SPDEs with space interactions and application to population modelling

TL;DR

This paper introduces a new class of non-local SPDEs with space interactions, proves their well-posedness, and applies the framework to optimize vaccine strategies in epidemic models.

Contribution

It develops existence, uniqueness, and positivity results for SPDEs with space interactions and derives maximum principles for their optimal control, with applications to epidemiology.

Findings

Existence and uniqueness of solutions for SPDEs with space interactions.

Solutions remain positive under certain conditions.

Application to optimal vaccine strategy in epidemic modeling.

Abstract

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.