An Epidemic Model of Computer Worms with Time Delay and Variable Infection Rate

Yu Yao,Qiang Fu,Wei Yang,Ying Wang,Chuan Sheng

doi:10.1155/2018/9756982

Abstract

With rapid development of Internet, network security issues become increasingly serious. Temporary patches have been put on the infectious hosts, which may lose efficacy on occasions. This leads to a time delay when vaccinated hosts change to susceptible hosts. On the other hand, the worm infection is usually a nonlinear process. Considering the actual situation, a variable infection rate is introduced to describe the spread process of worms. According to above aspects, we propose a time-delayed worm propagation model with variable infection rate. Then the existence condition and the stability of the positive equilibrium are derived. Due to the existence of time delay, the worm propagation system may be unstable and out of control. Moreover, the threshold τ0 of Hopf bifurcation is obtained. The worm propagation system is stable if time delay is less than τ0. When time delay is over τ0, the system will be unstable. In addition, numerical experiments have been performed, which can match the conclusions we deduce. The numerical experiments also show that there exists a threshold in the parameter a, which implies that we should choose appropriate infection rate β(t) to constrain worm prevalence. Finally, simulation experiments are carried out to prove the validity of our conclusions.

Highlights

With the deep application of the Internet, network security plays a more and more important role in recent years
Considering the fact that the intrusion detection system (IDS) may lead to time delay, Yao et al [13,14,15] obtained the threshold of time delay when Hopf bifurcation occurred
E∗ = (S∗, I∗, asymptotically stable and it is unstable when τ ≥ τ0, (2) awhHenopsfybstiefumrc(a2ti)osnataistfitehse(pHo3s)it,ivtheeeqsyusitleibmriuumndeEr∗goe=s (S∗, I∗, Q∗, V∗, D∗) when τ = τ0. This implies that when the time delay τ < τ0, the system will stabilize at its equilibrium point, which is beneficial for us to implement a containment strategy; when the delay τ ≥ τ0, the system will be unstable and worms cannot be effectively controlled

Summary

Introduction

With the deep application of the Internet, network security plays a more and more important role in recent years. As the number of infected nodes increases, the true infection rate will tend to be saturated, and it can be significantly nonlinear In this case, the linear assumption will overestimate the harmfulness of the worms and lead to great waste of resources. Feng et al [27] have proposed the SIRS model with a variable infection rate which plays an important role in the spread of the Internet worm. We try to establish a realistic worm propagation model, motivated by the works [7, 27] This model can give deep insight into predicting worm spread in networks.

Model Formulation

Stability of Equilibrium and Bifurcation Analysis

Numerical Simulations and Simulations Experiments

Conclusions