Two quarantine models on the attack of malicious objects in computer network. (English) Zbl 1264.68015

Summary: SEIQR (susceptible, exposed, infectious, quarantined, recovered) models for the transmission of malicious objects with simple mass action incidence and standard incidence rate in computer network are formulated. Threshold, equilibrium, and their stability are discussed for the simple mass action incidence and standard incidence rate. Global stability and asymptotic stability of endemic equilibrium for simple mass action incidence have been shown. With the help of the Poincaré-Bendixson property, asymptotic stability of endemic equilibrium for standard incidence rate has been shown. Numerical methods have been used to solve and simulate the system of differential equations. The effect of quarantine on recovered nodes is analyzed. We have also analyzed the behavior of the susceptible, exposed, infected, quarantine, and recovered nodes in the computer network.


68M10 Network design and communication in computer systems
34D20 Stability of solutions to ordinary differential equations
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[1] M. E. J. Newman, S. Forrest, and J. Balthrop, “Email networks and the spread of computer viruses,” Physical Review E, vol. 66, no. 3, Article ID 035101, 4 pages, 2002.
[2] R. M. Anderson and R. M. May, Infection Disease of Humans, Dynamics and Control, Oxford University Press, Oxford, UK, 1992.
[3] R. M. Anderson and R. M. May, “Population biology of infectious diseases,” Nature, vol. 180, pp. 361-367, 1999.
[4] S. Forrest, S. Hofmeyr, A. Somayaji, and T. Longstaff, “Self-nonself discrimination in a computer,” in Proceedings of the IEEE Symposium on Research in Security and Privacy, pp. 202-212, May 1994.
[5] E. Gelenbe, “Keeping viruses under control,” in Proceedings of the 20th International Symposium on Computer and Information Sciences (ISCIS ’05), vol. 3733 of Lecture Notes in Computer Science, pp. 304-311, Springer, October 2005.
[6] E. Gelenbe, “Dealing with software viruses: a biological paradigm,” Information Security Technical Report, vol. 12, no. 4, pp. 242-250, 2007.
[7] E. Gelenbe, V. Kaptan, and Y. Wang, “Biological metaphors for agent behavior,” in Computer Science, pp. 667-675, Springer-Verlag, 2004.
[8] J. R. C. Piqueira and F. B. Cesar, “Dynamical models for computer viruses propagation,” Mathematical Problems in Engineering, vol. 2008, Article ID 940526, 11 pages, 2008. · Zbl 1189.68036
[9] J. R. C. Piqueira, B. F. Navarro, and L. H. A. Monteiro, “Epidemiological models applied to virus in computer networks,” Journal of Computer Science, vol. 1, no. 1, pp. 31-34, 2005.
[10] W. T. Richard and J. C. Mark, “Modeling virus propagation in peer-to-peer networks,” in Proceedings of the 5th International Conference on Information, Communications and Signal Processing, pp. 981-985, December 2005.
[11] B. K. Mishra and D. K. Saini, “SEIRS epidemic model with delay for transmission of malicious objects in computer network,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1476-1482, 2007. · Zbl 1118.68014
[12] B. K. Mishra and D. K. Saini, “Mathematical models on computer viruses,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 929-936, 2007. · Zbl 1120.68041
[13] Z. Feng and H. R. Thieme, “Recurrent outbreaks of childhood diseases revisited: the impact of isolation,” Mathematical Biosciences, vol. 128, no. 1-2, pp. 93-130, 1995. · Zbl 0833.92017
[14] Z. Feng and H. R. Thieme, “Endemic models with arbitrarily distributed periods of infection I: general theory,” SIAM Journal on Applied Mathematics, vol. 61, no. 3, pp. 803-833, 2000. · Zbl 0991.92028
[15] Z. Feng and H. R. Thieme, “Endemic models with arbitrarily distributed periods of infection II: fast disease dynamics and permanent recovery,” SIAM Journal on Applied Mathematics, vol. 61, no. 3, pp. 983-1012, 2000. · Zbl 1016.92035
[16] L. I. Wu and Z. Feng, “Homoclinic bifurcation in an SIQR model for childhood diseases,” Journal of Differential Equations, vol. 168, no. 1, pp. 150-167, 2000. · Zbl 0969.34042
[17] K. L. Cooke and P. van den Driessche, “Analysis of an SEIRS epidemic model with two delays,” Journal of Mathematical Biology, vol. 35, no. 2, pp. 240-260, 1996. · Zbl 0865.92019
[18] M. Y. Li, J. R. Graef, L. C. Wang, and J. Karsai, “Global dynamics of a SEIR model with varying total population size,” Mathematical Biosciences, vol. 160, no. 2, pp. 191-213, 1999. · Zbl 0974.92029
[19] D. Greenhalgh, “Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity,” Mathematical and Computer Modelling, vol. 25, no. 2, pp. 85-107, 1997. · Zbl 0877.92023
[20] J. K. Hale, Ordinary Differential Equations, Krieger, Basel, Switzerland, 2nd edition, 1980. · Zbl 0433.34003
[21] H. W. Hethcote, H. W. Stech, and P. van dan Driessche, “Periodicity and stability in epidemic models: a survey,” in Differential Equations and Applications in Ecology, Epidemics and Population Problems, K. L. Cook, Ed., pp. 65-85, Academic Press, New York, NY, USA, 1981. · Zbl 0477.92014
[22] M. Y. Li and J. S. Muldowney, “Global stability for the SEIR model in epidemiology,” Mathematical Biosciences, vol. 125, no. 2, pp. 155-164, 1995. · Zbl 0821.92022
[23] M. Y. Li and L. Wang, “Global stability in some SEIR epidemic models,” in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory, C. Chavez, S. Blower, P. van den Driessche, D. Kirscner, and A. Yakhdu, Eds., vol. 126, pp. 295-311, Springer, New York, NY, USA, 2002. · Zbl 1022.92035
[24] Y. Michel, H. Smith, and L. Wang, “Global dynamics of an seir epidemic model with vertical transmission,” SIAM Journal on Applied Mathematics, vol. 62, no. 1, pp. 58-69, 2001. · Zbl 0991.92029
[25] H. Dong, Z. Wang, and H. Gao, “Observer-based H\infty control for systems with repeated scalar nonlinearities and multiple packet losses,” International Journal of Robust and Nonlinear Control, vol. 20, no. 12, pp. 1363-1378, 2010. · Zbl 1206.93035
[26] Z. Wang, D. W. C. Ho, H. Dong, and H. Gao, “Robust H\infty finite-horizon control for a class of stochastic nonlinear time-varying systems subject to sensor and actuator saturations,” IEEE Transactions on Automatic Control, vol. 55, no. 7, Article ID 5441037, pp. 1716-1722, 2010. · Zbl 1368.93668
[27] B. Shen, Z. Wang, and Y. S. Hung, “Distributed H\infty -consensus filtering in sensor networks with multiple missing measurements: the finite-horizon case,” Automatica, vol. 46, no. 10, pp. 1682-1688, 2010. · Zbl 1204.93122
[28] B. Shen, Z. Wang, H. Shu, and G. Wei, “Robust H\infty finite-horizon filtering with randomly occurred nonlinearities and quantization effects,” Automatica, vol. 46, no. 11, pp. 1743-1751, 2010. · Zbl 1218.93103
[29] B. Shen, Z. Wang, and X. Liu, “Bounded H-infinity synchronization and state estimation for discrete time-varying stochastic complex networks over a finite-horizon,” IEEE Transactions on Neural Networks, vol. 22, no. 1, pp. 145-157, 2011.
[30] H. Dong, Z. Wang, D. W. C. Ho, and H. Gao, “Variance-constrained H\infty filtering for a class of nonlinear time-varying systems with multiple missing measurements: the finite-horizon case,” IEEE Transactions on Signal Processing, vol. 58, no. 5, Article ID 5410142, pp. 2534-2543, 2010. · Zbl 1391.93233
[31] H. Hethcote, M. Zhien, and L. Shengbing, “Effects of quarantine in six endemic models for infectious diseases,” Mathematical Biosciences, vol. 180, pp. 141-160, 2002. · Zbl 1019.92030
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