The dynamics of heroin and illicit opioid use disorder, casual use, treatment, and recovery: A mathematical modeling analysis

Sandra Cole, M. Foster Olive, Stephen Wirkus

Research output: Contribution to journalArticlepeer-review


A leading crisis in the United States is the opioid use disorder (OUD) epidemic. Opioid overdose deaths have been increasing, with over 100,000 deaths due to overdose from April 2020 to April 2021. This paper presents a mathematical model to address illicit OUD (IOUD), initiation, casual use, treatment, relapse, recovery, and opioid overdose deaths within an epidemiological framework. Within this model, individuals remain in the recovery class unless they relapse back to use and due to the limited availability of specialty treatment facilities for individuals with OUD, a saturation treatment function was incorporated. Additionally, a casual user class and its corresponding specialty treatment class were incorporated. We use both heroin and all-illicit opioids datasets to find parameter estimates for our models. Bistability of equilibrium solutions was found for realistic parameter values for the heroin-only dataset. This result implies that it would be beneficial to increase the availability of treatment. An alarming effect was discovered about the high overdose death rate: by 2046, the disorder-free equilibrium would be the only stable equilibrium. This consequence is concerning because it means the epidemic would end due to high overdose death rates. The IOUD model with a casual user class, its sensitivity results, and the comparison of parameters for both datasets, showed the importance of not overlooking the influence that casual users have in driving the all-illicit opioid epidemic. Casual users stay in the casual user class longer and are not going to treatment as quickly as the users of the heroin epidemic. Another result was that the users of the all-illicit opioids were going to the recovered class by means other than specialty treatment. However, the change in the relapse rate has more of an influence for those individuals than in the heroin-only epidemic. The results above from analyzing this model may inform health and policy officials, leading to more effective treatment options and prevention efforts.

Original languageEnglish (US)
Pages (from-to)3165-3206
Number of pages42
JournalMathematical biosciences and engineering : MBE
Issue number2
StatePublished - Feb 1 2024


  • compartmental model
  • dynamical systems
  • mathematical epidemiology
  • opioid drug addiction
  • population biology

ASJC Scopus subject areas

  • Modeling and Simulation
  • General Agricultural and Biological Sciences
  • Computational Mathematics
  • Applied Mathematics


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