本文選題：基本再生數(shù)　切入點：時滯模型　出處：《大連交通大學(xué)》2012年碩士論文　論文類型：學(xué)位論文

【摘要】：傳染病歷來是危害人類健康的大敵。是指由病菌、細(xì)菌、真菌等病原體或原蟲、蠕蟲等寄生感染人或其他生物體后所產(chǎn)生且能在人群或相關(guān)生物種群中引起流行的疾病。世界衛(wèi)生組織(WHO)報告表明,傳染病仍是人類的第一殺手。在傳染病研究方法中,傳染病動力學(xué)的建模與研究是一種重要方法。傳染病動力學(xué)的里程碑式的工作是1927年Kermack和Mekendrick的倉室建模方法。從臨床以及可以觀測的實驗中我們發(fā)現(xiàn),一些疾病的傳染是需要一段時間的,即其是具有潛伏期的。時滯微分方程比常微分方程存在更復(fù)雜更豐富的動力學(xué)性態(tài),我們利用Routh-Hurwitz定理判定其局部漸近穩(wěn)定,在此基礎(chǔ)上構(gòu)造Lyapunov函數(shù),利用Lyapunov-LaSalle不變集原理證明傳染病動力學(xué)模型的全局穩(wěn)定性,從而預(yù)測傳染病的發(fā)展情況。 第一部分主要介紹了傳染病動力學(xué)模型近幾年的發(fā)展歷史、主要的內(nèi)容以及近年來對于傳染病動力學(xué)模型主要方法和成果。首先考慮了一類無潛伏期的傳染病模型,其次,論述了含潛伏期的傳染病模型研究概況,得到了疾病滅絕與否的基本再生數(shù)。最后,根據(jù)對離散時滯和連續(xù)時滯的歸因分析,定下本文采用方式方法。 第二部分我們在Kermack和Mckendrick的SIR倉室模型[8]的基礎(chǔ)上,為了更加全面精確的研究病毒的性態(tài)在原模型的基礎(chǔ)上引入時滯,得到了豐富的動力學(xué)性態(tài),有助于人們進一步理解和控制疾病。我們通過構(gòu)造Lyapunov泛函的方法,確立了無病平衡點的全局穩(wěn)定性,進而得出疾病的持久性。 第三部分首先對H1N1的疾病發(fā)生情況進行了分析,其次討論了一類具有隔離項Q的H1N1傳染病模型。結(jié)合上述兩點,我們構(gòu)建了傳染病H1N1的模型,研究了具有時滯SIQR傳染病模型解,并對其性態(tài)進行分析。根據(jù)得到基本再生數(shù),確立了無病平衡點的全局穩(wěn)定性和疾病的持久性。進而判斷疾病滅絕還是形成地方病。通過研究,我們在持久性的基礎(chǔ)上,得到了地方病平衡點的局部漸近穩(wěn)定性、全局漸近穩(wěn)定性的充分條件。
[Abstract]:Infectious diseases have always been major enemies of human health. They refer to pathogens or protozoa, such as germs, bacteria, fungi, etc. Parasitic diseases such as worms that infect people or other organisms and cause epidemics in people or related populations. The World Health Organization (WHO) report shows that infectious diseases are still the number one killer of human beings. The modeling and research of infectious disease dynamics is an important method. The landmark work of infectious disease dynamics is the chamber modeling method of Kermack and Mekendrick in 1927. The infection of some diseases takes a period of time, that is, it has latent period. The delay differential equation has more complex and abundant dynamics than ordinary differential equation. We use Routh-Hurwitz theorem to determine its local asymptotic stability. On this basis, the Lyapunov function is constructed, and the global stability of infectious disease dynamic model is proved by using Lyapunov-LaSalle invariant set principle, and the development of infectious disease is predicted. The first part mainly introduces the history of the dynamics model of infectious diseases in recent years, the main contents and the main methods and achievements of the dynamics model of infectious diseases in recent years. In this paper, the general situation of infectious disease model with latent period is discussed, and the basic regenerative number of disease extinction is obtained. Finally, according to the attribution analysis of discrete and continuous delays, the method of this paper is proposed. In the second part, on the basis of the SIR chamber model of Kermack and Mckendrick, in order to study the virus behavior more comprehensively and accurately, we introduce time delay on the basis of the original model, and obtain a rich dynamic state. We establish the global stability of disease-free equilibrium by constructing Lyapunov functional, and then obtain the persistence of disease. In the third part, we first analyze the incidence of H1N1, then we discuss a class of H1N1 infectious disease models with isolation Q. Combined with the above two points, we construct the H1N1 model of infectious disease. In this paper, the solution of SIQR infectious disease model with time delay is studied, and its behavior is analyzed. The global stability of disease-free equilibrium and the persistence of disease are established, and then the extinction of disease or the formation of endemic disease is determined. On the basis of the study, we obtain the local asymptotic stability of endemic equilibrium. Sufficient conditions for global asymptotic stability.
【學(xué)位授予單位】：大連交通大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2012
【分類號】：O175;R311

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