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# Compartmental Model Exposes Reality of the U.S. Opioid Epidemic

The United States is presently in the midst of an opioid epidemic. The number of overdose-incurred fatalities has increased steadily since the 1990s, with 91 Americans dying from overdose every day. Unlike other substance abuse crises, wherein young generations comprise the majority of abusers, the opioid epidemic claims victims of all ages. According to the Centers for Disease Control and Prevention (CDC), 45-54 year-olds are responsible for the highest concentration of overdose deaths, with 11.7 deaths per 100,000 people in 2014 (see Figure 1). In contrast, heroin overdose boasts only 4.7 deaths for the same age range.

Figure 1. Comparison of heroin and opioid overdose deaths in 2014. Data courtesy of the Centers for Disease Control and Prevention.
While most epidemics of this type stem from illicit drug use, opioid addiction is rooted in legal prescription medication. The legality and availability of opioids significantly increases the number of exposed individuals, making it fairly easy for addicts to obtain drugs. During the 2018 SIAM Conference on Mathematics of Planet Earth, which took place earlier this month in Philadelphia, Pa., Christopher Strickland of the University of Tennessee, Knoxville presented a sobering model of prescription opioid addiction and treatment that examines the feasibility of an addiction-free equilibrium. “This is not the same as the heroin addition or the crack epidemic,” he said. “This is not the rebellious teenager epidemic. This is your-mother-who-did-everything-right-her-entire-life epidemic.”

Despite the prevalence of news coverage and the epidemic’s status as a national public health emergency, Strickland was surprised to find a dearth of mathematical models pertaining to opioid addiction. In response, he created an ordinary differential equation model to explore the dynamics that distinguish the opioid crisis from traditional illicit drug epidemics. Strickland’s compartmental model (in the style of a classic susceptible-infectious-recovered model) accounts for four population classes, labeled $$S$$, $$P$$, $$A$$, and $$R$$: susceptible (non-opioid users), prescribed (those with prescription opioids), addicted (those chronically abusing opioids), and rehabilitated (those in rehab/treatment).

Each of the above classes has a background death rate (due to natural causes) and an enhanced death rate (due to overdose). To ensure a constant total population, Strickland included a birth rate that assumes the number of births and deaths are equal. He also considered multiple addiction terms that account for (i) people whose addiction stems from their own prescriptions, (ii) people who abuse the leftover prescriptions of others, and (iii) people who purchase black market opioids illegally. Finally, Strickland identified rehab and relapse dynamics in his model, which represent entry into rehab/treatment programs, completion of rehab, normal relapse, and relapse due to addiction culture. “In this model we assume that there is no permanent safe space,” he said. “You’re never immune.”

After establishing his model, Strickland investigated the possibility of a completely existence-free state, which translates to an addiction-free society. “Very stringent controls are necessary for it to even exist,” he said. First, medically-prescribed users would have to stop getting addicted to their prescriptions. Additionally, opioids would have to be available solely through illicit means, rather than directly prescribed or lying around in medicine cabinets. The satisfaction of these two requirements would convert Strickland’s model to a traditional illicit drug epidemic.

Figure 2. The results of Sobol sensitivity analysis, measured with 10-year values of susceptible, prescribed, addicted, and rehabilitated population classes.

Assuming the aforementioned requirements and existence of a disease-free equilibrium, Strickland evaluated stability via the basic reproduction number ($$R_0$$), a next-generation matrix, and Jacobian analysis. The resulting equation includes secondary usage rate, rehab rate, and rehab completion rate. According to estimated parameter values from existing literature, an illicit-only opioid epidemic would likely not be self-sustaining. “This tells you immediately where your focal point should be for control,” Strickland said.

He then used Sobol sensitivity analysis—a variance-based method—to determine parameter sensitivity. “This concisely quantifies first-order and total-order effects, and anything in between,” Strickland said. “You really get a whole lot more information this way.” A 10-year run with Sobol sensitivity, measured with 10-year values of $$S$$, $$P$$, $$A$$, and $$R$$, yielded graphs of first-order and total-order indices (see Figure 2). Each bar represents a different rate, including rate of opioid prescription by healthcare professionals, addict admission to treatment facilities, prescribed user addiction, successful rehabilitation, and prescription completion without addiction; taller bars indicate heightened sensitivity and a more impactful parameter. Strickland attempted to connect these results with national, publically-available data from institutions like the CDC. “Because of the way they conduct the surveys, some of this data is hard to find,” he said. “But the takeaway is that the model certainly produces feasible values as outputs for the expected parameter values.”

Figure 3. Limiting prescription-induced opioid availability and increasing rehab/treatment effectiveness could reduce addition and overdose levels.
Strickland conducted parameter analysis to examine the parameter space in more detail. “An increase in illicit opioids can increase the amount of rehab and prescription control needed,” he said. “Decreasing illicit opioids won’t have a major effect.” He also found that increasing the rate of successful rehab and treatment efforts can have a significant impact on getting people into treatment, particularly when the starting treatment rate is low. Because his model is not sensitive to small changes, the reduction of intrinsic relapse rate is much more valuable than the reduction of illicit-based relapse. In short, decreasing prescription-induced addictions (by managing risk and prescribing fewer opioids) while simultaneously increasing treatment entry (and success) rate is realistically the only valid way to drastically reduce addiction and overdose levels (see Figure 3).

In summation, the elimination of opioid addiction is all but impossible in Strickland’s model without stringent controls on medication prescription. At some point, he intends to include heroin and fentanyl in his model, incorporate state-level statistics to improve parameter estimates and time-series data, examine the impact of user age, and study control strategies for best management solution. For the time being, Strickland recommends prioritizing the control of patients’ prescription-induced addiction. With careful prescription limitations, rehab efforts can have a profound impact in minimizing addiction, even if per-case treatment success rates remain somewhat low. “This is in no way going away anytime soon,” Strickland said frankly of the opioid epidemic. “This is really driven by the prescriptions. The illicit factors are a submodel.”

 Lina Sorg is the associate editor of SIAM News.