Skip to main content
Log in

The pulse treatment of computer viruses: a modeling study

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Unlike new medical procedures, new antivirus software can be disseminated rapidly through the Internet and takes effect immediately after it is run. As a result, a considerable number of infected computers can be cured almost simultaneously. Consequently, it is of practical importance to understand how pulse treatment affects the spread of computer viruses. For this purpose, an impulsive malware propagation model is proposed. To the best of our knowledge, this is the first computer virus model that takes into account the effect of pulse treatment. The dynamic properties of this model are investigated comprehensively. Specifically, it is found that (a) the virus-free periodic solution is globally asymptotically stable when the basic reproduction ratio (BRR) is less than unity, (b) infections are permanent when the BRR exceeds unity, and (c) a locally asymptotically stable viral periodic solution bifurcates from the virus-free periodic solution when the BRR goes through unity. A close inspection of the influence of different model parameters on the BRR allows us to suggest some feasible measures of eradicating electronic infections.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Szor, P.: The Art of Computer Virus Research and Defense. Addison-Wesley Education Publishers Inc., Boston, MA (2005)

    Google Scholar 

  2. Cohen, F.: Computer viruses: theory and experiments. Comput. Secur. 6(1), 22–35 (1987)

    Article  Google Scholar 

  3. Murray, W.H.: The application of epidemiology to computer viruses. Comput. Secur. 7(2), 130–150 (1988)

    Article  Google Scholar 

  4. 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)

  5. Billings, L., Spears, W.M., Schwartz, I.B.: A unified prediction of computer virus spread in connected networks. Phys. Lett. A 297(3–4), 261–266 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Ren, J., Yang, X., Zhu, Q., Yang, L.X., Zhang, C.: A novel computer virus model and its dynamics. Nonlinear Anal. Real World Appl. 13(1), 376–384 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Zhu, Q., Yang, X., Ren, J.: Modeling and analysis of the spread of computer virus. Commun. Nonlinear Sci. Numer. Simulat. 17(12), 5117–5124 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gan, C., Yang, X., Liu, W., Zhu, Q., Zhang, X.: Propagation of computer virus under human intervention: a dynamical model. Discrete Dyn. Nat. Soc. 2012 (2012). Article ID 106950

  9. Gan, C., Yang, X., Zhu, Q., Jin, J., He, L.: The spread of computer virus under the effect of external computers. Nonlinear Dyn. 73(3), 1615–1620 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gan, C., Yang, X., Liu, W., Zhu, Q.: A propagation model of computer virus with nonlinear vaccination probability. Commun. Nonlinear Sci. Numer. Simulat. 19(1), 92–100 (2014)

    Article  MathSciNet  Google Scholar 

  11. Yuan, H., Chen, G.: Network virus-epidemic model with the point-to-group information propagation. Appl. Math. Comput. 206(1), 357–367 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Yuan, H., Chen, G., Wu, J., Xiong, H.: Towards controlling virus propagation in information systems with point-to-group information sharing. Decis. Support Syst. 48(1), 57–68 (2009)

    Article  Google Scholar 

  13. Mishra, B.K., Pandey, S.K.: Dynamic model of worms with vertical transmission in computer network. Appl. Math. Comput. 217(21), 8438–8446 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Mishra, B.K., Saini, D.K.: SEIQRS model for the transmission of malicious objects in computer network. Appl. Math. Model. 34(3), 710–715 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Yang, X., Yang, L.X.: Towards the epidemiological modeling of computer viruses. Discrete Dyn. Nat. Soc. 2012, 1–11 (2012). Article ID 259671

    MATH  Google Scholar 

  16. Yang, L.X., Yang, X., Wen, L., Liu, J.: Propagation behavior of virus codes in the situation that infected computers are connected to the Internet with positive probability. Discrete Dyn. Nat. Soc. 2012, 1–13 (2012). Article ID 693695

    MATH  Google Scholar 

  17. Yang, L.-X., Yang, X., Wen, L., Liu, J.: A novel computer virus propagation model and its dynamics. Int. J. Comput. Math. 89(17), 2307–2314 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  18. Yang, M., Zhang, Z., Li, Q., Zhang, G.: An SLBRS model with vertical transmission of computer virus over Internet. Discrete Dyn. Nat. Soc. 2012 (2012). Article ID 693695

  19. Yang, L.-X., Yang, X.: The spread of computer viruses under the influence of removable storage devices. Appl. Math. Comput. 219(8), 3914–3922 (2012)

    Article  MathSciNet  Google Scholar 

  20. Yang, L.-X., Yang, X., Zhu, Q., Wen, L.: A computer virus model with graded cure rates. Nonlinear Anal. Real world Appl. 14(1), 414–422 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  21. Yang, L.-X., Yang, X., Liu, J., Zhu, Q., Gan, C.: Epidemics of computer viruses: a complex-network approach. Appl. Math. Comput. 219(16), 8705–8717 (2013)

    Article  MathSciNet  Google Scholar 

  22. Zhang, C., Liu, W., Xiao, J., Zhao, Y.: Hopf bifurcation of an improved SLBS model under the influence of latent period. Math. Probl. Eng. 2013 (2013) Article ID 196214

  23. Yang, L.X., Yang, X.: A new epidemic model of computer viruses. Commun. Nonlinear Sci. Numer. Simulat. 19(6), 1935–1944 (2014)

    Google Scholar 

  24. Muroya, Y., Li, H., Kuniya, T.: On global stability of a nonresident computer virus model. Acta Math. Sci. 19(6), 1935–1944 (2014)

    Google Scholar 

  25. Zhu, Q., Yang, X., Yang, L.-X., Zhang, X.: A mixing propagation model of computer viruses and countermeasures. Nonlinear Dyn. 73(3), 1433–1441 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  26. Yang, L.-X., Yang, X.: The effect of infected external computers on the spread of viruses: a compartment modeling study. Phys. A 392(24), 6523–6535 (2013)

    Article  MathSciNet  Google Scholar 

  27. Kephart, J.O., White, S.R.: Measuring and modeling computer virus prevalence. In: Proceedings IEEE Symposium on Security and Privacy, pp. 2–15 (1993)

  28. Piqueira, J.R.C., de Vasconcelos, A.A., Gabriel, C.E.C.J., Araujo, V.O.: Dynamic models for computer viruses. Comput. Secur. 27(7–8), 355–359 (2008)

    Article  Google Scholar 

  29. Piqueira, J.R.C., Araujo, V.O.: A modified epidemiological model for computer viruses. Appl. Math. Comput. 213(2), 355–360 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  30. Toutonji, O.A., Yoo, S.-M., Park, M.: Stability analysis of VEISV propagation modeling for network worm attack. Appl. Math. Model. 36(6), 2751–2761 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  31. Mishra, B.K., Jha, N.: Fixed period of temporary immunity after run of anti-malicious software on computer nodes. Appl. Math. Comput. 190(2), 1207–1212 (2007)

    Article  MATH  Google Scholar 

  32. Mishra, B.K., Saini, D.K.: SEIRS epidemic model with delay for transmission of malicious objects in computer network. Appl. Math. Comput. 188(2), 1476–1482 (2007)

    Google Scholar 

  33. Han, X., Tan, Q.: Dynamical behavior of computer virus on Internet. Appl. Math. Comput. 217(6), 2520–2526 (2010)

    Google Scholar 

  34. Ren, J., Yang, X., Yang, L.-X., Xu, Y., Yang, F.: A delayed computer virus propagation model and its dynamics. Chaos Solitons Fractals 45(1), 74–79 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  35. Feng, L., Liao, X., Li, H., Han, Q.: Hopf bifurcation analysis of a delayed viral infection model in computer networks. Math. Comput. Model. 56(7–8), 167–179 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  36. Zhu, Q., Yang, X., Yang, L.-X., Zhang, C.: Optimal control of computer virus under a delayed model. Appl. Math. Comput. 218(23), 11613–11619 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  37. Dong, T., Liao, X., Li, H.: Stability and Hopf bifurcation in a computer virus model with multistate antivirus. Abstr. Appl. Anal. 2012 (2012). Article ID 841987

  38. Yang, X., Mishra, B.K., Liu, Y.: Computer viruses: theory, model, and methods. Discrete Dyn. Nat. Soc. 2012 (2012). Article ID 473508

  39. Zhang, C., Zhao, Y., Wu, Y., Deng, S.: A stochastic dynamic model of computer viruses. Discrete Dyn. Nat. Soc. 2012 (2012). Article ID 264874

  40. Dequadros, C., Andrus, J., Olive, J.: Eradication of poliomyelitis: progree. Pediatr. Inf. Dis. J. 10, 222–229 (1991)

    Article  Google Scholar 

  41. Sabin, A.: Measles, killer of millions in developing countries: strategies of elimination and continuing control. Eur. J. Epidemiol. 7(1), 1–22 (1991)

    Article  MathSciNet  Google Scholar 

  42. Agur, Z., Cojocaru, L., Mazor, G., Anderson, R., Danon, Y.: Pulse mass measles vaccination across age cohorts. Proc. Natl. Acad. Sci. USA 90(24), 11698–11702 (1993)

    Article  Google Scholar 

  43. Zhou, Y., Liu, H.: Stability of periodic solutions for an SIS model with pulse vaccination. Math. Comput. Model. 38(3–4), 299–308 (2003)

    Article  MATH  Google Scholar 

  44. Liu, X., Takeuchi, Y., Iwami, S.: SVIR epidemic models with vaccination strategies. J. Theor. Biol. 253(1), 1–11 (2008)

    Article  MathSciNet  Google Scholar 

  45. Zhang, T., Teng, Z.: An SIRVS epidemic model with pulse vaccination strategy. J. Theor. Biol. 250(2), 375–381 (2008)

    Article  MathSciNet  Google Scholar 

  46. Li, Y., Cui, J.: The effect of constant and pulse vaccination on SIS epidemic models incorporating media coverage. Commun. Nonlinear Sci. Numer. Simulat. 14(5), 2353–2365 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  47. Jiang, G., Yang, Q.: Bifurcation analysis in an SIR epidemic model with birth pulse and pulse vaccination. Appl. Math. Comput. 215(3), 1035–1046 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  48. Wei, H., Jiang, Y., Song, X., Su, G.H., Qiu, S.Z.: Global attractivity and permanence of a SVEIR epidemic model with pulse vaccination and time delay. J. Comput. Appl. Math. 229(1), 302–312 (2009)

    Google Scholar 

  49. Meng, X., Chen, L., Wu, B.: A delay SIR epidemic model with pulse vaccination and incubation times. Nonlinear Anal. Real World Appl. 11(1), 88–98 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  50. Yao, Y., Guo, L., Guo, H., Yu, G., Gao, F., Tong, X.: Pulse quarantine strategy of internet worm propagation: modeling and analysis. Comput. Electr. Eng. 38(5), 1047–1061 (2012)

    Article  Google Scholar 

  51. Zhang, C., Zhao, Y., Wu, Y.: An impulse model for computer viruses. Discrete Dyn. Nat. Soc. 2012 (2012). Article ID 260962

  52. Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)

    Book  MATH  Google Scholar 

  53. Bainov, D.D., Simeonov, P.S.: Impulsive Differential Equations: Periodic Solutions and Applications. Longman Scientific & Technical, New York (1993)

    MATH  Google Scholar 

  54. Lakmeche, A., Arino, O.: Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment. Dyn. Continuous Discrete Impuls. Syst. 7(2), 265–287 (2000)

    MATH  MathSciNet  Google Scholar 

  55. Faloutsos, M., Faloutsos, P., Faloutsos, C.: On power-law relationships of the Internet topology. ACM SIGCOMM Comput. Commun. Rev. 29(4), 251–262 (1999)

    Article  Google Scholar 

  56. Albert, R., Barabasi, A.-L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74(1), 47–97 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  57. Pastor-Satorras, R., Vespignani, A.: Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86(14), 3200–3203 (2001)

    Article  Google Scholar 

  58. Pastor-Satorras, R., Vespignani, A.: Epidemic dynamics and endemic states in complex networks. Phys. Rev. E 63(6), (2001). Article ID 066117

  59. Yang, L.X., Yang, X.: The spread of computer viruses over a reduced scale-free network. Phys. A. 396, 173–184 (2014)

    Google Scholar 

Download references

Acknowledgments

The authors wish to express sincere gratitude to the two anonymous reviewers for their valuable suggestions that have greatly improved the quality of this paper. This work is supported by the Natural Science Foundation of China (Grant No. 10771227) and Doctorate Foundation of Educational Ministry of China (Grant No. 20110191110022).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Xiaofan Yang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yang, LX., Yang, X. The pulse treatment of computer viruses: a modeling study. Nonlinear Dyn 76, 1379–1393 (2014). https://doi.org/10.1007/s11071-013-1216-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-013-1216-x

Keywords

Navigation