Approaching epidemiological dynamics of COVID-19 with physics-informed neural networks

TL;DR

This work embeds the Susceptible-Infected-Recovered (SIR) framework into physics-informed neural networks (PINNs) to study epidemic dynamics, validating on synthetic SAIRD data and real-world Germany COVID-19 records. By coupling data-informed loss with ODE residuals, the approach learns both network parameters and key epidemiological parameters ($\beta$, $\gamma$) while enforcing the SIR/SAIRD dynamics. Experiments show that PINNs with a simple SIR prior can closely track synthetic trajectories and provide robust predictions for real data, with the physics regularization improving stability and peak accuracy. The method offers a data-efficient pathway to infer and forecast epidemic dynamics under limited or noisy observations, suggesting extensions to richer compartmental models and PINN architectures in future work.

Abstract

A physics-informed neural network (PINN) embedded with the susceptible-infected-removed (SIR) model is devised to understand the temporal evolution dynamics of infectious diseases. Firstly, the effectiveness of this approach is demonstrated on synthetic data as generated from the numerical solution of the susceptible-asymptomatic-infected-recovered-dead (SAIRD) model. Then, the method is applied to COVID-19 data reported for Germany and shows that it can accurately identify and predict virus spread trends. The results indicate that an incomplete physics-informed model can approach more complicated dynamics efficiently. Thus, the present work demonstrates the high potential of using machine learning methods, e.g., PINNs, to study and predict epidemic dynamics in combination with compartmental models.

Figures (14)

• Figure 1: Schematic illustration of the interactions between the compartments in the SIR model.
• Figure 2: Diagram for the SAIRD model which is inspired from angeli2022modeling illustrating the interactions of compartment.
• Figure 3: A schematic diagram of physics-informed neural networks (PINNs). The black dashed-line block is a common neural network that takes a time $t$ as input and the output is $U$, $\beta$ and $\gamma$ are weights and biases, respectively. The orange dashed-line block stands for the calculation of residual loss. The loss function consists of a mismatch of boundaries and initial conditions for the observed data (data loss). The residuals of the ODE is a set of random points in the spatial-temporal domain (residuals). The parameters of the PINNs can be optimized by minimising the loss $MSE = MSE_{{data }}+MSE_{{residuals}}$.
• Figure 4: Schematic diagram of the SIR-dynamics informed neural network. The black-dashed frame represents the dense neural network used here. The green-dashed frame, on the other hand, represents the SIR-informed neural network, which takes time $t$ as input and outputs the susceptible-$(S)$, infected-$(I)$ and removed $(R)$ populations. The box labeled ‘ODEs’ represents the computation of the residual, with respect to the SIR model. The label ‘Loss’ is comprised of two parts: the mismatch between the available data and the network output, on one hand and the physical residual, on the other hand. The NN fits simultaneously both, the data and infers the dynamic parameters $\beta$ and $\gamma$, by satisfying the ODE dynamics, by minimizing loss function.
• Figure 5: A Mathematical model generated time evolution of an example SAIRD model. The solid yellow line represents the number of susceptible people in the population, the solid grey line represents the number of asymptomatic infected people, the solid blue line represents the number of recovered people, and the solid red line stand for the number of active infected person. The solid black line is the death population, and The population is assumed to be constant (N=1000).