Stability analysis of SEIR model with general contact rate

A type of SEIR epidemic model with different general contact rates β1(N), β2(N) and β3(N), having infective force in all the latent, infected and immune periods, was studied. And the threshold, basic reproductive number R0 which determines whether a disease is extinct or not, was obtained. By using the Liapunov function method, it was proved that the disease-free equilibrium E0 is globally asymptotically stable and the disease eventually goes away if R0<1. It was also proved that in the case where R0>1, E0 is unstable and the unique endemic equilibrium E* is locally asymptotically stable by Hurwitz criterion theory. It is shown that when disease-induced death rate and elimination rate are zero, the unique endemic equilibrium E* is globally asymptotically stable and the disease persists.