一类SIR传染病模型的分岔分析
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摘要
研究一类连续SIR传染病模型的分岔性质.通过中心流形定理研究模型的跨临界分岔和音叉分岔并计算出其规范形,同时进一步给出分岔生物学解释.
In this paper,the bifurcation properties of a class of continuous SIR epidemic models are studied. The center manifold theorem is used to study the model's transcritical bifurcation and pitchfork bifurcation,and calculate its normal form. In addition,the biological explanation of bifurcation is given.
In this paper,the bifurcation properties of a class of continuous SIR epidemic models are studied. The center manifold theorem is used to study the model's transcritical bifurcation and pitchfork bifurcation,and calculate its normal form. In addition,the biological explanation of bifurcation is given.
引文
[1] KERMACK W O,MCKENDRICK A G. Contributions to the mathematical theory of epidemics[J]. P Roy Soc A,1927,115(772):700-721.
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[3]ESQUIVEL E R,VALES E A,ALMEIDA G G. Stability and bifurcation analysis of a SIR model with saturated incidence rate and saturated treatment[J]. Math Comput Simulat,2016,2016(121):109-132.
[4]ZHANGX,LIU X N. Backward bifurcation of an epidemic model with saturated treatment function[J]. J Math Anal Appl,2008,348(1):433-443.
[5]WANG J L,LIU S Q,ZHENG B W,et al. Qualitative and bifurcation analysis using an SIR model with a saturated treatment function[J]. Math Comput Model,2012,55(3/4):710-722.
[6]ZHOU T T,ZHANG W P,LU Q Y. Bifurcation analysis of an SIS epidemic model with saturated incidence rate and saturated treatment function[J]. Appl Math Comput,2014,226(1):288-305.
[7]HUZ X,LIU S,WANG H. Backward bifurcation of an epidemic model with standard incidence rate and treatment rate[J]. Nonlinear Anal:RWA,2008,9(5):2302-2312.
[8]LI M S,LIU X M,ZHOU X L. The dynamic behavior of a discrete Vertical and horizontal transmitted disease model under constant vaccination[J]. International J Modern Nonlinear Theory and Application,2016(5):171-184.
[9]李明山,张渝曼,黄晓玉,等.一类具有水平和垂直传播的连续SIR传染病模型的分岔性质[J].应用数学进展,2017,6(2):218-224.
[10]商宁宁,王辉,胡志兴,等.一类具有饱和发生率和饱和治愈率的SIR传染病模型的分支分析[J].昆明理工大学学报(自然科学版),2015,40(3):139-148.
[11]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.
[12]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.
[13]陈兰荪.数学生态学模型与研究方法[M].北京:科学出版社,1988.
[14]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.
[15]WIGGINS S. Introduction to Applied Nonlinear Dynamical Systems and Chaos[M]. New York:Springer-Verlag,1990.
[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]ESQUIVEL E R,VALES E A,ALMEIDA G G. Stability and bifurcation analysis of a SIR model with saturated incidence rate and saturated treatment[J]. Math Comput Simulat,2016,2016(121):109-132.
[4]ZHANGX,LIU X N. Backward bifurcation of an epidemic model with saturated treatment function[J]. J Math Anal Appl,2008,348(1):433-443.
[5]WANG J L,LIU S Q,ZHENG B W,et al. Qualitative and bifurcation analysis using an SIR model with a saturated treatment function[J]. Math Comput Model,2012,55(3/4):710-722.
[6]ZHOU T T,ZHANG W P,LU Q Y. Bifurcation analysis of an SIS epidemic model with saturated incidence rate and saturated treatment function[J]. Appl Math Comput,2014,226(1):288-305.
[7]HUZ X,LIU S,WANG H. Backward bifurcation of an epidemic model with standard incidence rate and treatment rate[J]. Nonlinear Anal:RWA,2008,9(5):2302-2312.
[8]LI M S,LIU X M,ZHOU X L. The dynamic behavior of a discrete Vertical and horizontal transmitted disease model under constant vaccination[J]. International J Modern Nonlinear Theory and Application,2016(5):171-184.
[9]李明山,张渝曼,黄晓玉,等.一类具有水平和垂直传播的连续SIR传染病模型的分岔性质[J].应用数学进展,2017,6(2):218-224.
[10]商宁宁,王辉,胡志兴,等.一类具有饱和发生率和饱和治愈率的SIR传染病模型的分支分析[J].昆明理工大学学报(自然科学版),2015,40(3):139-148.
[11]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.
[12]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.
[13]陈兰荪.数学生态学模型与研究方法[M].北京:科学出版社,1988.
[14]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.
[15]WIGGINS S. Introduction to Applied Nonlinear Dynamical Systems and Chaos[M]. New York:Springer-Verlag,1990.