Mathematical Modeling of the Opioid Epidemic...Mathematical Modeling of the Opioid Epidemic Cullen...
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Mathematical Modeling of the Opioid Epidemic Cullen Asaro 1 , Jordan Briscoe 2 , Nina Dorfner 3 1 Center for Environmental Research and Education, 2 Dept of Mathematics and Computer Science, 3 Department of Engineering Faculty Mentor: Dr. Rachael Neilan, Department of Mathematics and Computer Science References Model Discussion 1. = โ โ SA- + + โ + โ + โ + โ + + + + 2. = โ ++ 3. = + + โ โ + + + + + + + + 4. = โ โ + + 5. = โ โ + + 6. = โ โ + + 7. = โ โ + + 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), 2258โ 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 () (Susceptible): Individuals who are not using opioids or in treatment for addiction () (Prescribed): Individuals who have been prescribed opioids () (Addicted): Individuals who are addicted to opioids () (Recovery): Individuals who are in one of the four sub classes below receiving some form of addiction treatment โ Assisted with Pharmaceutics โ Inpatient โ Outpatient โ Residential Housing Abstract Background Prescription Opioids Prescription Painkiller Sales and Deaths 1999-2013 Community Partner Allegheny County Department of Human Services provided the number of opioid overdose deaths and the frequency in which individuals used various treatment options in Allegheny County over the past 5 years. 4x the number of overdose deaths in 2018 compared to end of the 20 th 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 โขMedically Assisted Treatment (MAT) โขUses pharmaceutics to relieve withdrawal symptoms and cravings โขEX: Methadone, Naltrexone Inpatient โข Short-term stay in facility (30-60 days) โข Includes counseling โข Professionals monitor withdrawal symptoms โข Sometimes includes medical detox Outpatient โข Patients reside in own home with 3-6 hrs. per day at treatment center โข Includes family services, counseling, and holistic options โข Individualized plan that lasts 6-12 months Residential Housing โขLong-term stay in a non-hospital setting โขMore freedom than inpatient but still has structured services โขEX: Halfway Houses Which treatment options are most impactful in reducing overdose deaths? We developed a mathematical model describing opioid addiction and recovery in Allegheny County. Our model tracks the number of susceptible individuals, prescribed opioid users, addicted users, and individuals receiving addiction treatment over time. Individuals who enter addiction treatment are placed in one of four recovery classes. Rates of movement into and out of each of the four recovery classes are estimated from data provided by the Allegheny County Department of Human Services. Model simulations demonstrate the impact of diverse treatment options and the opioid prescription rate on overdose fatalities. โOpiatesโ that are naturally found in the opium plant EX: Oxycodone, Morphine, Codeine Manufactured to mimic effects of the opium plant EX: Fentanyl, Methadone Synthetic Results Statistical software R was used to solve equations 1 โ (7) with the following initial conditions: 0 = 0.90, 0 = 0.10, 0 = O 0 = P 0 = Q 0 = R 0 =0 Figure 2. Model solutions show the values of , , , O , P , Q , and R over time. Model simulations were performed to determine the cumulative number of opioid overdose deaths for different prescription rates. Results show that even a large reduction in prescription rates still yields an increasing number of overdose deaths. Figure 3. Impact of decreasing the opioid prescription rate on overdose deaths over a 15-year period. Impact of Different Treatment Options on Deaths Model simulations were performed for different combinations of treatment options. The graph below indicates that the R 3 treatment class (Outpatient) is critical in reducing overdose deaths. Figure 4. Bar plot shows the average annual overdose death rate in Allegheny county predicted by the model for 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 Information Cullen Asaro: [email protected] Jordan Briscoe: [email protected] Nina Dorfner: [email protected] Special thank you to Dr. Benedict Kolber of the Department of Biological Sciences and Peter Jhon from the Allegheny County DHS for sharing their expertise on the opioid epidemic and assisting with our research. The results from our model can be implemented to devise an optimal treatment plan in Allegheny County: โข Reductions in prescription rate correspond to reductions in overdose deaths. However, the epidemic 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. โข Outpatient treatment services are an essential component in any successful plan to minimize the impact of the opioid epidemic. โข These results could be attributed to the willingness of addicts to enroll in outpatient programs as opposed to residential or in-patient programs. Future Work Future studies may include model parameters that vary by location within Allegheny County. Figure 5. Map of opioid overdose deaths per township illustrates geographical differences in the epidemic. Description Treatment entry rate, 0.034685 0.200436 1.302917 0.057650 Successful treatment rate, 0.816445 0.188742 0.562118 0.713349 Relapse rate, 0.583396 1.760260 0.843970 0.673344 Death rate, โ 0.003759 0.008344 0.004232 0.006610 Table 1. Parameter values for treatment classes.
Transcript of Mathematical Modeling of the Opioid Epidemic...Mathematical Modeling of the Opioid Epidemic Cullen...
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. ๐ ๐บ๐๐ญ = ๐ฎ๐ท โ ๐ถ๐บ โ ๐ท๐จSA- ๐ท๐๐บ๐ท + ๐๐ท+๐๐โ๐๐ +๐๐โ๐น๐ +๐๐โ๐น๐ +๐๐โ๐น๐ +๐น๐๐น๐ +๐น๐๐น๐ +๐น๐๐น๐ +๐น๐๐น๐
2. ๐ ๐ท๐ ๐ = ๐ถ๐บ โ ๐ท ๐บ + ๐ธ + ๐
3. ๐ ๐จ๐ ๐ = ๐ธ๐ท +๐ท๐๐บ๐จ +๐ท๐๐บ๐ท โ ๐จ ๐โ + ๐ป๐ + ๐ป๐ + ๐ป๐ + ๐ป๐ +๐๐๐น๐ +๐๐๐น๐ +๐๐๐น๐ + ๐๐๐น๐
4. ๐ ๐น๐๐ ๐ = ๐ป๐๐จ โ ๐น๐ ๐๐โ +๐๐ +๐น๐
5. ๐ ๐น๐๐ ๐ = ๐ป๐๐จ โ ๐น๐ ๐๐
โ +๐๐ +๐น๐
6. ๐ ๐น๐๐ ๐ = ๐ป๐๐จ โ ๐น๐ ๐๐
โ +๐๐ +๐น๐
7. ๐ ๐น๐๐ ๐ = ๐ป๐๐จ โ ๐น๐ ๐๐
โ +๐๐ +๐น๐
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), 2258โ2289. 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
๐บ(๐) (Susceptible): Individuals who are not using opioids or in treatment for addiction๐ท(๐) (Prescribed): Individuals who have been prescribed opioids๐จ(๐) (Addicted): Individuals who are addicted to opioids๐น(๐) (Recovery): Individuals who are in one of the four sub classes below receiving some form of addiction treatment๐น๐ ๐ โAssisted with Pharmaceutics๐น๐ ๐ โInpatient ๐น๐ ๐ โ Outpatient๐น๐ ๐ โ 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
โขMedically Assisted Treatment (MAT)โขUses pharmaceutics to relieve withdrawal symptoms and cravingsโขEX: Methadone,Naltrexone
Inpatientโข Short-term stay in
facility (30-60 days)โข Includes counselingโข Professionals monitor
withdrawal symptomsโข Sometimes includes
medical detox
Outpatientโข Patients reside in own
home with 3-6 hrs. perday at treatment center
โข Includes family services, counseling, and holistic options
โข Individualized plan that lasts 6-12 months
Residential Housing
โขLong-term stay in a non-hospital settingโขMore freedom than inpatient but still has structured services โขEX: 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.
โOpiatesโ 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 โ (7)with the following initial conditions:
๐ 0 = 0.90, ๐ 0 = 0.10,๐ด 0 = ๐ O 0 = ๐ P 0 = ๐ Q 0 = ๐ R 0 = 0
Figure 2. Model solutions show the values of ๐,๐,๐ด, ๐ O,๐ P, ๐ Q, and ๐ R 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:โข 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.
โข Outpatient treatment services are an essentialcomponent in any successful plan to minimize theimpact of the opioid epidemic.
โข 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 ๐น๐ ๐น๐ ๐น๐ ๐น๐
Treatment entryrate, ๐ 0.034685 0.200436 1.302917 0.057650
Successful treatment rate, ๐ฟ 0.816445 0.188742 0.562118 0.713349
Relapse rate, ๐ 0.583396 1.760260 0.843970 0.673344Death rate, ๐โ 0.003759 0.008344 0.004232 0.006610
Table 1. Parameter values for treatment classes.
