两类具有常数输入率的SIRS模型的稳定性与最优控制

发布时间：2019-01-01 16:10

【摘要】：人们利用常微分方程所建立的数学模型来研究疾病的传播规律,预测疾病发展机制.但大多数成果在假设种群数量不变的基础上得到的.事实上,人口流动性较大,人口输入率、输出率等对人口数量变化有较大影响.本文对SIRS模型进行改进,建立了两类具有常数输入率的SIRS模型,并作了稳定性分析与最优控制.全文分为三章.第一章简要介绍了传染病模型的发展背景与SIRS模型的研究现状.第二章本章研究具有常数输入率与标准传染率的SIRS模型首先,利用Hurwitz判别法分别讨论了无染病者输入时和有染病者输入时平衡点的局部渐近稳定性.其次,通过构造李亚普诺夫函数法对平衡点的全局渐近稳定性进行分析,最后利用极值原理得到了疾病暴发早期的最优控制方案.第三章本章研究具有常数输入率与阶段结构的SIRS模型首先,利用F-v函数法找到疾病消除的临界阈值即基本再生数R0.其次,通过构造李亚普诺夫函数法对平衡点的全局渐近稳定性进行分析,最后通过构造性能指标函数对疾病爆发后期设计控制方案,得到了性能指标最优时的控制方案,为疾病得以有效控制提供了有力参考.
[Abstract]:The mathematical model established by ordinary differential equation is used to study the law of disease transmission and to predict the mechanism of disease development. But most of the results are based on the assumption that the population size is constant. In fact, population mobility, population input rate and output rate have great influence on population change. In this paper, the SIRS model is improved, two kinds of SIRS models with constant input rate are established, and the stability analysis and optimal control are made. The full text is divided into three chapters. The first chapter briefly introduces the development background of infectious disease model and the research status of SIRS model. In Chapter 2, we study the SIRS model with constant input rate and standard infection rate. Firstly, by using Hurwitz criterion, we discuss the local asymptotic stability of the equilibrium point of the input-free and disease-free input-point respectively. Secondly, the global asymptotic stability of the equilibrium point is analyzed by constructing Lyapunov function method, and the optimal control scheme in the early stage of disease outbreak is obtained by using the principle of extreme value. In Chapter 3, we study the SIRS model with constant input rate and stage structure. Firstly, we use F-v function method to find the critical threshold of disease elimination, that is, the basic reproduction number R0. Secondly, the global asymptotic stability of the equilibrium point is analyzed by constructing Lyapunov function method. Finally, the control scheme is designed for the later stage of disease outbreak by constructing the performance index function, and the control scheme when the performance index is optimal is obtained. It provides an effective reference for the effective control of the disease.
【学位授予单位】：曲阜师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175;O232

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