Stability a nd Bifurcation Analysis in A SEIR Epidemic Model with Nonlinear Incidence Rates

In this paper, a special SEIR epidemic model with nonlinear incidence rates is considered. By analyzing the associated characteristic transcendental equation, it is found that Hopf bifurcation occurs when these delays pass through a sequence of critical value. Some explicit formulae for deter- mining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are ob- tained by using the normal form theory and center manifold theory. Some numerical simulation for justifying the theoretical analysis are also presented. Finally, biological explanations and main conclusions are given. Index Terms—SEIR epidemic model; stability; Hopf bifurca- tion; periodic solution