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

Giulia Bertaglia, Walter Boscheri, Giacomo Dimarco, Lorenzo Pareschi (15/6/2021 preprint arXiv:2106.07262)

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.

Hyperbolic compartmental models for epidemic spread on networks with uncertain data: application to the emergence of Covid-19 in Italy


Giulia Bertaglia, Lorenzo Pareschi (31/5/2021, preprint arXiv:2105.14258)

The importance of spatial networks in the spread of an epidemic is an essential aspect in modeling the dynamics of an infectious disease. Additionally, any realistic data-driven model must take into account the large uncertainty in the values reported by official sources, such as the amount of infectious individuals. In this paper we address the above aspects through a hyperbolic compartmental model on networks, in which nodes identify locations of interest, such as cities or regions, and arcs represent the ensemble of main mobility paths. The model describes the spatial movement and interactions of a population partitioned, from an epidemiological point of view, on the basis of an extended compartmental structure and divided into commuters, moving on a suburban scale, and non-commuters, acting on an urban scale.

Moment preserving Fourier-Galerkin spectral methods and application to the Boltzmann equation

 Lorenzo Pareschi, Thomas Rey (28/5/2021 preprint arXiv:2105.13158)

Spectral methods, thanks to the high accuracy and the possibility of using fast algorithms, represent an effective way to approximate collisional kinetic equations in kinetic theory. On the other hand, the loss of some local invariants can lead to the wrong long time behavior of the numerical solution. We introduce in this paper a novel Fourier-Galerkin spectral method that improves the classical spectral method by making it conservative on the moments of the approximated distribution, without sacrificing its spectral accuracy or the possibility of using fast algorithms.

Binary interaction methods for high dimensional global optimization and machine learning

Alessandro Benfenati, Giacomo Borghi, Lorenzo Pareschi (7/5/2021 preprint arXiv:2105.02695)

In this work we introduce a new class of gradient-free global optimization methods based on a binary interaction dynamics governed by a Boltzmann type equation. In each interaction the particles act taking into account both the best microscopic binary position and the best macroscopic collective position. In the mean-field limit we show that the resulting Fokker-Planck partial differential equations generalize the current class of consensus based optimization (CBO) methods. For the latter methods, convergence to the global minimizer can be shown for a large class of functions.

Anisotropic Diffusion in Consensus-based Optimization on the Sphere

Massimo Fornasier, Hui Huang, Lorenzo Pareschi, Philippe Sünnen (1/4/2021 preprint arXiv:2104.00420)

In this paper we are concerned with the global minimization of a possibly non-smooth and non-convex objective function constrained on the unit hypersphere by means of a multi-agent derivative-free method. The proposed algorithm falls into the class of the recently introduced Consensus-Based Optimization. In fact, agents move on the sphere driven by a drift towards an instantaneous consensus point, which is computed as a convex combination of agent locations, weighted by the cost function according to Laplaces principle, and it represents an approximation to a global minimizer. The dynamics is further perturbed by an anisotropic random vector field to favor exploration.