A delayed computer virus propagation model and its dynamics. (English) Zbl 1343.34186

Summary: We propose a delayed computer virus propagation model and study its dynamic behaviors. First, we give the threshold value \(R_0\) determining whether the virus dies out completely. Second, we study the local asymptotic stability of the equilibria of this model and it is found that, depending on the time delays, a Hopf bifurcation may occur in the model. Next, we prove that, if \(R_0=1\), the virus-free equilibrium is globally attractive; and when \(R_0 <1\), it is globally asymptotically stable. Finally, a sufficient criterion for the global stability of the virus equilibrium is obtained.


34K60 Qualitative investigation and simulation of models involving functional-differential equations
92D30 Epidemiology
34K20 Stability theory of functional-differential equations
34K18 Bifurcation theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
34K25 Asymptotic theory of functional-differential equations
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[1] Cohen, F., Computer virus: theory and experiments, Comput Secur, 6, 22-35 (1987)
[2] Wierman, J. C.; Marchette, D. J., Modeling computer virus prevalence with a susceptible-infected-susce-ptible model with reintroduction, Comput Stat Data Anal, 45, 3-23 (2004) · Zbl 1429.68037
[3] Piqueira, J. R.C.; Araujo, V. O., A modified epidemiological model for computer viruses, Appl Math Comput, 213, 355-360 (2009) · Zbl 1185.68133
[4] Mishra, B. K.; Jha, N., Fix period of temporary immunity after run of anti-malicious software on computer nodes, Appl Math Comput, 190, 1207-1212 (2007) · Zbl 1117.92052
[5] Mishra, B. K.; Jha, N., An SEIQRS model for the transmission of malicious objects in computer network, Appl Math Model, 34, 710-715 (2010) · Zbl 1185.68042
[6] Mishra, B. K.; Pandey, S. K., Dynamic model of worms with vertical transmission in computer network, Appl Math Model, 34, 710-715 (2010) · Zbl 1185.68042
[7] Song, L. P.; Jin, Z.; Sun, G. Q.; Zhang, J.; Han, X., Influence of removable devices on computer worms: dynamic analysis and control strategies, Comput Math Appl Comput Math, 61, 1823-1829 (2011) · Zbl 1219.37065
[8] Ren, J.; Yang, X.; Zhu, Q.; Yang, L. X.; Zhang, C., A novel computer model and its dynamics, Nonlinear Anal Real World Appl, 13, 376-384 (2012) · Zbl 1238.34076
[9] Kafai, Y. B.; White, S., Understanding virtual epidemics: children’s folk conceptions of a computer virus, J Sci Edu Tech, 6, 523-529 (2009)
[10] Hale, J., Theory of functional differential equations (1977), Springer-Verlag: Springer-Verlag Heidelberg · Zbl 0352.34001
[11] Thieme, H. R.; van den Driessche, P., Global stability in cyclic epidemic models with disease fatalities, Fields Inst Commun, 21, 459-472 (1999) · Zbl 0924.92018
[12] Brauer, F.; Castillo-Chavez, C., Mathematical models in population biology and epidemiology (2001), Springer · Zbl 0967.92015
[13] Li, G.; Wang, W.; Jin, Z., Global stability of an SEIR epidemic model with constant immigration, Chaos, Solitons Fractals, 30, 4, 1012-1018 (2006) · Zbl 1142.34352
[14] Li, X. Z.; Zhou, L. L., Global stability of SEIR epidemic model with vertical transmission and saturating contact rate, Chaos, Solitons Fractals, 40, 2, 874-884 (2009) · Zbl 1197.34077
[15] Cai, L. M.; Li, X. Z., Analysis of a SEIV epidemic model with a nonlinear incidence rate, Appl Math Model, 33, 2919-2926 (2009) · Zbl 1205.34049
[16] Li, G.; Jin, Z., Global stability of a SEIR epidemic model with infectious force in latent, infected and immune period, Chaos, Solitons Fractals, 25, 5, 1177-1184 (2005) · Zbl 1065.92046
[17] Gakkhar, S.; Negi, K., Pulse vaccination in SIRS eidemic model with non-monotonic incidence rate, Chaos, Solitons Fractals, 35, 3, 626-638 (2008) · Zbl 1131.92052
[18] Liu, J.; Zhou, Y., Global stability of an SIRS epidemic model with transport-related infection, Chaos, Solitons Fractals, 40, 1, 145-158 (2009) · Zbl 1197.34098
[19] Cooke, K. L.; van den Driessche, P., Analysis of an SEIRS epidemic model with two delays, J Math Biol, 35, 240-260 (1996) · Zbl 0865.92019
[20] Ma, W.; Song, M.; Takeuchi, Y., Global stability of an SIR epidemic model with time delay, Appl Math Lett, 17, 1141-1145 (2004) · Zbl 1071.34082
[21] Schwartz Ira, B.; Smith, H. L., Infinite subharmonic bifurcation in an SEIR epidemic model, J Math Biol, 18, 233-253 (1993) · Zbl 0523.92020
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