Stability and Hopf Bifurcation in a Delayed HIV Infection Model with General Incidence Rate and Immune Impairment.

Fuxiang Li,Wanbiao Ma,Dan Li,Zhichao Jiang

doi:10.1155/2015/206205

Abstract

We investigate the dynamical behavior of a delayed HIV infection model with general incidence rate and immune impairment. We derive two threshold parameters, the basic reproduction number R 0 and the immune response reproduction number R 1. By using Lyapunov functional and LaSalle invariance principle, we prove the global stability of the infection-free equilibrium and the infected equilibrium without immunity. Furthermore, the existence of Hopf bifurcations at the infected equilibrium with CTL response is also studied. By theoretical analysis and numerical simulations, the effect of the immune impairment rate on the stability of the infected equilibrium with CTL response has been studied.

Highlights

Mathematical models have been proved to be valuable in understanding the dynamics of viral infection
Cytotoxic T lymphocyte (CTL) cells play a significant role in antiviral defense by attacking virus-infected cells
In order to study the role of the population dynamics of the viral infection with cytotoxic T lymphocyte (CTL) response, Nowak and Bangham et al proposed a basic viral infection model describing the interactions between a replicating virus population and a specific antiviral CTL response, which takes into account four populations: uninfected cells, actively infected cells, free virus, and CTL cells

Summary

Introduction

Mathematical models have been proved to be valuable in understanding the dynamics of viral infection (see, e.g., [1,2,3,4,5,6,7,8]). In order to study the role of the population dynamics of the viral infection with CTL response, Nowak and Bangham et al proposed a basic viral infection model describing the interactions between a replicating virus population and a specific antiviral CTL response, which takes into account four populations: uninfected cells, actively infected cells, free virus, and CTL cells (see, e.g., [1,2,3,4, 9, 10]). The infected cells are separated into two distinct compartments, latently infected and actively infected. These latently infected cells do not produce virus and can evade from viral cytopathic effects and host immune mechanisms (see, e.g., [17,18,19,20]). The following model with latent infection and CTL response has been proposed (see, e.g., [11]):

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