Dynamic Properties Of The Two Types Of Infectious Disease Model Studies

Epidemiological dynamics is an important method for research on epidemic, which emphasis on the dynamic behavior of epidemic model, such as the existence and global stability of equilibrium (periodic solution), the persistence and extinction of the infectious disease, and so on. Studing epidemic models have not only the theory significance, but also very important practical application value. In this paper, we consider two types of models: an epidemic model with saturated treatment rate and vertical transmission, an SIRS epidemic model with seasonal variability and mixed vaccination strategy.The main contents of this paper can be summarized as follows:In the first chapter, the background and the works of relational epidemic models are introduced. At the same time, the main work of this paper are given.In the second chapter, an SIR epidemic model with saturated treatment rate and vertical transmission is studied. By calculating, the basic reproduction number is obtained. Using Poincare -Bendixson theorem, centre manifold theorem, Hopf bifurcation theorem and LaSalle invariant principle, we analyze the dynamic behavior of models.In the third chapter, an SIRS epidemic model with seasonal contact rate and mixed vaccination strategy (including first vaccination and pulse vaccination strategy) is investigated. The effects of the variation of dependent on the season of the contact rate and the vaccination strategy to eradicate infectious diseases are studied and discussed. A threshold for a disease to be extinct or endemic is established. By using Floquet multipliers theory and comparison theorem of differential equations, the existence and global asymptotic stability of disease-free periodic solution and the permanence of the disease are illustrated. Finally, our theoretical results are confirmed by numerical simulations.