Modeling the Prescription Opioid Epidemic

Abstract

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

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

“…Note that the control strategy appears linearly in the model and, as a result of this, we interpreted such an admissible control strategy as the rate at which the susceptible individuals are effectively removed from the population due to an intervention strategy or when the dynamics of addicted is influenced due to an accessible addiction treatment facility. In the simulation results, we used literature based parameter values (that are given in Table 2) for the prescription opioid epidemic model (e.g., see also Battista et al (2018) for additional discussions). Here, we are mainly interested in the addiction-free equilibrium case, i.e., for γ = 0, ξ = 0 and β > 0, when the linearized prescription opioid epidemic model (corresponding to the deterministic model, i.e.,Ẋ(t) = F(X(t))) becomes an addiction-free equilibrium, with the following steady state conditions 4…”

“…respectively (e.g., see also Battista et al (2018) and Befekadu and Zhu (2018) for additional discussions on the detailed model derivation). In the above prescription opioid epidemic dynamical population model, we assume that no new addictive opioid drug users are introduced from outside, but there is an external small random perturbing noise that enters through the dynamics of the susceptible group and then its effect is subsequently propagated to the other subsystems.…”

Agent-based and continuous models of hopper bands for the Australian plague locust: How resource consumption mediates pulse formation and geometry

“…space chosen via Saltelli's extension of the Sobol sequence [49,50]. Sobol indices represent a global, variance-based sensitivity analysis for nonlinear models that has become extremely popular in recent years for examining the performance of mathematical models in the context of data (e.g., [51,52]). One of its strengths is the ability to calculate not just first-order (one-ata-time) parameter sensitivity, but also second-order (two-at-a-time) and total-order (all possible combinations of parameters that include the given parameter) indices [50].…”

Optimal control of diffusion processes pertaining to an opioid epidemic dynamical model with random perturbations

“…In this subsection, we consider an opioid epidemic dynamical model that describes the interplay between regular prescription opioid use, addictive use, and the process of rehabilitation and relapsing into opioid drug use (e.g., see [2] for a detailed discussion). To this end, we introduce the following population groups (i) Susceptible group -S: This group in the compartmental model includes those who are susceptible to opioid addiction, but they are not currently using opioids.…”

“…In response to this, a number of federal and state agencies throughout the United States have implemented a wide range of opioid-related policies 1 that are primarily aimed at curbing prescription opioid abuse, establishing guidelines to prevent inappropriate prescribing practices, developing abuse deterrents or preventing drug diversion mechanisms [8], [27] and [12]. On the other hand, only a few studies have been reported on the need for effective intervention strategies, based on mathematical optimal control theory of epidemiology for infectious diseases, with the intent of better understanding the dynamics of the current serious opioid epidemic (e.g., see [2], [18] and [4] in context of exploring the dynamics of drug abuse epidemics, focusing on the interplay between the different opioid user groups and the process of rehabilitation and treatment from addiction; see also [15], [24] or [17] for additional studies, but in the context of heroin epidemics that resembling the classic susceptible-infected-recovered (SIR) model, based on the work of [28]). Here, we would like to point out that the roots to opioid crisis are complex and tangled with social and political issues; and therefore, only systemic research and evidence-based strategies can identify the most effective ways for intervention of the current opioid crisis.…”