Stability analysis of a virus dynamics model with general incidence rate and two delays☆
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 equationsThe susceptible host cells are produced at a rate , die at a rate dx and become infected by virus at a rate . 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 systemThe first delay, , represents the time needed for infected cells to produce virions after viral entry. We assume that virus production lags by a delay 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 and are still alive at time t. We assume that the death rate for infected but not yet virus-producing cells is . Therefore, the probability of surviving from time to time t is . In addition, the delay 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 and the average life time of an immature virus is given by .
The incidence function is assumed to be continuously differentiable in the interior of and satisfies the following hypotheses:
The fundamental characteristics of our model are:
(() The incidence rate is dependent on the concentrations of uninfected cells, infected cells and virus. Moreover, It generalize most famous forms such as, and because these functions satisfy the above hypotheses on f.
() 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 and prove that the values of completely determine the global dynamics of system. If , the disease-free equilibrium is globally asymptotically stable and the disease always dies out, whereas if , 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 . In Section 4, we establish the sufficient conditions for global stability of the infected steady state when 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 be the Banach space of continuous functions mapping the interval into with the topology of uniform convergence, where . By the fundamental theory of functional differential equations
Global stability when
For an arbitrary equilibrium , the characteristic equation is given byThe characterization of the local stability of the disease-free equilibrium is given by the following theorem. Theorem 3.1 Let us define . If , then is locally asymptotically stable. If , then is unstable.
Proof
At , (14) reduces toClearly, is a root
Global stability when
Note that the disease-free equilibrium is unstable when . Now, we establish a set of conditions which are sufficient for the global stability of the chronic infection equilibrium . Here, we assume that and the function f satisfies the following: Theorem 4.1 Assume and (19) hold. Then the chronic infection equilibrium is globally asymptotically stable. Proof Consider the following Lyapunov functional
Application
In this section, we consider the following model as an application.We distinguish three particular cases of this model which are very used in the literature.
Case I: . 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, , is globally asymptotically stable if the basic reproduction number satisfies . In this case, all positive solutions converge to 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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