A spatial SEIRS reaction-diffusion model in heterogeneous environment

We propose a susceptible-exposed-infected-recovered-susceptible (SEIRS) reaction-diffusion model, where the disease transmission and recovery rates can be spatially heterogeneous. The basic reproduction number (R₀) is connected with the principal eigenvalue of a linear cooperative elliptic system. Threshold-type results on the global dynamics in terms of R₀ are established. The monotonicity of R₀ with respect to the diffusion rates of the exposed and infected individuals, which does not hold in general, is established in several cases. Finally, the asymptotic profile of the endemic equilibrium is investigated when the diffusion rate of the susceptible individuals is small. Our results reveal the importance of the movement of the exposed and recovered individuals in disease dynamics, as opposed to most of previous works which solely focused on the movement of the susceptible and infected individuals.