Mathematical Modeling of the Opioid EpidemicCullen Asaro1, Jordan Briscoe2, Nina Dorfner31Center for Environmental Research and Education, 2Dept of Mathematics and Computer Science, 3Department of Engineering
Faculty Mentor: Dr. Rachael Neilan, Department of Mathematics and Computer Science
References
Model Discussion
1. 𝒅𝑺𝐝𝐭 = 𝓮𝑷 − 𝜶𝑺 − 𝜷𝑨SA- 𝜷𝝆𝑺𝑷 + 𝝁𝑷+𝝁𝟏∗𝐑𝟏 +𝝁𝟐∗𝑹𝟐 +𝝁𝟑∗𝑹𝟑 +𝝁𝟒∗𝑹𝟒 +𝜹𝟏𝑹𝟏 +𝜹𝟐𝑹𝟐 +𝜹𝟑𝑹𝟑 +𝜹𝟒𝑹𝟒
2. 𝒅𝑷𝒅𝒕 = 𝜶𝑺 − 𝑷 𝜺 + 𝜸 + 𝝁
3. 𝒅𝑨𝒅𝒕 = 𝜸𝑷 +𝜷𝒂𝑺𝑨 +𝜷𝝆𝑺𝑷 − 𝑨 𝝁∗ + 𝜻𝟏 + 𝜻𝟐 + 𝜻𝟑 + 𝜻𝟒 +𝝈𝟏𝑹𝟏 +𝝈𝟐𝑹𝟐 +𝝈𝟑𝑹𝟑 + 𝝈𝟒𝑹𝟒
4. 𝒅𝑹𝟏𝒅𝒕 = 𝜻𝟏𝑨 − 𝑹𝟏 𝝁𝟏∗ +𝝈𝟏 +𝜹𝟏
5. 𝒅𝑹𝟐𝒅𝒕 = 𝜻𝟐𝑨 − 𝑹𝟐 𝝁𝟐
∗ +𝝈𝟐 +𝜹𝟐
6. 𝒅𝑹𝟑𝒅𝒕 = 𝜻𝟑𝑨 − 𝑹𝟑 𝝁𝟑
∗ +𝝈𝟑 +𝜹𝟑
7. 𝒅𝑹𝟒𝒅𝒕 = 𝜻𝟒𝑨 − 𝑹𝟒 𝝁𝟒
∗ +𝝈𝟒 +𝜹𝟒
Figure 1. Visualization of the mathematical model
Model Diagram
Differential Equations
Battista, N. A., Pearcy, L. B., & Strickland, W. C. (2019). Modeling the Prescription Opioid Epidemic. Bulletin of Mathematical Biology, 81(7), 2258–2289. doi: 10.1007/s11538-019-00605-0
Long-Term Follow-Up of Medication-Assisted Treatment for Addiction to Pain Relievers Yields "Cause for Optimism". (2015, November 30). Retrieved March 23, 2020, from https://archives.drugabuse.gov/news-events/nida-notes/2015/11/long-term-follow-up-medication-assisted-treatment-addiction-to-pain-relievers-yields-cause-optimism
Drug Rehab Success Rates. (2020, February 6). Retrieved March 23, 2020, from https://www.therecoveryvillage.com/treatment-program/related/drug-rehab-success-rates/#gref
𝑺(𝒕) (Susceptible): Individuals who are not using opioids or in treatment for addiction𝑷(𝒕) (Prescribed): Individuals who have been prescribed opioids𝑨(𝒕) (Addicted): Individuals who are addicted to opioids𝑹(𝒕) (Recovery): Individuals who are in one of the four sub classes below receiving some form of addiction treatment𝑹𝟏 𝒕 −Assisted with Pharmaceutics𝑹𝟐 𝒕 −Inpatient 𝑹𝟑 𝒕 − Outpatient𝑹𝟒 𝒕 − Residential Housing
Abstract
Background
Prescription OpioidsPrescriptionPainkillerSalesandDeaths1999-2013
Community PartnerAllegheny County Department of Human Servicesprovided the number of opioid overdose deaths and thefrequency in which individuals used various treatmentoptions in Allegheny County over the past 5 years.
4x the number of overdose deaths in 2018 compared to end of the 20th century
70% of overdose deaths in 2018 involved an opioid
128 people die everyday from an overdose
Quick Facts: The Deadly Spread of the Opioid Epidemic
Addiction Treatment
Prescription VS
Prescription opioids serve as the gateway for 4 out of every 5
people suffering from addiction.
Can we reduce opioid deaths by
writing fewer opioid prescriptions?
Assisted with Pharmaceutics
•Medically Assisted Treatment (MAT)•Uses pharmaceutics to relieve withdrawal symptoms and cravings•EX: Methadone,Naltrexone
Inpatient• Short-term stay in
facility (30-60 days)• Includes counseling• Professionals monitor
withdrawal symptoms• Sometimes includes
medical detox
Outpatient• Patients reside in own
home with 3-6 hrs. perday at treatment center
• Includes family services, counseling, and holistic options
• Individualized plan that lasts 6-12 months
Residential Housing
•Long-term stay in a non-hospital setting•More freedom than inpatient but still has structured services •EX: Halfway Houses
Which treatment
options are most impactful
in reducing overdose deaths?
We developed a mathematical model describing opioidaddiction and recovery in Allegheny County. Our modeltracks the number of susceptible individuals, prescribedopioid users, addicted users, and individuals receivingaddiction treatment over time. Individuals who enteraddiction treatment are placed in one of four recoveryclasses. Rates of movement into and out of each of thefour recovery classes are estimated from data providedby the Allegheny County Department of HumanServices. Model simulations demonstrate the impact ofdiverse treatment options and the opioid prescription rateon overdose fatalities.
“Opiates” that are naturally found in the opium plantEX: Oxycodone,
Morphine, Codeine
Manufactured to mimic effects of the opium plantEX: Fentanyl,
Methadone
Synthetic
ResultsStatistical software R was used to solve equations 1 − (7)with the following initial conditions:
𝑆 0 = 0.90, 𝑃 0 = 0.10,𝐴 0 = 𝑅O 0 = 𝑅P 0 = 𝑅Q 0 = 𝑅R 0 = 0
Figure 2. Model solutions show the values of 𝑆,𝑃,𝐴, 𝑅O,𝑅P, 𝑅Q, and 𝑅R over time.
Model simulations were performed to determine thecumulative number of opioid overdose deaths fordifferent prescription rates. Results show that even alarge reduction in prescription rates still yields anincreasing number of overdose deaths.
Figure 3. Impact of decreasing the opioid prescriptionrate on overdose deaths over a 15-year period.
Impact of Different Treatment Options on Deaths
Model simulations were performed for differentcombinations of treatment options. The graph belowindicates that the R3 treatment class (Outpatient) iscritical in reducing overdose deaths.
Figure 4. Bar plot shows the average annual overdosedeath rate in Allegheny county predicted by the modelfor different combinations of treatment options.
Long-Term Impact of Decreasing Opioid Prescriptions
The following variables track the proportion of the population in each of the seven classes over time.
Variables and Parameters
The model is written as seven differential equations:
Acknowledgements
Contact InformationCullen Asaro: [email protected]
Jordan Briscoe: [email protected] Dorfner: [email protected]
Special thank you to Dr. Benedict Kolber of the Department of BiologicalSciences and Peter Jhon from the Allegheny County DHS for sharing theirexpertise on the opioid epidemic and assisting with our research.
The results from our model can be implemented todevise an optimal treatment plan in Allegheny County:• Reductions in prescription rate correspond to
reductions in overdose deaths. However, theepidemic still exists.
Opioid deaths will rise over the next 15 years
despite a decrease in the prescription rate.
Therefore, reducing the prescription rate is not a satisfactory prevention
measure alone.
• Outpatient treatment services are an essentialcomponent in any successful plan to minimize theimpact of the opioid epidemic.
• These results could be attributed to the willingnessof addicts to enroll in outpatient programs asopposed to residential or in-patient programs.
Future Work
Future studies may include model parameters that varyby location within Allegheny County.
Figure 5. Map of opioid overdose deaths per township illustrates geographical differences in the epidemic.
Description 𝑹𝟏 𝑹𝟐 𝑹𝟑 𝑹𝟒
Treatment entryrate, 𝜁 0.034685 0.200436 1.302917 0.057650
Successful treatment rate, 𝛿 0.816445 0.188742 0.562118 0.713349
Relapse rate, 𝜎 0.583396 1.760260 0.843970 0.673344Death rate, 𝜇∗ 0.003759 0.008344 0.004232 0.006610
Table 1. Parameter values for treatment classes.
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