一类具时滞的周期logistic传染病模型空间动力学研究
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摘要
利用动力系统的理论研究一类具有时滞的周期logistic反应扩散传染病模型的动力学.首先证明了周期解半流对应ω算子的全局吸引子的存在性.然后利用次代算子方法引入了模型的基本再生数.最后,利用持久性理论结合比较原理,得到了疾病持久或灭绝的阈值条件:若基本再生数小于1,无病周期解是全局渐近稳定的,疾病将逐渐消失;若基本再生数大于1,系统一致持久,疾病将继续流行并最终形成地方病.
The dynamics of periodic reaction-diffusion epidemic models with delay and logistic growth was investigated based on the theory of dynamic systems. Firstly,the existence of the global attractor of theω operator associated with the periodic semiflow was proved. Next,the basic reproduction number of the model was introduced via the next generation operator. Finally,by means of the persistence theory and the comparison principle,the sufficient conditions for the disease persistence and extinction were obtained.If the basic reproduction number is less than 1,the disease-free periodic solution will be globally asymptotically stable and the disease will go extinct. If the basic reproduction number is greater than 1,the system will be uniformly persistent and the disease will become endemic.
The dynamics of periodic reaction-diffusion epidemic models with delay and logistic growth was investigated based on the theory of dynamic systems. Firstly,the existence of the global attractor of theω operator associated with the periodic semiflow was proved. Next,the basic reproduction number of the model was introduced via the next generation operator. Finally,by means of the persistence theory and the comparison principle,the sufficient conditions for the disease persistence and extinction were obtained.If the basic reproduction number is less than 1,the disease-free periodic solution will be globally asymptotically stable and the disease will go extinct. If the basic reproduction number is greater than 1,the system will be uniformly persistent and the disease will become endemic.
引文
[1]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,2004:1-24.(M A Z hien,Z HO U Yicang,W AN G W endi,et al.Mathematics Modeling and R esearch of Infectious D isease D ynamics[M].Beijing:Science Press,2004:1-24.(in C hinese))
[2]王拉娣.传染病动力学模型及控制策略研究[D].博士学位论文.上海:上海大学,2005:1-9.(WANG Ladi.Infectious disease dynamics and controlling strategy[D].Ph D Thesis.Shanghai:Shanghai U niversity,2005:1-9.(in C hinese))
[3]谢英超,程燕,贺天宇.一类具有非线性发生率的时滞传染病模型的全局稳定性[J].应用数学和力学,2015,36(10):1107-1116.(XIE Yingchao,CHENG Yan,HE Tianyu.Global stability of a class of delayed epidemic models w ith nonlinear incidence rates[J].A pplied Mathematics and Mechanics,2015,36(10):1107-1116.(in C hinese))
[4]PENG Rui,ZHAO Xiaoqiang.A reaction-diffusion SIS epidemic model in a time-periodic environment[J].N onlinearity,2012,25(5):1451-1471.
[5]VAN DEN DRIESSCHE P,WATMOUGH J.Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission[J].Mathematical Biosciences,2002,180(1):29-48.
[6]杨亚莉,李建全,刘万萌,等.一类具有分布时滞和非线性发生率的媒介传染病模型的全局稳定性[J].应用数学和力学,2013,34(12):1291-1299.(YANG Yali,LI Jianquan,LIU Wanmeng,et al.Global stability of a vector-borne epidemic model w ith distributed delay and nonlinear incidence[J].A pplied Mathematics and Mechanics,2013,34(12):1291-1299.(in C hinese))
[7]THIEME H R.Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity[J].SIA M Journal on A pplied Mathematics,2009,70(1):188-211.
[8]BACAR N.Genealogy with seasonality,the basic reproduction number,and the influenza pandemic[J].Journal of Mathematical Biology,2011,62(5):741-762.
[9]WANG Wendi,ZHAO Xiaoqiang.Threshold dynamics for compartmental epidemic models in periodic environments[J].Journal of D ynamics and D ifferential Equations,2008,20(3):699-717.
[10]王智诚,王双明.一类时间周期的时滞反应扩散模型的空间动力学研究[J].兰州大学学报(自然科学版),2013,49(4):535-540.(WANG Zhicheng,WANG Shuangming.Spatial dynamics of a class of delayed nonlocal reaction-diffusion models w ith a time period[J].Journal of L anz hou University(Natural Sciences),2013,49(4):535-540.(in Chinese))
[11]WANG Shuangming,ZHANG Liang.Dynamics of a time-periodic and delayed reaction-diffusion model w ith a quiescent stage[J].Electronic Journal of Q ualitative T heory of D ifferential Equations,2016,47:1-25.
[12]ZHANG Liang,WANG Zhicheng.Spatial dynamics of a diffusive predator-prey model with stage structure[J].D iscrete and C ontinuous D ynamical Systems—Series B,2015,20(6):1831-1853.
[13]王双明.一类具有时滞的周期流行病模型的动力学分析[J].山东大学学报(理学版),2017,52(1):81-87.(WANG Shuangming.Dynamical analysis of a class of periodic epidemic model with delay[J].Journal of Shandong University(Natural Science),2017,52(1):81-87.(in Chinese))
[14]ZHAO Xiaoqiang.Basic reproduction ratios for periodic compartmental models with time delay[J].Journal of D ynamics and D ifferential Equations,2015,29(1):1-16.
[15]ZHANG Liang,WANG Zhicheng,ZHAO Xiaoqiang.Threshold dynamics of a time periodic reaction-diffusion epidemic model w ith latent period[J].Journal of D ifferential Equations,2015,258(9):3011-3036.
[16]MARTIN R H,SMITH H L.Abstract functional-differential equations and reaction-diffusion systems[J].Transactions of the American Mathematical Society,1990,321(1):1-44.
[17]ZAOH Xiaoqiang.Dynamical Systems in Population Biology[M].New York:Springer-Verlag,2003:1-65.
[18]Hess P.Periodic-Parabolic Boundary Value Problems and Positivity[M].UK:Longman Scientific and Technical,1991:91-93.
[19]MAGAL P,ZHAO Xiaoqiang.Global attractors and steady states for uniformly persistent dynamical systems[J].SIA M J Math A nal,2005,37:251-275.
[20]LOU Yijun,ZHAO Xiaoqiang.Threshold dynamics in a time-delayed periodic SIS epidemic model[J].Discrete and Continuous Dynamical Systems—Series B,2009,12:169-186.
[2]王拉娣.传染病动力学模型及控制策略研究[D].博士学位论文.上海:上海大学,2005:1-9.(WANG Ladi.Infectious disease dynamics and controlling strategy[D].Ph D Thesis.Shanghai:Shanghai U niversity,2005:1-9.(in C hinese))
[3]谢英超,程燕,贺天宇.一类具有非线性发生率的时滞传染病模型的全局稳定性[J].应用数学和力学,2015,36(10):1107-1116.(XIE Yingchao,CHENG Yan,HE Tianyu.Global stability of a class of delayed epidemic models w ith nonlinear incidence rates[J].A pplied Mathematics and Mechanics,2015,36(10):1107-1116.(in C hinese))
[4]PENG Rui,ZHAO Xiaoqiang.A reaction-diffusion SIS epidemic model in a time-periodic environment[J].N onlinearity,2012,25(5):1451-1471.
[5]VAN DEN DRIESSCHE P,WATMOUGH J.Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission[J].Mathematical Biosciences,2002,180(1):29-48.
[6]杨亚莉,李建全,刘万萌,等.一类具有分布时滞和非线性发生率的媒介传染病模型的全局稳定性[J].应用数学和力学,2013,34(12):1291-1299.(YANG Yali,LI Jianquan,LIU Wanmeng,et al.Global stability of a vector-borne epidemic model w ith distributed delay and nonlinear incidence[J].A pplied Mathematics and Mechanics,2013,34(12):1291-1299.(in C hinese))
[7]THIEME H R.Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity[J].SIA M Journal on A pplied Mathematics,2009,70(1):188-211.
[8]BACAR N.Genealogy with seasonality,the basic reproduction number,and the influenza pandemic[J].Journal of Mathematical Biology,2011,62(5):741-762.
[9]WANG Wendi,ZHAO Xiaoqiang.Threshold dynamics for compartmental epidemic models in periodic environments[J].Journal of D ynamics and D ifferential Equations,2008,20(3):699-717.
[10]王智诚,王双明.一类时间周期的时滞反应扩散模型的空间动力学研究[J].兰州大学学报(自然科学版),2013,49(4):535-540.(WANG Zhicheng,WANG Shuangming.Spatial dynamics of a class of delayed nonlocal reaction-diffusion models w ith a time period[J].Journal of L anz hou University(Natural Sciences),2013,49(4):535-540.(in Chinese))
[11]WANG Shuangming,ZHANG Liang.Dynamics of a time-periodic and delayed reaction-diffusion model w ith a quiescent stage[J].Electronic Journal of Q ualitative T heory of D ifferential Equations,2016,47:1-25.
[12]ZHANG Liang,WANG Zhicheng.Spatial dynamics of a diffusive predator-prey model with stage structure[J].D iscrete and C ontinuous D ynamical Systems—Series B,2015,20(6):1831-1853.
[13]王双明.一类具有时滞的周期流行病模型的动力学分析[J].山东大学学报(理学版),2017,52(1):81-87.(WANG Shuangming.Dynamical analysis of a class of periodic epidemic model with delay[J].Journal of Shandong University(Natural Science),2017,52(1):81-87.(in Chinese))
[14]ZHAO Xiaoqiang.Basic reproduction ratios for periodic compartmental models with time delay[J].Journal of D ynamics and D ifferential Equations,2015,29(1):1-16.
[15]ZHANG Liang,WANG Zhicheng,ZHAO Xiaoqiang.Threshold dynamics of a time periodic reaction-diffusion epidemic model w ith latent period[J].Journal of D ifferential Equations,2015,258(9):3011-3036.
[16]MARTIN R H,SMITH H L.Abstract functional-differential equations and reaction-diffusion systems[J].Transactions of the American Mathematical Society,1990,321(1):1-44.
[17]ZAOH Xiaoqiang.Dynamical Systems in Population Biology[M].New York:Springer-Verlag,2003:1-65.
[18]Hess P.Periodic-Parabolic Boundary Value Problems and Positivity[M].UK:Longman Scientific and Technical,1991:91-93.
[19]MAGAL P,ZHAO Xiaoqiang.Global attractors and steady states for uniformly persistent dynamical systems[J].SIA M J Math A nal,2005,37:251-275.
[20]LOU Yijun,ZHAO Xiaoqiang.Threshold dynamics in a time-delayed periodic SIS epidemic model[J].Discrete and Continuous Dynamical Systems—Series B,2009,12:169-186.