Ensemble Bootstrap Methodology for Forecasting Dynamic Growth Processes Using Differential Equations: Application to Epidemic Outbreaks

Background: Ensemble modeling aims to boost the forecasting performance by systematically integrating the predictive accuracy across individual models. Here we introduce a simple-yet-powerful ensemble methodology for forecasting the trajectory of dynamic growth processes, such as the spread of infectious diseases, that are defined by a system of non-linear differential equations. Methods: : We propose and assess the performance of two ensemble modeling schemes with different parametric bootstrapping procedures for trajectory forecasting and uncertainty quantification. Specifically, we conduct sequential probabilistic forecasts to evaluate their forecasting performance using simple dynamical growth models with good track records including the Richards model, the generalized-logistic growth model, and the Gompertz model. We first demonstrate the functionality of the method using simulated data from phenomenological models and a mechanistic transmission model. Next, the performance of the method is demonstrated using a diversity of epidemic datasets including scenario outbreak data of the Ebola Forecasting Challenge and real-world epidemic data outbreaks of including influenza, plague, Zika, and COVID-19. Results: : We found that the ensemble method that randomly selects a model from the set of individual models for each time point of the trajectory of the epidemic frequently outcompetes the individual models as well as an alternative ensemble method based on the weighted combination of the individual models and yields broader and more realistic uncertainty bounds for the trajectory envelope and achieves not only better coverage rate of the 95% prediction interval but also improved mean interval scores across a diversity of epidemic datasets. Conclusion: We introduced new methodology for ensemble forecasting that outcompetes component models and an alternative ensemble model that differ in how the variance is evaluated for the generation of the prediction intervals of the forecasts.