Physics-informed deep learning for infectious disease forecasting

Abstract

Accurate forecasting of contagious illnesses has become increasingly important to public health policymaking, and better prediction could prevent the loss of millions of lives. To better prepare for future pandemics, it is essential to improve forecasting methods and capabilities. In this work, we propose a new infectious disease forecasting model based on physics-informed neural networks (PINNs), an emerging area of scientific machine learning. The proposed PINN model incorporates dynamical systems representations of disease transmission into the loss function, thereby assimilating epidemiological theory and data using neural networks (NNs). Our approach is designed to prevent model overfitting, which often occurs when training deep learning models with observation data alone. In addition, we employ an additional sub-network to account for mobility, cumulative vaccine doses, and other covariates that influence the transmission rate, a key parameter in the compartmental model. To demonstrate the capability of the proposed model, we examine the performance of the model using state-level COVID-19 data in California. Our simulation results show that predictions of PINN model on the number of cases, deaths, and hospitalizations are consistent with existing benchmarks. In particular, the PINN model outperforms naive baseline forecasts and various sequence deep learning models, such as Recurrent Neural Networks (RNNs), Long Short-Term Memory (LSTM) networks, Gated Recurrent Units (GRUs), and Transformer models. We also show that the performance of the PINN model is comparable to a sophisticated Gaussian infection state forecasting model that combines the compartmental model, a data observation model and a regression model for inferring parameters in the compartmental model. Nonetheless, the PINN model offers a simpler structure and is easier to implement. In summary, we perform a systematic study of the predictive capability of the PINN model in forecasting the dynamics of infectious diseases and our results showcase the potential of the proposed model as an efficient computational tool to enhance the current capacity of infectious disease forecasting.

Figure 7:

MASE for the predictions of PINNs models on cases, deaths, and hospitalizations when trained with varied lengths of historical data ranging from 4 weeks to 24 weeks.

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 $L_{\mathit{\text{data}}}$ 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 [22]. 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.

Figure 6:

MASE comparison with Recurrent Neural Networks (RNNs), Long Short-Term Memory (LSTM) networks, Gated Recurrent Units (GRUs), and Transformers on the number of cases, deaths, and hospitalizations for the following 1 – 4 weeks.