Global Stability of HIV Infection of CD4+ T Cells and Macrophages with CTL Immune Response and Distributed Delays

We study the global stability of a human immunodeficiency virus (HIV) infection model with Cytotoxic T Lymphocytes (CTL) immune response. The model describes the interaction of the HIV with two classes of target cells, CD4+ T cells and macrophages. Two types of distributed time delays are incorporated into the model to describe the time needed for infection of target cell and virus replication. Using the method of Lyapunov functional, we have established that the global stability of the model is determined by two threshold numbers, the basic reproduction number and the immune response reproduction number . We have proven that, if , then the uninfected steady state is globally asymptotically stable (GAS), if , then the infected steady state without CTL immune response is GAS, and, if , then the infected steady state with CTL immune response is GAS. 1. Introduction One of the most diseases that have attracted the attention of many mathematicians is the acquired immunodeficiency syndrome (AIDS) caused by human immunodeficiency virus (HIV). HIV infects the CD4+ T cell which plays the central role in the immune system. Mathematical modeling and model analysis of HIV dynamics are important to discover the dynamical behaviors of the viral infection process and estimating key parameter values which leads to development of efficient antiviral drug therapies. Several mathematical models have been proposed to describe the HIV dynamics with CD4+ T cells [1–15]. In these papers, the Cytotoxic T Lymphocytes (CTL) immune response was not taken into account. The role of CTL is universal and necessary to eliminate or control the disease during viral infections. In particular, as a part of innate response, CTL plays a particularly important rate in antiviral defense by attacking infected cells. The basic HIV infection model which takes into consideration the CTL immune response has been proposed in [16] as The state variables describe the plasma concentrations of , the uninfected CD4+ T cells; , the infected CD4+ T cells; , the free virus particles; and , the CTL cells at time . Here, (1) describes the population dynamics of the uninfected CD4+ T cells, where represents the rate of new uninfected cells that are generated from sources within the body, is the death rate constant, and is the infection rate constant at which a target cell becomes infected via contacting with virus. Equation (2) describes the population dynamics of the infected CD4+ T cells and shows that they die with rate constant and are killed by the CTL immune response with rate constant . Equation (3)