Global dynamics of a novel multi-group model for computer worms

doi:10.1088/1674-1056/22/4/040204

Abstract In this paper, we study worm dynamics in computer networks composed of many autonomous systems. A novel multi-group SIQR (susceptible-infected-quarantined-removed) model is proposed for computer worms by explicitly considering anti-virus measures and the network infrastructure. Then, the basic reproduction number of worm R₀ is derived and the global dynamics of the model is established. It is shown that if R₀ is less than or equal to 1, the disease-free equilibrium is globally asymptotically stable and the worm dies out eventually, whereas, if R₀ is greater than 1, there exists one unique endemic equilibrium and it is globally asymptotically stable, thus the worm persists in the network. Finally, numerical simulations are given to illustrate the theoretical results.

Keywords: computer worm multi-group model Lyapunov function global dynamics

Received: 04 September 2012
Revised: 16 October 2012
Accepted manuscript online:

PACS:	02.40.Vh	(Global analysis and analysis on manifolds)
	89.20.Ff	(Computer science and technology)
	02.60.Lj	(Ordinary and partial differential equations; boundary value problems)

Fund: Project supported by the Natural Science Foundation of Jiangsu Province, China (Grant No. BK2010526), the Six Projects Sponsoring Talent Summits of Jiangsu Province, China (Grant No. SJ209006), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20103223110003), the Ministry of Education Research in the Humanities and Social Sciences Planning Fund of China (Grant No. 12YJAZH120), and the Graduate Student Innovation Research Project of Jiangsu Province, China (Grant Nos. CXLX11_-0417 and CXLX11_-0404).

Corresponding Authors: Gong Yong-Wang
E-mail: gong_yw@126.com

Cite this article:

Gong Yong-Wang (巩永旺), Song Yu-Rong (宋玉蓉), Jiang Guo-Ping (蒋国平) Global dynamics of a novel multi-group model for computer worms 2013 Chin. Phys. B 22 040204

[1]	Toutonji O A, Yoo S M and Park M 2012 Appl. Math. Model. 36 2751
[2]	Moore D, Shannon C and Brown J 2002 Proceedings of the 2nd ACM SIGCOMM Workshop on Internet Measurement, Novermber 6-8, 2002, Marseille, France, p. 273
[3]	Moore D, Paxson V, Savage S, Staniford S and Weaver N 2003 IEEE Secur. Priv. 1 33
[4]	Kim J, Radhakrishnan S and Dhall S K 2004 Proceedings of the 13th International Conference on Computer Communications and Networks, October 11-13, 2004, Chicago, USA, p. 495
[5]	Li J and Knickerbocker P 2007 Comput. Secur. 26 338
[6]	Mishra B K and Saini D 2007 Appl. Math. Comput. 187 929
[7]	Li Y, Liu Y, Shan X M, Ren Y, Jiao J and Qiu B 2005 Chin. Phys. 14 2153
[8]	Mishra B K and Jha N 2007 Appl. Math. Comput. 190 1207
[9]	Ren J, Yang X, Zhu Q, Yang L X and Zhang C 2012 Nonlinear Anal-Real. 13 376
[10]	Zou C C, Gong W and Towsley D 2003 Proceedings of the 2003 ACM Workshop on Rapid Malcode, October 27, 2003, Washington DC, USA, p. 51
[11]	Wang F, Zhang Y, Wang C, Ma J and Moon S 2010 Comput. Secur. 29 410
[12]	Liljenstam M, Yuan Y, Premore B and Nicol D 2002 Proceedings of 10th IEEE/ACM Symposium on Modeling, Analysis and Simulation of Computer and Telecommunication Systems (MASCOTS), October 11-16, 2002, Fort Worth, USA, p. 109
[13]	Huang W, Cooke K L and Castillo-Chavez C 1992 SIAM J. Appl. Math. 52 835
[14]	Xiao D and Ruan S 2007 Math. Biosci. 208 419
[15]	Ji C, Jiang D and Shi N 2011 Physica A 390 1747
[16]	Lu J and Chen G 2005 IEEE Trans. Automat. Control 50 841
[17]	Lu J, Yu X, Chen G and Cheng D 2004 IEEE Trans. Circ. Syst. I 51 787
[18]	Zhou J, Lu J A and Lu J 2006 IEEE Trans. Automat. Control 51 652
[19]	Lin P, Qin K and Wu H 2011 Chin. Phys. B 20 108701
[20]	Zhang W and Liu J 2011 Chin. Phys. B 20 030701
[21]	Wang H, Men F, He X and Wei Q 2012 Chin. Phys. B 21 060501
[22]	Gong Y W, Song Y R and Jiang G P 2012 Acta Phys. Sin. 61 110205 (in Chinese)
[23]	Wang P, Lu J and Ogorzalek M J 2012 Neurocomputing 78 155
[24]	Guo H, Li M Y and Shuai Z 2006 Can. Appl. Math. Q. 14 259
[25]	Guo H, Li M Y and Shuai Z 2008 Proc. Amer. Math. Soc. 136 2793
[26]	Van den Driessche P and Watmough J 2002 Math. Biosci. 180 29
[27]	Freedman H I, Tang M X and Ruan S G 1994 J. Dyn. Diff. Eq. 6 583
[28]	Li M Y, Graef J R, Wang L and Karsai J 1999 Math. Biosci. 160 191
[29]	Smith H L and Waltman P 1995 The Theory of the Chemostat: Dynamics of Microbial Competition (Cambridge: Cambridge University Press)

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