Qualitative analysis of a reaction-diffusion SIRS epidemic model with nonlinear incidence rate and partial immunity.

In this paper, a reaction-diffusion SIRS epidemic model with nonlinear incidence rate and partial immunity in a spatially heterogeneous environment is proposed. The well-posedness of the solution is firstly established. Then the basic reproduction number is defined and a threshold dynamics is obtained. That is, when  < 1, the disease-free steady state is locally stable, which implies that the disease is extinct, when  > 1, the disease is permanent, and there exists at least one positive steady state solution. Finally, the asymptotic profiles of the positive steady state solution as individuals disperse at small and large rates are investigated. Furthermore, as an application of theoretical analysis, a numerical example involving the spread of influenza is discussed. Based on the numerical simulations, we find that the increase of transmission rate and spatial heterogeneity can enhance the risk of influenza propagation, and the increase of diffusion rate, saturation incidence for susceptible and recovery rate can reduce the risk of influenza propagation. Therefore, we propose to reduce the flow of people to lower the effect of spatial heterogeneity, increase the transfer of infected individuals to hospitals in surrounding areas to increase the diffusion rate, and increase the construction of public medical resources to increase the recovery rate for controlling influenza propagation.