On the asymptotic exit control problem for stochastically perturbed prescription opioid epidemic models

Abstract

In this paper, we consider the problem of asymptotic exit control for a prescription opioid epidemic that describes the interaction between the regular prescription or addictive use of opioid drugs, and the process of rehabilitation and that of relapsing into the opioid drug use. In particular, our interest is in the situation, when the optimal control effort appearing linearly in the opioid epidemic model is interpreted as the rate at which the susceptible individuals are effectively removed from the population due to an opioid-related intervention policy, while a small perturbing noise enters through the dynamics of the susceptible group in the population compartmental model. To this end, we introduce a mathematical apparatus that minimizes the asymptotic exit-rate with which the solution for such stochastically perturbed prescription opioid epidemics exits from a given bounded open domain. Moreover, under certain conditions, we provide an admissible optimal control for the corresponding optimal control problem that optimally effected removal of the susceptible or recovered individuals from the population dynamics.