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; Sandra Cole; M. Foster Olive; Stephen Wirkus

doi:10.3934/mbe.2024141

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 2: 3165-3206. doi: 10.3934/mbe.2024141

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

1.
Arizona State University, School of Mathematical and Natural Sciences, Glendale, AZ, USA
2.
Arizona State University, Department of Psychology, Tempe, AZ, USA
3.
The University of Texas at San Antonio, Department of Mathematics, San Antonio, TX, USA

Academic Editor: Ruijun Zhao

Received: 28 November 2023 Revised: 15 January 2024 Accepted: 17 January 2024 Published: 01 February 2024

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.
- population biology,
- dynamical systems,
- mathematical epidemiology,
- compartmental model,
- opioid drug addiction
Citation: Sandra Cole, M. Foster Olive, Stephen Wirkus. The dynamics of heroin and illicit opioid use disorder, casual use, treatment, and recovery: A mathematical modeling analysis[J]. Mathematical Biosciences and Engineering, 2024, 21(2): 3165-3206. doi: 10.3934/mbe.2024141

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Abstract

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.

References

[1]	Center for Disease Control and Prevention (CDC), Drug Overdose Deaths in the U.S. Top 100,000 Annually, 2021, Access date: 3-May-2022. Available from: https://www.cdc.gov/nchs/pressroom/nchs_press_releases/2021/20211117.htm.
[2]	D. Ciccarone, The rise of illicit fentanyls, stimulants and the fourth wave of the opioid overdose crisis, Curr. Opin. Psychiatry, 34 (2021), 344.
[3]	M. Heilig, J. MacKillop, D. Martinez, J. Rehm, L. Leggio, L. J. Vanderschuren, Addiction as a brain disease revised: why it still matters, and the need for consilience, Neuropsychopharmacology, 46 (2021), 1715–1723. https://doi.org/10.1038/s41386-020-00950-y doi: 10.1038/s41386-020-00950-y
[4]	S. Satel, S. O. Lilienfeld, Addiction and the brain-disease fallacy, Front. Psychiatry, 4 (2013), 141. https://doi.org/10.3389/fpsyt.2013.00141 doi: 10.3389/fpsyt.2013.00141
[5]	J. Upadhyay, N. Maleki, J. Potter, I. Elman, D. Rudrauf, J. Knudsen, et al., Alterations in brain structure and functional connectivity in prescription opioid-dependent patients, Brain, 133 (2010), 2098–2114. https://doi.org/10.1093/brain/awq138 doi: 10.1093/brain/awq138
[6]	B. L. Thompson, M. Oscar-Berman, G. B. Kaplan, Opioid-induced structural and functional plasticity of medium-spiny neurons in the nucleus accumbens, Neurosci. Biobehav. Rev., 120 (2021), 417–430. https://doi.org/10.1016/j.neubiorev.2020.10.015 doi: 10.1016/j.neubiorev.2020.10.015
[7]	H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599–653. https://doi.org/10.1137/S0036144500371907 doi: 10.1137/S0036144500371907
[8]	M. Martcheva, An Introduction to Mathematical Epidemiology, Springer, 61 (2015).
[9]	I. M. Wangari, L. Stone, Analysis of a heroin epidemic model with saturated treatment function, J. Appl. Math., 2017 (2017).
[10]	S. Cole, S. Wirkus, Modeling the dynamics of heroin and illicit opioid use disorder, treatment, and recovery, Bull. Math. Biol., 84 (2022), 1–49. https://doi.org/10.1007/s11538-022-01002-w doi: 10.1007/s11538-022-01002-w
[11]	N. A. Battista, L. B. Pearcy, W. C. Strickland, Modeling the prescription opioid epidemic, Bull. Math. Biol., 81 (2019), 2258–2289. https://doi.org/10.1007/s11538-019-00605-0 doi: 10.1007/s11538-019-00605-0
[12]	L. Böttcher, T. Chou, M. R. D'Orsogna, Forecasting drug overdose mortality by age in the united states at the national and county levels, medRxiv, 2023. https://doi.org/10.1101/2023.09.25.23296097
[13]	L. Böttcher, T. Chou, M. R. D'Orsogna, Modeling and forecasting age-specific drug overdose mortality in the united states, Eur. Phys. J. Spec. Top., 232 (2023), 1743–1752. https://doi.org/10.1140/epjs/s11734-023-00801-z doi: 10.1140/epjs/s11734-023-00801-z
[14]	G. Vanden Boss, American Psychological Association (APA), Dictionary of Psychology, Washington, 2006.
[15]	K. S. Kendler, H. Ohlsson, J. Sundquist, K. Sundquist, A contagion model for within-family transmission of drug abuse, Am. J. Psychiatry, 176 (2019), 239–248. https://doi.org/10.1176/appi.ajp.2018.18060637 doi: 10.1176/appi.ajp.2018.18060637
[16]	C. Morris, The Social Contagion Effect of Addiction, on Ipsos News & Events: News, Access date: November 1, 2023. Available from: https://www.ipsos.com/en-us/social-contagion-addiction.
[17]	National Institute on Drug Abuse, Notice of Special Interest (NOSI): Modeling Social Contagion of Substance Use Epidemics, 2020. Available from: https://grants.nih.gov/grants/guide/notice-files/NOT-DA-20-009.html.
[18]	American Psychiatric Association, Diagnostic and Statistical Manual of Mental Disorders: DSM-5, American Psychiatric Association Washington, DC, 5 (2013).
[19]	Center for Behavioral Health Statistics and Quality, Results from the 2019 National Survey on Drug Use and Health: Detailed tables. Rockville, MD: Substance Abuse and Mental Health Services Administration, 2020. Available from: https://datafiles.samhsa.gov/.
[20]	L. Perko, Differential Equations and Dynamical Systems, Springer Science & Business Media, 7 (2013).
[21]	H. R. Thieme, Math. Popul. Biol., Princeton University Press, 2018.
[22]	Centers for Disease Control and Prevention, National Center for Health Statistics, Multiple Cause of Death 1999–2019 on CDC WONDER Online Database, Released 12/2020, 2020. Available from: https://www.drugabuse.gov/drug-topics/trends-statistics/overdose-death-rates.
[23]	United Nations, Department of Economic and Social Affairs, Population Division, World Population Prospects, Online Edition. Rev. 1., 2019. Available from: https://www.worldometers.info/world-population/us-population/.
[24]	Center for Behavioral Health Statistics and Quality, Results from the 2017 National Survey on Drug Use and Health: Detailed Tables. Rockville, MD: Substance Abuse and Mental Health Services Administration, 2018. Available from: https://datafiles.samhsa.gov/.
[25]	Center for Behavioral Health Statistics and Quality, Results from the 2015 National Survey on Drug Use and Health: Detailed Tables. Rockville, MD: Substance Abuse and Mental Health Services Administration, 2016. Available from: https://datafiles.samhsa.gov/.
[26]	Center for Behavioral Health Statistics and Quality, Results from the 2014 National Survey on Drug Use and Health: Detailed Tables. Rockville, MD: Substance Abuse and Mental Health Services Administration, 2015. Available from: https://datafiles.samhsa.gov/.
[27]	Center for Behavioral Health Statistics and Quality, Results from the 2013 National Survey on Drug Use and Health: Detailed Tables. Rockville, MD: Substance Abuse and Mental Health Services Administration, 2014. Available from: https://www.samhsa.gov/data/population-data-nsduh.
[28]	R. N. Lipari, A. Hughes, Trends in Heroin Use in the United States: 2002 to 2013, 2015. Available from: https://www.samhsa.gov/data/sites/default/files/report_1943/ShortReport-1943.html.
[29]	Center for Behavioral Health Statistics and Quality, Results from the 2012 National Survey on Drug Use and Health: Summary of National Findings (HHS Publication No. SMA 13-4795, NSDUH Series H-46). Rockville, MD: Substance Abuse and Mental Health Services Administration, 2013. Available from: https://www.samhsa.gov/data/population-data-nsduh.
[30]	Substance Abuse and Mental Health Services Administration, Results from the 2010 National Survey on Drug Use and Health: Summary of National Findings. Rockville, MD: Substance Abuse and Mental Health Services Administration, 2011. Available from: https://datafiles.samhsa.gov/.
[31]	Substance Abuse and Mental Health Services Administration, Results from the 2009 National Survey on Drug Use and Health: Summary of National Findings. Rockville, MD: Substance Abuse and Mental Health Services Administration, 2010. Available from: https://datafiles.samhsa.gov/.
[32]	Substance Abuse and Mental Health Services Administration, Results from the 2007 National Survey on Drug Use and Health: National Findings (Office of Applied Studies, NSDUH Series H-34, DHHS Publication No. SMA 08-4343). Rockville, MD., 2008. Available from: https://datafiles.samhsa.gov/.
[33]	Substance Abuse and Mental Health Services Administration, Results from the 2005 National Survey on Drug Use and Health: National Findings (Office of Applied Studies, NSDUH Series H-30, DHHS Publication No. SMA 06-4194). Rockville, MD., 2006. Available from: https://datafiles.samhsa.gov/.
[34]	NIDA, How Long Does Drug Addiction Treatment Usually Last?, 2020, Access date: 20-May-2022. Available from: https://nida.nih.gov/publications/principles-drug-addiction-treatment-research-based-guide-third-edition/frequently-asked-questions/how-long-does-drug-addiction-treatment-usually-last.
[35]	R. D. Weiss, V. Rao, The prescription opioid addiction treatment study: what have we learned, Drug Alcohol Depend., 173 (2017), S48–S54. https://doi.org/10.1016/j.drugalcdep.2016.12.001 doi: 10.1016/j.drugalcdep.2016.12.001
[36]	G. L. Bailey, D. S. Herman, M. D. Stein, Perceived relapse risk and desire for medication assisted treatment among persons seeking inpatient opiate detoxification, J. Subst. Abuse Treat., 45 (2013), 302–305. https://doi.org/10.1016/j.jsat.2013.04.002 doi: 10.1016/j.jsat.2013.04.002
[37]	B. P. Smyth, J. Barry, E. Keenan, K. Ducray, Lapse and relapse following inpatient treatment of opiate dependence, Ir. Med. J., 103 (2010), 176–179.
[38]	M. Gossop, L. Green, G. Phillips, B. Bradley, Lapse, relapse and survival among opiate addicts after treatment: A prospective follow-up study, Br. J. Psychiatry, 154 (1989), 348–353. https://doi.org/10.1192/bjp.154.3.348 doi: 10.1192/bjp.154.3.348
[39]	NIDA, National Drug Overdose (OD) Deaths, 1999–2020, 2022, Access date: 22-May-2022. Available from: https://nida.nih.gov/publications/drugs-brains-behavior-science-addiction/treatment-recovery.
[40]	Northern Illinois Recovery, Treatment and Recovery, 2021, Access date: 20-May-2022. Available from: https://www.northernillinoisrecovery.com/heroin-addiction-recovery-rate/.
[41]	NIDA, Treatment and Recovery, 2022, Access date: 20-May-2022. Available from: https://nida.nih.gov/publications/drugs-brains-behavior-science-addiction/treatment-recovery.
[42]	C. D. C. Wonder, National Drug Overdose (OD) Deaths, 1999–2020, 2020, Access date: 18-May-2022. Available from: https://www.google.com/search?client = safari & rls = en & q = NIDA+number+of+national+drug+overdose+deaths+involving+select+prescription+and+illicit+drugs+NIH+CDC+wonder & ie = UTF-8 & oe = UTF-8.
[43]	P. Van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. https://doi.org/10.1016/S0025-5564(02)00108-6 doi: 10.1016/S0025-5564(02)00108-6
[44]	P. Van den Driessche, Reproduction numbers of infectious disease models, Infect. Dis. Modell., 2 (2017), 288–303. https://doi.org/10.1016/j.idm.2017.06.002 doi: 10.1016/j.idm.2017.06.002
[45]	G. Albi, G. Bertaglia, W. Boscheri, G. Dimarco, L. Pareschi, G. Toscani, et al., Kinetic modelling of epidemic dynamics: social contacts, control with uncertain data, and multiscale spatial dynamics, in Predicting Pandemics in a Globally Connected World, Volume 1: Toward a Multiscale, Multidisciplinary Framework through Modeling and Simulation, Springer, (2022), 43–108.
[46]	S. Marino, I. B. Hogue, C. J. Ray, D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theor. Biol., 254 (2008), 178–196. https://doi.org/10.1016/j.jtbi.2008.04.011 doi: 10.1016/j.jtbi.2008.04.011

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