QUALITATIVE ANALYSIS OF AN SEIR EPIDEMIC MODEL WITH A RELAPSE RATE

In all previous works, the dynamics of the SEIR and SEIRI epidemic models demonstrate a threshold phenomenon as follows: If the basic reproduction number R0 is below unity the diseasefree equilibrium is globally asymptotically stable and the disease always dies out. If R0 > 1; a unique endemic equilibrium is globally asymptotically stable and the disease will persist at the endemic equilibrium if it is initially present. In this paper, we show that the de nition of the basic reproduction number can modify this threshold phenomenon as follows: the disease-free equilibrium is globally asymptotically stable when the basic reproduction number is less than a critical value (di erent to unity) and the unique endemic equilibrium is globally asymptotically stable when the basic reproduction number is greater than this critical value. The global stability results are obtained by constructing suitable Lyapunov functionals and using the Lyapunov-LaSalle invariance principle. Our results show that the latent period plays a positive role to control disease development. Also some numerical simulations are given to compare the results of the SEIR and SEIRI epidemic models.