Advertisement

SIRA Computer Viruses Propagation Model: Mortality and Robustness

  • Cristiane Mileo Batistela
  • José Roberto Castilho Piqueira
Technical Note
  • 57 Downloads

Abstract

Among the several works based on the classical Kermack–Mckendrick’s SIR (Susceptible–Infected–Removed) epidemiological model, applied to the context of computer viruses propagation, the introduction of antidotal elements has provided the dynamics of the networks when anti-virus programs are used. The SIRA (Susceptible–Infected–Removed–Antidotal) model has shown good qualitative fitness regarding to real operation of the networks, when the mortality rate is considered zero and all the infected nodes being recovered. Here, the SIRA model is studied, considering the mortality rate as a parameter and the conditions for the existence of a disease-free equilibrium state are derived, helping the design of robust networks.

Keywords

Bifurcation Disease free Endemic Equilibrium SIRA Stability 

Mathematics Subject Classification

34C23 34D30 

Notes

Compliance with ethical standard

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this article.

References

  1. 1.
    Amador, J.: The stochastic SIRA model for computer viruses. Appl. Math. Comput. 232, 1112–1124 (2014)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Amador, J., Artalejo, J.R.: Stochastic modeling of computer virus spreading with warning signals. J. Frankl. Inst. 350, 1112–1138 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cohen, F.: Computer viruses, theory and experiments therapies. Comput. Secur. 6, 22–35 (1987)CrossRefGoogle Scholar
  4. 4.
    Cohen, F.: A short course of computer viruses. Comput. Secur. 8, 149–160 (1990)CrossRefGoogle Scholar
  5. 5.
    Gan, C., Yang, X., Liu, W., Zhu, Q.: A propagation model of computer virus with nonlinear vaccination probability. Commum. Nonlinear Sci. Numer. Simul. 19, 92–100 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamic Systems and Bifurcation of Vector Fields. Springer, New York (1983)CrossRefGoogle Scholar
  7. 7.
    Kemar, U.: Stochastic model on computer virus. Int. J. Syst. Signal Control Eng. Appl. 4(4–6), 74–79 (2011)Google Scholar
  8. 8.
    Kephart, J.O., White, S.R.: Directed-graph epidemiological models of computer viruses. In: Proceedings of IEEE Computer Society Symposium on Research in Security and Privacy, pp. 343–359 (1991)Google Scholar
  9. 9.
    Kephart, J.O., White, S.R.: Measuring and modeling computer virus prevalence. In: Proceedings of IEEE Computer Society Symposium on Research in Security and Privacy, pp. 2–15 (1993)Google Scholar
  10. 10.
    Kephart, J.O., White, S.R., Chess, D.M.: Computers and epidemiology. IEEE Spectr. 30, 20–26 (1993)CrossRefGoogle Scholar
  11. 11.
    Kermack, W., Mckendrick, A.: Contributions of mathematical theory to epidemics. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. A115, 700–721 (1927)CrossRefGoogle Scholar
  12. 12.
    Kermack, W., Mckendrick, A.: Contributions of mathematical theory to epidemics. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. A38, 55–83 (1932)CrossRefGoogle Scholar
  13. 13.
    Kermack, W., Mckendrick, A.: Contributions of mathematical theory to epidemics. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. A141, 94–122 (1933)CrossRefGoogle Scholar
  14. 14.
    Mishra, B.K., Jha, N.: SEIQRS model for the transmission of malicious objects in computer network. Appl. Math. Comput. 34, 710–715 (2010)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Mishra, B.K., Keshri, N.: Mathematical model on the transmission of worms in wireless sensor network. Appl. Math. Comput. 37, 4103–4111 (2013)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Mishra, B.K., Saini, D.K.: SEIRS epidemic model with delay for transmission of malicious objects in computer network. Appl. Math. Comput. 188, 1476–1482 (2007)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Mishra, A.K., Verma, M., Sharma, A.: Capturing the interplay between malware and anti-malware in a computer network. Appl. Math. Comput. 229, 340–349 (2014)zbMATHGoogle Scholar
  18. 18.
    Moler, C.B.: Numerical Computing with MATLAB. SIAM, Philadelphia (2004)CrossRefGoogle Scholar
  19. 19.
    Murray, J.D.: Mathematical Biology, 3rd edn. Springer, New York (2002)zbMATHGoogle Scholar
  20. 20.
    Peng, S., Yu, S., Yang, A.: Smartphone malware and its propagation modeling: a survey. IEEE Commun. Surv. Tutor. 16(2), 925–941 (2014)CrossRefGoogle Scholar
  21. 21.
    Piqueira, J.R.C., Navarro, B.F., Monteiro, L.H.A.: Epidemiological models applied to viruses in computer networks. J. Comput. Sci. 1(1), 31–34 (2005)CrossRefGoogle Scholar
  22. 22.
    Piqueira, J.R.C., de Vasconcelos, A.A., Gabriel, C.E.C.J., Araujo, V.O.: Dynamic models for computer viruses. Comput. Secur. 27(7), 355–359 (2008)CrossRefGoogle Scholar
  23. 23.
    Piqueira, J.R.C., Araujo, V.O.: A modified epidemiological model for computer viruses. Appl. Math. Comput. 213, 355–360 (2009)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Wang, Y., Wen, S., Xiang, Y., Zhou, W.: Modeling the propagation of worms in networks: a survey. IEEE Commun. Surv. Tutor. 16(2), 942–960 (2014)CrossRefGoogle Scholar
  25. 25.
    Weiss, G.H., Dishon, M.: On the asymptotic behavior of the stochastic and deterministic models of an epidemic. Math. Biosci. 11, 261–265 (1971)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Yang, L.-X., Yang, X., Wu, Y.: The impact of patch forwarding on the prevalence of computer virus: a theoretical assessment approach. Appl. Math. Model. 43, 110–125 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Yang, L.-X., Yang, X., Tang, Y.Y.: A bi-virus competing spreading model with generic infection rates. IEEE Trans. Netw. Sci. Eng. 5(1), 2–13 (2018)CrossRefGoogle Scholar

Copyright information

© Springer Nature India Private Limited 2018

Authors and Affiliations

  1. 1.Escola Politécnica da Universidade de São PauloSão PauloBrazil

Personalised recommendations