几类具有饱和发生率的传染病模型的研究

发布时间：2018-06-21 05:17

本文选题：传染病模型 + 饱和发生率　；参考：《山西师范大学》2015年硕士论文

【摘要】：本文研究了几类具有饱和发生率的传染病模型的动力学性质,全文共分为四章：第一章,绪论,主要介绍传染病的研究背景和意义,国内外研究现状,本文的主要工作以及所用到的预备知识.第二章,研究了一类具有饱和发生率和饱和治疗函数的SEIR传染病模型的动力学行为.给出了决定疾病灭绝与持久的基本再生数,得到了各类平衡点以及后向分支存在的阈值条件.利用Routh-Hurwitz判据和特征根方法得到了平衡点的局部渐近稳定性,通过构造Lyapunov函数讨论了无病平衡点的全局稳定性,利用自治收敛定理证明了地方病平衡点的全局渐近稳定性.最后用数值模拟验证了本章的主要理论结果.第三章,研究了一类具有时滞,信息变量和饱和发生率的传染病模型.借助特征值理论分析了无病平衡点和地方病平衡点的局部稳定性.同时以时滞为分支参数,得出Hopf分支存在的条件,并应用规范型理论和中心流形定理得到了Hopf分支周期解的稳定性及分支方向.最后利用数值模拟验证了本章的主要理论结果.第四章,讨论了一类具有周期接触率和时变脉冲接种率的传染病模型.通过计算得到判别疾病流行与否的阈值,当基本再生数R01时,系统的无病周期解是全局稳定的,即疾病根除,当R0-α*A/d1时,疾病持久.最后利用数值模拟验证了本章的主要理论结果.
[Abstract]:In this paper, the dynamic properties of several infectious disease models with saturated incidence are studied. The full text is divided into four chapters: chapter 1, introduction, mainly introduces the research background and significance of infectious diseases, the current research situation at home and abroad. The main work of this paper and the preparatory knowledge used. In chapter 2, the dynamic behavior of a seir infectious disease model with saturation incidence and saturation treatment function is studied. The basic regenerative numbers which determine the extinction and persistence of disease are given, and the threshold conditions for the existence of equilibrium points and backward branches are obtained. The local asymptotic stability of the equilibrium is obtained by using Routh-Hurwitz criterion and eigenvalue method. The global stability of disease-free equilibrium is discussed by constructing Lyapunov function, and the global asymptotic stability of endemic equilibrium is proved by using the autonomous convergence theorem. Finally, the main theoretical results of this chapter are verified by numerical simulation. In chapter 3, we study a class of infectious disease models with delay, information variable and saturation incidence. The local stability of disease-free equilibrium and endemic equilibrium is analyzed by means of eigenvalue theory. The existence condition of Hopf bifurcation is obtained by taking time delay as bifurcation parameter. The stability and bifurcation direction of Hopf bifurcation periodic solution are obtained by using normal form theory and center manifold theorem. Finally, the main theoretical results of this chapter are verified by numerical simulation. In chapter 4, we discuss a class of infectious disease models with periodic contact rate and time-varying pulse vaccination rate. By calculating the threshold of disease prevalence or not, the disease-free periodic solution of the system is globally stable, that is, disease eradication, when the basic reproduction number is R01, and the disease is persistent when R0- 伪 + A / D ~ (-1). Finally, the main theoretical results of this chapter are verified by numerical simulation.
【学位授予单位】：山西师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175

【共引文献】