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We consider optimal control of a new type of non-local stochastic partialdifferential equations (SPDEs). The SPDEs have space interactions, in the sensethat the dynamics of the system at time $t$ and position in space x also dependon the space-mean of values at neighbouring points. This is a model with manyapplications, e.g. to population growth studies and epidemiology. We prove theexistence and uniqueness of solutions of a class of SPDEs with spaceinteractions, and we show that, under some conditions, the solutions arepositive for all times if the initial values are. Sufficient and necessarymaximum principles for the optimal control of such systems are derived.Finally, we apply the results to study an optimal vaccine strategy problem foran epidemic by modelling the population density as a space-mean stochasticreaction-diffusion equation.

SPDEs with space interactions and application to population modelling