Elsevier

Applied Mathematics and Computation

Volume 221, 15 September 2013, Pages 514-521
Applied Mathematics and Computation

Stability analysis of a virus dynamics model with general incidence rate and two delays

https://doi.org/10.1016/j.amc.2013.07.005Get rights and content

Abstract

The aim of this work is to study the dynamical behavior of a virus dynamics model with general incidence rate and two delays. The first delay represents the time from the virus entry to the production of new viruses and the second delay corresponds to the time necessary for a newly produced virus to become infectious. Lyapunov functionals are constructed and LaSalle invariance principle for delay differential equations is used to establish the global asymptotic stability of the disease-free and the chronic infection equilibria. The results obtained show that the global dynamics are completely determined by the value of a certain threshold parameter called the basic reproduction number R0 and under some assumptions on the general incidence function. Our results extend the known results on delay virus dynamics considered in the other papers and suggest useful methods to control virus infection. These results can be applied to a variety of possible incidence functions that could be used in virus dynamics model as well as epidemic models.

Introduction

Recently, many mathematical delayed models have been proposed to study the dynamics of viral infections such as the human immunodeficiency virus (HIV) and the hepatitis B virus (HBV). These delayed models used different form of incidence rate. For example, in form as mass action process [6], [13], as standard incidence function [17], [18], [19], as saturated mass action [15], [16] and as Beddington–DeAngelis functional response [12]. In this paper, we consider the virus dynamics model with general incidence rate presented by Hattaf et al. [21] and described by the following nonlinear system of differential equationsẋ=λ-dx-f(x,y,v)v,ẏ=f(x,y,v)v-ay,v̇=ky-uv.The susceptible host cells (x) are produced at a rate λ, die at a rate dx and become infected by virus at a rate f(x,y,v)v. Infected cells die at a rate ay. Free virus is produced by infected cells at a rate ky and decays at a rate uv.

In this paper we incorporate two delays in model (1), obtaining the following systemẋ(t)=λ-dx(t)-fx(t),y(t),v(t)v(t),ẏ(t)=fx(t-τ1),y(t-τ1),v(t-τ1)v(t-τ1)e-α1τ1-ay(t),v̇(t)=ky(t-τ2)e-α2τ2-uv(t).The first delay, τ1, represents the time needed for infected cells to produce virions after viral entry. We assume that virus production lags by a delay τ1 behind the infection of a cell. This implies that recruitment of virus-producing cells at time t is given by the number of cells that were newly infected at time t-τ1 and are still alive at time t. We assume that the death rate for infected but not yet virus-producing cells is α1. Therefore, the probability of surviving from time t-τ1 to time t is e-α1τ1. In addition, the delay τ2 represents the time necessary for the newly produced virions to become mature and then infectious particles. The probability of survival of immature virions is given by e-α2τ2 and the average life time of an immature virus is given by 1α2.

The incidence function f(x,y,v) is assumed to be continuously differentiable in the interior of R+3 and satisfies the following hypotheses:f(0,y,v)=0,for ally0andv0,fx(x,y,v)>0,for allx>0,y0andv0,fy(x,y,v)0andfv(x,y,v)0,for allx0,y0andv0.

The fundamental characteristics of our model are:

  • ((C1) The incidence rate f(x,y,v) is dependent on the concentrations of uninfected cells, infected cells and virus. Moreover, It generalize most famous forms such as, βx,βxx+y and βx1+ax+bv+abxv because these functions satisfy the above hypotheses on f.

  • (C2) Many recent papers such as [10], [5], [12], [6], [13], [14], [17], [18], [19], [15], [16] proposed the viral models only with one delay and ignored the other delay. In our setting, the two delays in cell infection and in virus production are incorporated in our model.

By mathematical analysis, we derive a threshold value R0 and prove that the values of R0 completely determine the global dynamics of system. If R01, the disease-free equilibrium is globally asymptotically stable and the disease always dies out, whereas if R0>1, the infection persists.

This paper is organized as follows. Section 2 deals with some basic results, e.g., positivity and boundedness of solutions, basic reproduction number and existence of equilibria. In Section 3, we present the global dynamics when R01. In Section 4, we establish the sufficient conditions for global stability of the infected steady state when R0>1 In Section 5, we apply our results to virus dynamics model with a specific nonlinear incidence rate and two delays. Finally, a discussion and conclusion are given in Section 6.

Section snippets

Positivity and boundedness

In this subsection, we establish the positivity and boundedness of solutions of model (2) because this model describes the evolution of a cell population. Hence the cell numbers should remain non-negative and bounded. These properties imply the global existence of solutions.

Let C=C([-τ,0],R3) be the Banach space of continuous functions mapping the interval [-τ,0] into R3 with the topology of uniform convergence, where τ=max(τ1,τ2). By the fundamental theory of functional differential equations

Global stability when R01

For an arbitrary equilibrium E(x,y,v), the characteristic equation is given by-d-fxv-ξ-fyv-fvv-f(x,y,v)fxve-(ξ+α1)τ1fyve-(ξ+α1)τ1-a-ξfvv+f(x,y,v)e-(ξ+α1)τ10ke-(ξ+α2)τ2-u-ξ=0.The characterization of the local stability of the disease-free equilibrium is given by the following theorem.

Theorem 3.1

Let us define R0=kaufλd,0,0e-α1τ1-α2τ2.

  • If R0<1, then Ef is locally asymptotically stable.

  • If R0>1, then Ef is unstable.

Proof

At Ef, (14) reduces to(ξ+d)[ξ2+(a+u)ξ+au(1-R0e-ξ(τ1+τ2))]=0,Clearly, ξ1=-d is a root

Global stability when R0>1

Note that the disease-free equilibrium Ef is unstable when R0>1. Now, we establish a set of conditions which are sufficient for the global stability of the chronic infection equilibrium E. Here, we assume that R0>1 and the function f satisfies the following:1-f(x,y,v)f(x,y,v)f(x,y,v)f(x,y,v)-vv0,for allx,y,v>0.

Theorem 4.1

Assume R0>1 and (19) hold. Then the chronic infection equilibrium E is globally asymptotically stable.

Proof

Consider the following Lyapunov functionalW(t)=x(t)-x-xx(t)f(x,y,v)f(s,y

Application

In this section, we consider the following model as an application.ẋ(t)=λ-dx(t)-βx(t)v(t)1+δ1x(t)+δ2v(t)+δ3x(t)v(t),ẏ(t)=βe-α1τ1x(t-τ1)v(t-τ1)1+δ1x(t-τ1)+δ2v(t-τ1)+δ3x(t-τ1)v(t-τ1)-ay(t),v̇(t)=ky(t-τ2)e-α2τ2-uv(t).We distinguish three particular cases of this model which are very used in the literature.

Case I: δ1=δ2=δ3=0. In this case we obtain the basic model of virus dynamics with mass action presented by Zhu and Zou in [9]. The authors Zhu and Zou have identified the basic reproduction

Conclusion and discussion

In this paper, we have developed a virus dynamics model with general incidence rate and two delays. The global asymptotic stability of the disease-free equilibrium and the chronic infection equilibrium have been established by using suitable Lyapunov functionals and LaSalle invariance principle. We have shown that the disease-free equilibrium, Ef, is globally asymptotically stable if the basic reproduction number satisfies R01. In this case, all positive solutions converge to Ef and the virus

Acknowledgments

The authors thank the anonymous referees for very helpful suggestions and comments which led to improvement of our original paper.

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