## Modeling computer virus prevalence with a susceptible-infected-susceptible model with reintroduction.(English)Zbl 1429.68037

Summary: Computer viruses are an extremely important aspect of computer security, and understanding their spread and extent is an important component of any defensive strategy. Epidemiological models have been proposed to deal with this issue, and we present one such here. We consider a modification of the Susceptible-Infected-Susceptible (SIS) epidemiological model as a model of computer virus spread. This model includes a reintroduction parameter, which models the rerelease of a computer virus, or the introduction of a new virus. This is a more realistic model of computer virus spread than the standard SIS model, and can be used to understand the behavior of the quasi-stationary regime of the SIS model.

### MSC:

 68M25 Computer security 60J85 Applications of branching processes

Mathematica
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### References:

 [1] Abreu, E.M., 2001. Computer virus costs reach $$10.7b this year. The Washington Post, Sept 1, 2001. available at http://www.washtech.com/news/netarch/12267-1.html$$ [2] Ball, F., The threshold behavior of epidemic models, J. appl. probab., 20, 227-241, (1983) · Zbl 0519.92023 [3] Ball, F., A threshold theorem for the Reed-frost chain-binomial epidemic, J. appl. probab., 20, 153-157, (1983) · Zbl 0505.92020 [4] Ball, F., Stochastic and deterministic models for SIS epidemics among a population partitioned into households, Math. biosci., 156, 41-67, (1999) · Zbl 0979.92033 [5] Ball, F.; Donnelly, P., Strong approximations for epidemic models, Stoch. proc. appl., 55, 1-21, (1995) · Zbl 0823.92024 [6] Bartholomew, D.J., Continuous time diffusion models with random duration of interest, J. math. sociology, 4, 187-199, (1976) · Zbl 0339.92013 [7] Caraco, T., 1979. Ecological response of animal group size frequencies. International Co-operative Publishing House, Fairland, MD, pp. 371-386. [8] Cavender, J.A., Qausi-stationary distributions of birth-and-death processes, Adv. appl. probab., 10, 570-586, (1978) · Zbl 0381.60068 [9] Darroch, J.N.; Seneta, E., On quasi-stationary distributions in absorbing continuous-time finite Markov chains, J. appl. probab., 4, 192-196, (1967) · Zbl 0168.16303 [10] Jacquez, J.A.; Simon, C.P., The stochastic SI model with recruitment and deaths. I. comparison with the closed SIS model, Math. biosci., 117, 77-125, (1993) · Zbl 0785.92025 [11] Johnson, N.L.; Katz, S.; Kemp, A.W., Univariate discrete distributions, (1992), Wiley New York [12] Kendall, D.G., 1956. Deterministic and stochastic epidemics in closed populations. In: J. Neyman (Ed.), Third Berkeley Symposium on Mathematical Statistics and Probability, Vol. 4. University of California Press, California, pp. 149-165. · Zbl 0070.15101 [13] Kephart, J.O., White, S.R., 1991. Directed – graph epidemiological models of computer viruses. In: 1991 IEEE Computer Society Symposium on Research in Security and Privacy, Oakland, California, pp. 343-359. [14] Kephart, J.O., White, S.R., 1993. Measuring and modeling computer virus prevalence. In: 1993 IEEE Computer Society Symposium on Research in Security and Privacy, Oakland, California, pp. 2-15. [15] Kephart, J.O.; White, S.R.; Chess, D.M., Computers and epidemiology, IEEE spectrum, 30, 20-26, (1993) [16] Kryscio, R.J.; Lefèvre, C., On the extinction of the SIS stochastic logistic epidemic, J. appl. probab., 27, 685-694, (1989) · Zbl 0687.92012 [17] Martin-Löf, A., Symmetric sampling procedures, general epidemic processes, and their threshold limit theorems, J. appl. probab., 23, 265-282, (1986) · Zbl 0605.92009 [18] Murray, W., The application of epidemiology to computer viruses, Comput. security, 7, 139-150, (1988) [19] Nåsell, I., The quasi-stationary distribution of the closed endemic SIS model, Adv. appl. probab., 28, 895-932, (1996) · Zbl 0854.92020 [20] Nåsell, I., On the time to extinction in recurrent epidemics, J. R. statist. soc. B, 61, 309-330, (1999) · Zbl 0917.92023 [21] Oppenheim, I.; Shuler, K.E.; Weiss, G.H., Stochastic theory of nonlinear rate processes with multiple stationary states, Physica A, 88, 191-214, (1977) [22] Ross, R., Some a priori pathometric equations, Br. med. J., 1, 546, (1915) [23] Ross, S., Stochastic processes, (1996), Wiley New York [24] Scalia-Tomba, G., Asymptotic final size distribution for some chain-binomial processes, Adv. appl. probab., 17, 477-495, (1985) · Zbl 0581.92023 [25] von Bahr, B.; Martin-Löf, A., Threshold limit theorems for some epidemic processes, Adv. appl. probab., 12, 319-349, (1980) · Zbl 0425.60074 [26] Weiss, G.H.; Dishon, M., On the asymptotic behavior of the stochastic and deterministic models of an epidemic, Math. biosci., 11, 261-265, (1971) · Zbl 0224.92018 [27] Wolfram, S., The Mathematica book, (1996), Cambridge University Press New York · Zbl 0878.65001
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