一类具有非线性发生率的无线传感网络蠕虫传播模型的延迟动力学行为（英文）
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摘要
研究了一类具有不同发生率的无线传感网络蠕虫传播模型的延迟动力学行为。由于在监测节点隔离不稳定节点需要消耗一定的时间,在模型中考虑了处理时滞。通过分析相应特征方程根的分布情况,得到了平衡点存在性、模型局部稳定性和Hopf分岔存在的充分性条件。通过构造合适的李雅普诺夫函数,证明了蠕虫病毒平衡点的全局稳定性。数值仿真实验验证了理论分析结果的正确性。仿真结果表明,当处理时滞的值越过关键值时,网络中的蠕虫传播将失去控制,发现无线传感网络的覆盖范围是控制蠕虫传播和保证无线传感网络安全最为重要的因素之一。并通过仿真发现消除模型混沌状态的一些关键参数,其中非线性发生率βSI/I+1是控制蠕虫病毒传播、保证无线传感网络安全的最佳选择。
In this paper, delay dynamics of a worm propagation model has been investigated with different incidence rates in wireless sensor network. Processing time delay occurs in the proposed model due to time consumed during monitoring the erratic behaviors of the nodes and isolating it from the network. Sufficient conditions for the existence of equilibrium points, stability analysis and Hopf bifurcation of the system are derived by analyzing distribution of roots of an associated characteristic equation. Global stability for worm-induced equilibrium is derived by constructing a suitable Lyapunov function.To verify analytical results, numerical simulations are carried out. In the case that the processing time delay exceeds the critical value, the worms in the network is beyond the control. There are many significant features of wireless sensor networks, among them coverage area is the most effective factor with respect to worm control and security purposes. The value of some influential parameters of sensor network are carefully selected so that the oscillations can be reduced and removed from the network. According to comparative study of model systems, it can be concluded that the incidence rate βSI/I + 1 is one of the best selection for worm control and security purposes in wireless sensor network.
In this paper, delay dynamics of a worm propagation model has been investigated with different incidence rates in wireless sensor network. Processing time delay occurs in the proposed model due to time consumed during monitoring the erratic behaviors of the nodes and isolating it from the network. Sufficient conditions for the existence of equilibrium points, stability analysis and Hopf bifurcation of the system are derived by analyzing distribution of roots of an associated characteristic equation. Global stability for worm-induced equilibrium is derived by constructing a suitable Lyapunov function.To verify analytical results, numerical simulations are carried out. In the case that the processing time delay exceeds the critical value, the worms in the network is beyond the control. There are many significant features of wireless sensor networks, among them coverage area is the most effective factor with respect to worm control and security purposes. The value of some influential parameters of sensor network are carefully selected so that the oscillations can be reduced and removed from the network. According to comparative study of model systems, it can be concluded that the incidence rate βSI/I + 1 is one of the best selection for worm control and security purposes in wireless sensor network.
引文
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[21] KADDAR A. Stability analysis in a delayed SIR epidemic model with saturated incidence rate[J].Nonlinear Analysis:Modelling and Control, 2010,15(3):299-306.
[22] WANG F W, HUANG W Y, SHEN Y L, et al.Analysis of SVEIR worm attack model with saturated incidence and partial immunization[J].Journal of Communications and Information Networks, 2016,1(4):105-115.DOI:10.1007/bf03391584
[23] ZHANG Z Z, WANG Y G, GUERRINI L.Bifurcation analysis of a delayed worm propagation model with saturated incidence[J].Advances in Mathematical Physics, 2018, Article ID 7619074.DOI:10.1155/2018/7619074
[24] CAPASSO V, SERIO G. A generalization of Kermack-McKendrick deterministic epidemic model[J].Mathematical Bioscience, 1978, 42(1/2):43-61.DOI:10.1016/0025-5564(78)90006-8
[25] WEI C, CHEN L. A delayed epidemic model with pulse vaccination[J].Discrete Dynamics in Nature and Society,2008, 2008(1):1-12.DOI:10.1155/2008/746951
[26] ZHANG Z Z, BI D J. Dynamical analysis of a computer virus propagation model with delay and infectivity in latent period[J].Discrete Dynamics in Nature and Society, 2016(1):Article ID 3067872.DOI:10.1155/2016/3067872
[27] REN J G, XU Y H, ZHANG C M. Optical control of a delay-varying computer virus propagation model[J].Discrete Dynamics in Nature and Society, 2013,Article ID 210291.DOI:10.1155/2013/210291
[28] ZHU Y L, SONG J J, DONG F Z. Applications of wireless sensor network in the agriculture environment monitoring[J].Procedia Engineering, 2011, 16:608-614.DOI:10.1016/j.proeng.2011.08.1131
[29] VAN D D P, WATMOUGH J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission[J].Mathematical Biosciences, 2002, 180(1/2):29-48.DOI:10.1016/s0025-5564(02)00108-6
[30] HASSARD B D, KAZARINOFF N D, WAN Y H.Theory and Applications of Hopf Bifurcation[M]. Cambridge:Cambridge University Press, 1981.DOI:10.1090/conm/445
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[2] KEPHART J O, WHITE S R. Directed-graph epidemiological models of computer viruses[C]//1991Proceedings of the IEEE Computer Society Symposium on Research in Security and Privacy.Oakland:IEEE, 1991:343-359. DOI:10.1109/risp.1991.130801
[3] KEPHART J O, WHITE S R. Measuring and modeling computer virus prevalence[C]//1993 Proceedings of the IEEE Computer Society Symposium on Research in Security and Privacy.Oakland:IEEE, 1993:2-15.DOI:10.1109/risp.1993.287647
[4] REN J G, YANG X F, YANG L X, et al. A delayed computer virus propagation model and its dynamics[J].Chaos, Soliton and Fractals, 2012, 45(1):74-79.DOI:10.1016/j.chaos.2011.10.003
[5] PENG M, HE X, HUANG J J, et al. Modelling computer virus and its dynamics[J].Mathematical Problems in Engineering, 2013, Article ID:842614.DOI:10.1155/2013/842614
[6] GUILLéN J D H, REY A M D, ENCINAS L H.Study of the stability of a SEIRS model for computer worm propagation[J].Physica A:Statistica Mechanics and Its Applications, 2017, 479(1):411-421. DOI:10.1016/j.physa.2017.03.023
[7] HOSSEINI S, AZGOMI M A, RAHMANI A T.Malware propagation modeling considering software diversity and immunization[J].Journal of Computational Science, 2016, 13:49-67. DOI:10.1016/j.jocs.2016.01.002
[8] BADSHAH Q. Global stability of SEIQRS computer virus propagation model with nonlinear incidence function[J].Applied Mathematics, 2015, 6(11):1926-1938.DOI:10.4236/am.2015.611170
[9] LI Y, THAI M T.Wireless Sensor Networks and Applications[M]. Germany:Springer Science and Business Media,2008.
[10] UPADHYAY R K, KUMARI S, MISRA A K.Modeling the virus dynamics in computer network with SVEIR model and nonlinear incident rate[J].Journal of Applied Mathematics and Computing,2017, 54(1/2):485-509. DOI:10.1007/s12190-016-1020-0
[11] XIAO X, FU P, DOU C S, et al. Design and analysis of SEIQR worm propagation model in mobile internet[J].Communications in Nonlinear Science and Numerical Simulation, 2017, 43:341-350.DOI:10.1016/j.cnsns.2016.07.012
[12] KHANH N H. Dynamics of a worm propagation model with quarantine in wireless sensor networks[J].Applied Mathematics and Information Sciences,2016, 10(5):1739-1746.DOI:10.18576/amis/100513
[13] MISHRA B K, JHA N. SEIQRS model for the transmission of malicious objects in computer network[J].Applied Mathematical Modelling, 2010, 34(3):710-715.DOI:10.1016/j.apm.2009.06.011
[14] ZIA T A, ZOMAYA A Y. A lightweight security framework for wireless sensor networks[J].Journal of Wireless Mobile Networks, Ubiquitous Computing,and Dependable Applications, 2011, 2(3):53-73.DOI:10.1109/ICC.2014.6883886
[15] ZOU C C, GONG W, TOWSLEY D. Worm propagation modeling and analysis under dynamic quarantine defense[C]//2003 Proceedings of the ACM CCS Workshop on Rapid Malcode. Washington DC:ACM, 2003:51-60.DOI:10.1145/948187.948197
[16] MOORE D, SHANNON C, VOELKER G M, et al. Internet quarantine:Requirements for containing self-propagating code[C]//IEEE INFOCOM 2003.Twenty-second Annual Joint Conference of the IEEE Computer and Communications Societies.San Francisco:IEEE, 2003:1901-1910. DOI:10.1109/infcom.2003.1209212
[17] CHENTM,JAMILN.Effectivenessofquarantineinworm epidemics[C]//2006 IEEE International Conference on Communications.Istanbul:IEEE,2006:2142-2147.DOI:10.1109/icc.2006.255087
[18] BAYANI M, ALPIZAR Y, GAMBOA R, et al. Radio communication range effects on a flat wireless sensor network performance[C]//Central American and Panama Convention.San Jose:IEEE, 2016:1-10.DOI:10.1109/concapan.2016.7942365
[19] FENG L P, LIAO X F, LI H Q, et al. Hopf bifurcation analysis of a delayed viral infection model in computer networks[J].Mathematical and Computer Modelling, 2012, 56(7/8):167-179. DOI:10.1016/j.mcm.2011.12.010
[20] OJHA R P, SANYAL G, SRIVASTAVA P K, et al. Design and analysis of modified SIQRS model for performance study of wireless sensor network[J].Scalable Computing, 2017, 18(3):229-241. DOI:10.12694/scpe.v18i3.1303
[21] KADDAR A. Stability analysis in a delayed SIR epidemic model with saturated incidence rate[J].Nonlinear Analysis:Modelling and Control, 2010,15(3):299-306.
[22] WANG F W, HUANG W Y, SHEN Y L, et al.Analysis of SVEIR worm attack model with saturated incidence and partial immunization[J].Journal of Communications and Information Networks, 2016,1(4):105-115.DOI:10.1007/bf03391584
[23] ZHANG Z Z, WANG Y G, GUERRINI L.Bifurcation analysis of a delayed worm propagation model with saturated incidence[J].Advances in Mathematical Physics, 2018, Article ID 7619074.DOI:10.1155/2018/7619074
[24] CAPASSO V, SERIO G. A generalization of Kermack-McKendrick deterministic epidemic model[J].Mathematical Bioscience, 1978, 42(1/2):43-61.DOI:10.1016/0025-5564(78)90006-8
[25] WEI C, CHEN L. A delayed epidemic model with pulse vaccination[J].Discrete Dynamics in Nature and Society,2008, 2008(1):1-12.DOI:10.1155/2008/746951
[26] ZHANG Z Z, BI D J. Dynamical analysis of a computer virus propagation model with delay and infectivity in latent period[J].Discrete Dynamics in Nature and Society, 2016(1):Article ID 3067872.DOI:10.1155/2016/3067872
[27] REN J G, XU Y H, ZHANG C M. Optical control of a delay-varying computer virus propagation model[J].Discrete Dynamics in Nature and Society, 2013,Article ID 210291.DOI:10.1155/2013/210291
[28] ZHU Y L, SONG J J, DONG F Z. Applications of wireless sensor network in the agriculture environment monitoring[J].Procedia Engineering, 2011, 16:608-614.DOI:10.1016/j.proeng.2011.08.1131
[29] VAN D D P, WATMOUGH J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission[J].Mathematical Biosciences, 2002, 180(1/2):29-48.DOI:10.1016/s0025-5564(02)00108-6
[30] HASSARD B D, KAZARINOFF N D, WAN Y H.Theory and Applications of Hopf Bifurcation[M]. Cambridge:Cambridge University Press, 1981.DOI:10.1090/conm/445
[31] UPADHYAY R K, AGRAWAL R. Dynamics and response of a predator-prey system with competitiveinterference and time delay[J].Nonlinear Dynamics, 2016, 83(1/2):821-837. DOI:10.1007/s11071-015-2370-0
[32] XU C J. Delay-induced oscillations in a competitorcompetitor-mutualist Lotka-Volterra model[J].Complexity, 2017, Article ID:2578043.DOI:10.1155/2017/2578043
[33] BIANCA C, FERRARA M, GUERRINI L. The cai model with time delay:Existence of periodic solutionsand asymptotic analysis[J].Applied Mathematics and Information Science, 2013, 7(1):21-27. DOI:10.12785/amis/070103
[34] HerijavecGroup.2017CybercrimeReport[R/OL].http://cyberseurityventures.com/2015-wp/wp-content/uploads/2017/10/2017-Cybercrime-Report.pdf.
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