Condensation effects in kinetic models for consensus dynamics: finite-time blow-up and regularity aspects

G. Toscani, M. Zanella

Preprint arXiv, 2024

We study the main properties of the solution of a Fokker-Planck equation characterized by a variable diffusion coefficient and a polynomial superlinear drift, modeling the formation of consensus in a large interacting system of individuals.

The Fokker-Planck equation is derived from the kinetic description of the dynamics of a quantum particle system, and in presence of a high nonlinearity in the drift operator, mimicking the effects of the mass in the alignment forces, allows for steady states similar to a Bose-Einstein condensate. The analysis shows that the regularity of the solution is strongly linked to the degree of nonlinearity in the drift, and that finite-time blow-up of the solution can occur when the degree of nonlinearity is sufficiently high. However, the presence of diffusion prevents the solution from forming condensation after the blow-up time.

Derivation of macroscopic epidemic models from multi-agent systems

M. Zanella

Preprint arXiv, 2024

We present a systematic review of some basic results on the derivation of classical epidemiological models from simple agent-based dynamics. The evolution of large populations is coupled with the dynamics of the contact distribution, providing insights into how individual behaviors impact macroscopic epidemiological trends.

The resulting set of equations incorporates local characteristics of the operator governing the emergence of a family of contact distributions. To validate the proposed approach, we provide several numerical results based on asymptotic preserving methods, demonstrating their effectiveness in capturing the multi-scale nature of the problem and ensuring robust performance across different regimes.

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

G. Albi, E. Calzola, G. Dimarco, M. Zanella.

Preprint arXiv, 2024

After the recent COVID-19 outbreaks, it became increasingly evident that individuals’
thoughts and beliefs can have a strong impact the 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.

Emerging properties of the degree distribution in large non-growing networks

J. Franceschi, L. Pareschi, M. Zanella.

Preprint arXiv, 2024

The degree distribution is a key statistical indicator in network theory, often used to understand how information spreads across connected nodes.

In this paper, we focus on non-growing networks formed through a rewiring algorithm and develop kinetic Boltzmann-type models to capture the emergence of degree distributions that characterize both preferential attachment networks and random networks. Under a suitable mean-field scaling, these models reduce to a Fokker-Planck-type partial differential equation with an affine diffusion coefficient, that is consistent with a well-established master equation for discrete rewiring processes. We further analyze the convergence to equilibrium for this class of Fokker-Planck equations, demonstrating how different regimes – ranging from exponential to algebraic rates – depend on network parameters. Our results provide a unified framework for modeling degree distributions in non-growing networks and offer insights into the long-time behavior of such systems.

Predictability of viral load kinetics in the early phases of SARS-CoV-2 through a model-based approach

A. Bondesan, A. Piralla, E. Ballante, A. M. G. Pitrolo, S. Figini, F. Baldanti, M. Zanella

Preprint arXiv, 2024.

A pipeline to evaluate the evolution of viral dynamics based on a new model-driven approach has been developed in the present study. The proposed methods exploit real data and the multiscale structure of the infection dynamics to provide robust predictions of the epidemic dynamics. We focus on viral load kinetics whose dynamical features are typically available in the symptomatic stage of the infection. Hence, the epidemiological evolution is obtained by relying on a compartmental approach characterized by a varying infection rate to estimate early-stage viral load dynamics, of which few data are available. We test the proposed approach with real data of SARS-CoV-2 viral load kinetics collected from patients living in an Italian province. The considered database refers to early-phase infections, whose viral load kinetics are not affected by mass vaccination policies in Italy. Our contribution is devoted to provide an effective computational pipeline to evaluate in real time the evolution of infectivity. Comprehending the factors influencing the in-host viral dynamics represents a fundamental tool to provide robust public health strategies. This pilot study could be implemented in further investigations involving other respiratory viruses, to better clarify the process of viral dynamics as a preparatory action for future pandemics.

Emergence of condensation patterns in kinetic equations for opinion dynamics

E. Calzola, G. Dimarco, G. Toscani, M. Zanella

Physica D: Nonlinear Phenomena, 470 Part A: 134356, 2024. (Preprint arXiv)

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.

Uncertainty quantification for charge transport in GNRs through particle Galerkin methods for the semiclassical Boltzmann equation

A. Medaglia, G. Nastasi, V. Romano, M. Zanella

Preprint arXiv, 2024.

In this article, we investigate some issues related to the quantification of uncertainties associated with the electrical properties of graphene nanoribbons. The approach is suited to understand the effects of missing information linked to the difficulty of fixing some material parameters, such as the band gap, and the strength of the applied electric field. In particular, we focus on the extension of particle Galerkin methods for kinetic equations in the case of the semiclassical Boltzmann equation for charge transport in graphene nanoribbons with uncertainties. To this end, we develop an efficient particle scheme which allows us to parallelize the computation and then, after a suitable generalization of the scheme to the case of random inputs, we present a Galerkin reformulation of the particle dynamics, obtained by means of a generalized polynomial chaos approach, which allows the reconstruction of the kinetic distribution. As a consequence, the proposed particle-based scheme preserves the physical properties and the positivity of the distribution function also in the presence of a complex scattering in the transport equation of electrons. The impact of the uncertainty of the band gap and applied field on the electrical current is analyzed.

Measure-valued death state and local sensitivity analysis for Winfree models with uncertain high-order couplings

S.-Y. Ha, M. Kang, J. Yoon, M. Zanella

Preprint arXiv, 2024.

We study the measure-valued death state and local sensitivity analysis of the Winfree model and its mean-field counterpart with uncertain high-order couplings. The Winfree model is the first mathematical model for synchronization, and it can cast as the effective approximation of the pulse-coupled model for synchronization, and it exhibits diverse asymptotic patterns depending on system parameters and initial data. For the proposed models, we present several frameworks leading to oscillator death in terms of system parameters and initial data, and the propagation of regularity in random space. We also present several numerical tests and compare them with analytical results.

Breaking consensus in kinetic opinion formation models on graphons

B. Düring, J. Franceschi, M.-T. Wolfram, M. Zanella

Journal of Nonlinear Science, 34:79,2024 . (Preprint arXiv)

In this work we propose and investigate a strategy to prevent consensus in kinetic models for opinion formation. We consider a large interacting agent system, and assume that agent interactions are driven by compromise as well as self-thinking dynamics and also modulated by an underlying static social network.

This network structure is included using so-called graphons, which modulate the interaction frequency in the corresponding kinetic formulation. We then derive the corresponding limiting Fokker Planck equation, and analyze its large time behavior. This microscopic setting serves as a starting point for the proposed control strategy, which steers agents away from mean opinion and is characterised by a suitable penalization depending on the properties of the graphon. We show that this minimalist approach is very effective by analyzing the quasi-stationary solutions mean-field model in a plurality of graphon structures. Several numerical experiments are also provided the show the effectiveness of the approach in preventing the formation of consensus steering the system towards a declustered state.

Reduced variance random batch methods for nonlocal PDEs

L. Pareschi, M. Zanella

Acta Appl. Math., 191, 4, 2024. (Preprint arXiv)

Random Batch Methods (RBM) for mean-field interacting particle systems enable the reduction of the quadratic computational cost associated with particle interactions to a near-linear cost. The essence of these algorithms lies in the random partitioning of the particle ensemble into smaller batches at each time step.

The interaction of each particle within these batches is then evolved until the subsequent time step. This approach effectively decreases the computational cost by an order of magnitude while increasing the amount of fluctuations due to the random partitioning. In this work, we propose a variance reduction technique for RBM applied to nonlocal PDEs of Fokker-Planck type based on a control variate strategy. The core idea is to construct a surrogate model that can be computed on the full set of particles at a linear cost while maintaining enough correlations with the original particle dynamics. Examples from models of collective behavior in opinion spreading and swarming dynamics demonstrate the great potential of the present approach.