题目：具有垂直传染的两类SIRS传染病模型的研究

[摘 要]

传染病动力学模型的分析与研究是当今的热点问题之一，本文构造了两类具有垂直传染的SIRS传染病模型。一类是具有免疫接种，饱和传染率 ，总人口恒定的SIRS传染病模型。另一类是总人口非恒定，具有一般接触率且考虑疾病对染病者的生育能力造成一定影响的SIRS传染病模型。
本文应用微分方程理论知识对两类模型进行了系统地分析与研究，讨论了无病平衡点和地方病平衡点存在的条件；通过将系统线性化，讨论其特征根，并利用 判别法，证明了无病平衡点和地方病平衡点的局部渐近稳定性；通过构造 函数及根据 不变集原理证明了无病平衡点和地方病平衡点的全局渐近稳定性；得到了疾病消除与否的阈值，并进行了相应的数值模拟；最终，根据实际情况，提出了相应的防控思路与方法。

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the study of two kinds of SIRS infectious disease models with vertical infection

Abstract
It becomes one of the hot issues to analyse and study the infectious disease dynamic model.Two kinds of SIRS infectious disease models with vertical infection were constructed in the article.One features immunization,saturated infection rate and constant total population.The other features non-constant total population,general infection rate,and it considered the effect that the diseases have on the fertility of the infectives.
The differential equation was used to systematically analyse and study the two models in the article,and the existense condition of the disease-free equilibrium point and endemic equilibrium point were discussed.Also,the latent root was discussed by linearizing the system,and the disease-free equilibrium point and endemic equilibrium point were proved with local asymptotic stability by use of the discriminance.Then the disease-free equilibrium point and endemic equilibrium point were proved with global asymptotic stability by constructing functions and basing on the invariant set principle.The threshold of the disease-eliminating or not was obtained,and corresponding numerical simulation was done.Finally,the corresponding prevention and control thread and method according to the practical situation.