兰州大学机构库 >数学与统计学院
三类不同的传染病模型及其动力学分析
Alternative Titlethree different kinds of epidemic model and their dynamics analysis
吕陇
Thesis Advisor孙红蕊
2013-12-07
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name硕士
Keyword传染病模型 连续 脉冲 时滞 持久性 全局稳定性
Abstract现今利用数学模型分析传染病的流行过程已是数学应用的一个重要手段, 对疾病的控制和预防有着重要贡献.其中以微分方程为模型的研究主要集中在连续动力系统,脉冲动力系统和时滞动力系统上.随着研究的深入, 对模型的建立要求更高, 希望能够真实反映客观事实.过去主要以连续动力系统为工具进行研究, 但人们很快发现在现实疾病的免疫防疫与流行过程中都存在瞬时骤增或骤减, 以及滞后的情况出现, 仅用一种动力系统无法精确描述.因此本文分别就连续、连续与脉冲、脉冲与时滞三种动力系统进行了研究. 首先, 本文介绍了问题的研究背景、发展概论及研究问题结论的同时, 也介绍了常微分方程, 脉冲微分方程与时滞微分方程的一些基本理论和相关的预备知识. 其次, 建立了一类具有非线性传染率的SEIRV传染病模型, 通过使用Routh-Hurwitz定理、构造Liapunov函数与Bendixson判据得到了疾病灭绝与否的基本再生数R_0. 当R_0<1时, 无病平衡点全局渐近稳定且疾病最终消亡;当R_0>1时, 唯一的地方病平衡点全局渐近稳定. 再次, 研究了一类具有Logistic死亡率的连续和脉冲接种的SIRVS传染病模型.利用频闪映射方法, 得到了系统的无病周期解.运用Floquet乘子理论和比较定理, 证明了周期解的全局渐近稳定性, 并获得了系统一致持续存在的充分条件, 还讨论了连续接种率, 脉冲接种率, 免疫接种周期对疾病防治的影响. 最后, 介绍了一类具有脉冲接种时滞的SEIRS传染病模型, 利用频闪映射的方法得到模型的无病周期解.通过运用Floquet乘子理论和比较定理, 证明了模型的无病周期解是全局吸引的, 并且给出了疾病持久存在的充分条件.
Other AbstractNowadays, it is an important means to analysis the popular process of the infectious diseases by using the mathematics model and it have made important contributions to the control and prevention of the diseases. So the study of differential equation mainly focuses on the continuous dynamic system, pulsed dynamic system and time delay dynamic system. With the further study, there is a higher request that hopes to reflect objective truth for the model. In the past, people mainly study the continuous dynamical system, but people soon discovered that it can not illustrate it precisely by using only one power system, because the situation of instantaneous increase or decrease, and delay, exist in the prevention and popular process of the disease.This thesis studies three kinds of infectious disease model,that is continuous differential dynamical system, continuous and pulse differential dynamic system, pulse and delay differential dynamical systems and impact on them. First, the paper introduces the research background, overviews and the conclusion of research questions, and also introduces some definitions and fundamental theory of differential equations and impulsive differential equations. Second, a kind of non-linear incidence rate SEIRV model is investigated, the threshold which determines whether a disease is extinct or not, is obtained by using the Liapunov function and Bendixson criterion. The main result show that the disease-free equilibrium is globally asymptotically stable when R_0<1, the unique endemic equilibrium is globally asymptotically stable when R_0>1. Third, A SIRVS epidemic model with generalized Logistic death rate, continuous and pulse vaccination is formulated. The dynamical behavior of the model is analyzed. Using stroboscopic mapping, the existence of an infection-free periodic solution is obtained. Based on Floquet theory and the comparison theorem of impulsive differential equation, it proved global asymptotic stability of the infection-free periodic solution. And the sufficient condition for the permanence of the system is obtained. Moreover, the thesis discusses the effect of the strategies to control and eliminate infectious disease. At last, A SEIRS epidemic model with delay and pulse vaccination is formulated. By the analysis of the equivalent system of the model, the existence condition for the global attractive of infection-free periodic solution is obtained. And the thesis also put forward the necessity of...
URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/224750
Collection数学与统计学院
Recommended Citation
GB/T 7714
吕陇. 三类不同的传染病模型及其动力学分析[D]. 兰州. 兰州大学,2013.
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