The Stability Analysis Of Two Kinds Of Epidemic Models With Pulse Vaccination And Time Delay

In recent years, impulsive differential equation and delay differential equa-tion have been widely used to study the dynamics of epidemic. Impulsive and delay differential equations take into account not only instantly change but also the past state being considered, which provide a more reasonable and accurate reflect of the population changes. In this paper, based on the incidence law and propagation of infectious diseases in reality, considering some dynamic behaviors mutation in a short period of time due to outside interference (such as pulse birth or pulse vaccination), the traditional classic continuous dynamic system is con-verted into a more reasonable impulse one. At the same time, delays and impulses are introduced to make the model more in line with the actual situation. We study the existence of disease-free periodic solution (DFP) of the model, its locally and globally stability, the permanence of disease and so on.In the first chapter, we mainly elaborate the significance of the research, the basic model and its development process. We also give an overview of the research status and the main work done in this article.Next, we analysis a new SIRS epidemic model with varying population size and pulse vaccination. By using the Normalization method, the three-dimensional problem can be converted into a two-dimensional one. According to the pulse differential equations and Floquet theorem, we can prove the existence of disease-free periodic solution, locally and globally stability and persistence of disease, drawing that if vaccination period less than a critical value, or vaccination ratio greater than a certain value, the disease will eventually go extinction. It also provides many useful suggestions for making pulse vaccination strategy. Finally, a numerical simulation of the system is conducted to verify these theoretical results.In the following, we consider a model with double delays and double pulses. Using discrete dynamical systems stroboscopic map, the disease-free period solu- tion can be obtained. We discuss the globally attractive of the periodic solution, give a theoretical basis for the eradication of disease. Beside, using the quali-tative analysis method, the permanence of the system is proved. The sufficient conditions of disease persistence can be obtained by introducing the Lyapunov function. The main conclusion is that enhancement the vaccination proportion or prolongation the delayed period is more conducive to the extinction of the disease.Shortcomings as well as points need to be improved are overviewed in the fourth chapter, and the directions of future study of the topic are also pointed out.