Most epidemiological models are rooted in the pioneering compartmental modeling proposed by Kermack and McKendrick, based on systems of ODEs, which describe only the temporal evolution of the spread of infectious diseases, neglecting the spatial component in favor of an assumption of population and territorial homogeneity, and taking into account a deterministic framework.
Generally, the concept of the average behavior of a population is sufficient to have a first reliable description of an epidemic scenario, but the inclusion of the spatial component becomes crucial when it is necessary to consider spatially heterogeneous interventions. Moreover, initial conditions and modeling parameters, particularly in social sciences, are always affected by uncertainty and must be so considered when attempting to solve the problem numerically.
In this talk, drawing inspiration from kinetic theory, we present recent advances in the development of stochastic hyperbolic transport models proposed to study the propagation of an epidemic phenomenon described by the spatial movement and interactions of a population divided into commuters, moving on a wide scale with an advective behavior, and non-commuters, acting only on urban scales in a diffusive regime.
The resulting models are solved numerically through a suitable stochastic Asymptotic-Preserving IMEX Runge-Kutta Finite Volume Collocation Method, which ensures consistent treatment of the system of equations, without loss of accuracy when entering the stiff, diffusive regime. Application studies on the spread of the COVID-19 pandemic in Italy assess the validity of the proposed methodology.