几类具有时滞的传染病动力学模型研究

发布时间：2018-03-19 20:02

本文选题：基本再生数　切入点：时滞模型　出处：《大连交通大学》2012年硕士论文　论文类型：学位论文

【摘要】：传染病历来是危害人类健康的大敌。是指由病菌、细菌、真菌等病原体或原虫、蠕虫等寄生感染人或其他生物体后所产生且能在人群或相关生物种群中引起流行的疾病。世界卫生组织(WHO)报告表明,传染病仍是人类的第一杀手。在传染病研究方法中,传染病动力学的建模与研究是一种重要方法。传染病动力学的里程碑式的工作是1927年Kermack和Mekendrick的仓室建模方法。从临床以及可以观测的实验中我们发现,一些疾病的传染是需要一段时间的,即其是具有潜伏期的。时滞微分方程比常微分方程存在更复杂更丰富的动力学性态,我们利用Routh-Hurwitz定理判定其局部渐近稳定,在此基础上构造Lyapunov函数,利用Lyapunov-LaSalle不变集原理证明传染病动力学模型的全局稳定性,从而预测传染病的发展情况。第一部分主要介绍了传染病动力学模型近几年的发展历史、主要的内容以及近年来对于传染病动力学模型主要方法和成果。首先考虑了一类无潜伏期的传染病模型,其次,论述了含潜伏期的传染病模型研究概况,得到了疾病灭绝与否的基本再生数。最后,根据对离散时滞和连续时滞的归因分析,定下本文采用方式方法。第二部分我们在Kermack和Mckendrick的SIR仓室模型[8]的基础上,为了更加全面精确的研究病毒的性态在原模型的基础上引入时滞,得到了丰富的动力学性态,有助于人们进一步理解和控制疾病。我们通过构造Lyapunov泛函的方法,确立了无病平衡点的全局稳定性,进而得出疾病的持久性。第三部分首先对H1N1的疾病发生情况进行了分析,其次讨论了一类具有隔离项Q的H1N1传染病模型。结合上述两点,我们构建了传染病H1N1的模型,研究了具有时滞SIQR传染病模型解,并对其性态进行分析。根据得到基本再生数,确立了无病平衡点的全局稳定性和疾病的持久性。进而判断疾病灭绝还是形成地方病。通过研究,我们在持久性的基础上,得到了地方病平衡点的局部渐近稳定性、全局渐近稳定性的充分条件。
[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.
【学位授予单位】：大连交通大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：O175;R311

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