Predicting the Future Course of Opioid Overdose Mortality: An Example from Two US States

Abstract

Background:

The rapid growth of opioid abuse and related mortality across the United States has spurred the development of predictive models for the allocation of public health resources. These models should characterize heterogeneous growth across states using a drug epidemic framework that enables assessments of epidemic onset, rates of growth, and limited capacities for epidemic growth.

Methods:

We used opioid overdose mortality data for 146 North and South Carolina counties from 2001 through 2014 to compare the retrodictive and predictive performance of a logistic growth model that parameterizes onsets, growth, and carrying capacity within a traditional Bayesian Poisson space–time model.

Results:

In fitting the models to past data, the performance of the logistic growth model was superior to the standard Bayesian Poisson space–time model (deviance information criterion: 8088 vs. 8256), with reduced spatial and independent errors. Predictively, the logistic model more accurately estimated fatality rates 1, 2, and 3 years in the future (root mean squared error medians were lower for 95.7% of counties from 2012 to 2014). Capacity limits were higher in counties with greater population size, percent population age 45 to 64, and percent white population. Epidemic onset was associated with greater same-year and past-year incidence of overdose hospitalizations.

Conclusion:

Growth in annual rates of opioid fatalities was capacity limited, heterogeneous across counties, and spatially correlated, requiring spatial epidemic models for the accurate and reliable prediction of future outcomes related to opioid abuse. Indicators of risk are identifiable and can be used to predict future mortality outcomes.

Introduction

In the wake of surging opioid-related mortality rates, a key public health concern is the timely identification of high-risk communities that require greater prevention efforts and resources. Ideally, we would like to predict which communities are likely to present high mortality rates at some future time. Unfortunately, there are substantial time lags (often several years) in the availability of mortality data, rendering retrospective examinations of these data less than timely for targeting prevention efforts, particularly for deaths requiring further investigation (1). Effective predictive models that go beyond available data to predict future outcomes are needed. While the retrospective assessment of dynamic disease processes may be adequately captured using standard statistical models, assessment of future behaviors demands more firmly grounded theoretical models (2).

Public health researchers have employed numerous strategies to model historic trends in opioid use and overdose, including polynomial and segmented linear regression (3–7), nonlinear logistic regression (8–9), Poisson and negative binomial regression (10–11), ARIMA modeling (12–13), and differences-in-differences regression (14). While these approaches can successfully characterize past opioid abuse trends, they exhibit two flaws in predicting future outcomes. First, such predictions are unbounded, allowing unlimited growth over time. Intuitively, the incidence of fatal opioid overdoses will be limited in capacity to, at most, the available pool of people who use opioids in any given year. Second, these approaches provide no meaningful measure of epidemic onset. Identification of the initial onset of an epidemic is conditional on an estimate of the at-risk population (e.g., the point at which 10% of the at-risk population is affected). Theoretically grounded approaches to epidemic processes are needed to address these concerns.

While research in this area is limited, two recent studies have used infectious disease frameworks to predict future opioid overdose trends. Darakjy et al. (15) used Farr’s Law (16–19) to estimate the likely peak of the US opioid epidemic. They assumed that (a) overdose incidence would rise and fall in a bell-shaped curve over time, (b) the US opioid overdose epidemic was nearing its peak in 2011, and (c) the population used to calculate mortality rates would not substantively change in the near term. They predicted that the epidemic would peak around 2016–2017; tragically, there would be a 500% increase in opioid deaths fueled by the dramatic rise in synthetic opioid (e.g. fentanyl) overdoses between 2013 and 2017 (20). Chen et al. (21) used a three-compartment dynamic systems model to project future opioid overdose deaths, assuming that the population of persons who use opioids could be partitioned into those who use nonmedical prescription opioids, those with opioid use disorders, and those who use other illicit opioids; such persons could transition between the three compartments, each with its own unique mortality risks. The model was calibrated using historical data on opioid overdose deaths and the annual prevalence of each of the three types of opioid use and predicted a 147% increase in overdose deaths (predominantly from illicit opioids) that would peak in 2025. Importantly, both studies adopted models that assumed unbounded growth over time (Farr’s Law presumes an epidemic peak; the size of the pool of those who use opioids constrains growth in the three-compartment model). Neither study accounted for the variability in space and time of characteristics and environments of people who use opioids, thus overlooking the substantial regional and local heterogeneity in epidemic growth that characteristically underlies the opioid epidemic (22). It has been demonstrated that demographic, economic, and geographic characteristics are associated with opioid-related outcomes, and these vary geographically across different levels of resolution (e.g., states, counties, and ZIP codes) with heterogeneous effects across places (23–28). There is a large gap between retrodictive and predictive models of the opioid epidemic; looking back in time, we see great heterogeneity in covariate relations and spatiotemporal patterns of growth between areas; looking forward in time, predictive models express little of this detail.

A geographically specific capacity-limited approach

To identify capacity limits and heterogeneous growth between populations across states, we will assume that county populations are distinct from each other, and opioid overdose mortality is proportional to the size of an unobserved at-risk population of people who use opioids. This unobserved population is a subgroup of people who misuse opioids who both use frequently and are in circumstances where heavy use can lead to overdose. Mortality will be proportional to the at-risk pool, and the opioid-related overdose deaths will not typically exceed this capacity in any given year. We also assume that the onset of epidemic growth is best characterized with respect to this capacity and will differ between areas reflecting local conditions that drive epidemic growth. Thus, an appropriate qualitative model for these data will allow us to estimate capacity limits, which vary across places. Several nonlinear models are suitable to this task (see 29–31), but none have been applied to the prediction of opioid overdose fatalities. Here, we use a logistic function to provide this assessment (31–33):

$$\frac{C}{1+\text{exp}\left(-r\left(t-t0\right)\right)}$$ #(1)

where C represents the carrying capacity, r is the growth rate, t is the current time, and t0 is the time at which 50% of the carrying capacity is reached. With these three parameters estimated, an early point in the growth of the epidemic can be calculated.

Applying this model to counties within states, we capture heterogeneity in patterns of growth by assuming that carrying capacities, C_i, midpoints, t0_i, and growth rates, r_i, vary across counties (random effects), and that correlated growth between nearby areas will arise through population movements and the geographic spread of drug sales and use (spatial autocorrelation). Critically, calculated midpoints of onsets may be early or late, and mortality rates may grow (r_i > 0) or decline (r_i < 0) relative to the capacity limits of each county. Additionally, capacity limitations and midpoints may be related to population characteristics (e.g., area income, proportion of blue-collar workers) and indicators of high-risk use (e.g., hospital discharges related to opioid). Finally, represented as a stochastic spatial process, opioid mortality may appear over or under capacity limitations (34). While challenging to implement, stochastic logistic models enable the analysis of data with unexpected changes in growth rates and capacities (35), random effects related to differential midpoints (36), and a broad class of birth–death processes (37).

The current study

In this paper, we use county-level annual opioid mortality data from North and South Carolina for 2001–2014 to compare the retrodictive and predictive performance of standard log–linear vs. log–logistic Bayesian hierarchical Poisson conditionally autoregressive (CAR) spatial models. Retrodictive evaluations use data from all years. Predictive evaluations use models fit to the first 11 years of data to predict mortality in the latter three years. Finally, we extend the log–logistic formulation to include determinants of carrying capacity and epidemic “onset,” defined for the remainder of this paper as that point at which 10% of the at-risk population has died from an opioid overdose.

Methods

We used the National Vital Statistics System (NVSS) multiple cause-of-death mortality files to assess overdose deaths (38). International Classification of Disease, Tenth Revision (ICD-10) cause-of-death codes related to drug overdose included unintentional overdose (X40–44), suicide by drug self-poisoning (X60–64), homicide by drug poisoning (X85), deaths of undetermined intent (Y10–14), and several multiple cause-of-death codes: natural/semisynthetic opioids (T40.2), methadone (T40.3), synthetic opioids other than methadone (T40.4), and heroin (T40.1). Annual counts were aggregated across 146 counties in the U.S. states of North Carolina and South Carolina for the years 2001–2014. These states were selected because they have a substantial but manageable number of counties, regions with high and low rates of opioid mortality, and a mix of urban and rural populations. Because mortality records did not always record county of residence, annual counts were computed based on county of death.

We obtained annual sociodemographic estimates from GeoLytics (39), indexing population sizes, economic measures, race, and age distributions. We calculated the number of blue-collar businesses using U.S. Census County Business Patterns data (40). A dummy variable indicating whether GeoLytics population estimates were based on the 2000 or 2010 Census was also included.

The log–logistic formulation allows in-depth assessment of the determinants of carrying capacity and onset by including covariates specific to these parameters (our primary interest here). However, these models are substantially more complex than standard log–linear formulations, and covariates must be parsimoniously selected. With this in mind, we included six carrying capacity determinants (percent population age 0–19 and 45–64 years old, number of blue collar businesses per 100 square miles, and percent population living below 150% of the poverty line, unemployed, and white), and one determinant of onset lagged up to 2 years (county-specific proportions of opioid-related overdoses relative to total hospitalizations in the same year, and 1 and 2 years before).

North Carolina hospitalizations represented county-level counts of community inpatient hospital discharges related to prescription opioids or heroin obtained through the Healthcare Cost and Utilization Project’s State Inpatient Database (41). South Carolina hospitalizations were obtained directly from the state government (42). Relevant International Classification of Diseases, Ninth Revision, Clinical Modification (ICD-9-CM) diagnostic codes included opium poisoning (965.00), methadone poisoning (965.02), poisoning by other opioids (965.09), accidental poisoning by methadone (E850.1), heroin poisoning (965.01), and accidental poisoning by other opiates (E850.2).

The study protocol was reviewed and approved by the University of California Davis and New York University Langone Health Institutional Review Boards.

Statistical approach

We compared log–linear and logistic Poisson models within a Bayesian space–time framework that incorporated CAR spatial random effects (43) to account for spatial dependence among nearby units coupled with independent random effects to address small-area effects and over-dispersion (25, 44–46). We assumed overdose counts to be Poisson distributed with mean μ_i,t for each county, i = 1, …,146, and year, t = 1, …,14:

$$Y_{i,t}|{\mu }_{i,t}~\mathit{\text{Poisson}}\left({\mu }_{i,t}\right).$$ #(2)

The standard model, Model 1, was Poisson with a log–linear link:

$$\text{log}\left({\mu }_{i,t}\right)=b_0+b_1·{x^{\prime }}_{i,t}+b_2·{\text{x}}_{i,t}^{\text{''}}+b_3·\text{t}+{\beta }_i·t+{\xi }_i+{\varphi }_{i,t}+{\theta }_{i,t},$$ #(3)

The logistic model, Model 2, was also Poisson with a log–linear link, but with the parameterization suggested by equation 1 above:

$$\text{log}\left({\mu }_{i,t}\right)=\frac{k_0+k_1·{x^{\prime }}_{i,t}+k_2·{x^{″}}_{i,t}+{\kappa }_i}{1+\text{exp}\left(-\left(r_0+{\rho }_i\right)\left(\text{t}-t_0+{\tau }_i\right)\right)}+{\varphi }_{i,t}+{\theta }_{i,t}.$$ #(4)

Model 1 is a log–linear Poisson model with a global intercept b₀, fixed (b₃) and county-specific (β_i) random trend effects across years (t), county-specific random intercepts (ξ_i), and fixed effects (b_1, b₂) for population size (x′_i,t) and census year (x″_i,t). Model 2 is the log–logistic counterpart to Model 1. It includes intercepts k₀, r₀, and t₀, and county-level random effects κ_i, ρ_i, and τ_i for capacity limits, growth rate, and time of onset, respectively. Here, population size and the census year dummy variable are determinants of carrying capacity (the numerator, k₀ + k₁ · x′_i,t + k₂ · x″_i,t + k_i). ϕ_i,t and θ_i,t are CAR spatial and non-spatial random effects, respectively, in both models. Flat prior distributions are assumed for all fixed intercepts, and normal distributions with a mean of zero and high variance are assigned to all other fixed effects. Non-spatial random effects are assigned normal distributions with mean zero and variance following inverse gamma distributions.

Population size is assumed to be a fixed determinant of opioid mortality in both models to ensure comparability. While population size may be modeled as an offset in the log–linear Poisson model, we include it here as an explicit covariate in order to maintain comparability with the log–logistic model; there the carrying capacity is scaled relative to population size (see Equation 4).

Finally, Model 3 added covariates to Model 2:

$$\text{log}\left({\mu }_{i,t}\right)=\frac{k_0+\mathit{k}{\mathit{x}}_{i,t}+{\kappa }_i}{1+\text{exp}\left(-\left(r_0+{\rho }_i\right)\left(\text{t}-(t_0+\mathit{t}{\mathit{x}}_{i,t}^1)+{\tau }_i\right)\right)}+{\varphi }_{i,t}+{\theta }_{i,t}.$$ #(5)

Here, k is a vector of coefficients of carrying capacity determinants (population size, age distributions, poverty, and unemployment) with corresponding observed values x_i,t. x¹_i,t is the analogous matrix of onset determinants (hospital discharge data opioid-related non-fatal overdose rates).

We used the WinBUGS 1.4.3 software (47) to fit all models with two chains, each having different initial values. A burn-in period of 50,000 Markov chain-Monte Carlo (MCMC) iterations was used to estimate parameters for Models 1 and 2, by which point estimates had converged as assessed using Brooks–Gelman–Rubin and trace plots. We then used two posterior chains of 40,000 iterations to estimate posterior distributions for retrodictive and predictive models.

We estimated the much more complex logistic Poisson model with determinants of carrying capacity and onset using a burn-in period of 2,000,000 iterations followed by two chains of 3,000,000 posterior iterations to obtain posterior effective sample sizes for all parameter estimates of at least 300 (or 100 in the case of intercepts). Minimum effective sample sizes vary by author (e.g., 48) and goals of analyses. As the goal of this work is in comparing the predictive value of different models, we did not place excessive weight on characterizing the tails of parameter distributions. Run times for stochastic log–logistic models are characteristically long and increase substantially with the addition of each new covariate (49). While we have also run these models using standardized covariates, based on observation of trace plots, convergence did not improve upon having done so.

We used the deviance information criterion (DIC) and root mean squared error (RMSE) of predicted vs. observed counts to compare model performance. The DIC is more specifically geared toward Bayesian hierarchical models. RMSEs of estimated vs. observed counts were used to compare performance of the models when fit to all years of data (retrodictive) vs. when fit to years 2001–2011 and used to predict data from 2012 through 2014.

Results

Table 1 presents a comparison of the retrodictive performance of models 1 and 2. The DIC difference between models (8256 vs. 8088; ΔDIC=168) indicates that the performance of the logistic model was far better than that of the standard log–linear model (ΔDICs greater than 3–7 are generally considered substantial among models with common likelihoods; 50). RMSE distributions for the two models (medians and [95% CIs] of 2.5 [2.4, 2.6] both for the log–linear and the log–logistic) indicate that the two models performed similarly in mortality estimation. However, the magnitude of CAR was reduced by 31% and independent errors by 25% from the log–linear to the logistic formulation, suggesting that more variation was explained by the systematic component of the logistic model.

Table 1.

Comparison of the standard log–linear and log–logistic models using 2001–2014 mortality data (ln(relative rates) and 95% credible intervals).

	Model 1	Model 2
Variable	Standard (log–linear)	Log–logistic
Fixed effects
Standard Model
Global intercept	−0.29[−0.069, 0.17]
Year	0.094 [0.082, 0.11]a
Census 2010	−0.40 [−0.48, −0.32]a
Population size (x100,000)	0.55 [0.41, 0.70]a
Log–logistic Model
Carrying capacity intercept		0.48 [0.23, 0.78]a
Carrying capacity determinants
Census 2010		−0.16 [−0.23, −0.088]a
Population size (x100,000)		1.2 [1.0, 1.4]a
Growth rate intercept		0.41 [0.35, 0.49]a
Onset intercept		1.6 [0.35, 2.8]a
Random effects
County level (std. dev.)
Year intercept	1.0 [0.92, 1.1]
Year slope	0.028 [0.037, 0.047]
Carrying capacity		1.0 [0.94, 1.1]
Growth rate		0.34 [0.25, 0.45]
Onset		4.3 [3.5, 5.2]
Observation level (std. dev.)
Spatial CAR process	0.17 [0.12, 0.22]	0.12 [0.063, 0.17]
Non-spatial	0.19 [0.14, 0.23]	0.14 [0.069, 0.19]
Spatial proportion	0.45 [0.22, 0.68]	0.41 [0.13, 0.83]
RMSE	2.5 [2.4, 2.6]	2.5 [2.4, 2.6]
DIC	8256	8088

^aWell-supported as indicated by 95% credible intervals that exclude 0.

ln = natural log

CAR = conditional autoregressive

RMSE = root mean squared error

DIC = deviance information criterion

HDD = hospital discharge data

As expected, population size was positively associated with opioid mortality in the standard model with an analogous association with carrying capacity in the log–logistic model. Annual growth and heterogeneity in growth across counties were positive and substantive in both models, indicating that the two models were capturing similar information about growth trends in these data.

Excluding four counties for which extreme estimates were obtained from the log–logistic model (discussed below), Figure 1 shows the distribution of 2014 carrying capacities and 10% onsets for the overdose epidemic for 142 counties. Carrying capacities are presented proportional to each county’s total population in 2014 (Figure 1a). These percentages were low, ranging from 0.005% to 0.038% of the total population (median 0.009%), reflecting the small size of the estimated population at-risk for opioid-related mortality. Ten percent onsets (Figure 1b) ranged from 1989 to 2011, with a median of 1997, indicating that the epidemic was well underway by the turn of this century.

Figure 1.

Histograms of carrying capacities (left) and 10% onsets (right) for 142 North Carolina and South Carolina counties as estimated by the log–logistic model using 2001–2014 data. Carrying capacities are plotted as percents of each county’s total population in 2014. The 2014 mean observed opioid mortality incidence rate (per total population) is plotted as a dashed vertical line.

A comparison of the predictive performance of standard and log–logistic models is presented in Table 2. The distribution of 2012–2014 RMSEs for the standard predictive model had a median of 3.0 and a 95% CI of [2.7, 3.4]; for the logistic, the median was 2.8 with a 95% CI of [2.5, 3.1]. Overall, the log–logistic model had lower RMSEs for 96% of all counties from 2012 to 2014. Broken down by year, this percentage increased with time, and in 2014, the log–logistic model predicted mortality counts more accurately in all counties. For combined years, RMSEs were also assessed for counties across five population quintiles. Log–logistic predictions were consistently more accurate in all quintiles.

Table 2.

RMSEs for the standard (log–linear) vs. the log–logistic models using 2001–2011 data to predict opioid-related mortality in 2012, 2013, 2014, and combined years 2012–2014; the 2012–2014 predictions are then broken down by population quintile.

Number of counties with lower RMSEs by model type and year
Model	2012	2013	2014	2012–2014
Standard, number of counties	13	6	0	19
Log–logistic, number of counties	133	140	146	419
Total, number of counties	146	146	146	438
Counties in which the log–logistic model had lower RMSEs, %	91	96	100	956
Binomial test P-value	<0.001	<0.001	<0.001	<0.001
Number of counties with lower RMSEs by model type and population counties
Model	Quintile 1	Quintile 2	Quintile 3	Quintile 4	Quintile 5
Standard, number of counties	0	1	1	2	15
Log–logistic, number of counties	85	81	86	81	86
Total, number of counties	85	82	87	83	101
Counties in which the log–logistic model had lower RMSEs, %	100	99	99	98	85
Binomial test P-value	<0.001	<0.001	<0.001	<0.001	<0.001

Figure 2 presents a selection of plots representing the fit of the standard and log–logistic models to data from 2001–2011 and projections through 2014. For confidentiality reasons, county names, mortality counts, and population counts are suppressed. Selected counties represent each population quintile, labeled in the title of each plot. Importantly, the figures demonstrate the degree to which the standard model over-estimates future growth of the epidemic in most counties.

Figure 2.

Standard (log–linear) and log–logistic predictive opioid mortality estimates (using 2001–2011 data to predict counts in 2012–2014) are plotted from 2001 through 2014 for a sample of two counties per population quintile. Observed counts are plotted as dots, gray solid lines represent standard model estimates, and black dashed lines represent log–logistic estimates. Horizontal gray dashed lines are carrying capacity estimates from the log–logistic model. Each plot’s title notes the population quintile and the standard vs. log–logistic RMSE for 2012–2014. For confidentiality reasons, county names and y-axis labels revealing exact mortality counts are omitted from this figure.

Table 3 presents parameter estimates for covariates related to carrying capacity and onset for the log–logistic model estimated concurrently and for two temporal lags. Greater carrying capacities were related to greater population size, greater proportions in the 0–19 and 45–64 age groups, and greater percent white populations (Model 3a). The onset of the opioid epidemic was earlier in counties with greater rates of hospitalizations related to opioid overdoses. These rates were useful for the estimation of onsets one year (Model 3b) but not two years (Model 3c) in the future.

Table 3.

The log–logistic model with the addition of covariates for carrying capacity and onset (ln(relative rates) and 95% credible intervals). Model 3a uses data from 2001–2014. Due to inclusion of time-lagged data, Model 3b and Model 3c use data from years 2002–2014 and 2003–2014, respectively. HDD non-fatal opioid overdose rates are proportions of respective year- and county-specific total hospitalization counts multiplied by 1000.

Variable	Model 3a	Model 3b	Model 3c
Fixed effects
Carrying capacity intercept	−3.4 [−5.2, −1.6]a	−2.4 [−4.4, −0.078]a	−2.1 [−4.0, 0.18]
Growth rate intercept	0.42 [0.30, 0.60]a	0.40 [0.072, 0.60]a	0.45 [0.29, 0.68]a
Onset intercept	1.7 [0.036, 3.0]a	1.2 [−0.55, 16.8]	−1.3 [−3.6, 0.33]
Carrying capacity determinants:
Population size (x100,000)	0.91 [0.74, 1.1]a	0.96 [0.79, 1.1]a	0.95 [0.78, 1.13]a
Age 0–19, %	0.0090 [−0.017, 0.035]	0.0088 [−0.061, 0.036]	0.015 [−0.011, 0.042]
Age 45–64, %	0.078 [0.031, 0.12]a	0.047 [−0.004, 0.126]	0.029 [−0.021, 0.078]
Below 150% of poverty level, %	−0.0049 [−0.026, 0.016]	−0.0089 [−0.061, 0.012]	−0.014 [−0.036, 0.0073]
Unemployed, %	−0.0069 [−0.018, 0.0030]	−0.0085 [−0.030, 0.0020]	−0.0076 [−0.019, 0.0024]
Blue collar, %	0.053 [−0.077, 0.19]	9.3E-5 [−0.0012, 0.0056]	−2.3E-5 [−0.0013, 0.0013]
White, %	0.028 [0.018, 0.038]a	0.027 [0.017, 0.051]a	0.028 [0.019, 0.039]a
Census 2010	0.057 [−0.033, 0.15]	0.0030 [−0.012, 0.025]	0.038 [−0.060, 0.13]
Onset determinants:
HDD non-fatal opioid overdose rate [t]	−0.62 [−1.05, −0.26]a	-	-
HDD non-fatal opioid overdose rate [t-1]	-	−1.2 [−2.7, −0.21]a	-
HDD non-fatal opioid overdose rate [t-2]	-	-	0.090 [−0.75, 0.75]
Random effects
County level (std. dev.)
Carrying capacity	0.90 [0.80, 1.0]	0.87 [0.78, 0.99]	0.83 [0.73, 0.95]
Growth rate	0.21 [0.15, 0.32]	0.20 [0.045, 0.33]	0.24 [0.15, 0.40]
Onset	5.2 [4.1, 6.7]	5.1 [0.20, 6.9]	4.9 [3.8, 6.7]
Observation level (std. dev.)
Spatial CAR process	0.11 [0.045, 0.17]	0.11 [0.040, 0.18]	0.13 [0.060, 0.18]
Non-spatial	0.14 [0.070, 0.18]	0.14 [0.076, 0.19]	0.14 [0.065, 0.19]
Spatial proportion	0.44 [0.11, 0.82]	0.37 [0.041, 0.80]	0.46 [0.12, 0.86]
RMSE	2.5 [2.4, 2.6]	2.6 [2.4, 2.7]	2.6 [2.5, 2.8]

^aWell-supported as indicated by 95% credible intervals that exclude 0.

Four counties with extreme estimates

The fit of the log–logistic model to four high-population counties indicated very high carrying capacities (ranging from 1,192 to 26,332) with widely differing onsets (from year 1781 to 2193), which appeared somewhat implausible. These counties had substantially higher population counts than other counties, and some experienced large population jumps (e.g., an increase of approximately 20%) from 2010 to 2014. Carrying capacity estimates were more reasonable in earlier years. Nevertheless, in 2012–2014, the log–logistic and the standard model offered similar errors in prediction; the log–logistic model produced lower error in exactly 50% of cases.

Discussion

Considering the enduring use of standard generalized linear modeling approaches for analysis of mortality data from drug epidemics, the results of the current study provide both good and bad news. The good news is that standard spatial Poisson models perform well in retrospective analyses of county-level mortality data in comparison to one theoretically motivated model of epidemic growth. The bad news is that the standard model inadequately predicts future mortality outcomes 1 to 3 years in the future. The log–logistic model more accurately predicts future opioid mortality for counties in North and South Carolina; estimates are available for each county, indicate the onset of the epidemic, and provide useful projections into the near future.

The key ingredient that explains the effectiveness of log–logistic models is their incorporation of carrying capacities, “hidden numbers” that estimate the sizes of year- and county-specific at-risk populations. The benefits of estimating these, particularly for predictions further out in time, is visually apparent in Figure 2. There, the standard model predicts growth well beyond what theory and data indicate. This is also substantiated by improved accuracy in predicting opioid mortality counts (Table 2). Thus, the qualitative application of the log–logistic model here, used to identify and estimate carrying capacity limits, shows that predictive models must incorporate carrying capacity limits to estimate future mortality outcomes. That these limits are associated with population characteristics, which also change over time, further supports the value of these models for predictive purposes.

This approach also helps elucidate epidemic phases, suggesting the point at which onset began (e.g., 10% of capacity limits) and the point at which epidemic peaks may be reached (e.g., 90% of capacity limits). This information would assist in effective resource allocation to counties just beginning to experience epidemic onset vs. those where opioid use has become endemic. And, again, assessing covariates directly related to carrying capacity and onset allows us to discern correlates of the population size of at-risk populations and provide early indicators of risks for rapid epidemic growth. Past year and concurrent non-fatal opioid overdose hospital discharge data rates are related to earlier epidemic onsets (Table 3). Identification of community characteristics connected to carrying capacity and onset is essential to exploring the timing and extent of future overdose risk in different communities.

These models can also be used to predict the future of epidemic growth across states. Extending the predictive model to years 2015 through 2018 and aggregating across counties, the model predicts that opioid mortality had reached 90% of capacity by 2018 (77%, 86%, and 92% in 2016–2018, respectively). This estimate is conditional on carrying capacities not changing substantively over time. Such future projections are therefore conditional on accurate assessment of future demography.

Finally, the results also emphasize both the heterogeneity and the spatial dependence of epidemic growth across geographic regions; accounting for both is indispensable to obtaining accurate estimates. These processes explain observations of opioid mortality counts in excess of estimated carrying capacity limits. County boundaries artificially divide populations, and county populations are not self-contained. This has many side effects—for example, if populations among different counties interact, it is possible that mortality counts in some counties will exceed their carrying capacities (34).

Limitations

There are three critical limitations to the analyses presented in this paper. First, log–logistic models are more complex, and their estimation comes with greater computational costs. Adding many covariates can quickly overwhelm Markov-chain-Monte Carlo capabilities for model estimation. Second, not all county-level estimates appear plausible. Four counties with large populations presented implausible estimates of carrying capacity and onsets, although the log–logistic model still had similar predictive value relative to the standard model. Future work should nonetheless aim to resolve this issue. Third, the theoretical motivation for the application of the log–logistic model is qualitatively strong but quantitatively weak; specific mathematical models for the current purpose must now be developed.

Conclusion

Compared to the standard model, the log–logistic model exhibited considerably improved retrodictive and predictive capabilities of opioid mortality by both theoretical and practical standards, thus laying the groundwork for future exploration and extension of these models. Other model forms, such as those incorporating spatially correlated carrying capacities should be explored. Bearing heterogeneity of populations in mind, analyses should be carried out in other U.S. states. Standard models grossly overestimate the growth curve of opioid mortality trends. The improved predictive value of the log–logistic models is instrumental to aiding public health intervention efforts in understanding and assessing the future risk of opioid mortality.