Flexible Modeling of Epidemics with an Empirical Bayes Framework

Abstract

Seasonal influenza epidemics cause consistent, considerable, widespread loss annually in terms of economic burden, morbidity, and mortality. With access to accurate and reliable forecasts of a current or upcoming influenza epidemic's behavior, policy makers can design and implement more effective countermeasures. This past year, the Centers for Disease Control and Prevention hosted the "Predict the Influenza Season Challenge", with the task of predicting key epidemiological measures for the 2013-2014 U.S. influenza season with the help of digital surveillance data. We developed a framework for in-season forecasts of epidemics using a semiparametric Empirical Bayes framework, and applied it to predict the weekly percentage of outpatient doctors visits for influenza-like illness, and the season onset, duration, peak time, and peak height, with and without using Google Flu Trends data. Previous work on epidemic modeling has focused on developing mechanistic models of disease behavior and applying time series tools to explain historical data. However, tailoring these models to certain types of surveillance data can be challenging, and overly complex models with many parameters can compromise forecasting ability. Our approach instead produces possibilities for the epidemic curve of the season of interest using modified versions of data from previous seasons, allowing for reasonable variations in the timing, pace, and intensity of the seasonal epidemics, as well as noise in observations. Since the framework does not make strict domain-specific assumptions, it can easily be applied to some other diseases with seasonal epidemics. This method produces a complete posterior distribution over epidemic curves, rather than, for example, solely point predictions of forecasting targets. We report prospective influenza-like-illness forecasts made for the 2013-2014 U.S. influenza season, and compare the framework's cross-validated prediction error on historical data to that of a variety of simpler baseline predictors.

Fig 1. Examples of possible peak week, peak height, and pacing transformations, and different noise levels.

Thick black, original curve; red, possible peak week transformations; thick red, a random peak week transformation; purple, possible peak height transformations; thick purple, a random peak height transformation; blue, possible pacing transformations; thick blue, a random pacing transformation; dotted green, 5th and 95th (pointwise) percentiles of noise distribution for possible noise levels; dashed green, percentiles for a random noise level; thick green, one possible trajectory for the selected transformations and noise level. (A) Peak week transformations. Peak weeks of historical smoothed curves occurred between weeks 51 and week 10 of the next year, so we limit transformations to give peak weeks roughly within this range. (B) Peak height transformations. Peak heights of historical smoothed curves were between 2% and 8%, so we limit transformations to give peak heights roughly within this range. (C) Pacing transformations. We stretch the curve by a factor between 75% and 125% about the peak week. (D) Noise levels. We randomly select one of 15 noise levels from the fitting procedure and add this level of Gaussian noise to the transformed curve.

Fig 2. 2013–2014 national forecast, retrospectively, using the final revisions of wILI values, using revised wILI data through epidemiological weeks (A) 47, (B) 51, (C) 1, and (D) 7.

Fig 3. Point predictions and observed values of the forecasting targets for the 2013–2014 season.

Black, observed target value; blue, our current framework’s predictions using revised ILINet wILI data; red, our submitted point predictions using ILINet data only; green, our submitted point predictions that used both ILINet and GFT data. Historical target value ranges exclude the 2009–2010 and 2013–2014 seasons.

Fig 4. Cross-validated mean absolute error estimates and standard error bars for point predictions for (A) onset, (B) peak week, (C) peak height, and (D) duration.

(The onset and duration were defined based on the 2% national threshold set by CDC for the 2013–2014 season.)