Introduction

In recent years, prescription drug abuse has hit record levels. The CDC reports that the number of opioid overdose deaths has quadrupled since 1999, with half a million deaths from 2000 to 2015. In the same time period, drug prescription rates have also quadrupled (\citet{syfxu}). This so called opioid epidemic has been featured in medical journals, but also in the news and political campaigns because of the startling rates of death and the socioeconomic changes that go alongside it.
This problem first arose in the mid 1990's, when a number of factors led to greater opioid use. There was growing concern that the medical field was not treating pain properly, and a small number of physicians lobbied for greater regulation around treating reported pain (Battista et al. (2017)). Around the same time, there were a handful of articles published in medical journals promoting prescription opioids for non-cancer pain, while reporting low risks of addiction.
Soon pharmaceutical companies began to market their products more aggressively for long-term non-cancer pain. Then in 2001, the Joint Commission issued pain management standards that required medical organizations to measure pain on a scale and report treatment methods \cite{epidemic}.
As physicians responded to these pressures, prescription rates for opioids began to increase dramatically, and addiction rates followed closely behind. By 2012, there were as many as 81.3 opioid prescriptions per 100 people in the United States. Since then, that number has fallen to 66.5, but addiction and overdose rates continue to climb. \cite{c51a95}
While the opioid "epidemic" is not truly an epidemic - that is, drug abuse is not an infection - we hypothesize that methods used for mathematical epidemiology can be applied similarly to the phenomenon of opioid addiction. As we will discuss, traditional models for epidemics are based on the assumption that susceptible individuals become infected through exposure to infected individuals. Then, the infected population increases and the probability of infection spreads in a vicious cycle. The field of mathematical epidemiology has developed a number of models for such phenomena. 
Similarly, susceptible individuals may be more prone to opioid addiction when opioids are more readily available, such as through accessing excess prescriptions from friends or family. The more individuals obtain prescriptions for opioids and become addicted, the higher the likelihood that susceptible individuals will become addicted through accessing these drugs. Thus, we expect similarities to infections in these dynamics.
Very little work has been done to apply these types of epidemiological methods to drug addiction or to the opioid epidemic specifically, so we believe there is rich opportunity to create foundational models in this area. In this paper, we examine one such approach for modeling the opioid epidemic. Our goals are to provide improvements to existing models for opioid addiction, to better understand the driving factors and leverage points of the model, and to compare our model to real world behavior.
We will first review the models from literature and suggest a number of modifications to the model from Battista et al. (2017). Next, we present methods for solving this system with initial guesses for the model parameters drawn from available data. Because data is not available for all parameters and initial conditions, and existing data may be subject to inaccuracies, we emphasize methods for understanding the importance of each parameter and estimating their values. Therefore, we will report the sensitivity of each parameter to equilibrium addiction and recovery states, and then finally examine methods for optimizing the model fit to real world data.

Background

SIR Model

There is substantial literature surrounding methods for using differential equations to model epidemiological phenomena. The simplest and most well known model is the SIR model first described in \citet{Kermack_1927} which characterizes the dynamics of an infection moving through a population.