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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References
Bhunu CP, Mushayabasa S (2013) Assessing the effects of drug misuse on HIV/AIDS prevalence. Theory Biosci 133–2:83–92
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
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
Cross GW (1978) Three types of matrix stability. Linear Algebra Appl 20:253–263
Driessche VD, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180(1):29–48
Han X, Tan QL (2010) Dynamical behavior of computer virus on internet. Appl Math Comput 217:2520–2526
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
Kephart J, White S (1991) Directed-graph epidemiological models of computer viruses, in. IEEE Symp Secur Pri 1991:343–359
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
Li J, Knickerbocker P (2007) Functional similarities between computer worms and biological pathogens. Comput Secur 26(4):338–347
Liu S, Ruan S, Zhang X (2015) On avian influenza epidemic models with time delay. Theory Biosci 134–3:75–82
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
Mishra BK, Saini DK (2007) SEIRS epidemic model with delay for transmission of malicious objects in computer network. Appl Math Comput 188:1476–1482
Murray WH (1988) The application of epidemiology to computer viruses. Comput Secur 7(2):130–150
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
Rinaldi F (1990) Global stability results for epidemic models with latent period. IMA J Math Appl Med Biol 7:69–75
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
Wang J, Liao S (2012) A generalized cholera model and epidemic-endemic analysis. J Biol Dynam 6:568–589
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
Xu R, Ma Z (2010) Global stability of a delayed SEIRS epidemic model with saturation incidence rate. Nonlinear Dyn 61(1):229–239
Yang LX, Yang X (2015) The impact of nonlinear infection rate on the spread of computer virus. Nonlinear Dyn 82(1):85–95
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
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Appendix
Appendix
Proof of \(det(-P) > 0\):
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