一个具有双线性发生率的随机SIR传染病模型的动力学性质
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摘要
研究一个具有双线性发生率的随机SIR传染病模型的动力学性质.首先证明系统对任意的正初值具有唯一的全局正解,然后通过分析的方法得到传染病消亡的充分条件和均值意义下传染病持久性的条件,最后给出传染病的一些防控措施和生物学意义.
In this paper,the dynamical property of a stochastic SIR epidemic model with the bilinear incidence rate is researched.Firstly,we prove that the system has a unique global positive solution for any positive initial value. Then we get the sufficient condition for extinction and the condition for persistence in the mean of disease. Finally,we give some ways to control disease and the biological mean of epidemic model.
In this paper,the dynamical property of a stochastic SIR epidemic model with the bilinear incidence rate is researched.Firstly,we prove that the system has a unique global positive solution for any positive initial value. Then we get the sufficient condition for extinction and the condition for persistence in the mean of disease. Finally,we give some ways to control disease and the biological mean of epidemic model.
引文
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[3]ZHOU X L,LI X P,WANG W S. Bifurcations for a deterministic SIR epidemic model in discrete time[J]. Advances in Difference Equations,2014,2014(1):1-16.
[4]NIE L F,TENG Z D,ANGELA T. Dynamic analysis of an SIR epidemic model with state dependent pulse vaccination[J]. Nonlinear Analysis:RWA,2012,13(4):1621-1629.
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[6]JI C Y,JIANG D Q,SHI N Z,et al. The behavior of an SIR epidemic model with stochastic perturbation[J]. Stochastic Analysis and Applications,2012,30(5):755-773.
[7]JIANG D Q,YU J J,JI C Y,et al. Asymptotic behavior of global positive solution to a stochastic SIR model[J]. Math Comput Model,2011,54(1/2):221-232.
[8]LIN Y G,JIANG D Q. Long-time behavior of perturbed SIR model by white noise[J]. Discrete and Continuous Dynamical Systems Series,2013,A18(7):1873-1887.
[9]ZHOU Y L,ZHANG W G,YUAN S L,et al. Survival and stationary distribution of a SIR epidemic model with stochastic perturbations[J]. Appl Math Comput,2014,244:118-131.
[10]JI C Y,JIANG D Q. Threshold behaviour of a stochastic SIR model[J]. Math Comput Model,2014,38(21/22):5067-5079.
[11]WANG F L,WANG X Y,ZHANG S W,et al. On pulse vaccine strategy in a periodic stochastic SIR epidemic model[J]. Chaos Soliton Fract,2014,66:127-135.
[12]WANG J J,ZHANG J Z,JIN Z,et al. Analysis of an SIR model with bilinear incidence rate[J]. Nonlinear Analysis:RWA,2010,11(4):2390-2402.
[13]ZHAO Y N,JIANG D Q. The threshold of a stochastic SIRS epidemic model with saturated incidence[J]. Applied Mathematics Letters,2014,34:90-93.
[14]ZHANG X B,HUO H F,XIANG H,et al. Dynamics of the deterministic and stochastic SIQS epidemic model with nonlinear incidence[J]. Appl Math Comput,2014,243:546-558.
[15]LIU Q,CHEN Q M. Dynamics of a stochastic SIR epidemic model with saturated incidence[J]. Appl Math Comput,2016,282:155-166.
[16]MAO X R. Stochastic Differential Equations and Applications[M]. Chichester:Horwood Publishing,1997.
[17]HASMINSKII R. Stochastic Stability of Differential Equations[M]. Alphen aan den Rijn:Sijthoff and Noordhoff,1980.
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[19]ARNOLD L,HORSTHEMKE W,STUXKI J. The influence of external real and white noise on the Lotka-Volterra model[J]. Biometrical J,1979,1979(21):451-471.
[20]MAY R M. Stability and Complexity in Model Ecosystems[M]. New Jersey:Princeton University Press,2001.
[21]ANDERSON R M,ROBERT M M,ANDERSON B. Infectious Diseases of Humans Dynamics and Control[M]. Oxford:Oxford University Press,1992.
[22]SHULGIN B,STONE L,AGUR Z. Pulse vaccination strategy in the SIR epidemic model[J]. Bull Math Biol,1998,1998(60):1-26
[23]黄玉梅,李树勇,潘杰.具有阶段结构的SIRS传染病模型[J].四川师范大学学报(自然科学版),2005,28(1):31-33.
[24]樊爱军,王开发.一类具有非线性接触率的种群力学流行病模型分析[J].四川师范大学学报(自然科学版),2002,25(3):261-263.
[25]苟清明,余沛.一类有迁移的传染病模型的闽值[J].四川师范大学学报(自然科学版),2006,29(3):273-276.
[26]马知恩,周义仓,王稳地,等.传染病动力学的数学建模与研究[M].北京:科学出版社,2004.
[27]马知恩,周义仓,吴建宏.传染病建模与动力学[M].北京:高等教育出版社,2009.
[28]王克.随机生物数学模型[M].北京:科学出版社,2010.
[2]MEMG X Z,CHEN L S. The dynamics of a new SIR epidemic model concerning pulse vaccination strategy[J]. Appl Math Comput,2008,197(2):582-597.
[3]ZHOU X L,LI X P,WANG W S. Bifurcations for a deterministic SIR epidemic model in discrete time[J]. Advances in Difference Equations,2014,2014(1):1-16.
[4]NIE L F,TENG Z D,ANGELA T. Dynamic analysis of an SIR epidemic model with state dependent pulse vaccination[J]. Nonlinear Analysis:RWA,2012,13(4):1621-1629.
[5]LIAO X D,WANG H B,HUANG X H,et al. The dynamic properties of a deterministic SIR epidemic model in discrete time[J].Applied Mathematics,2015,6(10):1665-1675.
[6]JI C Y,JIANG D Q,SHI N Z,et al. The behavior of an SIR epidemic model with stochastic perturbation[J]. Stochastic Analysis and Applications,2012,30(5):755-773.
[7]JIANG D Q,YU J J,JI C Y,et al. Asymptotic behavior of global positive solution to a stochastic SIR model[J]. Math Comput Model,2011,54(1/2):221-232.
[8]LIN Y G,JIANG D Q. Long-time behavior of perturbed SIR model by white noise[J]. Discrete and Continuous Dynamical Systems Series,2013,A18(7):1873-1887.
[9]ZHOU Y L,ZHANG W G,YUAN S L,et al. Survival and stationary distribution of a SIR epidemic model with stochastic perturbations[J]. Appl Math Comput,2014,244:118-131.
[10]JI C Y,JIANG D Q. Threshold behaviour of a stochastic SIR model[J]. Math Comput Model,2014,38(21/22):5067-5079.
[11]WANG F L,WANG X Y,ZHANG S W,et al. On pulse vaccine strategy in a periodic stochastic SIR epidemic model[J]. Chaos Soliton Fract,2014,66:127-135.
[12]WANG J J,ZHANG J Z,JIN Z,et al. Analysis of an SIR model with bilinear incidence rate[J]. Nonlinear Analysis:RWA,2010,11(4):2390-2402.
[13]ZHAO Y N,JIANG D Q. The threshold of a stochastic SIRS epidemic model with saturated incidence[J]. Applied Mathematics Letters,2014,34:90-93.
[14]ZHANG X B,HUO H F,XIANG H,et al. Dynamics of the deterministic and stochastic SIQS epidemic model with nonlinear incidence[J]. Appl Math Comput,2014,243:546-558.
[15]LIU Q,CHEN Q M. Dynamics of a stochastic SIR epidemic model with saturated incidence[J]. Appl Math Comput,2016,282:155-166.
[16]MAO X R. Stochastic Differential Equations and Applications[M]. Chichester:Horwood Publishing,1997.
[17]HASMINSKII R. Stochastic Stability of Differential Equations[M]. Alphen aan den Rijn:Sijthoff and Noordhoff,1980.
[18]GARD T. Introduction to Stochastic Differential Equations[M]. New York:Marcel Dekker,1988.
[19]ARNOLD L,HORSTHEMKE W,STUXKI J. The influence of external real and white noise on the Lotka-Volterra model[J]. Biometrical J,1979,1979(21):451-471.
[20]MAY R M. Stability and Complexity in Model Ecosystems[M]. New Jersey:Princeton University Press,2001.
[21]ANDERSON R M,ROBERT M M,ANDERSON B. Infectious Diseases of Humans Dynamics and Control[M]. Oxford:Oxford University Press,1992.
[22]SHULGIN B,STONE L,AGUR Z. Pulse vaccination strategy in the SIR epidemic model[J]. Bull Math Biol,1998,1998(60):1-26
[23]黄玉梅,李树勇,潘杰.具有阶段结构的SIRS传染病模型[J].四川师范大学学报(自然科学版),2005,28(1):31-33.
[24]樊爱军,王开发.一类具有非线性接触率的种群力学流行病模型分析[J].四川师范大学学报(自然科学版),2002,25(3):261-263.
[25]苟清明,余沛.一类有迁移的传染病模型的闽值[J].四川师范大学学报(自然科学版),2006,29(3):273-276.
[26]马知恩,周义仓,王稳地,等.传染病动力学的数学建模与研究[M].北京:科学出版社,2004.
[27]马知恩,周义仓,吴建宏.传染病建模与动力学[M].北京:高等教育出版社,2009.
[28]王克.随机生物数学模型[M].北京:科学出版社,2010.
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