Optimal Control Strategies for Virus Spreading in Inhomogeneous Epidemic Dynamics

Shang, Yilun

Canadian mathematical bulletin, Tome 56 (2013) no. 3, pp. 621-629

Résumé

In this paper, we study the spread of virus/worm in computer networks with a view to addressing cyber security problems. Epidemic models have been applied extensively to model the propagation of computer viruses, which characterize the fact that infected machines may spread malware to other hosts connected to the network. In our framework, the dynamics of hosts evolves according to a modified inhomogeneous Susceptible-Infectious-Susceptible $\left( \text{SIS} \right)$ epidemic model with time-varying transmission rate and recovery rate. The infection of computers is subject to direct attack as well as propagation among hosts. Based on optimal control theory, optimal attack strategies are provided by minimizing the cost (equivalently maximizing the profit) of the attacker. We present a threshold function of the fraction of infectious hosts, which captures the dynamically evolving strategies of the attacker and reflects the persistence of virus spreading. Moreover, our results indicate that if the infectivity of a computer worm is low and the computers are installed with antivirus software with high reliability, the intensity of attacks incurred will likely be low. This agrees with our intuition.

Détail

DOI : 10.4153/CMB-2012-007-2

Mots-clés : 49J15, 92D30, computer virus, epidemic dynamics, optimal control, network security

Bibliographie

[1] [1] Bailey, M., Cooke, E., Jahanian, F., Xu, Y. and Karir, M., A survey of botnet technology and defenses. In: Proc. Cybersecurity Applications & Technology Conference For Homeland Security (CATCH ‘09), Washington, DC, March 2009, 299–304. Google Scholar

[2] [2] Ball, F., D, Mollison and G, Scalia-Tomba, Epidemics with two levels of mixing. Ann. Appl. Prob. 7 (1997, 46–89. Google Scholar | DOI

[3] [3] Ball, F., D. Sirl and Trapman, P., Analysis of a stochastic SIR epidemic on a random network incorporating household structure. Math. Biosci. 224 (2010, 53–73. Google Scholar | DOI

[4] [4] Bensoussan, A., Kantarcioglu, M. and Hoe, C., A game-theoretical approach for finding optimal strategies in a botnet defense model. In: Proc. GameSec ‘10, Berlin, Germany, 2010, 135–148. Google Scholar

[5] [5] Berger, N., Borgs, C., Chayes, J. T. and Saberi, A., On the spread of viruses on the internet. In: Proc. 16th Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, 2005, 301–310. Google Scholar

[6] [6] Bertsekas, D. P., Dynamic Programming and Optimal Control. Vol. 1. Third edition. Athena Scientific, Belmont, MA, 2005. Google Scholar

[7] [7] Britton, T., Stochastic epidemic models: a survey. Math. Biosci. 225 (2010, 24–35. Google Scholar | DOI

[8] [8] Britton, T., Kypraios, T. and O'Neill, P. D., Inference for epidemics with three levels of mixing: methodology and application to a measles outbreak. Scand. J. Stat. 38 (2011, 578–599. Google Scholar

[9] [9] Cohen, F., Computer viruses: theory and experiments. Computer and Security 6 (1987, 22–35. Google Scholar

[10] [10] Diekmann, O. and Heesterbeek, J. A. P. , Mathematical Epidemiology of Infectious Disease. JohnWiley & Sons, Chichester, 2000. Google Scholar

[11] [11] Fultz, N. and Grossklags, J., Blue versus red: towards a model of distributed security attacks. Lecture Notes in Computer Science 5628 (2009, 167–183. Google Scholar

[12] [12] Higgins, K. J., Conficker botnet ‘dead in the water’, researcher says. Technical Report, http://www. darkreading.com/vulnerability management/security/attacks/showArticle.jhtml?articleID=224201115. Google Scholar

[13] [13] Lelarge, M., Economics of malware: epidemic risks model, network externalities and incentives. In: Proc. 47th Annual Allerton Conference on Communication, Control, and Computing, IEEE Press, Piscataway, NJ, 2009, 1353–1360. Google Scholar

[14] [14] Li, Z., Liao, Q. and Striegel, A., Botnet economics: uncertainty matters. In: Managing Information Risk and the Economics of Security, Springer, New York, 2009, 245–267. Google Scholar

[15] [15] Piqueira, J. R. C. and Araujo, V. O., A modified epidemiological model for computer viruses. Appl. Math. Comput. 213 (2009, 355–360. Google Scholar | DOI

[16] [16] Provos, N. and Holz, T., Virtual Honeypots—From Botnet Tracking to Intrusion Detection. Pearson Education Inc., Boston, 2008. Google Scholar

[17] [17] Shang, Y., Optimal attack strategies in a dynamic botnet defense model. Appl. Math. Inf. Sci. 6 (2012, 29–33. Google Scholar

[18] [18] Shang, Y., Likelihood estimation for stochastic epidemics with heterogeneous mixing populations. Int. J. Comput. Math. Sci. 6 (2012, 34–38. Google Scholar

[19] [19] Shang, Y., Multi-agent coordination in directed moving neighborhood random networks. Chinese Phys. B 19 (2010, 070201. Google Scholar

[20] [20] van den Broek, J. and Heesterbeek, J. A. P., Nonhomogeneous birth and death models for epidemic outbreak data. Biostatistics 8 (2007, 453–467. Google Scholar

[21] [21] van den Broek, J. and Nishiura, H., Using epidemic prevalence data to jointly estimate reproduction and removal. Ann. Appl. Stat. 3 (2009, 1505–1520. Google Scholar | DOI

[22] [22] Yeung, D. and Petrosyan, L., Cooperative Stochastic Differential Games. Springer, New York, 2006. Google Scholar

[23] [23] Zou, C., Duffield, N., D. Towsley andGong, W., Adaptive defense against various network attacks. IEEE Journal on Selected Areas in Communications 24 (2006, 1877–1888. Google Scholar