时滞疟疾模型和一类SIR传染病模型的稳定性与分支

发布时间：2018-05-17 01:34

本文选题：疟疾传播模型 + 时滞　；参考：《新疆大学》2017年硕士论文

【摘要】：全文共分为四章.内容可概述如下:第一章,为本文引言,介绍具有时滞的蚊子偏爱疟疾传播模型和具有饱和治疗、logistic增长和双线性发生率的SIR传染病模型的研究背景及其目前的研究现状,并简洁地给出本文的主要研究内容.第二章,研究了具有时滞和蚊子偏爱疟疾传播模型的动力学行为,讨论了该模型平衡点的存在性.借助于线性化方法、Hopf分支理论和中心流形定理,讨论了平衡点的局部稳定性、Hopf分支的存在性、稳定性和分支方向.最后给出了一些数值例子来验证理论结果,并且发现时滞让模型变得不稳定,导致了混沌吸引子的产生.第三章,讨论一个具有饱和治疗、logistic增长和双线性发生率的SIR传染病模型.治疗函数由一个连续可微的函数所刻画,其描述了当染病者数量太大、医疗条件有限时延迟治疗的现象.针对该模型,我们得到了地方病平衡点存在性、局部渐近稳定性的充分条件.另外,通过分支分析,证明了该模型存在后向分支、稳定的极限环、Hopf分支和Bogdanov-Takens分支.最后给出一些数值例子验证理论结果,也观察到一些有趣的现象,比如,双周期解、稳定的极限环.第四章,对本文所研究的两类传染病模型进行讨论与总结,并提出未来可以研究的问题.
[Abstract]:The full text is divided into four chapters. The contents can be summarized as follows: chapter 1, introduction to this paper, introduces the research background and current research status of the mosquito preference malaria transmission model with time delay and the SIR infectious disease model with saturation treatment logistic growth and bilinear incidence. The main research contents of this paper are given succinctly. In chapter 2, the dynamic behavior of malaria transmission model with time delay and mosquito preference is studied, and the existence of equilibrium point of the model is discussed. By means of the Hopf bifurcation theory and the central manifold theorem, the existence, stability and bifurcation direction of the local stability of the equilibrium point are discussed. Finally, some numerical examples are given to verify the theoretical results, and it is found that the delay makes the model unstable, which leads to the production of chaotic attractors. In chapter 3, we discuss a model of SIR infectious disease with logistic growth and bilinear incidence of saturation therapy. The treatment function is characterized by a continuous differentiable function which describes the phenomenon of delayed treatment when the number of infected persons is too large and the medical conditions are limited. For this model, we obtain sufficient conditions for the existence of endemic equilibrium and local asymptotic stability. In addition, the existence of backward bifurcation, stable limit cycle Hopf bifurcation and Bogdanov-Takens bifurcation is proved by bifurcation analysis. Finally, some numerical examples are given to verify the theoretical results, and some interesting phenomena are observed, such as biperiodic solutions, stable limit cycles. Chapter four discusses and summarizes the two kinds of infectious disease models studied in this paper, and puts forward some problems that can be studied in the future.
【学位授予单位】：新疆大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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