Spatiotemporal Dynamics of an HIV Infection Model with Delay in Immune Response Activation

Mehdi Maziane,Noura Yousfi,Khalid Hattaf

doi:10.1155/2018/3294268

Abstract

We propose and analyse an human immunodeficiency virus (HIV) infection model with spatial diffusion and delay in the immune response activation. In the proposed model, the immune response is presented by the cytotoxic T lymphocytes (CTL) cells. We first prove that the model is well-posed by showing the global existence, positivity, and boundedness of solutions. The model has three equilibria, namely, the free-infection equilibrium, the immune-free infection equilibrium, and the chronic infection equilibrium. The global stability of the first two equilibria is fully characterized by two threshold parameters that are the basic reproduction number R0 and the CTL immune response reproduction number R1. The stability of the last equilibrium depends on R0 and R1 as well as time delay τ in the CTL activation. We prove that the chronic infection equilibrium is locally asymptotically stable when the time delay is sufficiently small, while it loses its stability and a Hopf bifurcation occurs when τ passes through a certain critical value.

Highlights

human immunodeficiency virus (HIV) is a virus that attacks the CD4+ T cells and reduces their number in the body
We have studied an HIV infection model including infected cells in eclipse stage and delay in the activation of cytotoxic T lymphocytes (CTL) immune response
The model is governed by reaction diffusion equations and the transmission process is modeled by a specific nonlinear incidence rate that includes many types of special incidence functions as special cases

Summary

Introduction

HIV is a virus that attacks the CD4+ T cells and reduces their number in the body. It is known that when the number of these cells is less than 200 cells per μl, the patient enters the phase of acquired immunodeficiency syndrome (AIDS). Rong et al [3] extended the model of [2] by including the infected cells in eclipse stage (unproductive infected cells) and considered that a portion of these cells returns to the uninfected state. In 2014, Hu et al [4] replaced the bilinear incidence rate in [3] by a saturated infection rate and they studied the global stability of equilibria. In 2016, Maziane et al [11] generalized and extended the model of Lv et al [10] by considering the mobility of cells and virus. Where T(x, t), E(x, t), I(x, t), V(x, t), and C(x, t) represent the densities of uninfected CD4+ T cells, unproductive infected cells, productive infected cells, and free virus particles and CTL cells at location x and time t, respectively.

Well-Posedness and Equilibria

Stability Analysis and Hopf Bifurcation

Conclusion