Dynamics of a diffusive vaccination model with therapeutic impact and non-linear incidence in epidemiology.

In this paper, we study a more general diffusive spatially dependent vaccination model for infectious disease. In our diffusive vaccination model, we consider both therapeutic impact and nonlinear incidence rate. Also, in this model, the number of compartments of susceptible, vaccinated and infectious individuals are considered to be functions of both time and location, where the set of locations (equivalently, spatial habitats) is a subset of $R^n$ with a smooth boundary. Both local and global stability of the model are studied. Our study shows that if the threshold level $R_0\le 1,$ the disease-free equilibrium $E_0$ is globally asymptotically stable. On the other hand, if $R_0>1$ then there exists a unique stable disease equilibrium $E^{\ast }$. The existence of solutions of the model and uniform persistence results are studied. Finally, using finite difference scheme, we present a number of numerical examples to verify our analytical results. Our results indicate that the global dynamics of the model are completely determined by the threshold value $R_0$.