时滞SVIR传染病模型的稳定性与Hopf分支

发布时间：2018-07-08 11:12

本文选题：时滞 + Hopf分支　；参考：《湖北师范大学》2016年硕士论文

【摘要】：本文主要研究了一类具有不同时滞和不同传染率的SVIR传染病模型,运用时滞微分方程的稳定性理论得到正平衡点局部稳定和Hopf分支存在的充分条件,运用标准型理论和中心流形定理,分析了分支周期解的方向和稳定性等性质.揭示了时滞和传染率对正平衡点稳定性的影响.第一章,介绍了传染病数学模型的研究背景,以及其理论成果对于控制和预防传染病传播有着重要现实意义.通过分析国内外研究现状,从而引出SVIR传染病模型的形成过程.第二章,研究了一类具有潜伏期时滞和双线性传染率的SVIR传染病模型.当基本再生数10R?时模型存在正平衡点,得到正平衡点局部稳定的条件,当时滞经过某些特定临界值时模型出现Hopf分支,导出分支周期解的属性公式,用Matlab等软件计算一组数值模拟验证了所得的理论结果.第三章,研究了一类具有免疫期时滞和非线性传染率的SVIR传染病模型.其存在正平衡点,且是局部渐近稳定的.同时,研究了时滞对正平衡点稳定性的影响,证明了当时滞经过某些特定临界值时模型出现Hopf分支.第四章,总结全文,展望SVIR传染病模型的研究前景.
[Abstract]:In this paper, we study a class of SVIR infectious disease models with different time delays and different infection rates. By using the stability theory of delay differential equations, we obtain the sufficient conditions for the existence of local stability and Hopf bifurcation of positive equilibrium points. By using the theory of canonical form and the theorem of center manifold, the direction and stability of the bifurcation periodic solution are analyzed. The effects of time delay and infection rate on the stability of positive equilibrium point are revealed. The first chapter introduces the research background of the mathematical model of infectious diseases and its theoretical results have important practical significance to control and prevent the spread of infectious diseases. By analyzing the current research situation at home and abroad, the formation process of SVIR infectious disease model is introduced. In chapter 2, we study a class of SVIR infectious disease models with latency delay and bilinear infection rate. When the basic regeneration number is 10R? The local stability condition of the positive equilibrium point is obtained. The Hopf bifurcation appears when the model passes through certain critical values at that time, and the attribute formula of the periodic solution of the bifurcation is derived. The theoretical results are verified by a set of numerical simulations calculated by Matlab and other software. In chapter 3, we study a class of SVIR infectious disease models with immune delay and nonlinear infection rate. There exists a positive equilibrium and it is locally asymptotically stable. At the same time, the influence of time delay on the stability of positive equilibrium point is studied, and it is proved that Hopf bifurcation occurs when the time delay passes through certain critical values. Chapter 4 summarizes the full text and looks forward to the future of SVIR infectious disease model.
【学位授予单位】：湖北师范大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O175

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