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Bifurcation analysis of a model for network worm propagation with time delay. (English) Zbl 1201.68060
Summary: Considering that reassembly of a system and/or using anti-virus software will take a period of time, we introduce a time delay for modeling this period of time. Also, considering that at different times the propagation of a worm shows different characteristics, we build a section model for Internet worm propagation depending on a two-factor model. We first consider the stability of the positive equilibrium and the existence of a local Hopf bifurcation. In succession, using the normal form theory and center manifold argument, we derive explicit formulas determining the stability, direction and other properties of bifurcation periodic solutions. Finally, a numerical simulation is presented. The techniques of analysis of the mathematical model provide a theoretical foundation for control and forecasting for Internet worms.
MSC:
68Q25Analysis of algorithms and problem complexity
68M11Internet topics
34K18Bifurcation theory of functional differential equations
34K20Stability theory of functional-differential equations
References:
[1]Moore, David; Shannon, Colleen; Voelker, Geoffrey M.; Savage, Stefan: Requirements for containing self-propagating code, IEEE Internet quarantine (2003)
[2]Daley, D. J.; Gani, J.: Epidemic modelling: an introduction, (1999)
[3]Y. Wang, C.X. Wang, Modeling the effects of timing parameters on virus propagation, in: S. Staniford (Ed.) Proc. of the ACM CCS Workshop on Rapid Malcode, WORM 2003, Washington, 2003.
[4]Frauenthal, J. C.: Mathematical modeling in epidemiology, (1980)
[5]C.C. Zou, W. Gong, D. Towsley, Code red worm propagation modeling and analysis, in: Proc. of the 9th ACM Symp. on Computer and Communication Security, Washington, 2002, pp. 138–147.
[6]Kuang, Y.: Delay differential equations: with applications in population dynamics, (1993) · Zbl 0777.34002
[7]Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H.: Theory and applications of Hopf bifurcation, (1981)