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 immu- nizing 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 al- lowing 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 cov- erage inducing global immunization against disease by bringing down the basic reproduction number. Keywords-Transmission potential, minimum pro- portion 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 vacci- nation 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 per- fect vaccine and vertical transmission: In equilib- rium, 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 ex- pectancy of newborn. We analyse the basic re- production 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 ”pre- infection” 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 (de- pending 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 vari- able u ( and then ”locally” Lipschitz in u) pertur- bation (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 in- tegrals): ∫ 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 so- lution (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 func- tions: 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 with- out vertical transmission and RΛ ′ 6=0 0 (Ψ) is the ba- sic 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 com- pared 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 corre- sponds to a ”massive and agressive” campaign (Φ huge) of vaccination that reverse the effect of verti- cal transmission (λ0 very small). C2) naturally tra- duces 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 min- imum proportion p∗ to be covered if vaccination takes place at age V : Theorem 6: A formula showing minimum pro- portion 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 mini- mal proportion for vaccination: reduce Λ′ (verti- cal transmission), adds chances to satisfy crite- rion (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 vacci- nation (in biological ranges) reduces the minimum vaccination coverage p∗ inducing immunization. A further work will consider an imperfect vac- cine and differential infectivity as [3, (Castillo- Chavez 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 De- velopment 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 ex- pressed 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 (AIMS- NEI 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 struc- ture, 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 op- timal 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. 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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