A computer virus spreading model with nonlinear infectivity on scale-free network

Abstract

In this paper, a novel epidemic model of computer virus on scale-free network with nonlinear infectivity is proposed. The spreading dynamics of the virus was analyzed. The spreading critical threshold for the model was presented. Theoretical analyses indicate that the outbreak of the virus is entirely determined by the threshold. Numerical simulations confirmed the analytical results. Introduction Internet provides conveniently various services and information to people. Meanwhile, the easy access and wide usage of Internet make it a primary target for viruses. In the past, outbreaks of computer viruses have brought about huge financial losses. As the major means of defending against viruses, antivirus software is more a technique than a science; it cannot predict the evolution of viruses and hence, cannot provide global prevention/control policies. With the dramatic increase in the number of virus, relying solely on measures to clear the viruses can not meet the ultimate needs. Facing the severe situation of virus’ destroy, it’s urgent for people to establish epidemic model, explore virus propagation characteristics and advance controlling schemes in the network information security field. The computer viruses bear a striking resemblance to their biological counterparts. Cohen [1] and Murray [2] inventively suggested to exploit the tools developed in epidemic dynamics of infectious diseases to study the spreading behavior of computer viruses. Following this idea, Kephart and White [3] established the first epidemic model of computer viruses. From then on, most epidemic models usually have been proposed based on the assumption that Internet is homogeneous network [4-6]. At the end of last century empirical studies report that Internet has scale-free properties [7-9]. Studies on epidemic spreading in SF networks present us with surprising result, such as absence of any epidemic threshold, hierarchical spread of epidemic outbreaks [10-12], and so on. Most previous work toward this direction was limited to three simplistic models: the SI model [13-14], the SIS model [15-16] and the SIR model [17-18]. However, most previous works mainly focus on the impact of the underlying topology and assume that the transmission is uniformly distributed among all links, that is, each infected individual will try to contact all its susceptible neighbors once within one time step. In fact, this kind of uniform transmission is induced from the assumption that each node’s potential infectivity, counted by its possible maximal contribution to the propagation process within one time step, is strictly equal to its degree, i.e., ‘infectivity≡degree’. The node with very large degree is called hub in computer networks, whereas the node with great infectivity in epidemic contact networks is termed super-spreader. Thus, the above assumption means hub≡super-spreader. This assumption fails to consider that the infectivity of an infected computer is limited, because computer’s processing capacity and network bandwidth are limited. In this paper, we aim to overlook the above defects and understand the spreading behaviors of computer viruses in Internet. We present a novel SIDS (susceptible-infected-detached-susceptible) epidemic model of computer virus to investigate the impact of nonlinear infectivity in Internet with the scale-free property. We obtain the corresponding spreading threshold and analyze the globally dynamic behaviors of the virus. Then, the numerical simulations are given, which are well consistent with the theoretical results. The remainder of this paper is structured as follows. In Section 2, we will briefly formulate this novel epidemic model. Then in Section 3, we determine the threshold value and analyze the International Conference on Information Sciences, Machinery, Materials and Energy (ICISMME 2015) © 2015. The authors Published by Atlantis Press 1684 globally dynamic behaviors of the virus. In Section 4, we carry out extensive numerical simulations to verify the theoretical analysis in Section 3. At last, in Section 5 we summarize this work. Model formulation It is well known [11-13] that the node degrees of Internet asymptotically follow a power law distribution, ( ) P k k γ − � , where ( ) P k stands for the probability that a node chosen randomly from Internet is of degree k . For this model in Internet, each computer is represented by a vertex of the network and the edges are communication links between them, along which the infection may spread. Let ( ) k S t , ( ) k I t and ( ) k D t be the densities of susceptible, infected, and quarantined vertexes of degree k at time t . Obviously, they must satisfy the normalized condition ( ) ( ) ( ) 1 k k k S t I t D t + + = if the total number of the node is fixed. The infection transmission is defined by the spreading rateλ , at which each susceptible computer acquires the infection from an infected neighbor during one time step. An infected computer goes through a quarantine period firstly before becoming susceptible again. The rate constant of quarantine for infected computers is denoted by μ . τ is the average quarantine period. All parameters are positive. Then by applying the mean-field technique to the above assumptions, we have the following dynamics model based on delay differential equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) k k k