Dynamics of classical solutions to a diffusive epidemic model with varying population demographics

We study the asymptotic dynamics of solutions to a diffusive epidemic model with varying population dynamics. The large-time behavior of solutions is completely described in spatially homogeneous environments. When the environment is spatially heterogeneous, it is shown that there exist two critical numbers $1\le {\sigma }_{⁎}\le {\sigma }^{⁎}<\infty$ such that if the ratio $\frac{d_I}{d_S}$ of the infected population diffusion rate and the susceptible population rate either exceeds ${\sigma }^{⁎}$ or is less than ${\sigma }_{⁎}$, then the epidemic model has an endemic equilibrium (EE) solution if and only if the basic reproduction number (BRN) exceeds one. The unique EE is non-degenerate if $\frac{d_I}{d_S}\ge {\sigma }^{⁎}$. Furthermore, results on the global dynamics of solutions are established when ${\sigma }^{⁎}=1$. Our results shed some light on the differences on disease predictions for constant total population size models versus varying population size models. Results on the asymptotic profiles of the EEs for small population diffusion rates are also established.