一类带时滞且具有预防接种免疫力的SIR传染病模型的稳定性与Hopf分岔分析

发布时间：2018-01-04 09:35

本文关键词：一类带时滞且具有预防接种免疫力的SIR传染病模型的稳定性与Hopf分岔分析　出处：《天津大学》2012年硕士论文　论文类型：学位论文

【摘要】：本文所研究的问题是一类带时滞且具有预防接种免疫力的SIR传染病模型的稳定性及Hopf分岔情况. 全文的内容分为两部分：首先是建模的过程,其次是分析该模型无病平衡点和地方病平衡点的稳定性和Hopf分岔的情况. 在建模过程中,我们充分考虑了一定的种群增长、疾病的传染速度、预防接种、自然死亡、因病死亡、治愈情况、疾病潜伏期等多种因素,使得我们所建模型能更贴切地反应本文所研究问题的真实情况. 本文所建模型为：建模后,全文关心的问题是潜伏期对模型稳定性和分岔的影响,所利用的知识和方法有：稳定性和分岔理论、特征值法、中心流形、规范型方法. 最后,通过严密的论证,得出了： (1)模型无病平衡点和地方病平衡点存在的条件； (2)模型无病平衡点的稳定性和Hopf分岔的情况； (3)模型地方病平衡点的稳定性和Hopf分岔的情况； (4)模型地方病平衡点的Hopf分岔周期解的稳定性和分岔方向的计算公式.
[Abstract]:The problem studied in this paper is the stability and Hopf bifurcation of a class of SIR infectious disease models with time delay and immunity from inoculation. The content of this paper is divided into two parts: first, the process of modeling, and secondly, the stability of the disease-free equilibrium and endemic equilibrium and the Hopf bifurcation of the model are analyzed. In the process of modeling, we fully considered a certain number of factors, such as population growth, disease transmission rate, vaccination, natural death, disease death, cure situation, disease incubation period and so on. The model can reflect the real situation of the problem studied in this paper. The models in this paper are as follows: After modeling, the effect of incubation period on the stability and bifurcation of the model is concerned. The knowledge and methods are as follows: stability and bifurcation theory, eigenvalue method, center manifold, canonical form method. Finally, through rigorous argumentation, it is concluded that: 1) the conditions for the existence of disease-free equilibrium and endemic equilibrium in the model; (2) the stability of the disease-free equilibrium point and the Hopf bifurcation of the model; (3) the stability of the endemic equilibrium point and the Hopf bifurcation of the model; 4) the stability and bifurcation direction of the Hopf bifurcation periodic solution of the endemic equilibrium point of the model.
【学位授予单位】：天津大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：R311;R181.3

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