Study On The Stability And The Bifurcation Of SEIR Epidemic Model

In this article, two types of SEIR epidemic model with the nonlinear contact rate and the standard contact rate had been studied. The disease-free equilibrium and endemic equilibrium of the stability, backward bifurcation, existence of periodic solutions and the stability of the positive periodic solution had been discussed, Main results are as follows:Firstly, a kind of SEIR epidemic model with nonlinear contact rate was built to consider the effect of the factor of incubation period, population of constant input rate, mortality rate etc. By using the properties of inequality, the basic reproduction number of the model was given. Through analyzing the thresholds of the model, the locally stability global stability of the disease-free equilibrium were obtained. By the bifurcation theory, the conditions of forward bifurcation and backward bifurcation were given. Using the method of composite matrix, the global stability condition of endemic equilibrium was Proved. By using MATLAB numerical simulation, the accuracy of the model was proved. At the same time, the effect on spreading epidemic of model’s parameters were researched.Secondly, the incubation period of the disease, changing population rate and standard contact rate etc factors were considered. On the basis of autonomous epidemic model, a kind nonautonomous SEIR epidemic model was built, after involving standard contact rate. The existence of positive solution was proved. By using the comparison principle, the sufficient conditions of persistence and illnesses extinction were obtained. The existence of periodic solution was proved by using the fixed point theorem. By constructing Liapunov functional, the sufficient condition of model globally asymptotically stability were obtained.The accuracy of the model’s conclusion was proved by MATLAB numerical simulation.