一类具有时滞的SIR传染病模型的分析
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摘要
研究了一类具有时滞的SIR传染病模型,确定了模型的基本再生数R0,并利用线性化、特征理论和LaSalle不变原理分析了模型无病平衡点和地方病平衡点的稳定性.以时滞为参数,得到了在地方病平衡点处Hopf分支存在的条件.
The SIR epidemic model with time delay was studied and the basic reproductive number R0 of the model was obtained. By using the linearization,characteristic theory and La Salle invariant principle,the stability of the disease-free equilibrium and the endemic equilibrium was analyzed. Taking time delay as a parameter,the existence condition of Hopf bifurcation at the endemic equilibrium was obtained.
The SIR epidemic model with time delay was studied and the basic reproductive number R0 of the model was obtained. By using the linearization,characteristic theory and La Salle invariant principle,the stability of the disease-free equilibrium and the endemic equilibrium was analyzed. Taking time delay as a parameter,the existence condition of Hopf bifurcation at the endemic equilibrium was obtained.
引文
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[14]WANG Wendi,MA Zhien.Global dynamics of an epidemic model with time delay[J].Nonlinear Analysis:Real World Applications,2002,3(3):365-373.
[15]TAKEUCHI Y,MA Wanbiao,BERETTA E.Global asymptotic properties of a delay SIR epidemic model with finite incubation times[J].Nonlinear Analysis:Theory,Methods and Applications,2000,42(6):931-947.
[16]WANG Jianjun,ZHANG Jinzhu,JIN Zhen.Analysis of an SIR model with bilinear incidence rate[J].Nonlinear Analysis:Real World Applications,2010,11(4):2390-2402.
[2]LIU Qun,JIANG Daqing,SHI Ningzhong,et al.Asymptotic behavior of a stochastic delayed SEIR epidemic model with nonlinear incidence[J].Physica A:Statistical Mechanics and Its Applications,2016,462:870-882.
[3]LIU Panping.Periodic solutions in an epidemic model with diffusion and delay[J].Applied Mathematics and Computation,2015,265(15):275-291.
[4]BERNOULLI D.Essai d'une nouvelle analyse de la mortalite causee par la petite verole et des avantages de l'inoculation pour al prevenir in memoires de mathematiques et de physique[D].Paris:Academic Royale des Science,1760.
[5]KERMACK W O,MCKENDRICK A G.Contributions to the mathematical theory of epidemics[J].Proc Roy Soc A,1927,115:700-721.
[6]KERMACK W O,MCKENDRICK A G.Contributions to the mathematical theory of epidemic.Ⅱ.The problem of endemicity[J].Proc Roy Soc A,1932,138(834):55-83.
[7]SHULGIN B,STONE L,AGUR Z.Pulse vaccination strategy in the SIR epidemic model[J].Bulletin of Mathematical Biology,1998,60(6):1123-1148.
[8]BALL F.Stochastic and deterministic models for SIS epidemics among a population partitioned into households[J].Mathematical Biosciences,1999,156(1/2):41-67.
[9]CAO Boqiang,SHAN Meijing,ZHANG Qimin,et al.A stochastic SIS epidemic model with vaccination[J].Physica A:Statistical Mechanics and Its Applications,2017,486(15):127-143.
[10]KUNIYA T,WANG Jinliang.Global dynamics of an SIR epidemic model with nonlocal diffusion[J].Nonlinear Analysis:Real World Applications,2018,43:262-282.
[11]CAI Yongli,KANG Yun,WANG Weiming.A stochastic SIRS epidemic model with nonlinear incidence rate[J].Applied Mathematics and Computation,2017,305(c):221-240.
[12]XIAO Dongmei,RUAN Shigui.Global analysis of an epidemic model with nonmonotone incidence rate[J].Mathematical Biosciences,2007,208(2):419-429.
[13]LIU Qun,JIANG Daqing.The threshold of a stochastic delayed SIR epidemic model with vaccination[J].Physica A:Statistical Mechanics and Its Applications,2016,461(1):140-147.
[14]WANG Wendi,MA Zhien.Global dynamics of an epidemic model with time delay[J].Nonlinear Analysis:Real World Applications,2002,3(3):365-373.
[15]TAKEUCHI Y,MA Wanbiao,BERETTA E.Global asymptotic properties of a delay SIR epidemic model with finite incubation times[J].Nonlinear Analysis:Theory,Methods and Applications,2000,42(6):931-947.
[16]WANG Jianjun,ZHANG Jinzhu,JIN Zhen.Analysis of an SIR model with bilinear incidence rate[J].Nonlinear Analysis:Real World Applications,2010,11(4):2390-2402.