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 dif- fusion 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 at- tacks 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 Col- letotrichum species (acutatum, capsici, gloeospo- rioides, 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 con- ducive 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 includ- ing 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 chronol- ogy 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 distigu- ished: 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 car- bohydrates, dissolve cell walls, and hydrolyze cuticle. Some of those enzymes are polyglactur- onases, 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-acetoxy- 2-hydroxy-4-oxo-heneicosa-12, 15-diene) that in- hibits the growth of Colletotrichum gloeospori- oides 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 environ- mental factors affecting host sensitivity. DAN- NEBERGER et al. in [9] have developed a fore- casting model for the annual bluegrass anthrac- nose 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 relation- ship 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 spatio- temporal 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 mod- els in literature [12], [17], [18], [19], [20], [21], [28]. C. Controlling anthracnose There are many approaches to controlling anthracnose diseases. The genetic approach in- volves selection or synthesis of more resistant cul- tivars [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 pol- icy should schedule different approaches to opti- mize 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 subsec- tion 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 infec- tion 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, de- noted 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 environ- mental 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 corre- sponding 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 condi- tions 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 diffi- cult. 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 alter- native control method is to slow down the within- host 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 fol- lowing 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 Theo- rem, 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 asymp- totic 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 en- vironmental 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 se- quence 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 nonnega- tive 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 within- host model We performed simulations in order to demon- strate the practical controllability of the system, For these simulations we used an inhibition pres- sure 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 rep- resented 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 ef- fectiveness 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 well- posed. 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 co- ercive bilinear form on H1 (Ω). The Lax- Milgram 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 Hille- Yosida theorem there is a unique function θ ∈ C1 ( [0,T] ; L2 (Ω) )⋂ C ([0,T] ; E) which satis- fies (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 gener- ated 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 non- negative 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 nonnegli- gible 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 max- imal monotone and S£ is a contraction semi- group. 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 fol- lows 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 con- ditionally 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 ge- ographical 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 neigh- borhood 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 semi- group 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 ad- missible 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 maxi- mum 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 corre- sponding 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 exten- sively 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 ex- plicitly 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 condi- tions of smoothness of parameters, and the control strategy is also very regular. In practice parame- ters 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 con- tinuous 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 con- trol Problem arising from dengue disease transmission mode, Math. Biosci., 242, p9-16, 2013. [2] Anita S., Arnautu V., Capasso V., An Introduction to Op- timal Control Problems in Life Scienes and Economics, Springer Science+Business Media, New York, 2011. 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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