arXiv subjects

Elisa Calzola

Publications and source records attributed to Elisa Calzola.

12 recordsLinked to original sources

arXiv2026

How opinions shape epidemics: a graphon-based kinetic approach

Understanding the mutual influence between social behavior and physical health is crucial for designing effective epidemic mitigation strategies. Individual interactions drive the evolution of opinions, which in turn shape how infectious diseases are perceived and consequently how they spread within a population, for instance through the adoption or rejection of preventive measures. At the same time, the distribution and dynamics of physical contacts play a fundamental role in determining transmission patterns. To this end, we develop a mathematical framework to analyze the coupled dynamics of opinion formation, disease transmission, and physical contacts by employing graphon-based networks, which capture heterogeneous and large-scale connectivity patterns typical of realistic social structures. The epidemic compartmental model further incorporates a kinetic description of microscopic level physical contacts, allowing for a consistent multiscale representation of interaction patterns. Starting from a microscopic description governed by interpersonal compromise and intrinsic self-thinking processes, we derive a kinetic compartmental epidemic model on graphons via a mean-field limit. This formulation allows us to investigate the joint evolution of the disease state and the opinion distribution, with a particular focus on the role of social networks and physical contacts. Numerical experiments demonstrate that the graphon-kinetic approach provides a comprehensive representation of the coupled opinion-epidemic dynamics, revealing new possibilities for controlling disease spread by shaping population opinion patterns.

physics.soc-ph↗

arXiv2026

Discovery of interaction and diffusion kernels in particle-to-mean-field multi-agent systems

We propose a data-driven framework to learn interaction kernels in stochastic multi-agent systems. Our approach aims at identifying the functional form of nonlocal interaction and diffusion terms directly from trajectory data, without any a priori knowledge of the underlying interaction structure. Starting from a discrete stochastic binary-interaction model, we formulate the inverse problem as a sequence of sparse regression tasks in structured finite-dimensional spaces spanned by compactly supported basis functions, such as piecewise linear polynomials. In particular, we assume that pairwise interactions between agents are not directly observed and that only limited trajectory data are available. To address these challenges, we propose two complementary identification strategies. The first based on random-batch sampling, which compensates for latent interactions while preserving the statistical structure of the full dynamics in expectation. The second based on a mean-field approximation, where the empirical particle density reconstructed from the data defines a continuous nonlocal regression problem. Numerical experiments demonstrate the effectiveness and robustness of the proposed framework, showing accurate reconstruction of both interaction and diffusion kernels even from partially observed. The method is validated on benchmark models, including bounded-confidence and attraction-repulsion dynamics, where the two proposed strategies achieve comparable levels of accuracy.

cs.LG↗

arXiv2026

Control of kinetic opinion dynamics in popularity-adaptive social networks

This paper presents a mathematical model for opinion dynamics in popularity-adaptive social networks, where both opinion spreading and the evolution of social media contacts depend on agents' popularity and the prominence of their views. While previous approaches accounted for the influence of popularity on opinion dynamics, we introduce a novel feedback mechanism in which opinion affects the formation of contacts. Within a kinetic modeling framework, we describe the evolution of the coupled dynamics of opinions and network structure, incorporating a class of control laws in order to promote interactions with popular individuals and amplify dominant opinions. Such control strategies are introduced to influence both opinion formation and connectivity, representing interventions such as awareness campaigns or moderation policies. Numerical results show how control strategies can mitigate polarization, foster consensus, or guide opinion distributions in dynamically evolving networks.

math.OC↗

arXiv2025

A second-order numerical scheme for optimal control of non-linear Fokker-Planck equations and applications in social dynamics

In this work, we present a second-order numerical scheme to address the solution of optimal control problems constrained by the evolution of nonlinear Fokker-Planck equations arising from socio-economic dynamics. In order to design an appropriate numerical scheme for control realization, a coupled forward-backward system is derived based on the associated optimality conditions. The forward equation, corresponding to the Fokker-Planck dynamics, is discretized using a structure preserving scheme able to capture steady states. On the other hand, the backward equation, modeled as a Hamilton-Jacobi-Bellman problem, is solved via a semi-Lagrangian scheme that supports large time steps while preserving stability. Coupling between the forward and backward problems is achieved through a gradient descent optimization strategy, ensuring convergence to the optimal control. Numerical experiments demonstrate second-order accuracy, computational efficiency, and effectiveness in controlling different examples across various scenarios in social dynamics. This approach provides a reliable computational tool for the study of opinion manipulation and consensus formation in socially structured systems.

math.NA↗

arXiv2025

A data-driven kinetic model for opinion dynamics with social network contacts

Opinion dynamics is an important and very active area of research that delves into the complex processes through which individuals form and modify their opinions within a social context. The ability to comprehend and unravel the mechanisms that drive opinion formation is of great significance for predicting a wide range of social phenomena such as political polarization, the diffusion of misinformation, the formation of public consensus, and the emergence of collective behaviors. In this paper, we aim to contribute to that field by introducing a novel mathematical model that specifically accounts for the influence of social media networks on opinion dynamics. With the rise of platforms such as Twitter, Facebook, and Instagram and many others, social networks have become significant arenas where opinions are shared, discussed, and potentially altered. To this aim after an analytical construction of our new model and through incorporation of real-life data from Twitter, we calibrate the model parameters to accurately reflect the dynamics that unfold in social media, showing in particular the role played by the so-called influencers in driving individual opinions towards predetermined directions.

physics.soc-ph↗

arXiv2025

A PINN approach for the online identification and control of unknown PDEs

Physics-Informed Neural Networks (PINNs) have revolutionized solving differential equations by integrating physical laws into neural networks training. This paper explores PINNs for open-loop optimal control problems (OCPs) with incomplete information, such as sparse initial and boundary data and partially unknown system parameters. We derive optimality conditions from the Lagrangian multipliers and use PINNs to predict the state, adjoint, and control variables. In contrast with previous methods, our approach integrates these elements into a single neural network and addresses scenarios with consistently limited data. In addition, we address the study of partially unknown equations identifying underlying parameters online by searching for the optimal solution recurring to a 2-in-series architecture of PINNs, in which scattered data of the uncontrolled solution is used. Numerical examples show the effectiveness of the proposed method even in scenarios characterized by a considerable lack of information.

math.OC↗

arXiv2024

Impact of opinion formation phenomena in epidemic dynamics: kinetic modeling on networks

After the recent COVID-19 outbreaks, it became increasingly evident that individuals' thoughts and beliefs can have a strong impact on disease transmission. It becomes therefore important to understand how information and opinions on protective measures evolve during epidemics. To this end, incorporating the impact of social media is essential to take into account the hierarchical structure of these platforms. In this context, we present a novel approach to take into account the interplay between infectious disease dynamics and socially-structured opinion dynamics. Our work extends a conventional compartmental framework including behavioral attitudes in shaping public opinion and promoting the adoption of protective measures under the influence of different degrees of connectivity. The proposed approach is capable to reproduce the emergence of epidemic waves. Specifically, it provides a clear link between the social influence of highly connected individuals and the epidemic dynamics. Through a heterogeneity of numerical tests we show how this comprehensive framework offers a more nuanced understanding of epidemic dynamics in the context of modern information dissemination and social behavior.

physics.soc-ph↗

arXiv2024

Emergence of condensation patterns in kinetic equations for opinion dynamics

In this work, we define a class of models to understand the impact of population size on opinion formation dynamics, a phenomenon usually related to group conformity. To this end, we introduce a new kinetic model in which the interaction frequency is weighted by the kinetic density. In the quasi-invariant regime, this model reduces to a Kaniadakis-Quarati-type equation with nonlinear drift, originally introduced for the dynamics of bosons in a spatially homogeneous setting. From the obtained PDE for the evolution of the opinion density, we determine the regime of parameters for which a critical mass exists and triggers blow-up of the solution. Therefore, the model is capable of describing strong conformity phenomena in cases where the total density of individuals holding a given opinion exceeds a fixed critical size. In the final part, several numerical experiments demonstrate the features of the introduced class of models and the related consensus effects.

nlin.AO↗

arXiv2023

A high-order scheme for mean field games

In this paper we propose a high-order numerical scheme for time-dependent mean field games systems. The scheme, which is built by combining Lagrange-Galerkin and semi-Lagrangian techniques, is consistent and stable for large time steps compared with the space steps. We provide a convergence analysis for the exactly integrated Lagrange-Galerkin scheme applied to the Fokker-Planck equation, and we propose an implementable version with inexact integration. Finally, we validate the convergence rate of the proposed scheme through the numerical approximation of two mean field games systems.

math.NA↗

arXiv2023

Exponential integrators for mean-field selective optimal control problems

In this paper we consider mean-field optimal control problems with selective action of the control, where the constraint is a continuity equation involving a non-local term and diffusion. First order optimality conditions are formally derived in a general framework, accounting for boundary conditions. Hence, the optimality system is used to construct a reduced gradient method, where we introduce a novel algorithm for the numerical realization of the forward and the backward equations, based on exponential integrators. We illustrate extensive numerical experiments on different control problems for collective motion in the context of opinion formation and pedestrian dynamics.

math.OC↗

arXiv2021

A semi-Lagrangian scheme for Hamilton-Jacobi-Bellman equations with oblique boundary conditions

We investigate in this work a fully-discrete semi-Lagrangian approximation of second order possibly degenerate Hamilton-Jacobi-Bellman (HJB) equations on a bounded domain with oblique boundary conditions. These equations appear naturally in the study of optimal control of diffusion processes with oblique reflection at the boundary of the domain. The proposed scheme is shown to satisfy a consistency type property, it is monotone and stable. Our main result is the convergence of the numerical solution towards the unique viscosity solution of the HJB equation. The convergence result holds under the same asymptotic relation between the time and space discretization steps as in the classical setting for semi-Lagrangian schemes. We present some numerical results that confirm the numerical convergence of the scheme.

math.NA↗

arXiv2020

Second order fully semi-Lagrangian discretizations of advection-diffusion-reaction systems

We propose a second order, fully semi-Lagrangian method for the numerical solution of systems of advection-diffusion-reaction equations, which employs a semi-Lagrangian approach to approximate in time both the advective and the diffusive terms. Standard interpolation procedures are used for the space discretization on structured and unstructured meshes. The proposed method allows for large time steps, while avoiding the solution of large linear systems, which would be required by an implicit time discretization technique. Numerical experiments demonstrate the effectiveness of the proposed approach and its superior efficiency with respect to more conventional explicit and implicit time discretizations

math.NA↗