×

zbMATH — the first resource for mathematics

A novel computer virus model and its dynamics. (English) Zbl 1238.34076
Summary: We propose a novel computer virus propagation model and study its dynamic behaviors; to our knowledge, this is the first time the effect of anti-virus ability has been taken into account in this way. In this context, we give the threshold for determining whether the virus dies out completely. Then, we study the existence of equilibria, and analyze their local and global asymptotic stability. Next, we find that, depending on the anti-virus ability, a backward bifurcation or a Hopf bifurcation may occur. Finally, we show that under appropriate conditions, bistable states may be around. Numerical results illustrate some typical phenomena that may occur in the virus propagation over computer network.

MSC:
34C23 Bifurcation theory for ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
37N25 Dynamical systems in biology
34D23 Global stability of solutions to ordinary differential equations
68M10 Network design and communication in computer systems
68M11 Internet topics
92D30 Epidemiology
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Cohen, F., Computer virus: theory and experiments, Computers and security, 6, 22-35, (1987)
[2] Kephart, J.O.; Hogg, T.; Huberman, B.A., Dynamics of computational ecosystems, Physical review A, 40, 1, 404-421, (1998)
[3] Wierman, J.C.; Marchette, D.J., Modeling computer virus prevalence with a susceptible-infected-susceptible model with reintroduction, Computational statistics & data analysis, 45, 3-23, (2004) · Zbl 1429.68037
[4] Piqueira, J.R.C.; Araujo, V.O., A modified epidemiological model for computer viruses, Applied mathematics and computation, 213, 355-360, (2009) · Zbl 1185.68133
[5] Mishra, B.K.; Jha, N., Fixed period of temporary immunity after run of anti-malicious software on computer nodes, Applied mathematics and computation, 190, 1207-1212, (2007) · Zbl 1117.92052
[6] Mishra, B.K.; Jha, N., SEIQRS model for the transmission of malicious objects in computer network, Applied mathematical modeling, 34, 710-715, (2010) · Zbl 1185.68042
[7] Mishra, B.K.; Pandey, S,K., Fuzzy epidemic model for the transmission of worms in computer network, Nonlinear analysis: real world applications, 11, 4335-4341, (2010) · Zbl 1203.94148
[8] Mishra, B.K.; Pandey, S.K., Dynamic model of worms with vertical transmission in computer network, Applied mathematics and computation, 217, 8438-8446, (2011) · Zbl 1219.68080
[9] Yuan, H.; Chen, G.Q., Network virus-epidemic model with the point-to-group information propagation, Applied mathematics and computation, 206, 357-367, (2008) · Zbl 1162.68404
[10] Han, X.; Tan, Q., Dynamical behavior of computer virus on Internet, Applied mathematics and computation, (2010) · Zbl 1209.68139
[11] Wang, F.G.; Zhang, Y.K.; Wang, C.G.; Ma, J.F.; Moon, S.J., Stability analysis of a SEIQV epidemic model for rapid spreading worms, Computers & security, 29, 410-418, (2010)
[12] Yao, Y.; Xie, X.W.; Guo, H.; Yu, G.; Gao, F.X.; Tong, X.J., Hopf bifurcation in Internet worm propagation with time delay in quarantine, Mathematical and computer modelling, (2011)
[13] Song, L.P.; Jin, Z.; Sun, G.Q.; Zhang, J.; Han, X., Influence of removable devices on computer worms: dynamic analysis and control strategies, Computers and mathematics with applications, 61, 1823-1829, (2011) · Zbl 1219.37065
[14] Forrest, S.; Hofmayer, S.A.; Somayaji, A., Computer immunology, Communications of the ACM, 40, 88-96, (1997)
[15] Wang, J.J.; Zhang, J.Z.; Jin, Z., Analysis of an SIR model with bilinear incidence rate, Nonlinear analysis: real world applications, 11, 2390-2402, (2010) · Zbl 1203.34136
[16] Li, L.; Sun, G.Q.; Jin, Z., Bifurcation and chaos in an epidemic model with nonlinear incidence rates, Applied mathematics and computation, 216, 1226-1234, (2010) · Zbl 1187.92073
[17] Hadeler, K.P.; Driessche, P., Backward bifurcation in epidemic control, Mathematical biosciences, 146, 15-35, (1997) · Zbl 0904.92031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.