Mathematical analysis of a vaccination epidemic model with nonlocal diffusion

We study the global dynamics of a susceptible-vaccinated-infected-recovered model that incorporates nonlocal diffusion. By identifying the basic reproduction number Script capital R0$$ {\mathrm{\mathcal{R}}}_0 $$ of the model, we obtain the following threshold-type results: (i) If Script capital R0<1$$ {\mathrm{\mathcal{R}}}_0, then the epidemic becomes extinct in the sense that the infection-free equilibrium is globally attractive; (ii) if Script capital R0>1$$ {\mathrm{\mathcal{R}}}_0>1 $$ and the diffusion coefficients are the same for all classes, then the epidemic persists in the sense that the system is uniformly persistent; and (iii) if Script capital R0>1$$ {\mathrm{\mathcal{R}}}_0>1 $$, the diffusion coefficients for susceptible, and the vaccinated classes are zero, then the system admits a unique endemic equilibrium, and the omega-limit set is included in the singleton of the endemic equilibrium. Our results show that Script capital R0$$ {\mathrm{\mathcal{R}}}_0 $$ is an essential value for determining global epidemic dynamics in our model.