带扩散的具有常数迁入率和非线性传染率的SI模型分析
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摘要
对一种具有种群动力和非线性传染率的传染病模型的研究,建立了带扩散的具有常数迁入率和非线性传染率βIpSq的SI模型.以往的具有非线性传染率的传染病模型是运用常微分方程稳定性理论分析了模型平衡解的存在性和稳定性,该模型运用偏微分方程理论在狄氏边条件和诺依曼边条件下分析模型平衡解的存在性和稳定性,得出疾病消除平衡点和地方病平衡点的存在性和稳定性.
本文共分四章.
第一章简单介绍传染病的研究背景,给出传染病的相关概念、模型和研究意义,及前人运用常微分方程理论研究已得到的结论.
第二章首先介绍了本章所需要的引理,然后分别用两种方法研究狄氏边条件下模型平衡解的存在性.
第三章首先介绍本章所需要的引理,然后研究诺依曼边条件下模型平衡解的稳定性.
第四章总结了第一、二、三章的内容,并给出了最新研究结果以及今后的研究前景.
Diffused with constant rate and non-linear transmission rateβIp Sq of the SI model is constructed for a non-linear population dynamics and the transmission rate of infec-tious disease model study. Previous non-linear model of the transmission rate of infec-tious disease use stability theory of ordinary differential equations model analysis of the existence of equilibrium and stability,and the use of partial differential equations of the model theory in Dirichlet boundary conditions and Neumann boundary conditions analysis of the model under the existence of equilibrium and stability. In this paper,we conclude the equilibrium point to eliminate diseases and endemic diseases of the exis-tence of equilibrium and stability.
There are four chapters in this thesis.
In Chapter 1, we introduce the research background of infectious diseases,given of the concept of infectious diseases.model and research significance. And the conclusion using ordinary differential equations study by predecessors.
In Chapter 2, we firstly introduce the needed theorems, and then study the existence of equilibrium solution of the model under the conditions of Dirichlet in two ways.
In Chapter 3, we firstly introduce the needed theorems,and then study the stability of equilibrium solution of the model under the conditions of Neumann.
In Chapter 4, the main conclusions of this thesis are summarized.It gives the latest research results and the further research prospect.
本文共分四章.
第一章简单介绍传染病的研究背景,给出传染病的相关概念、模型和研究意义,及前人运用常微分方程理论研究已得到的结论.
第二章首先介绍了本章所需要的引理,然后分别用两种方法研究狄氏边条件下模型平衡解的存在性.
第三章首先介绍本章所需要的引理,然后研究诺依曼边条件下模型平衡解的稳定性.
第四章总结了第一、二、三章的内容,并给出了最新研究结果以及今后的研究前景.
Diffused with constant rate and non-linear transmission rateβIp Sq of the SI model is constructed for a non-linear population dynamics and the transmission rate of infec-tious disease model study. Previous non-linear model of the transmission rate of infec-tious disease use stability theory of ordinary differential equations model analysis of the existence of equilibrium and stability,and the use of partial differential equations of the model theory in Dirichlet boundary conditions and Neumann boundary conditions analysis of the model under the existence of equilibrium and stability. In this paper,we conclude the equilibrium point to eliminate diseases and endemic diseases of the exis-tence of equilibrium and stability.
There are four chapters in this thesis.
In Chapter 1, we introduce the research background of infectious diseases,given of the concept of infectious diseases.model and research significance. And the conclusion using ordinary differential equations study by predecessors.
In Chapter 2, we firstly introduce the needed theorems, and then study the existence of equilibrium solution of the model under the conditions of Dirichlet in two ways.
In Chapter 3, we firstly introduce the needed theorems,and then study the stability of equilibrium solution of the model under the conditions of Neumann.
In Chapter 4, the main conclusions of this thesis are summarized.It gives the latest research results and the further research prospect.
引文
[1]叶其孝,李正元,反应扩散方程引论,北京：科学出版社,1990.
[2]石磊,俞军,姚洪兴,具有常数迁入率和非线性传染率βIpSq的SI模型分析,高校应用数学学报,2008,23(1)：7-12.
[3]张汉姜,李艳玲,一类捕食-食饵模型正平衡解的整体分歧,系统科学与数学,J. Sys. Sci.& Math. Scis.27(5) (2007,10),791-800.
[4]Smoller J.Shock Waves and Reaction-diffusion Equations. New York,Spring-Verlag,1983.
[5]胡志兴,王辉,管克英,具有非线性接触率和时滞的SIRS流行病模型[J],应用数学学报,1999,22(1)：10-16.
[6]廖晓听,稳定性的数学理论及应用[M],北京：科学出版社,武昌：华中师范大学出版社,1988.
[7]Ruan Shigui,Wang Wendi,Dynamical behavior of an epidemic model with a non-linear incidence rate[J].J Diff Eqs,2003,188:135-163.
[8]王明新,非线性抛物型方程,北京：科学出版社,1993.
[9]Li Zhengyuan,Liu Yingdong.Ye Qixiao,The Global of Constant Equilibrium of Reaction-Diffusion Systems-Differential Inequalities and Liapunov Function-als[J] Journal of systems Science and Complexity,2002,15(2):89-101.
[10]马知恩,周义仓,王稳地,等,传染病动力学的数学建模研究[M],北京：科学出版社,2004,Ma Zhi-en,Zhou Yi-Cang,Wang Wen-di,et al,Mathematical Modelling and Study in Dynamics of Infectious Diseases[M],BeiJing:Science Press,2004.(in Chinese)
[11]Peng R,Wang M X,On multiplicity and stability of positive solutions of diffusive prey-predator model,Math. Anal. Appl.,2006,316:256-268.
[12]Li L and Logan R,Positive solutions to general elliptic competition mod-els,Differential and Interal Equations,1991,4(4):817-834.
[13]张芷芬,微分方程定性理论[M],北京：科学出版社,1985.
[14]Hethcote H W,Van de Driessche P,Some epidemiological models with nonlinear incidence[J],J Math Biol,1991,29:271-287.
[15]Liu Weimin,Hethcote H W,Levin S A,Dynamical behavior of epidemiological model with nonlinear incidence rates [J].J Math Biol,1987,25:359-380.
[16]郭大钧,非线性泛函分析,山东科学技术出版社,2001.
[17]Dancer,E.N.,On the indexes of fixed point of mapping in a cones and applications,J.Math. Anal. Appl.,91 (1983),131-151.
[18]Dancer,E.N.,On positive solutions of some pairs of differential eqs.,Trans,Amer.Math.Soc.,284(1984).729-743.
[19]Li,Lige,coexistence theorems of steady states for predator-prey interacting systems.Trans.Amer.Math.Soc.,305(1988).143-166.
[2]石磊,俞军,姚洪兴,具有常数迁入率和非线性传染率βIpSq的SI模型分析,高校应用数学学报,2008,23(1)：7-12.
[3]张汉姜,李艳玲,一类捕食-食饵模型正平衡解的整体分歧,系统科学与数学,J. Sys. Sci.& Math. Scis.27(5) (2007,10),791-800.
[4]Smoller J.Shock Waves and Reaction-diffusion Equations. New York,Spring-Verlag,1983.
[5]胡志兴,王辉,管克英,具有非线性接触率和时滞的SIRS流行病模型[J],应用数学学报,1999,22(1)：10-16.
[6]廖晓听,稳定性的数学理论及应用[M],北京：科学出版社,武昌：华中师范大学出版社,1988.
[7]Ruan Shigui,Wang Wendi,Dynamical behavior of an epidemic model with a non-linear incidence rate[J].J Diff Eqs,2003,188:135-163.
[8]王明新,非线性抛物型方程,北京：科学出版社,1993.
[9]Li Zhengyuan,Liu Yingdong.Ye Qixiao,The Global of Constant Equilibrium of Reaction-Diffusion Systems-Differential Inequalities and Liapunov Function-als[J] Journal of systems Science and Complexity,2002,15(2):89-101.
[10]马知恩,周义仓,王稳地,等,传染病动力学的数学建模研究[M],北京：科学出版社,2004,Ma Zhi-en,Zhou Yi-Cang,Wang Wen-di,et al,Mathematical Modelling and Study in Dynamics of Infectious Diseases[M],BeiJing:Science Press,2004.(in Chinese)
[11]Peng R,Wang M X,On multiplicity and stability of positive solutions of diffusive prey-predator model,Math. Anal. Appl.,2006,316:256-268.
[12]Li L and Logan R,Positive solutions to general elliptic competition mod-els,Differential and Interal Equations,1991,4(4):817-834.
[13]张芷芬,微分方程定性理论[M],北京：科学出版社,1985.
[14]Hethcote H W,Van de Driessche P,Some epidemiological models with nonlinear incidence[J],J Math Biol,1991,29:271-287.
[15]Liu Weimin,Hethcote H W,Levin S A,Dynamical behavior of epidemiological model with nonlinear incidence rates [J].J Math Biol,1987,25:359-380.
[16]郭大钧,非线性泛函分析,山东科学技术出版社,2001.
[17]Dancer,E.N.,On the indexes of fixed point of mapping in a cones and applications,J.Math. Anal. Appl.,91 (1983),131-151.
[18]Dancer,E.N.,On positive solutions of some pairs of differential eqs.,Trans,Amer.Math.Soc.,284(1984).729-743.
[19]Li,Lige,coexistence theorems of steady states for predator-prey interacting systems.Trans.Amer.Math.Soc.,305(1988).143-166.