Research On Propagating Laws And Containing Policies Of Network Virus

The so-called network viruses are malware that can propagate through the Internet. As a double-edged sword, while the Internet has enhanced the efficiency of our work and improved the quality of our life greatly, it has become the paradise of virus. Network virus has long been a nightmare of mankind, leading to enormous financial losses as well as huge negative social effect. With the advent of the mobile Internet era as well as the Internet of Things era, viruses are becoming more harmful or even deadly. It is an extremely challenging subject we are faced with to minimize the loss caused by virus at lower cost; its solution helps cleanse the network environment and hence has tremendous commercial value.Virus patches are the natural enemy of network viruses; a patch can clear all viruses it can recognize that stay in individual smart devices(nodes, for short). However, it is far from enough to contain viruses only by patches; indeed, the continuously evolving viruses can tactfully circumvent the detection done by outdated patches, and the release of a new patch always lags behind the appearance of a cunning virus it can recognize. For the purpose of containing the prevalence of virus, various measures, ranging from technological means and quality training to self-regulation and network legislation, must be taken simultaneously.The propagating dynamics of network virus is aimed at establishing dynamical models capturing the propagating rules of virus by taking into full account various factors, especially countermeasures, that have significant impact on the spread of virus, understanding the propagating laws of virus and evaluating the cost and effectiveness of different virus-containing policies through in-depth model analysis; thereby presenting a feasible set of cost-effective virus-containing policies.This thesis is devoted to the understanding of propagating laws of network virus and the evaluation of virus-containing policies. The major contributions are listed below.① SLBS models based on homogeneously mixed networkThe so-called destructive viruses are a class of network viruses that are aimed at wreaking havoc. For that purpose, a destructive virus staying in a node usually spends an incubation period before outburst, so that more nodes can be infected. It is the first time that the propagating laws of destructive virus are studied in this thesis(the related virus propagation models are all referred to as SLBS models).First, in the absence of external infecting source, a set of two SLBS model based on homogeneously mixed network is proposed. Studies show that for either of the two models, a nearly globally stable viral equilibrium bifurcates from an originally globally stable virus-free equilibrium when the basic reproduction number exceeds the unity. As a result, the virus on the network tends to extinction or persists according as the basic reproduction number is below or above the unity.Second, in the presence of external infecting sources, a set of four SLBS models based on homogeneously mixed network is suggested, where the fourth model considers a generic nonlinear infection rate rather than the conventional linear one. Studies show that for any of the four models, there is always a globally stable viral equilibrium. As a result, the virus on the network always persists.The above results show that the prevalence of virus can be contained effectively by adjusting some model parameters.② SIPS models based on homogeneously mixed networksAll previous models assume that some central node distributes patches directly to all other online nodes. Due to the limitation on the traffic of each online node, however, this centralized patch distribution strategy won’t work. Rather, patches should be distributed to all online nodes in a decentralized way; every online node that has received a patch forwards it to some neighboring nodes. This is the first time that the propagating laws of virus under the decentralized patch distribution strategy are studied in this thesis(the related virus propagation models are all referred to as SIPS models).First, in the presence of infected removable storage media, a SIPS model based on homogeneously mixed network is formulated. Studies show that a nearly globally stable viral equilibrium bifurcates from an originally globally stable viral equilibrium when a bifurcation parameter goes across a critical value. As a result, the virus on the network always persists.Second, in the presence of both infected removable storage media and offline infected nodes, a SIPS model based on homogeneously mixed network is advised. Studies show that the model always admits a globally stable viral equilibrium. As a result, the virus on the network always persists.The above results show that the prevalence of virus can be contained effectively by adjusting some model parameters.③ Virus propagation models based on scale-free networkScale-free networks are a class of networks for which the node degree obeys a power-law distribution. The Internet and World Wide Web are scale-free networks approximately. The propagating laws of virus on scale-free network are studied in this thesis.First, in the absence of external infecting source, a SLBS model based on scale-free network is presented. Criteria for the eventual extinction or persistence of destructive virus are given, respectively.Second, in the presence of infected removable storage media, a SLBS model based on scale-free network is established. A criterion for the global stability of a viral equilibrium is given.Third, a SIS model based on reduced scale-free network is introduced. Studies show that the model always admits a globally stable viral equilibrium. As a result, the virus on the network always persists.The above results show that the prevalence of virus can be contained effectively by adjusting some model parameters or the network structure.④ Node-level virus propagation modelsIn the mobile Internet era, networks are time-varying and, hence, their structures cannot be figured out via technical means. The propagating laws of virus on arbitrary network are studied in this thesis(the related virus propagation models are all referred to as node-level models).First, a node-level SLBS model is put forward. Studies show that the virus on a network tends to extinction or persists according as the maximum eigenvalue of the adjacency matrix of the network is below or above a critical value.Second, a node-level SIPS model is considered. Studies show that the virus on a network tends to extinction or persists according as the maximum eigenvalue of a matrix associated with the network is below or above a critical value.Third, a heterogeneous node-level SIRS model is recommended, where what heterogeneity means is that every node has personalized parameters. Studies show that the virus on a network tends to extinction or persists according as the maximum eigenvalue of a matrix associated with the network.The above results show that the prevalence of virus can be contained effectively by adjusting the network structure.⑤ An impulsive SIRS modelIn view of the fact that a large proportion of online infected nodes can be cured in a very short period of time, an impulsive SIRS model is described. Studies show that an asymptotically stable viral periodic solution bifurcates from an originally globally stable virus-free periodic solution when the basic reproduction number exceeds the unity. As a result, the virus on the network tends to extinction or persists according as the basic reproduction number is below or above the unity. The above results show that the prevalence of virus can be contained effectively by shortening the development period of new patches.⑥ Virus-containing strategiesThe problem of how to minimize the loss caused by network virus at lower cost is studied in this thesis.First, an optimal control problem is formulated based on a controlled heterogeneous node-level SIRS model and a quadratic objective functional. The existence of an optimal control for the problem is shown, the optimality system for the problem is presented, and a few numerical examples are given.Second, by differentiating the patch distribution network from the virus propagation network, an optimal control problem is formulated based on a controlled heterogeneous SIPS model and a generic objective functional. The structural feature of an optimal control for the problem is described provided that the Lagrange is concave or strictly convex, and a few numerical examples are given.The above results show that based on the personalized feature of every node, new patches can be distributed in a cost-effective way.