Stability analysis of SEIR model with general contact rate

A type of SEIR epidemic model with different general contact ratesβ1(N),β2(N)andβ3(N),having infective force in all the latent,infected and immune periods,was studied.And the threshold,basic reproductive number R0 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 E0 is globally asymptotically stable and the disease eventually goes away if R01.It was also proved that in the case where R01,E0 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 equilibriumE*is globally asymptotically stable and the disease persists.