featuring both in the news political campaigns, and academic journals.
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 a large literature of 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 modeling parameters they suggest.
Another work of significance to us is \citet{2017arXiv171103658B}, which goes into much farther detail in terms of trying to find