Spatial dynamics of a diffusive susceptible-infectious-recovered-susceptible epidemic model with transfer from infectious to susceptible

To explore the effects of individual diffusion patterns, transient immunity and acquired immunity, this paper focuses on a reaction-diffusion Susceptible-Infectious-Recovered-Susceptible epidemic model with transfer from infectious to susceptible and logistic source in a heterogeneous environment. First, the uniform bounds of solutions are obtained by using Moser–Alikakos iteration method and then the threshold dynamics is established based on the basic reproduction number. After investigating the asymptotic profiles of endemic steady states for small or large motility rates, it is found that regardless of whether the diffusion rates are minimal or maximal, the disease may persist globally, which means that the restriction on the migration of population is not an effective strategy. Next, some bifurcation analysis is conducted by taking transient/acquired immunity coefficient as bifurcation parameter. Finally, the impact of spatial heterogeneity, large/small diffusion rates on the disease transmission as well as bifurcation dynamics are explored via some numerical simulations for system on a one-dimensional domain.