基于分数阶微分方程的木马病毒传播规律
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摘要
建立并研究了一类基于分数阶微分方程的木马病毒传播模型,利用分数阶微分方程的相关理论,详细证明了该模型非负解的有界性、存在唯一性,分析了平衡点的存在性及其局部稳定性,并通过数值试验验证了理论结果的正确性。得到:在基本再生数小于1的情况下,未感染平衡点是局部渐近稳定的,病毒会消亡;在基本再生数大于1时,感染平衡点局部渐近稳定,病毒将扩散。根据所得到的理论结果,给出了控制木马病毒传播的有效措施。
In this paper,an epidemic model of trojan virus based on the fractional differential equation was studied.By means of the theory of fractional differential equations,the existence and uniqueness of the positive solution were established,then the existence and stability condition of the equilibrium of the model were analyzed.It was showed that if the basic reproduction number is less than 1,the infection free equilibrium is locally asymptotically stable,virus will die out,and if the basic reproductive number is greater than 1,the infection equilibrium is stable and the virus will spread.Some numerical experiments are carried out to confirm the obtained results.In addition,some effective measures are given to control the spread of the trojan virus.
In this paper,an epidemic model of trojan virus based on the fractional differential equation was studied.By means of the theory of fractional differential equations,the existence and uniqueness of the positive solution were established,then the existence and stability condition of the equilibrium of the model were analyzed.It was showed that if the basic reproduction number is less than 1,the infection free equilibrium is locally asymptotically stable,virus will die out,and if the basic reproductive number is greater than 1,the infection equilibrium is stable and the virus will spread.Some numerical experiments are carried out to confirm the obtained results.In addition,some effective measures are given to control the spread of the trojan virus.
引文
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[2]REN J,YANG X,ZHU Q,et al.A novel computer virus model and its dynamics[J].Nonlinear Analysis Real World Applications,2012,13(1):376-384.DOI:10.1016/j.nonrwa.2011.07.048.
[3]MISHRA B K,SAINI D K.SEIRS epidemic model with delay for transmission of malicious objects in computer network[J].Applied Mathematics & Computation,2007,188(2):1476-1482.DOI:10.1016/j.amc.2006.11.012.
[4]尹礼寿,梁娟.一类计算机病毒SEIR模型稳定性分析[J].生物数学学报,2017,32(3):403- 407.
[5]EL-SAKA H A A.The fractional-order SIR and SIRS epidemic models with variable population size[J].Saka,2013,2(3):195-200.DOI:10.12785/ms1/020308.
[6]肖丽,包骏杰,冯丽萍.一种新的计算机病毒模型的稳定性分析[J].湘潭大学自然科学学报,2012,34(2):94-96.DOI:10.13715/j.cnki.nsjxu.2012.02.018.
[7]AGUSTO F B.Optimal isolation control strategies and cost-effectiveness analysis of a two-strain avian influenza model[J].Biosystem,2013,113(3):155-164.DOI:10.1016/j.biosystems.2013.06.004.
[8]陈永雪.一类计算机病毒传播模型的数学分析[J].福建农林大学学报(自然科学版),2014,43(5):551-555.
[9]HANERT E,SCHUMACHER E,DELEERSNIJDER E.Front dynamics in fractional-order epidemic models[J].Journal of Theoretical Biology,2011,279(1):9-16.
[10]GONZáLEZ-PARRA G,ARENAS A J,CHEN-CHARPENTIER B M.A fractional order epidemic model for the simulation of outbreaks of influenza A(H1N1)[J].Mathematical Methods in the Applied Sciences,2015,37(15):2218-2226.
[11]DIETHELM K.A fractional calculus based model for the simulation of an outbreak of dengue fever[J].Nonlinear Dynamics,2013,71(4):613-619.
[12]PODLUBNY I.Fractional differential equations[M].New York:Academic Press,1999.
[13]ODIBAT Z M,SHAWAGFEH N T.Generalized Taylor's formula[J].Applied Mathematics & Computation,2007,186(1):286-293.
[14]LI H L,ZHANG L,HU C,et al.Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge[J].Journal of Applied Mathematics & Computing,2017,54(1/2):435- 449.
[15]DIETHELM K.The analysis of fractional differential equations[M].Berlin:Springer-Verlag,2010.
[16]AHMED E,EL-SAYED A M A,EL-SAKA H A A.Equilibrium points,stability and numerical solutions of fractional-order predator-prey and rabies models[J].Journal of Mathematical Analysis & Applications,2007,325(1):542-553.