Physics-informed deep learning for infectious disease forecasting

TL;DR

The paper tackles infectious-disease forecasting under evolving dynamics by fusing epidemiological compartmental models with neural networks through physics-informed constraints, formalized with a loss $\mathcal{L}(\theta)=\mathcal{L}_{data}(\theta)+w_{ODE}\mathcal{L}_{ODE}(\theta)$. A two-subnetwork PINN estimates the state variables of a nine-state COVID-19 model and a time-varying transmission rate $\beta_t$ driven by mobility and vaccination covariates. On California COVID-19 data, PINNs outperform naive baselines and many sequence models, achieving competitive performance with GISST while offering a simpler implementation and real-time updating potential. The work demonstrates the value of embedding epidemiological principles into neural forecasting and outlines future directions for uncertainty-aware extensions via Bayesian PINNs.

Abstract

Accurate forecasting of contagious diseases is critical for public health policymaking and pandemic preparedness. We propose a new infectious disease forecasting model based on physics-informed neural networks (PINNs), an emerging scientific machine learning approach. By embedding a compartmental model into the loss function, our method integrates epidemiological theory with data, helping to prevent model overfitting. We further enhance the model with a sub-network that accounts for covariates such as mobility and cumulative vaccine doses, which influence the transmission rate. Using state-level COVID-19 data from California, we demonstrate that the PINN model accurately predicts cases, deaths, and hospitalizations, aligning well with existing benchmarks. Notably, the PINN model outperforms naive baseline forecasts and several sequence deep learning models, including Recurrent Neural Networks (RNNs), Long Short-Term Memory (LSTM) networks, Gated Recurrent Units (GRUs), and Transformers. It also achieves performance comparable to a sophisticated Gaussian infection state forecasting model that combines compartmental dynamics, a data observation model, and parameter regression. However, the PINN model features a simpler structure and is easier to implement. In summary, we systematically evaluate the PINN model's ability to forecast infectious disease dynamics, demonstrating its potential as an efficient computational tool to strengthen forecasting capabilities.

Figures (7)

• Figure 1: A. Schematic of the proposed PINNs model for infectious disease forecasting. The model comprises two sub-networks: the upper sub-network predicts the state variables in the compartmental model, while the lower sub-network estimates the time-dependent model parameters. The output $u_1$ represents the nine compartmental state variables, including $X$, $L$, $Z$, $Z_r$, $H$, $A$, $D$, $D_r$. The output $u_2$ represents factors including mobility, cumulative vaccine doses, and transmission rate. The data loss $L_{data}$ is a weighted sum of observable state variables and factors. B. Schematic of the rolling window approach. As training data accumulates over time, the PINNs model continuously updates and generates forecasts for the subsequent 1–4 weeks.
• Figure 2: Original dataset (upper panel) and preprocessed (lower panel) dataset used for training and testing the proposed PINNs model.
• Figure 3: PINNs' point predictions on the number of cases, deaths, and hospitalizations for the following 1 - 4 weeks. GISST represents a mathematical model named Gaussian infection state space with time dependence o2022semi. The naive model uses data from the previous weeks as predictions for the following weeks.
• Figure 4: PINNs' quantile predictions on the number of cases, deaths, and hospitalizations for the following 1 - 4 weeks. The red dots represent the ground truth. The blue points indicate PINNs' predictions and the light blue region represents the associated uncertainty in the forecasts. Each polygon spans a continuous four-week prediction interval, encompassing the 1-, 2-, 3-, and 4-week forecasts made from a time point one week prior to the start of the polygon
• Figure 5: A MASE comparison of Recurrent Neural Networks (RNNs), Long Short-Term Memory (LSTM) networks, Gated Recurrent Units (GRUs), and Transformer models, with training window sizes ranging from 3 weeks to 15 weeks.