Some Research On Dynamics Of Epidemic Models With Time Delay

Infectious diseases have always been the great harmful enemy against human health. They are such kinds of diseases that infect human or other organisms and cause epidemic diseases in the populations or biological populations by pathogens such as germs, bacteria, fungi or protozoon, worms or other parasites.The world health organization (WHO) reported that infectious diseases were still the largest killer of people. In the field of infectious disease research, modeling the dynamics of infectious diseases is one of the most important methods. In1927, Kermack and Mekendrick, who found the warehouse room modeling, completed milestone work in this field. Through clinical and experimental observation, the process of disease infections was found preclinical, which means a period of time before the onset of diseases. Compared with ordinary differential equation, delay differential equation has more comprehensive dynamic properties. Therefore, in order to predict the development of infectious diseases, we use Routh-Hurwitz theorem to determine its local asymptotic stability and construct Lyapunov function on this basis. Then the global stability of the dynamic model of infectious diseases is proved according to Lyapunov-LaSalle invariance principle.The thesis is divided into three chapters. A complete survey about developments, results and methods of dynamic model in recent years is given in Chapter1. Firstly, a class of infectious disease models without incubation period is considered. Then we introduce the research situation on infectious disease model with incubation period and obtain the basic reproductive number associated with whether disease extinct or not. Finally, based on the attribution analysis of the discrete and continuous delay, we determine the methods used in this article.In the second Chapter, on the basis of SIR warehouse room model by Kermack and Mckendrick [8], the introduction of time delay on the original model helps get the rich dynamic nature for comprehensive and accurate virus research. This contributes to further understanding and control of disease. By constructing Lyapunov functional, we obtain the global stability of disease-free equilibrium point and the persistence of the disease.In chapter3, while how H1N1disease takes place is analyzed, a class of H1N1epidemic models with quarantine strategy Q is studied then. Based on the above two, we construct a new H1N1epidemic model and study solution of the SIQR epidemic model with time delay. According to the basic reproductive number, the global stability of disease-free equilibrium point and the persistence of the disease are determined. And we can predict whether disease will disappear or become endemic. Through the study, sufficient conditions for local asymptotic stability and global asymptotic stability of the endemic equilibrium are obtained.