A Physics-Informed Deep Learning Framework for Solving Inverse Problems in Epidemiology

Sammanfattning: This thesis develops and evaluates a physics-informed neural network (PINN) modelling framework for solving inverse problems in epidemiology. The PINN works by modifying the standard mean squared error loss function of the neural network, by adding a term penalizing deviations from a given compartmental model's system of ordinary differential equations. To find estimates for the unknown parameters in the compartmental model, such as the transmission rate, this compound loss function is then minimized with respect to both the neural network's inherent parameters and the unknown parameters in the compartmental model. The following question guided the study: Given time-series data consisting of the 7-day rolling average of the daily incidence of new infectious individuals, and a compartmental model for that data, can a PINN learn the corresponding time-dependent transmission rate parameter? The PINN framework was first validated on simulated (synthetic) epidemiological data, where the PINN was tasked o retrieve the unknown parameters in a given three-compartment SIR (Susceptible-Infectious-Recovered) model. It was then tested on real Covid-19 case data, and tasked to retrieve a time-dependent transmission rate parameter in an SEIR (Susceptible-Exposed-INfectious-Recovered) model. Two different approaches to learning a time-dependent transmission rate based were compared: one assumed a sigmoidal transmission rate with three unknown parameters (model IIa); the other allowed the transmission rate to be aprameterized by the neural netowrk, by adding it as an additional output node (Model IIb). The findings were that the PINN was able to reliably retrieve unknown constant parameters in an SIR model based on simulated data. However, it was also found that the PINN's parameter estimates can be sensitive to noise. Moreover, when learning a time-dependent transmission rate with Model IIb, an important finding was that the PINN would struggle to converge to the true transmission rate in regions of time when there were a relatively low number of total infections. Nevertheless, when employed on Covid-19 data from Stockholm county corresponding to the first wave, the PINN was still able to extract a time-dependent transmission rate in the given SEIR model, largely consistent with the 7-day rolling average of the incidence of new cases, without imposing any a priori assumptions on the shape of the transmission rate other than it should be positive.