A Stochastic Viral Infection Model with General Functional Response

Transcription

1 Nonlinear Analysis and Differential Equations, Vol. 4, 16, no. 9, HIKARI Ltd, A Stochastic Viral Infection Model with General Functional Response Marouane Mahrouf a,1, El Mehdi Lotfi a, Mehdi Maziane a Khalid Hattaf a,b and Noura Yousfi a a Department of Mathematics and Computer Science, Faculty of Sciences Ben M sik, Hassan II University, P.O Box 7955 Sidi Othman, Casablanca, Morocco b Centre Régional des Métiers de l Education et de la Formation CRMEF, 34 Derb Ghalef, Casablanca, Morocco Copyright c 16 Marouane Mahrouf et al. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we propose a stochastic viral infection model with general functional response. The well posedness of the proposed model is investigated. Also, the extinction of the infection is fully determined by the basic reproduction number R. Furthermore, the dynamical behavior around the chronic infection equilibrium is established. Finally, Numerical simulations are given to illustrate our theoretical results. Keywords: Viral infection, stochastic differential equations, extinction, Lyapunov functional. 1 Corresponding author

2 436 Marouane Mahrouf et al. 1 Introduction In this paper, we consider the following viral infection model dx = λ δxt f xt, vt vt, dy = f xt, vt vt ayt, dv = kyt µvt, where xt, yt and vt denote the concentrations of uninfected cells, infected cells and free virus at time t, respectively. λ is the recruitment rate of uninfected cells. The parameters δ, a and µ are the death rates of uninfected cells, infected cells and free virus, respectively. The constant k is the production rate of the free virus by an infected cell. In system 1, the incidence rate of infection is modeled by a general functional response cf. K. Hattaf, N. Yousfi βx [1] of the form fx, v =, where β is the infection α + α 1 x + α v + α 3 xv coefficient and α, α 1, α, α 3 are nonnegative constants. We recall that this incidence rate includes the common types such as the mass action called also the bilinear incidence function when α = 1 and α 1 = α = α 3 = ; the saturation incidence rate when α = 1 and α 1 = α 3 = ; the Beddington-DeAnglis functional response [, 3] when α = 1 and α 3 = ; the Crowley-Martin functional response introduced in [4] and used by Zhou et al. [5] when α = 1 and α 3 = α 1 α ; the specific functional response proposed by Hattaf et al. see Section 5, [6] and used in [7, 8, 9, 1] when α = 1 and the incidence function was used by Zhuo in [11] when α = α 3 = and α 1 = α = 1. Based on the results given by Hattaf et al. [6], we deduce that the basic βkλ reproduction number of system 1 is R = aµα δ + α 1 λ. Further, sys- λ tem 1 has a unique infection-free equilibrium E f δ,, which is globally asymptotically stable if R 1. When R > 1, E f becomes unstable and there exists an other biological equilibrium, namely the chronic infection equilibrium E x, λ δx a, kλ δx aµ 1 that is globally asymptotically stable, where x is the unique solution on, + of the equation f kλ δx aµ x, aµ k =. To incorporate the effect of environmental noise, we assume that the clearance rate of free virus is subject to random fluctuations. Hence, we replace µ by µ σdbt, where Bt is an one-dimensional standard Brownian motion defined on a complete probability space Ω, F, P with a filtration {F t } t

3 Stochastic viral infection model 437 satisfying the usual conditions i.e., it is increasing and right continuous while F contains all P-null sets and σ represents the intensity of Bt. Therefore, model 1 will be changed into the following stochastic viral infection model with incidence rate dxt = λ δxt f xt, vt vt, dyt = f xt, vt vt ayt, dvt = kyt µvt σvtdbt. The paper is organized as follows. In the next section, we investigate the well posedness of the stochastic model by showing the existence and uniqueness of the positive solution. In Section 3, we establish the extinction of the infection in terms of the basic reproduction number R. In Section 4, we study the dynamical behavior around the chronic infection equilibrium E. In Section 5, we give some numerical simulations to illustrate our main results. The paper ends with a brief discussion and conclusion in Section 6. Existence and uniqueness of the positive solution Let φx be a function defined on, + with φx = x 1 lnx and { } IR 3 + = x 1, x, x 3 IR 3 x i >, i = 1,, 3. Lemma.1. For all x >, we have x φx + ln. 3 Proof. Consider hx = x + 1 lnx. Clearly, hx has a global minimum at x = and h = 3 ln. Hence, 3 holds. Theorem.. For any initial value X IR 3 +, there exists a unique solution Xt of system defined on [, + and this solution remains in IR 3 + with probability one, namely Xt IR 3 + for all t almost surely briefly a.s.. Proof. From [1], we deduce that system has a unique local solution Xt on t [, τ e, where τ e is the explosion time. To prove that this solution is

4 438 Marouane Mahrouf et al. global, we need { to prove that τ e = a.s. For this, we define } the stopping time by τ + = inf t [, τ e : xt or yt or vt, with the traditional setting inf =, where denotes the empty set. It is clear that τ + τ e a.s. Now, we need to prove that τ + = a.s. Assume that this statement is false, then there is a constant T > such that P {τ + < T } >. We define a C -function V : IR 3 + IR + as follows V X = α x α + α 1 x φ x x + φy + a k φv, where x = λ δ. Using Itô s formula for all t [, τ +, we obtain dv Xt = [ α δ α fx, vv x x + 1 fx, vv α + α 1 x x α + α 1 x y α x + α + α 1 x x fx, vv aµ k v ay v + a + aµ k + a ] k σ a σv 1dBt k a + aµ k + a k σ + fx, v a σv 1dBt. k According to Lemma.1, we have fx, v lnfx, + k a fx, V Xt. Hence, dv Xt C 1 + C V Xt a k σv 1dBt, where C 1 = a + aµ k + a k σ + lnfx, and C = k a fx,. Integrating both sides of the above inequality from to t, we get V Xt V X + C 1 t + C V Xs ds a σv 1dBs. 4 k Noticing some components of Xτ + equal. Then lim t τ + V Xt = +. Letting t τ + in 4 leads to the contradiction + V X + C 1 τ + + τ + C V Xs ds τ + a σv 1dBs < +. k Therefore, τ + = a.s., which means that Xt IR 3 + a.s. for all t. From the first equation of system, we get dxt λ δxt. Then xt max { x, λ δ }.

5 Stochastic viral infection model 439 Adding the two first equations of system, we obtain d xt + yt λ δxt ayt. { Hence, xt + yt max x + y, Γ = } λ. Consequently, the region min{δ, a} } { xt, yt, vt IR 3 + : xt λ δ, xt + yt λ min{δ, a} is positively invariant set of system. In the remainder of this paper, we assume that X Γ. 3 Extinction of the infection Obviously, system has also the infection-free equilibrium E f λ,,. δ Theorem 3.1. Let Xt = xt, yt, vt be the solution of system with initial value X Γ. If R < 1, then yt and vt are almost surely exponentially stable in the sense that yt and vt will tend to their equilibrium value zero exponentially with probability one. Moreover, the system converges to their infection-free equilibrium E f λ,,. δ Proof. Using Itô s formula, we obtain d lny + a k v µr 1 a σv k y + a k vdbt. Integrating the above inequality and using the fact that lim sup t Bt t = see 1 [1], we get lim sup t t ln y + a k v µr 1 < a.s. Then yt and vt tends to zero exponentially with probability one. From the first and the second equations of system, we have Thus, d xte δt dxt dyt + δxt = λ + ayt. = e δt λ e at dyteat. By integration, we get xt = xe δt + λ δ 1 e δt yt + ye δt + δ a yse δs t ds. 5

6 44 Marouane Mahrouf et al. According to Lemma 3.3 in [13], we have Since lim sup t + lim yt =, we have lim t + From 5, we deduce that yse δs t ds 1 δ t + lim t + xt = λ δ lim sup yt. t + yse δs t ds =. a.s.. This completes the proof. 4 Dynamics around the chronic infection equilibrium Also, the chronic infection equilibrium E is not a steady state of system. For this reason, we will discuss the dynamics around E when the noise is sufficiently small. Theorem 4.1. Let Xt be the solution of system with initial value X Γ. Assume that R > 1. If σ < µ, then 1 t lim sup E[ Xs E ]ds a η 1 + v µ σ v σ, 6 t t kξ k where ξ = min {δ, a, aµ } σ 4k Proof. Consider a Lyapunov functional as follows η 1 = λα + α 1 x + α v + α 3 x v a + δ. aδ α + α v Ṽ X = η 1 Ṽ 1 X + ṼX + η Ṽ 3 X, with Ṽ 1 X = x x x fx, v x fs, v ds + y φ y y + k 1v φ v v, Ṽ X = 1 [ x x + y y ] and Ṽ 3 X = 1 v v, where x = λ δ and η 1, η, k 1, are positive constants to be determined later. Using Itô s formula, we have dṽ1 = LṼ1 k 1 σv v dbt, where LṼ1 = 1 fx, v λ δx + fx, v fx, vv fx, vv + k fx, v fx, v 1 k a y y ky v k 1µv + a + µ + k 1σ v.

7 Stochastic viral infection model 441 Choosing k 1 = a k and using λ = δx + fx, v v, fx, v v = ay and y = µv, we get k LṼ1 = δ α + α v λα + α 1 x + α v + α 3 x v x x +ay 4 fx, v fx, v y vfx, v yv fx, v yv y v fx, v fx, v ay α + α 1 xα + α 3 xv v v α + α 1 x + α v + α 3 xvα + α 1 x + α v + α 3 xv + aσ v k. By using the arithmetic-geometric inequality, we have 4 fx, v fx, v y vfx, v yv fx i, v yv y v fx, v. fx, v δ α + α v Therefore, LṼ1 λα + α 1 x + α v + α 3 x v x x + aσ v k. Further, we have dṽ = LṼ, where LṼ = δx x ay y δ + ax x y y δ + a δ x x 3a a 4 y y. Moreover, we have dṽ3 = LṼ3 σvv v db 1 t, where Therefore, LṼ3 = v v ky y µv v + σ v k µ σ y y µ σ v v + σ v. LṼ = η 1LṼ1 + LṼ + η LṼ3 [ ] δ α + α v η 1 λα + α 1 x + α v + α 3 x v x x + aσ v k δ + a 3a δ x x a 4 η k y y µ σ η µ σ v v + η σ v.

8 44 Marouane Mahrouf et al. Choosing η = aµ σ, we obtain k LṼ [ δ α + α v η 1 λα + α 1 x + α v + α 3 x v a y y aµ σ v v + 4k a + δ + δ a η1 a ] x x k + aµ σ v k v σ. Choosing η 1 = λα + α 1 x + α v + α 3 x v a + δ, we get aδ α + α v LṼ ξ[ x x + y y + v v ] η1 a + k + aµ σ v v σ. k where ξ = min {δ, a, aµ } σ. Thus, 4k E[Ṽ Xt ] Ṽ X ξ η1 a + Consequently, lim sup t 1 t [ xs E x + ys y + vs v ] k + aµ σ v k E[ Xs E ]ds 5 Numerical simulations v σ t. a η 1 + µ σ v v σ. kξ k In this section, we give some numerical simulations to illustrate our analytical results. Firstly, we choose the following parameter values: λ = 1, δ =.139, β =.4 1 5, a =.7, k =.5, µ = 3, α = 1, α 1 =.1, α =.1 and α 3 =.1. By calculation, we have R =.357 < 1. Hence, system has a infection-free equilibrium E ,,. By Theorem 3.1, the solution of system converges to E f see Figure 1. In this case, the virus is cleared and the infection dies out. ds Secondly, we choose β = and we keep the other parameter values. We have R = > 1 and the solution of system fluctuates around the chronic infection equilibrium E 85.78, , The figure demonstrates this result.

9 Stochastic viral infection model Uninfected cells Infected cells Virus 6 4 Virus Uninfected cells Infected cells Figure 1: Demonstration of extinction of the infection when R < 1. 6 Discussion and conclusion In this work, we have proposed a stochastic viral infection models with general functional response. We first proved the existence and uniqueness of the positive solution with positive initial value in order to ensure the well-posedness of the problem. In the absence of perturbations σ =, the model has two equilibrium namely: the infection-free equilibrium E f and the chronic infection equilibrium E. In this case, the global stability of these two equilibria are fully determined by threshold value R. In the presence of perturbations, E f is also an equilibrium of system which is globally asymptotically stable when R < 1, for all intensities of white noise. Therefore, we deduce that the white noise has no effect on the dynamics of if R < 1. When R > 1 and σ, the chronic infection equilibrium E is not an equilibrium of system. In this case, the solution of system fluctuates around E see Fig..

10 444 Marouane Mahrouf et al Uninfected cells Infected cells Virus 4 Virus Uninfected cells Infected cells References Figure : Dynamics of system around E when R > 1. [1] K. Hattaf, N. Yousfi, A class of delayed viral infection models with general incidence rate and adaptive immune response, International Journal of Dynamics and Control, 4 16, [] J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., , [3] D. L. DeAngelis, R. A. Goldstein, R. V. O Neill, A model for trophic interaction, Ecology, , [4] P. H. Crowley, E. K. Martin, Functional responses and interference within and between year classes of a dragonfly population, J. North. Am. Benth. Soc., ,

11 Stochastic viral infection model 445 [5] X. Zhou, J. Cui, Global stability of the viral dynamics with Crowley-Martin functional response, Bull. Korean Math. Soc., 48 11, no. 3, [6] K. Hattaf, N. Yousfi, A. Tridane, Stability analysis of a virus dynamics model with general incidence rate and two delays, Applied Mathematics and Computation, 1 13, [7] J. Adnani, K. Hattaf, N. Yousfi, Stability analysis of a stochastic SIR epidemic model with specific nonlinear incidence rate, International Journal of Stochastic Analysis, 13 13, [8] J. Adnani, K. Hattaf, N. Yousfi, Analysis of a stochastic SIRS epidemic model with specific functional response, Applied Mathematical Sciences, 1 16, [9] E. M. Lotfi, M. Maziane, K. Hattaf, N. Yousfi, Partial differential equations of an epidemic model with spatial diffusion, International Journal of Partial Differential Equations, 14 14, [1] M. Maziane, E. M. Lotfi, K. Hattaf, N. Yousfi, Dynamics of a class of HIV infection models with cure of infected cells in eclipse stage, Acta Biotheoretica, 63 15, no. 4, [11] X. Zhuo, Analysis of a HBV infection model with non-cytolytic cure process, IEEE 6th International Conference on Systems Biology, 1, [1] X. Mao, Stochastic Differential Equations and Applications, Second edition, Horwood Publishing Chichester, UK, 7. [13] N. Yousfi, K. Hattaf, A. Tridane, Modeling the adaptive immune response in HBV infection, Journal of Mathematical Biology, 63 11, no. 5, Received: July 9, 16; Published: August 8, 16