Abstract
Cybersecurity dynamical system model is a promising tool to describe and understand virus spreading in networks. The modelling comprises of two issues: the state transition diagram and the infection graph. Most works focus on proposing models (the state transition diagram) and studying the relationship between dynamics and the infection graph topology. In this paper, We propose the SEIC model and illustrate how the model transition diagram influence the dynamics, in particular, the epidemic threshold by calculating and comparing their thresholds in a class of Secure-Exposed-Infectious-Cured (SEIC) models. We show that as a new state enters the state transition diagram in the fashion of the SEIC model, the epidemic threshold increases, which implies that the model has a larger region of parameters to be stabilized. Numerical examples are presented to verify the theoretical results.
This work is jointly supported by the National Natural Sciences Foundation of China under Grant No. 61673119.
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References
Ball, F., Sirl, D., Trapman, P.: Threshold behaviour and final outcome of an epidemic on a random network with household structure. Adv. Appl. Probab. 41(3), 765–796 (2009)
Chakrabarti, D., Wang, Y., Wang, C., Leskovec, J., Faloutsos, C.: Epidemic thresholds in real networks. ACM Trans. Inf. Syst. Secur. 10(4), 1:1–1:26 (2008)
Cohen, F.: Computer viruses: theory and experiments. Comput. Secur. 6(1), 22–35 (1987)
d’Onofrio, A.: A note on the global behaviour of the network-based SIS epidemic model. Nonlinear Anal.: Real World Appl. 9(4), 1567–1572 (2008)
Ganesh, A., Massoulie, L., Towsley, D.: The effect of network topology on the spread of epidemics. In: Proceedings IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies, vol. 2, pp. 1455–1466, March 2005
Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42(4), 599–653 (2000)
Kang, H., Fu, X.: Epidemic spreading and global stability of an SIS model with an infective vector on complex networks. Commun. Nonlinear Sci. Numer. Simul. 27(1), 30–39 (2015)
Kephart, J.O., White, S.R.: Directed-graph epidemiological models of computer viruses. In: Proceedings of the 1991 IEEE Computer Society Symposium on Research in Security and Privacy, pp. 343–359, May 1991
Kephart, J.O., White, S.R., Chess, D.M.: Computers and epidemiology. IEEE Spectr. 30(5), 20–26 (1993)
Kim, J., Radhakrishnan, S., Dhall, S.K.: Measurement and analysis of worm propagation on internet network topology. In: Proceedings of the 13th International Conference on Computer Communications and Networks (IEEE Cat. No. 04EX969), pp. 495–500, October 2004
Murray, W.H.: The application of epidemiology to computer viruses. Comput. Secur. 7(2), 139–145 (1988)
Pastor-Satorras, R., Vespignani, A.: Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200–3203 (2001)
Shi, H., Duan, Z., Chen, G.: An SIS model with infective medium on complex networks. Phys. A: Stat. Mech. Appl. 387(8), 2133–2144 (2008)
Wang, Y., Chakrabarti, D., Wang, C., Faloutsos, C.: Epidemic spreading in real networks: an eigenvalue viewpoint. In: Proceedings of the 22nd International Symposium on Reliable Distributed Systems, pp. 25–34, October 2003
Kermack, W.O., Mckendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 115(772), 700–721 (1927)
Kermack, W.O., Mckendrick, A.G.: Contributions to the mathematical theory of epidemics. II.—the problem of endemicity. Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 138(834), 55–83 (1932)
Xu, S., Lu, W., Li, H.: A stochastic model of active cyber defense dynamics. Internet Math. 11(1), 23–61 (2015)
Xu, S., Lu, W., Xu, L.: Push- and pull-based epidemic spreading in networks: thresholds and deeper insights. ACM Trans. Auton. Adapt. Syst. 7(3), 32:1–32:26 (2012)
Xu, S., Lu, W., Xu, L., Zhan, Z.: Adaptive epidemic dynamics in networks: thresholds and control. ACM Trans. Auton. Adapt. Syst. 8(4), 19:1–19:19 (2014)
Yang, M., Chen, G., Fu, X.: A modified SIS model with an infective medium on complex networks and its global stability. Phys. A: Stat. Mech. Appl. 390(12), 2408–2413 (2011)
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Qiang, H., Lu, W. (2018). A Note on Dependence of Epidemic Threshold on State Transition Diagram in the SEIC Cybersecurity Dynamical System Model. In: Liu, F., Xu, S., Yung, M. (eds) Science of Cyber Security. SciSec 2018. Lecture Notes in Computer Science(), vol 11287. Springer, Cham. https://doi.org/10.1007/978-3-030-03026-1_4
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DOI: https://doi.org/10.1007/978-3-030-03026-1_4
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