Optimal control of diffusion processes pertaining to an opioid epidemic dynamical model with random perturbations

TL;DR

This paper develops an optimal control framework for a degenerate, hypoelliptic diffusion model of the opioid epidemic, aiming to minimize the rate of the epidemic leaving a specified domain under random perturbations.

Contribution

It introduces a novel control approach for a hypoelliptic diffusion model of opioid dynamics, deriving the Hamilton-Jacobi-Bellman equation and a verification theorem for optimal control.

Findings

Derived the HJB equation for the control problem

Established a connection to a nonlinear eigenvalue problem

Provided a verification theorem for optimal control

Abstract

In this paper, we consider the problem of controlling a diffusion process pertaining to an opioid epidemic dynamical model with random perturbation so as to prevent it from leaving a given bounded open domain. Here, we assume that the random perturbation enters only through the dynamics of the susceptible group in the compartmental model of the opioid epidemic dynamics and, as a result of this, the corresponding diffusion is degenerate, for which we further assume that the associated diffusion operator is hypoelliptic. In particular, we minimize the asymptotic exit rate of such a controlled-diffusion process from the given bounded open domain and we derive the Hamilton-Jacobi-Bellman equation for the corresponding optimal control problem, which is closely related to a nonlinear eigenvalue problem. Finally, we also prove a verification theorem that provides a sufficient condition for optimal control.