Stability and Hopf Bifurcation of a Computer Virus Model with Infection Delay and Recovery Delay

Haitao Song,Qiaochu Wang,Weihua Jiang

doi:10.1155/2014/929580

Abstract

A computer virus model with infection delay and recovery delay is considered. The sufficient conditions for the global stability of the virus infection equilibrium are established. We show that the time delay can destabilize the virus infection equilibrium and give rise to Hopf bifurcations and stable periodic orbits. By the normal form and center manifold theory, the direction of the Hopf bifurcation and stability of the bifurcating periodic orbits are determined. Numerical simulations are provided to support our theoretical conclusions.

Highlights

In recent years, the computer networks have become more and more popular, and people can find many useful things through computer networks
We show that the time delay can destabilize the virus infection equilibrium and give rise to Hopf bifurcations and stable periodic orbits
By the Kermack and McKendrick SIR epidemic model [3], the computer virus models were proposed in [4,5,6,7] which analyzed the spread of computer virus by epidemiological models

Summary

Introduction

The computer networks have become more and more popular, and people can find many useful things through computer networks. For τ1 > 0 and τ2 > 0, Ren et al [12] constructed a Lyapunov function by linearized equations of E∗ and obtained the sufficient conditions for global stability of the virus infection equilibrium. We construct a suitable Lyapunov function and obtain the sufficient conditions of global stability for the virus infection equilibrium when τ1 > 0 and τ2 > 0. Analyzed the global stability of a SIR epidemic model with a time delay.

Preliminaries

Global Asymptotic Stability

Conclusions