一类具有时滞的传染病数学模型的Hopf分支和稳定性分析
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摘要
为了研究媒体播报的滞后性对传染病传播的影响,建立了一类具有时滞的传染病模型,给出了基本再生数的计算公式,分析了平衡点的稳定性,证明了Hopf分支的存在性,并通过数值模拟验证了:媒体播报时滞和部分有意识易感人群也有可能被感染这两种情形都会使感染者增多,并且推迟达到峰值的时间.
In order to study the influence of the time delay resulting from media coverage on the prevalence of any infectious disease,an epidemic model with delay is established and the calculation formula of the basic reproduction number is provided. Both the stability of equilibria and the existence of Hopf bifurcation are proved. Finally,numerical simulations are carried out and the results show that the time delay in execution of media coverage,as well as some people who are aware susceptible may also be infected, will increase the number of the infected and postpone the peak time.
In order to study the influence of the time delay resulting from media coverage on the prevalence of any infectious disease,an epidemic model with delay is established and the calculation formula of the basic reproduction number is provided. Both the stability of equilibria and the existence of Hopf bifurcation are proved. Finally,numerical simulations are carried out and the results show that the time delay in execution of media coverage,as well as some people who are aware susceptible may also be infected, will increase the number of the infected and postpone the peak time.
引文
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[2] DEL V S,HETHCOTE H,HYMAN J M,et al.Effects of behavioral changes in a smallpox attack model[J].Mathematical Biosciences,2005,195(2):228-251.
[3] LIU Y,CUI J.The impact of media coverage on the dynamics of infectious disease[J].International Journal of Biomathematics,2014,1(1):0800002.
[4] XIAO Y,XU X,TANG S.Sliding mode control of outbreaks of emerging infectious diseases[J].Bulletin of Mathematical Biology,2012,74(10):2403-2422.
[5] ZHANG Y,JIA J.Hopf bifurcation of an epidemic model with a nonlinear birth in population and stage structure[J].Applied Mathematics & Computation,2014,230(4):164-173.
[6] SHI X,CUI J,ZHOU X.Stability and Hopf bifurcation analysis of an eco-epidemic model with a stage structure[J].Nonlinear Analysis,2011,74(4):1088-1106.
[7] SUN C,YANG W,ARINO J,et al.Effect of media-induced social distancing on disease transmission in a two patch setting[J].Mathematical Biosciences,2011,230(2):87-95.
[8] MISRA A K,SHARMA A,SINGH V.Effect of awareness programs in controlling the prevalence of an epidemic with time delay[J].Journal of Biological Systems,2011,19(2):389-402.
[9] HASSARD B D,KAZARINOFF N D,WAN Y H.Theory and applications of Hopf bifurcation[M].Cambridge,Cambridge University Press,1981.