Skip to main content
Log in

On the global stability of an epidemic model of computer viruses

  • Original Article
  • Published:
Theory in Biosciences Aims and scope Submit manuscript

Abstract

In this paper, we study the global properties of a computer virus propagation model. It is, interesting to note that the classical method of Lyapunov functions combined with the Volterra–Lyapunov matrix properties, can lead to the proof of the endemic global stability of the dynamical model characterizing the spread of computer viruses over the Internet. The analysis and results presented in this paper make building blocks towards a comprehensive study and deeper understanding of the fundamental mechanism in computer virus propagation model. A numerical study of the model is also carried out to investigate the analytical results.

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

Similar content being viewed by others

References

  • Bhunu CP, Mushayabasa S (2013) Assessing the effects of drug misuse on HIV/AIDS prevalence. Theory Biosci 133–2:83–92

    Article  Google Scholar 

  • Bradley JT, Gilmore ST, Hillston J (2008) Analysing distributed Internet worm attacks using continuous state-space approximation of process algebra models. J Comput Syst Sci 74(6):1013–1032

    Article  Google Scholar 

  • Britten NF (2003) Essential mathematical biology. Springer-Verlag, Berlin

  • Chavez CC, Feng Z, Huang W (2002) On the computation of \(R_0\) and its role on global stability. Math Approaches Emerg Reemerg Infect Dis Intro IMA 125:229–250

  • Chong NS, Tchuenche JM, Smith RJ (2014) A mathematical model of avian influenza with half-saturated incidence. Theory Biosci 133–1:23–38

    Article  Google Scholar 

  • Cross GW (1978) Three types of matrix stability. Linear Algebra Appl 20:253–263

    Article  Google Scholar 

  • Driessche VD, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180(1):29–48

    Article  PubMed  Google Scholar 

  • Han X, Tan QL (2010) Dynamical behavior of computer virus on internet. Appl Math Comput 217:2520–2526

    Google Scholar 

  • Imran M, Rafique H, Khan A, Malik T (2014) A model of bi-mode transmission dynamics of hepatitis C with optimal control. Theory Biosci 133:91–109

    Article  CAS  PubMed  Google Scholar 

  • Kephart J, White S (1991) Directed-graph epidemiological models of computer viruses, in. IEEE Symp Secur Pri 1991:343–359

    Google Scholar 

  • Li MY, Graef JR, Wang L, Karsai J (1999) Global dynamics of a SEIR model with varying total population size. Math Biosci 160:191–213

    Article  CAS  PubMed  Google Scholar 

  • Li J, Knickerbocker P (2007) Functional similarities between computer worms and biological pathogens. Comput Secur 26(4):338–347

    Article  CAS  Google Scholar 

  • Liu S, Ruan S, Zhang X (2015) On avian influenza epidemic models with time delay. Theory Biosci 134–3:75–82

    Article  Google Scholar 

  • Ma J, Ying X, Guodong R, Wang C (2016) Prediction for breakup of spiral wave in a regular neuronal network. Nonlinear Dyn 84:497–509

    Article  Google Scholar 

  • Mishra BK, Saini DK (2007) SEIRS epidemic model with delay for transmission of malicious objects in computer network. Appl Math Comput 188:1476–1482

    Google Scholar 

  • Murray WH (1988) The application of epidemiology to computer viruses. Comput Secur 7(2):130–150

    Article  Google Scholar 

  • Piqueira JRC, Devasconcelos AA, Gabriel CECJ, Araujo VO (2008) Dynamic models for computer viruses. Comput Secur 27(7–8):355–359

  • Redheffer R (1985) Volterra multipliers I. SIAM J Algebraic Discrete Methods 6:592–611

  • Redheffer R (1985) Volterra multipliers II. SIAM J Algebraic Discrete Methods 6:612–623

  • Ren JG, Yang XF, Zhu QY, Yang LX, Zhang CM (2012) A novel computer virus model and its dynamics. Nonlinear Anal Real World Appl 13:376–384

    Article  Google Scholar 

  • Rinaldi F (1990) Global stability results for epidemic models with latent period. IMA J Math Appl Med Biol 7:69–75

    Article  CAS  PubMed  Google Scholar 

  • Satorras RP, Castellano C, Mieghem PV, Vespignani A (2015) Epidemic processes in complex networks. Rev Mod Phys 87(3):925. doi:10.1103/RevModPhys.87.925

  • Song X, Wang C, Ma J, Ren G (2016) Collapse of ordered spatial pattern in neuronal network. Phys A 451:95–112

    Article  Google Scholar 

  • Wang J, Liao S (2012) A generalized cholera model and epidemic-endemic analysis. J Biol Dynam 6:568–589

    Article  Google Scholar 

  • Wang FW, Zhang YK, Wang CG, Ma JF, Moon SJ (2010) Stability analysis of a SEIQV epidemic model for rapid spreading worms. Comput Secur 29:410–418

    Article  Google Scholar 

  • Xu R, Ma Z (2010) Global stability of a delayed SEIRS epidemic model with saturation incidence rate. Nonlinear Dyn 61(1):229–239

    Article  Google Scholar 

  • Yang LX, Yang X (2015) The impact of nonlinear infection rate on the spread of computer virus. Nonlinear Dyn 82(1):85–95

    Article  Google Scholar 

  • Yang LX, Yang X (2017) The effect of network topology on the spread of computer viruses: a modelling study. Int J Comput Mathe 94(8):1591–1608

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

  • Yang LX, Yang X, Wu Y (2017) The impact of patch forwarding on the prevalence of computer virus: a theoretical assessment approach. Appl Math Modell 43:110–125

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

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to Mohammad Reza Parsaei or Hassan Saberi Nik.

Ethics declarations

Conflict of interest

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

Appendix

Appendix

Proof of \(det(-P) > 0\):

$$\begin{aligned} det(-P)=\;& {} (\beta L+\beta B+\delta )[(\gamma _2+\delta )(-\beta S^*+\gamma _1+\alpha +\delta )-\alpha \beta S^*]\\&+(\beta L+\beta B) [(\gamma _2+\delta )(\beta S^*-\gamma _1)+\alpha (\beta S^*-\gamma _2)]. \end{aligned}$$

Since \(\widetilde{P^{-1}}_{11}=0,\) and \(\widetilde{P^{-1}}_{12}<0\), we have \(det(-P) > 0\).

Hence, it is clear to see \(det(-E) > 0\). The proof is then complete. \(\square\)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Parsaei, M.R., Javidan, R., Shayegh Kargar, N. et al. On the global stability of an epidemic model of computer viruses. Theory Biosci. 136, 169–178 (2017). https://doi.org/10.1007/s12064-017-0253-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12064-017-0253-2

Keywords

Navigation