NUMERICAL-ANALYTIC SOLUTION OF ONE MODELING PROBLEM OF FRACTIONAL-DIFFERENTIAL DYNAMICS OF COMPUTER VIRUSES

Abstract

The paper considers the problem of modeling the dynamics of computer viruses spreading using a model based on the mathematical theory of biological epidemics. The urgency of the considered problem arises from the need to build effective anti-virus protection systems for computer networks based on the results of mathematical modeling of the spread of malicious software. We consider the SIES-model (Gan C., Yang X., Zhu Q.), that studies spread dynamics of computer viruses separating the influence of the action of computers accessible and unavailable on the Internet. In order to take into account non-local effects in this model, in particular memory effects, its modification on the ideas of the theory of fractional-order integro-differentiation is proposed. The technique of obtaining a numerical-analytical solution of the problem of modeling of computer viruses spread dynamics on the base of the fractional-differential counterpart of the SIES-model is presented. Closed forms solutions of the problems for the number of vulnerable and external computers are obtained, and a finite-difference scheme of the fractional Adams method for the problem of determining the number of infected computers is constructed. The results of computational experiments based on the developed technique of numerical-analytical solution show that there is a subdiffusion evolution of the system to the steady state. At the same time, for the number of external computers, a fast short-term growth is observed at the initial stages of process development with subsequent smooth and slow decrease towards the steady state. For medium and large values of the time variable, the evolution of the number of infected computers to the steady state occurs in an ultra-slow mode. Thus, the proposed technique makes it possible to study the families of dynamic reactions in the process of computer viruses spreading, including fast transient processes and ultra-slow evolution of systems with memory.