Analysis Of An SIRS Epidemic Model With Nonlinear Incidence Rate An Diffusion

Human beings has been struggling with the epidemics for a long time, and the epidemics have always been doing harm to humans’ health. The analysis of pathogenesis, rule of infect, prevention and control strategy of epidemic has been a more and more vital topic.During the past20years, the research of dynamics of epidemics made a rapidly progress. Plenty of mathematical models are applied in the study of epi-demics, and those models are reasonable to the general regulations of all kinds of epidemics. Here, we discuss an SIRS epidemic model with diffusion and nonlin-ear incidence rate, we obtained the traveling wave solutions which connected the disease-free equilibrium and the endemic equilibrium.First of all, with the use of the classical results of differential equations, we get the existence and uniqueness of the disease-free equilibrium and the endemic equilibrium, and then we construct a Lyapunov function, give the global asymp-totic stability of the disease-free equilibrium and the endemic equilibrium with the aid of Lyapunov-LaSalle asymptotical stable theory.The second, we discuss the existence and nonexistence of the traveling wave solutions of the system. Assume that the basic reproductive number R0>1, with the use of Schauder’s fixed point theory and upper-lower solutions, we obtained the existence and nonexistence of the traveling wave solutions when c> c*and c∈(0, c*) separately.