【作者】 张丹；

【导师】 宋燕；

【作者基本信息】 渤海大学 ， 基础数学， 2012， 硕士

【摘要】 本文在以往传染病模型研究的基础上提出了三类具有饱和发生率的SIQR传染病动力学模型，并证明了模型的稳定性.本文首先对传染病模型的研究背景、本文涉及研究领域的研究现状及与本文有关的基本概念和定理、引理等作了简要介绍.同时，给出了本文所要做的主要工作.其次，提出了一类具有常数输入及饱和传染率的SIQR传染病模型，应用微分方程理论知识对这一类模型进行了系统地分析与研究，利用特征根法、构造Liapunov函数、LaSalle不变集原理及Bendixsen-Dulac判别法，分别证明了平衡点的局部渐近稳定性和全局渐近稳定性，得到了疾病消除与否的阈值，从而讨论了无病平衡点和地方病平衡点存在的条件。然后，根据实际情况，提出了相应的防控思路与方法.接着，提出了一类具有垂直传染和预防接种的SIQR传染病模型,给出了疾病持续和灭绝的阈值.利用Hurwith判据、Lasalle不变集原理、LiaPunov函数证明了无病平衡点的全局渐近稳定性，以及地方病平衡点的局部渐近稳定性.最后，建立并分析了一类具有垂直传染及脉冲接种的SIQR传染病模型，利用脉冲微分方程的Floquet乘子定理和比较定理，证明了无病周期解的存在性和全局稳定性；并且得到了系统一致持续生存的充分条件.更多还原

【Abstract】 This article in the previous study of infectious disease model is proposed on the basis of thethree class with saturation incidence of infectious disease dynamics model, and proves the modelstability.Based on the model of infectious disease research background, this paper addresses theresearch present situation and the basic concepts and theorems, lemmas are briefly presented. Atthe same time, we have to do the main work.Secondly, it presentes with constant input and saturation incidence SIQR infectious diseasemodel，applies differential equation theories knowledge to this model carried on analysis andresearch systematically, makes use of the characteristic root method, construct whether Liapunovfunction and LaSalle invariance principle and Bendixsen-Dulac criterion, in order to prove thelocally stability and the global stability,and gets the liminal value,discussed the existencecondition of the disease-free equilibrium and the endemic equilibrium. Then, according to theactual situation, it puts forward the corresponding prevention and control method.Then, it puts forward a class of vertical infection and vaccination in the SIQR epidemicmodel, gives the liminal value of the disease persistence and extinction thresholds. Using Lasalleinvariance principle, LiaPunov function, Hurwith criterion is proved that the disease-freeequilibrium and the endemic equilibrium is global stability.Finally, establishment and analysis of a class of vertical transmission and pulse vaccinationin the SIQR epidemic model, the Floquet multiplier theorem and comparison theorem, provesthat the disease-free periodic solution is existence and global stability; and obtain sufficientconditions for the permanence of the system.更多还原