Dynamical behavior of a reaction-diffusion SEIR epidemic model with mass action infection mechanism in a heterogeneous environment

In this paper, we focus on a reaction-diffusion SEIR (susceptible-exposed-infected-recovered) epidemic model with mass action infection mechanism. We first give the definition of the basic reproduction number of the SEIR model, and then we study the properties of the basic reproduction number under the situations of low mobilities of the susceptible, exposed/infected populations, respectively. The global asymptotic stabilities of the disease-free equilibrium and endemic equilibrium are determined by the basic reproduction number in the homogeneous environment, and the asymptotic behaviors of the endemic equilibrium (when it exists) in a spatially heterogeneous environment with low migration rates of susceptible, exposed or infected populations are established. Our results suggest that strategies to control population migration rates will not eradicate the disease even in the case of permanent immunization, but reducing disease transmission rates or increasing recovery rates will reduce the possibility of disease persistence, which may be an effective disease control strategy.