一类具有饱和发生率和复发的随机SIRI模型的稳定性
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摘要
本文研究一类具有饱和发生率和复发的随机SIRI传染病模型.首先,我们证明随机系统存在唯一的全局正解.然后讨论无病平衡点的稳定性,并利用Lyapunov函数法证明流行病的灭绝.随后,我们得到疾病持久性的充分条件.最后,通过数值模拟说明结论的正确性.
In this paper, a stochastic SIRI epidemiological model with saturation incidence and relapse is investigated. Firstly, we show that there exists a unique global positive solution of the stochastic system. Then we discuss the stability of the disease-free equilibrium state and show the extinction of epidemics by using Lyapunov functions. Subsequently, a sufficient condition for persistence has been established in the mean of the disease. Finally, some numerical simulations are carried out to confirm the analytical results.
In this paper, a stochastic SIRI epidemiological model with saturation incidence and relapse is investigated. Firstly, we show that there exists a unique global positive solution of the stochastic system. Then we discuss the stability of the disease-free equilibrium state and show the extinction of epidemics by using Lyapunov functions. Subsequently, a sufficient condition for persistence has been established in the mean of the disease. Finally, some numerical simulations are carried out to confirm the analytical results.
引文
[1]VAN DEN DRIESSCHE P,ZOU X.Modeling relapse in infectious diseases[J].Mathematical Biosciences,2007,207(1):89-103.
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[7]LIU Q,JIANG D,SHI N,et al.The threshold of a stochastic SIS epidemic model with imperfect vaccination[J].Mathematics and Computers in Simulation,2018,144:78-90.
[8]LIU Q,CHEN Q.Dynamics of a stochastic SIR epidemic model with saturated incidence[J].Applied Mathematics and Computation,2016,282:155-166.
[9]ZHAO D,ZHANG T,YUAN S.The threshold of a stochastic SIVS epidemic model with nonlinear saturated incidence[J].Physica A:Statistical Mechanics and It’s Applications,2016,443:372-379.
[10]MAO X.Stochastic Differential Equations and Applications[M].Chichester:Horwood Publishing,1997.
[11]MAO X,MARION G,RENSHAW E.Environmental Brownian noise suppresses explosions in population dynamics[J].Stochastic Processes and Their Applications,2002,97:95-110.
[12]JI C,JIANG D.Threshold behaviour of a stochastic SIR model[J].Applied Mathematical Modelling,2014,38:5067-5079.
[13]HIGHAM,DESMOND J.An algorithmic introduction to numerical simulations of stochastic differential equations[J].SIAM Review,2001,43:525-546.
[2]VAN DEN DRIESSCHE P,WANG L,ZOU X.Modeling diseases with latency and relapse[J].Mathematical Biosciences and Engineering,2007,4(2):205-19.
[3]AGUSTO F B.Mathematical model of Ebola transmission dynamics with relapse and reinfection[J].Mathematical Biosciences,2017,283:48-59.
[4]VARGAS-De-León C.On the global stability of infectious diseases models with relapse[J].Abstraction and Application,2013(9):50-61.
[5]ZHANG Y,LI Y,ZHANG Q,et al.Behavior of a stochastic SIR epidemic model with saturated incidence and vaccination rules[J].Physica A:Statistical Mechanics and Its Applications,2018,501:178-187.
[6]FATINI M E,LAHROUZ A,PETTERSSON R,et al.Stochastic stability and instability of an epidemic model with relapse[J].Applied Mathematics and Computation,2018,316:326-341.
[7]LIU Q,JIANG D,SHI N,et al.The threshold of a stochastic SIS epidemic model with imperfect vaccination[J].Mathematics and Computers in Simulation,2018,144:78-90.
[8]LIU Q,CHEN Q.Dynamics of a stochastic SIR epidemic model with saturated incidence[J].Applied Mathematics and Computation,2016,282:155-166.
[9]ZHAO D,ZHANG T,YUAN S.The threshold of a stochastic SIVS epidemic model with nonlinear saturated incidence[J].Physica A:Statistical Mechanics and It’s Applications,2016,443:372-379.
[10]MAO X.Stochastic Differential Equations and Applications[M].Chichester:Horwood Publishing,1997.
[11]MAO X,MARION G,RENSHAW E.Environmental Brownian noise suppresses explosions in population dynamics[J].Stochastic Processes and Their Applications,2002,97:95-110.
[12]JI C,JIANG D.Threshold behaviour of a stochastic SIR model[J].Applied Mathematical Modelling,2014,38:5067-5079.
[13]HIGHAM,DESMOND J.An algorithmic introduction to numerical simulations of stochastic differential equations[J].SIAM Review,2001,43:525-546.