几类离散传染病模型的稳定性与分支研究
发布时间：2018-11-19 11:28
【摘要】：本文利用稳定性理论和中心流形理论研究了几类离散传染病模型的稳定性与分支问题.全文共分为四章.第一章简要介绍了问题的研究背景和本文的主要工作.第二章研究了具有饱和发生率βSI1+αI的离散SIR模型的稳定性和分支问题,利用局部稳定性理论和Jury判据,讨论了无病平衡点和地方病平衡点的局部稳定性,并证明了当基本再生数R01时,无病平衡点是全局稳定的,最后利用中心流形定理给出了当参数h在小邻域内变化时,地方病平衡点处产生Flip分支和Hopf分支的充分条件.第三章研究了具有饱和发生率βSI1+αS的离散SIR模型的稳定性和分支问题,通过对模型特征方程的分析,得到了无病平衡点和地方病平衡点局部稳定的充分条件,并证明了当基本再生数R01时,无病平衡点是全局稳定的,最后给出了在地方病平衡点处产生Flip分支和Hopf分支的充分条件.第四章研究了具有logistic增长率的离散SI模型首先通过分析模型的特征方程,得到了无病平衡点和有病平衡点的局部渐近稳定性,其次给出了当参数h在小邻域内变化时,在地方病平衡点处产生Flip分支和Hopf分支的充分条件.
[Abstract]:In this paper, the stability and bifurcation of several discrete infectious disease models are studied by using the stability theory and the center manifold theory. The full text is divided into four chapters. The first chapter briefly introduces the research background and the main work of this paper. In chapter 2, the stability and bifurcation of discrete SIR model with saturation incidence 尾 SI1 伪 I are studied. By using local stability theory and Jury criterion, the local stability of disease-free equilibrium and endemic equilibrium is discussed. It is proved that the disease-free equilibrium point is globally stable when the basic reproducing number R _ (01). Finally, by using the central manifold theorem, the sufficient conditions for the Flip bifurcation and Hopf bifurcation at the endemic equilibrium point are given when the parameter h changes in the small neighborhood. In chapter 3, the stability and bifurcation of discrete SIR model with saturation incidence 尾 SI1 伪 S are studied. By analyzing the characteristic equations of the model, sufficient conditions for local stability of disease-free equilibrium and endemic equilibrium are obtained. It is proved that the disease-free equilibrium is globally stable when the basic reproduction number is R01. At last, the sufficient conditions for producing Flip bifurcation and Hopf bifurcation at endemic equilibrium point are given. In chapter 4, the discrete SI model with logistic growth rate is studied. Firstly, by analyzing the characteristic equations of the model, the local asymptotic stability of disease-free equilibrium and diseased equilibrium is obtained. Secondly, when the parameter h changes in the small neighborhood, the local asymptotic stability of the disease-free equilibrium and the diseased equilibrium is obtained. Sufficient conditions for generating Flip bifurcation and Hopf bifurcation at endemic equilibrium point.
【学位授予单位】：曲阜师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175
本文编号：2342172
[Abstract]:In this paper, the stability and bifurcation of several discrete infectious disease models are studied by using the stability theory and the center manifold theory. The full text is divided into four chapters. The first chapter briefly introduces the research background and the main work of this paper. In chapter 2, the stability and bifurcation of discrete SIR model with saturation incidence 尾 SI1 伪 I are studied. By using local stability theory and Jury criterion, the local stability of disease-free equilibrium and endemic equilibrium is discussed. It is proved that the disease-free equilibrium point is globally stable when the basic reproducing number R _ (01). Finally, by using the central manifold theorem, the sufficient conditions for the Flip bifurcation and Hopf bifurcation at the endemic equilibrium point are given when the parameter h changes in the small neighborhood. In chapter 3, the stability and bifurcation of discrete SIR model with saturation incidence 尾 SI1 伪 S are studied. By analyzing the characteristic equations of the model, sufficient conditions for local stability of disease-free equilibrium and endemic equilibrium are obtained. It is proved that the disease-free equilibrium is globally stable when the basic reproduction number is R01. At last, the sufficient conditions for producing Flip bifurcation and Hopf bifurcation at endemic equilibrium point are given. In chapter 4, the discrete SI model with logistic growth rate is studied. Firstly, by analyzing the characteristic equations of the model, the local asymptotic stability of disease-free equilibrium and diseased equilibrium is obtained. Secondly, when the parameter h changes in the small neighborhood, the local asymptotic stability of the disease-free equilibrium and the diseased equilibrium is obtained. Sufficient conditions for generating Flip bifurcation and Hopf bifurcation at endemic equilibrium point.
【学位授予单位】：曲阜师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175
【参考文献】
相关期刊论文 前1条
1 李建全;娄洁;娄梅枝;;Some discrete SI and SIS epidemic models[J];Applied Mathematics and Mechanics(English Edition);2008年01期
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