Asymptotic-Preserving Neural Networks for inverse and forward problems in multiscale epidemic dynamics

The aim of this talk is to present some recent results in the mathematical modeling of epidemic phenomena through the use of kinetic equations and their numerical solution using physics-informed machine learning techniques. To account for aspects of spatial heterogeneity, the spatial spread of an infectious disease can be described by means of a class of multiscale systems of partial differential equations, in which a part of the population acting on an urban scale is characterized by parabolic diffusive behavior and the rest, acting on an extra-urban scale, follows a hyperbolic transport mechanism. However, the model parameters required to simulate the predictive dynamics of the propagation of the virus of interest entail a delicate calibration phase, often made even more challenging by the scarcity and uncertainty of data from official sources. Moreover, the initial and boundary conditions of the problem are always difficult to determine. In this context, Asymptotic-Preserving Neural Networks (APNNs) for hyperbolic transport models of epidemic spread were designed to solve the inverse problem (of estimating model parameters) and the forward problem (of predicting epidemic evolution) even in the face of sparse and incomplete observed data, and without losing the ability to describe the multiscale dynamics of the phenomenon, thanks to an appropriate AP formulation of the physics-informed neural network loss function. A series of numerical tests performed considering different epidemic scenarios confirms the validity of the proposed approach, highlighting the importance of the AP property of the neural network for the study of multiscale systems, especially when partially observed.