  Opioid addiction has become a global epidemic and a national health crisis in
recent years, with the number of opioid overdose fatalities steadily increasing
since the 1990s. In contrast to the dynamics of a typical illicit drug or
disease epidemic, opioid addiction has its roots in legal, prescription
medication - a fact which greatly increases the exposed population and provides
additional drug accessibility for addicts. In this paper, we present a
mathematical model for prescription drug addiction and treatment with
parameters and validation based on data from the opioid epidemic. Key dynamics
considered include addiction through prescription, addiction from illicit
sources, and treatment. Through mathematical analysis, we show that no
addiction-free equilibrium can exist without stringent control over how opioids
are administered and prescribed, effectively transforming the dynamics of the
opioid epidemic into those found in a purely illicit drug model. Numerical
sensitivity analysis suggests that relatively low states of endemic addiction
can be obtained by primarily focusing on medical prevention followed by
aggressive treatment of remaining cases - even when the probability of relapse
from treatment remains high. Further empirical study focused on understanding
the rate of illicit drug dependence versus overdose risk, along with the
current and changing rates of opioid prescription and treatment, would shed
significant light on optimal control efforts and feasible outcomes for this
epidemic and drug epidemics in general.
