Kinetic modelling of epidemic dynamics: social contacts, control with uncertain data, and multiscale spatial dynamics

Giacomo Albi, Giulia Bertaglia, Walter Boscheri, Giacomo Dimarco, Lorenzo Pareschi, Giuseppe Toscani, Mattia Zanella (4/10/2021 to appear in Predicting Pandemics in a Globally Connected World, Vol. 1, N. Bellomo and M. Chaplain Editors, Springer-Nature (2021). Preprint arXiv:2110.00293)

In this survey we report some recent results in the mathematical modeling of epidemic phenomena through the use of kinetic equations. We initially consider models of interaction between agents in which social characteristics play a key role in the spread of an epidemic, such as the age of individuals, the number of social contacts, and their economic wealth. Subsequently, for such models, we discuss the possibility of containing the epidemic through an appropriate optimal control formulation based on the policy maker's perception of the progress of the epidemic. The role of uncertainty in the data is also discussed and addressed.

Spreading of fake news, competence, and learning: kinetic modeling and numerical approximation

 Jonathan Franceschi, Lorenzo Pareschi (28/9/2021 preprint arXiv:2109.14087)

The rise of social networks as the primary means of communication in almost every country in the world has simultaneously triggered an increase in the amount of fake news circulating online. This fact became particularly evident during the 2016 U.S. political elections and even more so with the advent of the COVID-19 pandemic. Several research studies have shown how the effects of fake news dissemination can be mitigated by promoting greater competence through lifelong learning and discussion communities, and generally rigorous training in the scientific method and broad interdisciplinary education. The urgent need for models that can describe the growing infodemic of fake news has been highlighted by the current pandemic.

Mean-field particle swarm optimization

 

Sara Grassi, Hui Huang, Lorenzo Pareschi, Jinniao Qiu (3/8/2021 to appear in Modeling and Simulation for Collective Dynamics, IMS Lecture Note Series, World Scientific, preprint arXiv:2108.00393)

In this work we survey some recent results on the global minimization of a non-convex and possibly non-smooth high dimensional objective function by means of particle based gradient-free methods. Such problems arise in many situations of contemporary interest in machine learning and signal processing. After a brief overview of metaheuristic methods based on particle swarm optimization (PSO), we introduce a continuous formulation via second-order systems of stochastic differential equations that generalize PSO methods and provide the basis for their theoretical analysis. Subsequently, we will show how through the use of mean-field techniques it is possible to derive in the limit of large particles number the corresponding mean-field PSO description based on Vlasov-Fokker-Planck type equations.

A bi-fidelity stochastic collocation method for transport equations with diffusive scaling and multi-dimensional random inputs

 

Liu Liu, Lorenzo Pareschi, Xueyu Zhu (20/07/2021 preprint arXiv:2107.09250)

In this paper, we consider the development of efficient numerical methods for linear transport equations with random parameters and under the diffusive scaling. We extend to the present case the bi-fidelity stochastic collocation method introduced in [33,50,51]. For the high-fidelity transport model, the asymptotic-preserving scheme [29] is used for each stochastic sample. We employ the simple two-velocity Goldstein-Taylor equation as low-fidelity model to accelerate the convergence of the uncertainty quantification process.

On the construction of conservative semi-Lagrangian IMEX advection schemes for multiscale time dependent PDEs

Walter Boscheri, Maurizio Tavelli, Lorenzo Pareschi (14/07/2021 preprint arXiv:2107.06956)

This article is devoted to the construction of a new class of semi-Lagrangian (SL) schemes with implicit-explicit (IMEX) Runge-Kutta (RK) time stepping for PDEs involving multiple space-time scales. The semi-Lagrangian (SL) approach fully couples the space and time discretization, thus making the use of RK strategies particularly difficult to be combined with. First, a simple scalar advection-diffusion equation is considered as a prototype PDE for the development of a high order formulation of the semi-Lagrangian IMEX algorithms. The advection part of the PDE is discretized explicitly at the aid of a SL technique, while an implicit discretization is employed for the diffusion terms. Second, the SL-IMEX approach is extended to deal with hyperbolic systems with multiple scales, including balance laws, that involve shock waves and other discontinuities.