Applying Farr’s Law to project the drug overdose mortality epidemic in the United States

In mid-nineteenth century England, William Farr, one of the founders of modern epidemiology, analyzed mortality rates attributed to a smallpox outbreak and derived a “law of epidemics” from the observed patterns, postulating that the dynamics of epidemic outbreaks generally follow a symmetric curve (Farr [1840]). Since that time, the theory that became known as “Farr’s Law” has been applied to projecting the trajectories of the 1885 cattle plague epidemic (Brownlee [1915a]), the 1901 outbreak of smallpox (Brownlee [1915b]), and the AIDS epidemic (Bregman and Langmuir [1990]), with mixed results.

More recently, several authors cited data that demonstrate increases in drug overdose deaths over the past decade (Johnson et al. [2014]; Jones et al. [2013]) or longer (Unick et al. [2013]). Such observations precipitated interventions aimed at limiting opportunities for abuse of controlled substances, particularly prescription opioids. Two of these interventions, the expansion of Prescription Drug Monitoring Programs (PDMPs) across the United States and increased availability and training on rescue treatments that reverse an opioid overdose, have coincided with the stabilization of the annual death rate due to opioid overdoses since 2007 (Brady et al. [2014]). Opioid analgesics account for approximately two thirds of the total drug overdose mortality (CDC [2011]). Temporal trends in drug overdoses closely follow those in prescription drug consumption: as the morphine kilogram equivalents per capita increase, so does the rate of drug overdose mortality. Likewise, both measures began to plateau in 2007 (Brady et al. [2014]). This observation motivated our hypothesis that the drug overdose epidemic is leveling off, and the mortality from drug overdose should start to decline soon and eventually return to a baseline level. Underlying this hypothesis is the fundamental principle of Farr’s Law: epidemics follow a natural sequence in which they spread and subside in a pattern that approximates a symmetric curve (Bregman and Langmuir [1990]). Our objective is to apply Farr’s Law to the drug overdose epidemic and to evaluate over the next several years whether our projections hold true.

A drug overdose epidemic curve for the United States was generated by combining observed mortality rates (1980–2011) with mortality rates projected for 2012–2035 following the principles of Farr’s Law. For the observed period (1980–2011), data came from the National Center for Health Statistics (Warner et al. [2011]). Through 1998, unintentional drug overdose deaths were classified using the International Classification of Diseases, Ninth Revision, and included underlying cause of death codes: E850–E858, E950.0–E950.5, E962.0, and E980.0–E980.5. Starting in 1999, deaths were classified using the International Classification of Diseases, Tenth Revision, and included underlying cause of death codes: X40–X44, X60–X64, X85, and Y10–Y14. Crude annual drug poisoning mortality rates were calculated using the annual population for the years observed and the 2011 population for the rates projected for 2012–2035 (CDC WONDER [2014]).

For future projections on the number of drug overdose deaths expected under the principles of Farr’s Law, we applied the method described by Bregman and Langmuir ([1990]) and calculated two ratios: 1) the percent increase in the mortality rate for each year of observed data, and 2) the acceleration—or rate of change—of the first ratio. To fit a symmetric curve, we took the mean acceleration (=0.9876) for the last five years of observed data (2007–2011), and extrapolated the first ratio followed by the number of deaths expected if the principles of Farr’s Law hold for the drug overdose epidemic.

Our projections based on Farr’s Law indicate that drug overdose mortality in the United States will decrease from its peak in 2016–2017 to the 1980 baseline level in about 18 years. Our projections are made under the assumption that the United States drug overdose epidemic is nearing its peak, as suggested by recent mortality data and opioid prescription data (Brady et al. [2014]; Li et al. [2014]). The shape of the drug overdose mortality curve depends on two key elements: 1) the constant (i.e., the second ratio) chosen for approximating the symmetric curve; and 2) the population count used to calculate the mortality rate. Our decision to use 0.9876 as the second ratio constant was derived from synthesizing information about annual per capita consumption of opioid analgesics and the observed mortality rate accelerations, both of which indicate a shift around 2007 in that opioid sales and deaths stabilized thereafter. The more controversial choice in our projected calculations is the population denominator. Initially, we used the annual population data from the United States Census Bureau in an attempt to factor changes in the number of people at risk. The mortality curve initially generated under the assumption of a dynamic (i.e., increasing) population very closely resembled the one presented in Figure 1. However, since the population susceptible to drug overdose may remain relatively stable in the projected 24-year period, with adults aging both into and out of the highest risk group (45–54 year-olds) (Warner et al. [2011]), we opted for a more conservative method and approximated overdose mortality rates based on the 2011 United States population.

This study represents our first attempt to project the trajectory of the epidemic of drug overdose deaths in the United States. Although this epidemic has been studied extensively through statistical models using predictive variables (Dasgupta et al. [2012]; Li et al. [2014]; Paulozzi and Stier [2010]), we are unaware of any previous study predicting the trajectory of a non-infectious epidemic using Farr’s Law. Our projections, if partially accurate, may help assess intermediate outcomes to gauge whether interventions are working and guide long-term planning and management of public health resources and prevention efforts. When access to prescription medication becomes more difficult with enforcement of PDMPs, there is a hazard of diversion whereby people shift to using alternative substances such as heroin. To avoid this substitution effect would require multifaceted interventions such as the expansion of prescription limits, increased drug screening, requiring pain contracts, and the distribution of naloxone to first responders.

There are several limitations to our study. First, this is a crude approximation of an epidemic curve based on assumptions that guided our choices of the acceleration constant and population at risk. Although the method we applied originated from studies of infectious diseases, it is unknown whether Farr’s Law applies to epidemics of a non-infectious origin. It is plausible that a non-communicable disease, such as drug overdose, can follow infectious patterns. Indeed, a theory of social contagion has been proposed for a myriad of behavioral disorders (Eisenberg et al. [2013]) as well as obesity (Christakis and Fowler [2007]), albeit with some controversy in the latter example. As with behavioral disorders, the social mechanism of transmission may underlie drug-use initiation, continuation, and transition such that those activities then mimic infectious disease patterns, proliferating through the population until some natural threshold or intervention prevents further spread. Second, for simplicity, we estimated drug overdose mortality for the projection period without considering changes in the size of the population or the age distribution of the population. This may bias our estimates somewhat because the United States population is projected to gradually grow in size and get older in age composition. Furthermore, these results are limited to overdose mortality data and are for the United States as a whole. Thus, the results may not reflect trends in non-fatal drug overdose incidents and in specific geographic areas. Despite these limitations, the predicted epidemic curve guided by Farr’s Law suggests that annual drug overdose mortality is likely to decline in the coming years and probably return to the baseline level by the year 2034.

Mortality data over the next two decades will ultimately test the accuracy of our projections. If the drug overdose epidemic is indeed waning, it may imply that the intensified efforts in recent years, such as enhanced prescription drug monitoring, are working and should be continued.

SD reviewed the literature, analyzed the data, and wrote the first draft of the manuscript. JEB acquired the data, supervised the data analysis, and contributed to the critical revision of the manuscript. CJD contributed to the study design, statistical approach, and critical revision of the manuscript. GL conceived of the study, secured funding, oversaw the implementation of the research plan, and contributed to the critical revision of the manuscript. All authors read and approved the final manuscript.