A Study of Computer Virus Propagation on Scale Free Networks Using Differential Equations

A Study of Computer Virus Propagation on Scale Free Networks Using Differential Equations

Mohammad S. Khan (Texas A&M University – Kingsville, USA)
Copyright: © 2016 |Pages: 19
DOI: 10.4018/978-1-4666-9964-9.ch008
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Abstract

The SIR model is used extensively in the field of epidemiology, in particular, for the analysis of communal diseases. One problem with SIR and other existing models is that they are tailored to random or Erdos type networks since they do not consider the varying probabilities of infection or immunity per node. In this paper, we present the application and the simulation results of the pSEIRS model that takes into account the probabilities, and is thus suitable for more realistic scale free networks. In the pSEIRS model, the death rate and the excess death rate are constant for infective nodes. Latent and immune periods are assumed to be constant and the infection rate is assumed to be a function of the size of the total population and the size of the infected population. A node recovers from an infection temporarily with a probability p and dies from the infection with probability (1-p).
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1. Introduction

The growth of the internet has created several challenges and one of these challenges is cyber security. A reliable cyber defense system is therefore needed to safeguard the valuable information stored on a system and the information in transit. To achieve this goal, it becomes essential to understand and study the nature of the various forms of malicious entities such as viruses, trojans, and worms, and to do so on a wider scale. It also becomes essential to understand how they spread throughout computer networks.

A computer virus [1]is a malicious computer code that can be of several types, such as a trojan, worm, and so on (Aron, O’Leary, Gove, Azadegan, & Schneider, 2002; Perdisci, Lanzi, & Lee, 2008). Although each type of malicious entity has a different way of spreading over the network, they all have common properties such as infectivity, invisibility, latency, destructibility and unpredictability (Kafai, 2008).

More recently, we have also started witnessing viruses that can spread on social networks. These viruses spread by infecting the accounts of social network users, who click on any option that may trigger the virus’s takeover of the user’s sharing capabilities, resulting in the spreading of these malicious programs without the knowledge of the user. While their inner coding might be different and while their triggering and spreading mechanisms (on physical vs. virtual social networks) may be different, both traditional viruses such as worms and social network viruses share, in common, the critical property of proliferation via spreading through a network (physical or social).

These malicious programs behave similarly to an infection in a human population. This, in turn, allows us to draw comparisons between the study of epidemiology, in particular the mathematical aspect of infectious diseases(Bailey, 1987) and the behavior of a computer virus in a computer network. This is generally studied via mathematical models of the spread of a virus or disease. Any mathematical model’s ability to mimic the behavior of the infection largely depends on the assumptions made during the modeling process.

Most of the existing epidemiology models are modified versions of the classical Kermack and McKendrick’s (W.O.Kermack & McKendrick, 1927) model, more commonly known as the SIR (Susceptible/Infected/Recovered) model. For example, Hethcote (Hethcote, 1976) proposed a version of the SIR model, in which it was assumed that the total population was constant. But in real world scenarios, the population will change in time. Thus, this model was later improved by Diekmann and Heersterbeek(Diekmann & Heesterbeek, 2000) by assuming that:

  • I.

    The population size changes according to an exponential demographic trend.

  • II.

    The infected individuals cannot reproduce.

  • III.

    The individuals acquire permanent immunity to further infection when removed from the infected class.

The eigenvalue approach has recently emerged as one of the popular techniques to analyze virus or malicious entity propagation in a computer network. Wang et al. (Yang, Chakrabarti, Chenxi, & Faloutsos, 2003) have associated the epidemic threshold parameter for a network with the largest eigenvalue of its adjacency matrix. This technique works only with the assumption that the eigenvalues exist. There are also certain restrictions on the size of the adjacency matrix, since calculating eigenvalues may not be easy or even possible for large matrices.

In this paper, we apply the malicious object transmission model (Mishra & Saini, 2007) in complete and scale free networks, that assumes a variable population and with constant latent and immune periods. The current model extends the classical SIR model proposed in (Lloyd & May, 2001) to a probabilistic SEIRS (Mishra & Saini, 2007) {Mishra, 2007 #38}type model in several directions:

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