  In this paper, we consider an optimal control problem for a prescription
opioid epidemic model that describes the interaction between the regular
prescription or addictive use of opioid drugs, and the process of
rehabilitation and that of relapsing into opioid drug use. In particular, our
interest is in the situation, where the control appearing linearly in the
opioid epidemics is interpreted as the rate at which the susceptible
individuals are effectively removed from the population due to an
opioid-related intervention policy or when the dynamics of the addicted is
strategically influenced due to an accessible addiction treatment facility,
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 assumptions, we also
provide an admissible optimal Markov control for the corresponding optimal
control problem that optimally effected removal of the susceptible or recovered
individuals from the population dynamics.
