As physicians responded to these pressures, prescription rates for opioids began to increase dramatically, and addiction rates followed 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.
This problem has not been studied extensively:
“very little has been published applying mathematical epidemiology to the problem of drug use in general. “
“White and Comiskey [68] published perhaps the first such model, mathematically describing the heroin epidemic as a system of differential equations resembling the classic SIR model of Kermack and McKendrick [31].”
“no one has developed a compartmental differential equation model specifically for prescription opioids with the intent of better understanding the dynamics involved.”
Is the opioid epidemic truly an epidemic? While drug abuse is not an infection per se, at first look it appears to be a compelling way to reason about the phenomenon. Individuals that are "infected" seem to cluster together, and having an "infected" region seems to increase the probability of more drug abuse in a vicious cycle.
Related Work
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 describes the dynamics of susceptible, infectious and removed (immune, sometimes called "recovered") groups within a compartmentalized population. During an epidemic, individuals move from one group to the other: Everyone starts as susceptible, until some individuals move into the infectious category. More infectious individuals increase the probability of further infections, and the recovery rate competes against the infection rate. Once individuals move into the removed category, they are immune and cannot be re-infected.
Current models often build on these ideas by removing 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.
The opioid epidemic specifically has also been studied in the context of mathematical epidemiology. \citet{nielsen2013epidemic} is an early work that creates a System Dynamics model (a field that is related to Systems theory that uses differential equations to do computer simulations of real-world phenomena). The model they have is very detailed, however it is impractical to find empirical values for many of the large number of modeling parameters they suggest, e.g. "average number of friends" or "leftover medication available". They also do not provide the mathematical equations behind the models.
Another work of significance to us is \citet{2017arXiv171103658B}, which has a more high-level view that resembles the SIR-derived approaches, but which replaces the infected category with "Prescribed Users" and "Addicted Users". It also replaces the removed category with "Recovered", which has a pathway back into the addicted population. They then do sensitivity analysis which provides ... but we don't like .... about this paper, and we want to change .... .
- The way they assign real world parameters back into the model is weird (nu, etc), and also the ranges in the sensitivity analysis might not be realistic because of this
Methods
SIR Model