  We consider optimal control of a new type of non-local stochastic partial
differential equations (SPDEs). The SPDEs have space interactions, in the sense
that the dynamics of the system at time $t$ and position in space x also depend
on the space-mean of values at neighbouring points. This is a model with many
applications, e.g. to population growth studies and epidemiology. We prove the
existence and uniqueness of solutions of a class of SPDEs with space
interactions, and we show that, under some conditions, the solutions are
positive for all times if the initial values are. Sufficient and necessary
maximum principles for the optimal control of such systems are derived.
Finally, we apply the results to study an optimal vaccine strategy problem for
an epidemic by modelling the population density as a space-mean stochastic
reaction-diffusion equation.
