Deep neural networks for endemic measles dynamics: Comparative analysis and integration with mechanistic models

Measles is an important infectious disease system both for its burden on public health and as an opportunity for studying nonlinear spatio-temporal disease dynamics. Traditional mechanistic models often struggle to fully capture the complex nonlinear spatio-temporal dynamics inherent in measles outbreaks. In this paper, we first develop a high-dimensional feed-forward neural network model with spatial features (SFNN) to forecast endemic measles outbreaks and systematically compare its predictive power with that of a classical mechanistic model (TSIR). We illustrate the utility of our model using England and Wales measles data from 1944-1965. These data present multiple modeling challenges due to the interplay between metapopulations, seasonal trends, and nonlinear dynamics related to demographic changes. Our results show that while the TSIR model yields similarly performant short-term (1 to 2 biweeks ahead) forecasts for highly populous cities, our neural network model (SFNN) consistently achieves lower root mean squared error (RMSE) across other forecasting windows. Furthermore, we show that our spatial-feature neural network model, without imposing mechanistic assumptions a priori, can uncover gravity-model-like spatial hierarchy of measles spread in which major cities play an important role in driving regional outbreaks. We then turn our attention to integrative approaches that combine mechanistic and machine learning models. Specifically, we investigate how the TSIR can be utilized to improve a state-of-the-art approach known as Physics-Informed-Neural-Networks (PINN) which explicitly combines compartmental models and neural networks. Our results show that the TSIR can facilitate the reconstruction of latent susceptible dynamics, thereby enhancing both forecasts in terms of mean absolute error (MAE) and parameter inference of measles dynamics within the PINN. In summary, our results show that appropriately designed neural network-based models can outperform traditional mechanistic models for short to long-term forecasts, while simultaneously providing mechanistic interpretability. Our work also provides valuable insights into more effectively integrating machine learning models with mechanistic models to enhance public health responses to measles and similar infectious disease systems.Author summary: Mechanistic models have been foundational in developing an understanding of the transmission dynamics of infectious diseases including measles. In contrast to their mechanistic counterparts, machine learning techniques including neural networks have primarily focused on improving forecasting accuracy without explicitly inferring transmission dynamics. Effectively integrating these two modeling approaches remains a central challenge. In this paper, we first develop a high-dimensional neural network model to forecast spatiotemporal endemic measles outbreaks and systematically compare its predictive power with that of a classical mechanistic model (TSIR). We illustrate the utility of our model using a detailed dataset describing measles outbreaks in England and Wales from 1944–1965, one of the best-documented and most-studied nonlinear infectious disease systems. Our results show that overall, our neural network model outperforms the TSIR in all forecasting windows. Furthermore, we show that our neural network model can uncover the mechanism of hierarchical spread of measles where major cities drive regional outbreaks. We then develop an integrative approach that explicitly and effectively combines mechanistic and machine learning models, improving simultaneously both forecasting and inference. In summary, our work offers valuable insights into the effective utilization of machine learning models, and integration with mechanistic models, for enhancing outbreak responses to measles and similar infectious disease systems.