Bifurcation Analysis of a Delayed Worm Propagation Model with Saturated Incidence

Zizhen Zhang,Yougang Wang,Luca Guerrini

doi:10.1155/2018/7619074

Abstract

This paper is concerned with a delayed SVEIR worm propagation model with saturated incidence. The main objective is to investigate the effect of the time delay on the model. Sufficient conditions for local stability of the positive equilibrium and existence of a Hopf bifurcation are obtained by choosing the time delay as the bifurcation parameter. Particularly, explicit formulas determining direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are derived by using the normal form theory and the center manifold theorem. Numerical simulations for a set of parameter values are carried out to illustrate the analytical results.

Highlights

Worms, as one kind of malicious codes, have become one of the main threats to the security of networks
The dynamical behaviors of a delayed SVEIR worm propagation model with saturated incidence are discussed based on the work in literature [29]
The dynamical behaviors of the model are investigated from the point of view of local stability and Hopf bifurcation both analytically and numerically

Summary

Introduction

As one kind of malicious codes, have become one of the main threats to the security of networks. SIR model considered the immunity of the nodes in which the worms have been cleaned, it assumes that the recovered hosts have permanent immunity This is not consistent with the reality in networks, because they may be infected by some new emerging worms again. Based on this consideration, the SEIR (Susceptible-Exposed-InfectiousRecovered) model [13, 14] and the SEIRS (SusceptibleExposed-Infectious-Recovered-Susceptible) model [11, 15] are proposed to describe the dynamics of worm propagation in networks. In this paper, we extend system (1) by incorporating the time delay due to the latent period of the worms in the exposed hosts into system (1) and obtain the following delayed worm propagation model: p) A μS (t) γV (t) , γ) V (t) , σβV (t) I μE (t).

Existence of Hopf Bifurcation

Properties of the Hopf Bifurcation

Numerical Simulation

Conclusions

A: Recruitment rate of the susceptible host p