  In this paper, we consider an opioid epidemic dynamical model with random
perturbation that typically describes the interplay between regular
prescription use, addictive use, and the process of rehabilitation from
addiction and vice-versa. In particular, we provide two-sided bounds on the
solution of the transition density function for the Fokker-Planck equation that
corresponds to the opioid epidemic dynamical model, when a random perturbation
enters only through the dynamics of the susceptible group in the compartmental
model. Here, the proof for such bounds basically relies on the interpretation
of the solution for the transition density function as the value function of a
certain optimal stochastic control problem. Finally, as a possible interesting
development in this direction, we also provide an estimate for the attainable
exit probability with which the solution for the randomly perturbed opioid
epidemic dynamical model exits from a given bounded open domain during a
certain time interval. Note that such qualitative information on the first
exit-time as well as two-sided bounds on the transition density function are
useful for developing effective and fact-informed intervention strategies that
primarily aim at curbing opioid epidemics or assisting in interpreting outcome
results from opioid-related policies.
