Seir Model With Strong Nonlinear Infection Rate Overall Behavior

The SEIR model with nonlinear incidence rates βIpSq in epidemiology is studied. Firstly, under certain scope of parameters only an infection-equilibrium P0 can be obtained, and the condition for the global stability of the infection-equilibrium P0 is gained by the stability theory of ordinary differential equation. Secondly , an infection-equilibrium P0and an endemic equilibrium P* are existed. Global analyse of the endemic equilibrium P* is proved using a general criterion for the orbital stability of periodic orbits associated with higher-dimensional nonlinear autonomous systems as well as the theory of competitive systems of differential equations , and the scope of parameters is divided into three parts .To the first part, a one dimensional central manifold is obtained; To the second part, a two dimensional central manifold is obtained; To the third part, the direction of orbits are acquired by dividing the positive invariant into eight portions, in addition, periodic orbit may be exsited. Lastly, the infection-equilibrium P0 and two endemicequilibria P1, P2 are existed, and Hopf bifurcation is gained through analyzing the stability of endemic equilibrium P1. Besides,we illustrated them by Matlab with special parameters and gave the according ecological significances.