Control of Time-Varying Epidemic-Like Stochastic Processes and Their Mean-Field Limits

TL;DR

This paper develops methods for controlling time-varying epidemic-like stochastic processes, linking them to stochastic shortest path problems and mean-field limits, and demonstrates convergence of optimal controls to a dynamical system.

Contribution

It introduces a framework connecting control of epidemic-like stochastic processes with time-varying parameters to mean-field dynamical systems, providing convergence results for optimal controls.

Findings

Optimal control solutions for stochastic epidemic processes with time-varying parameters.

Convergence of stochastic process controls to controls of the limiting dynamical system.

Mathematical properties of the controlled dynamical systems are analyzed.

Abstract

The optimal control of epidemic-like stochastic processes is important both historically and for emerging applications today, where it can be especially important to include time-varying parameters that impact viral epidemic-like propagation. We connect the control of such stochastic processes with time-varying behavior to the stochastic shortest path problem and obtain solutions for various cost functions. Then, under a mean-field scaling, this general class of stochastic processes is shown to converge to a corresponding dynamical system. We analogously establish that the optimal control of this class of processes converges to the optimal control of the limiting dynamical system. Consequently, we study the optimal control of the dynamical system where the comparison of both controlled systems renders various important mathematical properties of interest.