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On the global stability of an epidemic model of computer viruses

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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.

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Correspondence to Mohammad Reza Parsaei or Hassan Saberi Nik.

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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. \(\square\)

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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

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

Keywords

  • Dynamical models
  • Computer viral propagation
  • Global stability
  • Computer virus
  • Lyapunov function
  • Volterra–Lyapunov stability