Stability Analysis for Viral Infection Model with Multitarget Cells, Beddington-DeAngelis Functional Response, and Humoral Immunity

Xinxin Tian,Jinliang Wang

doi:10.1155/2015/654507

Abstract

We formulate a (2n+2)-dimensional viral infection model with humoral immunity,nclasses of uninfected target cells and nclasses of infected cells. The incidence rate of infection is given by nonlinear incidence rate, Beddington-DeAngelis functional response. The model admits discrete time delays describing the time needed for infection of uninfected target cells and virus replication. By constructing suitable Lyapunov functionals, we establish that the global dynamics are determined by two sharp threshold parameters:R0andR1. Namely, a typical two-threshold scenario is shown. IfR0≤1, the infection-free equilibriumP0is globally asymptotically stable, and the viruses are cleared. IfR1≤1<R0, the immune-free equilibriumP1is globally asymptotically stable, and the infection becomes chronic but with no persistent antibody immune response. IfR1>1, the endemic equilibriumP2is globally asymptotically stable, and the infection is chronic with persistent antibody immune response.

Highlights

In the study of mathematical models of infectious diseases in vivo, it is an important problem to predict whether the infection disappears or the pathogens and the immune system persist
CTLs can kill infected cells or they can secrete soluble factors, which can inhibit viral replication. Another immune response to viral infection is antibody immune response, which is widely developed to analyze the dynamics of infections agents such as HIV and malaria
The global stability result for the equilibria is new for in-host models with Beddington-DeAngelis functional response, antibody immune response, and intracellular delays

Summary

Introduction

In the study of mathematical models of infectious diseases in vivo, it is an important problem to predict whether the infection disappears or the pathogens and the immune system persist. Discrete Dynamics in Nature and Society models is the Volterra-type function which was applied in literatures [4,5,6,7,8,9] to prove global stability of the steady states of viral infections models with discrete intracellular delay and distributed delay They all ignore the antibody immune response. From system (2), the rate of infection of these viral dynamics models is assumed to be bilinear in the virus V and the uninfected CD4+ T cells xi. The global stability result for the equilibria is new for in-host models with Beddington-DeAngelis functional response, antibody immune response, and intracellular delays.

Preliminaries

Main Results

Numerical Example

Conclusions and Discussion