Spatial spread of COVID-19 outbreak in Italy using multiscale kinetic transport equations with uncertainty

Abstract

In this paper we introduce a space-dependent multiscale model to describe the spatial spread of an infectious disease under uncertain data with particular interest in simulating the onset of the COVID-19 epidemic in Italy. While virus transmission is ruled by a SEIAR type compartmental model, within our approach the population is given by a sum of commuters moving on a extra-urban scale and non commuters interacting only on the smaller urban scale. A transport dynamic of the commuter population at large spatial scales, based on kinetic equations, is coupled with a diffusion model for non commuters at the urban scale. Thanks to a suitable scaling limit, the kinetic transport model used to describe the dynamics of commuters, within a given urban area coincides with the diffusion equations that characterize the movement of non-commuting individuals. Because of the high uncertainty in the data reported in the early phase of the epidemic, the presence of random inputs in both the initial data and the epidemic parameters is included in the model. A robust numerical method is designed to deal with the presence of multiple scales and the uncertainty quantification process. In our simulations, we considered a realistic geographical domain, describing the Lombardy region, in which the size of the cities, the number of infected individuals, the average number of daily commuters moving from one city to another, and the epidemic aspects are taken into account through a calibration of the model parameters based on the actual available data. The results show that the model is able to describe correctly the main features of the spatial expansion of the first wave of COVID-19 in northern Italy.

Optimal control of epidemic spreading in presence of social heterogeneity

“…and Q i (f i (x, t)) remains the same as in (7). Note in particular that in (21) the control term does not depend on the time scale τ of the Fokker-Planck operator.…”

“…Here, we do not consider this aspect and we refer to recent attempts to overcome this issue based on tools of uncertainty quantification [1,34] that can be successfully employed in this context. For related approaches we mention also [6,7,11,31].…”

“…is mass and momentum preserving. In (7) σ and µ are positive constants which characterize the intensities of the diffusion and drift terms. The above description through Fokker-Planck operators is obtained thanks to an upscaling of an underlying microscopic dynamics which takes into account the typical aspects of human behaviour for what concerns the social attitudes [14].…”

“…where B(x) = 1/x is the interaction kernel assigning a low probability to interactions to individuals exhibiting a large number of contacts and assigning a high probability to interactions when x is small. Letting finally → 0 + (the limit of grazing interactions) one shows that (12) converges, up to extraction of a subsequence, to the Fokker-Planck equation (7), see e.g. [14,18].…”

Scalable Domain Decomposition Methods for Nonlinear and Time-Dependent Stochastic Systems

“…• The inclusion of uncertainty in the model (as in [158,167]) resulting from considering parameters as random variables or random fields leads to stochastic PDEs which can be solved by using sampling (Monte Carlo or quadrature) methods or sampling-free stochastic Galerkin methods [13]. Slow convergence of Monte Carlo or large stochastic dimensions can increase the number of deterministic sample evaluations which in turn increases the computational cost for sampling approaches.…”