【摘要】：傳染病的防制是關(guān)系到人類健康和國計民生的重大問題,對疾病流行規(guī)律的定量研究是防制工作的重要依據(jù).長期以來,人類與各種傳染病進行了不屈不撓的斗爭并且征服了很多對人類生命和財產(chǎn)造成重大損失的傳染病.但還有一些新老傳染病依然無法有效地控制,艾滋病也是威脅人類生命的罪魁禍?zhǔn)字?盡管傳播途徑有限,艾滋病繼續(xù)在人類社會上驚人地蔓延.為了研究人類行為和流動現(xiàn)象對艾滋病傳播的影響,本文討論了人口遷入率為常數(shù)、且將預(yù)防作為主要控制策略的艾滋病模型.本文包括以下五個部分:引言部分介紹了人口移民、人口流動對一些傳染病的影響,且概括了具有人口遷移的傳染病模型的研究現(xiàn)狀和建模意義;第二部分是預(yù)備知識;第三和第四部分研究兩種艾滋病模型的建立、穩(wěn)定性和最優(yōu)控制,其主要內(nèi)容為:一、研究具有染病者遷移的艾滋病模型.首先,討論該模型的基本性質(zhì)包括系統(tǒng)解的正性、有界性及正平衡點的存在性.其次,利用幾何方法研究模型唯一正平衡點的全局漸近穩(wěn)定性.最后,為了研究預(yù)防艾滋病策略對系統(tǒng)動力學(xué)行為的影響,把原模型推廣為具有最優(yōu)控制的模型.這部分主要利用最優(yōu)控制理論、Pontriagin最大原理及相關(guān)知識來證明最優(yōu)控制的存在性,并給出了原系統(tǒng)和最優(yōu)控制系統(tǒng)相互對比的數(shù)值模擬圖像.引入控制模型的目的是,制定一系列定期的控制、預(yù)防策略,使得有限時間段內(nèi)以最少的成本把染病者數(shù)量控制在較低的狀態(tài).二、在第一個模型的基礎(chǔ)上引入標(biāo)準(zhǔn)發(fā)生率,還考慮了易感者被艾滋病病毒感染到首次被確診為艾滋病的時間差.首先,討論模型解的正性及有界性,給出系統(tǒng)解的正向不變集.其次,研究模型正平衡點的存在性.當(dāng)時滯τ= 0時,構(gòu)造適當(dāng)?shù)腖yapunov函數(shù)證明平衡點的全局漸近穩(wěn)定性.最優(yōu)控制部分研究τ≠0時,該模型的最優(yōu)控制.這部分主要利用最優(yōu)控制理論、Pontriagin最大原理及相關(guān)知識.最后,通過原系統(tǒng)和最優(yōu)控制系統(tǒng)相互對比的數(shù)值模擬圖來說明結(jié)論的正確性和方法的有效性.第五部分作為結(jié)論總結(jié)了整篇論文的主要內(nèi)容.染病者人口的遷入讓艾滋病模型不存在無病平衡點,使得艾滋病更難以消除.當(dāng)我們實行篩選和加大宣傳力度的方法時,能讓染病者數(shù)量保持較低的水平.目前在全世界范圍內(nèi)仍缺乏根治HIV感染的有效藥物,對于這樣的疾病來說,本文提出的控制策略更符合實際而且非常有效.
[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，

本文編號：2167102