The Propagation Mechanism Of Computer Virus Across The Internet: Three Novel Dynamic Models

With the advance of computer software and hardware technology and communicationtechnology, the number and sort of computer viruses have increased dramatically, whichcauses huge losses to the human society. Therefore, how to prevent and kill computervirus is an urgent task in the network security field. The most commonly used anti-virustechnology is, probably, to install effective anti-virus software and firewall. Because ofthe hysteresis and incompleteness of anti-virus technology, the use anti-virus software isnot enough to forecast and suppress computer virus. To control the spread of computervirus over the Internet, we must understand the rules governing its propagation.Therefore, establishing reasonable computer virus propagation models by consideringthe characteristics of computer virus and, by model analysis, understanding the spreadlaw of the virus over the network, are a currently hot topic of research.This thesis aims to establish and research dynamic system model of computer viruspropagation, and then puts forword the effective prevention measures. The maincontributions made are summarized as follows:1: By considering the effect of anti-virus ability, we propose a novel computer viruspropagation model and analyze its dynamical behavior. First, we study the existence ofequilibria and investigate their local asymptotic stability. We find that, depending on theanti-virus ability, the system may undergo two local bifurcations: backward bifurcationand Hopf bifurcation, and a global B-T bifurcation, which are instructive for us tochoose appropriate virus-controlling strategy. Next, we give some criteria for the globalasymptotic stability of the equilibria and a bistable state (one stable virus-freeequilibrium and one stable virus equilibrium, or two stable virus equilibria) byemploying the Dulac’s criteria and the Poincaré-Bendixson theorem. In this case, theinitial condition is critical for the eventual steady state of the system.2. By considering latent and immuse characteristics of computer virus, we propose adelayed computer virus propagation model and study its dynamic behaviors. First, wegive the threshold value R0determining whether the virus dies out completely, study thelocal asymptotic stability of the equilibria of this model and it is found that, dependingon the threshold value R0, the stability of virus-free equilibrium may change; whereasdepending on delays, the stability of virus equilibrium may change, i.e., a Hopfbifurcation may occur in the model. Next, we prove that, if R0=1, the virus-free equilibrium is globally attractive; and when R0<1, it is globally asymptotically stable,which indicates that making threshold value R0less than one is a feasible strategy toprevent the computer from spread over the network. Finally, a sufficient criterion for theglobal stability of the virus equilibrium is obtained.3. By considering the ignitionability of computer virus, we propose a model withvarying delay for computer virus propagation in network. Under this model, we give thethreshold value determining whether or not the virus finally dies out, and study thecorresponding stabilities of the virus-free and virus equilibrium. It is found the modelmay undergo two kinds of bifurcations, one being transcritical bifurcation and the otherbeing Hopf bifurcation, which may provide theoretical basis to stop spread of computervirus. Next, from economic and security perspective, an optimal control in our model istaken into account and the objective functional of this system is described, the existenceof an optimal control is shown. Finally, we derive the optimal control and the optimalitysystem. It is comparatively shown that the optimal control is rather effective forcontrolling the spread of virus.