具有人口迁移的艾滋病模型的稳定性分析及最优控制
发布时间：2018-08-06 08:18
【摘要】：传染病的防制是关系到人类健康和国计民生的重大问题,对疾病流行规律的定量研究是防制工作的重要依据.长期以来,人类与各种传染病进行了不屈不挠的斗争并且征服了很多对人类生命和财产造成重大损失的传染病.但还有一些新老传染病依然无法有效地控制,艾滋病也是威胁人类生命的罪魁祸首之一.尽管传播途径有限,艾滋病继续在人类社会上惊人地蔓延.为了研究人类行为和流动现象对艾滋病传播的影响,本文讨论了人口迁入率为常数、且将预防作为主要控制策略的艾滋病模型.本文包括以下五个部分:引言部分介绍了人口移民、人口流动对一些传染病的影响,且概括了具有人口迁移的传染病模型的研究现状和建模意义;第二部分是预备知识;第三和第四部分研究两种艾滋病模型的建立、稳定性和最优控制,其主要内容为:一、研究具有染病者迁移的艾滋病模型.首先,讨论该模型的基本性质包括系统解的正性、有界性及正平衡点的存在性.其次,利用几何方法研究模型唯一正平衡点的全局渐近稳定性.最后,为了研究预防艾滋病策略对系统动力学行为的影响,把原模型推广为具有最优控制的模型.这部分主要利用最优控制理论、Pontriagin最大原理及相关知识来证明最优控制的存在性,并给出了原系统和最优控制系统相互对比的数值模拟图像.引入控制模型的目的是,制定一系列定期的控制、预防策略,使得有限时间段内以最少的成本把染病者数量控制在较低的状态.二、在第一个模型的基础上引入标准发生率,还考虑了易感者被艾滋病病毒感染到首次被确诊为艾滋病的时间差.首先,讨论模型解的正性及有界性,给出系统解的正向不变集.其次,研究模型正平衡点的存在性.当时滞τ= 0时,构造适当的Lyapunov函数证明平衡点的全局渐近稳定性.最优控制部分研究τ≠0时,该模型的最优控制.这部分主要利用最优控制理论、Pontriagin最大原理及相关知识.最后,通过原系统和最优控制系统相互对比的数值模拟图来说明结论的正确性和方法的有效性.第五部分作为结论总结了整篇论文的主要内容.染病者人口的迁入让艾滋病模型不存在无病平衡点,使得艾滋病更难以消除.当我们实行筛选和加大宣传力度的方法时,能让染病者数量保持较低的水平.目前在全世界范围内仍缺乏根治HIV感染的有效药物,对于这样的疾病来说,本文提出的控制策略更符合实际而且非常有效.
[Abstract]:The prevention and control of infectious diseases is a major problem related to human health and the national economy and people's livelihood, and the quantitative study of disease prevalence law is an important basis for prevention and control work. For a long time, human beings have fought indefatigably against various infectious diseases and conquered many infectious diseases which have caused great losses to human life and property. But there are still new and old infectious diseases can not be effectively controlled, AIDS is also a major threat to human life. Despite the limited means of transmission, AIDS continues to spread alarmingly in human society. In order to study the effect of human behavior and mobility on the spread of HIV / AIDS, this paper discusses the HIV / AIDS model with a constant migration rate and prevention as the main control strategy. This paper includes the following five parts: the introduction part introduces the population migration, the impact of population mobility on some infectious diseases, and summarizes the research status and modeling significance of infectious disease model with population migration, the second part is the preparatory knowledge; The third and fourth parts study the establishment, stability and optimal control of two AIDS models. The main contents are as follows: 1. Firstly, the basic properties of the model are discussed, including the positivity, boundedness and existence of positive equilibrium. Secondly, the global asymptotic stability of the unique positive equilibrium of the model is studied by geometric method. Finally, in order to study the effect of AIDS prevention strategy on system dynamics behavior, the original model is extended to one with optimal control. In this part, the existence of optimal control is proved by using Pontriagin's maximum principle and related knowledge, and the numerical simulation images of the original system and the optimal control system are given. The purpose of introducing the control model is to formulate a series of periodic control and prevention strategies so as to keep the number of infected persons in a lower state at a minimum cost in a limited period of time. Secondly, based on the first model, the standard incidence rate was introduced, and the time lag between HIV infection and first diagnosis of AIDS was taken into account. Firstly, the positive and boundedness of the model solution are discussed, and the positive invariant set of the system solution is given. Secondly, the existence of positive equilibrium is studied. When 蟿 = 0, a proper Lyapunov function is constructed to prove the global asymptotic stability of the equilibrium point. The optimal control of the model with 蟿 鈮，
本文编号：2167102
[Abstract]:The prevention and control of infectious diseases is a major problem related to human health and the national economy and people's livelihood, and the quantitative study of disease prevalence law is an important basis for prevention and control work. For a long time, human beings have fought indefatigably against various infectious diseases and conquered many infectious diseases which have caused great losses to human life and property. But there are still new and old infectious diseases can not be effectively controlled, AIDS is also a major threat to human life. Despite the limited means of transmission, AIDS continues to spread alarmingly in human society. In order to study the effect of human behavior and mobility on the spread of HIV / AIDS, this paper discusses the HIV / AIDS model with a constant migration rate and prevention as the main control strategy. This paper includes the following five parts: the introduction part introduces the population migration, the impact of population mobility on some infectious diseases, and summarizes the research status and modeling significance of infectious disease model with population migration, the second part is the preparatory knowledge; The third and fourth parts study the establishment, stability and optimal control of two AIDS models. The main contents are as follows: 1. Firstly, the basic properties of the model are discussed, including the positivity, boundedness and existence of positive equilibrium. Secondly, the global asymptotic stability of the unique positive equilibrium of the model is studied by geometric method. Finally, in order to study the effect of AIDS prevention strategy on system dynamics behavior, the original model is extended to one with optimal control. In this part, the existence of optimal control is proved by using Pontriagin's maximum principle and related knowledge, and the numerical simulation images of the original system and the optimal control system are given. The purpose of introducing the control model is to formulate a series of periodic control and prevention strategies so as to keep the number of infected persons in a lower state at a minimum cost in a limited period of time. Secondly, based on the first model, the standard incidence rate was introduced, and the time lag between HIV infection and first diagnosis of AIDS was taken into account. Firstly, the positive and boundedness of the model solution are discussed, and the positive invariant set of the system solution is given. Secondly, the existence of positive equilibrium is studied. When 蟿 = 0, a proper Lyapunov function is constructed to prove the global asymptotic stability of the equilibrium point. The optimal control of the model with 蟿 鈮，
本文编号：2167102
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