Stability and Hopf Bifurcation in an HIV-1 Infection Model with Latently Infected Cells and Delayed Immune Response

Haibin Wang,Rui Xu

doi:10.1155/2013/169427

Abstract

An HIV-1 infection model with latently infected cells and delayed immune response is investigated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria is established and the existence of Hopf bifurcations at the CTL-activated infection equilibrium is also studied. By means of suitable Lyapunov functionals and LaSalle’s invariance principle, it is proved that the infection-free equilibrium is globally asymptotically stable if the basic reproduction ratio for viral infectionR0≤1; if the basic reproduction ratio for viral infectionR0>1and the basic reproduction ratio for CTL immune responseR1≤1, the CTL-inactivated infection equilibrium is globally asymptotically stable. If the basic reproduction ratio for CTL immune responseR1>1, the global stability of the CTL-activated infection equilibrium is also derived when the time delayτ=0. Numerical simulations are carried out to illustrate the main results.

Highlights

Mathematical and computational models of the human immune response under HIV-1 infection have received great attention in recent years [1,2,3,4,5,6,7,8,9]
It is a useful tool of better understanding disease dynamics and making prediction of disease outbreak and evaluations of prevention strategies and drug therapy strategies used against HIV-1 infection
We study the global stability of the infection-free equilibrium and the cytotoxic T lymphocytes (CTLs)-inactivated infection equilibrium and discuss the global stability of the CTL-activated infection equilibrium when τ = 0

Summary

Introduction

Mathematical and computational models of the human immune response under HIV-1 infection have received great attention in recent years [1,2,3,4,5,6,7,8,9]. A basic mathematical model describing HIV-1 infection dynamics that has been studied in [1, 2] is of the form ẋ (t) = λ − dx (t) − βx (t) V (t) ,. We assume that the free virus interacts with the uninfected cells to produce actively infected cells at rate qβx(t)y(t)/[1 + αy(t)] and latently infected cells at rate (1−q)βx(t)y(t)/[1+αy(t)] due to the saturation response of the infection rate, where 0 < q < 1 and α > 0. A brief remark is given to conclude our work

Equilibria and Their Local Stability

Stability and Hopf Bifurcation at the CTL-Activated Infection Equilibrium

Global Stability

Findings

Numerical Simulations and Conclusions