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In this paper, we consider an opioid epidemic dynamical model with randomperturbation that typically describes the interplay between regularprescription use, addictive use, and the process of rehabilitation fromaddiction and vice-versa. In particular, we provide two-sided bounds on thesolution of the transition density function for the Fokker-Planck equation thatcorresponds to the opioid epidemic dynamical model, when a random perturbationenters only through the dynamics of the susceptible group in the compartmentalmodel. Here, the proof for such bounds basically relies on the interpretationof the solution for the transition density function as the value function of acertain optimal stochastic control problem. Finally, as a possible interestingdevelopment in this direction, we also provide an estimate for the attainableexit probability with which the solution for the randomly perturbed opioidepidemic dynamical model exits from a given bounded open domain during acertain time interval. Note that such qualitative information on the firstexit-time as well as two-sided bounds on the transition density function areuseful for developing effective and fact-informed intervention strategies thatprimarily aim at curbing opioid epidemics or assisting in interpreting outcomeresults from opioid-related policies.

A further study on the opioid epidemic dynamical model with random perturbation