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.