  This contribution mainly focuses on the finite horizon optimal control
problems of a susceptible-infected-vaccinated(SIV) epidemic system governed by
reaction-diffusion equations and Markov switching. Stochastic dynamic
programming is employed to find the optimal vaccination effort and economic
return for a stochastic reaction diffusion SIV epidemic model. To achieve this,
a key step is to show the existence and uniqueness of invariant measure for the
model. Then, we obtained the necessary and sufficient conditions for the
near-optimal control. Furthermore, we give an algorithm to approximate the
Hamilton-Jacobi Bellman (HJB) equation. Finally, some numerical simulations are
presented to confirm our analytic results.
