Threshold dynamics of a reaction-diffusion epidemic model with stage structure

A time-delayed reaction-diffusion epidemic model with stage structure and spatial heterogeneity is investigated, which describes the dynamics of disease spread only proceeding in the adult population. We establish the basic reproduction number $\mathcal{R}_0$ for the model system, which gives the threshold dynamics in the sense that the disease will die out if $\mathcal{R}_0＜1$ and the disease will be uniformly persistent if $\mathcal{R}_0>1.$ Furthermore, it is shown that there is at least one positive steady state when $\mathcal{R}_0>1.$ Finally, in terms of general birth function for adult individuals, through introducing two numbers $\check{\mathcal{R}}_0$ and $\hat{\mathcal{R}}_0$ , we establish sufficient conditions for the persistence and global extinction of the disease, respectively.