Bifurcation analysis of a computer virus propagation model

Year 2021, Volume: 50 Issue: 5, 1384 - 1400, 15.10.2021

Kenneth Dukuza

https://doi.org/10.15672/hujms.747872

Cited By: 1

Abstract

We propose a mathematical model for investigating the efficacy of Countermeasure Competing (CMC) strategy which is a method for reducing the effect of computer virus attacks. Using the Centre Manifold Theory, we determine conditions under which a subcritical (backward) bifurcation occurs at Basic Reproduction Number $R_{0}=1$. In order to illustrate the theoretical findings, we construct a new Nonstandard Finite Difference Scheme (NSFD) that preserves the bifurcation property at $R_{0}=1$ among other dynamics of the continuous model. Earlier results given by Chen and Carley [The impact of countermeasure propagation on the prevalence of computer viruses, IEEE Trans. Syst., Man, Cybern. B. Cybern. 2004] show that the CMC strategy is effective when the countermeasure propagation rate is higher than the virus spreading rate. Our results reveal that even if this condition is not met, the CMC strategy may still successfully eradicate computer viruses depending on the extent of its effectiveness. 

Keywords

basic reproduction number, bifurcation, computer virus, countermeasure competing, nonstandard finite difference scheme

References

• [1] R. Anguelov and J.M-S. Lubuma, Contributions to the mathematics of the nonstandard finite difference method and applications, Numer. Methods Partial Differential Equations, 17, 518–543, 2001.
• [2] R. Anguelov and J.M-S. Lubuma, Nonstandard finite difference method by nonlocal approximation, Math. Comput. Simulation, 61 (3-6), 465–475, 2003.
• [3] R. Anguelov, K. Dukuza and J.M-S. Lubuma, Backward bifurcation analysis for two continuous and discrete epidemiological models, Math. Methods Appl. Sci. 41 (18), 8784– 8798, 2018.
• [4] R.M. Anderson and R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, USA, 1992.
• [5] C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applica- tions, Math. Biosci. Engin. 1, 361–404, 2004.
• [6] L.C. Chen and K.M. Carley, The impact of countermeasure propagation on the preva- lence of computer viruses, IEEE Trans. Syst., Man, Cybern. B. Cybern. 34 (2), 823–833, 2004.
• [7] CSI/FBI; Computer crime and security survey. www.gocsi.com, 2008. Accessed 26 March 2020.
• [8] J.D. Crawford, Introduction to bifurcation theory, Rev. Modern Phys. 63 (4), 991, 1991.
• [9] Q.A. Dang and M.T. Hoang, Positivity and global stability preserving NSFD schemes for a mixing propagation model of computer viruses, J. Comput. Appl. Math. 374, 112753, 2020.
• [10] D.T. Dimitrov and H.V. Kojouharov, Positive and elementary stable nonstandard numerical methods with applications to predator-prey models, J. Comput. Appl. Math. 189 (1-2), 98–108, 2006.
• [11] N.K.K. Dukuza, Centre Manifold Theory for some Continuous and Discrete Epidemi- ological Models, University of Pretoria, PhD Thesis, South Africa, 2019.
• [12] J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bi- furcations of vector fields, 42, Springer Science & Business Media, 2013.
• [13] J.M. Heffernan, R.J. Smith and L.M. Wahl, Perspectives on the basic reproductive ratio, J. R. Soc. Interface, 2 (4), 281–293, 2005.
• [14] P. Kama, Non-standard finite difference methods in dynamical systems, University of Pretoria, PhD Thesis, South Africa, 2009.
• [15] W.O. Kermack and A.G. McKendrick, A contribution to the mathematical theory of epidemics, The Royal Society London, Proc. R. Soc. Lond. 115 (772), 700–721, 1927.
• [16] W.O. Kermack and A.G. McKendrick, Contributions to the mathematical theory of epidemics. II.The problem of endemicity. The Royal Society London, Proc. R. Soc. Lond. 138 (834), 55–83, 1932.
• [17] R. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Hackensack, NJ, 1994.
• [18] B.K. Mishra and D.K. Saini, SEIRS epidemic model with delay for transmission of malicious objects in computer network, Appl. Math. Comput. 188 (2), 1476–1482, 2007.
• [19] W.H. Murray,The application of epidemiology to computer viruses, Comput. Secur. 7 (2), 130–150, 1988.
• [20] J.R.C. Piqueira, A.A. de Vasconcelos, C.E.C.J. Gabriel and V.O. Araujo, Dynamic models for computer viruses, Comput. Secur. 27 (7-8), 355–359, 2008.
• [21] P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (1-2), 29–48, 2002.
• [22] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, Berlin, 1990.
• [23] H. Yuan, G. Chen, J. Wu and H. Xiong, Towards controlling virus propagation in information systems with point-to-group information sharing, Decis. Support Syst. 48 (1), 57–68, 2009.
• [24] Q. Zhu, X. Yang, L. Yang and X. Zhang, A mixing propagation model of computer viruses and countermeasures, Nonlinear Dynam. 73, 1433–1441, 2013.