent the time lag in the growth to maturity of the prey, and the time lag in the growth to maturity of the predator, respectively. we show how the dynamics of the system change depending on certain conditions on τ1 and on another bifurcation parameter r. we also point out conditions for the system to go through stability changes when both delays τ1 and τ2 are non-zero. we present necessary conditions for the system to go through a hopf bifurcation for τ1 > 0 and τ2 = 0. finally we show numerical results illustrating the theoretical results. ii. one dimensional field a. one equation with one delay consider the one dimensional delay differential equation with the time delay τ, and the parameter µ as bifurcation parameters: dx dt = f(x(t),x(t− τ),µ), (1) where f is assumed to be smooth enough to guarantee the existence and uniqueness of solutions to (1) under the initial condition (r. bellman and k. l. cooke [4]) x(θ) = φ(θ), θ ∈ [−τ, 0]. unfortunately equation (1) is too general to analyze. therefore we will consider a more special form: dx dt = f1(x(t),µ) + f2(x(t− τ),µ). (2) this form has the advantage that it simplifies the analytical work and also it is the form present in many population dynamical models involving delays [6], [7], [9], [10]. the dde (2) may or may not have equilibrium points (or steady states) and these will depend on the values of µ. let µ∗ ∈ dµ = {µ ∈ r : f(x∗,x∗,µ) = 0 exists} ,that is µ∗ is in the range of values of µ for which the dde has an equilibrium point x∗, i.e., f(x∗,x∗,µ∗) = 0. we are interested in studying the stability of such equilibrium point. in particular, in studying the effect of the parameter µ and of the discrete time τ on its stability. to do this we linearize the dde around the equilibrium point. the characteristic equation is : λ− df1 dx |(x∗,µ∗) − df2 dx |(x∗,µ∗)e −λτ = 0, (3) and the stability of the equilibrium point (x∗,µ∗) is determined by the sign of the real part of the eigenvalues λ of equation (3). biomath 1 (2015), 1505201, http://dx.doi.org/10.11145/j.biomath.2015.05.201 page 2 of 17 http://dx.doi.org/10.11145/j.biomath.2015.05.201 i. diakite et al., effects of discrete time delays and parameters ... 1) stability of the steady state: if τ = 0 then the characteristic equation (3) becomes λ− df1 dx |(x∗,µ∗) − df2 dx |(x∗,µ∗)=0. the stability of the steady state then depends only on values of µ∗ within dµ. we have two cases: (a) the steady state (x∗,µ∗) is stable if df1 dx |(x∗,µ∗) + df2 dx |(x∗,µ∗) < 0. (b) the steady state (x∗,µ∗) is unstable if df1 dx |(x∗,µ∗) + df2 dx |(x∗,µ∗) > 0. assume that condition (a) holds, namely the steady state (x∗,µ∗) is stable when there is no delay (τ = 0). we want to know if there exists τ > 0 for which the steady state will lose stability. so for τ ≥ 0, let λ(τ) = α(τ) + iω(τ). the characteristic equation (3) becomes: α + iω = df1 dx |(x∗,µ∗) + df2 dx |(x∗,µ∗)e −ατcosωτ+ i df2 dx |(x∗,µ∗)e −ατsinωτ, (4) where, for clarity in the notation, we have not explicitly shown the dependence on τ. separating the real and imaginary parts, we have: α = df1 dx |(x∗,µ∗) + df2 dx |(x∗,µ∗)e −ατcosωτ, (5) ω = df2 dx |(x∗,µ∗)e −ατsinωτ. (6) the steady state will lose stability when the real part of the eigenvalue λ crosses the zero axis from negative to positive as τ passes a critical value. by rouche’s theorem (dieudonne[11], theorem 9.17.4) and by the continuity in τ, the transcendental equation (3) has roots with positive real parts if and only if it has pure imaginary roots. therefore, we look at when the real part of the eigenvalue λ becomes zero. in other words, we want to find if there exists a τc > 0 such that α(τc) = 0. since α(0) = df1 dx |(x∗,µ∗) + df2 dx |(x∗,µ∗), and α(0) < 0 by assumption (a) , therefore if τc > 0 exists such that α(τc) = 0 then by the continuity (michael y. li and hogying shu [10]) of α we have: • α(τ) < 0 for any 0 ≤ τ < τc, • α(τ) > 0 for any τ > τc. namely the steady state (x∗,µ∗) will lose stability as the delay parameter τ crosses a critical value τc. such τc exists if and only if α(τc) = 0 and ω(τc) = ωc satisfies : df1 dx |(x∗,µ∗) = − df2 dx |(x∗,µ∗)cosωcτc (7) ωc = df2 dx |(x∗,µ∗)sinωcτc. (8) squaring equations (7) and (8), and adding them up, we obtain: ω2c = [ df2 dx |(x∗,µ∗)] 2 − [ df1 dx |(x∗,µ∗)] 2. (9) if equation (9) has at least a positive root ωc, then there exists a τc > 0 such that α(τ) > 0 whenever τ > τc (see proof in appendix a). an important question we want to address is, since equation (9) depends on the bifurcation parameter µ∗, can one chose µ∗ within dµ so that equation (9) does not have a positive root ωc? that is, are there values of µ∗ within dµ such that the delay does not have any effect on the stability of the steady state (x∗,µ∗)? this question motivates the following propositions (see appendix b for the proof). proposition 1: consider the one dimensional delay differential equation dx dt = f1(x(t),µ) + f2(x(t− τ),µ). and assume that the steady state (x∗,µ∗) is stable for τ = 0 then we have (i) if df1 dx |(x∗,µ∗) < 0 and df2 dx |(x∗,µ∗) > 0 then the steady state (x∗,µ∗) remains stable for all τ ≥ 0. (ii) if df1 dx |(x∗,µ∗) > 0 and df2 dx |(x∗,µ∗) < 0 then there exists a critical value of the delay such that the steady state loses stability as the delay crosses its critical value. (iii) if df1 dx |(x∗,µ∗) < 0 and df2 dx |(x∗,µ∗) < 0 then: biomath 1 (2015), 1505201, http://dx.doi.org/10.11145/j.biomath.2015.05.201 page 3 of 17 http://dx.doi.org/10.11145/j.biomath.2015.05.201 i. diakite et al., effects of discrete time delays and parameters ... (a) the steady state remains stable for all τ ≥ 0 if | df2 dx |(x∗,µ∗)| < | df1 dx |(x∗,µ∗)|, (b) there exists a τc > 0 such that the steady state becomes unstable for all τ > τc if | df2 dx |(x∗,µ∗)| > | df1 dx |(x∗,µ∗)|. proposition 2: consider the one dimensional delay differential equation dx dt = f1(x(t),µ) + f2(x(t− τ),µ). and assume that the steady state (x∗,µ∗) is stable for τ = 0, that conditions of proposition 1(iii) hold, and that further more for some µ∗ within dµ we have: df1 dx |(x∗,µ∗) = g(x ∗)o(µ∗), df2 dx |(x∗,µ∗) = h(x ∗)o( 1 µ∗ ), then there exists a critical value for µ∗ within dµ such that the steady state (x∗,µ∗) will stay stable for all τ ≥ 0 when µ∗ > µc. 2) example: consider the one dimensional dde{ dy dt = µ y (t) y (t)+1 − 1 µ y (t− τ)2, if µ 6= 0 y (t) = 0, if µ = 0 where µ is a bifurcation parameter and τ ≥ 0 is a discrete time delay. for µ ∈ dµ = r, the equation has two non-negative equilibrium points: the trivial one y ∗0 = 0, and the positive equilibrium point y ∗1 = −1+ √ 1+4µ2 2 . the characteristic equation is given as λ−µ 1 (y ∗ + 1)2 − 2 µ y ∗e−λτ = 0. (10) • for the trivial equilibrium point y ∗ = 0, its stability only depends on µ since equation (10) evaluated at y ∗ = 0 becomes λ = µ. the trivial equilibrium is unstable for µ > 0 and all τ ≥ 0. the trivial equilibrium is stable for µ < 0 and all τ ≥ 0. • at y ∗1 = −1+ √ 1+4µ2 2 , equation (10) becomes: λ− 4µ (1+ √ 1+4µ2)2 + √ 1+4µ2−1 µ e−λτ = 0, (11) then the stability of y ∗1 depends on both µ and τ. 1) if τ = 0 then equation (11) becomes λ = − 4µ √ 1 + 4µ2 (1 + √ 1 + 4µ2)2 then λ < 0 if µ > 0, therefore the equilibrium y ∗1 is stable (fig 2) λ > 0 if µ < 0, therefore the equilibrium y ∗1 is unstable (fig 2). remark: to better understand the situation, the stability of both equilibria when there is no delay is shown in the following table: table i stability regions case y ∗0 = 0 y ∗ 1 = −1+ √ 1+4µ2 2 µ < 0 stable unstable µ = 0 stable stable µ > 0 unstable stable at the equilibrium (y,µ) = (0, 0), there is an exchange of stability. this is a transcritical bifurcation (guckenheimer[12]). geometrically, there are two curves of equilibria which intersect at the origin and lie on both sides of µ = 0. stability of the equilibrium changes along either curve on passing through µ = 0. 2) if τ > 0 and λ(τ) = α(τ) + iω(τ), there exists a critical τc such that α(τc) = 0 and λ(τc) = ±iω(τc) = ±iωc (a pair of pure biomath 1 (2015), 1505201, http://dx.doi.org/10.11145/j.biomath.2015.05.201 page 4 of 17 http://dx.doi.org/10.11145/j.biomath.2015.05.201 i. diakite et al., effects of discrete time delays and parameters ... fig. 1. transcritical bifurcation around µ = 0. unstable equilibrium (red), and stable equilibrium (blue). imaginary eigenvalues) is solution of (8) if and only if ω2c = 16[µ2(1+ √ 1+4µ2)2−1] (1+ √ 1+4µ2)4 has a positive root ωc, and that is the case if and only if µ2(1 +√ 1 + 4µ2)2 − 1 > 0. (2a) if µ ≥ 1 2 then µ2(1+ √ 1 + 4µ2)2−1 > 0 therefore there exists τc > 0 such that the equilibrium loses stability whenever τ > τc (fig 3 left). (2b) if µ ≤−1 2 then µ2(1+ √ 1 + 4µ2)2−1 > 0 therefore there exists τc > 0 such that the equilibrium gains stability whenever τ > τc . (2c) if −1 2 < µ < 1 2 then µ2(1 +√ 1 + 4µ2)2 − 1 < 0 therefore the delay has no effect on the stability of the equilibrium. for µ ≥ 1 2 , the equilibrium is unstable for all τ > 0.55, and for 0 < µ < 1 2 the equilibrium remains stable for all τ. b. one equation with multiple delays consider the one dimensional delay differential equation with the time lags τk, k = 1, 2, ..., and µ as bifurcation parameters: dx dt = f1(x(t),µ) + f2(x(t− τ1), ...,x(t− τk),µ) (12) fig. 2. the positive equilibrium is stable for τ = 0 and µ = 2, top graph. the equilibrium still remains stable for τ = 0.4 (τ < τc = 0.55) and µ = 2, bottom graph let (x∗,µ∗) = (x∗,x∗, ...,x∗,µ∗) be the steady state of equation (12), i.e., f1(x∗,µ∗) + f2(x ∗,x∗, ...,x∗,µ∗) = 0. to study the stability of the steady state we compute the characteristic equation: λ− df1 dx |(x∗,µ∗) − k∑ j=1 df2 dx |(x∗,µ∗)e −λτj = 0. (13) for clarity of the presentation we consider the case of only two delays. therefore the characteristic equation is written as λ− df1 dx |(x∗,µ∗) − df2 dx |(x∗,µ∗)(e −λτ1 + e−λτ2 ) = 0 (14) biomath 1 (2015), 1505201, http://dx.doi.org/10.11145/j.biomath.2015.05.201 page 5 of 17 http://dx.doi.org/10.11145/j.biomath.2015.05.201 i. diakite et al., effects of discrete time delays and parameters ... note that if τ1 = τ2 = τ or τ1 = 0 or τ2 = 0 then we are back to the previous case of one equation with one delay. we will assume that τ1 is in its stable domain, i.e., 0 < τ1 < τ1c and τ2 > 0. we now examine how variation of τ2 and µ affects the stability of the steady state. consider λ(τ2) = α(τ2) + iω(τ2) as solution of equation (14). we look for a critical value τ2c of τ2 such that α(τ2c) = 0 and λ(τ2c) = iω(τ2c) = iω2c is solution of equation (14). such τ2c exists if and only if: iω2c − df2 dx |(x∗,µ∗)(cos ω2cτ1 − i sin ω2cτ1) − df2 dx |(x∗,µ∗)(cos ω2cτ2c − i sin ω2cτ2c) − df1 dx |(x∗,µ∗) (15) separate real and imaginary parts: − df2 dx |(x∗,µ∗) cos ω2cτ2c = df2 dx |(x∗,µ∗) cos ω2cτ1, + df1 dx |(x∗,µ∗) df2 dx |(x∗,µ∗) sin ω2cτ2c = − df2 dx |(x∗,µ∗) sin ω2cτ1 −ω2c. (16) adding the square of (16) and (ii-b) we have [ df2 dx |(x∗,µ∗)] 2 = (ω2c + df2 dx |(x∗,µ∗) sin ω2cτ1) 2 +( df1 dx |(x∗,µ∗) + df2 dx |(x∗,µ∗) sin ω2cτ1) 2 (17) clearly τ2c exists if and only the function: h(ω2c) = (ω2c + df2 dx |(x∗,µ∗) sin ω2cτ1) 2 + ( df1 dx |(x∗,µ∗) + df2 dx |(x∗,µ∗) cos ω2cτ1) 2 − [ df2 dx |(x∗,µ∗)] 2 (18) has at least a positive root. fig. 3. for µ = 2 and τ = 0.9 the equilibrium is unstable, top graph. for µ = 0.2 and τ = 1 the equilibrium is stable, bottom graph proposition 3: consider the one dimensional delay differential equation with the time lag τ1, τ2, and µ as bifurcation parameters: dx dt = f1(x(t),µ) + f2(x(t− τ1),x(t− τ2),µ). (19) assume that the steady state (x∗,µ∗) = (x∗,x∗,µ∗) of (19) is stable for 0 < τ1 < τ1c. if df2 dx |(x∗,µ∗) > 0 and df1 dx |(x∗,µ∗) < 0, then there exists a critical value τ2c > 0 for τ2 such that (x∗,µ∗) losses stability as τ2 crosses τ2c. proof: such τ2c exists if and only if equation h(ω2c) = 0 has at least a positive equation. or biomath 1 (2015), 1505201, http://dx.doi.org/10.11145/j.biomath.2015.05.201 page 6 of 17 http://dx.doi.org/10.11145/j.biomath.2015.05.201 i. diakite et al., effects of discrete time delays and parameters ... if df2 dx |(x∗,µ∗) > 0 and df1 dx |(x∗,µ∗) < 0 then h(0) = ( df1 dx |(x∗,µ∗) + df2 dx |(x∗,µ∗)) 2 − ( df2 dx |(x∗,µ∗)) 2 < 0 (20) and also h(ω2c) → ∞ as ω2c → ∞. then the intermediate value theorem assures that equation h(ω2c) = 0 has at least a positive root. � we now extend our analysis to a system of ndelay differential equations with multiple discrete time delays τ1,τ2, ...,τk, and a local bifurcation parameter µ. iii. n dimensional field consider the following system non-linear delay differential equations: dx dt = f(x(t),x(t− τ1), ...,x(t− τk),µ), (21) where x ∈ rn, τj ≥ 0, 1 ≤ j ≤ k are constant discrete times, f : rn+1 × ck → rn is assumed to be smooth enough to guarantee existence and uniqueness of solutions to (21) under the initial value condition (r. bellman and k. l. cooke [4] and j. k. hale and s. m. verduyn lunel [13]) x(θ) = φ(θ), θ ∈ [−τ, 0], where c = c([−τ, 0],rn), τ = max 1≤j≤k τj. suppose f(x∗,x∗, ...,x∗,µ∗) = 0, that is, (x∗,µ∗) is a steady state of system (21). we are interested in studying the stability of such equilibrium point. in particular studying the effect of the parameter µ and the discrete time delays τ1,τ2, ...,τk on its stability. the linearization of (21) at (x∗,µ∗) has the form (ruan [9]): dx dt = a0(µ ∗)x(t) + k∑ j=1 aj(µ ∗)x(t− τj), (22) where x ∈ rn, each aj(µ∗) (0 ≤ j ≤ k) is an n×n constant matrix that depends on values of µ∗ within dµ. the transcendental equation associated with system (21) is given as : det [ λi −a0(µ∗) − k∑ j=1 aj(µ ∗)e−λτj ] = 0 (23) equation (23) has been studied by many researchers (ruan [9], r. bellman and k. l. cooke [4] and j. k. hale and s. m. verduyn lunel [13]). the following result, which was proved by chin [14] for k = 1 and by datko [15] and hale et al. [13] for k ≥ 1, gives a necessary and sufficient condition for the absolute stability of system (22). lemma 1: system (22) is stable for all delays τj(1 ≤ j ≤ k) if and only if (i) reλ( ∑k j=0 aj(µ ∗)) < 0; (ii) det[iωi−a0(µ∗)− ∑k j=1 aj(µ ∗)e−iωτj ] 6= 0 for all ω > 0 clearly, the stability of the steady state (x∗,µ∗) and the effects of the discrete times τj on its stability depend on values of µ∗ within dµ. to further investigate the effects of µ, and the discrete time delays τj on the stability of (x∗,µ∗), the exact entries of the matrices aj(µ∗) are needed to avoid doing a large number of cases. note that the difficulty of the analysis is not due to the number of delays but to the number of equations. even in the case of two equations with one delay, one needs to consider: det[λi −a0(µ∗) −a1(µ∗)eλτ = 0], where ai(µ ∗) = ∂f ∂xi |(x∗,µ∗), i = 0, 1. so the stability depends on all the entries of the ai, i = 0, 1, we have many different cases. therefore to present the ideas we consider a specific example with n = 2, k = 2, that is a two dimensional delay differential equations with two discrete time delays, and a local bifurcation parameter. biomath 1 (2015), 1505201, http://dx.doi.org/10.11145/j.biomath.2015.05.201 page 7 of 17 http://dx.doi.org/10.11145/j.biomath.2015.05.201 i. diakite et al., effects of discrete time delays and parameters ... a. two dimensional field example consider the non-kolmogorov type (holling) predator-prey model dx dt = r1x(t− τ1) −a1 x(t)y(t) x(t) + 1 , (24) dy dt = −r2y(t) + a2 x(t− τ2)y(t− τ2) x(t− τ2) + 1 (25) where the parameters are described in the following table: table ii parameter values parameters description values x(t) the prey population y(t) the predator population r1 the growth rate of the prey in the absence of predators 0.5 r2 the death rate of predators in the absence of the prey 0.5 a1 the predation rate of the prey by the predators 0.5 a2 the conversion rate for the predators 5 τ1 the time lag in the growth to maturity of the prey varies τ2 the time lag in the growth to maturity of the predators varies note that r1 > 0, r2 > 0, a1 > 0, a2 > 0, τ1 > 0, τ2 > 0. proposition 4: if the basic reproductive ratio (ameh[16]) r > 1, the system has two nonnegative steady states: (x∗0,y ∗ 0) = (0, 0), and (x ∗ 1,y ∗ 1) = ( 1 r−1, rr′ r−1 ), where r = a2 r2 r′ = a1 r1 . we consider r, τ1 and τ2 as the bifurcation parameters for the system (24-25) since changes of them may affect the existence and stability of the equilibrium points. b. stability analysis proposition 5: there exists a critical value for τ1 such that (i) the steady state (x∗0,y ∗ 0) is unstable for τ1 = 0, and all τ2 ≥ 0. (ii) the steady state (x∗0,y ∗ 0) is stable for τ1 ≥ τ1c, and all τ2 ≥ 0. proposition 6: if [(b−d)2−r21−2a1f] < 0 and ∆ = [(b−d)2 −r21 − 2a1f] 2 − 4a21f 2 ≥ 0 then there exists a critical τ ′1c such that (i) the steady state (x∗1,y ∗ 1) = ( 1 r−1, rr′ r−1 ) is unstable for 0 ≤ τ1 < τ ′1c and τ2 = 0. (ii) the steady state (x∗1,y ∗ 1) is stable for τ1 > τ ′ 1c and τ2 = 0. note that τ1 affects the stability of the positive equilibrium only for values of r such that conditions c(0) are satisfied. remark: for our parameter values, we have [(b−r2)2 −r21 − 2a1f] = −0.9475 < 0 and ∆ = [(b−r2)2−r21−2a1f] 2−4a21f 2 = 0.0878 > 0 proposition 7: consider system (24-25) with τ1 in its unstable interval (0 ≤ τ1 < τ ′1c). if a1 ≥ 2, then there exists a critical τ2 > 0, such that the positive equilibrium becomes stable for τ2 > τ2c. note that the effect of τ2 on the stability of the positive equilibrium does not depend on the values of r. c. hopf bifurcation analysis according to the hopf bifurcation theorem (culshaw [6]), the discrete time delay τ1 will cause the system to go through a hopf bifurcation near the steady state (x∗1,y ∗ 1), if the following transversality condition is satisfied: dα(τ1) dτ1 |τ1=τ′1c 6= 0. (26) to check this condition we recall that the characteristic equation of the system at (x∗1,y ∗ 1) when τ2 = 0 is given as : λ2 + (b−r2)λ−r1e−λτ1λ + a1f = 0. (27) substituting λ(τ1) = α(τ1) + iω(τ1) in equation (27), we have : α2 −ω2 + 2αωi + (b−r2)α + (b−r2)ωi −r1eατ1 (cos ωτ1 − i sin ωτ1)(α + iω) + a1f = 0. (28) biomath 1 (2015), 1505201, http://dx.doi.org/10.11145/j.biomath.2015.05.201 page 8 of 17 http://dx.doi.org/10.11145/j.biomath.2015.05.201 i. diakite et al., effects of discrete time delays and parameters ... we equate the real and the imaginary parts to zero, and we have : α2 −ω2 + a1f + r1eατ1 (α cos ωτ1 + ω sin ωτ1) +(b−r2)α = 0, (29) 2αω −r1eατ1 (ω cos ωτ1 −α sin ωτ1) +(b−r2)ω = 0. (30) we differentiate equations (29) and (30) with respect to τ1 and evaluate at τ1 = τ ′1c for which α(τ ′1c) = 0 and ω(τ ′ 1c) = ω ′ c. we obtain a dω(τ1) dτ1 |τ1=τ′1c −b dα(τ1) dτ1 |τ1=τ′1c = c cos ω′cτ ′ 1c + d sin ω ′ cτ ′ 1c (31) b dω(τ1) dτ1 |τ1=τ′1c + a dα(τ1) dτ1 |τ1=τ′1c = c sin ω′cτ ′ 1c −d cos ω ′ cτ ′ 1c (32) where a := 2ω′c −r1τ ′ 1c sin ω ′ cτ ′ 1c, b := (b−r2) + r1τ ′1c cos ω ′ cτ ′ 1c c := r1τ ′ 1c(ω ′2 c + ω ′ cτ ′ 1c), d := r1τ ′ 1cω ′ c sin ω ′ cτ ′ 1c. by solving equations (31) and (32) we have: dα(τ1) dτ1 |τ1=τ′1c = (ac −bd) sin ω′cτ ′1c − (ad + bc) cos ω ′ cτ ′ 1c a2 + b2 . (33) the system undergoes through a hopf bifurcation near (x∗1,y ∗ 1) if: (ac−bd) sin ω′cτ ′ 1c−(ad+bc) cos ω ′ cτ ′ 1c 6= 0. d. numerical results to illustrate the effect of the parameter r and the discrete time delay on the stability of the steady state (x∗,y∗), and to support the theoretical predictions discussed above, we conducted numerical simulations for the system (24-25). we used dde-biftool (engelborghs[17]) for the stability and bifurcation analysis and also used the matlab solvers ode23 and dde23 (shampine[18],shampine[19]) to see the behavior of the predator and prey populations through time. all the parameter values are given in table ii. fig. 4. the positive equilibrium is unstable for τ1 = τ2 = 0 and r = 10 > 1.the system exhibits a spiral out from the equilibrium (x∗1,y ∗ 1) = (0.111,1.111). biomath 1 (2015), 1505201, http://dx.doi.org/10.11145/j.biomath.2015.05.201 page 9 of 17 http://dx.doi.org/10.11145/j.biomath.2015.05.201 i. diakite et al., effects of discrete time delays and parameters ... for the given parameters values we have r = 10 > 1, and a positive equilibrium exists and is given as (x∗1,y ∗ 1) = (0.111, 1.111). when there is no delay the prey and predator populations variation through time is shown on figure 4. for our parameter values we have: [(b−d)2 −r21 − 4f] = −0.9475 < 0 and ∆ = [(b−d)2 −r21 − 4f] 2 − 16f2 = 0.0878 > 0. then there exists a τ1c = 6 such that the steady state remains unstable for 0 ≤ τ1 < τ1c and τ2 = 0 (see figure 5), it becomes stable as τ1 crosses τ1c and τ2 = 0 as shown on figure 6. fig. 5. the positive equilibrium remains unstable for τ1 = 1 < τ1c = 6 and τ2 = 0. fig. 6. the positive equilibrium is stable for τ1 = 7 > τ1c = 6 and τ2 = 0. we examine closely the stability switch introduces by τ1. we use dde-biftool to compute the eigenvalues of the characteristic equation (38) for τ2 = 0 and 0 ≤ τ1 ≤ 10. in figure 7 we plot the real parts versus the imaginary parts of these eigenvalues. we see that the equilibrium (x∗1,y ∗ 1) stabilizes as τ1 crosses the critical value τ ′1c = 6. we also plot in figure 7 the eigenvalues of equation (38) for τ1 = τ ′1c = 6 and observe a pair of two pure imaginary eigenvalues. the system undergoes through a hopf bifurcation as τ1 crosses τ ′1c. we compute the hopf bifurcations branches using matlab and show them in figure 8. biomath 1 (2015), 1505201, http://dx.doi.org/10.11145/j.biomath.2015.05.201 page 10 of 17 http://dx.doi.org/10.11145/j.biomath.2015.05.201 i. diakite et al., effects of discrete time delays and parameters ... fig. 7. the eigenvalues of the characteristic equation (38) for τ1 = 3 (left) and τ1 = 8(center) with τ2 = 0.at τ1 = τ′1c = 6 we can clearly observe a pair of 2 pure imaginary eigenvalues (right). note τ ′1c = 6 and τ2c = 2.5 for τ1 = 2 and τ2 = 0.5 the equilibrium is unstable as shown in figure 9. fig. 8. global hopf bifurcations branches as we vary τ1 and a1 (same as varying r). fig. 9. the positive equilibrium is unstable for τ1 = 2 and τ2 = 0.5. biomath 1 (2015), 1505201, http://dx.doi.org/10.11145/j.biomath.2015.05.201 page 11 of 17 http://dx.doi.org/10.11145/j.biomath.2015.05.201 i. diakite et al., effects of discrete time delays and parameters ... fig. 10. the positive equilibrium is stable for τ1 = 7 and τ2 = 1.2.the system exhibits a spiral in toward the equilibrium (x∗1,y ∗ 1) = (0.111,1.111). for τ1 = 7 and τ2 = 1.2 the equilibrium becomes stable as shown in figure 10. for the case of two non-zero delays, we use matlab to compute numerical simulations illustrating the effects of the two delays. the analysis is summarized in table iii table iii stability regions in case of two non-zero delays unstable stable stable unstable{ 0 ≤ τ1 < τ′1c, 0 ≤ τ2 < τ2c { τ1 > τ ′ 1c, 0 ≤ τ2 < τ2c { 0 ≤ τ1 < τ′1c, τ2 > τ2c { τ1 > τ ′ 1c, τ2 > τ2c see fig 4 and fig9 see fig6 and fig10 see fig11 see fig12 fig. 11. the positive equilibrium is stable for τ1 = 0.7 and τ2 = 8. biomath 1 (2015), 1505201, http://dx.doi.org/10.11145/j.biomath.2015.05.201 page 12 of 17 http://dx.doi.org/10.11145/j.biomath.2015.05.201 i. diakite et al., effects of discrete time delays and parameters ... fig. 12. the positive equilibrium is unstable for τ1 = 7 and τ2 = 3.1.the system exhibits an unstable periodic solutions. for τ1 = 7 and τ2 = 3.1 the equilibrium becomes unstable again as shown in figure 12. iv. conclusions and discussion it is well known that changes in the parameters play a crucial role in understanding dynamical systems. there is a need to incorporate discrete time delays in dynamical systems (biological systems, physical systems,...) as has been shown and studied by many researchers (perelson[1],bellen[3],..). published papers have shown that the incorporation of discrete time delays can highly impact the dynamics of the system, since they can cause stability switches of a steady state point, and can also cause the system to go through a hopf bifurcation near that steady state point (culshaw[6], bellen[3],...). the highlight of this paper is on how a local bifurcation parameter of the system may modify the stability changes caused by the delay(s).to understand the effects of discrete time delays and parameter variations on certain biological system models, we carried out a bifurcation analysis of a system of delay differential equations in detail for n=1 with specific examples, gave the procedure for higher n, and did a concrete example for n=2. we investigated the stability of the steady states as both bifurcation parameters, the discrete time delay τ and a local bifurcation parameter µ, cross critical values. our analysis shows that while both parameters can destabilize the steady state, the discrete time delay can cause stability switches of the steady state only upon certain values of µ. the local bifurcation parameter effects on the stability of the steady state do not depend on the value of τ. we also showed that both parameters act differently in term of bifurcation. while the discrete time delay may only introduce a hopf bifurcation, the parameter µ can introduce other type of bifurcations. biomath 1 (2015), 1505201, http://dx.doi.org/10.11145/j.biomath.2015.05.201 page 13 of 17 http://dx.doi.org/10.11145/j.biomath.2015.05.201 i. diakite et al., effects of discrete time delays and parameters ... v. appendix a theorem 1: consider the transcendental equation λn + n∑ i=1 an−iλ n−i + n∑ i=1 bn−ie −λτλn−i = 0, (34) if there exists a τc > 0 such that λ(τc) is a purely imaginary eigenvalue of (34), then for τ > τc the transcendental equation (34) has at least one eigenvalue with a strictly positive real part. before we prove the above theorem , let just first consider a much simpler case. consider the analytic function h(λ,a) = λ + e−λτ + a, with τ ≥ 0, and a ∈ r. then h(λ, 0) = 0 if and only if λ = −e−λτ. (35) equation (35) has purely imaginary roots if and only if τ = τc = 2jπ + π2 , j = 0, 1, 2, . . . the proof of the following lemma can be found in cooke and van den driessche [20]; see also bellman and cooke [4]. lemma 2: if τ ∈ [0, π 2 ), then all roots of equation (35) have strictly negative real parts. if τ ∈ ( 2jπ + π 2 , (2j + 1)π + π 2 ] , then equation (35) has exactly 2j + 1 roots with strictly positive real parts. we have h(λ,a) is an analystic function in λ, a. when τ 6= 2jπ + π 2 , the function h(λ, 0) has no zeros on the boundary of ω, where ω = {λ, |re(λ) ≥ 0, |λ| ≤ ρ}. thus, rouche’s theorem (dieudonne[11], theorem 9.17.4) implies that there exists a δ > 0 such that : (1) for any a < δ, h(λ,a) has no zero on the boundary of ω (2) for any a < δ, h(λ,a) and h(λ, 0) have the same sum of the orders of zeros belonging to ω. it follows from lemma 2 that when τ > π 2 , the sum of the orders of the zeros of h(λ, 0) belonging to ω is at least 1. thus when τ > π 2 , τ 6= 2jπ+ π 2 , and a < δ then h(λ,a) has at least a root with strictly positive real part. now we can prove the more general form which is theorem 1 proof : consider the analytic function in λ, a h(λ,a) =λn+ n∑ i=1 an−iλ n−i+ n∑ i=1 bn−ie −λτλn−i, (36) where λ ∈ c, and a = (an−1, ...,a1,a0,bn−1, ...,b1) ∈ rn×(n−1). then h(λ,a0) = λ n + b0e −λτ where a0 = (0, ..., 0) is the null vector. h(λ,a0) has purely imaginary roots if and only if τ = τjc = 2jπ b 1/n 0 j = 1, 2, ... when n is even, or τ = τjc = (4j + 1)π 2b 1/n 0 j = 0, 1, 2, ... when n is odd, and here we assume that b0 > 0, otherwise we multiple by a − sign. when τ 6= τjc the function h(λ,a0) has no zero on the boundary of ω, where ω = {λ, |re(λ) ≥ 0, |λ| ≤ ρ}. thus, rouche’s theorem implies that there exists a δ > 0 such that : (1) when ‖a‖∞ < δ, h(λ,a) has no zero on the boundary of ω (2) when ‖a‖∞ < δ, h(λ,a) and h(λ,a0) have the same sum of the orders of zeros belonging to ω. it follows from lemma 2 that when τ > τc = 2π b 1/n 0 and τ 6= 2jπ b 1/n 0 (or τ > τc = 1π 2b 1/n 0 and τ 6= (4j+1)π 2b 1/n 0 ), the sum of the orders of the zeros of h(λ,a0) belonging to ω is at least 1. thus when τ > τc , τ 6= τ j c and ‖a‖∞ < δ then h(λ,a) has at least a root with strictly positive real part. � biomath 1 (2015), 1505201, http://dx.doi.org/10.11145/j.biomath.2015.05.201 page 14 of 17 http://dx.doi.org/10.11145/j.biomath.2015.05.201 i. diakite et al., effects of discrete time delays and parameters ... vi. appendix b a. proof of proposition 1 the characteristic equation of the one dimensional dde is given by (3) and because the steady state is assumed to be stable at τ = 0 then α(0) = df1 dx |(x∗,µ∗) + df2 dx |(x∗,µ∗) < 0. (37) if (i) holds then equation (37) implies |df2 dx |(x∗,µ∗)| < | df1 dx |(x∗,µ∗)| therefore equation (9) has no positive root meaning the steady state remains stable for all τ ≥ 0. if (ii) holds then equation (37) implies |df2 dx |(x∗,µ∗)| > | df1 dx |(x∗,µ∗)| therefore equation (9) has a positive root then there exists a τc > 0 such that α(τ) > 0 whenever τ > τc. if (iii)(a) holds then again equation (9) has no solution therefore α(τ) < 0 for all τ ≥ 0 meaning the steady state remains stable. if (iii)(b) holds then equation (9) has a positive root then there exists a τc > 0 such that α(τ) > 0 whenever τ > τc. � b. proof of proposition 2 if conditions of proposition 1(iii)(a) hold then there is nothing to prove. assume that conditions of proposition 1(iii)(b) hold then equation (9) has a positive solution, therefore the delay can affect the stability of the equilibrium point. but if for some µ∗ in dµ we have the extra condition df1 dx |(x∗,µ∗) = g(x ∗)o(µ∗) and df2 dx |(x∗,µ∗) = h(x ∗)o( 1 µ∗ ), then one can rewrite equation (9) as ω2c = [ h(x∗) µ∗ ]2 − [g(x∗)µ∗]2. then there exists a critical value µc ∈ dµ of µ such that h(x∗) µ∗ ≈ 0 as µ∗ → µc. therefore equation (9) becomes ω2c = −[g(x ∗)µc] 2 < 0, which has no real positive root ωc, therefore α(τ) < 0 for all τ ≥ 0. this implies the delay does not have any effect on the stability of the equilibrium point when µ∗ > µc. � c. proof of proposition 5 the jacobian matrix of the system (24-25) is given by : j = [ r1e −λτ1 − a1y ∗ (x∗+1)2 − a1x ∗ x∗+1 a2y ∗ (x∗+1)2 e−λτ2 −r2 + a2x ∗ x∗+1 e−λτ2 ] . evaluating at (x∗,y∗) = (0, 0), the characteristic equation is given as (λ−r1e−λτ1 )(λ + r2) = 0. we note that the stability of (x∗,y∗) = (0, 0) depends only on τ1. • if τ1 = 0 then the eigenvalues are : λ = r1 > 0 and λ = −r2 < 0. therefore the (0, 0) is unstable. • if τ1 > 0, we have λ = r1e−λτ1 , let λ(τ) = α(τ) + iω(τ) then we have λ = r1e −ατ1 (cos ωτ1 − i sin ωτ1). one can choose ωcτ1c = π(2n+1) 2 (n=0,1,2,...) or τ1c = π(2n+1) 2ωc such that the real part of λ(τ) = α(τ) + iω(τ) at τ1c is zero (α(τ1c) = 0) and the imaginary part ω(τ1c) = ωc is a solution of the characteristic equation. then by the continuity of α we have : – α(τ) > 0 for τ1 < τ1c, – α(τ) < 0 for τ1 > τ1c. � d. proof of proposition 6 the characteristic equation of the system evaluating at (x∗1,y ∗ 1) is given by λ2 + (b−r1e−λτ1 −r2e−λτ2 )λ + (f −f1e−λτ1 )+ (a1 − 1)fe−λτ2 + f1e−λ(τ1+τ2) = 0, (38) biomath 1 (2015), 1505201, http://dx.doi.org/10.11145/j.biomath.2015.05.201 page 15 of 17 http://dx.doi.org/10.11145/j.biomath.2015.05.201 i. diakite et al., effects of discrete time delays and parameters ... where b = r2 + r1 x∗1 + 1 , f = r1r2 x∗1 + 1 , f1 = r1r2. • if τ1 = τ2 = 0 we have: λ2 − (b−r1 −r2)λ + a1f = 0 with b−r1 −r2 = − r1 r < 0, and a1f > 0. then the characteristic equation has at least a positive eigenvalue (if the eigenvalues are real) λ = r1 2r + √ δ 2 where δ = (b−r1 −d)2 − 4a1f, or, all its eigenvalues (if complex) have a positive real part ( r1 2r ). therefore the steady state (x∗1,y ∗ 1) is unstable. • if τ1 > 0 and τ2 = 0 then the characteristic equation becomes: λ2 + (b−r2)λ−r1e−λτ1λ + a1f = 0. since we know that the steady state is unstable when τ1 = τ2 = 0, the question becomes: does there exist a τ ′1c such that the steady state stabilizes as τ1 crosses τ ′1c? in other words if λ(τ1) = α(τ1) + iω(τ1), does there exist τ ′1c such that α(τ ′1c) = 0 and ω(τ ′ 1c) = ω ′ c which satisfies −ω′2c + i(b−r2)ω ′ c− ir1ω ′ c(cos ω ′ cτ ′ 1c − i sin ω ′ cτ ′ 1c) + a1f = 0. (39) setting the real and imaginary parts equal zero, we obtain: −ω′2c + a1f = r1ω ′ c sin ω ′ cτ ′ 1c (40) (b−d)ω′c = r1ω ′ c cos ω ′ cτ ′ 1c. (41) adding the square of both equations, we obtain: ω′4c +[(b−r2) 2 −r21 − 2a1f]ω ′2 c + a 2 1f 2 = 0 (42) such τ ′1c exists if and only if the above equation has at least a positive root ω′c. let m = ω′2c , then we have the quadratic equation: m2 + [(b−r2)2 −r21 − 2a1f]m + a 2 1f 2 = 0 (43) which has at least a positive root if: c(0) : [(b − r2)2 − r21 − 2a1f] < 0, and ∆ = [(b − r2)2 − r21 − 2a1f] 2 − 4a21f 2 ≥ 0, consequently, equation (42) has at least a positive root ω′c. which implies there exist a τ ′1c > 0 such that the steady state changes stability as τ1 crosses τ ′1c for τ2 = 0. in fact τ ′1c is the smallest of : τ ′j 1c = 1 ω′c arccos b−r2 r1 + 2πj ω′c , j = 1, 2, ...� references [1] a. s. perelson, d. e. kirschner, r. de boer, dynamics of hiv infection of cd4+ t cells,math biosci. 114 (1993) 81-125 [2] l. allen, an introduction to mathematical biology, pearson-prentice hall, upper saddle river, nj, 2007. 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[9] s. ruan, on nonlinear dynamics of predator-prey models with discrete delay, math. model. nat. phenom. 4 (2009) 140-188. biomath 1 (2015), 1505201, http://dx.doi.org/10.11145/j.biomath.2015.05.201 page 16 of 17 http://dx.doi.org/10.11145/j.biomath.2015.05.201 i. diakite et al., effects of discrete time delays and parameters ... [10] m. y. li, h. shu, joint effects of mitosis and intracellular delay on viral dynamics:two-parameter bifurcation analysis, math. biol. 64 (2011) 1005-20. [11] j. dieudonne, foundations of modern analysis, academic press, new york. 1960 [12] j. guckenheimer, p. holmes, nonlinear oscillations, dynamical systems, and bifurcations of vector fields, springer-verlag, new york, 1997. [13] j. k. hale, s. m. v. lunel, introduction to functional differential equations, springer verlag, new york, 1993. [14] y. s. chin, unconditional stability of systems with timelags., acta math. sinica, 1 (1960) 138-155. [15] r. datko, a procedure for determination of the exponential stability of certain differential difference equations. quart. appl. math. 36 (1978) 279-292. [16] j. e. ameh and r. ouifki, the basic reproductive number: bifurcation and stability, thesis, african institute for mathematical sciences. 2009. [17] k. engelborghs, t. luzyanina, g. samaey, ddebiftool v. 2.00: a matlab package for bifurcation analysis of delay differential equations, report tw 330, 2001. [18] l. f. shampine, i. gladwell, s. thompson,solving odes with matlab, cambridge university press, cambridge 2003. [19] l. f. shampine, s. thompson, solving ddes in matlab, applied numerical mathematics 37 (2001) 441458. [20] k.l. cooke, p.vanden driessche, and x. zou, interaction of maturation delay and nonlinear birth in population and epidemic models, j. math. biol, 39 (1999) 332-352. biomath 1 (2015), 1505201, http://dx.doi.org/10.11145/j.biomath.2015.05.201 page 17 of 17 http://dx.doi.org/10.11145/j.biomath.2015.05.201 introduction one dimensional field one equation with one delay stability of the steady state example one equation with multiple delays n dimensional field two dimensional field example stability analysis hopf bifurcation analysis numerical results conclusions and discussion appendix a appendix b proof of proposition 1 proof of proposition 2 proof of proposition 5 proof of proposition 6 references original article biomath 1 (2012), 1211119, 1–7 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum nonlinear filters and characterization of the discrete pulse transform of images inger fabris-rotelli department of statistics, university of pretoria pretoria, south africa inger.fabris-rotelli@up.ac.za received: 15 july 2012, accepted: 11 november 2012, published: 29 december 2012 abstract—the discrete pulse transform (dpt) for images and videos has been developed over the past few years and provides a theoretically sound setting for a nonlinear decomposition of an image or video. in [1] the theoretical basis of the dpt was presented. in this paper we now present a sound characterization of this useful nonlinear hierarchical decomposition by referring to its ability as a separator, the consistency of the decomposition, as well as the smoothing ability of the decomposition. keywords-lulu; discrete pulse transform; dpt; nonlinear decomposition; feature detection i. introduction the discrete pulse transform (dpt) is a nonlinear hierarchical decomposition obtained by the successive application of the lulu operators ln and un where n increases from 1 to n, where n is the number of data points in the signal. for a concise overview of the onedimensional lulu operators see [2] as well as further collaboration with laurie and wild. the lulu operators were extended in detail to multidimensional arrays in [1], which provides a framework for the obvious areas of image processing in two dimensions, as well as video processing in three dimensions. we provide a short overview of these operators and the dpt here for completeness. a. lulu operators the concept of morphological connectivity was introduced by j. serra and g. matheron in the 1980’s. they recognised the need for the concept of an axiomatic connectivity and thus the axiomatic approach to connectivity was introduced. in 1988 serra and matheron, [3], introduced the concept of a connectivity class, for use in mathematical morphology. definition 1: c is a connectivity class or a connection on the power set p(e) if the following axioms hold: (i) ∅ ∈ c (ii) {x} ∈ c for each x ∈ e (iii) for each family {ci} in c such that ⋂ ci 6= ∅, we have ⋃ ci ∈c. a set c ∈c is called connected. we define the operators ln and un on a(zd), where a(zd) is the vector lattice of all real functions defined on zd with respect to the usual point-wise defined addition, scalar multiplication and partial order. definition 2: let f ∈a(zd) and n ∈ n. then ln(f)(x) = max v∈nn(x) min y∈v f(y), x ∈ zd, un(f)(x) = min v∈nn(x) max y∈v f(y), x ∈ zd, where nn(x) = {v ∈c : x ∈ v,card(v ) = n + 1}. it is important to notice that here the collection of nneighbourhoods, nn(x), the lulu operators act on, can take on any shape as the only restriction is on their size. this is the important advantage of the lulu operators, which are only concerned about size, and morphological filters, which operate in conjunction with a specified structuring element with a specified size. this allows for open investigation of the image structures instead of searching for structures of a specific shape. citation: i fabris-rotelli, nonlinear filters and characterization of the discrete pulse transform of images, biomath 1 (2012), 1211119, http://dx.doi.org/10.11145/j.biomath.2012.11.119 page 1 of 7 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.11.119 i fabris-rotelli, nonlinear filters and characterization of the discrete pulse transform of images b. the discrete pulse transform the operators ln and un smooth the input signal by removing peaks (the application of ln) and pits (the application of un). these peaks and pits are defined mathematically in definition 3 and 4. definition 3: let v ∈ c. a point x /∈ v is called adjacent to v if v ∪ {x} ∈ c. the set of all points adjacent to v is denoted by adj(v ), that is, adj(v ) = {x ∈ zd : x /∈ v,v ∪{x}∈c}. definition 4: a connected subset v of zd is called a local maximum (minimum) set of f ∈a(zd) if sup(inf)y∈adj(v )f(y) < (>)inf(sup)x∈v f(x). the results proved in [1] and [4] provide the following summary of the effect of the operators ln and un on a function f ∈a(zd): (1) the application of ln (un) removes local maximum (minimum) sets of size smaller or equal to n. (2) no new local minimum (maximum) sets are created where there were none, however the action of ln (un) may enlarge existing local minimum (maximum) sets or join two or more local minimum (maximum) sets of f into one local minimum (maximum) set of ln(f) (un(f)). (3) ln(f) = f (un(f) = f) if and only if f does not have local maximum (minimum) sets of size n or less. the dpt provides a representation of an image (when d = 2) and higher dimensional arrays at all the scale levels. we obtain a decomposition of a function f ∈ a(zd), with finite support. let n = card(supp(f)). we derive the dpt of f ∈a(zd) by applying iteratively the operators ln,un with n increasing from 1 to n as follows dpt(f) = (d1(f),d2(f), ...,dn(f)), (1) where the components of (1) are obtained through d1(f) = (i−p1)(f), dn(f) = (i−pn)◦qn−1(f), n = 2, ...,n, pn = ln ◦ un or pn = un ◦ ln and qn = pn ◦ ... ◦ p1, n ∈ n. this decomposition results in components dn which are each a sum of discrete pulses φns,s = 1,2, ...,γ(n) with disjoint supports of size n, where in this setting a discrete pulse is defined as follows, definition 5: a function φ ∈a(zd) is called a pulse if there exists a connected set v and a real number α such that φ(x) = { α if x ∈ v 0 if x ∈ zd \v . the set v is the support of the pulse φ, that is supp(φ) = v . we can then reconstruct the original signal as f = n∑ n=1 dn(f) = n∑ n=1 γ(n)∑ s=1 φns. ii. characterization of the dpt a. linear versus nonlinear as discussed in [4], the nonlinearity of the lulu smoothers makes theoretical development more complicated than for linear operators. however, taking on the additional complexity is justified since in two dimensions an image is basically the transformation of data by a human eye or measuring instrument. this transformation is significantly complicated to be considered nonlinear [5]. thus taking this stance the analysis of images via nonlinear operators is more logical than that of linear. linear operators are also notorious for blurring edges. linear processing techniques are however a natural starting point for analysis due to the simplicity of their application and theoretical backbone available. examples of linear filters are the fourier transform, hadamard transform, the discrete cosine transform, and wavelets. they also provide sufficient results in most applications, but there are problems in which a nonlinear process would prove more viable and efficient. pitas and venetsanopoulos [6] provide examples of such cases, such as signal dependent noise filtering e.g. photoelectron noise of photosensing devices; multiplicative noise appearing as speckle noise in ultrasonic imaging and laser imaging; and nonlinear image degradations e.g. when transmission occurs through nonlinear channels. advantages of nonlinear filters are 1) the ability to handle various noise types, 2) edge preservation, 3) fine detail preservation, 4) unbiasedness (directional and illumination based) or invariance, and 5) computational complexity [6]. nonlinear filtering techniques can be broadly classified accordingly in the following areas: order statistic filters, homomorphic filters, polynomial filters, mathematical morphology, neural networks, and nonlinear image restoration [6]. the lulu operators fit nicely into the areas of mathematical morphology, due to their similarity to area operators therein, as well as order statistics, two areas which have been integrated quite efficiently in literature [6]. examples of order statistics, discussed in detail in [6], are the median, rank-order filters, maxmin filters, lp-mean filters, and α-trimmed mean filters. the lulu operators are examples of max-min filters but with the disadvantages listed in [6] improved upon. biomath 1 (2012), 1211119, http://dx.doi.org/10.11145/j.biomath.2012.11.119 page 2 of 7 http://dx.doi.org/10.11145/j.biomath.2012.11.119 i fabris-rotelli, nonlinear filters and characterization of the discrete pulse transform of images the basic filters of mathematical morphology are the erosion and dilation, and subsequently the morphological opening and closing, to which the lulu filters are again closely related. b. separators a common requirement for a filter p , linear or nonlinear, is its idempotence, that is p ◦p = p . for linear operators the idempotence of p implies the idempotence of the complementary operator i −p , where i denotes the identity operator. for nonlinear filters this implication generally does not hold, so the idempotence of i − p , also called co-idempotence, [7], can be considered as an essential measure of consistency. for every a ∈ zd the operator ea : a(zd) → a(zd) given by ea(f)(x) = f(x − a), x ∈ zd, is called a shift operator. we now define a separator which mimics the actions required of an operator p . the first three properties in definition 6 define a smoother. more detail on smoothers can be found in [4]. definition 6: an operator p : a(zd) → a(zd) is called a separator if (i) p ◦ ea = ea ◦ p, a ∈ zd (horizontal shift invariance) (ii) p(f + c) = p(f) + c, f,c ∈a(zd), c a constant function (vertical shift invariance) (iii) p(αf) = αp(f), α ∈ r, α ≥ 0, f ∈ a(zd) (scale invariance) (iv) p ◦p = p (idempotence) (v) (i −p)◦ (i −p) = i −p (co-idempotence) figure 1 illustrates the action of a separator p . it shows how a separator will separate the signal into noise and the true signal without the need for recursive smoothing, that is, it does the separation on the first filter application so that there is no ‘signal’ left in the ‘noise’ nor any ‘noise’ left in the ‘signal’. the median smoother is an example of a filter which requires recursive application. the lulu operators ln and un and all their compositions are separators thus providing a strong separating capability of a signal. c. nonlinear decompositions the structure of a hierarchical decomposition is as follows in general. the operator f1 is applied to the input image f to obtain a decomposition of f into f1, the smoother image, and d1, the noise component removed. this process is repeated with f2, f3,...,fn until there is nothing left to remove except the constant image dn . the decomposition then has the form f = β1d1 + β2d2 + ... + βndn , for some βi, i = 1, ...,n. such a fig. 1. the action of a separator p hierarchical decomposition has been investigated intensively in literature, see [8], [9], [10] for some nonlinear cases. however, in no literature have we found a unified theoretical backbone to connect such nonlinear hierarchical decompositions and provide methods of comparison nor methods of testing the capability of the structure of the decomposition. in tadmor et al [8], for example, a decomposition f = ∑k j=1 uj + vk is obtained, where vk is the noise component and the uj’s the decomposition components, by functional minimization. tadmor et al discuss convergence of the minimizer, localization and adaptability, but nothing to indicate the strength of the decomposition save numerical visual examples. similarly wong et al [10] do not provide a theoretical indication of the strength of their decomposition obtained as a probabilistic scale-space derived from the nonlinear diffusion equation in [11]. in [9], the authors even state that comparisons with their proposed nonlinear scale-space and other nonlinear hierarchical decompositions ‘are to be made with care’. d. consistent decompositions as stated in [6] and above in section ii-c, the main limitations of nonlinear decompositions is the lack of a theory with which to compare the ability of various decompositions. the highlight theorem first conjectured in 2007 [12], and later proved in 2010 [13], provides this much needed backbone. the quality of a nonlinear hierarchical decomposition, such as the discrete pulse transform given in (1), can be characterized through the concept of consistent decomposition (also called strong consistency, [12]). for a linear decomposition of two signals f and g, namely d(f) = ∑ i di(f) and d(g) =∑ j dj(g), the equality d(αf + βg) = α ∑ i di(f) + β ∑ j dj(g) ∀ α,β ∈ r biomath 1 (2012), 1211119, http://dx.doi.org/10.11145/j.biomath.2012.11.119 page 3 of 7 http://dx.doi.org/10.11145/j.biomath.2012.11.119 i fabris-rotelli, nonlinear filters and characterization of the discrete pulse transform of images ? f (image) j j j f1 �� d1 @@? � � � ��� α1d1 j j j f2 �� d2 @@? � � ��� α2d2 . . . . . . . . . . . . . . . . . . ? j j j fn �� dn� � � ��� αndn ? @ @ @ @ @ @ @ @ @ @ @ @ @ j j j f1 �� α1d1 @@? j j j f2 �� α2d2 @@? . . . . . . . . . . . . . . . . . . ? j j j fn �� αndn fig. 2. illustration of the highlight theorem always holds. this is clearly a desirable property but for a nonlinear decomposition this will in general not be satisfied. however, a weaker form of linearity is provided by the highlight theorem. it shows that the multidimensional discrete pulse transform in (1) is strongly consistent, in the sense that the above holds for α,β > 0. the highlight theorem [13] states the following: given a basis of pulses identified by the dpt for a signal f, form a new function g as any linear combination of the pulses of f with heights the same sign as before. then the dpt of g will identify the same basis of pulses, and recover the new heights. figure 2 provides an illustration of the highlight theorem. the name of theorem indicates its usefulness. besides the ‘weak’ linearity it presents for a nonlinear decomposition, which we shall define as highlightlinearity, it allows for the highlighting or emphasizing of specific pulses deemed to be important, without destroying the structure of the decomposition. we state the theorem more precisely in the following form. theorem 7: for a dpt decomposition of f, let g =∑n n=1 ∑γ(n) s=1 αnsφns where the constants αns are positive. then the dpt decomposition of g is obtained as dpt(g) = ∑n n=1 ∑γ(n) s=1 αnsφns, so that the pulses of g are obtained as αnsφns. if αns = αn for each n then dpt(g) = ∑n n=1 αndn(f), so that dn(g) = αndn(f). the proof of theorem 7 by laurie for any dimension can be found in [13]. therein the theory is described by considering the signal as a graph where the data points are the graph vertices and with the edges between the vertices based on the geometry of the of the signal (i.e. connectivity). laurie also provides an implementation algorithm for the dpt which is o(m) where m is the number of edges in the graph. the dpt decomposition is also total variation preserving. we assume c on zd is a graph connectivity, for example for images the pulses of the dpt are based on 4or 8-connectivity, the individual pulses can be viewed as a graph g = (vn,em) with the data points as the n vertices vn and the neighbour relation between data points as the m edges em. the connectivity of such a graph g can be defined via a relation r ⊂ zd × zd, where p ∈ zd is connected (by an edge) to q ∈ zd iff (p,q) ∈ r. the relation r reflects what we consider neighbours of a point in the given context. for example, 4-connectivity and 8-connectivity. let r be a relation on zd. we call a set c ⊆ zd connected, with respect to the graph connectivity defined by r, if for any two points p,q ∈ c there exists a set of points {p1,p2, ...,pk}⊆ c such that each point is neighbour to the next one, p is neighbour to p1 and pk is neighbour to q. here we assume that r is reflexive, symmetric and shift invariant, (p,p + ek) ∈ r for all k = 1,2, ...,d and p ∈ zd, where ek ∈ zd is defined by (ek)i = { 0 if i 6= k 1 if i = k . biomath 1 (2012), 1211119, http://dx.doi.org/10.11145/j.biomath.2012.11.119 page 4 of 7 http://dx.doi.org/10.11145/j.biomath.2012.11.119 i fabris-rotelli, nonlinear filters and characterization of the discrete pulse transform of images fig. 3. full total variation distribution and log histogram total variation distribution of cat image the total variation of f ∈a(zd) is given by tv (f) = ∑ p∈zd d∑ i=1 |f(p + (ek)i)−f(p)|. although the importance of total variation preservation for separators cannot be doubted, it is even more so for hierarchical decompositions like the discrete pulse transform, due to the fact that they involve iterative applications of separators. since the operators ln, un, n = 1,2, ...,n, and all their compositions, are total variation preserving, it is easy to obtain the following result, which shows that, irrespective of the length of the vector or the number of terms in the sum , no additional total variation, or noise, is created via the decomposition, namely tv (f) = n∑ n=1 γ(n)∑ s=1 tv (φns). (2) the proof of this result can be found in [1]. we should remark that representing a function as a sum of pulses can be done in many different ways. however, in general, such decompositions increase the total variation, that is, we might have strict inequality in (2) instead of equality. based on this result we can construct the total variation distribution of images. more precisely, this is the distribution of the total variation of an image among the different layers of the dpt. that is, essentially the plot of tv (dn(f)) vs. n. in figure 3 we present the total variation distribution of an image, where one can observe how the total variation is distributed over the pulse size. in the graph a log scale is used on the vertical axis and the pulse size values are grouped to form a histogram. the different character of images naturally manifests through different forms of total variation distributions. using the total variation distribution as a guide for the content of the image we can determine, firstly, that the image is relatively ‘denoised’ when the total variation graph stabalizes, that is, the very little information is removed after this scale by the dpt. in figure 4 various scales of the dpt are picked out based on the total variation graph. scales 1 to 17859 indicate the smoothed image, that is, when the total variation removal stabalizes. this smoothed image appears very similar to the original (with an mssim value of 0.9736 (see section ii-e)) but has been smoothed by removing the remaining scale levels, 17860 to 58571, which contain the large undetailed pulses. scales 1 to 880 provide the texture or detail of the image, scales 4385 to 4395 represent the large eye of the cat and scale 11420 the eyes, nose and mouth of the cat. e. measuring the smoothing ability of the lulu operators the ability of an operator to effectively remove noise and smooth the signal is usually measured by its output variance or the rate of success in the noise removal [6]. other measures used to assess the performance are the mean square error (mse) and signal-noise-ratio (snr) [6]. in this section we shall present a method in which to measure the quality of the smoothed image as the question of the smoothing ability of the dpt arises. the aim of a smoother is of course to remove the noise element present. the noise can be due to a number of factors, for example, acquisition, processing, compression, storage, transmission and reproduction of the image [14]. the easiest method of evaluation is purely subjective namely, human visual investigation. in order for evaluation to be objective, quantitative methods need to be used instead. quantitative methods can be divided into three categories [14]. first, full-reference, where the complete reference (undistorted) image is known with certainty, secondly, no-reference, where this reference image is not known at all, and third, reduced-reference, where only part of the original reference image is known, for example, a set of extracted features. we measure the similarity of the smoothed images pn(f) to the original unsmoothed image f with the structural similarity index [14], ssim(x,y) = (2µxµy + c1)(2σxy + c2) (µ2x + µ 2 y + c1)(σ 2 x + σ 2 y + c2) , for two corresponding sets of pixels, x and y, in each image, where µi, i = x,y is the mean of the pixel values in i, σ2i , i = x,y is the variance of the pixel values in i, σxy the covariance between x and y, cj = (kjl) 2 ,j = 1,2 constants to stabilize the division by the weak denominator, l = 255 for greyscale images and biomath 1 (2012), 1211119, http://dx.doi.org/10.11145/j.biomath.2012.11.119 page 5 of 7 http://dx.doi.org/10.11145/j.biomath.2012.11.119 i fabris-rotelli, nonlinear filters and characterization of the discrete pulse transform of images fig. 4. chelsea image using only scales 1 to 17859, 17860 to 58571, and 1 to 880 in the first row, and 4385 to 4395 and 11420 in the second row, respectively fig. 5. mssim values comparing pn(f) with f values plotted against scale for the cat image and an ocean image kj � 1 constants (we used kj = 0.05). this measure is a full-reference measure which provides a useful framework since we are comparing a smoothed version of the original distorted image with the original distorted image. the most widely used such measures are the mean-square-error (mse) and the peak signal-noise-ratio (psnr), but these measures do not compare well with the perceived visual quality of the human visual system. this measure is introduced in order to penalize errors based on their visibility, that is, to simulate the hvs as much as possible. this measure is applied to 8 × 8 windows in the image for each pixel and a final mean structural similarity index is calculated as the average of these ssim values, called the mssim. an mssim value closer to 1 indicates stronger similarity. wang et al provide matlab code for the implementation of the mssim as a free download, which was made use of. figure 5 provides mssim values for various images as the lulu smoothing progresses through the dpt from scale n = 1 up to n. notice how, based on the content of the images, the reduction in the mssim values as the dpt progresses varies from image to image. the graphs provide a mechanism to determine where visual structure is in the image, that is, when the hvs would pick out structures of significance. notice how the ocean image in figure 5 contains very little structure and the mssim plot decreases gradually through the application of the dpt without any ‘occurrences’. the cat image however presents a number of phenomenons in its mssim plot which indicate structure. figure 6 indicates what is present at these scales. scales 1 to 4030 represent the detail and the remaining scales the large relatively unimportant scales. the eye, face and forehead are represented at scales 4234 to 4235, 4325 to 4335 and 14565 to 14575 respectively. iii. conclusion we have presented on overview of the lulu operators and the resulting discrete pulse transform for multidimensional arrays. as a new hierarchical decomposition the status of the dpt within the image processing community needs to be justified, thus we provide a characterization of the theoretical backbone of this nonlinear decomposition. this also provides a method of comparison for other nonlinear decompositions, which does not currently exist. the opportunity for further measures biomath 1 (2012), 1211119, http://dx.doi.org/10.11145/j.biomath.2012.11.119 page 6 of 7 http://dx.doi.org/10.11145/j.biomath.2012.11.119 i fabris-rotelli, nonlinear filters and characterization of the discrete pulse transform of images fig. 6. cat image at scales 1 to 4030, 4031 to 58571, 4234 to 4235, 4325 to 4335, and 14565 to 14575 respectively of capability should not escape the reader, for example, the ability of noise removal for different types of noise contamination (independent as well as dependent noise structures), total variation preservation, and the nature of the pulse shapes and what the shapes mean relative to the image structure, image patterns and image content. such work is currently under investigation. acknowledgment the author acknowledges dr s van der walt whose collaboration ensured the implementation of lulu and the dpt in python [15]. references [1] r. anguelov and i. n. fabris-rotelli, “lulu operators and discrete pulse transform for multi-dimensional arrays,” ieee transactions on image processing, vol. 19, no. 11, pp. 3012– 3023, 2010 http://dx.doi.org/10.1109/tip.2010.2050639. [2] c. rohwer, nonlinear smoothers and multiresolution analysis. birkhäuser, 2005. [3] j. serra, image analysis and mathematical morphology, volume ii: theoretical advances, j. serra, ed. london: academic press, 1988. [4] i. fabris-rotelli, “lulu operators on multidimensional arrays and applications,” masters dissertation, university of pretoria, november 2009. [5] c. rohwer and m. wild, “lulu theory, idempotent stack filters, and the mathematics of vision of marr,” advances in imaging and electron physics, vol. 146, pp. 57–162, 2007 http://dx.doi.org/10.1016/s1076-5670(06)46002-x. [6] i. pitas and a. n. venetsanopoulos, “order statistics in digital image processing,” proceedings of the ieee, vol. 80, no. 12, pp. 1893–1921, 1992 http://dx.doi.org/10.1109/5.192071. [7] m. wild, “idempotent and co-idempotent stack filters and minmax operators,” theoretical computer science, vol. 299, pp. 603–631, 2003 http://dx.doi.org/10.1016/s0304-3975(02)00540-6. [8] e. tadmor, s. nezzar, and l. vese, “multiscale hierarchical decomposition of images with applications to deblurring, denoising and segmentation,” communications in mathematical sciences, vol. 6, no. 2, pp. 281–307, 2008. [9] l. florack, a. salden, b. ter haar romeny, and j. k. andma viergever, “nonlinear scale-space,” image and vision computing, vol. 13, no. 4, pp. 279–294, 1995 http://dx.doi.org/10.1016/0262-8856(95)99716-e. [10] a. wong and a. mishra, “generalized probabilistic scale space for image restoration,” ieee transactions on image processing, vol. 19, no. 10, pp. 2774–2780, 2010 http://dx.doi.org/10.1109/tip.2010.2048973. [11] p. perona and j. malik, “scale-space and edge detection using anisotropic diffusion,” ieee transactions on pattern analysis and machine intelligence, vol. 12, pp. 629–639, 1990 http://dx.doi.org/10.1109/34.56205. [12] d. laurie and c. rohwer, “the discrete pulse transform,” siam journal of mathematical analysis, vol. 38, no. 3, 2007. [13] d. laurie, “the roadmaker’s algorithm for the discrete pulse transform,” ieee transactions on image processing, vol. 20, no. 2, pp. 361–371, 2011 http://dx.doi.org/10.1109/tip.2010.2057255. [14] z. wang, a. bovik, h. sheik, and e. somincelli, “image quality assessment: from error visibility to structural similarity,” ieee transactions on image processing, vol. 13, no. 4, pp. 600–612, 2004. [15] s. j. van der walt, “super-resolution imaging,” phd thesis, stellenbosch university, december 2010. biomath 1 (2012), 1211119, http://dx.doi.org/10.11145/j.biomath.2012.11.119 page 7 of 7 http://dx.doi.org/10.1109/tip.2010.2050639 http://dx.doi.org/10.1016/s1076-5670(06)46002-x http://dx.doi.org/10.1109/5.192071 http://dx.doi.org/10.1016/s0304-3975(02)00540-6 http://dx.doi.org/10.1016/0262-8856(95)99716-e http://dx.doi.org/10.1109/tip.2010.2048973 http://dx.doi.org/10.1109/34.56205 http://dx.doi.org/10.1109/tip.2010.2057255 http://dx.doi.org/10.11145/j.biomath.2012.11.119 introduction lulu operators the discrete pulse transform characterization of the dpt linear versus nonlinear separators nonlinear decompositions consistent decompositions measuring the smoothing ability of the lulu operators conclusion references original article biomath 1 (2012), 1209031, 1–7 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum exploring family relations between international patent applications peter hingley european patent office erhardtstrasse 27, munich, germany email: phingley@epo.org received: 12 july 2012, accepted: 3 september 2012, published: 10 october 2012 abstract—in the international system for granting patents for inventions, first patent filings can be followed by subsequent filings at other patent offices within one year. each such group of related filings constitutes a patent family. tests are developed as to whether the observed number of first filings that leads to subsequent filings (r) is in agreement with a random process of assignment of the hits from the subsequent filings. an exact expression for the random distribution can be used for small sized data sets. its behaviour and also the behaviour of an asymptotic poisson approximation as well as a censored binomial distribution for r are assessed. the approach is stimulated by the fisher-wright model in population genetics and possible parallel applications to other biological processes are sought, such as transformations of stem cells and cancer. keywords-censored binomial; genetics; patents; random assignment i. introduction usually an inventor starts a quest for intellectual property protection by making a first patent filing at the local national patent office. then, within one year, subsequent filings quoting the priority of that first filing can be made at any patent office. these are termed subsequent filings. unlike most national patent offices, the applications that are received at the european patent office (epo) are mostly subsequent filings, due to its supranational character as an umbrella office for the european patent convention contracting states (epc), which also have their own national offices to which applications can be made [1]. to aid the statistical description of the flows of such (provisional) patent rights, the concept of patent families is useful. these are explained in ii. the pri database is a patent families file that is extracted from a worldwide patent database at epo called docdb, that itself contains data on patent publications from all the main offices around the world [2]. a subset of the documents in docdb represents published patent filings that can be identified as representing either first filings or subsequent filings, depending on whether or not they contain priority references to earlier first filings. in pri the data are re-ordered and compacted so that each record is indexed by a priority reference. information is also given on the activities of subsequent filings that quote that priority, such as the major geographical blocs in which subsequent filings took place. studying the international spread of patent filings combines concepts from several streams. mainly since the 1960s there have been studies of patent economics and statistics, starting with examples of the patenting process as motivators for econometric models, but later on centring more on elements of the system itself that has become an important economic driver [3]. in parallel the subjects bibliometrics and scientometrics have been developed. network theory can also be relevant [4] because the relationship of subsequent filings to first filings is not one-to-one, even when considering just a single first filing office / subsequent filing office pair. that is, one first filing can be quoted as a priority in more than one subsequent filing, and a subsequent filing can quote several first filings as priorities. citation: p. hingley, exploring family relations between international patent applications, biomath 1 (2012), 1209031, http://dx.doi.org/10.11145/j.biomath.2012.09.031 page 1 of 7 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.09.031 p. hingley, exploring family relations between international patent applications fig. 1. structure of patenting in terms of a generation of first filings, followed by subsequent filings up to one year later. here some probabilistic aspects will be considered, concentrating on subsequent filing activities at the epo. the concepts to be explored are inspired by tests for random mating in population genetics. patent family populations do not directly exhibit so many directly analogous characteristics to biological populations, but nevertheless it is interesting to study the parallels and differences. it will be suggested that the methods might have some as yet unrecognised ability to model cellular processes in biology, or at least be able to give fresh insight into experiments and models that could be tried out. ii. the structure of patenting and patent families first patent filings (ffs) lead on to subsequent filings (sfs) in other countries up to one year later. imagine two generations as set up in fig. 1, rather like the fisherwright model in population genetics [5]. the ffs are the f0 generation and the sfs are the f1 generation. the members of f0 do not all reproduce, but some do to give one or more offspring in f1. each member of f1 however must have at least one parent in f0. the parallels with biology do not go much further than this in any strict sense, because after f1 the patents are examined and granted if considered worthy, then maintained against the payment of appropriate fees for up to 20 years before they lapse. this means that no further reproduction of this cohort can normally take place. the generational pair of populations f0 and f1 can be said to be renewed over and over again every year. (this could be paralleled perhaps in population genetics by a model for pets.) beyond the limited set-up considered here however, the population of inventors and fig. 2. single priority patent families arising from first filings in 2006 (2005 in brackets), indicating first filings and flows, which are the counts of first filings referenced as priorities in subsequent filings in other blocs. from [7]. firms that make patent applications persists over time as well as with dynamic entries and departures each year [6]. the act of filing for patents can be considered as a possible survival tactic in a competitive world. there are no genes or dna in patent families, although there are technical classification systems to describe the areas covered by a patent that play some kind of analogous role. there are various types of patent families according to different definitions. for single priority families, which will be used here, each family constitutes one ff together with all the sfs that lead from it. thus each ff from which a priority filing emanates can initiate one family only. but the sfs in f1, that are the offspring of the ffs, can belong to more than one family. more extensive definitions of patent families are possible, that include for instance composite patent families, where each family consists of a complete interconnected network of ffs and sfs. this has the advantage of making every family unique, because no patent publication can belong to more than one family. however this may not be such an important consideration. in a study involving the whole population of recorded publications with earliest priorities in the period from 1991 to 1999, it was found that the nine most common family structures relate to a single priority and make up more than 77 per cent of patent families [8]. also, 29 per cent of families consisted of only a pair of one ff with one sf. single priority families can be used to describe patent biomath 1 (2012), 1209031, http://dx.doi.org/10.11145/j.biomath.2012.09.031 page 2 of 7 http://dx.doi.org/10.11145/j.biomath.2012.09.031 p. hingley, exploring family relations between international patent applications fig. 3. random hits model. filings flows between countries (fig. 2), although the subtleties of multiple assignments between ffs and sfs should also be taken into account to give a complete description. in order to do this, the concept of links is important. that is, each priority reference from a sf forms a link to the ff that is referenced. in a set of p ffs and y sfs emanating from those ffs, involving in each case filings at one or more patent offices, let there be l links. say that the average number of sfs that are linked to a ff is φ, while the average number of ffs that are linked to a sf is θ. these averages relate to l as follows [2]. l = p.φ = y.θ iii. random hits model for priorities fig. 3 shows that the setup of p priority filings in the f0 generation with l links to y sfs in the f1 generation can be represented by a surjective directed graph. for the idealised model to be considered here, the occasional groupings of several members of l into sets that represent sfs with several priority references will be ignored. let r be the number of members of p that are hit by the l links. if p and l are considered to be fixed, r can be modelled by a random process of hits on p by l. a. exact distribution feller [9] developed the following formula for the discrete probability distribution of r, under the hypothesis of a process of independent random hits. p r(r) = 1 pl ( p p−r ) ∑r ν=0 (−1) ν ( rν )(r − ν)l this is valid for any values of l and p that are positive integers. no explicit expression is given for the moments of this distribution. feller’s examples concentrate on the case l > p, such as where r is the number of days in a year when there is at least one birthday in a village of 2000 people (l = 2000, p = 365). in our case here l < p, because only a proportion of ffs lead to sfs. for the following calculations, routines were written in r. p r(r) is easily computable only when r is small. fig. 4 shows p r(r) for the case p = 30 and l = 20. direct evaluation gives a mean of 14.77 and a standard deviation (square root of variance) of 1.49. the distribution was checked by constructing the r values obtained in one million simulated sets of data, where each set was formed by sampling randomly with replacement the first p integers l times. the resulting histogram is indistinguishable visually from fig. 4, with a mean of 14.76 and a standard deviation of 1.50. this shows good agreement with the exact distribution. b. poisson approximation feller [9] argues for a poisson approximation for p r(r) as p and l −→ ∞. say t is the number of members of p that do not lead to subsequent filings. he asserts that, if λ = pe− l p remains bounded, p r(t) −→ e−λ. λ t t! | [0 < t < ∞] the support of this distribution is not bounded above. in fact p is finite and we are interested in p r(r = p − t). this can be approximated by a transform of the poisson distribution, bounded above at p but unbounded below 0. p r(r) ≈ e−λ. λ (p−r) (p−r)! | [−∞ < r ≤ p] fig. 5 shows p r(r) for the case p = 30 and l = 20, and can be compared to the exact distribution in fig. 4. direct evaluation gives a mean of 14.61 and the quantity p − λ is 14.60, showing good agreement with the mean according to feller’s argument, and not too far from the exact distribution mean of 14.76. however the standard deviation is 3.92, which is more than twice as high as 1.49 for the exact distribution. the shape is also different, and is essentially censored at the upper limit of 20, where r = l. c. censored binomial approximation since the poisson approximation does not work well at this sample size, other approximations can be tried. a censored binomial distribution is in some way equivalent biomath 1 (2012), 1209031, http://dx.doi.org/10.11145/j.biomath.2012.09.031 page 3 of 7 http://dx.doi.org/10.11145/j.biomath.2012.09.031 p. hingley, exploring family relations between international patent applications fig. 4. exact distribution with p = 30 and l = 20. to feller’s exact distribution, except that the hit probabilities are all considered independent and equivalent, and dependency due to the conditional probability chain is ignored. say that s is the number of hits from l to a member of p. this can be represented approximately as binomial(s, l, 1 p ), meaning the binomial probability of s successful outcomes when there are l independent trials, each with probability 1 p of success. p r(s) ≈ l! s!(l−s)! ( 1 p )s ( 1 − ( 1 p ) )(l−s) = binomial(s, l, 1 p ) under an assumption of independence, the probability that a particular member of p is hit is then as follows. 1 − binomial(0, l, 1 p ) = 1 − (1 − 1 p )l r is the number of distinct members of p that are hit. if l ≥ p, as in the birthdays example in iiia, p r(r) ≈ binomial(r,p,1−(1− 1 p )l) [1−binom(0,p,1−(1− 1 p )l)] this is a censored binomial that removes the zero class, because at least one member of p must be hit (p r(0) = 0). but in the patent families case, where l < p, the response range is restricted to r in (1, ..., l), so there is also censorship to remove all classes between and including l + 1 and p. p r(r) ≈ binomial(r,p,1−(1− 1 p )l) [1−binom(0,p,1−(1− 1 p )l)− pp j=l+1 binomial(j,p,1−(1− 1 p )l] fig. 5. poisson approximation with p = 30 and l = 20. fig. 6 shows p r(r) for the case p = 30 and l = 20, to compare with fig.s 4 and 5. the r routine in this case calculates the probabilities using the normal approximation to the binomial distribution. it was checked that this makes minimal difference to usage of the exact binomial expressions, even at this small population size. direct evaluation gives a mean of 14.63, again fairly close to the exact distribution mean of 14.76. this time the standard deviation is 2.64, which is closer than the poisson approximation to the 1.49 for the exact distribution. the shape is however still quite different to the exact distribution, although not as far away as the poisson was. iv. random hits model for patent families data the distributions in iii can be scaled up to give tests of random hits to patent families with sfs at epo. in the following examples, ffs at epo were ignored because they were already hit in a sense at the time of first filing. it should also be recognised that ffs and sfs at epo do not represent all the patenting activity in europe, because of the alternative possibility to file at the national patent offices in each epc contracting state. note also that the analysis will be monospecific, in that it is only the flows to epo that are considered, and not the spread of flows to all offices, as was considered in [8]. in order to scale up to the case of an annual data set of first filings (f0 generation) and the subsequent filings that quote them as priorities (f1 generation), consider p =1 052 420 worldwide first filings in the year 2002 and l =135 439 references to these priorities that were made in sfs to epo, mainly in the year 2003. the subset biomath 1 (2012), 1209031, http://dx.doi.org/10.11145/j.biomath.2012.09.031 page 4 of 7 http://dx.doi.org/10.11145/j.biomath.2012.09.031 p. hingley, exploring family relations between international patent applications fig. 6. censored binomial approximation with p = 30 and l = 20. of p that were referenced was of size r =120 701. how near was this to a random process of hits of l on p? feller’s exact distribution is not calculated in this case because it is no longer straightforward to do so with large numbers (the practical upper limit for the r routine is about p = 60). simulations are also more time consuming, but it is possible to make enough of these to get a good idea of the shape of p r(r). fig. 7 shows a first estimate of the distribution that was made with 1000 simulated data sets. what is being emulated here is presumably a unimodal discrete distribution like fig. 4, but due to a lack of binning we see a discretised approximation (p r(r) = 0.001 equivalent to one simulated outcome, p r(r) = 0.002 equivalent to two simulated outcomes, etc.). the mean is estimated as 127 079 and the standard deviation is 84. while these parameters are obviously not determined with great accuracy, due to the small number of simulations carried out, the shape of the distribution indicates that the observed value r = 120 701 is significantly lower than its expectation under the random hits model. the simulation results in fig. 7 lie in a very tight range around their mean, compared to the support. under a normal approximation, 95 per cent of the simulated r values are expected to be between 126 911 and 127 247. the distributions p r(r) according to feller’s poisson formula and the censored binomial approximation are shown in fig.s 8a and 8b respectively. they are both centered close to the mean of the simulations, and p−λ from the poisson approximation is 127 086, which is also close to the mean of the simulations. but the spreads of both distributions are again too large, with standard deviation according to the poisson distribution at 962 and for the censored binomial formula at 334. however fig. 7. simulated distribution of r, based on 1 000 simulated data sets with p = 1 052 420 and l = 135 439. it can be seen in this figure that the observed value of r is still significantly too low to be entirely random, even for the poisson formula. so it seems that the number r of worldwide priorities in 2002 that were hit by epo sfs was lower than expected under a random hits model. the test was also carried out on priorities after separation into the main geographical blocs of origin (epc, japan, us, others) and over five priority years (2002 to 2006 inclusive). see figs. 9a to 9d. the expected values under the random hits model are represented in these diagrams by values of p − λ (triangles). the results are fairly consistent over the years that were studied. there are less hits than expected for priority references to us first filings, but more hits than expected for priority references to epc, which is the european home area for epo operations. this suggests that there is only a subset of us ffs that somehow qualify for filings as sfs later on at epo, which is reasonable for a large country with some of its own specific internal markets that are not relevant abroad. for europe, the contrary result means that priorities are better sampled than expected and rarely lead to multiple epo sfs. the results show r conforming more or less to its expectation under random hits for others origin and japan origin priorities (in the case of japan at least for the years 2004 to 2006). the values of l p (average number of links to priorities overall) differ between blocs (epc 35 per cent, japan 8 per cent, us 20 per cent, others 2 per cent, for priorities in 2002). it is interesting that in japan and others this was far less than in the other two blocs. perhaps the fact that the probability of a hit was lower has led to a better fit of the poisson biomath 1 (2012), 1209031, http://dx.doi.org/10.11145/j.biomath.2012.09.031 page 5 of 7 http://dx.doi.org/10.11145/j.biomath.2012.09.031 p. hingley, exploring family relations between international patent applications fig. 8. a) poisson approximation; b) censored binomial approximation; with p = 1 052 420 and l = 135 439. approximation or to the random hits model in general. v. possible applications in biology in ii it was suggested that temporal f0/f1 generations with annual replacement represents only a special case in population genetics. the schema lacks the attractive equilibrium properties that are interesting when making theoretical predictions about population dynamics and evolution over many generations. but there may be possibilities for applications of such models to special cases in biology. consider for example the transformation of an undifferentiated bank of stem cells into tissues and/or organs, such as the development of clones of immune cell against specific antigens [10]. how many of the stem cells will differentiate and into what tissue, if there are indeed different possible stem cell fates in development? in iii and iv an unexpected tightness of the distribution p r(r) was found around its mean. this suggests that the number of stem cells committing to become certain organs under a random differentiation model may be almost constant, even if random. aberrations in the process could perhaps occur in cancer. it may be useful to study competition between tissues as sinks for stem cells. in the patent world this is analogous to studying the fates of first filings in terms of priority references from subsequent filings in several other offices. for example such counts appear in [7] in terms of trilateral (epo, japan, us) and four office (epc, japan, korea, us) family subsets. another extension to the present model that can be beneficial to consider in both biological and patent regimes is the case where there are several conversions of the original entity via a sequence of transforming hits taking place in a temporal series. in the patent world there is the sequence of transformations of the priority forming first filing into a subsequent filing, followed by the possible grant of the patent and its eventual expiry, not to mention the collection of a cumulative set of fees at the patent office in lieu of these various steps. in biology there are sequences of cellular development that lead down limited paths of development under certain restrictions, such as colon crypt cell growth. this is a special case to which population genetics theory can be adapted, and brings us back towards schemes such as in fig. 1 [11]. vi. conclusions the development started with a description of the family relations of groups of patents in terms of population genetic parameters, and then continued by developing specific tests of random assignment of subsequent filings to priorities via distributions of hits. it turned out that the distributions are so tight that the outcomes almost appear to be fixed, even though the underlying process is random. there was a brief consideration of how biological models for certain special phenomena may be able to make use of the method. the exact formula in iiia gives the best representation of the effects of random hits, but it is not easily calculable when constructing a null distribution for large data sets. feller’s argument for a limiting poisson distribution does not apply well for the case that l < p, because its variance is too large. however the quantity p − λ is a good approximation for the mean. a censored binomial distribution is also well centered and has a lower variance than the poisson, but is still too disperse. a closer approximation to the exact distribution should be established, that can work with larger numbers and stays as close to the original formula as possible. for patent families that involve subsequent filings at epo, the observed number of priorities is less than that predicted by the random hits model. this is mainly due to less hits than expected from applicants in us, although there are more hits than expected from europe. to model these situations more explicilty, it may be beneficial to develop a weighted version of the exact formula, where combinations with fewer hits have higher weights (us case) or lower weights (epc case) than combinations having a greater number of hits. apart from the possible extensions that were mentioned in v, it will also be interesting to develop more intricate models of the international patenting system. this could include an extension of the model presented here to test independence of hits to a common set of first filings when subsequent filings are made to several biomath 1 (2012), 1209031, http://dx.doi.org/10.11145/j.biomath.2012.09.031 page 6 of 7 http://dx.doi.org/10.11145/j.biomath.2012.09.031 p. hingley, exploring family relations between international patent applications fig. 9. the results for apparent randomness of r depend on blocs of origin of the patent families. reading clockwise from top left: a) epc, b) japan, c) others, d) us. l: dotted line; r: boxes; p − λ: triangles. other patent offices. models could also be developed to consider the effects of having several priority references to different first filings from some of the subsequent filings. acknowledgment the author would like to thank colleagues for supplying and maintaining the pri data file, and marc nicolas for useful discussions on the methods. references [1] p. hingley, and m. nicolas, (eds), forecasting innovations, methods for predicting numbers of patent filings, springe, 2006. [2] p. hingley, patent families defined as priority forming filings and their descendents, http://forums.epo.org/students-2-students/topic720.html (2010). [3] z. griliches, patent statistics as economic indicators: a survey, journal of economic literature 28, 1661–1707 (1990). [4] j.c. vivar, and d. banks, models for networks: a crossdisciplinary science, wires comp. stat. 4, 13–27 (2012). http://dx.doi.org/10.1002/wics.184 [5] j. ewens, mathematical population genetics, vol. 1, 2nd edition, springer, 2004. http://dx.doi.org/10.1007/978-0-387-21822-9 [6] p. hingley, and s. bas, numbers and sizes of applicants at the european patent office, world patent information 31, 285–298 (2009). [7] european patent office, japan patent office, korean intellectual property office, united states patent and trademark office. four office statistics report, 2010 edition, http://www.trilateral.net/statistics/tsr/fosr2010.html (2011). [8] c. martinez, insight into different types of patent families, http://www.oecd.org/dataoecd/21/32/44604939.pdf (2010). [9] w. feller, an introduction to probability theory and its applications, vol. 1, wiley, 1968. [10] d. wodarz, killer cell dynamics, springer, 2007. http://dx.doi.org/10.1007/978-0-387-68733-9 [11] m.a. nowak, evolutionary dynamics, belknap harvard, 2006. biomath 1 (2012), 1209031, http://dx.doi.org/10.11145/j.biomath.2012.09.031 page 7 of 7 http://forums.epo.org/students-2-students/topic720.html http://dx.doi.org/10.1002/wics.184 http://dx.doi.org/10.1007/978-0-387-21822-9 http://www.trilateral.net/statistics/tsr/fosr2010.html http://www.oecd.org/dataoecd/21/32/44604939.pdf http://dx.doi.org/10.1007/978-0-387-68733-9 http://dx.doi.org/10.11145/j.biomath.2012.09.031 introduction the structure of patenting and patent families random hits model for priorities exact distribution poisson approximation censored binomial approximation random hits model for patent families data possible applications in biology conclusions references original article biomath 3 (2014), 1411171, 1–11 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum how compressibility influences the mechanical bidomain model kharananda sharma∗, bradley j. roth† department of physics, oakland university rochester, mi usa ∗ksharma@oakland.edu †roth@oakland.edu received: 28 july 2014, accepted: 17 november 2014, published: 31 december 2014 abstract—compressibility influences the mechanical bidomain model describing the elastic properties of tissue. displacements of the intracellular and extracellular spaces are analyzed individually, and differences in these displacements produce membrane forces. two length constants are associated with the membrane spring constant, one contains the shear moduli and the other contains the bulk moduli. the analytical solutions in these examples indicate that the monodomain part does not contribute to the membrane force. accounting for compressibility has its largest impact on the intracellular and extracellular pressures. the bidomain contribution to the pressure obeys the helmholtz equation rather than laplace’s equation. this model predicts membrane forces that might cause tissue remodeling or mechanotransduction. keywords-biomechanics; mechanical bidomain model; mechanotransduction; pressure; remodeling. i. introduction the mechanical bidomain model is a new mathematical description of the biomechanics of tissue, which distinguishes between displacements in the intracellular and extracellular spaces and focuses on forces across the cell membrane [1-5]. such membrane forces may affect transmembrane proteins, such as integrins, that are responsible for tissue remodeling and mechanotransduction [6-8]. membrane forces may also play a role in tissue engineering because mechanical stresses influence the growth of replacement tissue [9], in remodeling of blood vessels [10], and in development because mechanical stresses guide the growth of fetal tissue [11]. the unique feature of the mechanical bidomain model is that it is macroscopic (representing the tissue rather than individual cells) but because it tracks the intracellular and extracellular spaces individually it can predict membrane forces. the mechanical bidomain model is analogous to the electrical biodmain model [12], which is currently the most widely used model for simulating defibrillation. for a review of the mechanical bidomain model, see [4]. in a previous version of the mechanical bidomain model [4], both the intracellular and extracellular spaces were incompressible, so tissue displacements did not change the tissue volume (dilatation). the pressure distribution in a tissue is defined as the bulk modulus times the dilatation [13, 14]. the incompressible limit corresponds to citation: kharananda sharma, bradley j. roth, how compressibility influences the mechanical bidomain model, biomath 3 (2014), 1411171, http://dx.doi.org/10.11145/j.biomath.2014.11.171 page 1 of 11 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2014.11.171 k. sharma et al., how compressibility influences the mechanical ... the finite product of these quantities as the bulk modulus goes to infinity and the dilatation goes to zero. tissue is largely water, so the incompressible assumption is often valid. tissue bulk moduli are much larger than shear moduli. the bulk modulus is on the order of 2 × 109 pa, whereas the shear modulus is about 2 × 104 pa, a difference of a factor of 100,000 [15], so one can typically assume incompressibility. the mechanical bidomain model is complicated because there are two bulk moduli: one for the intracellular space, χ, and one for the extracellular space, δ. although one can easily imagine a limiting process to arrive at intracellular and extracellular pressures, p and q, similar to that outlined above, the ratio χ δ may impact the tissue behavior, even if χ and δ are both much greater than the intracellular and extracellular shear moduli, ν and µ. furthermore, the pressures contribute to the boundary conditions at the tissue surface, and the appropriate boundary conditions remain uncertain. the goal of this paper is to resolve questions surrounding the mechanical bidomain model by rederiving the model without the assumption of incompressibility. we solve several biomechanics problems, analyze their solutions, and then impose incompressibility to determine the correct behavior in that limit. this procedure introduces new qualitative behavior into the model. for instance, previous studies revealed a bidomain length constant that depended on the membrane spring constant k coupling the intracellular and extracellular spaces [1, 4]. in this paper, there are two length constants, both involving k; one containing the shear moduli and one the bulk moduli. our results change the way we calculate and interpret the intracellular and extracellular pressures, impact the predicted displacements, and affect the membrane force distribution. ii. methods a. mechanical bidomain model consider an isotropic tissue and ignore any active tension. to keep things simple, we consider a twodimensional cartesian coordinate system (x, fig. 1. a schematic diagram of the mechanical bidomain model for a two-dimensional sheet of tissue. the elastic properties of the intracellular space are depicted by the lower grid of springs (green), and the properties of the extracellular space by the upper grid (blue). the two spaces are coupled by the membrane, shown as an array of springs (red). because this is a two-dimensional model, stretching of the membrane springs is not caused by displacements in the z direction perpendicular to the sheet. rather, if the intracellular and extracellular spaces are displaced by different amounts in the x-y plane, the membrane springs will stretch causing forces on the membrane. y). the strains are related to the intracellular and extracellular displacements, u and w, by �ixx = ∂ux ∂x �exx = ∂wx ∂x (1) �iyy = ∂uy ∂y �eyy = ∂wy ∂y (2) �ixy = 1 2 ( ∂ux ∂y + ∂uy ∂x ) �exy = 1 2 ( ∂wx ∂y + ∂wy ∂x ) . (3) the relationships between intracellular and extracellular stresses and strains are τixx = χ(�ixx + �iyy) + 2ν�ixx (4) τexx = δ(�exx + �eyy) + 2µ�exx (5) τiyy = χ(�ixx + �iyy) + 2ν�iyy (6) τeyy = δ(�exx + �eyy) + 2µ�eyy (7) biomath 3 (2014), 1411171, http://dx.doi.org/10.11145/j.biomath.2014.11.171 page 2 of 11 http://dx.doi.org/10.11145/j.biomath.2014.11.171 k. sharma et al., how compressibility influences the mechanical ... τixy = 2ν�ixy (8) τexy = 2µ�exy. (9) the equations of mechanical equilibrium are ∂τixx ∂x + ∂τixy ∂y = k (ux −wx) (10) ∂τexx ∂x + ∂τexy ∂y = −k (ux −wx) (11) ∂τixy ∂x + ∂τiyy ∂y = k (uy −wy) (12) ∂τexy ∂x + ∂τeyy ∂y = −k (uy −wy) . (13) plugging (1-9) into (10-13) and rearranging, we obtain the equations governing the mechanical bidomain model χ ∂ ∂x ( ∂ux ∂x + ∂uy ∂y ) + 2ν ( ∂2ux ∂x2 + 1 2 ∂2ux ∂y2 + 1 2 ∂2uy ∂x∂y ) = k ( ux −wx ) (14) δ ∂ ∂x ( ∂wx ∂x + ∂wy ∂y ) + 2µ ( ∂2wx ∂x2 + 1 2 ∂2wx ∂y2 + 1 2 ∂2wy ∂x∂y ) = −k ( ux −wx ) (15) χ ∂ ∂y ( ∂ux ∂x + ∂uy ∂y ) + 2ν ( ∂2uy ∂y2 + 1 2 ∂2ux ∂x∂y + 1 2 ∂2uy ∂x2 ) = k ( uy −wy ) (16) δ ∂ ∂y ( ∂wx ∂x + ∂wy ∂y ) + 2µ ( ∂2wy ∂y2 + 1 2 ∂2wx ∂x∂y + 1 2 ∂2wy ∂x2 ) = −k ( uy −wy ) . (17) fig. 1 shows a schematic drawing of a twodimensional sheet of tissue representing the mechanical bidomain model. the intracellular and extracellular spaces are each represented by a grid of springs (green and blue), and their behavior is described by the left-hand sides of (14-17). these grids are connected across the cell membrane by an array of springs (red) coupling the two spaces. the force produced by these springs is represented by the right-hand sides of (14-17). the membrane force depends on the difference between the intracellular and extracellular displacements and the spring constant k. b. shear displacement we first analyze shear displacements using methods similar to those presented previously [4]. take the y-derivative of (14) and subtract the x-derivative of (16), and similarly take yderivative of (15) and subtract the x-derivative of (17). the results are simpler when expressed in terms of the z-components of the curl of the displacements, ωi = (∇×u)z = ∂uy ∂x − ∂ux ∂y and ωe = (∇×w)z = ∂wy ∂x − ∂wx ∂y , ν∇2ωi = k (ωi −ωe) (18) µ∇2ωe = −k (ωi −ωe) , (19) where ν and µ are the shear moduli (terms containing the bulk moduli χ and δ cancel out). if we add (18) and (19), we get ∇2 (νωi + µωe) = 0. (20) if we divide (18) by ν and (19) by µ and subtract, we obtain ∇2 (ωi −ωe) = 1 σ2 (ωi −ωe) , (21) where σ = √ νµ k(ν+µ) . biomath 3 (2014), 1411171, http://dx.doi.org/10.11145/j.biomath.2014.11.171 page 3 of 11 http://dx.doi.org/10.11145/j.biomath.2014.11.171 k. sharma et al., how compressibility influences the mechanical ... define α = ωi + µ ν ωe and β = ωi − ωe, so ωi = ν ν+µ (α + µ ν β) and ωe = νν+µ(α−β). these functions obey ∇2α = 0 and ∇2β = 1 σ2 β. the function α is a weighted sum of the intracellular and extracellular spaces, obeys laplace’s equation, and describes the “monodomain” behavior of the system; β is the difference between the intracellular and extracellular spaces, obeys the helmholtz equation so it will decay with distance by the length constant σ, and specifies the “bidomain” behavior. because β represents the difference between u and w, it is responsible for the membrane force. c. volume dilatation we next consider volume changes in the tissue, which is the new feature presented in this paper. take the x-derivative of (14) and add it to the yderivative of (16), and similarly take x-derivative of (15) and add it to the y-derivative of (17), and get (χ + 2ν)∇2ei = k(ei −ee) (22) (δ + 2µ)∇2ee = −k(ei −ee), (23) where ei = ∇·u and ee = ∇·w are the intracellular and extracellular dilatations. now assume that the bulk moduli are much greater than the shear moduli (χ,δ >> ν,µ) so χ∇2ei = k(ei −ee) (24) δ∇2ee = −k(ei −ee). (25) if we add the two equations, we get ∇2(χei + δee) = 0. (26) if we divide (24) by χ and (25) by δ and subtract, we obtain ∇2(ei −ee) = 1 ξ2 (ei −ee) , (27) where ξ = √ χδ k(χ + δ) . (28) define the intracellular and extracellular pressures as p = −χei and q = −δee. to examine the incompressible limit, let χ and δ go to infinity and ei and ee go to zero in such a way that p and q remain finite. furthermore, define two auxiliary pressures: p = p+q is the monodomain pressure, and q = p − χ δ q is a weighted difference of the pressures. the pressures p and q are expressed in terms of p and q as p = χ χ + δ ( p + δ χ q ) , (29) q = δ χ + δ (p −q) . (30) the pressures p and q obey the equations ∇2p = 0, ∇2q = 1 ξ2 q. p obeys laplaces equation and represents the monodomain behavior. q obeys the helmholtz equation, decays with distance by a length constant ξ, and represents the bidomain behavior. d. summary of the methods this analysis shows that the displacements caused by shearing and the dilatations caused by volume changes obey analogous equations. both can be separated into a monodomain part and a bidomain part. both introduce new length constants, σ and ξ, that characterize the behavior. the analysis of shear was presented previously [4] and results in a boundary layer near the tissue edge [5]. the analysis of volume dilatation is new: it represents an advance beyond the previous model. the equations above suggest that the pressure and displacement are independent. however, they are in fact coupled by the boundary conditions at the tissue surface. for instance, if the tissue is perfused by an adjacent bath, the boundary conditions are: the normal component of the extracellular stress is continuous with the stress in the bath, and the normal component of the intracellular stress is zero. if the tissue-bath surface were specified by the plane y = constant, then the boundary conditions would be τexx = −pbath and τexy = τixx = τixy = 0. here, pbath is the biomath 3 (2014), 1411171, http://dx.doi.org/10.11145/j.biomath.2014.11.171 page 4 of 11 http://dx.doi.org/10.11145/j.biomath.2014.11.171 k. sharma et al., how compressibility influences the mechanical ... fig. 2. a schematic diagram of example 1: a cylinder of tissue of radius b immersed in a bath at pressure p0. hydrostatic pressure in the bath, which we take as a static fluid. our next goal is to examine three examples, from the very simple to the increasingly complex, that illustrate the behavior of the mechanical bidomain model, with our focus primarily on the dilatation and pressure. iii. results a. example 1: cylinder in a bath consider a cylinder of isotropic tissue (of radius b) immersed in a fluid bath at pressure p0 (fig. 2). this might represent, for example, a cardiac papillary muscle lying in a superfusing bath. we examine this case first because shear forces play no role in the behavior; we need to determine only the pressure distribution in the tissue. in this case, the displacement is in the radial direction r and does not depend on the angle θ, so in cylindrical coordinates the strains are related to the displacement by �irr = ∂ur ∂r , �iθθ = ur r , �irθ = 0, �err = ∂wr ∂r , �eθθ = wr r , and �erθ = 0 [16]. the stresses and strains are related by τirr = χ(�irr + �iθθ)+2ν�irr, τiθθ = χ(�irr+�iθθ)+2ν�iθθ, τirθ = 0, τerr = δ(�err + �eθθ) + 2µ�err, τeθθ = δ(�err + �eθθ) + 2µ�eθθ, and τerθ = 0. the equations of mechanical equilibrium written in cylindrical coordinates are [16] ∂τirr ∂r + τirr − τiθθ r = k (ur −wr) (31) ∂τerr ∂r + τerr − τeθθ r = −k (ur −wr) . (32) at the surface of the tissue (r = b), the boundary conditions are τirr = 0 and τerr = −p0. the displacements that obey the equations of mechanical equilibrium and boundary conditions are ur = − p0 χ + δ ( r 2 − ξ i1(r/ξ) i0(b/ξ) ) (33) wr = − p0 χ + δ ( r 2 + ξ χ δ i1(r/ξ) i0(b/ξ) ) , (34) where i0 and i1 are modified bessel functions of the first kind. membrane forces arise due to the difference in displacements, u w. in this case, ur −wr = p0 ξδ i1(r/ξ) i0(b/ξ) . the membrane force is largest near the boundary r = b. however, the displacements go to zero as χ and δ go to infinity, so the membrane force vanishes for an incompressible tissue. the pressures p = −χei and q = −δee remain finite when χ and δ go to infinity p = p0 χ χ + δ ( 1 − i0(r/ξ) i0(b/ξ) ) (35) q = p0 δ χ + δ ( 1 + χ δ i0(r/ξ) i0(b/ξ) ) . (36) fig. 3 shows p and q as functions of r. these pressures can be recast in terms of p and q, p = p0 and q = −p0 χδ i0(r/ξ) i0(b/ξ) . the monodomain pressure p is just the bath pressure, while the bidomain pressure q decays with length constant ξ as one moves inward from the tissue surface. if ξ << b, then several length constants below the surface q is approximately zero and the intracellular and extracellular pressures are constants; p = χ χ+δ p0 and q = δ χ+δ p0. although the pressures remain finite as the bulk moduli go to infinity, they still depend on the ratio of the bulk moduli, χ δ . the biomath 3 (2014), 1411171, http://dx.doi.org/10.11145/j.biomath.2014.11.171 page 5 of 11 http://dx.doi.org/10.11145/j.biomath.2014.11.171 k. sharma et al., how compressibility influences the mechanical ... fig. 3. the pressures p and q as functions of r, for χ = 2δ and ξ = b 10 . the blue curve represents the intracellular pressure p and the red curve represents the extracellular pressure q. tissue behavior is independent of the shear moduli ν and µ. the length constant ξ depends on the bulk moduli, and as χ and δ go to infinity ξ grows. however, the value of the membrane spring constant k is not known but is expected to be large [4]. therefore, the size of ξ = √ χδ k(χ+δ) relative to the cylinder radius b is not obvious. if ξ << b, the pressure changes over a thin boundary layer near the tissue surface, like shown in fig. 3. if ξ >> b, p = 0 and q = p0, and are constant. the limit ξ >> b corresponds to the prediction of the mechanical bidomain model presented previously [4]. ultimately, the size of ξ is an experimental question that cannot be resolved until we have the necessary data. one virtue of this first example is that it highlights a key new feature of our model, the new length constant ξ, and indicates what quantities need to be measured to assess the model. b. example 2: blood vessel the next example is of a cylinder of fluid, of radius a, surrounded by tissue, which is a model for a blood vessel (fig. 4). this example is slightly more complicated than the first example, because the shear moduli now play a role in the solution. however, the approach is similar to example 1. the displacements that obey the equations of equifig. 4. a schematic diagram of example 2: a cylindrical vessel of radius a containing fluid at pressure p0, surrounded by tissue. librium and the boundary conditions are ur = p0 ν + µ ( a2 2r − ξ ν χ k1(r/ξ) k0(a/ξ) ) (37) wr = p0 ν + µ ( a2 2r + ξ ν δ k1(r/ξ) k0(a/ξ) ) , (38) where k0 and k1 are modified bessel functions of the second kind. these expressions each have two terms: a first term, p0 ν+µ a2 2r , that governs the monodomain behavior and does not go to zero as the bulk moduli go to infinity, and a second bidomain term containing bessel functions. because the monodomain terms are the same in the intracellular and extracellular spaces, they contribute nothing to the membrane force. the monodomain part (fig. 5a) implies that a high pressure inside the vessel causes it to expand. the difference between the intracellular and extracellular displacements is ur−wr = −p0ξ νν+µ k1(r/ξ) k0(a/ξ) ( 1 χ + 1 δ ) , which gives rise to membrane forces near the tissue boundary r = a that fall off with a length constant ξ (fig. 5b). the difference ur −wr vanishes when χ and δ go to infinity. therefore in the incompressible limit, there is no membrane force. the dilatations ei = p0 χ ν ν+µ k0(r/ξ) k0(a/ξ) and ee = −p0 δ ν ν+µ k0(r/ξ) k0(a/ξ) become zero for large values of χ and δ. the pressures, p = −p0 νν+µ k0(r/ξ) k0(a/ξ) and biomath 3 (2014), 1411171, http://dx.doi.org/10.11145/j.biomath.2014.11.171 page 6 of 11 http://dx.doi.org/10.11145/j.biomath.2014.11.171 k. sharma et al., how compressibility influences the mechanical ... fig. 5. a) the monodomain part of the displacement, b) the bidomain part of the displacement ur − wr, and c) the pressures p and q, all as functions of r, for ν = 2µ , χ = 2δ, and ξ = a 10 . the displacements and pressures have been normalized to their values at r = a. q = p0 ν ν+µ k0(r/ξ) k0(a/ξ) are independent of χ and δ except though their dependence on ξ (fig. 5c). the monodomain pressure is zero, p = 0, so there is only a bidomain contribution. in the limit when ξ >> a (as presented by [4]), the pressures become constant: p = −p0 νν+µ and q = p0 ν ν+µ . this example emphasizes two points. first, like example 1, it illustrates how the pressures fall off from the boundary with length constant ξ, the new parameter introduced in this paper. second, it shows how the displacement is divided into two parts, one of which gives rise to membrane forces (although in this case the membrane force goes to zero as χ and δ go to infinity). c. example 3: sheet of active tissue the last example is of a two-dimensional sheet of cardiac tissue undergoing an active contraction (fig. 6). this example is more complicated than the first two, but we include here it for several reasons. first, it shows how the two length constants, σ and ξ, can both contribute to the solution. second, it contains a non-zero membrane force even when the bulk moduli go to infinity. third, it was analyzed in detail using the incompressible model [5], so redoing the calculation with compressibility highlights those changes compressibility introduces. finally, the example shows how the active tension enters the equations, which is a crucial element when modeling cardiac tissue. we represent the active contraction as a uniform tension t added to the intracellular stress tensor [13, 14]. the tension acts along the myocardial fibers, which we assume are straight, uniform and oriented in the x direction. other than this active tension, we assume that the tissue is isotropic. in cartesian coordinates, the tension would be represented by a constant t added to τixx. in cylindrical coordinates, the tension enters the stress tensor in a more complicated way, shown below. in cylindrical coordinates, the strains are expressed in terms of the displacements by [16] �irr = ∂ur ∂r �err = ∂wr ∂r (39) �iθθ = ur r + 1 r ∂uθ ∂θ �eθθ = wr r + 1 r ∂wθ ∂θ (40) �irθ = 1 2 ( 1 r ∂ur ∂θ + ∂uθ ∂r − uθ r ) (41) �erθ = 1 2 ( 1 r ∂wr ∂θ + ∂wθ ∂r − wθ r ) . (42) the stress and strain are related by [5] τirr = χ(�irr+�iθθ)+2ν�irr+ t 2 (1+cos 2θ) (43) τerr = δ(�err + �eθθ) + 2µ�err (44) biomath 3 (2014), 1411171, http://dx.doi.org/10.11145/j.biomath.2014.11.171 page 7 of 11 http://dx.doi.org/10.11145/j.biomath.2014.11.171 k. sharma et al., how compressibility influences the mechanical ... fig. 6. a schematic diagram of example 3: a sheet of isotropic cardiac tissue of radius r with straight and uniform myocardial fibers oriented in the x-direction. τiθθ = χ(�irr+�iθθ)+2ν�iθθ+ t 2 (1−cos 2θ) (45) τeθθ = δ(�err + �eθθ) + 2µ�eθθ (46) τirθ = 2ν�irθ − t 2 sin 2θ (47) τerθ = 2µ�erθ. (48) the equations of equilibrium are [16] ∂τirr ∂r + 1 r ∂τirθ ∂θ + τirr − τiθθ r = k (ur −wr) (49) ∂τerr ∂r + 1 r ∂τerθ ∂θ + τerr − τeθθ r = −k (ur −wr) (50) ∂τirθ ∂r + 1 r ∂τiθθ ∂θ + 2 τirθ r = k (uθ −wθ) (51) ∂τerθ ∂r + 1 r ∂τeθθ ∂θ + 2 τerθ r = −k (uθ −wθ) . (52) at the surface of the tissue r = r, the boundary is stress free: τirr = τerr = τirθ = τerθ = 0. the displacements obeying these equations of equilibrium and boundary conditions are ur = t 2 { 1 ν + µ [ − r 2 − µ ν 1 2h i2(r/σ) r − ξ2µ χ ( 1 − g h ) di2(r/ξ) dr i2(r/ξ) ] cos 2θ+ 1 χ + δ [ − r 2 − ( δ χ ) ξ i1(r/ξ) i0(r/ξ) ]} (53) wr = t 2 { 1 ν + µ [ − r 2 + 1 2h i2(r/σ) r + ξ2µ δ ( 1 − g h ) di2(r/ξ) dr i2(r/ξ) ] cos 2θ+ 1 χ + δ [ − r 2 + ξ i1(r/ξ) i0(r/ξ) ]} (54) uθ = t 2 1 ν + µ [ r 2 + µ ν 1 4h di2(r/σ) dr + 2 ξ2µ χ ( 1 − g h ) i2(r/ξ) ri2(r/ξ) ] sin 2θ (55) wθ = t 2 1 ν + µ [ r 2 − 1 4h di2(r/σ) dr − 2 ξ2µ δ ( 1 − g h ) i2(r/ξ) ri2(r/ξ) ] sin 2θ (56) where h = 1 4 d2i2(r/σ) dr2 − 1 4r di2(r/σ) dr + i2(r/σ) r2 (57) g = 1 r di2(r/σ) dr − i2(r/σ) r2 . (58) both u and w contain a leading monodomain term that is in general larger than the subsequent bidomain terms (fig. 7a). the tissue contracts along the fiber direction, and expands perpendicular to the fiber direction. (note: fig. 7a appears different than fig. 1 in [5] because fig. 1 in [5] is incorrect. it should look exactly like our fig. 7a). biomath 3 (2014), 1411171, http://dx.doi.org/10.11145/j.biomath.2014.11.171 page 8 of 11 http://dx.doi.org/10.11145/j.biomath.2014.11.171 k. sharma et al., how compressibility influences the mechanical ... fig. 7. a) the monodomain part of the displacement, and b) the bidomain part of the displacement, as functions of position, for χ = 2δ, ν = 2µ, ξ = r 10 and σ = r 100 . the solid circle shows the tissue boundary with zero displacement, and the dashed oval shows how the tissue deforms when an active tension is present. the fibers are horizontal. the arrows in panels a) and b) are scaled differently; without this scaling the arrows in (b) would be much smaller. as χ and δ go to infinity, the difference in the displacements is ur −wr = − t 8ν 2 h i2(r/σ) r cos 2θ (59) uθ −wθ = t 8ν 1 h di2(r/σ) dr sin 2θ. (60) fig. 8. a) the intracellular pressure p, and b) the extracellular pressure q, as functions of position, for χ = 2δ, ν = 2µ, and ξ = r 10 . in general, σ << r. the membrane force is largest within a few length constants σ of the edge r = r, and the θ component is larger than the r component (fig. 7b). (note: our fig. 7b is not identical to fig. 2 in [5] because we use a different value of σ r ; [5] used σ = r 10 , whereas we use σ = r 100 .) taking the limit as χ and δ go to infinity and ei and ee go to zero, the pressures become p = t 2 { χ χ + δ ( 1 + δ χ i0(r/ξ) i0(r/ξ) ) + µ ν + µ ( 1 − g h ) i2(r/ξ) i2(r/ξ) cos 2θ } (61) q = t 2 { δ χ + δ ( 1 − i0(r/ξ) i0(r/ξ) ) − µ ν + µ ( 1 − g h ) i2(r/ξ) i2(r/ξ) cos 2θ } . (62) the pressures can be expressed as p = t 2 and q = t 2 { i0(r/ξ) i0(r/ξ) + χ + δ δ µ ν + µ ( 1 − g h ) i2(r/ξ) i2(r/ξ) cos 2θ } . (63) fig. 8 shows p and q as functions of position. these pressure distributions are very different than those shown in figs. 3 and 4 of [5]. away from the tissue edge the pressures in our fig. 8 are both constant, with deviations from this constant value only within a few length constants of the boundary. iv. discussion the analysis in the methods and results illustrates how compressibility affects the mechanical bidomain model. it is useful to compare the previous derivation of the model with the version developed in this paper. if we assume an isotopic tissue t = 0 and analyze the model developed previously [4] using the methods derived here, the mechanical bidomain equations would be ∇2α = 0, ∇2β = 1 σ2 β, ∇2p = 0, and ∇2q = 0. in our revised model, we obtain the same equations, except for the equation governing q: ∇2α = 0, ∇2β = 1 σ2 β, ∇2p = 0, and ∇2q = 1 ξ2 q. the monodomain part of the model is the same in both cases (∇2α = 0 and ∇2p = 0) but the bidomain part is different. in the original model, the bidomain behavior was characterized by one length constant σ = √ νµ k(ν+µ) . in the revised model, the bidomain behavior is characterized by biomath 3 (2014), 1411171, http://dx.doi.org/10.11145/j.biomath.2014.11.171 page 9 of 11 http://dx.doi.org/10.11145/j.biomath.2014.11.171 k. sharma et al., how compressibility influences the mechanical ... two length constants, σ and ξ = √ χδ k(χ+δ) . as discussed previously [4], the value of the membrane spring constant k is not known but is expected to be large, implying that σ is small. the size of ξ is not as clear. k is large, but so are χ and δ. in the truly incompressible limit, χ and δ go to infinity, so ξ becomes very large and the equation for q becomes ∇2q = 0, as it was in the previous model. note that in many of the expressions for the tissue displacement (e.g., (33)), a factor of ξ χ+δ is present, which goes to zero as χ and δ become very large, even if ξ is large. the length constant governing shearing, σ, is in general much smaller than the length constant for dilatation, ξ, because the bulk moduli are much greater than the shear moduli. the ratio ξ σ is about 300, independent of the value of k. we often speak of boundary layers of both pressure and displacement arising from the mechanical bidomain model. this view of a thin boundary layer is only useful in the limit when both σ and ξ are small compared to other length scales, such as the radius of the papillary muscle or blood vessel being modeled. the solutions for the displacement in all three examples have a similar form: a monodomain part that depends on some power of the radius and which is the same in the intracellular and extracellular spaces, and a bidomain part that falls off with length constant σ or ξ. the monodomain part is typically larger than the bidomain part, but because it is the same in both spaces it contributes nothing to the membrane force, which is proportional to the difference between the intracellular and extracellular spaces. the bidomain part is different in the two spaces, so it alone contributes to the membrane force. in these three examples, the bidomain part contains a modified bessel function, which solves the helmholtz equation in two dimensions. although these special functions are somewhat unfamiliar, at large values of their argument they behave qualitatively like exponentials (i as an increasing exponential, and k as a decaying exponential). so, if the bidomain length constants are small the bidomain part of the displacement falls off approximately exponentially with distance from the tissue boundary. example 3 is particularly useful because that problem was solved completely using the previous model [5]. the monodomain parts of the calculation are the same in both cases. the solution for the membrane force is similar but not identical. a term containing a modified bessel function is present in both, and implies that both calculations result in a boundary layer at the edge of the tissue r = r of thickness σ. however, in the previous calculation [5] the membrane force also contained a term proportional to r, implying that the membrane force had a small contribution far from the boundary. this term is not present in our calculation. thus, the effect of properly accounting for compressibility can make a difference in the model predictions, even in the incompressible limit. the largest difference in example 3 between the previous calculation and ours is the pressure. in the previous calculation [5], both the intracellular and extracellular pressures contained a term that varied as r2 cos 2θ. the intracellular pressure also contained a constant term t 2 , but the extracellular pressure did not. therefore, there were large differences between p and q throughout the tissue. in our calculation, p and q are both dominated by constant terms, and the only spatial dependence arises from terms containing a bessel function. if ξ << r these bessel function terms decay away from the surface, so p = tχ 2(χ+δ) and q = tδ 2(χ+δ) . however, if ξ >> r the bessel function i2 is approximately proportional to r2, so the pressure varies as r2 cos 2θ like in the previous calculation. the physical meaning of the intracellular and extracellular pressures has always been a confusing issue with the mechanical bidomain model. reference [4] discussed the connection between macroscopic and microscopic properties of the model. according to that discussion, p and q are macroscopic pressures, and are related to the microscopic pressures pmicro and qmicro by p = θipmicro and q = θeqmicro, where θi and θe are the intracellular and extracellular volume fractions. the shear moduli ν and µ are also macroscopic parameters. if we similarly take χ and δ to be biomath 3 (2014), 1411171, http://dx.doi.org/10.11145/j.biomath.2014.11.171 page 10 of 11 http://dx.doi.org/10.11145/j.biomath.2014.11.171 k. sharma et al., how compressibility influences the mechanical ... macroscopic parameters, we find by analogy that χ = θiχmicro and δ = θeδmicro. if χmicro and δmicro are the same (both the bulk modulus of water) then χ δ = ν µ and the microscopic pressure difference pmicro−qmicro is proportional to q and is zero except within a few length constants of the tissue edge. in other words, the bidomain pressure q represents the difference between the microscopic intracellular and extracellular pressures, and any fluid flow between the two spaces would be driven by q. if one wants to use the previous model to do calculations, the primary change is the boundary conditions on the pressures. instead of requiring that p = 0 at the boundary, one specifies that q = 0, so p = χ χ+δ p and q = δ χ+δ p . the resulting calculations using the previous model will be correct, except in a boundary layer of thickness ξ. v. conclusion in conclusion, consideration of tissue compressibility clarifies the behavior of the mechanical bidomain model. the monodomain contribution to the displacement and pressure are unchanged. the bidomain contribution to the displacement is similar to previous calculations, and it determines the membrane force. this paper shows that the bidomain contribution to the pressure is very different than was thought previously, and it determines the difference in microscopic pressure between the intracellular and extracellular spaces. the bidomain model predicts two new length constants: σ determines the width of the boundary layer for the membrane forces, and ξ determines the width of the boundary layer for the pressure. references [1] s. puwal and b. j. roth, “mechanical bidomain model of cardiac tissue,” phys. rev. e., vol. 82, article 041904, 2010. http://dx.doi.org/10.1103/physreve.82.041904. [2] v. m. punal and b. j. roth, “a perturbation solution of the mechanical bidomain model,” biomechanics and modeling in mechanobiology, vol. 11, pp. 995-1000, 2012. http://dx.doi.org/10.1007/s10237-011-0368-1. [3] s. puwal, “two-domain mechanics of a spherical, single chamber heart with applications to specific pathologies,” springerplus, vol. 2, article 187, 2013, http://dx.doi.org/10.1186/2193-1801-2-187. [4] b. j. roth, “the mechanical bidomain model: a review,” isrn tissue eng., vol. 2013, article 863689, 2013. http://dx.doi.org/10.1155/2013/863689. [5] b. j. roth, “boundary layers and the distribution of membrane forces predicted by the mechanical bidomain model,” mech. res. commun, vol. 50, pp. 12-16, 2013. http://dx.doi.org/10.1016/j.mechrescom.2013.02.004. [6] m. chiquet, l. gelman, r. lutz, and s. maier, “from mechanotransduction to extracellular matrix gene expression in fibroblasts,” biochim. biophys. acta., vol. 1793, pp. 911-920, 2009. http://dx.doi.org/10.1016/j.bbamcr.2009.01.012. [7] j. y. kresh, and a. chopra, “intercellular and extracellular mechanotransduction in cardiac myocytes,” pflugers archiv. european j. phys., vol. 462, pp. 75-87, 2011. http://dx.doi.org/10.1007/s00424-011-0954-1. [8] m. l. mccain and k. k. parker, “mechanotransduction: the role of mechanical stress, myocytes shape, and cytoskeletal architecture of cardiac function,” pflugers archiv. european j. phys., vol. 462 pp. 89-104, 2011. http://dx.doi.org/10.1007/s00424-011-0951-4. [9] r. g. katare, m. ando, y. kakinuma, and t. sato, “engineered heart tissue: a novel tool to study the ischemic changes of the heart in vitro,” plos one, vol. 5, article e9275, 2010. http://dx.doi.org/10.1371/journal.pone.0009275. [10] d. lu and g. s. kassab “role of shear stress and stretch in vascular mechanobiology,” j. r. soc. interface, vol. 8, pp. 1379-1385, 2011. http://dx.doi.org/10.1098/rsif.2011.0177. [11] t. mammoto, and d. e. ingber “mechanical control of tissue and organ development,” development, vol. 137, pp. 1407-1420, 2010. http://dx.doi.org/10.1242/dev.024166. [12] c. s. henriquez, “simulating the electrical behavior of cardiac tissue using the bidomain model,” crit. rev. biomed. eng., vol. 21, pp. 1-77, 1993. [13] r. s. chadwick, “mechanics of the left ventricle,” biophys. j., vol. 39, pp. 279-288, 1982. http://dx.doi.org/10.1016/s0006-3495(82)84518-9. [14] j. ohayon and r. s. chadwick, “effects of collagen microstructure on the mechanics of the left ventricle,” biophys. j., vol. 54, pp. 1077-1088, 1988. http://dx.doi.org/10.1016/s0006-3495(88)83044-3. [15] p. n. t. wells, and h.-d. liang, “medical ultrasound: imaging of soft tissue strain and elasticity,” j. r. soc. interface, vol. 8, pp. 1521-1549, 2011. http://dx.doi.org/10.1098/rsif.2011.0054. [16] a. e. h. love, a treatise on the mathematical theory of elasticity, dover, new york, 1944. biomath 3 (2014), 1411171, http://dx.doi.org/10.11145/j.biomath.2014.11.171 page 11 of 11 http://dx.doi.org/10.1103/physreve.82.041904 http://dx.doi.org/10.1007/s10237-011-0368-1 http://dx.doi.org/10.1186/2193-1801-2-187 http://dx.doi.org/10.1155/2013/863689 http://dx.doi.org/10.1016/j.mechrescom.2013.02.004 http://dx.doi.org/10.1016/j.bbamcr.2009.01.012 http://dx.doi.org/10.1007/s00424-011-0954-1 http://dx.doi.org/10.1007/s00424-011-0951-4 http://dx.doi.org/10.1371/journal.pone.0009275 http://dx.doi.org/10.1098/rsif.2011.0177 http://dx.doi.org/10.1242/dev.024166 http://dx.doi.org/10.1016/s0006-3495(82)84518-9 http://dx.doi.org/10.1016/s0006-3495(88)83044-3 http://dx.doi.org/10.1098/rsif.2011.0054 http://dx.doi.org/10.11145/j.biomath.2014.11.171 introduction methods mechanical bidomain model shear displacement volume dilatation summary of the methods results example 1: cylinder in a bath example 2: blood vessel example 3: sheet of active tissue discussion conclusion references www.biomathforum.org/biomath/index.php/biomath original article on the cyclic dna codes over the finite rings z4 + wz4 and z4 + wz4 + vz4 + wvz4 abdullah dertli1, yasemin cengellenmis2 1ondokuz mayıs university, faculty of arts and sciences mathematics department, samsun, turkey abdullah.dertli@gmail.com 2trakya university, faculty of sciences mathematics department, edirne, turkey ycengellenmis@gmail.com received: 8 may 2017, accepted: 16 december 2017, published: 22 december 2017 abstract—the structures of the cyclic dna codes of odd length over the finite rings r = z4 + wz4, w2 = 2 and s = z4 + wz4 + vz4 + wvz4,w2 = 2,v2 = v,wv = vw are studied. the links between the elements of the rings r, s and 16 and 256 codons are established, respectively. the cyclic codes of odd length over the finite ring r satisfy reverse complement constraint and the cyclic codes of odd length over the finite ring s satisfy reverse constraint and reverse complement constraint are studied. the binary images of the cyclic dna codes over the finite rings r and s are determined. moreover, a family of dna skew cyclic codes over r is constructed, its property of being reverse complement is studied. keywords-dna codes; cyclic codes; skew cyclic codes. i. introduction dna is formed by the strands and each strand is sequence consists of four nucleotides ; adenine (a), guanine (g), thymine (t) and cytosine (c). two strands of dna are linked with watson-crick complement. this is as a = t , t = a, g = c, c = g. for example if c = (atccg) then its complement is c = (taggc). a code is called a dna code if it satisfies some or all of the following conditions: i) the hamming contraint, for any two different codewords c1,c2 ∈ c, h(c1,c2) ≥ d ii) the reverse constraint, for any two different codewords c1,c2 ∈ c, h(c1,cr2) ≥ d iii) the reverse complement constraint, for any two different codewords c1,c2 ∈ c, h(c1,c rc 2 ) ≥ d iv) the fixed gc content constraint, for any codeword c ∈ c contains the some number of g and c element. the purpose of the i)-iii) constraints is to avoid undesirable hybridization between different strands. dna computing were started by leonhard adleman in 1994, in [3]. the special error correctcopyright: c© 2017 dertli et al. this article is distributed under the terms of the creative commons attribution license (cc by 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. citation: abdullah dertli, yasemin cengellenmis, on the cyclic dna codes over the finite rings z4 + wz4 and z4 + wz4 + vz4 + wvz4, biomath 6 (2017), 1712167, http://dx.doi.org/10.11145/j.biomath.2017.12.167 page 1 of 11 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2017.12.167 abdullah dertli, yasemin cengellenmis, on the cyclic dna codes over the finite rings ... ing codes over some finite fields and finite rings with 4n elements where n ∈ n were used for dna computing applications. in [12], the reversible codes over finite fields were studied, firstly. it was shown that c = 〈f(x)〉 is reversible if and only if f(x) is a self reciprocal polynomial. in [1], they developed the theory for constructing linear and additive cyclic codes of odd length over gf(4). in [13], they introduced a new family of polynomials which generates reversible codes over a finite field gf(16). in [2], the reversible cyclic codes of any length n over the ring z4 were studied. a set of generators for cyclic codes over z4 with no restrictions on the length n was found. in [17], the cyclic dna codes over the ring r = {0, 1,u, 1 + u} where u2 = 1 based on a similarity measure were constructed. in [9], the codes over the ring f2 + uf2,u 2 = 0 were constructed for using in dna computing applications. i. siap et al. considered the cyclic dna codes over the finite ring f2[u]/ 〈 u2 − 1 〉 in [18]. in [10], liang and wang considered the cyclic dna codes over f2 +uf2,u2 = 0. yıldız and siap studied the cyclic dna codes over f2[u]/ 〈 u4 − 1 〉 in [20]. bayram et al. considered codes over the finite ring f4 + vf4,v2 = v in [3]. zhu and chan studied the cyclic dna codes over the non-chain ring f2[u,v]/ 〈 u2,v2 −v,uv −vu 〉 in [21]. in [6], bennenni at al. studied the cyclic dna codes over f2[u]/ 〈 u6 〉 . pattanayak et al. considered the cyclic dna codes over the ring f2[u,v]/ < u 2 − 1,v3 − v,uv − vu > in [15]. pattanayak and singh studied the cyclic dna codes over the ring z4 + uz4,u2 = 0 in [14]. j. gao et al. studied the construction of the cyclic dna codes by cyclic codes over the finite ring f4[u]/ 〈 u2 + 1 〉 , in [11]. also, the construction of dna the cyclic codes has been discussed by several authors in [7,8,16]. we study families of dna cyclic codes of the finite rings z4 + wz4, w2 = 2 and z4 + wz4 + vz4 + wvz4,w2 = 2,v2 = v,wv = vw. the rest of the paper is organized as follows. in section 2, details about algebraic structure of the finite ring z4 + wz4, w2 = 2 are given. we define a gray map from r to z4. in section 3, the cyclic codes of odd length over r satisfy the reverse complement constraint are determined. in section 4, the cyclic codes of odd length over s satisfy the reverse complement constraint and the reverse contraint are examined. a linear code over s is represented by means of two linear codes over r. in section 5, the binary image of cyclic dna code over r is determined. in section 6, the binary image of cyclic dna code over s is determined. in section 7, by using a non trivial automorphism, the dna skew cyclic codes are introduced. in section 8, the design of linear dna code is presented. ii. preliminaries the algebraic structure of the finite ring r = z4 + wz4, w2 = 2 is given in [4]. r is the commutative, characteristic 4 ring z4 + wz4 = {a + wb : a,b ∈ z4} with w2 = 2. r can also be thought of as the quotient ring z4[w]/ 〈 w2 − 2 〉 . r is a principal ideal ring with 16 elements and finite chain ring. the units of the ring are 1, 3, 1 + w, 3 + w, 1 + 2w, 1 + 3w, 3 + 3w, 3 + 2w, and the non-units are 0, 2,w, 2w, 3w, 2 + w, 2 + 2w, 2 + 3w. r has 4 ideals: 〈0〉 = {0}, 〈1〉 = 〈3〉 = 〈1 + 3w〉 = ... = r, 〈w〉 = {0, 2,w, 2w, 3w, 2+w, 2+2w, 2+3w}, = 〈3w〉 = 〈2 + w〉 = 〈2 + 3w〉 , 〈2w〉 = {0, 2w}, 〈2〉 = 〈2 + 2w〉 = {0, 2, 2w, 2 + 2w}. we have 〈0〉⊂ 〈2w〉⊂ 〈2〉⊂ 〈w〉⊂ r. moreover r is a frobenious ring. we define φ : r −→ z24 as φ (a + wb) = (a,b) . biomath 6 (2017), 1712167, http://dx.doi.org/10.11145/j.biomath.2017.12.167 page 2 of 11 http://dx.doi.org/10.11145/j.biomath.2017.12.167 abdullah dertli, yasemin cengellenmis, on the cyclic dna codes over the finite rings ... the gray map is extended component wise to φ : rn −→ z2n4 (α1,α2, ...,αn) , = (a1, ...,an,b1, ...,bn), where αi = ai + biw with i = 1, 2, ...,n. φ is a z4 module isomorphism. a linear code c of length n over r is an rsubmodule of rn. an element of c is called a codeword. a code of length n is cyclic if the code is invariant under the automorphism σ which is σ (c0,c1, ...,cn−1) = (cn−1,c0, ...,cn−2) a cyclic code of length n over r can be identified with an ideal in the quotient ring r[x]/〈xn − 1〉 via the r–modul isomorphism rn −→ r[x]/〈xn − 1〉 (c0,c1, ...,cn−1) 7−→ c0 +c1x+...+cn−1xn−1 +〈xn − 1〉 theorem 1: let c be a cyclic code in r[x]/〈xn − 1〉 .then there exists polynomials g(x),a(x) such that a(x)|g(x)|xn − 1 and c = 〈g(x),wa(x)〉 . the ring r[x]/〈xn − 1〉 is a principal ideal ring when n is odd. so, if n is odd, then there exists s(x) ∈ r[x]/〈xn − 1〉 such that c = 〈s(x)〉, in [4,19]. iii. the reversible complement codes over r in this section, we study the cyclic code of odd length over r satisfies the reverse complement constraint. let {a,t,g,c} represent the dna alphabet. dna occurs in sequences with represented by sequences of the dna alphabet. dna code of length n is defined as a set of the codewords (x0,x1, ...,xn−1) where xi ∈{a,t,g,c}. these codewords must satisfy the four constraints which are mentioned in [21]. since the ring r is of cardinality 16, we define the map φ which gives a one to one correspondence between the elements of r and the 16 codons over the alphabet {a,t,g,c}2 by using the gray map as follows elements gray images dna double pairs 0 (0, 0) aa 1 (1, 0) ca 2 (2, 0) ga 3 (3, 0) ta w (0, 1) ac 2w (0, 2) ag 3w (0, 3) at 1 + w (1, 1) cc 1 + 2w (1, 2) cg 1 + 3w (1, 3) ct 2 + w (2, 1) gc 2 + 2w (2, 2) gg 2 + 3w (2, 3) gt 3 + w (3, 1) tc 3 + 2w (3, 2) tg 3 + 3w (3, 3) tt the codons satisfy the watson-crick complement. definition 2: for x = (x0,x1, ...,xn−1) ∈ rn, the vector (xn−1,xn−2, ...,x1,x0) is called the reverse of x and is denoted by xr. a linear code c of length n over r is said to be reversible if xr ∈ c for every x ∈ c. for x = (x0,x1, ...,xn−1) ∈ rn, the vector (x0,x1, ...,xn−1) is called the complement of x and is denoted by xc. a linear code c of length n over r is said to be complement if xc ∈ c for every x ∈ c. for x = (x0,x1, ...,xn−1) ∈ rn, the vector (xn−1,xn−2, ...,x1,x0) is called the reversible complement of x and is denoted by xrc. a linear code c of length n over r is said to be reversible complement if xrc ∈ c for every x ∈ c. definition 3: let f(x) = a0 +a1x+...+atxt ∈ r[x] ( s[x] ) with at 6= 0 be polynomial. the reciprocal of f(x) is defined as f∗(x) = xtf( 1 x ). it is easy to see that deg f∗(x) ≤ deg f(x) and if a0 6= 0, then deg f∗(x) = deg f(x). f(x) is called a self reciprocal polynomial if there is a constant m such that f∗(x) = mf(x). lemma 4: let f(x),g(x) be polynomials in r[x]. suppose deg f(x) − deg g(x) = m then, biomath 6 (2017), 1712167, http://dx.doi.org/10.11145/j.biomath.2017.12.167 page 3 of 11 http://dx.doi.org/10.11145/j.biomath.2017.12.167 abdullah dertli, yasemin cengellenmis, on the cyclic dna codes over the finite rings ... i) (f(x)g(x))∗ = f∗(x)g∗(x) ii) (f(x) + g(x))∗ = f∗(x) + xmg∗(x) lemma 5: for any a ∈ r, we have a + a = 3 + 3w. lemma 6: if a ∈{0, 1, 2, 3}, then we have (3+ 3w) −wa = wa. theorem 7: let c = 〈g(x),wa(x)〉 be a cyclic code of odd length n over r. if f(x)rc ∈ c for any f(x) ∈ c, then (1+w)(1+x+x2 +...+xn−1) ∈ c and there are two constants e,d ∈ z∗4 such that g∗(x) = eg(x) and a∗(x) = da(x). proof: suppose that c = 〈g(x),wa(x)〉 , where a(x)|g(x)|xn − 1 ∈ z4[x]. since (0, 0, ..., 0) ∈ c, then its reversible complement is also in c. (0, 0, ..., 0)rc = (3 + 3w, 3 + 3w,..., 3 + 3w) = 3(1 + w)(1, 1, ..., 1) ∈ c. this vector corresponds of the polynomial (3 + 3w) + (3 + 3w)x + ... + (3 + 3w)xn−1 = (3 + 3w) xn − 1 x− 1 ∈ c. since 3 ∈ z∗4, then (1+w)(1+x+...+x n−1) ∈ c. let g(x) = g0 + g1x + ... + gs−1xs−1 + gsxs. note that g(x)rc= (3+3w)+(3+3w)x+...+(3+3w)xn−s−2 +gsx n−s−1 +...+g1x n−2 +g0x n−1 ∈ c. since c is a linear code, then 3(1 + w)(1 + x + x2 + ... + xn−1) −g(x)rc ∈ c which implies that ((3 + 3w)−gs)xn−s−1 + ((3 + 3w)−gs−1)xn−s−2+...+((3+3w)−g0)xn−1 ∈ c. by using (3 + 3w) −a = a, this implies that xn−s−1(gs+gs−1x+...+g0x s) = xn−s−1g∗(x) ∈ c since g∗(x) ∈ c, this implies that g∗(x) = g(x)u(x) + wa(x)v(x) where u(x),v(x) ∈ z4[x]. since gi ∈ z4, for i = 0, 1, ...,s, we have that v(x) = 0. as deg g∗(x) = deg g(x), we have u(x) ∈ z∗4. therefore there is a constant e ∈ z∗4 such that g ∗(x) = eg(x). so, g(x) is a self reciprocal polynomial. let a(x) = a0 + a1x + ... + atxt. suppose that wa(x) = wa0 + wa1x + ... + watx t. then (wa(x))rc = (3 + 3w) + (3 + 3w)x + ... +watx n−t−1 + ... + wa1x n−2 +wa0x n−1 ∈ c as (3 + 3w)x n−1 x−1 ∈ c and c is a linear code, then −(wa(x))rc + (3 + 3w) xn − 1 x− 1 ∈ c hence, xn−t−1[(−(wat)+(3+3w))+(−(wat−1)+ (3 + 3w))x+...+ (−(wa0) + (3 + 3w))xt]. by the lemma 6, we get xn−t−1(wat + wat−1x + ... + wa0x t) xn−t−1wa∗(x) ∈ c. since wa∗(x) ∈ c, we have wa∗(x) = g(x)h(x) + wa(x)s(x) since w doesn’t appear in g(x), it follows that h(x) = 0 and a∗(x) = a(x)s(x). as deg a∗(x) = deg a(x), then s(x) ∈ z∗4. so, a(x) is a self reciprocal polynomial. theorem 8: let c = 〈g(x),wa(x)〉 be a cyclic code of odd length n over r. if (1+w)(1+x+x2+ ... + xn−1) ∈ c and g(x),a(x) are self reciprocal polynomials, then c(x)rc ∈ c for any c(x) ∈ c. proof: since c = 〈g(x),wa(x)〉 , for any c(x) ∈ c, there exist m(x) and n(x) in r[x] such that c(x) = g(x)m(x) + wa(x)n(x). by using the lemma 4, we have c∗(x) = (g(x)m(x) + wa(x)n(x)) = (g(x)m(x))∗ + xs(wa(x)n(x)) = g∗(x)m∗(x) + wa∗(x)(xsn∗(x)) since g∗(x) = eg(x),a∗(x) = da(x), we have c∗(x) = eg(x)m∗(x) + dwa(x)(xsn∗(x)) ∈ c. so, c∗(x) ∈ c. let c(x) = c0 + c1x + ... + ctxt ∈ c. since c is a cyclic code, we get xn−t−1c(x) = c0x n−t−1+c1x n−t+...+ctx n−1 ∈ c biomath 6 (2017), 1712167, http://dx.doi.org/10.11145/j.biomath.2017.12.167 page 4 of 11 http://dx.doi.org/10.11145/j.biomath.2017.12.167 abdullah dertli, yasemin cengellenmis, on the cyclic dna codes over the finite rings ... since (1 + w) + (1 + w)x + ... + (1 + w)xn−1 ∈ c and c is a linear code we have −(1 + w) xn − 1 x− 1 −xn−t−1c(x) = −(1 + w) − (1 + w)x + ... + (−c0 − (1 + w))xn−t−1 +... + (−ct − (1 + w))xn−1 ∈ c. by using a + (1 + w) = −a, this implies that −(1 + w) − ... + c0xn−t−1 + ... + ctxn−1 ∈ c this shows that (c∗(x))rc ∈ c. ((c∗(x))rc)∗ = ct + ct−1x + ... + (3 + 3w)x n−1 this corresponds this vector (ct,ct−1, ...,c0, ..., 0). since (c∗(x)rc)∗ = (xn−t−1c(x))rc, so c(x)rc ∈ c. example 9: let x3−1 = (x+ 3)(x2 +x+ 1) ∈ z4[x]. let c = 〈 x2 + x + 1 + w(x2 + x + 1) 〉 . c is a cyclic dna code of length 3 over r. the gray image of c under the gray map φ is a dna code of length 6, hamming distance 3. these codewords are as follows all 16 codewords of c cccccc tgtgtg gggggg gtgtgt tttttt gcgcgc aaaaaa cgcgcg gagaga ctctct agagag tctctc tatata acacac atatat cacaca example 10: let x7 − 1 = (x + 3)(x3 − 2x2 + x − 1)(x3 − x2 + 2x − 1) ∈ z4[x]. let c =< x6−3x5 +x4−3x3 +x2−3x+ 1 +w(x6−3x5 + x4 − 3x3 + x2 − 3x + 1) >. c is a cyclic dna code of length 7 over r. the gray image of c under the gray map φ is a dna code of length 14, hamming distance 7. these codewords are as follows all 16 codewords of c cccccccccccccc gggggggggggggg tttttttttttttt aaaaaaaaaaaaaa gagagagagagaga agagagagagagag tatatatatatata atatatatatatat tgtgtgtgtgtgtg gtgtgtgtgtgtgt gcgcgcgcgcgcgc cgcgcgcgcgcgcg ctctctctctctct tctctctctctctc acacacacacacac cacacacacacaca iv. the reversible and reversible complement codes over s throughout this paper, s denotes the commutative ring z4 + wz4 + vz4 + wvz4 = {b1 + wb2 + vb3 + wvb4 : bj ∈ z4, 1 ≤ j ≤ 4} with w2 = 2,v2 = v,wv = vw, with characteristic 4. s can also be thought of as the quotient ring z4[w,v]/ < w2 − 2,v2 −v,wv −vw > . let s = z4 + wz4 + vz4 + wvz4 = (z4 + wz4) + v(z4 + wz4) = r + vr we define the gray map φ1 from s to r as follows φ1 : s −→ r2 a + vb 7−→ (a,b) where a,b ∈ r. this gray map is extended compenentwise to φ1 : s n −→ r2n x = (x1, ...,xn) 7−→ (a1, ...,an,b1, ...,bn) where xi = ai + vbi,ai,bi ∈ r for i = 1, 2, ...,n. in this section, we study the cyclic codes of odd length n over s satisfy reverse and reverse biomath 6 (2017), 1712167, http://dx.doi.org/10.11145/j.biomath.2017.12.167 page 5 of 11 http://dx.doi.org/10.11145/j.biomath.2017.12.167 abdullah dertli, yasemin cengellenmis, on the cyclic dna codes over the finite rings ... complement constraint. since the ring s is of the cardinality 44, then we define the map φ1 which gives a one to one correspondence between the element of s and the 256 codons over the alphabet {a,t,g,c}4 by using the gray map. for example: 0 = 0 + v0 7−→ φ1(0) = (0, 0) −→ aaaa 2wv = 0 +v(2w) 7−→φ1(2wv) = (0,2w)−→aaag 1+3v+3wv =1+v(3+3w) 7−→φ1(1+v(3+3w)) = (1, 3 + 3w) −→ catt definition 11: let a1,a2 be linear codes. a1 ⊗a2 = {(a1,a2) : a1 ∈ a1,a2 ∈ a2} and a1 ⊕a2 = {a1 + a2 : a1 ∈ a1,a2 ∈ a2} let c be a linear code of length n over s. define c1 = {a : ∃ b ∈ rn,a + vb ∈ c} c2 = {b : ∃ a ∈ rn,a + vb ∈ c} where c1 and c2 are linear codes over r of length n. theorem 12: let c be a linear code of length n over s. then φ1(c) = c1 ⊗ c2 and |c| = |c1| |c2| . corollary 13: if φ1(c) = c1 ⊗ c2, then c = vc1 ⊕ (1 −v)c2. theorem 14: let c = vc1 ⊕ (1 − v)c2 be a linear code of odd length n over s. then c is a cyclic code over s if and only if c1,c2 are cyclic codes over r. proof: let (a10,a 1 1, ...,a 1 n−1) ∈ c1, (a 2 0,a 2 1, ...,a 2 n−1) ∈ c2. assume that mi = va 1 i + (1 − v)a 2 i for i = 0, 1, 2, ...,n − 1. then (m0,m1, ...,mn−1) ∈ c. since c is a cyclic code, it follows that (mn−1,m0,m1, ...,mn−2) ∈ c. note that (mn−1,m0, ...,mn−2) = v(a 1 n−1,a 1 0, ...,a 1 n−2) + (1 − v)(a2n−1,a 2 0, ...,a 2 n−2). hence (a1n−1,a 1 0, ...,a 1 n−2) ∈ c1, (a 2 n−1,a 2 0, ...,a 2 n−2) ∈ c2. therefore c1,c2 are cyclic codes over r. conversely, suppose that c1,c2 are cyclic codes over r. let (m0,m1, ...,mn−1) ∈ c, where mi = va1i + (1 − v)a 2 i for i = 0, 1, 2, ...,n − 1. then (a1n−1,a 1 0, ...,a 1 n−2) ∈ c1, (a 2 n−1,a 2 0, ...,a 2 n−2) ∈ c2. note that (mn−1,m0, ...,mn−2) = v(a 1 n−1,a 1 0, ...,a 1 n−2) + (1 −v)(a2n−1,a 2 0, ...,a 2 n−2) ∈ c. so, c is a cyclic code over s. theorem 15: let c = vc1 ⊕ (1 − v)c2 be a linear code of odd length n over s. then c is reversible over s iff c1,c2 are reversible over r. proof: let c1,c2 be reversible codes. for any b ∈ c,b = vb1 + (1 − v)b2, where b1 ∈ c1,b2 ∈ c2. since c1 and c2 are reversible, br1 ∈ c1,b r 2 ∈ c2. so, b r = vbr1 + (1 −v)b r 2 ∈ c. hence c is reversible. on the other hand, let c be a reversible code over s. so for any b = vb1 + (1−v)b2 ∈ c, where b1 ∈ c1,b2 ∈ c2, we get br = vbr1 +(1−v)b r 2 ∈ c. let br = vbr1 + (1−v)b r 2 = vs1 + (1−v)s2, where s1 ∈ c1,s2 ∈ c2. so c1 and c2 are reversible codes over r. lemma 16: for any c ∈ s, we have c + c = (3 + 3w) + v(3 + 3w). lemma 17: for any a ∈ s, a + 30 = 3a. theorem 18: let c = vc1 ⊕ (1 − v)c2 be a cyclic code of odd length n over s. then c is reversible complement over s iff c is reversible over s and (0, 0, ..., 0) ∈ c. proof: since c is reversible complement, for any c = (c0,c1, ...,cn−1) ∈ c,crc = (cn−1,cn−2, ...,c0) ∈ c. since c is a linear code, so (0, 0, ..., 0) ∈ c. since c is reversible complement, so (0, 0, ..., 0) ∈ c. by using the lemma 17, we have 3cr = 3(cn−1,cn−2, ...,c0) = (cn−1,cn−2, ...,c0) + 3(0, 0, ..., 0) ∈ c. so, for any c ∈ c, we have cr ∈ c. on the other hand, let c be reversible. so, for any c = (c0,c1, ...,cn−1) ∈ c, cr = (cn−1,cn−2, ...,c0) ∈ c. to show that c is reversible complement, for any c ∈ c, crc = (cn−1,cn−2, ...,c0) = 3(cn−1,cn−2, ...,c0) + (0, 0, ..., 0) ∈ c. biomath 6 (2017), 1712167, http://dx.doi.org/10.11145/j.biomath.2017.12.167 page 6 of 11 http://dx.doi.org/10.11145/j.biomath.2017.12.167 abdullah dertli, yasemin cengellenmis, on the cyclic dna codes over the finite rings ... so, c is reversible complement. lemma 19: for any a,b ∈ s, a + b = a + b− 3(1 + w)(1 + v). theorem 20: let d1 and d2 be two reversible complement cyclic codes of length n over s. then d1 + d2 and d1 ∩d2 are reversible complement cyclic codes. proof: let d1 = (c0,c1, ...,cn−1) ∈ d1,d2 = (c10,c 1 1, ...,c 1 n−1) ∈ d2. then, (d1 +d2) rc= ( (cn−1 +c 1 n−1), ..., (c1 +c 1 1), (c0 +c 1 0) ) = ( cn−1 +c 1 n−1−3(1+w)(1+v), ..., c0 + c 1 0 − 3(1 + w)(1 + v) ) =(cn−1 − 3(1 + w)(1 + v), ...,c0 −3(1+w)(1+v))+ ( c1n−1, ...,c 1 0 ) = ( drc1 − 3(1 + w)(1 + v) xn − 1 x− 1 ) +drc2 ∈ d1 + d2. this shows that d1 +d2 is reversible complement cyclic code. it is clear that d1 ∩d2 is reversible complement cyclic code. v. binary images of cyclic dna codes over r the 2-adic expansion of c ∈ z4 is c = α(c) + 2β(c) such that α(c) + β(c) + γ(c) = 0 for all c ∈ z4 c α(c) β(c) γ(c) 0 0 0 0 1 1 0 1 2 0 1 1 3 1 1 0 the gray map is given by ψ : z4 −→ z22 c 7−→ ψ(c) = (β(c),γ(c)) for all c ∈ z4 in [14]. define ŏ : r −→ z42 a + bw 7−→ ŏ(a + wb) = ψ (φ (a + wb)) = ψ(a,b) = (β(a),γ(a),β(b),γ(b)) let a + wb be any element of the ring r. the lee weight wl of the element of the ring r is defined as follows wl(a + wb) = wl(a,b) where wl(a,b) described the usual lee weight on z24. for any c1,c2 ∈ r the lee distance dl is given by dl(c1,c2) = wl(c1 − c2). the hamming distance d(c1,c2) between two codewords c1 and c2 is the hamming weight of the codewords c1 − c2. aa −→ 0000 ca −→ 0100 ga −→ 1100 ta −→ 1000 ac −→ 0001 ag −→ 0011 at −→ 0010 cc −→ 0101 cg −→ 0111 ct −→ 0110 gc −→ 1101 gg −→ 1111 gt −→ 1110 tc −→ 1001 tg −→ 1011 tt −→ 1010 lemma 21: the gray map ŏ is a distance preserving map from (rn, lee distance) to (z4n2 , hamming distance). it is also z2-linear. proof: for c1,c2 ∈ rn, we have ŏ(c1 − c2) = ŏ(c1) − ŏ(c2). so, dl(c1,c2) = wl(c1 − c2) = wh(ŏ(c1 − c2)) = wh(ŏ(c1) − ŏ(c2)) = dh(ŏ(c1), ŏ(c2)). so, the gray map ŏ is distance preserving map. for any c1,c2 ∈ rn,k1,k2 ∈ z2,we have ŏ(k1c1 +k2c2) = k1ŏ(c1) +k2ŏ(c2). thus, ŏ is z2-linear. proposition 22: let σ be the cyclic shift of rn and υ be the 4-quasi-cyclic shift of z4n2 . let ŏ be the gray map from rn to z4n2 . then ŏσ = υŏ. proof: let c = (c0,c1, ...,cn−1) ∈ rn, we have ci = a1i + wb2i with a1i,b2i ∈ z4, 0 ≤ i ≤ n− 1. by applying the gray map, we have ŏ(c)=  β(a10),γ(a10),β(b20),γ(b20),β(a11),γ(a11),β(b21),γ(b21), ...,β(a1n−1), γ(a1n−1),β(b2n−1),γ(b2n−1)   . hence υ(ŏ(c)) =  β(a1n−1),γ(a1n−1),β(b2n−1),γ(b2n−1),β(a10),γ(a10),β(b20),γ(b20), ...,β(a1n−2), γ(a1n−2),β(b2n−2),γ(b2n−2)  . biomath 6 (2017), 1712167, http://dx.doi.org/10.11145/j.biomath.2017.12.167 page 7 of 11 http://dx.doi.org/10.11145/j.biomath.2017.12.167 abdullah dertli, yasemin cengellenmis, on the cyclic dna codes over the finite rings ... on the other hand, σ(c) = (cn−1,c0,c1, ...,cn−2). we have ŏ(σ(c)) =  β(a1n−1),γ(a1n−1),β(b2n−1),γ(b2n−1),β(a10),γ(a10),β(b20),γ(b20), ..., β(a1n−2),γ(a1n−2),β(b2n−2),γ(b2n−2)  . therefore, ŏσ = υŏ. theorem 23: if c is a cyclic dna code of length n over r then ŏ(c) is a binary quasi-cyclic dna code of length 4n with index 4. vi. binary image of cyclic dna codes over s we define ψ̃ : s −→ z44 a0 + wa1 + va2 + wva3 7−→ (a0,a1,a2,a3) where ai ∈ z4, for i = 0, 1, 2, 3. now, we define θ : s −→ z82 as a0 + wa1 + va2 + wva3 7−→ θ(a0 + wa1 + va2 + wva3) = ψ(ψ̃(a0 + wa1 + va2 + wva3)) = (β(a0),γ(a0),β(a1),γ(a1),β(a2),γ(a2),β(a3),γ(a3)), where ψ is the gray map z4 to z22. let a0 + wa1 + va2 + wva3 be any element of the ring s. the lee weight wl of the element of the ring s is defined as wl(a0 +wa1 +va2 +wva3) = wl((a0,a1,a2,a3)) where wl((a0,a1,a2,a3)) described the usual lee weight on z44. for any c1,c2 ∈ s, the lee distance dl is given by dl(c1,c2) = wl(c1 − c2). the hamming distance d(c1,c2) between two codewords c1 and c2 is the hamming weight of the codewords c1 − c2. the binary images of cyclic dna codes; aaaa −→ 00000000 aaca −→ 00000100 aaga −→ 00001100 aata −→ 00001000 ... ... ... lemma 24: the gray map θ is a distance preserving map from (sn, lee distance) to (z8n2 , hamming distance). it is also z2-linear. proof: it is proved as in the proof of the lemma 21. proposition 25: let σ be the cyclic shift of sn and ′ υ be the 8-quasi-cyclic shift of z8n2 . let θ be the gray map from sn to z8n2 . then θσ = ′ υθ. proof: it is proved as in the proof of the proposition 22. theorem 26: if c is a cyclic dna code of length n over s then θ(c) is a binary quasi-cyclic dna code of length 8n with index 8. proof: let c be a cyclic dna code of length n over s. so, σ(c) = c. by using the proposition 25, we have θ(σ(c)) = ′ υ(θ(c)) = θ(c). hence θ(c) is a set of length 8n over the alphabet z2 which is a quasi-cyclic code of index 8. vii. skew cyclic dna codes over r we will use a non trivial automorphism, for all a + wb ∈ r, it is defined by θ : r −→ r a + wb 7−→ a−wb the ring r[x,θ] = {a0 +a1x+...+an−1xn−1 : ai ∈ r,n ∈ n} is called skew polynomial ring. it is non commutative ring. the addition in the ring r[x,θ] is the usual polynomial and multiplication is defined as (axi)(bxj) = aθi(b)xi+j. the order of the automorphism θ is 2. definition 27: a subset c of rn is called a skew cyclic code of length n if c satisfies the following conditions, i) c is a submodule of rn, ii) if c = (c0,c1, ...,cn−1) ∈ c, then σθ (c) = (θ(cn−1),θ(c0), ...,θ(cn−2)) ∈ c let f(x) + 〈xn − 1〉 be an element in the set řn = r [x,θ] /〈xn − 1〉 and let r(x) ∈ r [x,θ]. define multiplication from left as follows, r(x)(f(x) + 〈xn − 1〉) = r(x)f(x) + 〈xn − 1〉 for any r(x) ∈ r [x,θ]. theorem 28: řn is a left r [x,θ]-module where multiplication defined as in above. biomath 6 (2017), 1712167, http://dx.doi.org/10.11145/j.biomath.2017.12.167 page 8 of 11 http://dx.doi.org/10.11145/j.biomath.2017.12.167 abdullah dertli, yasemin cengellenmis, on the cyclic dna codes over the finite rings ... theorem 29: a code c over r of length n is a skew cyclic code if and only if c is a left r [x,θ]submodule of the left r [x,θ]-module řn. theorem 30: let c be a skew cyclic code over r of length n and let f(x) be a polynomial in c of minimal degree. if f(x) is monic polynomial, then c = 〈f(x)〉 , where f(x) is a right divisor of xn − 1. for all x ∈ r, we have θ(x) + θ(x) = 3 − 3w. theorem 31: let c = 〈f(x)〉 be a skew cyclic code over r, where f(x) is a monic polynomial in c of minimal degree. if c is reversible complement, the polynomial f(x) is self reciprocal and (3 + 3w) xn − 1 x− 1 ∈ c. proof: let c = 〈f(x)〉 be a skew cyclic code over r, where f(x) is a monic polynomial in c. since (0, 0, ..., 0) ∈ c and c is reversible complement, we have ( 0, 0, ..., 0 ) = (3 + 3w, 3 + 3w,..., 3 + 3w) ∈ c. let f(x) = 1 + a1x + ... + at−1xt−1 + xt. since c is reversible complement, we have frc(x) ∈ c. that is frc(x) = (3+3w)+(3+3w)x+...+(3+3w)xn−t−2 +(2+3w)xn−t−1 +at−1x n−t+ ... +a1x n−2 +(2+3w)xn−1. since c is a linear code, we have frc(x) − (3 + 3w) xn − 1 x− 1 ∈ c. this implies that −xn−t−1 + (at−1 − (3 + 3w))xn−t + ... + (a1 − (3 + 3w))xn−2 −xn−1 ∈ c. multiplying on the right by xt+1−n, we have −1 + (at−1 − (3 + 3w))θ(1)x + ... + (a1 − (3 + 3w))θt−1(1)xt−1 −θt(1)xt ∈ c. by using a + a = 3 + 3w, we have −1 −at−1x−at−2x2 − ...−a1xt−1 −xt = 3f∗(x) ∈ c. since c = 〈f(x)〉, there exist q(x) ∈ r [x,θ] such that 3f∗(x) = q(x)f(x). since deg f(x) = deg f∗(x), we have q(x) = 1. since 3f∗(x) = f(x), we have f∗(x) = 3f(x). so, f(x) is self reciprocal. theorem 32: let c = 〈f(x)〉 be a skew cyclic code over r, where f(x) is a monic polynomial in c of minimal degree. if (3 + 3w)x n−1 x−1 ∈ c and f(x) is self reciprocal, then c is reversible complement. proof: let f(x) = 1+a1x+...+at−1xt−1+xt be a monic polynomial of the minimal degree. let c(x) ∈ c. so, c(x) = q(x)f(x), where q(x) ∈ r[x,θ]. by using lemma 4, we have c∗(x) = (q(x)f(x))∗ = q∗(x)f∗(x). since f(x) is self reciprocal, so c∗(x) = q∗(x)ef(x), where e ∈ z4\{0}. therefore c∗(x) ∈ c = 〈f(x)〉. let c(x) = c0 + c1x + ... + ctx t ∈ c. since c is a cyclic code, we get c(x)xn−t−1 =c0x n−t−1 +c1x n−t+...+ctx n−1∈c. the vector corresponding to this polynomial is (0, 0, ..., 0,c0,c1, ...,ct) ∈ c. since (3 + 3w, 3 + 3w,..., 3 + 3w) ∈ c and c linear, we have (3+3w, 3+3w,..., 3+3w)−(0, 0, ..., 0,c0,c1, ...,ct) = (3+3w,..., 3+3w, (3+3w)−c0, ..., (3+3w)−ct)∈c. by using a + a = 3 + 3w, we get (3 + 3w, 3 + 3w,..., 3 + 3w,c0, ...,ct) ∈ c, which is equal to (c(x)∗)rc. this shows that ((c(x)∗) rc ) ∗ = c(x)rc ∈ c. viii. dna codes over s definition 33: let f1 and f2 be polynomials with deg f1 = t1, deg f2 = t2 and both dividing xn − 1 ∈ r[x]. let m = min{n − t1,n − t2} and f(x) = vf1(x) + (1 − v)f2(x) over s. the set l(f) is called a γ-set, where the automorphism γ : s −→ s is defined as follows: a+wb+vc+wvd 7−→a+b+w(b+d)−vc−wvdc. biomath 6 (2017), 1712167, http://dx.doi.org/10.11145/j.biomath.2017.12.167 page 9 of 11 http://dx.doi.org/10.11145/j.biomath.2017.12.167 abdullah dertli, yasemin cengellenmis, on the cyclic dna codes over the finite rings ... l(f) =   a0 a1 a2 · · · at 0 · · · · · · · · · 0 0 γ(a0) γ(a1) · · · · · · γ(at) 0 · · · · · · 0 0 0 a0 a1 · · · · · · at 0 · · · 0 0 0 0 γ(a0) γ(a1) · · · · · · γ(at) · · · 0 ... · · · · · · · · · ... · · · · · · · · · · · · ...   (1) the set l(f) is defined as l(f) = {e0,e1, ...,em−1}, where ei = { xif if i is even xiγ(f) if i is odd l(f) generates a linear code c over s denoted by c = 〈f〉γ. let f(x) = a0 + a1x + ... + atx t be over s and s-submodule generated by l(f) is generated by the matrix in eq. (1). theorem 34: let f1 and f2 be self reciprocal polynomials dividing xn − 1 over r with degree t1 and t2, respectively. if f1 = f2, then f = vf1 + (1 −v)f2 and |〈l(f)〉| = 256m. c = 〈l(f)〉 is a linear code over s and θ(c) is a reversible dna code. proof: it is proved as in the proof of the theorem 5 in [5]. corollary 35: let f1 and f2 be self reciprocal polynomials dividing xn − 1 over r and c = 〈l(f)〉 be a cyclic code over s. if x n−1 x−1 ∈ c, then θ(c) is a reversible complement dna code. example 36: let f1(x) = f2(x) = x − 1 dividing x7 − 1 over r. hence, c = 〈vf1(x) + (1 −v)f2(x)〉γ = 〈x− 1〉γ is a γ-linear code over s and θ(c) is a reversible complement dna code, because of x7 − 1 x− 1 ∈ c. acknowledgement 37: we wish to express sincere thanks to steven dougherty who gave helpful comments. references [1] abualrub t., ghrayeb a., zeng x., construction of cyclic codes over gf(4) for dna computing, j. franklin institute, 343, 448-457, 2006. [2] abualrub t., siap i., reversible quaternary cyclic codes, proc. of the 9th wseas int. conference on appl. math., istanbul, 441-446, 2006. [3] adleman l., molecular computation of the solution to combinatorial problems, science, 266, 1021-1024, 1994. [4] aydın n., dertli a., cengellenmis y., cyclic and constacyclic codes over z4 + wz4, preprint. [5] bayram a., oztas e., siap i., codes over f4 + vf4 and some dna applications, designs, codes and cryptography, doi: 10.107/s10623-015-0100-8, 2015. [6] bennenni n., guenda k., mesnager s., new dna cyclic codes over rings, arxiv: 1505.06263v1, 2015. [7] gaborit p., king o. d., linear construction for dna codes, theor. computer science, 334, 99-113, 2005. [8] guenda k., gulliver t. a., sole p,. on cyclic dna codes, proc., ieee int. symp. inform. theory, istanbul, 121125, 2013. [9] guenda k., gulliver t. a., construction of cyclic codes over f2 + uf2 for dna computing, aaecc, 24, 445459, 2013. [10] liang j., wang l., on cyclic dna codes over f2 +uf2, j. appl. math. comput., doi: 10.1007/s12190-015-08928, 2015. [11] ma f., yonglin c., jian g., on cyclic dna codes over f4[u]/ 〈 u2 + 1 〉 . [12] massey j. l., reversible codes, inf. control, 7, 369-380, 1964. [13] oztas e. s., siap i., lifted polynomials over f16 and their applications to dna codes, filomat, 27, 459-466, 2013. [14] pattanayak s., singh a. k., on cyclic dna codes over the ring z4 + uz4, arxiv: 1508.02015, 2015. [15] pattanayak s., singh a. k., kumar p., dna cyclic codes over the ring f2[u, v]/ 〈 u2 − 1, v3 − v, uv − vu 〉 , arxiv:1511.03937, 2015. [16] pattanayak s., singh a. k., construction of cyclic dna codes over the ring z4[u]/ 〈 u2 − 1 〉 based on deletion distance, arxiv: 1603.04055v1, 2016. [17] siap i., abualrub t., ghrayeb a., cyclic dna codes over the ring f2[u]/ ( u2 − 1 ) based on the delition distance, j. franklin institute, 346, 731-740, 2009. biomath 6 (2017), 1712167, http://dx.doi.org/10.11145/j.biomath.2017.12.167 page 10 of 11 http://dx.doi.org/10.11145/j.biomath.2017.12.167 abdullah dertli, yasemin cengellenmis, on the cyclic dna codes over the finite rings ... [18] siap i., abualrub t., ghrayeb a., similarity cyclic dna codes over rings, ieee, 978-1-4244-1748-3, 2008. [19] wan z. x., quaternary codes, vol.8., world scientific, 1997. [20] yıldız b., siap i., cyclic dna codes over the ring f2[u]/ ( u4 − 1 ) and applications to dna codes, comput. math. appl., 63, 1169-1176, 2012. [21] zhu s., chen x., cyclic dna codes over f2 + uf2 + vf2 + uvf2, arxiv: 1508.07113v1, 2015. biomath 6 (2017), 1712167, http://dx.doi.org/10.11145/j.biomath.2017.12.167 page 11 of 11 http://dx.doi.org/10.11145/j.biomath.2017.12.167 introduction preliminaries the reversible complement codes over r the reversible and reversible complement codes over s binary images of cyclic dna codes over r binary image of cyclic dna codes over s skew cyclic dna codes over r dna codes over s references www.biomathforum.org/biomath/index.php/biomath original article stability analysis of a schistosomiasis transmission model with control strategies mouhamadou diaby inria, université de lorraine, cnrs. isgmp bat. a, ile du saulcy, 57045 metz cedex 01, france. umi-ird-209 ummisco and lani université gaston berger, saint-louis, sénégal. diabloss84@yahoo.fr received: 24 november 2014, accepted: 16 april 2015, published: 20 may 2015 abstract—we have established and rigorously analyzed a new mathematical model that describes the dynamics schistosomiasis infection. this model incorporates several realistic features including density-dependent births rate of snails and reduced fecundity in snail hosts. our qualitative analysis of the deterministic model is made with respect to the stability of the disease free equilibrium and the unique endemic equilibrium. some biological consequences and control strategies are discussed. we have derived the basic reproduction number above which the infection will be controlled under certain levels. we have shown that the disease free equilibrium is globally asymptotically stable when the basic reproduction number r0 is less than one. we have proved the existence and global asymptotic stability of an endemic steady state when r0 > 1. this mathematical analysis of the model gives insight about the effects of the reduced fecundity and intermediate host density-dependent birth rate. finally, numerical simulations are performed to illustrate the main results. keywords-epidemic models; nonlinear dynamical systems; monotone systems; global stability; reproduction number; schistosomiasis. i. introduction mathematical modeling of the spread of infectious diseases is an important instrument in the comprehension of the dynamics of diseases and in decision making processes regarding intervention programs for controlling these diseases in many countries. the high prevalence of infectious diseases as schistosomiasis has prompted the mathematical modelers to deploy multiple infectious disease models in recent years, and different mechanisms have been suggested to explain their occurrence. schistosomiasis is transmitted by human contact with contaminated fresh water (lakes and ponds, rivers, dams) inhabited by snails carrying the parasite. many people die from schistosomiasis disease every year. schistosomiasis is the most deadly ntd (neglected tropical diseases), killing an estimated 280, 000 people each year [20]. it can result in liver damage and anemia, especially among children [22]. it is found mostly in rural areas in tropical and subtropical countries and infects humans and other vertebrates, using snails in most cases as intermediate hosts. infection takes place when citation: mouhamadou diaby, stability analysis of a schistosomiasis transmission model with control strategies, biomath 1 (2015), 1504161, http://dx.doi.org/10.11145/j.biomath.2015.04.161 page 1 of 13 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2015.04.161 m. diaby, stability analysis of a schistosomiasis transmission model ... larvae of the parasite released by the intermediate hosts, penetrate the skin of an individual when in contact with infested water. in the body, the larvae develop and cross over into adult schistosomes. these parasites live in the blood vessels, where the females lay their eggs. some eggs out of the body via the feces or urine and the parasitic life cycle continues. others are trapped in the tissues of the body, causing an immune reaction in active lesions and organs. a possible method of struggle against schistosomiasis is to treat water bodies with a molluscicide to minimize the number of snail intermediate hosts, thus breaking the cycle of disease transmission [23], [24]. beside this one, there are other such that biological control, chemical molluscicides, chemotherapy, and more permanent methods such as the provision of safe water and sanitary facilities [1], [6]. control strategies focused on intermediate hosts which are the sites of intense proliferation of this parasite are seen as a priority of the reduction of schistosomiasis transmission [28]. many researchers studied the dynamics of schistosomiasis using systems similar to the one we are interested [2], [7], [13], [14], [15], [16], [21]. that systems depend on the epidemiological formulation, but also on the demographic process incorporated into the model under different control strategies. almost all of the work to date on schistosomiasis transmission has assumed constant immigration in the snail population [25], [36] or linear birth rates [26], [1], [27] not allowing the possibility of the reduced fecundity of the infected population snails. a drawback of the models with birth and death rates proportional to the size of the population are that the population size decreases or increases exponentially, except in the special case where births exactly balance deaths. moreover, researchers in laboratory reported the influence of a trematode on life history traits of adult lymnaea elodes snails. it has been displayed reduced fecundity relative to uninfected snails [29]. to our knowledge, due to various considerations of the factors related to the transmission of the disease, impaired fertility and density-dependent birth rate are not taken account simultaneously with chemotherapy and chemical molluscicides as control strategies. we shall introduce and analyze a compartmental model, which considers a host (human), intermediate host (snail) as well as free-living cercariae and miracidia and their interaction. the model focuses essentially on the aquatic life stages of the parasite. a full mathematical analysis of the model is derived. the aim focus of this paper is to study the effect of chemotherapy combined with chemical molluscicide as control strategies on the dynamics of the model with density-dependent birth rate and impaired fertility in snail host. the paper is organized as follows. in section ii we start by defining the mathematical framework we use and focus on the different processes of transmission that might be appropriate to understand schistosomiasis. the main results are developed in section iii, including the determination of r0 the basic reproductive number of the model and the analysis of its stability properties. we show that the disease-free equilibrium is global asymptotic stability when the basic reproductive number is less than one. when the basic reproductive number is larger than one, we prove the global asymptotic stability of a unique endemic equilibrium by using some properties of k-monotone systems (see [17]). section iv is devoted to the numerical analysis and control strategies. finally, in section v, concluding remarks close the paper. ii. model derivation we build an evolutionary outcomes model of interactions between a complex life-cycle parasite schistosoma and its hosts (humans and snails). the parasite populations at the free-living stages are modeled explicitly through miracidia and cercarie. the model consists of a system of ordinary differential equations. furthermore, it is admitted that the infected snails did not recover from schistosomiasis that their lifetimes are short compared to that of humans. the model sub-divides the total human biomath 1 (2015), 1504161, http://dx.doi.org/10.11145/j.biomath.2015.04.161 page 2 of 13 http://dx.doi.org/10.11145/j.biomath.2015.04.161 m. diaby, stability analysis of a schistosomiasis transmission model ... population at time t, denoted by nh(t), into the following sub-populations of susceptible individuals (sh(t)) and individuals with schistosomiasis symptoms (ih(t)). so that nh(t) = sh(t) + ih(t). the total snails population at time t, denoted by ns(t), is sub-divided into susceptible snails (ss(t)) and infectious snails (is(t)). thus, ns(t) = ss(t) + is(t). let wm and wp be the population of miracidia and cercariae, respectively. this model assumes a constant per capita rate of exposure between hosts and sensitive parasites, where exposure and susceptibility to infection are regrouped in the transmission coefficient. we considered the mass action transmission model. we follow of the available model for schistosomiasis [30] by incorporating the human interaction. susceptible host snails increase through densitydependent births with maximum rate bs and competitive intensity c. assume that individuals are born uninfected. infected hosts suffer from reduced fecundity (0 ≤ ρ < 1). susceptible snails die at background death rate ds, due to parasite natural death rate ds� and the elimination rates θs of snails and become infected at the per capita infection rate βs, through contact with infectious miracidia. infected snails die at an elevated death rate, ds, due to parasite natural death rate ds� and the elimination rates θs of snails at which is added the rate α due to parasite virulence. the susceptible human population is increased by the recruitment of individuals in the population (assumed susceptible), at a rate λ. the population of individuals is further decreased by natural death (at a rate dh). it is also assumed that infected individuals have additional host mortality during the given short time considered at the rate µ. susceptible humans become infected only through contact with free-living pathogen cercariae in infested water at the per capita infection rate βh and recover at the per capita rate η. free-living miracidia are introduced into the aquatic environment at a per capita rate k, but they are depleted during the infection process at the per capita rate δ and die naturally if they do not find snails to infect at the rate dm. here, depletion of parasites through transmission depends on total host density. infected snails will then free up second form of larvae called cercariae at a rate γ to be able to infect humans. some cercariae also die at an elevated death rate dc due to the natural death rate dc� and cercariae elimination θc by the chemical molluscicide. the time evolution of the different populations is governed by the following system of equations:   dss dt = bs (ss + ρis)(1 − c (ss + is)) − ds︷ ︸︸ ︷ (ds� + θs) ss −βs ss wm, dis dt = βs ss wm − (ds� + θs + α) is, dwm dt = k ih −δ (ss + is) wm −dm wm, dwc dt = γ is− dc︷ ︸︸ ︷ (dc� + θc) wc, dsh dt = λ −βh wc sh −dh sh + η ih, dih dt = βh wc sh − (dh + µ + η) ih. (1) for convenience of simplicity, we denote ds = ds� + θs, dc = dc� + θs. the dynamics of the total snails population (ns = ss + is) is governed by dns dt = bs (ss + ρis) (1 − cns) −ds ns −αis ≤ bs ns − bs cn2s −ds ns ≤ ( bs −ds bs c −ns ) bs cns. the dynamics of the total humans population (nh = sh + ih) is governed by dnh dt = λ −dh nh −µih ≤ λ dh . biomath 1 (2015), 1504161, http://dx.doi.org/10.11145/j.biomath.2015.04.161 page 3 of 13 http://dx.doi.org/10.11145/j.biomath.2015.04.161 m. diaby, stability analysis of a schistosomiasis transmission model ... thus the region d = {(ss,is,wm,wc,sh,ih) ∈ r6+ : ns ≤ bs −ds cbs , wm ≥ 0, wc ≥ 0, nh ≤ λ dh } is a positively invariant set of system (1). furthermore, the model (1) is well-posed epidemiologically and we will consider dynamic behavior of model (1) on d. iii. main results in this section, we firstly derive the basic reproduction number for system (1) by the method of next generation matrix formulated in [9], [5]. the basic reproduction number r0 is used to assess the stability of the disease free equilibrium and the endemic equilibrium. then we derive the asymptotic behavior of the system (1) by its limiting system. discussions of the relation of the dynamics between (1) and its limiting system can be found in [32], [33], [34]. in particular, there holds the following significant result : lemma iii.1. consider the following two systems dx dt = f(t,x), dy dt = g(y), where x,y ∈ rn, f and g are continuous, satisfy a local lipschitz condition in any compact set ω ∈ rn, and f(t,x) → g(x) as t → ∞, so that the second system is the limit system for the first system. let φ(t,t0,x0 and φ(t,x) be solutions of theses systems, respectively. suppose that p ∈ ω is a locally asymptotically stable equilibrium of the limit system and its attractive region is w(p) = {y ∈ ω|φ(t,t0,y) → p,t → +∞}. let wφ be the omega limit set of φ(t,t0,x0). if wφ ∩w(p) 6= ∅, then limt→+∞φ(t,t0,x0) = p. this limiting system is the key to understanding the global dynamics of system (1). by a method due to castillo-chavez et al [31], the stability of the disease free equilibrium of the proposed limiting system is discussed. moreover, the stability of positive equilibrium of a proposed limiting system of model (1) is analyzed theoretically thanks to monotone dynamical system theory [11]. a. the reproduction number the reproduction number is the expected number of secondary cases produced in a completely susceptible population by a typical infective individual. it is easy to see that system (1) admits always a disease-free equilibrium, e0 =( bs −ds bs c , 0, 0, 0, λ µ ) . let x = (is,ih,wm,wp,ss,sh)t . then system (1) can be written as x′ = f(x) −v(x), where f =   βs ss wm βh sh wc 0 0 0 0   , and v =   (ds + α) is (µ + dh + η) ih −k ih + δ (ss + is) wm + dm wm −γ is + dc wc −bs (ss −ρis)(1 − c (ss + is)) +ds ss + βs ss wm −λ + βh wc sh + µih + η ih   . we can get f =   0 0 βs (bs −ds) bs c 0 0 0 0 βh λ µ 0 0 0 0 0 0 0 0   and v =   α + ds 0 0 0 0 η + µ 0 0 0 −k δ (bs −ds) bs c + dm 0 −γ 0 0 dc   f v −1 is the next generation matrix for model (1). it then follows that the spectral radius biomath 1 (2015), 1504161, http://dx.doi.org/10.11145/j.biomath.2015.04.161 page 4 of 13 http://dx.doi.org/10.11145/j.biomath.2015.04.161 m. diaby, stability analysis of a schistosomiasis transmission model ... of matrix f v −1 is ρ(f v −1). according to theorem 2 in [18], the basic reproduction number of model (1) is r0 = ρ(f v −1) = √ β γ k λ (bs −ds) βh dh dc (ds + α) (η + µ + dh) (bs (cdm + δ) −δds) b. the local stability of the dfe before proceeding with the mathematical analysis of the model (1), the following simplification would be made. to make the mathematical analysis more tractable, we will consider the case where the the disease-induced death and the parasite virulence parameters in the model will be set to zero. that is, from now on, it is assumed that α = µ = 0 (it should be mentioned that such assumption may not be appropriate in modeling the transmission dynamics in hardest hit areas). however, this assumption will be relaxed in the numerical simulations section. without loss of generality (see [33], [32]), we assume that our population of humans has reached its limiting value, i.e, n∗h ≡ λ dh (with α = µ = 0). by eliminating the equation for dsh dt , we get from (1) the equivalent limiting system :  dss dt = bs (ss + ρis)(1 − c (ss + is)) −ds ss −βs ss wm, dis dt = βs ss wm −ds is, dwm dt = k ih −δ (ss + is) wm −dm wm, dwc dt = γ is −dc wc, dih dt = βh wc (n ∗ h − ih) − (dh + η) ih. (2) defined on d0 = {(ss,is,wm,wc,ih) ∈ r5+ : ns ≤ bs −ds cbs , wm ≥ 0, wc ≥ 0, ih ≤ λ dh } . we now study the local behavior of the disease free equilibrium for system (2), which also gives the analogous behavior for system (1). proposition iii.1. the dfe for limiting system (2) e0 is las (locally asymptotically stable) if r0 < 1 with ρ ≤ 1 −ds/bs and is unstable if r0 > 1. proof: the jacobian matrix of (2) at e0 is j0 = ( j11 j12 j21 j22 ) where, j11 = ( ds − bs (ρ + 1)ds − bs 0 −ds ) , j12 =   βs (ds − bs) bs c 0 0 βs (bs −ds) bs c 0 0   ,j21 =   0 00 γ 0 0   , j22 =   δ ds − bs (δ + cdm) bs c 0 k 0 −dc 0 0 nhβh −η −dh   . let us do the following transformation: js0 = t.j0.t where, t =   1 0 0 0 0 0 −1 0 0 0 0 0 −1 0 0 0 0 0 −1 0 0 0 0 0 −1   . then, js0 and j0 are similar. js0 is a metzler matrix and we can write js0 = f + v with f =   0 bs −ds βs (bs −ds) bs c 0 0 0 0 βs (bs −ds) bs c 0 0 0 0 0 0 k 0 γ 0 0 0 0 0 0 nh βh 0   , biomath 1 (2015), 1504161, http://dx.doi.org/10.11145/j.biomath.2015.04.161 page 5 of 13 http://dx.doi.org/10.11145/j.biomath.2015.04.161 m. diaby, stability analysis of a schistosomiasis transmission model ... v=   ds − bs−ρds 0 0 0 0 −ds 0 0 0 0 0 δ ds−bs(δ+cdm) bs c 0 0 0 0 0 −dc 0 0 0 0 0 −η −dh   . we have f > 0 and v is metzler stable, see [3], [12], [10], [4]. thanks to varga’s theorem in [19]: s(j0) ≤ 0 iff ρ(−f v −1) ≤ 1. since ρ(−f v −1) = r40, we then deduce that e0 is las if r0 < 1 and is unstable if r0 > 1. c. global stability of the dfe the object of this subsection is to prove the gas result using the second generation approach given in castillo-chavez et al [31]. herein stated to be self-contained. theorem iii.1. if the system (2) can be written in the form dx dt = f(x,z), dz dt = g(x,z),g(x,z) = 0, where the components of x ∈ rm denotes the number of uninfected individuals and the components of z ∈ rn the number of infected individuals. let e0 = (x∗, 0) be the disease free equilibrium of the system. assume that • for dx dt = f(x, 0), x∗ is globally asymptotically stable (gas), • g(x,z) = az − ĝ(x,z), ĝ(x,z) ≥ 0 for (x,z) ∈ d, where a = dz g(x∗, 0) is an m-matrix (the off diagonal of a are nonnegative) and d the region where the model makes biological sense. then the fixed e0 = (x∗, 0) is a globally asymptotically stable equilibrium of model system (2) provided that r0 < 1. applying theorem iii.1 to system (2) gives ĝ(x,z) =   wm βs (s ∗ −ss) wm δ (is + ss −s∗s ) 0 wc ih βh   . since, s ≤ s∗ ≡ n∗h thus ĝ(x,z) ≥ 0, and by theorem iii.1, e0 is gas. we summarize the result in the following theorem: theorem iii.2. if r0 < 1 then the dfe is gas. d. local stability of the endemic equilibrium we proceed to investigate the local stability of the model. we use the following limiting system (3) to obtain the information for the whole system. since the population size, ns(t) (with α = µ = 0) also satisfies ns → n∗s := bs −ds cbs as t →∞. using results from lemma iii.1 , we can get analytical results by considering the limiting system of (3) in which the total population of snails and humans both have reached the limiting states. then, we obtain the reduced limiting dynamical system:  dis dt = βs (ns − is) wm −ds is, dwm dt = k ih −δ n∗s wm −dm wm, dwc dt = γ is −dc wc, dih dt = βh wc (n ∗ h − ih) − (dh + η) ih. (3) the dynamics of system (1) can be focused on this restricted region d1 = {(is,wm,wc,ih) ∈ r4+ : is ≤ bs −ds cbs , wm ≥ 0, wc ≥ 0, ih ≤ λ dh } clearly, the system (1) is asymptotic to the limiting system (3). thus, the dynamics of system (1) are qualitatively equivalent to the dynamics of its limiting system. the variation matrix of system (3) at e∗ an equilibrium point is j(e∗) =   j11(e∗) j12(e∗) j21(e ∗) j22(e ∗)   , where, j11(e ∗) = ( −w∗m βs −ds (n∗s − i∗s )βs 0 −n∗s δ −dm ) , biomath 1 (2015), 1504161, http://dx.doi.org/10.11145/j.biomath.2015.04.161 page 6 of 13 http://dx.doi.org/10.11145/j.biomath.2015.04.161 m. diaby, stability analysis of a schistosomiasis transmission model ... j12(e ∗) = ( 0 0 0 k ) , j21(e ∗) = ( γ 0 0 0 ) , j22(e ∗) = ( −dc 0 (n∗h − i ∗ h)βh −η −w ∗ c βh −dh ) theorem iii.3. if r0 > 1, then the positive endemic equilibrium e∗ of the limiting system (3) is locally asymptotically stable. before proving theorem iii.3, we state the following useful lemma : lemma iii.2. (kamkang [35], proposition 3.1) let m be a square metzler matrix written in block form m = ( a b c d ) , with a and d square matrices. m is metzler stable if only if matrices a and d −c a−1 b are metzler stable. proof of theorem iii.3: j(e∗) is a metzler matrix and we can write : j(e∗) = ( a b c d ) , where a=  −w∗m βs −ds (n∗s − i∗s )βs 0 −n∗s δ −dm   , d=   −dc 0 (n∗h − i ∗ h)βh −η −w ∗ c βh −dh   . b= ( 0 0 0 k ) ,c= ( γ 0 0 0 ) . clearly, a is a stable metzler matrix. then, after some computations, we get d −c a−1 b =   m11 m12 m21 m22   . where, m11 = −dc, m12 = k ( n∗s − i∗h)β γ w∗m n ∗ s β δ + n ∗ s ds δ + w ∗ m β dm + ds dm m21 = (n ∗ h −i ∗ h)βh, m22 = −η−w ∗ c βh−dh. d −c a−1 b is a stable metzler matrix iff χ:=β γ k βh(i ∗ h −n ∗ s )(n ∗ h − i ∗ h) + dc (dm + δ n ∗ s ) (ds + β w ∗ m) . (w∗c βh + η + dh) > 0. the endemic equilibrium satisfies  βs (is −ns) wm = −ds is, wc = γ dc is. k ih = (δ ns + dm) wm. (4) hence, χ=−γ k βh ds wm is(n ∗ h − i ∗ h) + dc (dm + δ n ∗ s ) . (ds + βs w ∗ m) (w ∗ c βh + η + dh) >−γ k βh ds wm is(n ∗ h − i ∗ h) +dc (dm + δ n ∗ s ) ds βh γ is dc > ds βh γ is [ − k wm (n∗h − i ∗ h) + (dm + δ n ∗ s ) ] >ds βh γ is [ − k wm i∗h + (dm + δ n ∗ s ) ] >ds βh γ is [ − wm (δ ns + dm) wm i∗h + (dm + δ n ∗ s ) ] > 0, which implies j(e∗) is a metzler stable matrix. thus the unique endemic equilibrium is locally assymptoticaly stable. e. global stability of the endemic equilibrium in this section we will establish the global stability of the unique endemic equilibrium point when r0 > 1. we shall use the properties of kmonotone systems for the analysis of our system (see [17]). biomath 1 (2015), 1504161, http://dx.doi.org/10.11145/j.biomath.2015.04.161 page 7 of 13 http://dx.doi.org/10.11145/j.biomath.2015.04.161 m. diaby, stability analysis of a schistosomiasis transmission model ... theorem iii.4. if r0 > 1, then the positive endemic equilibrium state e∗ of the limiting system (3) is globally asymptotically stable in the interior of the set d1. proof: it is useful to be able to test monotonicity directly in terms of vector fields. we recall here a wider criterion to define monotonicity. a system x = f(x) is said to be monotone in the set k if there exists a diagonal matrix h = diag(�1,�2, · · · ,�n) where each �i is 1 or −1 such that the matrix product h df(x) h has only non positive values outside the diagonal for any x (see smith [17], lemma 2.1). in the above statement, df(x) is the jacobian matrix of f and k is a convex set. the jacobian of the system (3), j(e∗), is a metzler matrix and is irreducible, which implies the strong monotonicity of the system under a usual order defined by the orthant k = {(is,wm,wc,ih) ∈ r4+ : is, wm, wc, ih > 0} the significant result of convergence proved by hirsh in the late 1980s can be described in our framework as follows : given an autonomous system that is strongly monotone with respect to some proper cone and assuming that there is a unique equilibrium in an open set of points with compact orbit closures, every initial condition with bounded solution converges to the unique equilibrium; (see hirsh [11], theorem 10.3). thanks hirsch’s theorem and the fact that we have only one endemic equilibrium e∗ in d̊1 which is locally asymptotically stable when r0 > 1 we can conclude that e∗ is globally asymptotically stable in d̊1 when r0 > 1. iv. numerical analysis and control strategies in this section, we present some numerical simulation results to confirm our analytical predictions on the global dynamics of the schistosomiasis models. the two selected control strategies identified in this study against the spread of schistosomiasis are snails reduction strategies and chemotherapy treatments. snail reduction strategies include the elimination of snails as well as the elimination of free-living cercariae using appropriate biological agents. the strategy of applying chemotherapy treatments requires an effective drug delivery as praziquantel to infected humans. in order to examine the effects of these two classes of anti-schistosomiasis strategies, the model (1) is simulated using the set of parameter values given in table i. the effect of chemical molluscicide is incorporated in our model by increasing the death rate parameter of the snails and free-living cercariae at the rate θs and θc, respectively. in other words, a higher values in θs and θc serve to reduce the number of snails and free-living cercariae, and thus can be used to investigate the effects of chemical molluscicide on the epidemic. similarly, the effects of the drug administration method are modelled by increasing a recover rate η to the schistosomiasis. the basic reproduction number can be applied to measuring the control efforts needed to reduce or eliminate an infection. furthermore, since r0 is a decreasing function of θs, θc and η, the use of any preventive strategy that can increase θs, θc or η results in a reduction of schistosomiasis infections. we, consequently, seek to compare the effects of chemical molluscicide and drug administration on the control of the spread of schistosomiasis in humans. we first consider the original no-control model by setting η = θs = θc = 0 in system (1). we set the population size of initial conditions as ss(0) = 104; is(0) = 5 × 104; wm(0) = 5×103; wc(0) = 9×103; sh(0) = 2×103; ih(0) = 550, and take the values of the parameters from table i. it should be mentioned that some of these parameters (e.g. θs, θc, η) are estimated in the convenience use. we pick the following conclusions for the parameter values used. the results in fig. 1 show a marked increase in the number of asymptotically infected snails and infected humans at steady state with no control strategies (θs ≈ θc ≈ η ≈ 0). with the same configurations, we now add control measures and simulate the control model (1). the simulation biomath 1 (2015), 1504161, http://dx.doi.org/10.11145/j.biomath.2015.04.161 page 8 of 13 http://dx.doi.org/10.11145/j.biomath.2015.04.161 m. diaby, stability analysis of a schistosomiasis transmission model ... table i parameter values estimed parameter definition default value literature source bs maximum birth rathe of hosts 0.06 day−1 [30] ρ relative fecundity of infected hosts 0.75 day−1 −− c strenght of density dependence on host birth rate 0.025 l day−1 −− dh nartural mortality rate of humans 0.0000384 day −1 −− ds natural mortality rate of snails 0.003 day−1 −− µ parasite dependent mortality of humans 0.0039 day−1 −− dm loss rate of free-living miracidia 2.5 day−1 −− dc loss rate of free-living cerscariae 2 day −1 −− βs rate at which the miracidia successfully infects a snail 0.615 l mir−1 day−1 −− βh rate at which susceptible humans get infected 0.406 l mir −1 day−1 [36] α parasite virulence on survival 0.007 day−1 [30] γ per capita production rate of cercariae by infected snails 100 cerc host−1 day−1 [36] k per capita production rate of miracidia by infected humans 0.00232 day−1 [36] η treatment rate of infected human 0.03 day−1 estimated δ per capita depletion rate of miracidia 0.0039 day−1 [30] θs, θc elimination rates of snails and cercariae, respectively 0.05 day−1 estimated λ recruitment rate for humans 8000 day−1 [36] shown in figure 2 demonstrate the effectiveness of the control strategies adopted. we observe that an epidemic outbreak occurs for some period of time. for instance, if human recover at the rate η = 0.9 and snails be eliminated at the rate η = 0.05, then the number of asymptomatically infected humans tend to zeros after a period of time t ≈ 1050 days. mention should be made here of the fact that, in the last scenario, we have values for each of the following parameters η, θs, θc, so that r0 < 1. fig. 3 and 4 also show that the efficacy of snail elimination can be important for such strategies to make a meaningful impact in combatting schistosomiasis in humans. v. summary and conclusions in this paper, we have presented a stability analysis of a schistosomiasis infection model that explicitly includes density dependent births rate and impaired fecundity in the snail host population. the model also captures the effect of two control strategies: chemotherapy and snail elimination by molluscicides. six sub population sizes were considered: human host susceptible and infected, snail intermediate host susceptible and infected, free-living miracidia and cercariae. it is shown that the combination of chemotherapy and snail elimination can be effective control strategy that is there will not be an epidemic. mathematical properties of the model are analyzed and used to reduce the dimension of the system under consideration. the reproductive number r0 is then analytically and explicitly computed. we proved that the disease-free steady state e0 is globally asymptotically stable if r0 ≤ 1. we have also established the global asymptotic stability of the endemic equilibrium e∗ when it exists i.e., when r0 > 1 using some properties of monotone systems. biomath 1 (2015), 1504161, http://dx.doi.org/10.11145/j.biomath.2015.04.161 page 9 of 13 http://dx.doi.org/10.11145/j.biomath.2015.04.161 m. diaby, stability analysis of a schistosomiasis transmission model ... 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 0.5 1 1.5 2 2.5 3 3.5 x 10 6 time in fe c te d s n a il s (a) 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 0.5 1 1.5 2 2.5 x 10 5 time in fe c te d h u m a n s (b) fig. 1. time evolution of the number of infected snails population (a) and humans population (b) without control strategies with parameter values defined in table i: θc = 0, θs = 0, η = 0. these parameters correspond to r0 > 1. the initial condition is ss = 10000, is = 50000, wm = 5000, wc = 9000, sh = 2000, ih = 550. 0 50 100 150 200 250 300 350 400 450 0 1 2 3 4 5 6 x 10 4 time in fe c te d s n a il s (a) 0 200 400 600 800 1000 1200 1400 1600 0 2 4 6 8 10 12 14 16 18 x 10 4 time in fe c te d h u m a n s (b) fig. 2. effect of combined human treatment and snails elimination on snails population (a) and humans population behavior (b) with parameter values defined in table i: θc = 10, θs = 0.05, η = 0.90. these parameters correspond to r0 < 1. the initial condition is ss = 10000, is = 50000, wm = 5000, wc = 9000, sh = 2000, ih = 550. biomath 1 (2015), 1504161, http://dx.doi.org/10.11145/j.biomath.2015.04.161 page 10 of 13 http://dx.doi.org/10.11145/j.biomath.2015.04.161 m. diaby, stability analysis of a schistosomiasis transmission model ... 0 50 100 150 200 250 300 350 400 450 0 0.5 1 1.5 2 2.5 3 x 10 6 time (days) in fe c te d s n a il s (a) 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 0.5 1 1.5 2 2.5 x 10 5 time (days) in fe c te d h u m a n s (b) fig. 3. effect of human treatment only, by drug administration, on snails population (a) and humans population behavior (b) with parameter values defined in table i: θc = 0, θs = 0, η = 0.90. the initial condition is ss = 10000, is = 50000, wm = 5000, wc = 9000, sh = 2000, ih = 550. 0 50 100 150 200 250 300 350 400 450 0 1 2 3 4 5 6 x 10 4 time (days) in fe c te d s n a il s (a) 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 5 time (days) in fe c te d h u m a n s (b) fig. 4. effect of snails elimination only, by molluscicide, on snails population (a) and humans population behavior (b) with parameter values defined in table i: θc = 10, θs = 0.05, η = 0. the initial condition is ss = 10000, is = 50000, wm = 5000, wc = 9000, sh = 2000, ih = 550. biomath 1 (2015), 1504161, http://dx.doi.org/10.11145/j.biomath.2015.04.161 page 11 of 13 http://dx.doi.org/10.11145/j.biomath.2015.04.161 m. diaby, stability analysis of a schistosomiasis transmission model ... references [1] e. j. allen and h. d. j. victory. modelling and simulation of a schistosomiasis infection with biological control. acta trop, 87(2):251–267, 2003. 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[36] e. t. chiyaka and w. garira. mathematical analysis of the transmission dynamics of schistosomiasis in the human-snail hosts. journal of biological systems, 17(03):397–423, 2009. biomath 1 (2015), 1504161, http://dx.doi.org/10.11145/j.biomath.2015.04.161 page 13 of 13 http://dx.doi.org/10.11145/j.biomath.2015.04.161 introduction model derivation main results the reproduction number the local stability of the dfe global stability of the dfe local stability of the endemic equilibrium global stability of the endemic equilibrium numerical analysis and control strategies summary and conclusions references original article biomath 1 (2012), 1209032, 1–6 b f volume ░, number ░, 20░░ biomath issn 1314-684x editor–in–chief: roumen anguelov b f biomath h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / biomath forum model and simulations of a wood frog population nofe al-asuoad∗, roumen anguelov†, keith berven ‡ and meir shillor∗ ∗ department of math. and statistics, oakland university, rochester, mi, usa emails: nalasuoa@oakland.edu, shillor@oakland.edu † department of math. and applied mathematics, university of pretoria, pretoria, south africa email: roumen.anguelov@up.ac.za ‡ department of biological sciences, oakland university, rochester, mi, usa email: berven@oakland.edu received: 15 july 2012, accepted: 3 september 2012, published: 11 october 2012 abstract—this work presents and simulates a mathematical model for the dynamics of a population of wood frogs. the model consists of a system of five coupled impulsive differential equations for the larvae, juveniles (early, middle, and late) and the mature adult populations. a simulation result depicts possible dynamics of the frogs’ population when during one year the larvae population dies out. this provides a tool for the study of the resilience of the population and the conditions that may lead to its survival and flourishing or extinction. keywords-population dynamics; compartmental model; wood frog, impulsive odes; simulations i. introduction we present a model for a wood frog population and a preliminary simulation of its solutions. this research is motivated by more than two decades of field observations of a population of wood frogs, which has been recently reported in berven [3]. the aim is, once the model is validated by comparison with experimental data collected in [3], to study the conditions that allow for the survival, and possible flourishing of the population. the model is of the compartmental type (see, e.g., [1], [5], [6] and the references therein) and consists of a system of five nonlinear ordinary differential equations (odes), and includes impulses that describe the transitions from one population to the next. the equations describe the dynamics of the larval aquatic stage, and juvenile and adult stages, which are terrestrial, in the development of the frogs’ population. when the larvae metamorphose and become juveniles, they leave the pond over a period of two weeks, and it is found in [3] that there is considerable merit in dividing the juvenile population into three groups, those who leave the pond early, late, and in the middle. in this manner we obtain the five compartments with the associated impulsive odes. whereas the applied interest in the model, once it has been validated by comparison with the data from the field, is to study the conditions for the survival of the population, the mathematical interest lies in the facts that the aquatic larval stage is separate from the other stages, and the interactions are via transfer conditions at prescribed times, and the resulting impulses. the long-time interest in the model lies in its ability to provide for qualitative and quantitative predictions on the overall populations growth that will allow to better understanding and management of the populations. the model is described in the following section, then, we present the results of a typical computer simulation of the model which shows the dynamics and the recovery from a year without any larvae. in the conclusions section we also mention some unresolved questions that we plan to address in the future. ii. the model we construct a model for the dynamics of a wood frog population. the stages of the life cycle of the citation: n. al-asuoad , r. anguelov, k. berven, m. shillor, model and simulations of a wood frog population, biomath 1 (2012), 1209032, http://dx.doi.org/10.11145/j.biomath.2012.09.032 page 1 of 6 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.09.032 n. al-asuoad et al., model and simulations of a wood frog population frogs that we concentrate on are the larvae, juveniles, and mature adults. the first one is aquatic, and the other two are terrestrial, a fact that leads to a nonstandard model with impulses and possibly time dependent periodic coefficients. the field data obtained from the population study in berven [3] allows us to use odes. the model deals with the total populations because the spatial distribution of the populations is not taken into account. the frogs’ life cycle we model is as follows. the eggs are produced in very large numbers in the spring, over a period of two weeks, and those that survive become larvae. the larvae that survive undergo metamorphosis in the summer and become juveniles. these become mature adults over the next 1-4 years. we note that since the hatching rate of the eggs is constant, at about 90%, for the sake of simplicity we omit the eggs compartment. an interesting observation made in [3] leads us to split the juvenile population into three groups, those who leave the pond at the beginning of the first week, and the beginning and end of the second week, since these juveniles’ rates of growth and maturation are different. we denote by l and m the total numbers of larvae, and mature frogs, respectively, and by je, jm and jl the three subpopulations, early, middle, and late, of the juveniles, all functions of time t (measured in days). we let [0, t ] denote the time interval over which the populations grow, or have been under observation. we describe the rate of change of each population per day. the model is of the compartmental type, and is depicted schematically in fig. 1. l r µl q q qqs -� � ��3 je jl jm �µe r µm rµl h h h hj � � � �* m r µm fig. 1. compartmental structure and flow chart the eggs hatch within two weeks after fertilization. the larval period lasts about eight to ten weeks. so counting from the laying and fertilization of the eggs, the early larvae metamorphose into juveniles within 10 weeks, and the later ones one and two weeks later. since these periods are relatively short, compared to the rest of the dynamics of the population, e.g., the life span or growing to maturity, we model these changes using impulsive differential equations (see, e.g., [1] and also [4], [2] and the references therein). we start the time count t = 0 on the day of the eggs’ fertilization and assume that initially the number of larvae that successfully hatch is l0, and so the larvae population undergoes a discontinuous change on that day by jumping from no larvae to l0. then, at the same day at the year k (for k = 1, 2, ..., t ), that is at the times tk = 365k, all the eggs hatch and l jumps from zero larvae (before hatching) to the number that hatched, σm (tk), which is proportional to the mature population m at that time. the proportionality rate constant σ is the fertility rate of the mature female frogs (which are a third of the matures). the splitting of the juveniles into the three groups of early, middle, and late ones is based on the observation that when the larvae population is large, those who metamorphose and leave earlier develop and mature faster. we denote by τe, τm, and τl the respective days in the year on which the first second, and third group of larvae become early je, middle jm, and late jl juveniles, respectively. the data in [3] indicates that we may set approximately τe = 75, τm = 80, and τl = 85 days, but these choices are somewhat arbitrary, however, in the model we keep the general notation. we assume that at the times tk + τe the fraction δel becomes ‘early’ juveniles, at the times tk + τm the fraction δml becomes ‘middle’ juveniles, and at tk + τl the rest of the larvae become ‘late’ juveniles. at the exceptional times the larvae population jumps discontinuously, that is impulses take place. at the beginning of each year, at time tk, for k = 1, 2, 3, . . . , the larvae population is l(tk + 0) = σm (tk). at times tk + τe, tk + τm, and tk + τl we have, l(tk + τe + 0) = (1 − δe)l(tk + τe − 0), l(tk + τm + 0) = (1 − δm)l(tk + τm − 0), l(tk + τl + 0) = 0. the equation for the larvae population is dl dt = −µll, t 6= tk, tk + τe, tk + τm, tk + τl, for k = 1, 2, 3, . . . . here, the mortality rate is µl = µl(t, l) = µ1l + µ2l(t)l(t), where, following [7], we let µ1l represent the density independent part and µ2l(t)l(t) is the density dependent biomath 1 (2012), 1209032, http://dx.doi.org/10.11145/j.biomath.2012.09.032 page 2 of 6 http://dx.doi.org/10.11145/j.biomath.2012.09.032 n. al-asuoad et al., model and simulations of a wood frog population part that depends on the available resources and, hence, may be time dependent. we turn to the describe the rates of growth of the juvenile populations. in normal circumstances, e.g., when food is sufficient or the weather is mild, a juvenile is ready for reproduction in the next mating and egg laying season, that is the next spring. however, some, especially the late ones, may become fertile only in the second year. we assume that the different juvenile populations have different mortality rates, set as µr(t) = µ1r + µ2r(t)jr(t), for r = e, m, l, where µ1r represent the density independent rates, and µ2rjr are the density dependent parts. we denote by αe, αm, and αl the rates at which the juveniles mature and move to the m population, which may be time dependent, to take into account possible changes in the environmental conditions and, also, are likely to be periodic functions reflecting the availability of food and growth rates of the juveniles. we denote by βrs (r, s = e, m, l, r 6= s) the influence of population ls on lr, which may describe the competition for food. however, the experimental data is not clear about it, so we assume that these rate coefficients are small. thus, the population growth equations for je are dje dt = −µe(t)je(t) − αe(t)je(t) − βemjm(t)je(t) −βel jl(t)je(t), t 6= tk + τe, and similar equations hold for jm and jl. here, k = 0, 1, 2, . . . is the kth year. moreover, we assume that the competition for food between the juveniles and the mature frogs is negligible, since the mature frogs feed on larger insects. otherwise, a term of the form −γsm m , for r, s = e, m, l, has to be added to the equations that describe the dynamics of je, jm, and jl. the compartment of mature frogs is assumed to contain a homogeneous population the growth of which is governed by the equation dm dt = αeje + αmjm + αljl − µm m, where µm = µm (t, m ) = µ1m + µ2m (t)m (t) is the mortality rate consisting of density independent term µ1m , and density dependent term µ2m . collecting the equations and conditions above yields the following model consisting of five impulsive differential equations for the larvae, juvenile and mature populations. the model for the dynamics of the wood frog population is: find five functions: (l(t), je(t), jm(t), jl(t), m (t)), for 0 ≤ t ≤ t , such that, for tk = 365k, k = 0, 1, 2, . . . , dl(t) dt = −(µ1l + µ2l(t)l(t))l(t), t 6= tk, tk + τe, tk + τm, tk + τl, (1) l(0) = l0, (2) l(tk) = σm (tk), k 6= 0, (3) l(tk + τe + 0) = (1 − δe)l(tk + τe − 0), (4) l(tk + τm + 0) = (1 − δm)l(tk + τm − 0), (5) l(tk + τl + 0) = 0, (6) dje(t) dt = −(µ1e + µ2e(t)je(t))je(t) − αeje(t) −βemjm(t)je(t) − β e l jl(t)je(t), 0 < t 6= tk + τe, (7) djm(t) dt = −(µ1m + µ2m(t)jm(t))jm(t) − αmjm(t) −βml jl(t)jm(t) − β m e je(t)jm(t), 0 < t 6= tk + τm, (8) djl(t) dt = −(µ1l + µ2l(t)jl(t))jl(t) − αljl(t) −βleje(t)jl(t) − β l mjm(t)jl(t), 0 < t 6= tk + τl, (9) je(tk + τe + 0) = je(tk + τe − 0) +δel(tk + τe − 0), (10) jm(tk + τm + 0) = jm(tk + τm − 0) +δml(tk + τm − 0), (11) jl(tk + τl + 0) = jl(tk + τl − 0) +l(tkτl − 0), (12) dm (t) dt = αeje(t) + αmjm(t) + αljl(t) −(µ1m + µ2m m (t))m (t), (13) m (0) = m0, (14) je(0) = jm(0) = jl(0) = 0. (15) here, l0 is the number of larvae and m0 is the number of adult frogs at t = 0, i.e., at the beginning of the first year (k = 0). at that time there are no juveniles, (15). biomath 1 (2012), 1209032, http://dx.doi.org/10.11145/j.biomath.2012.09.032 page 3 of 6 http://dx.doi.org/10.11145/j.biomath.2012.09.032 n. al-asuoad et al., model and simulations of a wood frog population fig. 2. larvae vs. t for 41 years iii. simulations an algorithm for the numerical solutions of the model was constructed and implemented in maple, using the numerical solver dsolve. the main issue in designing the algorithm was the need to solve the equations between the various times of impulse or transfer, and over different time intervals in each year different equations were solved. in particular, the equation for the larvae, (1), was solved in year k only in the intervals 356k < t < 365k + 75, 356k + 75 < t < 365k + 82, and 356k + 82 < t < 365k + 90, and then l(t) = 0 for 356k + 90 < t < 365(k + 1), that is until the first day of the following year. the various input data were either taken or estimated from [3], or chosen reasonably, and taken as follows. µl1 = 2.8 10 −3, µl2 = 1 10 −7, µe1 = 6 10 −4, µe2 = 2 10 −9, µm1 = 6 10 −3, µm2 = 3.33 10 −9, µl1 = 6 10 −3, µl2 = 1 10 −8, µm 1 = 3.5 10 −3, µm 2 = 1.6 10 −8; αe = 6 10 −4, αm = 5 10 −4, αl = 4 10 −4, βe = 6 10 −6, βm = 5 10 −6, βl = 4 10 −6, δe = δm = δl = 0.5, τl = 75, τm = 82, τl = 90. the figures depict a typical run of 41 years, starting with l0 = 540, 000 larvae, no juveniles, and m0 = 3000 mature frogs. the number of eggs per mature female was fig. 3. matures vs. t for 41 years fig. 4. early juveniles vs. t for 41 years σ = 600 (a characteristic of wood frogs ([3])), and a third of the mature population was females. under these initial conditions, there is a large drop in year k = 1, and then the populations grow steadily, and in longer simulations (not presented here) they level off to what seems to be steady oscillations. the yearly oscillations are of interest since they cannot be observes directly. to study the effects of a year with harsh conditions, the larvae population was set to be zero in the year k = 21. the larvae population in fig. 2 is set to zero at day 90 of each year, since all the larvae leave the pond by then. then, on the first day of the next year a batch of biomath 1 (2012), 1209032, http://dx.doi.org/10.11145/j.biomath.2012.09.032 page 4 of 6 http://dx.doi.org/10.11145/j.biomath.2012.09.032 n. al-asuoad et al., model and simulations of a wood frog population fig. 5. middle juveniles vs. t for 41 years fig. 6. late juveniles vs. t for 41 years larvae that is proportional to the mature population, (3), appears in the pond from the fertilized eggs. similarly, on days 75, 82 and 90 portions of the early, middle and late juveniles, respectively, move from the pond to join the juveniles from the previous year. these impulses cause the solutions in figs. 2, 4–6 to be discontinuous. it is seen that there is leveling or stabilization of the populations due to the density dependent mortality rates in the equations. since in the year 21 there were no larvae, the other four populations clearly had considerable drops. nevertheless, the whole population recovered over the following 10 years, reaching a similar behavior as before the drop. unfortunately, the weather conditions in michigan this year were such that the larvae in the pond were wiped out completely, and this result allows us to hope that next year the population will begin to recover. the trends in the behavior of the system seem to be similar to what was observed in [3], however the details were not observed, and a number of the coefficients were chosen ‘reasonably.’ finally, there is strong indication that periodic solutions are possible. iv. conclusion the paper presents a new compartmental model, using impulsive odes, for the dynamics of a population of wood frogs, based on the field observations of berven [3]. then, it depicts computational results for the development of the larvae, juveniles (early, middle, and late) and the mature frogs populations. under the given choice of the parameters the populations grow and stabilize in about 20 years. the introduction of environmental adverse conditions that wiped out the larvae population in year 21 show that it takes about 10 years for the population to fully recover. the next stages in this research will be to validate the model by comparing the relevant predictions to the observations in [3]. once we have confidence in the model, we plan to use it to assess the possible behavior of the population when the environmental conditions are adverse as a result of bad weather. we plan to use the model and the numerical simulations to study various possible future scenarios for the population. the well-posedness of the model, its analysis and stability will be described elsewhere. moreover, is seems from the numerical results, that the model has periodic solutions, and it is of interest to establish this mathematically. acknowledgment the authors would like to thank the school of natural resources and environment at the university of michigan for access to the research site, and the anonymous referees for some useful suggestions. references [1] l.j.s. allen, an introduction to mathematical biology, pearson prentice-hall, 2007. [2] d. bainov, p.s. simeonov, impulsive differential equations: periodic solutions and applications, crc press, 1993. [3] k.a. berven, density dependence in the terrestrial stage of wood frogs: evidence from a 21-year population study, copeia, 2009 (2), 328–338 (2009). http://dx.doi.org/10.1643/ch-08-052 biomath 1 (2012), 1209032, http://dx.doi.org/10.11145/j.biomath.2012.09.032 page 5 of 6 http://dx.doi.org/10.1643/ch-08-052 http://dx.doi.org/10.11145/j.biomath.2012.09.032 n. al-asuoad et al., model and simulations of a wood frog population [4] v. lakshmikantham, d. bainov, p.s. simeonov, theory of impulsive differential equations, world scientific, 1989. http://dx.doi.org/10.1142/0906 [5] h.w. heathcote, the mathematics of infectious disease, siam review 42 (4), 599–653 (2000). http://dx.doi.org/10.1137/s0036144500371907 [6] h.r. thieme, mathematics in population biology, princeton university press, princeton, 2003. [7] j.r. vonesh and o. de la cruz, complex life cycles and density dependence: assessing the contribution of egg mortality to amphibian declines, oecologia 2002 133, 325–333 (2002). biomath 1 (2012), 1209032, http://dx.doi.org/10.11145/j.biomath.2012.09.032 page 6 of 6 http://dx.doi.org/10.1142/0906 http://dx.doi.org/10.1137/s0036144500371907 http://dx.doi.org/10.11145/j.biomath.2012.09.032 introduction the model simulations conclusion references www.biomathforum.org/biomath/index.php/biomath review article coupling within-host and between-host infectious diseases models maia martcheva∗, necibe tuncer†, colette m st. mary‡ ∗department of mathematics, university of florida, gainesville, maia@ufl.edu †department of mathematics, florida atlantic university, boca raton, ntuncer@fau.edu ‡department of biology, university of florida, gainesville, stmary@ufl.edu received: 6 february 2015, accepted: 9 october 2015, published: 30 october 2015 this article is dedicated to tanya kostova’s anniversary birthday abstract—biological processes occur at distinct but interlinked scales of organization. yet, mathematical models are often focused on a single scale. recently, there has been a significant interest in creating and using models that link the within-host dynamics and population level dynamics of infectious diseases. these types of multi-scale models, called immuno-epidemiological models, fall in four categories, dependent on the type of the epidemiological component of the model: network or individual based models (ibm), “nested” agesince-infection structured models, ordinary differential equation (ode) models, and “size-structured” models. immuno-epidemiological multi-scale models have been used to address a variety of questions, including what is the impact of within-host dynamics on population-level quantities such as reproduction number and prevalence, as well as questions related to evolution of the pathogen or co-evolution of the pathogen and the host. here we review the literature on immuno-epidemiological modeling as well as the main insights these models have created. keywords-within-host models, between-host models, immuno-epidemiology, mathematical models, differential equations, reproduction number, evolution. ams subject classification: 92d30, 92d40 i. introduction biological processes occur at nested scales of organization. in infectious diseases, the dynamical interplay between the microparasite and the host immune system has a strong impact on the epidemiological characteristics of the disease, such as pathogen shedding, population level transmission, disease-induced host mortality and recovery. yet, traditionally, differential equation modeling of infectious diseases has been strictly separated by biological scale of organization. within-host modeling of infectious diseases has been drawing significant attention in the last century. simple differential equation models, developed to describe a number of diseases such as hiv, hcv, malaria, citation: maia martcheva, necibe tuncer, colette m st. mary, coupling within-host and between-host infectious diseases models, biomath 4 (2015), 1510091, http://dx.doi.org/10.11145/j.biomath.2015.10.091 page 1 of 12 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2015.10.091 m. martcheva et al., coupling within-host and between-host infectious diseases models flu and others, have lead to dramatically improving our understanding of how microscopic processes develop and affect the host health. several books are devoted to within-host (immunological) modeling of infectious diseases [63], [50], [40] and multiple articles develop and use such models to answer an array of biological questions regarding the pathogen and its interplay with the immune system and target cells. at the same time numerous differential equation models have been developed to model the dynamics of specific diseases or in general the distribution of pathogens on the population level. between-host (epidemiological) models have addressed a variety of questions related to public health. multiple books focus on the contributions of mathematics to epidemiology ([10], [20], [18], [33] just to mention a few). some important public health questions that can be addressed with epidemic models include the fraction of the population that needs to be vaccinated to eradicate a disease, the reproduction number of various disease outbreaks and what efficacy do control measures have. pathogen reproduction, transmission and evolution are processes that span several scales of biological organization, i.e. intracellular, withinhost, and population scales. answering effectively public health questions at the population level often requires the understanding and the “lifting” of processes from within-host levels to the population level. unfortunately, very rarely do mathematical models encompass multiple scales of biological organization. here we will review the relatively recent models linking the withinhost scale with the between-host scale. the emergent area of linked data, models and knowledge is called immuno-epidemiology. hellriegell defines immuno-epidemiology as the area that “combines individualand population-oriented approaches to create new perspectives” [30]. we define the mathematical immuno-epidemiology as the area in which mathematical dynamical models of within-host disease processes are interlinked to population-level dynamical models of disease spread to allow for novel results. the concept of immuno-epidemiology is not new but in the last half of the 20th century it was primarily linked to the interaction of immunology and epidemiology of macroparasitic diseases and malaria [27], [54], [28]. until the 21st century little had been done linking immunological and epidemiological ideas in microparasitic diseases, such as viral pathogens. why is it important to develop and study multiscale infectious disease models? (1) because immunological considerations predict important epidemiological determinants, such as disease prevalence and reproduction number [19]. such important epidemiological quantities can be explicitly related (for arbitrary parameter values) to host pathogen load and immune responses. (2) data exist on both scales; linking them is essential for quantitatively coupling processes across scales. furthermore, the biological understanding gained from more data will be more comprehensive and accurate. (3) incorporating explicit immune responses is important in diseases, such as dengue, where disease severity depends on the strength of this response [14]. (4) these models are essential for elucidating the role of within-host disease dynamics for pathogen evolution [47], [15]. in this paper we review models linking withinhost and between-host processes and some of the insights that have resulted from them. first, we detail some immunological models used in the coupled frameworks. the variability of immunoepidemiological modeling techniques stems from biomath 4 (2015), 1510091, http://dx.doi.org/10.11145/j.biomath.2015.10.091 page 2 of 12 http://dx.doi.org/10.11145/j.biomath.2015.10.091 m. martcheva et al., coupling within-host and between-host infectious diseases models the various types of epidemiological models from which they have arisen. there are four basic types of epidemiological models being employed: network or individual based epidemiological models, reviewed in subsection 3.a, ode epidemiological models, reviewed in subsection 3.b, sizestructured epidemiological models, reviewed in section 3.c, and age-since-infection structured epidemiological models, reviewed in subsection 3.d. finally, in section 4, we provide further discussion of the immuno-epidemiological modeling and its implications to biology. ii. within-host models within-host models are dynamical models that represent, in caricature, the interaction of the pathogen with the host replication machinery or immune defenses within a single host individual. roughly speaking these models can be classified into three groups: models that depict the reproduction processes of the pathogen within the host; models that depict the pathogen with the immune responses; and models that include both the replication of the pathogen and the immune responses. the simplest within-host models are of the first two types. we introduce here an example of each of the first two types as these types are the ones typically used in linked models. a. within-host model of the pathogen replication cycle within-host models that represent the pathogen replication cycle assume a typical viral pathogen that replicates using the machinery of host cells, called target cells. to introduce such a model, let x(τ) be the number of pathogen-free (healthy) target cells (in the blood) and y(τ) be the number of infected target cells. the amount of pathogen is denoted by p(τ). the within-host replication fig. 1. the dynamics of the pathogen and target cells. parameter values are: r = 50000000, b = 0.000000000015, µ = 0.01, d = 0.5, δ = 3, ν = 250, s = 0.00008. model has been used for many viral diseases before [50]. x′ = r − bpx−µx, y′ = bpx−dy, p ′ = νdy − (δ + s)p, (ii.1) where r is the replication rate of target cells, b is the infection rate, µ is the clearance rate of healthy cells, d is the clearance rate of infected cells, ν is the number of pathogen particles released from lysis of an infected cell, δ is the clearance rate of pathogen, and s is the shedding rate. the dynamics of the pathogen and target cells is shown in figure 1. model (ii.1) has been completely analyzed [17]. the reproduction number of the pathogen is given by <0 = rνb µ(δ + s) . (ii.2) the model has two equilibria, an infection-free equilibrium e0 = (r/µ, 0, 0) and an infection equilibrium e∗ = (x∗,y∗,p∗) where x∗ = δ + s νb , y∗ = µ(δ + s) νbd (<0 − 1) , p∗ = µ b (<0 − 1) . (ii.3) biomath 4 (2015), 1510091, http://dx.doi.org/10.11145/j.biomath.2015.10.091 page 3 of 12 http://dx.doi.org/10.11145/j.biomath.2015.10.091 m. martcheva et al., coupling within-host and between-host infectious diseases models b. models with immune response the adaptive immune response includes a cellular component which includes various types of t cells and humoral response which consists of b cells and antibodies. simple pathogen-immune response models typically include only the pathogen p(τ) and one type immune response cells, bcells, b(τ). the pathogen replicates according to the malthus model or logistic model. b cells kill the pathogen and b cell production is stimulated by the pathogen. b cells are cleared at rate d: p ′ = rp ( 1 − p k ) − �pb, b′ = ap −db, (ii.4) where r is the parasite growth rate, k is its carrying capacity, � is the killing rate of the immune response, a is the activation rate of the immune response and d is the clearance of the immune response. the dynamics of model (ii.4) is illustrated in figure 2. fig. 2. the dynamics of the pathogen and the immune response. parameter values are r = 1, k = 1000, � = 0.1, a = 0.2. the model has two equilibria: (0, 0) which is always unstable and a coexistence equilibrium (p∗,b∗) where p∗ = rdk �ak + rd , b∗ = rak �ak + rd . (ii.5) immunological models vary immensely in complexity and detail. differences also stem from the particular processes in the disease being modeled. for our purposes here these two simple within-host models are sufficient to illustrate the concepts. in the next section, we introduce various epidemiological models. iii. immuno-epidemiological models the epidemiological component of the immunoepidemiological models can take several different forms based on existing modeling frameworks, or just epidemiologically relevant quantities, such as the reproduction number, expressed in terms of the immunological variables. in this section we consider four types of immuno-epidemiological models structured by the type of epidemic model. a. network epidemic models in network epidemic models, individuals are nodes in a network. each individual or node can exhibit its own within-host dynamics (see figure 3). the models can show the impact of the individual immune dynamics on populationlevel transmission of disease. tucknell [57] and kostova [34] introduced some of the first immunoepidemiological models where the epidemiological component is a network. kostova linked a network of n within-host models and showed that even if the immune response clears the infection in each individual when isolated, while these individuals are in a network, the pathogen persists in each one of them and on a “population level”. vickers and osgood [58] suggest that increased variance among people’s ability to respond to an infection, while maintaining the average immune responsiveness, may worsen the overall impact of an outbreak within a population. furthermore, high values for the network connectivity reduced biomath 4 (2015), 1510091, http://dx.doi.org/10.11145/j.biomath.2015.10.091 page 4 of 12 http://dx.doi.org/10.11145/j.biomath.2015.10.091 m. martcheva et al., coupling within-host and between-host infectious diseases models susceptible individuals infected individuals immune response immuno-epidemiological network model 1 fig. 3. network immuno-epidemiological schematic diagram. the timing between peak viral levels in neighboring individuals. a network based immunoepidemiological model was applied by vickers and osgood [59] to study treatment of chlamydia which suggested that treatment applied up to the third day post infection has significant chance of preventing transmission of the disease to the nearest neighbor. delivering treatment past the 3rd day post infection allows for infection of nearest neighbor as well as reinfection. lukens et al. [37] use a simple ode model of influenza a and link that model, through infectivity, to large-scale agent-based population-level model to study influenza a epidemics. the authors obtain a map of the parameters of the immune model that characterizes clinical phenotypes of influenza infection and immune response variability across the population. at the populationlevel, effectively the authors simulate epidemics in allegheny county, pennsylvania and consider both age-specific and age-independent severity assumptions. one of the serious drawbacks of network and agent-based immuno-epidemiological models is that very few population level quantities that describe the disease distribution can be computed analytically in closed form. in particular, in network models there are difficulties for computing the reproduction number and the prevalence of the disease. consequently, little can be learned of the effect of the immune response on these quantities outside of extensive simulations [52]. the next three classes of immunoepidemiological models remedy this shortcoming but some of them assume that all infected individuals undergo the same within-host dynamics, an assumption that is largely unrealistic. still these models have contributed immensely to our further understanding of the mutual impact of within-host and between-host processes. b. ode immuno-epidemiological models one way to obtain a simpler ode immunoepidemiological model for chronic diseases is to consider the immune model at infectious equilibrium. in this case one can make the parameters of a simple ode epidemic model dependent on the equilibria values of the pathogen and/or the immune response. for instance, a simple si model becomes: s′ = λ −β(p∗)si −m0s , i′ = β(p∗)si − (m0 + m1(p∗,b∗))i , (iii.1) where p∗ and b∗ are given by (ii.5) and m0 is the natural death rate, m1 is the disease-induced death rate, β is the transmission rate, and λ is the recruitment rate. one can then investigate how within-host parameters, pathogen load and immune response affect the epidemiological quantities, such as disease-induced mortality, prevalence and reproduction number. these conclusions are not necessarily equivalent to conclusions obtained biomath 4 (2015), 1510091, http://dx.doi.org/10.11145/j.biomath.2015.10.091 page 5 of 12 http://dx.doi.org/10.11145/j.biomath.2015.10.091 m. martcheva et al., coupling within-host and between-host infectious diseases models fig. 4. immuno-epidemiological modeling with environmental transmission. from other types of immuno-epidemiological models such as the nested models considered below. another opportunity to connect the withinhost and between host dynamical systems in an ode model emerges in environmentally transmitted disease. in this case a chronological-timestructured within-host ode system is linked to a chronological-time-structured ode epidemiological system through the pathogen load in the environment (see figure 4). using this novel modeling scenario feng at al [12], [21], [22] investigate the transmission of toxoplasma gondii. feng at al [21] link a dynamic within-host model of the type (ii.1), where the infection of target cells depends on population-level prevalence i, with an si epidemic model much like (iii.1), where transmission depends on viral load. within-host and between-host reproduction numbers are computed but for the linked model analysis suggests that long-term behavior of the infection may depend on the initial conditions. articles [12], [22] contain the environment as an explicit variable and find that infection may persist on the population level even if the isolated between-host reproduction number is less than one; a result that is facilitated by the within-host dynamics. c. size-structured pde immuno-epidemiological models size-structured pde immuno-epidemiological models are perhaps the most complex type of immuno-epidemiological models. in this case the epidemiological model consists of physiologically structured pdes in which the structural independent variables are the dynamical variables of the ode immune model. the first such model was proposed by [45] (see also [44]) where the “size structure” variable is the immune response. analysis of this model revealed similarities to age-sinceinfection structured models particularly because the structural variable is strictly increasing in time. this model was further extended by [23] to model where the population-level density of infected individuals is structured by both the viral load and the immune response. [23] also transforms the sizestructured model to an age-since-infection model where the independent variables are age-sinceinfection and initial pathogen inoculum. considering a specific within-host model that allows for both pathogen extinction and unbounded growth, the authors investigate the population level impact of the initial inoculum and of the isolation threshold. a somewhat different modeling approach to the same modeling scenario is given by [6], [7], [48]. the authors suggest coupling a classical hiv/hcv within-host model, given by equations (ii.1) with a size-structured sir epidemic model. the authors biomath 4 (2015), 1510091, http://dx.doi.org/10.11145/j.biomath.2015.10.091 page 6 of 12 http://dx.doi.org/10.11145/j.biomath.2015.10.091 m. martcheva et al., coupling within-host and between-host infectious diseases models establish well-posedness in the special case when the density of infected individuals is structured only by the viral load. furthermore, they develop numerical methods and discuss the impact of the (fixed) number of target cells and the burst size on the epidemic [6], [7]. the results allow them to determine the distribution of the density of the infected individuals by their viral load. in general, the size-structured approach presents interesting mathematical challenges, such as the potential for measure-valued solutions, modeling issues related to creating and linking the models, computational issues with the large number of independent variables [6], [7], [48]. because of these issues, creating the within-host model is an important step. furthermore, the computational and analytical problems with the large number of independent variables somewhat restrict the incorporation of significant realism into the immune system; a problem that can be addressed by incorporating fewer within-host model dependent variables as independent variables in the epidemiological model. there are still a lot of interesting and open questions, related to this approach. d. nested models nested models are a relatively recent class of models suggested for the first time by gilchrist and sasaki [25]. the main advantage of nested models relative to size-structured models is that these models allow for the use of very realistic and specific to a given disease models. establishing well-posedness for these models is generally not very problematic [51]. furthermore, since the number of independent variables in the pde part of the model is restricted to two, there are few difficulties with computation, independent of the complexity of the immunological or epidemiological components. because of their advantages, since the gilchrist and sasaki paper, nested models have acquired significant popularity. the nested models “nest” a time-post-infection structured immune dynamics model into a time-post-infection and chronological time structured epidemiological model. nested models also link the within-host model with the epidemiological model through the parameters of the epidemiological models that are expressed in terms of within-host dependent variables. to introduce a simple nested immuno-epidemiological model, let s(t) denote the number of susceptible humans and i(τ,t) be the density of infected humans. in the simplest case, we can use an si epidemiological model. the model takes the form: s′ = λ −s ∫ ∞ 0 β(τ)i(τ,t)dτ −m0s, iτ + it = −(m0 + m1(τ))i(τ,t), i(0, t) = s ∫ ∞ 0 β(τ)i(τ,t)dτ , (iii.2) where m0 is the natural death rate, m1 is the disease-induced death rate, λ is the recruitment rate and β is the transmission rate. model (iii.2) can be linked with either of the within-host models. the transmission rate β(τ) depends on the pathogen load. experimental evidence suggests that the transmission probability does not increase linearly with the pathogen load but in a saturating fashion [35]. we will use the following simple function of saturating growth: β(τ) = cp(τ) q + p(τ) where c is the contact rate and q is the halfsaturating constant. the disease-induced mortality can be linked in multiple ways to the immune system. it is thought that two distinct processes lead to disease-induced mortality in the host. on one side is the pathogen itself, and on the other is biomath 4 (2015), 1510091, http://dx.doi.org/10.11145/j.biomath.2015.10.091 page 7 of 12 http://dx.doi.org/10.11145/j.biomath.2015.10.091 m. martcheva et al., coupling within-host and between-host infectious diseases models the immune response. we take the disease-induced mortality generated by the pathogen proportional to the pathogen load. the disease-induced mortality generated by the immune response has been taken to be proportional to the growth of the immune response abp [25], [53]. we take here the disease-induced mortality proportional to b2. the square guarantees that at low values of b, the immune response almost has no impact on the disease-induced mortality while at high levels of b, it has significant impact. this way a tradeoff exists between the necessity of the immune response to be vigorous enough to clear the virus, but not too vigorous to kill the host. the diseaseinduced mortality is then given by m1(τ) = ν1rp(τ) + ξ1b 2 where ν and ξ are constants of proportionality. ξ1 = 0 if we are working with immune model (ii.1). one of the main disadvantages of the nested immuno-epidemiological models relative to the network and size-structured models is that they assume that all individuals exhibit the same immune dynamics. to remedy this disadvantage one may consider a multi-group immuno-epidemiological model where the different groups exhibit different immune dynamics. the multi-group model is somewhat complicated and obtaining analytical results on it is not easy. this weakens one of the great advantages of the nested models, namely that basic epidemiological quantities, such as the reproduction number and the prevalence, can be computed in analytical form. a schematic diagram of nested models is given in figure 5. the reproduction number of the immunoepidemiological model (iii.2) depends on the between host model susceptible population s(t) infected population i(t, ⌧) pathogen level in host (p (⌧)) antibody level in host (b(⌧)) within host model natural deaths births natural deaths + disease induced deaths transmission 1 fig. 5. nested immuno-epidemiological schematic diagram. within-host variables and parameters: r0 = λ m0 ∫ ∞ 0 β(τ)e−m0τe− ∫ τ 0 m1(σ)dσdτ. (iii.3) there are two types of questions being addressed with nested models: how does the withinhost pathogen dynamics affect the population-level reproduction number and prevalence? what are the evolutionary and co-evolutionary consequences of the pathogen and host within-host evolution? several articles have suggested that the dependence of the reproduction number r0 on the pathogen reproduction rate r may be nonmonotone [31], [53], [43]. as the within-host pathogen reproduction rate r grows, it should be increasing the population-level reproduction number r0; however the increased pathogen load is increasing host mortality, which in turn leads to decrease in the population-level reproduction number. this creates a hump-shaped form of r0 as a function of r, which is well-described in the literature. the dependence of r0 on immunological parameters has been further discussed in [32], [41], [34], [16], [8]. the dependence of prevalence on some of the within-host parameters may also be counter-intuitive. for instance, prevalence may decrease with increase of b in model (ii.1) [42]. this in turn implies that within-host medications biomath 4 (2015), 1510091, http://dx.doi.org/10.11145/j.biomath.2015.10.091 page 8 of 12 http://dx.doi.org/10.11145/j.biomath.2015.10.091 m. martcheva et al., coupling within-host and between-host infectious diseases models that lower the infection of target cells, that is decrease b, de facto increase the population-level prevalence of the disease. this paradox has been observed in practice in hiv, where medications lower within-host virus load and increase survivability of infected individuals which leads to increasing prevalence. furthermore, amplification of the hiv epidemic has been observed through nested models [55]. more mathematical questions related to immuno-epidemiological models, such as well-posedness and optimal control are addressed in [51]. the importance of multi-scale immunoepidemiological modeling is best highlighted by its role in studying evolution [47]. gilchrist and sasaki were among the first to address co-evolution [25] using multi-scale approaches but since then evolution of virulence has been attracting significant attention. because evolution involves multiple interacting strains, a number of approaches have been developed to handle the emergent complexity [3], [4], [5], [13], [15], [24], [26], [36], [38]. one possible way is to model the strains explicitly on within-host and/or between-host scales [42], [13]. in the absence of trade-off mechanisms, the strain that maximizes its between-host epidemiological reproduction number dominates; a result first established rigorously mathematically in [11]. in the presence of trade-offs there is coexistence between the strains and invasibility is governed by population-level invasion numbers. nested models further reveal [2] that the optimal virulence in a co-infection model increases with multiple infections and that in a linked within-host and between-host co-infection model, an evolutionary stable strategy (ess) can turn into a branching point [1]. an ess is a strategy which, if adopted fig. 6. pairwise invasibility plot. in the population, cannot be invaded by any other strategy. to see this from a pairwise invasibility plot (pip) (see figure 6), we draw a vertical line through the singular strategy and confirm that the vertical line lies entirely in the non-invasibility region. nested models have the potential to link in a natural way the virus reproduction rate, population-level fitness, and population level disease-induced mortality. for hiv, the reproduction rates of the virus increase at a moderate rate and the virulence is slightly higher than the level that maximizes the population-level transmissibility of the virus [39]. for hepatitis c, slowly replicating strains have a higher fitness and produce more population-level secondary infections while strains with higher replication rates dominate within a host [38]. for influenza a, the relative importance of virulence and viral clearance by the immune system on the viral fitness and persistence was found to depend on the temperature [29]. day et al [15] develop the mathematical theory that bridges the nested immuno-epidemiological models to quantitative genetics and evolution of biomath 4 (2015), 1510091, http://dx.doi.org/10.11145/j.biomath.2015.10.091 page 9 of 12 http://dx.doi.org/10.11145/j.biomath.2015.10.091 m. martcheva et al., coupling within-host and between-host infectious diseases models traits. using this framework, the authors show that the trade-off between transmission and virulence, studied early on in multi-scale models in [24], is an interplay of the genetic variation of the pathogen and the population-level dynamics of the disease. this framework is the backbone for future research at the interface of dynamic population modeling and quantitative genetic modeling. conclusions from multi-scale nested models for chronic diseases can be derived by writing the epidemiological quantities in term of the infected equilibrium of the within-host model [9], [26], [46]. this approach bridges to the ode immuno-epidemiological models, discussed in section iii.b. following this approach article [26] found that within-host selection favors viral production rates ν that maximize virulence but between-host pathogen fitness is maximized at some intermediate virulence and viral production rate. article [9] extends the results in [26] by incorporating superinfection. iv. discussion these new, more integrative modeling approaches, each have strengths and weaknesses mathematically and in terms of their levels of biological realism, but they have on the whole led to new and biologically counter intuitive insights. this is exemplified by even early examples of these models, such as kostova [34], which demonstrated population-level disease persistence even when individual immune responses are able to clear infection and result in immunity. a second example is the population level persistence of virus, even when the between host reproduction number is less than 1, as shown by feng et al.[22]. this result leads to the surprising interpretation that immune responses do not necessarily tip the balance of interactions in favor of the host and decrease disease prevalence. similarly, drug administration may mimic the host immune system and increase disease prevalence (e.g., references [42] and [54]). at another level, when we focus on the evolution of virulence, immuno-epidemiological models again lead to counter-intuitive results (e.g., [26]). the results of these analyses, that selection within an individual can favor different pathogen traits than selection among individuals, highlight that the within-host/among-host model structure characteristic of these models meets the requirements for trait-group selection to play a role in evolutionary dynamics [60], where virus in individuals constitutes trait groups from which virus emerge, intermix and infect susceptible individuals. wilson’s model has been criticized for unrealistic, or at least uncommon, assumptions of population structure (reviewed in [61]), however, infectious disease may represent a common context in which individualand group-level selection both act strongly and at times in conflict. ultimately, understanding the evolution of virulence might be informed by models designed to understand trait evolution in the context of multilevel selection such as those discussed in [56]. in general, because there are sometimes conflicts between within-host and among-host virulence optima, new insights will likely come from relaxing the biologically-unrealistic, albeit simplifying, assumption that virulence evolves to its within-host optimum prior to the epidemiological dynamics of the model. again, modeling approaches appropriate for relaxing this assumption might be informed by the body of literature focused in the modeling of group and individual level selection, especially those approaches that consider the continuous nature of the dynamics of within-individual and biomath 4 (2015), 1510091, http://dx.doi.org/10.11145/j.biomath.2015.10.091 page 10 of 12 http://dx.doi.org/10.11145/j.biomath.2015.10.091 m. martcheva et al., coupling within-host and between-host infectious diseases models among-individual processes [56]. acknowledgments the authors acknowledge support from nsf grants dms-1515661/dms-1515442. references [1] s. alizon, co-infection and super-infection models in evolutionary epidemiology, interface focus (in press). 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[63] d. wodarz, killer cell dynamics, springer, new york, 2007. biomath 4 (2015), 1510091, http://dx.doi.org/10.11145/j.biomath.2015.10.091 page 12 of 12 http://dx.doi.org/10.11145/j.biomath.2015.10.091 introduction within-host models within-host model of the pathogen replication cycle models with immune response immuno-epidemiological models network epidemic models ode immuno-epidemiological models size-structured pde immuno-epidemiological models nested models discussion references www.biomathforum.org/biomath/index.php/biomath original article modeling the dynamics of arboviral diseases with vaccination perspective hamadjam abboubakar∗, jean c. kamgang†, léontine n. nkamba‡, daniel tieudjo† and lucas emini¶ ∗department of computer science, uit–university of ngaoundéré, cameroon abboubakarhamadjam@yahoo.fr †department of mathematics and computer science, ensai–university of ngaoundéré, cameroon jckamgang@yahoo.fr, tieudjo@yahoo.com ‡department of mathematics, ens–university of yaoundé i, lnkague@gmail.com ¶department of mathematics, polytechnic–st. jerome catholic university, cameroon lemini@univ-catho-sjd.com received: 31 december 2014, accepted: 24 july 2015, published: 27 august 2015 abstract—in this paper, we propose a model of transmission of arboviruses, which takes into account a future vaccination strategy in human population. a qualitative analysis based on stability and bifurcation theory reveals that the phenomenon of backward bifurcation may occur; the stable disease-free equilibrium of the model coexists with a stable endemic equilibrium when the associated reproduction number, r0, is less than unity. we show that the backward bifurcation phenomenon is caused by the arbovirus induced mortality. using the direct lyapunov method, we prove the global stability of the trivial equilibrium. through a global sensitivity analysis, we determine the relative importance of model parameters for disease transmission. simulation results using a nonstandard qualitatively stable numerical scheme are provided to illustrate the impact of vaccination strategy in human communities. keywords-mathematical model; arboviral disease; vaccination; stability; backward bifurcation; sensitivity analysis; nonstandard numerical scheme. i. introduction arboviral diseases are affections transmitted by hematophagous arthropods. there are currently 534 viruses registered in the international catalogue of arboviruses and 25% of them have caused documented illness in humans [20], [49], [42]. examples of these kinds of diseases are dengue, yellow fever, saint louis fever, encephalitis, west nile fever and chikungunya. a wide range of arbovirus diseases are transmitted by mosquito bites and constitute a public health emergency of international concern. according to who, dengue, caused by any of four closely-related virus serotypes (den-1-4) of the genus flavivirus, causes 50–100 million infections worldwide every year, and the majority of patients worldwide are children aged 9 to 16 years [72], [84], [86]. the dynamics of arboviral diseases like dengue or chikungunya are influenced by many factors such as humans, the mosquito vector, the virus itself, as well as the environment which affects all the present mechanisms of control directly citation: hamadjam abboubakar, jean c. kamgang, léontine n. nkamba, daniel tieudjo, lucas emini, modeling the dynamics of arboviral diseases with vaccination perspective, biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 1 of 30 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... or indirectly. for all mentioned diseases, only yellow fever has a licensed vaccine. however, some works are underway for development of a vaccine for dengue [10], [11], [33], [50], [73], [85], japanese encephalitis [73], and chikungunya [53], [54], [55], [46]. moreover, the existence of different strains of dengue virus, for example, makes the developpement of an efficient vaccine a challenge for scientists. however, according to the french newspaper le figaro, the sanofi laboratory hopes to market in the second half of 2015, the first vaccine against dengue fever, with an overall efficacy of 61% vaccine among young people aged 9 to 16 years and the rate of protection against severe dengue 95.5% [39]. therefore, it is important to assess the potential impact of such vaccines in human communities. as part of the necessary multi–disciplinary research approach, mathematical models have been extensively used to provide a framework for understanding arboviral diseases transmission and control strategies of the infection spread in the host population. in the literature, considerable works can be found on the mathematical modeling of specific arboviral diseases, like west nile fever, yellow fever, dengue and chikungunya, see e.g. [2], [17], [24], [30], [35], [36], [38], [40], [56], [60], [61], [64], [68], [79]. although these models highlight the means to fight against these arbovirus, few papers deal with models that consider vaccination [40], [68], [79]. in this paper, we formulate a model, described by differential equations, in which we include two aspects: vaccination in the human population and the aquatic stage in the vectors population.we perform a qualitative analysis of the model, based on stability and bifurcation theory. in particular, we use an approach based on the center manifold theory [19], [31], [43] to evaluate the occurrence of a transcritical backward bifurcation and, as a consequence, the presence of multiple endemic equilibria when the basic reproduction number r0 is less than unity. under the point of view of disease control, the occurrence of backward bifurcation has relevant implications for disease control because the classical threshold condition r0 < 1, is no longer sufficient to ensure the elimination of the disease from the population. the global stability of the trivial equilibrium and the disease–free equilibrium (the equilibrium without disease in both populations), whenever the associated thresholds (the net reproductive number n and the basic reproduction number r0) are less than unity, is derived through the use of lyapunov stability theory and lasalle’s invariant set theorem, and the approach of kamgang and sallet [48], respectively. through global sensitivity analysis, we determine the relative importance of model parameters for disease transmission. the analysis of the model is completed by the construction of a nonstandard numerical scheme which is qualitatively stable. the rest of this paper is organized as follows. in section ii, we develop the mathematical model and give the invariant set in which the model is defined. in section iii, we compute two thresholds: the net reproductive number n and the basic reproduction number r0. depending of the values of these thresholds, we identify two disease–free equilibria: the trivial equilibrium which corresponds to the extinction of vectors, when n ≤ 1, and the disease-free equilibrium (dfe) when n > 1 and r0 < 1. the results concerning the local and global stability of these two equilibria are also given. the section iv is devoted to the existence of endemic equilibria and bifurcation analysis. vaccine impact is evaluated in section v. uncertainty and sensitivity analysis and the construction of a stable numerical scheme, are made in sections vi and vii respectively. a conclusion completes the paper. ii. model formulation, invariant region. in this section we describe the mathematical model that we shall study in this paper. the formulation is mostly inspired, with some exceptions, by the models proposed in [30], [40], [68], [80]. we assume that the human and vector populations biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 2 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... are divided into compartments described by time– dependent state variables. this said, the compartments in which the populations are divided are the following ones: –for humans, we consider susceptible (denoted by sh), vaccinated (vh), exposed (eh), infectious (ih) and resistant or immune (rh). humans susceptible population is recruited at a constant rates λh. each human subpopulation comes out from the dynamics at natural mortality rates µh. the human susceptible population decreased following infection, which can be acquired via effective contact with an exposed or infectious vector at a rate λh (the incidence rate of infection for humans), given by λh = aβ̃hv nv nh + m (ηvev + iv) nv = βhv (ηvev + iv) nh + m , (1) where m denote the alternatively sources of blood [1], [80], a is the biting rate per susceptible vector, β̃hv denotes the probability of transmission of infection from an infectious vector (ev or iv) to a susceptible human (sh or vh). we obtain the expression of λh as follows: the probability that a vector chooses a particular human or other source of blood to bite can be assumed as 1 nh + m . thus, a human receives in average a nv nh + m bites per unit of times. then, the infection rate per susceptible human is given aβ̃hv nv nh + m (ηvev + iv) nv . in expression (1), the modification parameter 0 < ηv < 1 accounts for the assumed reduction in transmissibility of exposed mosquitoes relative to infectious mosquitoes. it is worth emphasizing that, unlike many of the published modelling studies on dengue transmission dynamics, we assume in this study that exposed vectors can transmit dengue disease to humans. this is in line with some studies (see, for instance [34], [40], [87], [90]). however, it is significant to note that, in the case of chikungunya for example, the exposed vectors do not play any role in the infectious process, in this case ηv = 0. the vaccinated population is generated by vaccination of susceptible humans at a rate ξ. further, it is expected that any future dengue vaccine would be imperfect [40], [68]. thus, vaccinated individuals acquire infection at a rate (1 − �)λh where � is the vaccine efficacy. exposed humans develop clinical symptoms of disease, and move to the infectious class at rate γh. infectious humans may die as consequence of the infection, at a disease–induced death rate δ, or recover at a rate σ. it is assumed that individuals successfully acquire lifelong immunity against the virus. –vectors population is classified into four compartments: susceptible (sv), exposed (ev), infectious (iv) and aquatic (av). the aquatic state includes the eggs, larvae, and pupae. the vector population does not have an immune class, since it is assumed that their infectious period ends with their death. the population of vectors consists essentially of females which reached adulthood. a susceptible vector is generated by the adulthood females at rate θ. the susceptible vector population decreased following infection, which can be acquired via effective contact with an exposed or infectious human at a rate λv (the incidence rate of infection for vectors), given by λv = aβ̃vh (ηheh + ih) nh nh nh + m = βvh (ηheh + ih) nh + m (2) where β̃vh is the probability of transmission of infection from an infectious human (eh or ih) to a susceptible vector (sv), where the modification parameter 0 ≤ ηh < 1 accounts for the relative infectiousness of exposed humans in relation to infectious humans. here too, it is assumed that susceptible mosquitoes can acquire infection from exposed humans [23], [40]. exposed vectors move to the infectious class with the rate γv. as in the case of the outbreak of chikungunya on réunion island, it has been shown that lifespan of the infected mosquitoes is almost halved. this particular feature of infected mosquitoes therefore influences the dynamics of the disease [32], [30]. thus, following dumont and coworkers [29], [30], we assume in this work that the lifespan of a vector depends on its epidemiological status. so the average lifespan for susceptible mosquitoes is biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 3 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... table i the state variables of model (3). humans vectors sh: susceptible av aquatic vh: vaccined sv: susceptible eh: infected ev: exposed ih: infectious iv: infectious rh: resistant (immune) 1/µv days, the average lifespan of the exposed mosquitoes is 1/µ1 days and the average adult lifespan for infected vector is 1/µ2. thus, we have 1/µ2 ≤ 1/µ1 ≤ 1/µv (with equality for other arboviral diseases). we do not consider vertical transmission in this work, so only susceptible humans are recruited, and vectors are assumed to be born susceptible. we are now in position to write the model (the meaning of the state variables and parameters are summarized in table i and table ii:  ṡh = λh −λhsh − ξsh −µhsh v̇h = ξsh − (1 − �)λhvh −µhvh ėh = λh [sh + (1 − �)vh] − (µh + γh)eh i̇h = γheh − (µh + δ + σ)ih ṙh = σih −µhrh ȧv = µb ( 1 − av k ) (sv + ev + iv) − (θ + µa)av ṡv = θav −λvsv −µvsv ėv = λvsv − (µ1 + γv)ev i̇v = γvev −µ2iv (3) in model (3) the upper dot denotes the time derivative. k denote the carrying capacity of breeding sites. the parameter k is highly dependent on some factors such as (the location, temperature, season). in this work, we follow dumont and chiroleu [30], and consider, in our sensitive analysis, that k depend of the location, thus k = χnh, where χ is a positive integer which represent the location and nh the human population size. for example, in the year 2005 at saint-denis and saint-pierre in réunion island, χ = 2) [30]. µb represent the number of eggs at each deposit per capita and µa is the natural mortality of larvae. table ii description of parameters of model (3). parameter description λh recruitment rate of humans ξ vaccine coverage � the vaccine efficacy ηh, ηv modification parameters β̃hv probability of transmission of infection from an infectious vector to a susceptible human β̃vh probability of transmission of infection from an infectious humans to a susceptible vector γh progression rate from eh to ih γv progression rate from ev to iv µh natural mortality rate in humans µv natural mortality rate of susceptible vectors µa natural mortality of larvae µ−11 average lifespan of exposed mosquitoes µ−12 average lifespan of infected mosquitoes θ maturation rate from larvae to adult δ disease–induced death rate in humans σ recovery rate for humans a average number of bites m number of alternative source of blood k capacity of breeding sites µb number of eggs at each deposit per capita we set π = 1 − �, k1 = µh + ξ, k2 = µh + γh, k3 = µh + δ + σ, k4 = µ1 + γv, k6 = µa + θ. let nh the total human population and nv the total of adult vectors, i.e. nh = sh + vh + eh + ih + rh and nv = sv + ev + iv. system (3) can be rewritten in the following way dx dt = a(x)x + f (4) with x = (sh,vh,eh,ih,rh,av,sv,ev,iv) t , a(x) = (aij)1≤i,j≤9 were a1,1 = −(λh + k1), a2,1 = ξ, a2,2 = −(πλh + µh), a3,1 = λh, a3,2 = πλh, a3,3 = −k2, a4,3 = γh, a4,4 = −k3, a5,4 = σ, a5,5 = −µh, a6,7 = a6,8 = a6,9 = µb, a7,6 = θ, a7,7 = −(λv + µv), a8,7 = λv, a8,8 = −k4, a9,8 = γv, a9,9 = −µ2, a6,6 = − ( k6 + µb sv + ev + iv k ) and the other entries of a(x) are equal to zero; and f = (λh, 0, 0, 0, 0, 0, 0, 0, 0) t . note that a(x) is a metzler matrix, i.e. a matrix such that off diagonal terms are non negative [8], [47], for all x ∈ r9+. thus, using the fact that f ≥ 0, system (4) is positively invariant in r9+, biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 4 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... which means that any trajectory of the system starting from an initial state in the positive orthant r9+, remains forever in r 9 +. the right-hand side is lipschitz continuous: there exists an unique maximal solution. on the other hand, from the first four equations of model system (3), it follows that ṅh(t) = λh −µhnh −δih ≤ λh −µhnh. (5) so that 0 ≤ nh(t) ≤ λh µh + ( nh(0) − λh µh ) e−µht. (6) thus, at t −→∞, 0 ≤ nh(t) ≤ λh µh . from the last three equations of model system (3), it also follows that ṅv(t) = θav −µvsv −µ1ev −µ2iv ≤ θav −µvnv. (7) so that 0 ≤ nv(t) ≤ θav µv + ( nv(0) − θav µv ) e−µvt. (8) thus, at t −→ ∞, 0 ≤ nv(t) ≤ θk µv since av ≤ k. therefore, all feasible solutions of model system (3) enter the region: d = { (sh,vh,eh,ih,rh,av,sv,ev,iv) ∈ r9 : sh + vh + eh + ih + rh ≤ λh µh ; av ≤ k; sv + ev + iv ≤ θk/µv} . iii. the disease–free equilibria and stability analysis now let n the net reproductive number [25] given by n = µbθ µv(θ + µa) . (9) we prove the following result proposition 3.1: a) if n ≤ 1, then, system (3) has only one trivial disease–free equilibrium te := p0 = ( λh k1 , ξλh µhk1 , 0, 0, 0, 0, 0, 0, 0 ) . b) if n > 1, then, system (3) has a disease–free equilibrium p1 = ( s0h,v 0 h , 0, 0, 0,a 0 v,s 0 v, 0, 0 ) , with s0h = λh k1 , v 0h = ξλh k1µh , a0v = k ( 1 − 1 n ) , s0v = θ µv k ( 1 − 1 n ) . proof: see appendix a. in proposition 3.1, we have shown that system (3) have at least two equilibria depending of the value of treshold n and the basic reproduction number r0. precisely, we have proved that when n ≤ 1, model sytem (3) admits only one equilibrium called trivial equilibrium (te := p0). when n > 1, model sytem (3) admits additionally the disease–free equilibrium (dfe := p1). we prove, in the following, that the trivial equilibrium is locally and globally asymptotically stable whenever n ≤ 1. then, we prove that the trivial equilibrium is unstable when n > 1, and the disease–free equilibrium is locally asymptotically stable whenever r0 < 1. using kamgang and sallet approach [48], a necessary condition for the global stabilty of the disease–free equilibrium is also given. a. local stability analysis the local stability of the trivial equilibrium and the disease–free equilibrium is given in the following result: proposition 3.2: a) if n ≤ 1, then the trivial equilibrium te is locally asymptotically stable. b) if n > 1, then the trivial equilibrium is unstable and the disease free equilibrium p1 is locally asymptotically stable if r0 < 1 and unstable if r0 > 1, where r0 is the basic reproduction number [26], [82], given by r20 = βhvβvhkθ(πξ + µh)(k3ηh + γh) (µh + ξ)(µh + γh)(µh + δ + σ) × λhµh(µ2ηv + γv) µvµ2(λh + mµh) 2(µ1 + γv) ( 1 − 1 n ) . (10) proof: see appendix b. biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 5 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... b. global stability analysis 1) global asymptotic stability of the trivial equilibrium te := p0: proposition 3.3: if n ≤ 1, then, te := p0 is globally asymptotically stable on d. proof: see appendix c. 2) global asymptotic stability of the disease– free equilibrium : following [30], we prove that the disease–free equilibrium dfe := p1 is globally asymptotically stable under a certain threshold condition. to this aim, we use a result obtained by kamgang and sallet [48], which is an extension of some results given in [82]. using the property of dfe, it is possible to rewrite (3) in the following manner{ ẋs = a1(x)(xs −xdfe) + a12(x)xi ẋi = a2(x)xi (11) where xs is the vector representing the state of different compartments of non transmitting individuals (sh, vh, rh, av, sv) and the vector xi represents the state of compartments of different transmitting individuals (eh, ih, ev, iv). here, we have xs = (sh,vh,rh,av,sv)t , xi = (eh,ih,ev,iv) t , x = (xs,xi) and xdfe = ( s0h,v 0 h , 0, 0, 0,a 0 v,s 0 v, 0, 0 )t , with a1(x) =   −k1 0 0 0 0 ξ −µh 0 0 0 0 0 −µh 0 0 0 0 0 −k6 k7 0 0 0 θ −µv   , a12(x) =   0 0 −a13 −a14 0 0 0 −a23 −a24 0 0 σ 0 0 0 0 0 κ κ 0 −a41 −a42 0 0 0   , a2(x) =   −k2 0 b13 b14 γh −k3 0 0 b31 b32 −k4 0 0 0 γv −µ2   , with k6 = ( k6 + µb s0v k ) , k7 = µb ( 1 − av k ) , a13 = βhvηvsh nh + m , a14 = βhvsh nh + m , a23 = πβhvηvvh nh + m , a24 = πβhvvh nh + m , a41 = βvhηhsv nh + m , a42 = βvhsv nh + m , b13 = βhvηvh nh + m , b14 = βhvh nh + m , b31 = βvhηhsv nh + m , b32 = βvhsv nh + m , κ = µb ( 1 − av k ) and h = (sh + πvh). a direct computation shows that the eigenvalues of a1(x) are real and negative. thus the system ẋs = a1(x)(xs−xdfe) is globally asymptotically stable at xdfe. note also that a2(x) is a metzler matrix, i.e. a matrix such that off diagonal terms are non negative [8], [47]. following d, we now consider the bounded set g: g = { (sh,vh,eh,ih,rh,av,sv,ev,iv) ∈ r9 : sh ≤ nh,vn ≤ nh,eh ≤ nh,ih ≤ nh,rh ≤ nh, n̄h = λh/(µh + δ) ≤ nh ≤ n0h = λh/µh; av ≤ k; sv + ev + iv ≤ θk/µv} . let us recall the following theorem [48] theorem 3.1: let g ⊂ u = r5 × r4. the system (11) is of class c1, defined on u. if (1) g is positively invariant relative to (11). (2) the system ẋs = a1(x)(xs − xdfe) is globally asymptotically stable at xbrdfe. (3) for any x ∈g, the matrix a2(x) is metzler irreducible. (4) there exists a matrix ā2 , which is an upper bound of the set m = {a2(x) ∈m4(r) : x ∈g} with the property that if a2 ∈ m, for any x̄ ∈ g, such that a2(x̄) = ā2, then x̄ ∈ r5 ×{0}. (5) the stability modulus of ā2, α(a2) = max λ∈sp(a2) re(λ) satisfied α(a2) ≤ 0. then the dfe is gas in g. (see [48] for a proof). let us now verify the assumptions of the previous theorem: it is obvious that conditions (1–3) of the theorem are satisfied. an upper bound of the set of matrices m, which is the matrix ā2 is given biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 6 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... by ā2 =   −k2 0 b̄13 b̄14 γh −k3 0 0 βvhηhs̄v n̄h + m βvhs̄v n̄h + m −k4 0 0 0 γv −µ2   , where b̄13 = βhvηv(s̄h + πv̄h) n̄h + m , b̄14 = βhv(s̄h + πv̄h) n̄h + m , s̄h = s0h, v̄h = v 0 h , āv = k, s̄v = θ µv k, and n̄h = λh (µh + δ) . to check condition (5) in theorem 3.1, we will use the following useful lemma [48] which is the a characterization of metzler stable matrices: lemma 3.1: let m be a square metzler matrix written in block form ( a b c d ) with a and d square matrices. m is metzler stable if and only if matrices a and d −ca−1b are metzler stable. a necessary condition for a metzler matrix to be stable is that the elements on the diagonal are negative. note also that a is a metzler stable matrix is equivalent to a is invertible and −a−1 ≥ 0. lemma 3.1 allows to reduce the problem of metzler stability, by induction, to the stability of 2 × 2 metzler matrices [48]. in our case, we have a = ( −k2 0 γh −k3 ) , b =   βhvηv(s̄h + πv̄h)n̄h + m βhv(s̄h + πv̄h)n̄h + m 0 0  , c =   βvhηhs̄vn̄h + m βvhs̄vn̄h + m 0 0  , and d = ( −k4 0 γv −µ2 ) . clearly, a is a stable metzler matrix. then, after some computations, we obtain d − ca−1b is a stable metzler matrix if and only if r2g ≤ 1 (12) where r2g = βhvβvhkθλh(ηvµ2 + γv)(k3ηh + γh) µvµ2µhk1k2k3k4 × (µh + πξ)(µh + δ) 2 (λh + m(µh + δ)) 2 thus we claim the following result theorem 3.2: if n > 1 and r2g ≤ 1, then the disease–free equilibrium p1 is globally asymptotically stable in g. remark 3.1: note that r2g = r 2 0 (µh + δ) 2(λh + mµh) 2 µ2h(λh + m(µh + δ)) 2 ( n n − 1 ) and condition (12) is equivalent to r20 ≤ ( n − 1 n ) µ2h (µh + δ) 2 (λh + m(µh + δ)) 2 (λh + mµh) 2 (13) in absence of disease–induced death in human (δ = 0), inequality (13) becomes r20 ≤ ( n − 1 n ) < 1. (14) this shows that with the establishment of an effective treatment, it is possible to have r0 and rg less than 1. theorem (3.2) means that for r0 < rg < 1, the dfe is the unique equilibrium (no co-existence with an endemic equilibrium). if r0 ∈ [rg, 1], then it is possible to have co-existence with endemic equilibrium. to confirm whether or not the backward bifurcation phenomenon occurs in this case, one could use the approach developed in [19], [31], [82], which is based on the general center manifold theory [43]. iv. the endemic equilibria and bifurcation analysis a. existence of endemic equilibria we now turn to study the existence of an endemic equilibrium of model system (3). let r0 the basic reproduction number [26], [82] given by eq. (10). we claim the following proposition 4.1: let n > 1 and µv ≤ µ1 ≤ µ2. then biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 7 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... (i) there exists at least one endemic equilibrium whenever r0 > 1. (ii) the backward bifurcation phenomenon may occurs when r0 ≤ 1. (iii) the disease–induced death is responsible of the backward bifurcation phenomenon. (iv) in the absence of the disease–induced death (δ = 0 and µv = µ1 = µ2), system (4) have a unique endemic equilibrium whenever r0 > 1, and the backward bifurcation phenomenon not occurs whenever r0 ≤ 1 (see remark 4.1). proof: see appendix d. the backward bifurcation phenomenon, in epidemiological systems, indicate the possibility of existence of at least one endemic equilibrium when r0 is less than unity. thus, the classical requirement of r0 < 1 is, although necessary, no longer sufficient for disease elimination [6], [14], [40], [75]. in some epidemiological models, it has been shown that the phenomenon of backward bifurcation is caused by factors such as nonlinear incidence (the infection force), disease–induced death or imperfect vaccine [15], [16], [31], [40], [70], [75]. it is important to note that case (i) of proposition 4.1 suggests the possibility of a pithcfork (forward) bifurcation when r0 = 1. also, case (iv) of proposition 4.1 suggests that the principal cause of occurence of backward bifurcation phenomenon is the disease-induced death in both humans and vectors. in the following remark, we prove that, in absence of disease–induced death in both populations, the disease–free equilibrium is the unique equilibrium whenever n > 1 and r0|δ=0,µv=µ1=µ2 < 1. using the direct lyapunov method, we prove the global asymptotic stability of the disease–free equilibrium whenever r0|δ=0,µv=µ1=µ2 < 1. remark 4.1: assumed that n > 1. let k7 = λh + mµh, k8 = πξ + µh, k11 = k3ηh + γh and r1 = r0|δ=0,µv=µ1=µ2 . in the absence of disease-induced death, i.e, δ = 0 and µv = µ1 = µ2, eq. (44) (see appendix d) becomes λ∗h [ b02(λ ∗ h) 2 + b01λ ∗ h + b00 ] = 0 (15) with b02 = k2k3k27πµv + βvhk7k11λhµhπ > 0, b00 = k1k2k3k 2 7µhµv ( 1 −r21 ) and b01 = k1k2k3k 2 7µvπ ( 1 −µhr21 ) + k2k3k 2 7µhµv +βvhk7k8k11λhµh. equation (15) have only one positive solution whenever r1 > 1. if r1 ≤ 1, then coefficients b00, b01, b02 are always positive, and the diseasefree equilibrium is the unique equilibrium. from this we conclude that the disease–induced mortality is the possible cause for the occurrence of multiple endemic equilibria below the classical threshold r1 = 1. the global stability of the disease–free equilibrium may be achieved by lyapunov method. to this aim, let us consider the following lyapunov function [37], [40] y = 4∑ i=1 giii where i = (eh,ih,ev,iv) and gi, i = 1, . . . , 4 are positive weights given by g1 = 1; g2 = k2 (k3ηh + γh) , g3 = k2k3(n 0 h + m) βvhs 0 v(k3ηh + γh) , g4 = βhv [ s0h + πv 0 h ] µ2(n 0 h + m) . along the solutions of (3) we have: ẏ = 4∑ i=1 gii̇i = g1ėh + g2i̇h + g3ėv + g4i̇v = g1 [λh [sh + (1 − �)vh] − (µh + γh)eh] +g2 [γheh − (µh + δ + σ)ih] +g3 [λvsv − (µ1 + γv)ev] + g4 (γvev −µ2iv) = ( g1 βhvηv [sh + πvh] n0h + m + g4γv −g3k4 ) ev + ( g1 βhv [sh + πvh] n0h + m −g4µ2 ) iv + ( g3 βvhηhsv n0h + m + g2γh −g1k2 ) eh + ( g3 βvhsv n0h + m −g2k3 ) ih after replacing the constants gi, i = 1, . . . , 4 by their value, and using the fact that sh ≤ s0h, vh ≤ v 0h , av ≤ a 0 v, and sv ≤ s0v in d1 = {(sh,vh,eh,ih,rh,av,sv,ev,iv) ∈d : sh ≤ s0h,vh ≤ v 0 h ,av ≤ a 0 v,sv ≤ s0v } , biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 8 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... it follows that ẏ ≤ ( g1 βhvηv [ s0h + πv 0 h ] n0h + m + g4γv −g3k4 ) ev = k2k3k4(n 0 h + m) βvhs 0 v(k3ηh + γh) ( r21 − 1 ) ev we have ẏ ≤ 0 if r1 ≤ 1, with ẏ = 0 if r1 = 1 or ev = 0. whenever ev = 0, we also have eh = 0, ih = 0 and iv = 0 . substituting eh = ih = ev = iv = 0 in the first, second, fifth, sixth and seventh equations of eq. (3) with δ1 = 0 gives sh(t) → s0h, vh(t) → v 0 h , rh(t) → 0, av(t) → a0v, sv(t) → s0v as t →∞. thus [sh(t),vh(t),eh(t),ih(t),rh(t),av(t),sv(t),ev(t) ,iv(t)] → (s0h,v 0 h , 0, 0, 0,a 0 v,s 0 v, 0, 0) as t →∞. it follows from the lasalle’s invariance principle [45] that every solution of (3) (when r1 ≤ 1), with initial conditions in d1 converges to p1, as t → ∞. hence, the dfe (p1), of model (3), is gas in d1 if r1 ≤ 1. b. bifurcation analysis previous analysis state that multiple endemic equilibria may occur althougt r0 < 1. in order to better investigate the variation of model’s prediction as r0 varied, we perform a bifurcation analysis at the criticality, i. e. at the disease– free equilibrium dfe := p1 and r0 = 1. on one hand, this will provide a rigorous proof that the occurrence of multiple endemic equilibria comes from a backward bifurcation. on the other hand, we will get also information on the stability of equilibria near the criticality. in particular, on the stability of the endemic equilibrium points emerging from the criticality. we study the center manifold near the criticality by using the approach developed in [19], [31], [82], which is based on the general centre manifold theory [43]. in summary, this approach establishes that the normal form representing the dynamics of the system on the center manifold is given by u̇ = a?u2 + b?$u, where, u represent a real function of real variable, a? = v 2 ·dxxf(x0,$)w2 ≡ ≡ n∑ k,i,j=1 vkwiwj ∂2fk(0, 0) ∂xi∂xj (16) and b? = v ·dx$f(x0,$)w ≡ n∑ k,i=1 vkwi ∂2fk(0, 0) ∂xi∂$ . (17) note that the symbol $ denotes a bifurcation parameter to be chosen, fi’s denotes the right hand side of system (3), x denotes the state vector, x0 the disease–free equilibrium p1; v and w denote the left and right eigenvectors, respectively, corresponding to the null eigenvalue of the jacobian matrix of system (3) evaluated at the criticality. in order to apply this approach, let us choose βhv as bifurcation parameter. from r0 = 1 we get the critical value β∗hv = µvµ2k1k2k3k4(λh + mµh) 2 ( n n − 1 ) βvhλhµhkθ(πξ + µh)(µ2ηv + γv) [ηhk3 + γh]) . note also that in fk(0, 0), the first zero corresponds to the disease–free equilibrium, p1, for the system (3). since βhv = β∗hv is the bifurcation parameter, it follows from $ = βhv − β∗hv that $ = 0 when βhv = β ∗ hv which is the second component in fk(0, 0). the jacobian matrix of the system (4) evaluated at the disease–free equilibrium p1 with βhv = β∗hv is given by j(p1) =   −k1 0 0 0 0 ξ −µh 0 0 0 0 0 −k2 0 0 0 0 γh −k3 0 0 0 0 σ −µh 0 0 0 0 0 0 0 − βvhηhs 0 v h − βvhs 0 v h 0 0 βvhηhs 0 v h βvhs 0 v h 0 0 0 0 0 0 biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 9 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... 0 0 − β∗hvηvs 0 h h − β∗hvs 0 h h 0 0 − β∗hvπηvv 0 h h − β∗hvπv 0 h h 0 0 β∗hvηvs0 h β∗hvs0 h 0 0 0 0 0 0 0 0 −k1 k2 k2 k2 θ −µv 0 0 0 0 −k4 0 0 0 γv −µ2   , where we have set h = n0h + m, k1 = µbθ µv and k2 = k6µv θ . the eigenvalues of the jacobian matrix j(p1) are λ1 = −µh, λ2 = −k1, and the other eigenvalues are the eigenvalue of the following matrix j̄ =  −k2 0 0 0 β∗ hv ηvs0 h β∗ hv s0 h γh −k3 0 0 0 0 0 0 −k1 k2 k2 k2 − βvhηhs 0 v h − βvhs 0 v h θ −µv 0 0 βvhηhs 0 v h βvhs 0 v h 0 0 −k4 0 0 0 0 0 γv −µ2   . the characteristic polynomial of j̄ is given by f(λ) = λ6 +a5λ 5 +a4λ 4 +a3λ 3 +a2λ 2 +a1λ+a0 (18) with a0 = − k1k2k3k4k 2 7µ2µbµv(k6µv −µbθ) k1k 2 7µbµv ( 1 −r20 ) . the others coefficients a5, a4, a3, a2, and a1 are obtained after computations on maxima software [58], [89]. since at the criticality, we have r0 = 1, then a0 = 0, thus equation (18) becomes f(λ) = λ ( λ5 + a5λ 4 + a4λ 3 + a3λ 2 + a2λ + a1 ) . then, the jacobian j(p1) of the linearized system has a simple zero eigenvalue and therefore p1 is a nonhyperbolic equilibrium for r0 = 1. in order to get the coefficients (16) and (17), we need to calculate the right and the left eigenvectors corresponding to the zero eigenvalue. the right eigenvector of j(p1) is given by w = (w1,w2,w3,w4,w5,w6,w7,w8,w9) t where w1 = − β∗hvk9µhλh k21γvk7 w9 < 0, w2 = − ξλhβ ∗ hvk9(µh + k1π) k21µhγvk7 w9 < 0, w3 = β∗hvλhk9(µh + k1π) k1k2k7γv w9 > 0, w4 = β∗hvλhγhk9(µh + k1π) k1k2k3k7γv w9 > 0, w5 = β∗hvλhσγhk9(µh + k1π) k1k2k3k7µhγv w9 > 0, w7 = − βvhµhkθ µvk7 ( 1 − 1 n ) (ηhw3 + w4) < 0, w8 = βvhµhkθ µvk4k7 ( 1 − 1 n ) (ηhw3 + w4) > 0, w6 = µb k6n2 (w7 + w8 + w9) and w9 > 0. similarly, j(p1) has a left eigenvector v = (v1,v2,v3,v4,v5,v6,v7,v8,v9) where v1 = v2 = v5 = v6 = v7 = 0, v3 = µvk1k7 βvhλhk8 v9, v4 = βvhkθµh(ηvµv + γv) k3k4k7µv ( 1 − 1 n ) v9, v8 = (ηvµv + γv) k4 v9 and v9 > 0. a) computation of a?: using the non–zero components of v and the associated non–zero partial derivatives of f (at the dfe p1), for system (3), we obtain a? = 1 2 v3 9∑ i,j=1 wiwj ∂2f3(0, 0) ∂xi∂xj + 1 2 v8 9∑ i,j=1 wiwj ∂2f8(0, 0) ∂xi∂xj . we finally obtain (see the details in appendix e) a? = φ1 −φ2 biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 10 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... where φ1 = 1 2 v3 { β∗ hv µh k1(λh + mµh) 2 [(�ξλh + mµh) (w1 + π w2) ×(w8ηv + w9 + ηv + 1) −2λh(µh + πξ) (w3 + w4 + w5) (w8ηv + w9)]} − 1 2 v8 βvhµ 2 h kθ µv(λh + µhm) 2 ( 1 − 1 n ) w5 (ηhw3 + w4) + 1 2 v8 βvhµh (λh + µhm) (ηhw3 + w4) w7 − 1 2 v8 βvhµ 2 h kθ ( 1 − 1 n ) µv(λh + µhm) 2 × [ 2(ηh + 1)w3w4 + 2 ( ηhw 2 3 + w 2 4 )] < 0 and φ2 = 1 2 v8 βvhµ 2 h kθ µv(λh + µhm) 2 ( 1 − 1 n ) × [(ηhw3 + w4) (w1 + w2)] < 0 b) computation of b?: b? = v3 λh(µh + πξ) k1(λh + µhm) (ηvw8 + w9) > 0. since b? > 0 according to the sign of wi,vi, for i ∈ {1 . . . , 9}, we conclude that the backward bifurcation phenomenon may occurs if a? > 0. we can summarize the results obtained above in the following theorem: theorem 4.1: if a? > 0, then model (3) exhibits backward bifurcation at r0 = 1. if the reversed inequality holds, then the bifurcation at r0 = 1 is forward. this is illustrated by simulating the model with different set of parameter values (it should be stated that these parameters are chosen for illustrative purpose only, and may not necessarily be realistic epidemiologically): —using the parameters values in table ii, except µv = µ1 = µ2 = 1/14, λh = 200, � = 0.80, ξ = 0.475, δ = 0.6, β̃hv = 6, β̃vh = 50 and k = 1000 such that r0 = 0.6095 < 1 and a? = 1.0348 × 10−5 > 0, the numerical resolution of equation (44) (see appendix a), gives the following solution: λ∗1h = 0, λ ∗ 2h = 0.0083, λ ∗ 3h = 10.9412, λ∗4h = −0.0080 and λ ∗ 5h = −0.0001; note that the first solution λ∗1h = 0 corresponds to the disease free equilibrium. the second, and third solution, λ∗2h = 0.0083, λ ∗ 3h = 10.9412, correspond to endemic equilibria; λ∗2h = 0.0083 correspond to unstable endemic equilibrium and λ∗3h = 10.9412 corresponds to the stable endemic equilibrium. the fourth and fifth solution λ∗4h = −0.0080 and λ∗5h = −0.0001 are not biologically meaningful. 0 20 40 60 80 100 0 100 200 300 400 time (days) in fe c ti o u s h u m a n s , ih (t ) fig. 1. time profile of infectious humans using different initial conditions showing that the equilibrium λ∗3h = 10.9412 is stable even if r0 = 0.6095 < 1 . —using the parameters values in table ii, except µv = µ1 = µ2 = 1/14, λh = 100, � = 0.80, ξ = 0.475, δ = 0.6, β̃hv = 4.0385, β̃vh = 100 and k = 1000 such that r0 = 1 and a? = 2.3665×10−4 > 0 , the numerical resolution of equation (44), gives the following solution: λ∗11h = 0, λ ∗ 22h = 0.0114, λ ∗ 3h = 8.5310, and λ∗44h = −0.0111; the first solution λ ∗ 1h = 0 corresponds to the disease free equilibrium. the second, and third solution, λ∗2h = 0.0083, λ ∗ 33h = 8.5310, correspond to endemic equilibria; λ∗22h = 0.0114 correspond to unstable endemic equilibrium and λ∗33h = 8.5310 corresponds to the stable endemic equilibrium. the fourth solution λ∗4h = −0.0111 is not biologically meaningful. —in the absence to disease induced death (δ = 0) and choosing β̃hv = 4.0188 and k = 1000 such that r0 = 1, equation (44) admit only one solution λ∗h = 0 which corresponds to the disease–free equilibrium. in this case, the backward bifurcation phenomenon does not occurs. —choosing β̃hv = 10 and k = 1000 such that r0 = 1.630976 > 1 and a? = −1.8011 < 0, equation (44) admit only one positive solution given by λ∗1h = 0.0001, which correspond to the biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 11 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... 0 20 40 60 80 100 0 50 100 150 200 250 300 time (days) in fe c ti o u s h u m a n s , ih (t ) fig. 2. time profile of infectious humans using different initial conditions showing that the equilibrium λ∗33h = 8.5310 is stable even if r0 = 1 . endemic equilibria when the basic reproduction number, r0, is greater than 1. to conclude, depending to the values of parameters of model (3), the phenomenon of backward bifurcation may occurs when the classical basic reproduction number r0 is less than unity. v. threshold analysis and vaccine impact since a future dengue vaccine, for example, is expected to be imperfect, it is instructive to determine whether or not its widespread use in a community will be benefic (or not) [10], [40], [68]. now, consider the following model (model 3 without vaccination). ṡh = λh −λhsh −µhsh ėh = λhsh − (µh + γh)eh i̇h = γheh − (µh + δ + σ)ih ṙh = σih −µhrh ȧv = µb ( 1 − av k ) (sv + ev + iv) − (θ + µa)av ṡv = θav −λvsv −µvsv ėv = λvsv − (µ1 + γv)ev i̇v = γvev −µ2iv (19) with λh and λv defined at (1) and (2), respectively. following procedure in [26], [82], the corresponding basic reproduction number of model (19), rs, is given by r2s = βhvβvhkθ(k3ηh + γh)(γv + ηvµ2) µvµ2(µh + γh)(µh + δ + σ)(µ1 + γv) × λhµh (λh + mmuh) 2 ( 1 − 1 n ) (20) so we deduce that rvac := r0 = rs √ (πξ + µh) (µh + ξ) . (21) from eq. (21), it follows that, in the absence of vaccination (ξ = 0) or when the vaccine efficacy is very low (� → 0), we have rvac = rs. however, when humans vaccination comes to play, the basic reproductive number is reduced by a factor of√ (πξ + µh) (µh + ξ) < 1. since increasing vaccination efforts results in decreasing the magnitude of arboviruses infection, humans vaccination can contribute to control the spread of arboviral diseases. in the following, we use the set of parameters values given in table iii, which refer to dengue and chikungunya. figs. 3–5 show several simulations, by varying the vaccine efficacy and the percentage of population that is vaccinated. figure 3 shows simulations with different proportions of succeptible human vaccinated, using an imperfect vaccine, with a level of efficacy of 60%. both total number of infected humans and infected vectors reache a peack after 25 days approximatively. however, when � = 60%, the variation of vaccine coverage parameter have not a great impact in the number of infected humans and vectors. figure 4 illustrates the effect of vaccine efficacy in the reduction of the total number of infected humans and vectors. it is clear that the effectiveness of the vaccine has a great impact when � ≥ 90%. thus, it is suitable to add to vaccination (when � < 90%) another control, such as, treatment of infected individuals, personal protection, and vector control strategies to stop the spread of arboviral diseases. figure 5 shows the representation of the basic reproduction number r0 plotted as function of the vaccine efficacy parameter � and the proportion biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 12 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... table iii baseline values of parameters of model (3) and their sources. parameter baseline value sources λh 2.5 day −1 [40] ξ variable � variable ηh, ηv 0.35 assumed ˜βhv 0.75 day −1 [40] ˜βhv 0.75 day −1 [40] γh 1/3 day −1 [30] γv 1/2 day−1 [30] µh day−1 (71 × 365) [68] µv (1/14) day−1 [40] µa 1/5 day−1 [30] µ−11 10 days [30] µ−12 5 days [30] θ 0.08 day−1 [30], [68] δ 10−3 day−1 [40] σ 0.1428 day−1 [2], [40] a 1 day−1 [40], [61] m 100 assumed k 2 × 5000 assumed µb 6 day −1 [68], [60], [61] of susceptible population vaccinated ξ. the use of a vaccine with level of efficacy greather than 90% approximatively, dramaticaly decrease the basic reproduction number, when the proportion of susceptible humans vaccinated are greather than 85%. we observe the same result at figure 6. thus, the use of a vaccine with a high level of efficacy and a wide vaccine coverage has an impact on stopping the spread of the disease. however, if the vaccine efficacy is not high, it is important to add another control strategies. our sensitive analysis in later section will further support this result. vi. sensitivity analysis to determine the best way to fight against arboviruses, it is necessary to know the relative importance of the various factors responsible for their transmission in both the human population than in the vector population, as well as effective means to fight these diseases. the transmission of the disease is directly related to r0, and the 0 50 100 150 200 0 200 400 600 800 1000 time (days) t o ta l n u m b e r o f in fe c te d h u m a n s ,e h (t )+ i h (t ) ξ=0.05 ξ=0.25 ξ=0.5 xi=0.75 xi=1 0 50 100 150 200 0 500 1000 1500 2000 2500 time (days) t o ta l n u m b e r o f in fe c te d v e c to rs ξ=0.050 ξ=0.25 ξ=0.5 ξ=0.75 ξ=1 fig. 3. total number of infected humans and vectors varying the proportion of susceptible huamans vaccinated ξ = (0.05; 0.25; 0.5; 0.75; 1) with a vaccine simulating 60% of effectiveness (i.e. � = 0.60 or π = 1 − � = 0.4). 0 50 100 150 200 0 500 1000 1500 2000 2500 time (days) t o ta l n u m b e r o f in fe c te d v e c to rs ε=0.25 ε=0.50 ε=0.60 ε=0.70 ε=0.80 ε=0.90 ε=1 0 20 40 60 80 100 0 200 400 600 800 time (days) t o ta l n u m b e r o f in fe c te d h u m a n s ε=0.25 ε=0.50 ε=0.60 ε=0.70 ε=0.80 ε=0.90 ε=1 fig. 4. infected humans and vector varying the efficacy level of the vaccine � = (0.25; 0.50; 0.80; 0.90; 1) and considering that 85% of susceptible humans is vaccinated. prevalence of the disease is directly related to the infected states, especially for sizes of eh(t), ih(t), ev(t) and iv(t). these variables are relevant to biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 13 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 ξ ε r 0 fig. 5. the basic reproduction number r0 plotted as function of the vaccine efficacy parameter � and the proportion of susceptible population vaccinated ξ. the set of parameter values is given in table iii. 0 50 100 150 200 250 300 350 400 0 200 400 600 800 time (days) t o ta l n u m b e r o f in fe c te d h u m a n s without vaccination with vaccination:ξ=0.85, ε=0.60 with vaccination:ξ=0.85, ε=0.80 with vaccination:ξ=0.85, ε=1 0 50 100 150 200 250 300 350 400 0 500 1000 1500 2000 2500 time (days) t o ta l n u m b e r o f in fe c te d v e c to rs without vaccination with vaccination:ξ=0.85, ε=0.60 with vaccination:ξ=0.85, ε=0.80 with vaccination:ξ=0.85, ε=1 fig. 6. time profile of total number of infected human and vector without vaccination and with vaccination. the individuals (humans and vectors) who have some life stage of arboviruses in their bodies. the number of infectious humans, ih, is especially important because it represents the people who may be clinically ill, and is directly related to the total number of arboviral deaths [22]. we will perform a global sensitivity analysis. a. mean values of parameters and initial values of variables since we focus in this article, to a general model of arboviral diseases, we will, in this sectable iv parameter value ranges of model (3) used as input for the lhs method. parameter range parameter range λh [1 , 6 ] µa [1/10,1/4] ξ [0.05,1] µ1 [1/21,1/3] � [0.5,0.9] µ2 [1/21,1/3] ηh, ηv [0.1,0.8] θ [0.01,0.17] ˜βhv [0.375,1] δ [10 −5,10−2] ˜βvh [0.375,1] σ [0.1428,1/3] γh [1/12,1/2] a [1,3] γv [1/21,1/2] m [1,201] µh [ 1 78 × 365 , 1 45 × 365 ] k 103×[10,15] µv [1/21,1/10] µb [6,18] table v initial conditions. human initial value vector initial value sh: 1000 av 1000 vh: 0 sv: 500 eh: 20 ev: 20 ih: 10 iv: 40 rh: 0 tion, use the parameters values of two particular arboviruses, dengue and chikungunya. it is important to note that these two diseases are transmitted by the same mosquito: aedes albopictus. however, dengue is also transmited by aedes aegypti [30], [35], [36], [38], [40], [61], [68], [90]. the mean values of parameters are listed in table iii, the range values of parameters are in table iv and the initial conditions are given in table v. b. uncertainty and sensitivity analysis 1) sensitivity analysis of r0: we study the impact of each parameter of the model on the value of the basic reproduction number r0. following the approach of wu and colleagues [88], we perform the analysis by calculating the partial rank correlation coefficients (prcc) between each parameter of our model and the basic reproduction number, r0. table iii troughly estimates biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 14 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... the mean value for each parameter. it is important to notice that, variations of these parameters in our deterministic model lead to uncertainty to model predictions since the basic reproductive number varies with parameters. due to the absence of data on the distribution function, a uniform distribution is chosen for all parameters. the sets of input parameter values sampled using the latin hypercube sampling (lhs) method were used to run 1,000 simulations. with these 1,000 runs of latin hypercube sampling, the derived sampling distribution of r0 is shown in figure 7. from this sampling we get that the mean of r0 is 1.9304 and the standard deviation is 1.6128. hence, statistically we are very confidential that model (3) is in an endemic state since r0 > 1. 0 2 4 6 8 10 12 14 16 18 0 100 200 300 400 500 600 r 0 f re q u e n c ie s fig. 7. sampling distribution of r0 from 1,000 runs of latin hypercube sampling. the mean of r0 is 1.9304 and the standard deviation is 1.6128. from the previous sampling we compute the partial rank correlation coefficients between r0 and each parameter of model (3). the result is displayed in table vi. according to boloye gomero [13], the parameters with large prcc values (> 0.5 or < −0.5) as well as corresponding small p-values (< 0.05) are most influential in model (3). table vi show that the parameter � have the highest influence on the reproduction number r0. although � is the vaccine efficacy. this suggests that the development of a vaccine with high level of efficacy may potentially be an effective strategy to reduce r0. the other parameters with an important effect are θ, a, λh and µ2. the parameters table vi prcc between r0 and each parameter. parameter correlation coefficients p–values 1 λh *–0.6067 1.4578e−99 2 ξ 0.0529 0.0977 3 � ***–0.8043 2.6732e−223 4 ηh 0.2879 4.0576e−20 5 ˜βhv 0.4354 1.3609e−46 6 γh –0.2598 1.4099e−16 7 µh 0.2526 9.9492e−16 8 δ –0.0386 0.2274 9 σ –0.3269 7.7785e−26 10 ηv 0.2039 1.1635e−10 11 ˜βvh 0.4215 1.7130e−43 12 γv 0.2117 2.1787e−11 13 µv –0.3029 3.0015e−22 14 µa –0.0121 0.7049 15 µ1 –0.2948 4.2501e−21 16 µ2 *–0.5087 1.2669e−65 17 θ **0.7626 3.0823e−187 18 a **0.7134 3.4096e−153 19 m –0.0436 0.1724 20 k 0.3880 1.4683e−36 21 µb 0.0082 0.7973 which do not have almost any effect on r0 are ξ, δ, µa, m and µb. in particular, the least sensitive parameters is µb, the number of eggs at each deposit per capita. 2) sensitivity analysis of infected states of model (3): with 1,000 runs of latin hypercube sampling, we compute the prcc between infected states eh(t), ih(t), ev(t), and iv(t) and each parameter of model (3). the result is displayed in tables vii–x. as in table vi, the parameters with large prcc values (> 0.5 or < −0.5) as well as corresponding small p-values (< 0.05) are most influential in model (3). from tables vii–x, we can observe the following facts: –for the value of eh, the parameters with more influence are θ, k, a, �, λh and µ2. the parameters which do not have almost any effect on the variation of eh are µh, δ, µa, m and µb. in particular, the least sensitive parameters is µb, the number of eggs at each deposit per capita; –for the value of ih, the parameters with more influence are λh and γh. the least sensitive pabiomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 15 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... table vii prcc between infected humans eh and each parameter. parameter correlation coefficients p–values 1 λh **0.6842 3.2080e−136 2 ξ 0.4115 2.4590e−41 3 � ***0.7177 6.8762e−156 4 ηh –0.2457 6.1306e−15 5 β̃hv –0.4215 1.7187e−43 6 γh 0.2172 6.2865e−12 7 µh 0.0086 0.7879 8 δ -0.0259 0.4176 9 σ 0.3395 7.4246e−28 10 ηv –2378 4.5858e−14 11 β̃vh –0.4232 7.4972e−44 12 γv –0.2311 2.4083e−13 13 µv 0.2906 1.5881e−20 14 µa 0.0210 0.5122 15 µ1 0.3340 5.8090e−27 16 µ2 *0.5747 3.1691e−87 17 θ ***–0.7599 3.7832e−185 18 a ***–0.7597 4.9923e−185 19 m 0.0537 0.0931 20 k ***–0.7477 4.2124e−176 21 µb –0.0068 0.8328 table viii prcc between infectious humans ih and each parameter. parameter correlation coefficients p–values 1 λh ***0.8727 9.1342e−307 2 ξ 0.0078 0.8062 3 � –0.2887 2.8614e−20 4 ηh 0.0711 0.0261 5 β̃hv 0.0850 0.0078 6 γh ***–0.8722 5.9181e−306 7 µh –0.0363 0.2566 8 δ 0.0412 0.1978 9 σ –0.0531 0.0965 10 ηv 0.0310 0.3316 11 β̃vh 0.1297 4.6364e−5 12 γv –0.0179 0.5764 13 µv –0.0544 0.0886 14 µa –0.0222 0.4877 15 µ1 –0.0580 0.0697 16 µ2 –0.0423 0.1855 17 θ 0.1312 3.7931e−5 18 a 0.1428 2.8933e−6 19 m –0.0017 0.9586 20 k 0.1783 1.9260e−8 21 µb –0.0054 0.8648 table ix prcc between infected vectors ev and each parameter. parameter correlation coefficients p–values 1 λh –0.0186 0.5603 2 ξ –0.0111 0.7280 3 � 0.0135 0.6723 4 ηh –0.1086 6.6203e−4 5 β̃hv –0.0664 0.0375 6 γh 0.0560 0.0798 7 µh –0.0295 0.3563 8 δ 0.0116 0.7170 9 σ 0.0734 0.0215 10 ηv –0.0273 0.3928 11 β̃vh –0.0913 0.0043 12 γv 0.0069 0.8282 13 µv **-0.5923 7.6830e−94 14 µa 0.0157 0.6235 15 µ1 0.0331 0.3006 16 µ2 0.0043 0.8933 17 θ ***0.9225 0 18 a –0.0822 0.0100 19 m 0.0027 0.9324 20 k ***0.9199 0 21 µb 0.1125 4.1594e−4 table x prcc between infectious vectors iv and each parameter. parameter correlation coefficients p–values 1 λh 0.2254 9.3729e−13 2 ξ –0.0090 0.7785 3 � –0.1228 1.1697e−4 4 ηh 0.3126 1.1661e−23 5 β̃hv 0.0031 0.9216 6 γh –0.3233 2.7921e−25 7 µh 0.0381 0.2338 8 δ –0.0215 0.5015 9 σ –0.3869 2.4025e−36 10 ηv 0.0196 0.5402 11 β̃vh 0.5584 2.0585e−109 12 γv –0.6287 6.0859e−109 13 µv –0.4856 4.0722e−59 14 µa 0.0294 0.3583 15 µ1 –0.4380 3.3922e−47 16 µ2 –0.0103 0.7470 17 θ **0.8728 7.6088e−307 18 a *0.6011 2.5895e−97 19 m –0.0640 0.0451 20 k *0.8600 5.9602e−288 21 µb –0.0770 0.0159 biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 16 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... rameters is µb, the number of eggs at each deposit per capita; –for the value of ev, the parameters with more influence are the maturation rate from larvae to adult θ, and the capacity of breeding sites k. the other parameter is the natural mortality rate of vector µv. the least sensitive parameters is m, the number of alternative source of blood; –for the value of iv, the parameters with more influence are θ, k, γv, a and β̃vh. the least sensitive parameters is β̃hv, the probability of transmission of infection from an infectious vector to a susceptible human. although the model is sensitive to the variations of the vaccine efficacy parameter �, there are other parameters (such as θ, a, k, µv, µ2) which have a considerable impact on the value of the basic reproduction number r0 and the number of infected individuals. thus, it is important to take into account other control strategies in the fight against arboviral diseases. vii. numerical simulation in order to illustrate some of the results obtained in the previous sections, we provide here some numerical simulations. we use the nonstandard scheme given by (22). it is important to note that standard numerical methods may fail to preserve the dynamics of continuous models [4], [59], [81]. generally, compartmental models are solved using standard numerical methods, for example, euler or runge kutta methods included in software package such as scilab [76] or matlab [57]. these methods can sometimes lead to spurious behaviours which are not in adequacy with the continuous system properties that they aim to approximate. for example, they may lead to negative solutions, exhibit numerical instabilities, or even converge to the wrong equilibrium for certain values of the time discretization or the model parameters (see [3], [4], [5], [81] for further investigations). a. a nonstandard scheme for the model (3) following [30], system (3) is discretized as follows:  xk+1s −x k s φ(h) = a1(xk)(xks −xdfe) −d12(xki )x k+1 s + b12(x k)xki xk+1i −x k i φ(h) = a2(xk+1s )x k i (22) such that −d12(xi)xs + b12(x)xi = a12(x)xi (23) with d12(xi) =   λh 0 0 0 0 0 πλh 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 λv   , and b12(x) =  0 0 0 0 0 0 0 0 0 0 0 σ 0 0 0 0 0 µb ( 1 − a0v k ) µb ( 1 − av k ) 0 0 0 0 0 0   , which implies that the scheme is consistant with formulation (11). rearranging (22), we obtain the foollowing new expression { akxk+1 = bk xk ≥ 0 (24) with ak = ( i5 + φ(h)d12(x k i ) 0 0 i5 ) and bk =( xks + φ(h) [ a1(xk)(xks −xdf e) + b12(x k)xki ] xki [ i4 + φ(h)a2(xk+1s ) ] ) . where i4 and i5 are the identity matrix of order 4 and 5 respectively. thus, we claim the following result: biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 17 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... lemma 7.1: our non-standard numerical scheme (22) is positively stable, i.e. for xk ≥ 0 we obtain xk+1 ≥ 0, where xk = ( skh,v k h ,e k h,i k h,r k h,a k v,s k v ,e k v ,i k v )t . proof: we suppose xk ≥ 0. ak is a positive diagonal matrix and thus, a−1k ≥ 0. b12 is a positive matrix and we also have −a1(xk)xdfe ≥ 0. to show that bk is a positive matrix, it suffices to choose φ(h) such that id + φ(h)a1(x) ≥ id + φ(h)a1 ≥ 0, id + φ(h)a2(x) ≥ id + φ(h)a2 ≥ 0 where a1 and a2 are lower bounds for the sets {x ∈ d|a1(x)} and {x ∈ d|a2(x)} respectively. following [30], to have bk ≥ 0, it suffices to consider the following time-step function φ(h) = 1 −e−qh q (25) with q ≥ max (k1,k2,k3,µh,k4,k6,µv,µ2). we have proved that xk ≥ 0 implies xk+1 ≥ 0. concerning the equilibria of our numerical scheme, we have the following result lemma 7.2: our non-standard numerical scheme (22) and the continuous model (3) have the same equilibria. proof: see appendix f. the stability of the trivial equilibrium is given by the foollowing result lemma 7.3: if φ(h) has choosen as equation (25), then the tivial equilibrium te := p0 is locally asymptotically stable for our numerical scheme (22) whenever n ≤ 1. proof: see appendix g. now, we also have the following result concerning the stability of the disease–free equilibrium: lemma 7.4: if φ(h) has choosen as equation (25) and r0 < 1, then the disease–free equilibrium dfe := p1 is locally asymptotically stable for our numerical scheme (22) . proof: the proof of lemma 7.4 follows the proof of proposition 3.4 in [30]. see also [5] for a proof in a more general setting. b. simulation results we now provide some numerical simulations to illustrate the theoretical results (local stability, global stability and backward bifurcation). we use parameters values given in table iii with ξ = 0.475, � = 0.60, k = 1000 and initial conditions given in table v. figure 8 illustrates the asymptotic stability of the trivial equilbrium whenever the treshold n is less than unity. in figure 9, when n > 1 the trivial equilibrium is unstable and the disease– free equilibrium is stable (first panel). the phenomenon of backward bifurcation occurs in the second panel of figure 9, where the stable diseasefree equilibrium of the model co–exists with a stable endemic equilibrium when the associated reproduction number, r0, is less than unity. figures 10–11 show the existence of at least one endemic equilibrium whenever r0 is equal or greather than unity. it is important to mentione that the simulation results discussed in this work are subject to the uncertainties (see section vi) in the estimates of the parameter values (tabulated in table iii) used in the simulations. the effect of such uncertainties on the results obtained can be assessed using a sampling technique, such as latin hypercube sampling. viii. conclusions in this paper, we formulated a compartmental model which takes into account a future vaccination strategy in human population, the aquatic development stage of vector and the alternative sources of blood. the analysis has been performed by means of stability, bifurcation and sensitivity analysis. we have obtained that the disease–induced mortality may be the main cause for the occurrence of the backward bifurcation (see remark 4.1). this means that relatively high values of disease– induced mortality rate may induce stable endemic states also when the basic reproduction number r0 is below the classical threshold r0 = 1. the stability analysis reveals that for n ≤ 1, biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 18 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 1000 2000 3000 4000 5000 6000 times (days) n o n i n fe c te d h u m a n s h v h r h 0 100 200 300 400 0 200 400 600 800 1000 times (days) v e c to r p o p u la ti o n a v s v e v i v fig. 8. time profile of both population without vector (with θ = 0.0008, so n = 0.2679 < 1. in this case the trivial equilibrium is globally asymptotically stable. 0 500 1000 1500 2000 0 200 400 600 800 1000 1200 1400 time (days) in fe c te d h u m a n s a n d v e c to rs e h (t) i h (t) e v (t) i v (t) 0 500 1000 1500 2000 0 200 400 600 800 1000 time (days) in fe c te d h u m a n s a n d v e c to rs e h (t) i h (t) e v (t) i v (t) fig. 9. time profile infected humans and vectors. first panel:r0 =: 0.2377 < 1 and second panel: r0 = 0.9405 < 1. the backward bifurcation phenomenon is illustrate in second panel. the trivial equilibrium is globally asymptotically stable. when n > 1 and r0 < 1, the disease– free equilibrium is locally asymptotically stable. in 0 50 100 150 200 250 300 350 400 0 100 200 300 400 500 600 700 time (days) in fe c te d h u m a n s e h (t) i h (t) 0 50 100 150 200 250 300 350 400 0 200 400 600 800 time (days) in fe c te d h u m a n s e h (t) i h (t) fig. 10. time profile of infected humans with β̃hv = 42.9631, λh = 20, so that r0 = 1 (first panel) and β̃hv = β̃vh = 20, λh = 20, so that r0 = 3.5233 > 1 (second panel). 0 2000 4000 6000 8000 10000 0 200 400 600 800 1000 time (days) in fe c te d v e c to rs e v i v 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 4 0 200 400 600 800 1000 1200 1400 1600 time (days) in fe c te d v e c to rs e v i v fig. 11. time profile of infected vectors with β̃hv = 42.9631, λh = 20, so that r0 = 1 (first panel) and β̃hv = β̃vh = 20, λh = 20, so that r0 = 3.5233 > 1 (second panel). the absence of disease-induced death, the disease– free equilibrium is also globally asymptotically biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 19 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... stable. the reduced version of the model (3) (in the absence of disease–induced mortality in both human and vector populations) have a unique endemic equilibrium point whenever its associated reproduction number r1 exceeds unity. taking as cases study the dengue and chikungunya transmission, we used parameter values from the literature to estimate statistically the basic reproduction number, r0, and to perform a global sensitivity analysis on the basic reproduction number and infected states (eh, ih, ev, iv). using latin hypercube sampling, we obtain that the mean of r0 is 1.9304. hence, statistically we are very confident that our model (3) is in an endemic state. the global sensitivity analysis reveals that, apart from the parameters related to vaccination, particularly vaccine efficacy, other parameters ( such as parameters related to vector population) also have a great impact on the basic reproduction number (r0) and on the number of infected humans and vectors (eh,ih,ev,iv). numerical simulations of the model (3), using a nonstandard qualitatively stable scheme, show that the use of a vaccine with high level of efficacy has a proponderant role in the reduction of the disease spread. however, since the efficacy of the proposed vaccine for dengue, for example, has been around 60%, it is suitable to combine vaccination with other mechanisms of control. also, to be the best control strategy, the vaccination process must verify the following conditions: (a) the vaccine must be approved by the relevant agencies (such as who, cdc), before its use. (b) the vaccine efficacy should be high, as well as vaccine coverage. (c) the price of the vaccine must be low for countries affected by the disease. there are already governments, affected by the diseases, willing to use the vaccine before it is approved, which can have unpredictable consequences, so condition (a) does not hold. moreover, according to previous analysis and french laboratory sanofi, the condition (b) does not hold. thus it is important to know what happens when we combine vaccination with other mechanisms of control already studied in the literature, such as personal protection, chemical interventions and education campaigns [30], [40], [60], [61], [63], [64], [67], [68], [69]. this is the perspective of our work. acknowledgment hamadjam abboubakar and léontine nkague nkamba thank the direction of uit of ngaoundéré and ens of yaoundé i, respectively, for their financial help granted under research missions in the year 2014. the authors are very grateful to two anonymous referees, for valuable remarks and comments, which significantly contributed to the quality of the paper. appendix appendix a: proof of proposition 3.1 to find the equilibrium points of our system, we will solve the following system   λh −λhsh − (ξ + µh)sh = 0 ξsh − (1 − �)λhvh −µhvh = 0 λh [sh + (1 − �)vh] − (µh + γh)eh = 0 γheh − (µh + δ + σ)ih = 0 σih −µhrh = 0 µb ( 1 − av k ) (sv + ev + iv) − (θ + µa)av = 0 θav −λvsv −µvsv = 0 λvsv − (µ1 + γv)ev = 0 γvev −µ2iv = 0 (26) to this aim, let p∗ = (s∗h,e ∗ h,i ∗ h,r ∗ h,a ∗ v,s ∗ v,e ∗ v,i ∗ v ) represents any arbitrary endemic equilibrium of (3). further, let λ∗h = βhv(ηve ∗ v + i ∗ v ) (n∗h + m) , λ∗v = βvh(ηhe ∗ h + i ∗ h) (n∗h + m) , (27) be the forces of infection of humans and vectors at steady state, respectively. solving the first five biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 20 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... equations in (26) at steady state gives s∗h = λh k1 + λ ∗ h , v ∗h = ξλh (k1 + λ ∗ h)(πλ ∗ h + µh) , e∗h = λhλ ∗ h(πξ + µh + πλ ∗ h) k2(k1 + λ ∗ h)(πλ ∗ h + µh) , i∗h = γhλhλ ∗ h(πξ + µh + πλ ∗ h) k2k3(k1 + λ ∗ h)(πλ ∗ h + µh) , r∗h = σγhλhλ ∗ h(πξ + µh + πλ ∗ h) µhk2k3(k1 + λ ∗ h)(πλ ∗ h + µh) . (28) where π = 1 − �, k1 = µh + ξ, k2 = µh + γh and k3 = µh + σ + δ. solving the last three equations in (26) at steady state gives s∗v = θa∗v (µv + λ∗v) , e∗v = θa∗vλ ∗ v k4(µv + λ∗v) , i∗v = γvθa ∗ vλ ∗ v µ2k4(µv + λ∗v) . (29) where k4 = µ1 + γv. substituting (29) in the sixth equation of (26) gives a∗v { µbθ µ2k4 ( 1 − a∗v k )( µ2k4 + k5λ ∗ v µv + λ∗v ) −k6 } = 0 (30) with k5 = µ2 + γv and k6 = θ + µa. the trivial solution of (30) is a∗v = 0. substituting this solution in (29) gives s∗v = e ∗ v = i ∗ v = 0. when e∗v = i ∗ v = 0, we also have λ ∗ h = 0, thus e∗h = i ∗ h = r ∗ h = 0, s ∗ h = λh k1 and v ∗h = ξλh µhk1 . then we obtain the trivial equilibrium p0 =( λh k1 , ξλh µhk1 , 0, 0, 0, 0, 0, 0, 0 ) . now we suppose that a∗v 6= 0. the possible solution(s) of (30) is the solution(s) of the following equation µbθ µ2k4 ( 1 − a∗v k )( µ2k4 + k5λ ∗ v µv + λ∗v ) −k6 = 0 (31) the direct resolution of equation (31) gives a∗v = k   µ2µbθk4 ( 1 − 1 n ) + αλ∗v µbθ(µ2k4 + k5λ ∗ v)   (32) where n = µbθ µvk6 and α = µbθk5 −µ2k4k6. let us first compute the equilibrium without disease, i.e. λ∗h = λ ∗ v = 0 or eh = ih = ev = iv = 0. from (32), we obtain a0v := k ( 1 − µvk6 µbθ ) := k ( 1 − 1 n ) (33) thus, the existence of vector is regulated by the threshold n . if n ≤ 1, the system (3) correspond to human population of free vectors and the trivial equilibrium in this case is p0. now we suppose that n > 1. from (28) and (29) (with λ∗v = λ ∗ v = 0), we obtain the non trivial equilibrium or the disease–free equilibrium p1 =( s0h,v 0 h , 0, 0, 0,a 0 v,s 0 v, 0, 0 ) , where s0h = λh k1 , v 0h = ξλh k1µh , a0v = k ( 1 − 1 n ) , s0v = θ µv a0v. appendix b: proof of proposition 3.2 we consider the jacobian matrix associated to model (3) at the equilibrium te. we have j(te) =   −k1 0 0 0 0 0 ξ −µh 0 0 0 0 0 0 −k2 0 0 0 0 0 γh −k3 0 0 0 0 0 σ −µh 0 0 0 0 0 0 −k6 0 0 0 0 0 θ 0 0 0 0 0 0 0 0 0 0 0 0 0 − βhvηvs 0 h n0h + m − βhvs 0 h n0h + m 0 − πβhvηvv 0 h n0h + m − πβhvv 0 h n0h + m 0 βhvηvs0 n0h + m βhvs0 n0h + m 0 0 0 0 0 0 µb µb µb −µv 0 0 0 −k4 0 0 γv −µ2   , were s0 = s0h + πv 0 h . the eigenvalues of j(te) are given by λ1 = λ2 = −µh, λ3 = −k1, λ4 = −k2, λ5 = −k3, and λ6, λ7, λ8, λ9 are eigenvalues of the submatrix biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 21 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... j̄ =   −k6 µb µb µb θ −µv 0 0 0 0 −k4 0 0 0 γv −µ2   . the characteristic polynomial of j̄ is given by p(λ) = λ4 +a1λ3 +a2λ2 +a3λ+a4 = 0 (34) where a1 = µv + µ2 + k4 + k6, a2 = k6µv (1 −n) + (k4 + µ2) (µv + k6) + µ2k4, a3 = k6µv (1 −n) (k4 + µ2) + µ2k4 (µv + k6) and a4 = µ2k4 (1 −n). thus, it is clear that all coefficients are always positive since n < 1. now we just have to verify that the routh–hurwitz criterion holds for polynomial p(λ). to this aim, setting h1 = a1, h2 = ∣∣∣∣a1 1a3 a2 ∣∣∣∣, h3 = ∣∣∣∣∣∣ a1 1 0 a3 a2 a1 0 a4 a3 ∣∣∣∣∣∣, h4 = ∣∣∣∣∣∣∣∣ a1 1 0 0 a3 a2 a1 1 0 a4 a3 a2 0 0 0 a4 ∣∣∣∣∣∣∣∣ = a4h3. the routh-hurwitz criterion of stability of the trivial equilibrium te is given by  h1 > 0 h2 > 0 h3 > 0 h4 > 0 ⇔   h1 > 0 h2 > 0 h3 > 0 a4 > 0 (35) we have h1 = a1 = µv + µ2 + k4 + k6 > 0, h2 = a1a2 −a3 = (k6 + k4 + µ2) µ 2 v + ( µ2k6 ( 1 − µbθ µ2k6 ) +k26 + 2k4k6 + µ2k6 + k 2 4 + 2µ2k4 + µ 2 2 ) µv + µ2k 2 6 ( 1 − µbθ µ2k6 ) + k4k 2 6 + (k4 + µ2) 2 k6 + µ2k 2 4 + µ 2 2k4, h3 = a1a2a3 −a21a4 −a 2 3 = (k4 + µ2) (µv + k6) × ( k6µv (1 −n) + µ2µv + µ2k6 + µ22 ) × ( k6µv (1 −n) + k4µv + k4k6 + k24 ) , using inequality 1/µ2 ≤ 1/µ1 ≤ 1/µv, we obtain h2 > 0. h3 > 0 if n < 1; a4 > 0 if and only if n < 1. thus we conclude that the trivial equilibrium is locally asymptotically stable. now we assume that n > 1. following the procedure and the notation in [82], we may obtain the basic reproduction number r0 as the dominant eigenvalue of the next–generation matrix [26], [82]. observe that model (3) has four infected populations, namely eh, ih, ev, iv. it follows that the matrices f and v defined in [82], which take into account the new infection terms and remaining transfer terms, respectively, are given by f = 1 n0h + m ×   0 0 βhvηvs0 βhvs0 0 0 0 0 βvhηhs 0 v βvhs 0 v 0 0 0 0 0 0   , with n0h = λh µh , v =  (µh + γh) 0 0 0 −γh (µh + δ + σ) 0 0 0 0 (µ1 + γv) 0 0 0 −γv µ2   , and the dominant eigenvalue of the next– generation matrix fv −1 is given by eq. (10). the local stability of the disease–free equilibrium p1 is a direct consequence of theorem 2 of [82]. this ends the proof. appendix c: proof of proposition 3.3 setting y=x-te with x = (sh,vh,eh,ih,rh,av,sv,ev,iv) t , we can rewrite (3) in the following manner dy dt = b(y )y (36) biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 22 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... where b(y ) =   −λh −k1 0 0 0 0 ξ −πλh −µh 0 0 0 λh πλh −k2 0 0 0 0 γh −k3 0 0 0 0 σ −µh 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − βhvηvs 0 h nh + m − βhvs 0 h nh + m 0 0 − πβhvηvv 0 h nh + m − πβhvv 0 h nh + m 0 0 βhvηvs0 nh + m βhvs0 nh + m 0 0 0 0 0 0 0 0 −a66 µb µb µb θ −λv −µv 0 0 0 λv −k4 0 0 0 γv −µ2   , with s0 = (s0h + πv 0 h ), a66 =( k6 + µb sv + ev + iv k ) . it is clear that y = (0, 0, 0, 0, 0, 0, 0, 0, 0) is the only equilibrium. then it suffices to consider the following lyapunov function l(y ) =< g,y > were g = ( 1, 1, 1, 1, 1, 1, k6 θ , k6 θ , k6 θ , k6 θ ) . straightforward computations lead that l̇(y ) =< g,ẏ >def= < g,b(y )y > = −µh(y1 + y2 + y3 + y4 + y5) −δy4 + k6µv θ (n − 1)y7 + k6µ1 θ ( µbθ k6µ1 − 1 ) y8 −µb y6 k (y7 + y8 + y9) + k6µ2 θ ( µbθ k6µ2 − 1 ) y9 using the fact that 1/µ2 ≤ 1/µ1 ≤ 1/µv, we have µbθ k6µ1 − 1 ≤ 0 and µbθ k6µ2 − 1 ≤ 0, which implies that l̇(y ) ≤ 0 if n ≤ 1. moreover, the maximal invariant set contained in { l|l̇(y ) = 0 } is {(0, 0, 0, 0, 0, 0, 0, 0, 0)}. thus, from lyapunov theory, we deduce that (0, 0, 0, 0, 0, 0, 0, 0, 0) and thus, te := p0, is gas if n ≤ 1. appendix d: proof of proposition 4.1. we compute now the endemic equilibrium, i.e. we are looking for an equilibrium such that λ∗h 6= 0 and λ∗v 6= 0. we assume that n > 1. from the sixth equation of (26), at equilibrium, we have s∗v + e ∗ v + i ∗ v = kk6a ∗ v µb(k −a∗v) (37) from the last third equations of (26), at equilibrium, we have µvs ∗ v + µ1e ∗ v + µ2i ∗ v = θa ∗ v (38) we will observe the following two cases. a) absence of disease–induced death in vector: the absence of disease–induced death in vector is traduce by the relation µv = µ1 = µ2, then equation (38) becomes s∗v + e ∗ v + i ∗ v = θ µv a∗v (39) equalling eqs. (37) and (39) gives like before a0v := k ( 1 − µvk6 µbθ ) = k ( 1 − 1 n ) . (40) substituting a∗v by a 0 v in equation (29) gives s∗v = ( 1 − 1 n ) kθ (µv + λ∗v) , e∗v = ( 1 − 1 n ) kθλ∗v k4(µv + λ∗v) , i∗v = ( 1 − 1 n ) kθγvλ ∗ v µvk4(µv + λ∗v) . (41) replacing (41) in the expression of λ∗h gives λ ∗ h = βhv(ηve ∗ v + i ∗ v ) (n∗h + m) = k10 λ∗v (µv + λ∗v) ×( βhvµhk2k3(k1 + λ ∗ h)(πλ ∗ h + µh) k2k3k7(k1 + λ ∗ h)(πλ ∗ h + µh) − δγhλhλ ∗ h(k8 + πλ ∗ h) ) (42) where k7 = (λh + mµh), k8 = πξ + µh, k9 = µ2ηv + γv = ηvµv + γv and k10 = k9kθ µvk4 ( 1 − 1 n ) . replacing (28) in the expression of λ∗v gives λ ∗ v =( βvhλhµhk11λ ∗ h(k8 + πλ ∗ h) k2k3k7(k1 + λ ∗ h)(πλ ∗ h + µh) − δγhλhλ ∗ h(k8 + πλ ∗ h) ) (43) biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 23 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... where k11 = k3ηh + γh. substituting (43) in (42) gives the following equation f(λ ∗ h) := λ ∗ h [ b4(λ ∗ h) 4 + b3(λ ∗ h) 3 +b2(λ ∗ h) 2 + b1λ ∗ h + b0 ] = 0 (44) where b4 = π 2 [k7(µhk3 + γh(µh + σ)) + δγhmµh] ×{µv [k7(µhk3 + γh(µh + σ)) + δγhmµh] +βvhk11λhµh} b3 = 2πx{k2k3k7µh(1 + π) + δγhλhµh +πξx}µv + βvhπλhµhk11 {πk2k3y +k7 [k8(k2k3 − 2δγh) + µhk2k3]} b2 = µv [k1k2k3k7π −δλhγhπξ + xµh] 2 + 2k1k2k3k7πµhµvx + βvhλhµ 2 hπk2k3k11 {πk1k7 −βhvk10 [π(k8 + k1) + µh]} + βvhk8k11λhµh [k2k3k7πµh + k8x] b1 = 2k1k2k3k7µhµv [k8x + πµhk2k3k7] + k2k3k11βvhλhµ 2 h{k1k7k8 −βhvk10(µhk8 + k1π(k8 + µh))} with x = k2k3k7−δγhλh, y = k1k7−βhvµhk10; and b0 = µ 2 hµvk 2 1k 2 2k 2 3k 2 7 ( 1 −r20 ) we consider λ∗h 6= 0, otherwise we recover dfe. the positive endemic equilibria p∗ = (s∗h,v ∗ h ,e ∗ h,i ∗ h,r ∗ h,a ∗ v,s ∗ v,e ∗ v,i ∗ v ) are obtained by solving eq. (44) for λ∗h. the coefficient b4 is always positive and coefficient b0 is negative (resp. positive) whenever r0 > 1 (resp. r0 < 1). the number of possible nonnegative real roots of the polynomial of eq. (44) depends on the signs of b3, b2 and b1. the various possibilities for the roots of f(λ∗h) are tabulated in table xi and xii. from tables xi and xii , we deduce the following result which gives various possibilities of nonnegative solutions of (44). lemma a.1: assume that n > 1 and µv = µ1 = µ2. then, the arboviral-disease model (3) table xi total number of possible real roots of (44) when r0 > 1. number of cases b0 b1 b2 b3 b4 sign changes – + + + + 1 1 – – + + + 1 – – – + + 1 – – – – + 1 – + + – + 3 – + – + + 3 2 – + – – + 3 – – + – + 3 table xii total number of possible real roots of (44) when r0 < 1. number of cases b0 b1 b2 b3 b4 sign changes 1 + + + + + 0 + + + – + 2 + + – + + 2 + + – – + 2 2 + – + + + 2 + – – + + 2 + – – – + 2 3 + – + – + 4 1. has a unique endemic equilibrium when case 1 of table xi is satisfied and whenever r0 > 1. 2. could have more than one endemic equilibrium when case 2 of table xi is satisfied whenever r0 > 1. 3. could have more than one endemic equilibrium when case 2, 3 of table xii are satisfied and whenever r0 < 1. 4. has no endemic equilibrium when case 1 of table xii is satisfied and whenever r0 < 1. case 3 of lemma a.1 suggests that co-existence of the disease–free equilibrium and the endemic equilibrium for the arboviral-disease model (3) is possible, and hence the potential occurrence of the backward bifurcation phenomenon when r0 < 1. also, case 2 of lemma a.1 suggests the possibility of a pithcfork (forward) bifurcation when r0 = 1. biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 24 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... b) presence of disease–induced death in vector:: here, we will consider µv < µ1 < µ2 with µv 6= µ2. equation (27) becomes λ∗h = βhvµhµvk2k3k10(k1 + λ ∗ h)(µh + πλ ∗ h) k2k3k7(k1 + λ ∗ h)(µh + πλ ∗ h) − δγhλhλ ∗ h(k8 + πλ ∗ h) × ( µ2µbθk4n1 + αλ∗v µbθ(µ2k4 + k5λ∗v) )( λv µv + λv ) (45) with k12 = µvk10 k , n1 = ( 1 − 1 n ) and λ∗v = βvhµhλhk11λ ∗ h(k8 + πλ ∗ h) k2k3k7(k1 + λ ∗ h)(µh + πλ ∗ h) − δγhλhλ ∗ h(k8 + πλ ∗ h) (46) substituting (46) in (45) gives the following equation λ∗h 6∑ i=0 ci(λ ∗ h) i = 0 (47) where c0 = k31k 3 2k 3 3k4k 3 7θµ2µvµbµ 3 h ( r20 − 1 ) and c6 = −µbπ3θx (µ2k4x + βvhk5k11λhµh) ×(µvx + βvhk11λhµh) , with x = (k2k3k7 −δλhγh) > 0. the others coefficients c5, c4, c3, c2, and c1 are obtained after computations on maxima software. we also obtain the following result which gives various possibilities of solutions of eq. (47). lemma a.2: assume that n > 1. then, the arboviral-disease model (3) 1. could have a unique endemic equilibrium whenever r0 > 1. 2. could have more than one endemic equilibrium whenever r0 > 1. 3. haven’t endemic equilibrium whenever r0 < 1. 4. could have one or more than one endemic equilibrium whenever r0 < 1. case 4 of lemma a.2 suggests that co-existence of the disease–free equilibrium and endemic equilibrium for the arboviral-disease model (3) is possible, and hence the potential occurrence of a backward bifurcation phenomenon when r0 < 1. also, case 2 of lemma a.2 suggests the possibility of a pithcfork (forward) bifurcation when r0 = 1. appendix e: computation of a? of theorem 4.1. a? = 1 2 v3 9∑ i,j=1 wiwj ∂2f3(0, 0) ∂xi∂xj + 1 2 v8 9∑ i,j=1 wiwj ∂2f8(0, 0) ∂xi∂xj . (48) let a?3 = ∑9 i,j=1 wiwj ∂2f3(0, 0) ∂xi∂xj and a?8 = ∑9 i,j=1 wiwj ∂2f8(0, 0) ∂xi∂xj . after few computations, we obtain a ? 3 = β∗hvµh(�ξλh + mµh) k1(λh + mµh)2 w1 (ηvw8 + w9) + β∗hvπµh(�ξλh + mµh) k1(λh + mµh)2 w2 (ηvw8 + w9) − β∗hvµhλh(µh + πξ) k1(λh + mµh)2 w3 (w8ηv + w9) − β∗hvµhλh(µh + πξ) k1(λh + mµh)2 w4 (w8ηv + w9) − β∗hvµhλh(µh + πξ) k1(λh + mµh)2 w5 (w8ηv + w9) + β∗hvηvµh k1(λh + mµh)2 [(�ξλh + mµh) (w1 + π w2) −λh(µh + πξ)w8 (w3 + w4 + w5)] + β∗hvµh k1(λh + mµh)2 [(�ξλh + mµh) (w1 + π w2) −λh(µh + πξ)w9 (w3 + w4 + w5)] = β∗hvµh k1(λh + mµh)2 {(�ξλh + mµh) (w1 + π w2) −λh(µh + πξ) (w3 + w4 + w5)}(w8ηv + w9) + β∗hvµh k1(λh + mµh)2 {(�ξλh + mµh) (w1 + π w2) (ηv + 1) −λh(µh + πξ) (w3 + w4 + w5) (ηvw8 + w9)} = β∗hvµh k1(λh + mµh)2 × {(�ξλh + mµh) (w1 + π w2) (w8ηv + w9 + ηv + 1) −2λh(µh + πξ) (w3 + w4 + w5) (w8ηv + w9)} , biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 25 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... a ? 8 = w3 9∑ j=1 wj ∂2f8 ∂x3∂xj (x0, 0) + w4 9∑ j=1 wj ∂2f8 ∂x4∂xj (x0, 0) + w7 9∑ j=1 wj ∂2f8 ∂x5∂xj (x0, 0) + w8 9∑ j=1 wj ∂2f8 ∂x8∂xj (x0, 0) + w9 9∑ j=1 wj ∂2f8 ∂x9∂xj (x0, 0) = − βvhµ 2 hkθ ( 1 − 1 n ) µv(λh + µhm)2 [ ηh ( w3 + 5∑ i=1 wi ) + w4 ] w3 − βvhµ 2 hkθ ( 1 − 1 n ) µv(λh + µhm)2 [( w4 + 5∑ i=1 wi ) + ηhw3 ] w4 + βvhµh (λh + µhm) (ηhw3 + w4) w7 = − βvhµ 2 hkθ ( 1 − 1 n ) µv(λh + µhm)2 [(ηhw3 + w4) (w1 + w2 + w5) +2(ηh + 1)w3w4 + 2 ( ηhw 2 3 + w 2 4 )] + βvhµh (λh + µhm) (ηhw3 + w4) w7 using above results, eq. (48) becomes a? = φ1 −φ2 where φ1 = 1 2 v3 { β∗ hv µh k1(λh + mµh) 2 [(�ξλh + mµh) (w1 + π w2) ×(w8ηv + w9 + ηv + 1) −2λh(µh + πξ) (w3 + w4 + w5) (w8ηv + w9)]} − 1 2 v8 βvhµ 2 h kθ µv(λh + µhm) 2 ( 1 − 1 n ) w5 (ηhw3 + w4) + 1 2 v8 βvhµh (λh + µhm) (ηhw3 + w4) w7 − 1 2 v8 βvhµ 2 h kθ ( 1 − 1 n ) µv(λh + µhm) 2 [2(ηh + 1)w3w4 +2 ( ηhw 2 3 + w 2 4 )] < 0 and φ2 = 1 2 v8 βvhµ 2 hkθ µv(λh + µhm) 2 ( 1 − 1 n ) × [(ηhw3 + w4) (w1 + w2)] < 0 appendix f: proof of lemma 7.2 the kamgang-sallet approach used for (22) ensures that the trivial equilibrium(te := p0) and the disease–free equilibrium (dfe := p1) are the fixed point of (22). indeed, rewriting (22) gives  sk+1h = φ(h)λh + (1 −φ(h)k1)skh 1 + φ(h)λkh v k+1h = φ(h)ξskh + (1 −φ(h)µh)v k h 1 + φ(h)πλkh ek+1h = (1 −φ(h)k2)e k h + φ(h)λ k h ×(sk+1h + πv k+1 h ) ik+1h = φ(h)γhe k h + (1 −φ(h)k3)i k h rk+1h = φ(h)σi k h + (1 −φ(h)µh)r k h ak+1v = [ 1−φ(h) ( k6+µb skv+e k v+i k v k )] akv + φ(h)µb(s k v + e k v + i k v ) sk+1v = φ(h)θakv + (1 −φ(h)µv)skv 1 + φ(h)λkv ek+1v = (1 −φ(h)k4)ekv + φ(h)λkvsk+1v ik+1v = φ(h)γve k v + (1 −φ(h)µ2)ikv (49) if x∗ = (sh,v ∗ h ,e ∗ h,i ∗ h,r ∗ h,a ∗ v,s ∗ v,e ∗ v,i ∗ v ) t is an equilibrium of the discrete system (49), then we have after few simplifications  λh −λ∗hs ∗ h −k1s ∗ h = 0 ξs∗h −πλ ∗ hv ∗ h −µhv ∗ h = 0 λ∗h(s ∗ h + πv ∗ h ) −k2e ∗ h = 0 γhe ∗ h −k3i ∗ h = 0 σi∗h −µhr ∗ h = 0 µb(s ∗ v + e ∗ v + i ∗ v ) ( 1 − a∗v k ) −k6a∗v = 0 θa∗v −λ∗vs∗v + µvs∗v = 0 k4e ∗ v −λ∗vs∗v = 0 γve ∗ v −µ2i∗v = 0 (50) which is equivalent to{ a1(x∗)(x∗s −xdfe) + a12(x ∗)x∗i = 0 a2(x∗)x∗i = 0 (51) where a1, a12 and a2 are given at equation (11). appendix g: proof of lemma 7.3 the jacobian matrix associated with the righthand side of the numerical scheme (22) at the biomath 4 (2015), 1507241, http://dx.doi.org/10.11145/j.biomath.2015.07.241 page 26 of 30 http://dx.doi.org/10.11145/j.biomath.2015.07.241 h. abboubakar et al., modeling the dynamics of arboviral diseases ... tivial equilibrium te := p0 is given by jte = (jij)1≤i,j≤9 with j1,1 = 1 −k1φ(h); j1,8 = − φ(h)βhvηvλhµh k1(λh + µhm) ; j1,9 = − φ(h)βhvλhµh k1(λh + µhm) ; j2,1 = φ(h)ξ; j2,2 = 1 −µhφ(h); j2,8 = − φ(h)πβhvηvξλh k1(λh + µhm) , j2,9 = − φ(h)πβhvξλh k1(λh + µhm) , j3,3 = 1 −k2φ(h); j3,8 = φ(h)βhvηvλh(µh + πξ) k1(λh + µhm) ; j3,9 = φ(h)βhvλh(µh + πξ) k1(λh + µhm) ; j4,3 = φ(h)γh, j4,4 = 1 −k3φ(h); j5,4 = φ(h)σ; j5,5 = 1 −µhφ(h); j6,6 = 1 −φ(h)k6; j6,7 = j6,8 = j6,9 = φ(h)µb; j7,6 = φ(h)θ, j7,7 = 1 −φ(h)µv, j8,8 = 1 −φ(h)k4; j9,8 = φ(h)γv; j9,9 = 1 −µ2φ(h) the eigenvalues of jte are given by λ1 = λ2 = 1 −µhφ(h), λ3 = 1 −k1φ(h), λ4 = 1 −k2φ(h), λ5 = 1−k3φ(h), and λ6, λ7, λ8, λ9 are eigenvalues of the submatrix j̄ =   j6,6 φ(h)µb φ(h)µb φ(h)µb j7,6 j7,7 0 0 0 0 j8,8 0 0 0 j9,8 j9,9   since φ(h) > 0, it is clear that |λi| < 1, for i = 1, 2, . . . , 5. we need also to show that the characteristic polynomial associated with j̄ is schur polynomials, i.e. polynomials such that all roots λi verify |λi| < 1. the characteristic polynomial associated with j̄ is given by p(λ) = (λ + µ2φ(h) − 1) (λ + k4φ(h) − 1) h(λ) where h(λ) = λ 2 + (φ(h)(µv + k6) − 2) λ + 1 + φ(h) 2 (k6µv −µbθ) −φ(h)(µv + k6) the roots of p(λ) are λ6 = 1 − µ2φ(h), λ7 = 1 − k4φ(h) and the others roots are the roots of h(λ). note that |λ6| < 1 and |λ7| < 1. now, we need to show that h(λ) is a schur polynomial. to this aim, let q1 = (φ(h)(µv + k6) − 2) and q2 = 1 + φ(h)2(k6µv −µbθ)−φ(h)(µv + k6). using lemma 11 in [29], we just show that the following conditions hold: 1 +q1 +q2 > 0, 1−q1 +q2 > 0, 1−q2 > 0 (52) we compute 1 + q1 + q2 = φ(h)2k6µv(1 −n), 1 − q1 + q2 = 2 [(1 −φ(h)µv) + (1 −φ(h)k6)] + φ(h)2k6µv(1 −n) and 1 −q2 = φ(h) [µv + k6(1 −φ(h)µv) + φ(h)µbθ] if φ(h) has choosen as equation (25), then conditions (52) hold whenever n ≤ 1. thus, h(λ) is a schur polynomial. this ends the proof. references [1] ahmed abdelrazec, suzanne lenhart, huaiping zhu, transmission dynamics of west nile virus in mosquitoes and corvids and non-corvids, j. math. biol. 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