Global Dynamics of HIV Infection of CD4+T Cells and Macrophages

A M Elaiw,A S Alsheri

doi:10.1155/2013/264759

Abstract

We study the global dynamics of an HIV infection model describing the interaction of the HIV with CD4+T cells and macrophages. The incidence rate of virus infection and the growth rate of the uninfected CD4+T cells and macrophages are given by general functions. We have incorporated two types of distributed delays into the model to account for the time delay between the time the uninfected cells are contacted by the virus particle and the time for the emission of infectious (matures) virus particles. We have established a set of conditions which are sufficient for the global stability of the steady states of the model. Using Lyapunov functionals and LaSalle's invariant principle, we have proven that if the basic reproduction numberR0is less than or equal to unity, then the uninfected steady state is globally asymptotically stable (GAS), and if the infected steady state exists, then it is GAS.

Highlights

Many mathematical models have been developed to describe the interaction of human immunodeficiency virus (HIV) with the immune system [1]
We incorporate two types of distributed delays into the model to account for the time delay between the time the uninfected cells are contacted by the virus particle and the time for the emission of infectious virus particles
We have proposed an HIV dynamics model describing the interaction of the HIV with CD4+ T cells and macrophages

Summary

Introduction

Many mathematical models have been developed to describe the interaction of human immunodeficiency virus (HIV) with the immune system [1]. More accurate modeling was developed in 1997 when Perelson et al [19] observed that, after the rapid first phase of decay during the initial 1-2 weeks of antiretroviral treatment, plasma virus levels declined at a considerably slower rate This second phase of viral decay was attributed to the turnover of a longer-lived virus reservoir of infected cells. To model the second class of target cells, two additional equations describing the population dynamics of the uninfected and infected macrophages have to be added to the basic model (1)–(3) (see [20, 21]). Elaiw et al [26] studied the global stability of HIV model with BeddingtonDeAngelis functional response and one kind of discrete time delay. Elaiw [27] studied the global dynamics of a delay HIV model with saturated functional response. Using Lyapunov functionals and LaSalle’s invariant principle, we prove that if the basic reproduction number R0 is less than or equal to unity, the uninfected steady state is globally asymptotically stable (GAS), and if the infected steady state exists, it is GAS

The Model

Conclusion