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

A M Elaiw,R M Abukwaik,E O Alzahrani

doi:10.1155/2013/653204

Abstract

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 R 0 and the immune response reproduction number R 0 ∗. We have proven that, if R 0 ≤ 1, then the uninfected steady state is globally asymptotically stable (GAS), if R 0* ≤ 1 < R 0, then the infected steady state without CTL immune response is GAS, and, if R 0* > 1, then the infected steady state with CTL immune response is GAS.

Highlights

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)
We have proposed an HIV infection model describing the interaction of the HIV with two classes of target cells, CD4+ T cells and macrophages, taking into account the Cytotoxic T Lymphocytes (CTL) immune response
Two types of distributed time delays have been incorporated into the model to describe the time needed for infection of target cell and virus replication

Summary

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. Several mathematical models have been proposed to describe the HIV dynamics with CD4+ T cells [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15] In these papers, the Cytotoxic T Lymphocytes (CTL) immune response was not taken into account. The basic HIV infection model which takes into consideration the CTL immune response has been proposed in [16] as ẋ (t) = λ − dx (t) − βx (t) V (t) ,

Objectives

Conclusion