On the dynamics of a diffusive SEIRS epidemic model

Abstract This work investigates the dynamics of positive classical solutions to a diffusive susceptible-exposed-infected-recovered-susceptible epidemic model with a mass-action incidence mechanism in spatially heterogeneous environments. Under minimal assumptions on the initial data, the global existence of classical solutions is established. Moreover, the eventual boundedness of these solutions is proved when either the spatial domain has dimension five or lower or the susceptible and exposed subpopulations share the same diffusion rate. Next, we define the basic reproduction number, $\mathcal{R}_0$ , and demonstrate that the disease-free equilibrium is globally stable when $\mathcal{R}_0$ is sufficiently small. However, due to the complex interaction between population movement and spatial variation in transmission rates, we find that the disease may persist even when $\mathcal{R}_0$ is slightly less than one. In such cases, we show that the system admits at least two endemic equilibrium (EE) solutions, an outcome not observed under the frequency-dependent incidence mechanism. These results highlight the significant influence of the transmission mechanism on disease dynamics. Furthermore, we examine the spatial profiles of the EE solutions when diffusion rates are small. Our analysis suggests that limiting the movement of the susceptible population can significantly reduce disease prevalence, provided that the total population remains below a specific threshold. In contrast, restricting the movement of the infected, exposed, or recovered populations alone may not eradicate the disease. Overall, our findings provide important insights into the spatial dynamics of infectious diseases and may offer guidance for developing and implementing effective containment strategies.