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For office use only Team Control Number For office use only
T1 1901679 F1
T2 F2
T3 Problem Chosen F3
T4 C F4
2019
MCM/ICM
Summary Sheet
After exploring the data provided to our team describing drug use and socioeconomic factors between 2010 and
2016 inclusive, we sorted the 69 types of opioid substances into four drug categories based on their synthesis and
availability. Plotting use rates of each category of drugs over time revealed that use of mild painkillers and natural
alkaloids have stayed relatively constant over time, semi-synthetic drugs have declined slightly, and synthetic drugs
such as fentanyl and heroin have increased dramatically. These findings align with reports from the CDC. We also
selected 54 out of 149 socioeconomic variables based on their variance inflation factor score (a common measure
of multicollinearity) as well as their relevance based on the public health literature. To model the spread of the
opioid crisis across Kentucky, Ohio, Pennsylvania, West Virginia, and Virginia, we took a dual-pronged approach,
developing two completely different models and then comparing them at the end.
Our first model is founded on common modeling approaches in epidemiology: SIR/SIS models and stochastic
simulation. We designed an algorithm from scratch which simulates a random walk between six discrete classes,
each of which represent a different stage of the opioid crisis using thresholds for opioid abuse prevalence and rate of
change. We penalize transitions between certain classes differentially based on realistic expectations. Optimization
of parameters and coefficients for the model was guided by an error function which we also designed from scratch, and
was inspired by the global spatial autocorrelation statistic Moran’s I. Testing our model via both error calculation
and visual mapping illustrated high accuracy over many hundreds of trials. However, this model did not provide
much insight into the influence of socioeconomic factors on opioid abuse rates, because incorporating socioeconomic
factors did not significantly change the model results.
Our second model made up for this deficiency in socioeconomic factor analysis. By running a collection of spatial
regression models on our final collection of socioeconomic predictors (including total drug use rate), we explored
characterising the spatial patterning of the opioid crisis as the result of a spillover effect, as the result of spatially-
correlated risk factors, and as a combination of the two, using spatial lag, spatial error, and spatial Durbin models
respectively. While all models confirmed significant spatial signals, the spatial Durbin model always performed the
best. We also calculated the direct, indirect, and total impacts of each predictor variable on opioid abuse rate. Far
and away the most important variable in all models was the total drug use rate in each county. The average result
(across all seven years) was that, all else equal, a unit increase in total illicit drug use rate would raise the opioid
abuse rate by 52%. This is quite realistic given a CDC statistic that in 2014, 61% of drug overdose deaths involved
some type of opioid. By contrast, an ordinary linear regression reported only a 37% increase in opioid abuse rate per
unit increase in total drug use rate. Statistical measures such as the Akaike Information Criterion and Likelihood
Ratio Test verify the superiority of our spatial models.
To predict possible locations of origin of the opioid epidemic in each of the five states, we ran a Monte Carlo
simulation of our random walk model from 2000-2010. We map out these counties and discuss their arrangement in
the context of our other findings. The random walk finds that the opioid crises most likely starts in Montegomery,
KY which aligns with our research that opioid abuse is more prevalent in rural communities than urban [10].
To forecast spread of the opioid crisis from 2017-2020, we used both our random walk and spatial regression
models. The two models display surprisingly minimal deviance from one another, especially in 2019 and 2020. The
random walk predicts that the number of counties above the illicit opioid use threshold will go down naturally
within the next 7 years which also aligns with the idea that the opioid epidemic follows the spillover effect seen in
epidemiology.
Due to the assumption that the SES indicators will change linearly, the second model’s error will significantly
increase after about 4-5 years. The random walk on the other hand, operates on a healthy tension between wanting
to cluster together and randomly assigning classes. Near the initial date, it clusters more but the randomness starts
to compound rather quickly. For this reason, the random walk has lowest errors near the 4-7 year mark. This means
that the best strategy to predict the near future would be the spatial regression and the random walk for the 4-7
year range. Predicting anything beyond this point will have high error. Afterwards, we provide the the suggestion
to the government that reducing general drug usage will help decrease illicit opioid usage.
Random Walks and Rehab:
Analyzing the Spread of the Opioid Crisis
Control #1901679
January 29, 2019
2
CONTENTS Control #1901679
Contents
1 Introduction and Problem Statement 4
2 Background 4
2.1 A Brief Timeline of the Epidemic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 The Science of Addiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.1 Risk Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Common Epidemiological Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3.1 Compartmental Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3.2 Stochastic Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3.3 The Greenwood Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Foundations of the Models 6
3.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.3 Overarching Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.4 Exploring the Drug Report Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 Part I: Characteristics of the Epidemic 8
4.0.1 Discrete Classes for Prevalence and Rate of Change . . . . . . . . . . . . . . . . . . . . . . . . 8
4.1 How the Model Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.1.1 Evaluation of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2 Model Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5 Part II: Socioeconomic Factors 12
5.1 In the Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.2 American Community Survey Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.3 Model Optimization with Socioecononomic Status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.4 Error and Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.5 Possible Origin Locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
6 A Second Modeling Approach: Spatial Regression 15
6.1 Model Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6.2 Statistical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6.4 Future Trends and Comparison to First Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.4.1 Drug Identification Thresholds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.4.2 Concerns for the U.S. Government . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
7 Strategies to Combat the Opioid Epidemic 20
7.1 Reducing Illicit Drug Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
7.2 Direct Opioid Interventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
8 Limitations and Sources of Error 21
8.1 Overall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
8.2 Model I: The Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
8.3 Model II: Spatial Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
9 Conclusion 22
9.1 Future Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
10 Memo to the Chief Administrator 24
A Appendix 28
A.1 Predictor Variables for the 2014 Spatial Lag Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1. INTRODUCTION AND PROBLEM STATEMENT Control #1901679
1 Introduction and Problem Statement
The deadly consequences of abusing prescription narcotic pain relief medications, heroin, and synthetic opioids
are affecting people in all 50 states and across all socioeconomic classes. The opioid epidemic claims the lives of 115
people in the United States every day [46]. Through healthcare costs, rehabilitation treatment, lost productivity,
and criminal justice involvement, the opioid crisis is costing the US federal government an estimated 78.5 billion
dollars each year [47]. Our team was presented with the following modeling tasks: firstly, to characterize the spread
of the opioid epidemic throughout Kentucky, Ohio, Pennsylvania, Virginia, and West Virginia and analyze resulting
patterns; secondly, to incorporate socioeconomic factors in our model and analyze the associations, if any, between
them and opioid abuse rates; finally, to use results from these models to recommend public policy strategies to
combat the opioid epidemic. To perform these analyses, we were limited to sets of data covering the years 2010-2016
from the American Community Survey (ACS), which provided our socioeconomic indicators, and from the National
Forensic Laboratory Information System (NFLIS) on illicit drug use. All data was provided at the county level in
the aforementioned five states.
To characterize the spread of the opioid crisis throughout these five states, we developed two models. The first,
of our own design, simulates a random walk through stable, endemic, and epidemic stages. The second is a more
standard collection of spatial regression models. In this paper, after describing these two modeling approaches and
their results, we report forecasts from both models on the future spread of the opioid epidemic, and compare the
results. This dual-pronged approach provides diverse insights into the nature of the opioid crisis, and helps us to
identify strategies for government intervention.
2 Etiology of the Opioid Crisis
2.1 A Brief Timeline of the Epidemic
Previously restricted to use for chronic pain due to cancer, opioids emerged into the non-cancer pain market
following a series of studies published in the early 1990s indicating that pain was inadequately treated, with up
to 42% of sample populations reporting mismanaged pain. Around this same time, pharmaceutical companies and
medical societies began to relax their informal ban on opioids as they were reassured by somewhat erroneous studies
and articles that opioids were not at all addictive. Compounding the problem was the release in 1996 of OxyContin
by Purdue Pharma, along with a targeted marketing campaign that encouraged ongoing treatment for chronic pain
and implied minimal side effects for the semi-synthetic drug [24].
Thus the first great wave of opioid prescriptions began. The number of prescription opioids in the hands of
consumers continued to rise throughout the 1990s by as much as 2-3 million per year and into the early 2000s as the
government (Joint Commission) released new standards surrounding the monitoring and treatment of pain.
Experts believe the second wave of the epidemic occurred around 2010 as the true nature of the epidemic began
to surface. In an effort to curb addiction, government organizations placed limits on opioid prescriptions. Many of
those who were already addicted turned to heroin instead, as it was cheaper and could provide a more intense high
than standard prescription medication. However, due to illicit manufacturing, heroin was often impure or mixed with
other drugs without consumers’ knowledge, leading to increased deaths related to drug abuse in the United States
[48].
The third and hopefully final wave of the epidemic came in 2013 with the rise of illegally manufactured
synthetic alternatives to prescription opioids, such as fentanyl. However, efforts to combat the epidemic also inten-
sified. The US Centers for Disease Control declared the opioid problem an official epidemic in 2011 [49] and the US
federal government followed suit in 2017. Legislation calling the epidemic a “public health emergency,” freed the
Department of Health and Human Services to channel additional resources into treatment, distribution of overdose
antidotes, and prevention research [59].
2.2 The Science of Addiction
Addiction is a chronic biophysical disease with genetic predispositions and long-term consequences. As a chronic
disease, there is no cure and usually treatment must be continued throughout the life of the recovering addict. Note
that addiction to and physical dependence upon a substance are quite different. Physical dependence is a common
phenomenon and involves the biology of the body rather than the behavior of the individual. Addiction, on the other
2. BACKGROUND Control #1901679
hand, can interfere with a person’s self-control and challenge their ability to resist intense urges to use drugs [45].
This is why it is so common for addicted individuals to relapse, even years after “getting clean.”
Addiction begins simply: a person participates in a behavior or has an experience that triggers the production of
dopamine in their brain, making the individual feel happy. Usually these experiences are healthy and the release of
dopamine is intended to make the person repeat the behavior; natural causes of this include connecting with other
people in a community, sleeping, and eating. However, addictive substances like opioids and alcohol also activate
dopamine production, usually at an elevated level, causing an individual to feel elated. More often than not, the
individual will attempt to repeat this behavior to feel similar results [55].
Over time, the brain attempts to combat the addiction by reducing its number of dopamine receptors or producing
less dopamine. Thus a larger amount of the substance must be consumed to repeat the high; this is known as tolerance.
The decrease in dopamine receptors or in dopamine production can also lead to the individual finding little joy in
other activities they previously enjoyed. Craving elation, the person will turn again to the substance. Since dopamine
is also involved in the learning process, the brain will associate the drug with the high. Eventually, the desire for the
drug becomes more important than the high itself. And for some substances, such as cocaine and opioids, addicted
brains show fewer neurons in the frontal cortex, which means that cognitive reasoning and impulse control have been
impaired [55].
2.2.1 Risk Factors
Genetics and environmental factors both play large roles in determining whether a person will become addicted
to a substance. Some people are more susceptible to neurobiological changes that allow an addiction to take hold,
while others might have a genetic predisposition to addictive behavior. Additionally, those with poor social support
networks, individuals who have experienced trauma or abuse (especially at a young age), and people with some
mental illnesses are more likely to become addicted to a substance than members of the general US population.
Currently, age appears to have the largest impact on susceptibility to addiction. The younger a person is, the
more vulnerable she is to addiction. In fact, a federal study found that 74% of individuals admitted to treatment
programs between 18 and 30 years of age had started abusing drugs before the age of seventeen. However, this same
study discovered that the majority of people who were admitted for heroin and prescription painkiller addictions
started using drugs after the age of twenty-five [3].
This is only one of many ways in which opioid addiction is different from other kinds of drug addiction.
2.3 Common Epidemiological Models
We now switch gears to the mathematical modeling side of epidemiology. Several different types of models and
overarching principles in public health research helped to guide our approach to the given tasks.
2.3.1 Compartmental Models
A compartmental model is commonly used to simplify mathematical modeling of infectious diseases. Populations
are divided into compartments with assumptions about the nature of each compartment and the time rate of transfer
between them [27]. One such model is known as the SIR model, where the population is partitioned into three
groups: susceptible (S), infected (I), and removed (R). S’s are individuals who have not been infected, but who are
susceptible to infection; I’s are those who are infected and are capable of transmitting the disease; R’s are people
who can no longer contract the disease because they have recovered with immunity, been quarantined, or died [20].
In the simplest form of SIR model, individuals move directly from susceptible to infected to removed. Adaptations
of this model include the SIS, in which there is no removed category and individuals move from infected back to
susceptible, and the SIRS, in which immunity to the disease decays over time, eventually sending individuals in this
category back to the susceptible category.
While there is some contention that the spread of opioid abuse can be characterized by infectious disease diffusion,
dynamic compartmental models have already been applied to this topic. A 2018 study by researchers at Stanford
University used an SIR model to predict opioid overdose deaths as far into the future as 2026 and then to assess the
effects of different policies on the epidemic [2]. These investigators determined that reducing opioid prescriptions by
25% results in 2,500 fewer overdoses in the next ten years across the country. But this alone will not solve the problem,
as opioid abusers will turn to heroin or illicitly manufactured synthetics. “It’s like squeezing a balloon,” commented
addiction expert Keith Humphreys. “When you touch one aspect of the situation, an unexpected consequence often
pops up somewhere else.”
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