Stability And Traveling Waves Of SIRS Models With Diffusion

In this thesis, we consider the diffusive SIRS model with a bilinear incidence rate, and characterize the stability of equilibria and the existence of travelling waves.First of all, we study the globally asymptotic stability of equilibria by using Lya-punov function method. When the basic reproductive number R0>1, the endemic equilibrium E*is globally asymptotically stable, and the disease free equilibrium E0is unstable; when R0<1, E0is globally asymptotically stable.Further, in order to understand the spreading behavior of endemic diseases, we work on the existence of travelling wave solutions which connect equilibria E0and E*. Applying the Schauder’s fixed point theorem and the upper-lower solution method, the existence of travelling wave solutions are proved when R0>1and c> c*. Then the existence of travelling wave solutions are proved for the case c=c*by the Arzela-Ascoli theorem. Moreover using comparison principle we obtain the nonexistence of travelling waves when R0<1.