Anticipating infectious disease re-emergence and elimination: a test of early warning signals using empirically based models

Abstract

Timely forecasts of the emergence, re-emergence and elimination of human infectious diseases allow for proactive, rather than reactive, decisions that save lives. Recent theory suggests that a generic feature of dynamical systems approaching a tipping point-early warning signals (EWS) due to critical slowing down (CSD)-can anticipate disease emergence and elimination. Empirical studies documenting CSD in observed disease dynamics are scarce, but such demonstration of concept is essential to the further development of model-independent outbreak detection systems. Here, we use fitted, mechanistic models of measles transmission in four cities in Niger to detect CSD through statistical EWS. We find that several EWS accurately anticipate measles re-emergence and elimination, suggesting that CSD should be detectable before disease transmission systems cross key tipping points. These findings support the idea that statistical signals based on CSD, coupled with decision-support algorithms and expert judgement, could provide the basis for early warning systems of disease outbreaks.

Keywords: critical slowing down; early warning signals; epidemiology; infectious disease; measles.

Figure 1.

Locations of data sources and observed and predicted measles dynamics. (a) Locations and 1995–2005 population-size ranges (in parentheses) of our four focal cities in Niger. (b) Time series of weekly reported cases (incidence data; yellow solid lines) and the 68% prediction intervals (black ribbons) for one-week-ahead predictions from our fitted susceptible-exposed-infected-recovered (SEIR) models for each city.

Figure 2.

Accuracy of the fitted SEIR models and estimated seasonality. (a) Comparison of in-sample model predictions and observations for each city. Expected cases are one-week-ahead predictions from the fitted models. The dashed line shows 1 : 1. Coefficients of determination (R²) were calculated as the reduction in the sum-of-squared errors from model predictions relative to a null model of the mean number of cases (Material and methods). (b) The estimated seasonality of the basic reproductive ratio (${\mathcal{R}}_0$) for each city. ${\mathcal{R}}_0$ was approximated as: ηβ_t/((η + ν)(γ + ν)), where 1/η is the incubation period, 1/γ is the infectious period, β_t is the time-specific rate of transmission, and ν is the death rate. Only β_t is estimated by our model. We set $1/\eta =8\hspace{0.17em}\mathrm{days}$, $1/\gamma =5\hspace{0.17em}\mathrm{days}$, and ν = 0.05 for calculating ${\mathcal{R}}_0$ as shown in this figure. The white line is ${\mathcal{R}}_0$ calculated using the MLE parameters; shaded regions are the bootstrapped 95% confidence intervals. The dashed horizontal lines show the common range of measles ${\mathcal{R}}_0\hspace{0.17em}:\hspace{0.17em}12\hspace{0.17em}\mathrm{to}\hspace{0.17em}18$.

Figure 3.

Performance of early warning signals (EWS) over fixed windows on the approach to emergence. (a) A typical example of an emergence simulation for Maradi. The two vertical blue lines indicate the start (left-most line) and end (line for critical year) of the full window. The black line demarcates the division between the equal-length null and test intervals, in which we show the calculated variance. (b) Empirical densities of variance in the null and test intervals across 500 simulations and the associated area under the curve (AUC) statistic. (c) Heatmap of AUC statistics for each EWS at each level of susceptible depletion factor. AUC values closer to 0 or 1 indicate higher ability to distinguish among time series near and far from a critical transition. See electronic supplementary material, figure S8 for a visualization of how susceptible depletion factor maps to number of weeks in the null and test intervals.

Figure 4.

Performance of early warning signals (EWS) over fixed windows on the approach to elimination. (a) A typical example of an elimination simulation for Maradi. The two vertical blue lines indicate the start (left-most line) and end (line for critical year) of the full window. The black line demarcates the division between the equal-length null and test intervals, in which we show the calculated variance. (b) Empirical densities of variance in the null and test intervals across 500 simulations and the associated area under the curve (AUC) statistic. (c) Heatmap of AUC statistics for each EWS at each speed of approach to herd immunity. AUC values closer to 0 or 1 indicate higher ability to distinguish among time series near and far from a critical transition. See electronic supplementary material, figure S8 for a visualization of how vaccination speed maps to number of weeks in the null and test intervals.