Bulletin of Mathematical Biology

, Volume 81, Issue 7, pp 2258–2289 | Cite as

Modeling the Prescription Opioid Epidemic

  • Nicholas A. Battista
  • Leigh B. Pearcy
  • W. Christopher StricklandEmail author


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, in which case we estimate that the epidemic would cease to be self-sustaining. 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.


Population biology Dynamical systems Epidemiology Compartmental model Mathematical biology Prescription drug addiction 



The authors would like to thank Christina Battista, Robert Booth, Namdi Brandon, Kathleen Carroll, Jana Gevertz, Anne Ho, Shanda Kamien, Grace McLaughlin, Gianni Migliaccio, Matthew Mizuhara, and Laura Miller for comments, suggestions, and informative conversations. NAB would like to thank Patricia Clark of RIT, whose mathematical biology course gave the original motivation for this project in 2009. We would also like to thank the anonymous reviewers and the associate editor of Bulletin of Mathematical Biology for their helpful comments.


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Copyright information

© Society for Mathematical Biology 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsThe College of New JerseyEwingUSA
  2. 2.Department of Mathematics, CB 3250University of North Carolina at Chapel HillChapel HillUSA
  3. 3.Department of MathematicsUniversity of Tennessee at KnoxvilleKnoxvilleUSA
  4. 4.Department of MathematicsUniversity of Tennessee at KnoxvilleKnoxvilleUSA

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