没有合适的资源?快使用搜索试试~ 我知道了~
2019美赛O奖论文-MCM2019C-1901213.pdf
1.该资源内容由用户上传,如若侵权请联系客服进行举报
2.虚拟产品一经售出概不退款(资源遇到问题,请及时私信上传者)
2.虚拟产品一经售出概不退款(资源遇到问题,请及时私信上传者)
版权申诉
0 下载量 51 浏览量
2024-03-17
21:36:09
上传
评论
收藏 4.25MB PDF 举报
温馨提示
试读
30页
美国大学生数学竞赛获奖论文,历届,单项文件,内容丰富,大学生数学,数学竞赛,参考资料,极具参考价值
资源推荐
资源详情
资源评论
For office use only
T1
________________
T2
________________
T3
________________
T4
________________
Team Control Number
1901213
Problem Chosen
C
For office use only
F1
________________
F2
________________
F3
________________
F4
________________
2019
MCM/ICM
Summary Sheet
Take Me Home:
Preventing Journeys Down the Opioid Addiction Road
In recent years, overdose has been the leading cause of accidental deaths in the United
States, and prescription opioids and heroin are among the heaviest offenders in that category.
While many people need opioids to manage their chronic and severe pain, a common
consequence of these treatments are abuse, addiction, and escalation to worse substances. A
variety of strategies exist to combat the spread of drugs, like education, rehabilitation, and
law enforcement. However, a more targeted strategy is necessary for opioids, given their new
ubiquity in American society.
We have developed a model that robustly and accurately predicts the spread of
opioids within and between the states of Ohio, Pennsylvania, Virginia, West Virginia, and
Kentucky. In doing so, we:
● Visualized the movement and spread of opioids within the region between 2010 and
2016 by treating opioid addiction as a disease that spreads deterministically between
neighbors and assuming that the spread of it can be modeled in a Markovian fashion.
This will allow us to find a transition matrix that tells the influence of each county on
one another. We also factored into this matrix the distances between counties.
● Next, we modeled the effect of socioeconomic factors throughout the data set, and
correlated these changes with how opioid use in that county grows or shrinks over
time.
● We then combined the models in two ways - in a linear and parallel fashion. We used
this to estimate epicenters from where the drug problem emanates.
● Finally, we ran multiple simulations and predicted the drug problem well into the
future to develop a number of strategies to tackle the epidemic from different
perspectives, selecting from the variables that contribute most heavily to the spread of
opioids.
Our model treats illicit opioid use as a disease that is spread more frequently when
more people in a given area have it. This allowed us to design it such that it can be
generalized to a larger region in the future. By visualizing this spread, we were able to
witness predicted opioid use spread through and along major roads over longer distances than
simple adjacency would predict. The counties connected in this way include both the
epicenters and the vulnerable ones.
To evaluate our external model, we gave it the first two years of drug report data from
all the counties, and then allowed it to propagate through the year 2016. Our predictions have
an error on the order of 10
-5
drug reports per capita. When evaluating our internal model, we
realized that while socioeconomic factors are highly correlated, they could not accurately
predict opioid abuse.
After this, we modified various initial conditions in our model such as socioeconomic
factors and influence of the epicenters. By doing so, we were able to find effective and highly
targeted strategies which will drastically reduce and reverse the prevalence of opioids in the
region provided.
Team #1901213 Page 1 of 29
Contents
1 Introduction 2
1.1 Problem Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Our Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Background 3
3 Data 4
3.1 Given data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2 Additional data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4 Assumptions 5
4.1 As a disease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.2 The Markovian assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.3 Proximity to neighboring counties . . . . . . . . . . . . . . . . . . . . . . . . 5
4.4 All opioids are the same . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.5 Data homogeneity across counties . . . . . . . . . . . . . . . . . . . . . . . . 6
5 Model 6
5.1 Defining health . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
5.2 Modeling external factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
5.2.1 Understanding x
t
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
5.2.2 Understanding A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
5.2.3 Understanding Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
5.2.4 Estimating Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
5.3 Modeling internal factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5.3.1 Linear regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5.3.2 Logistic regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5.4 Combining the two models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5.4.1 A convex combination . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.4.2 A two stage model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
6 Solutions 10
6.1 Estimating δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
6.2 Estimating the influence of external factors . . . . . . . . . . . . . . . . . . . 12
6.3 Estimating the influence of internal factors . . . . . . . . . . . . . . . . . . . 12
6.4 Building a convex combination of models . . . . . . . . . . . . . . . . . . . . 13
6.5 Building the two stage model . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6.6 Epicenters and vulnerabilities . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6.6.1 Identifying epicenters . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
6.6.2 Identifying vulnerable counties . . . . . . . . . . . . . . . . . . . . . . 15
6.6.3 Understanding these counties . . . . . . . . . . . . . . . . . . . . . . 15
7 Sensitivity Analysis 16
Team #1901213 Page 2 of 29
7.1 Tweaking the external factors . . . . . . . . . . . . . . . . . . . . . . . . . . 16
7.2 Tweaking the internal factors . . . . . . . . . . . . . . . . . . . . . . . . . . 16
8 Strengths and Weaknesses 17
8.1 Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
8.2 Weaknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
9 Policy Strategies 18
10 Conclusion 19
11 Policy Letter Regarding Opioid Use in the United States 21
A Appendix: Epicenters and vulnerable counties 25
B Appendix: Top 30 highly correlated socioeconomic factors 28
C Appendix: Supplementary images 29
1 Introduction
In 2015, 33,091 people died of an overdose on heroin and other opioids, representing more
than 60 percent of the total overdose deaths in the United States that year [1]. Given that
overdose was also the leading cause of accidental deaths that year [1], it is clear that the
dispersion of opioids is a serious problem in this country. Opioids are a class of narcotic
painkillers, including the illicit drug heroin, that are derived from the poppy flower, or are
synthesized artificially and have a similar structure to other opioids. Some examples of opioid
prescription drugs are morphine, hyrocodone, oxycodone, dihydromorphone, and fentanyl.
While almost every drug in this class can be prescribed to manage chronic pain, it is clear
that the spread of opioid use has spread far beyond the reach of prescribed medicine and
presented itself as an epidemic. This plague, while deadly, is highly treatable with proper
targeting for law enforcement and strategies for potential victims. For this reason, it is of
the utmost importance that we understand and analyze the spread of opioids and figure out
how to reverse the trend. In this report, we do the same by building mathematically rigorous
models. In summary, we will take the data pertaining to counties of five different states -
Kentucky, West Virginia, Virginia, Ohio and Pennsylvania - and develop a predictive model
to analyze how drug abuse spreads and infects the counties. Then, we will attempt to find
solutions to the same problem using the model.
1.1 Problem Summary
•
Visualize drug report data for each county in the states of Ohio, Kentucky, Pennsylvania,
Virginia, and West Virginia for the years 2010-2016.
Team #1901213 Page 3 of 29
•
Develop a predictive model that anticipates the spread of opioid use over the given
time frame, and use it to find potential epicenters for the spread of opioids within those
states.
•
Analyze key socioeconomic factors to determine the relationship between these factors
and opioid use. Employ these relationships to find counties that are demographically
prone to abusing opioids.
•
Determine a course of action to prevent the spread of new addictions and make current
ones harder to maintain.
1.2 Our Model
•
Initially we treated opioid addiction as a disease that spreads deterministically between
neighbors, by assuming that the spread of it can be modeled in a Markovian fashion.
This will allow us to find a transition matrix that tells the influence of each county on
one another. We also factored into this matrix the distances between counties.
•
Next, we modeled the effect of socioeconomic factors throughout the data set, and
correlated these changes with how opioid use in that county grows or shrinks over time.
•
We then combined the models in two ways – in a linear and parallel fashion. We used
this to estimate epicenters from where the drug problem emanates.
•
Finally, we ran multiple simulations and predicted the drug problem well into the future
to develop a number of strategies to tackle the epidemic from different perspectives,
selecting from the variables that contribute most heavily to the spread of opioids.
In our model, we analyze the general effect of socioeconomic factors as well as the drug
problem of neighboring counties. Due to the general approach taken in model, the model will
very likely scale well to larger regions.
The rest of this report is organized as follows: Section 2 will provide background information
necessary for understanding the condition we are modeling. Section 3 will briefly discuss
the provided data for this analysis. In Section 4 we will make the necessary assumptions to
allow the situation to be modeled efficiently. Section 5 will encapsulate the details of our
models, both external and internal. In Section 6 we solve for the parameters of our model,
and discuss the results we get. In Section 7, we will modify key factors that appeared to
have great influence over the model, to determine its sensitivity to the initial conditions,
and give insight into which the most effective strategies for combating the drug problem. In
Section 8, we will evaluate the strengths and weaknesses of our model. In Section 9 will lay
out the strategies we have found will work best to reverse the drug crisis. Finally, we will
conclude the report in Section 10 by summarizing our key findings. Additionally, in Section
11, we have written a brief memo to the Drug Enforcement Administration suggesting policy
changes to combat the drug crisis.
2 Background
Before modeling the spread of opioids, it is important to understand the science behind
addiction. When an opioid is taken, it mimics natural neurotransmitters by locking onto and
Team #1901213 Page 4 of 29
preventing the re-uptake of these chemicals. This results in the brain being flooded with
dopamine, the chemical the brain normally releases as part of its reward system. This can
sometimes result in an unhealthy cycle of needing the drug that creates this feeling, and
taking the drug at higher and higher doses due to tolerance. When a person has taken this
cycle to the point where the drug becomes one of their basic needs, like food or water, and
they suffer extreme adverse effects in their life as a result, they have an addiction. Due to the
addiction potential of opioids and the frequency with which they are prescribed, the spread
of opioid use and abuse can be rapid.
In 2012, enough prescriptions for opioid medication were written for every American adult to
have their own bottle of pills [2]. It is important to understand how those around one will
factor into one’s likelihood to use opioids, due to how common opioids are in everyday life
now. Along with one’s surroundings, data supports that certain socioeconomic factors can be
correlated to an increase use in opioids [3]. Given these possibilities, it is paramount to find
if, and how much, these factors can predict future drug use, in order to stop the vicious cycle
of addiction for so many victims.
3 Data
In this section, we will give a brief overview of the data we used to develop the models
described in subsequent sections.
3.1 Given data
We were given 8 different data files. The first contained drug report counts from counties
of five states in USA (Kentucky, West Virginia, Virginia, Ohio, Pennsylvania). A drug is
reported in this case as part of an investigation by local police departments. The data provided
gives total drug reports for each county, and any counts of drug reports for prescription
opioids or heroin, courtesy of the National Forensics Laboratory Information System (NFLIS).
For our purposes, we used just the opioid and heroin counts. The other 7 data files contain
US Census data for these counties from years 2010-2016. These represent a common set of
about 134 socioeconomic factors.
While cleaning up the data, we realized that the number of counties differed in some years.
If a county was missing drug report data in the first file for any year, we set the drug reports
to zero. Further, we realized that Bedford City, VA was incorporated into Bedford County,
VA in 2013. So, we fixed this discrepancy by adding the data from Bedford City from years
2010-2012 into those of Bedford County. Also, since 2013, the US Census data divided a
few of the larger categories. So, during the cleanup process, we had to account for those
discrepancies manually.
3.2 Additional data
As we were not given any data about the distances between counties, but were allowed to get
such data from outside sources, we obtained the distance between counties, as measured in
剩余29页未读,继续阅读
资源评论
阿拉伯梳子
- 粉丝: 1560
- 资源: 5731
上传资源 快速赚钱
- 我的内容管理 展开
- 我的资源 快来上传第一个资源
- 我的收益 登录查看自己的收益
- 我的积分 登录查看自己的积分
- 我的C币 登录后查看C币余额
- 我的收藏
- 我的下载
- 下载帮助
安全验证
文档复制为VIP权益,开通VIP直接复制
信息提交成功