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. In particular, 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, i.e., such a hypoellipticity assumption also implies that the corresponding diffusion process has a transition probability density function with strong Feller property. Here, 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.
Keywords: Diffusion processes; Epidemiology; Exit probability; Markov controls; Minimum exit rates; Optimal control problem; Prescription drug addiction; Principal eigenvalues; SIR compartmental model.