As a result, current models often build on the SIR model by removing unrealistic assumptions and adding more nuance and parameters to different sections of the model depending on the type of the epidemic. Variations include asymptomatic carrier individuals, incubation periods, vital dynamics (births and deaths), seasonality, return from infectious back to susceptible (no immunity, e.g. influenza) or temporary immunity. There are also variations to the modeling approach itself, which include stochastic models, network models, spatial models. \citet{2008} provides a detailed survey of such methods.

Opioid-specific models

Very little has been published applying these methods to drug use, and even less has been published related to the opioid epidemic specifically.  \citet{nielsen2013epidemic} is one early work that creates a System Dynamics model for non-medical opioid use. They do not explicitly provide the system of equations behind the models, however. Furthermore, it is infeasible to determine empirical values for many of the large number of modeling parameters they suggest, such as "average number of friends" or "leftover medication available". 
Another work of significance to us is \citet{2017arXiv171103658B}, which presents a higher level view that resembles the SIR-derived approaches, but which replaces the infected class with "Prescribed Users" and "Addicted Users". It also replaces the removed category with "Recovered," which has a pathway back into the addicted population. We will discuss this work in more detail throughout the paper.

Methods

Modified SIR Model for Opioid Addiction