Study On The Three Classes Of Epidemic Models With Nonlinear Infection Rate

On the basis of previous studies, considering the affects of different infectious rate, and considering the factors of isolation, latent period, impulsive vaccination and vertical transmission, we discuss three classes of different epidemic models. The main contents are as follows:In the first part, we first introduce the history background and influence of infectious diseases on human development, then introduce the development course from simple to complex and trend of the infectious diseases model and the research achievements of recent years, and then introduce some basic definitions, theorems, sequiturs and lemmas, and finally introduce the brief works of this paper.In the second part, we mainly study the SEIQR epidemic model with constant input and nonlinear incidence rate, and obtain the threshold value which determines the outcome of the infectious disease. By using the method of eigenvalue and Lasalle invariable theory, we prove that the disease-free equilibrium is globally asymptotically stable when 10R<. By using the second additive compound matrix and the theory of the limit system, we prove that the unique endemic equilibrium is globally asymptotical stable when 11R>.In the third part, we deal with the SIQRS epidemic model of impulsive vaccination with constant input and nonlinearity of incidence rate. Using the Floquet theorem and compare theorem of impulsive differential equation, we obtain the sufficient condition under which the disease-free periodic solution is globally asymptotically stable. we prove that the system is uniform permanence under some conditions.In the fourth part, we discuss the SIQRS epidemic model with vertical transmission and nonlinear incidence rate and impulsive vaccination on susceptible. Using the Floquet theorem and compare theorem of impulsive differential equation, we obtain the sufficient condition under which the disease-free periodic solution is globally asymptotically stable. we prove that the system is uniform permanence under some conditions.