Learning COVID-19 Regional Transmission Using Universal Differential Equations in a SIR model

Highly-interconnected societies difficult to model the spread of infectious diseases such as COVID-19. Single-region SIR models fail to account for incoming forces of infection and expanding them to a large number of interacting regions involves many assumptions that do not hold in the real world. We propose using Universal Differential Equations (UDEs) to capture the influence of neighboring regions and improve the model's predictions in a combined SIR+UDE model. UDEs are differential equations totally or partially defined by a deep neural network (DNN). We include an additive term to the SIR equations composed by a DNN that learns the incoming force of infection from the other regions. The learning is performed using automatic differentiation and gradient descent to approach the change in the target system caused by the state of the neighboring regions. We compared the proposed model using a simulated COVID-19 outbreak against a single-region SIR and a fully data-driven model composed only of a DNN. The proposed UDE+SIR model generates predictions that capture the outbreak dynamic more accurately, but a decay in performance is observed at the last stages of the outbreak. The single-area SIR and the fully data-driven approach do not capture the proper dynamics accurately. Once the predictions were obtained, we employed the SINDy algorithm to substitute the DNN with a regression, removing the black box element of the model with no considerable increase in the error levels.