Hopf bifurcation of an epidemic model with a nonlinear birth in population and vertical transmission

In this paper, an epidemic model involving a nonlinear birth in population and vertical transmission was studied. When R0<1R0<1, the disease-free equilibrium was stable, while if R0>1R0>1, the disease-free equilibrium was unstable. We researched the existence of Hopf bifurcation and obtained the stability and direction of the Hopf bifurcation by using the normal theory and the center manifold theorem. Numerical simulations were carried out to illustrate the main theoretical results and a brief discussion was given to conclude this work.