Dynamic Model on the Transmission of Malicious Codes in Network

Full Text (PDF, 218KB), PP.17-23

Views: 0 Downloads: 0


Bimal Kumar Mishra 1,* Apeksha Prajapati 1

1. Department of Applied Mathematics, Birla Institute of Technology, Mesra, Ranchi-835215, India

* Corresponding author.

DOI: https://doi.org/10.5815/ijcnis.2013.10.03

Received: 2 Nov. 2012 / Revised: 17 Feb. 2013 / Accepted: 1 May 2013 / Published: 8 Aug. 2013

Index Terms

Computer network, Worms, Virus, Trojan horse, Epidemic Model, Reproduction number, Global stability


This paper introduces differential susceptible e-epidemic model S_i IR (susceptible class-1 for virus (S1) - susceptible class-2 for worms (S2) -susceptible class-3 for Trojan horse (S3) – infectious (I) – recovered (R)) for the transmission of malicious codes in a computer network. We derive the formula for reproduction number (R0) to study the spread of malicious codes in computer network. We show that the Infectious free equilibrium is globally asymptotically stable and endemic equilibrium is locally asymptotically sable when reproduction number is less than one. Also an analysis has been made on the effect of antivirus software in the infectious nodes. Numerical methods are employed to solve and simulate the system of equations developed.

Cite This Paper

Bimal Kumar Mishra, Apeksha Prajapati, "Dynamic Model on the Transmission of Malicious Codes in Network", International Journal of Computer Network and Information Security(IJCNIS), vol.5, no.10, pp.17-23, 2013. DOI:10.5815/ijcnis.2013.10.03


[2]N. T. J. Bailey, “The Mathematical Theory of Infectious Diseases”, second ed., Hafner, New York, 1975.
[3]W. O. Kermack, A. G. Mckendrick, “A contribution to the mathematical theory of epidemics”, Proc. Roy. Soc. Lond. Series- A, vol. 11, pp. 700– 721, 1927.
[4]W. O. Kermack, A.G. McKendrick, “Contributions of mathematical theory to epidemics”, Proc. R. Soc. Lon. Series- A, vol. 138, pp. 55– 83, 1932.
[5]W.O. Kermack, A.G. McKendrick, “Contributions of mathematical theory to epidemics”, Proc. R. Soc. Lon. Series-A, vol. 141, pp. 94– 122, 1933.
[6]J.O. Kephart, S.R.White, D.M. Chess, “Comput. and Epidemio” IEEE Spectrum, pp. 20 – 26, 1933.
[7]J.O. Kephart, “A biologically inspired immune system for computers”, Proceedings of International Joint Conference on Artificial Intelligence, 1995.
[8]N. Madar, T. Kalisky, R. Cohen, D. Ben Avraham, S. Havlin, “Immunization and epidemic dynamics in complex networks”, Eur. Phys. J. B, vol. 38, pp. 269– 276, 2004.
[9]R. Pastor-Satorras, A. Vespignani, “Epidemics and immunization in scale-free networks”, Handbook of Graphs and network, From the Genome to the Internet, Willey-VCH, Berlin, 2002.
[10]M.E.J. Newman, S. Forrest, J. Balthrop, “Email networks and the spread of computer virus”, Phys. Rev. E, vol. 66, pp. 035101-1-035101-4, 2002.
[11]C.C. Zou and W. Gong, D. Towsley, “Worm propagation modeling and analysis under dynamic quarantine defense”, Proceeding of the ACM CCS Workshop on Rapid Malcode, ACM, pp. 51 – 60, 2003.
[12]M.J. Keeling and K.T.D. Eames, “Network and epidemic models”, J. Roy. Soc. Interf., vol. 2, no. 4, pp. 295 – 307, 2005.
[13]C.C McCluskey “Global stability for an epidemic model with delay and nonlinear incidence”, Nolinear Analysis, Real World Application., vol. 11, pp. 3106– 3109, 2010.
[14]Bimal Kumar Mishra and Gholam Mursalin Ansari, “Differential epidemic model of virus and worms in computer network”, Int .J. of Net. Sec., vol. 14, no. 3, pp. 149-155, 2012.
[15]Bimal Kumar Mishra and Dinesh Saini, “Mathematical models on computer viruses”, Appl. Math. and Comput., vol. 187, no. 2, pp. 929– 936, 2007.
[16]Bimal Kumar Mishra and Navnit Jha, “SEIQRS model for the transmission of malicious objects in computer network”, App. Math. Modelling, Vol. 34, no. 3, pp. 710– 715, 2010.
[17]Bimal Kumar Mishra and Samir Kumar Pandey, “Fuzzy epidemic model for the transmission of worms in Computer network”, Nonlinear Analysis, Real World Application, vol. 11, pp. 4335– 4341, 2010.
[18]J. R. C Piqueira and V. O. Araujo, “A modified epidemiological model for computer viruses”, Appl. Math and Comput., vol. 213, no. 2, pp. 355– 360, 2009.
[19]J.R.C Piqueira, B.F. Navarro, L.H.A., “Monteiro, Epidemiological models applied to virus in computer network”, J.Comput. Sci., vol. 1, no. 1, pp. 31–34, 2005.