Lotka-Volterra-type kinetic equations for competing species

by Mattia Zanella

A. Bondesan, M. Menale, G. Toscani, M. Zanella.

Preprint arXiv, 2025.

In this work, we examine a kinetic framework for modeling the time evolution of size distribution densities of two populations governed by predator–prey interactions. The model builds upon the classical Boltzmann-type equations, where the dynamics arise from elementary binary interactions between the populations.

The model uniquely incorporates a linear redistribution operator to quantify the birth rates in both populations, inspired by wealth redistribution operators. We prove that, under a suitable scaling regime, the Boltzmann formulation transitions to a system of coupled Fokker–Planck-type equations. These equations describe the evolution of the distribution densities and link the macroscopic dynamics of their mean values to a Lotka–Volterra system of ordinary differential equations, with parameters explicitly derived from the microscopic interaction rules. We then determine the local equilibria of the Fokker–Planck system, which are Gamma-type densities, and investigate the problem of relaxation of its solutions toward these kinetic equilibria, in terms of their moments’ dynamics. The results establish a bridge between kinetic modeling and classical population dynamics, offering a multiscale perspective on predator–prey systems.

Control of Overpopulated Tails in Kinetic Epidemic Models

by Mattia Zanella

A. Medaglia, M. Zanella

Preprint arXiv, 2024.

We introduce model-based transition rates for controlled compartmental models in mathematical epidemiology, with a focus on the effects of control strategies applied to interacting multi-agent systems describing
contact formation dynamics.

In the framework of kinetic control problems, we compare two prototypical control protocols: one additive control directly influencing the dynamics and another targeting the interaction strength between agents. The emerging controlled macroscopic models are derived for an SIR compartmentalization to illustrate their impact on epidemic progression and contact interaction dynamics. Numerical results show the effectiveness of this approach in steering the dynamics and controlling epidemic trends, even in scenarios where contact distributions exhibit an overpopulated tail.

Understanding the Impact of Evaluation Metrics in Kinetic Models for Consensus-based Segmentation

by Mattia Zanella

R. F. Cabini, H. Tettamanti, M. Zanella.

Entropy, 27(2), 149, 2025. (Preprint arXiv)

In this article we extend a recently introduced kinetic model for consensus-based segmentation of images. In particular, we will interpret the set of pixels of a 2D image as an interacting particle system which evolves in time in view of a consensus-type process

obtained by interactions between pixels and external noise. Thanks to a kinetic formulation of the introduced model we derive the large time solution of the model. We will show that the choice of parameters defining the segmentation task can be chosen from a plurality of loss functions characterising the evaluation metrics.

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

by Mattia Zanella

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

by Mattia Zanella

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

by Mattia Zanella

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

by Mattia Zanella

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

by Mattia Zanella

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

by Mattia Zanella

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

by Mattia Zanella

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.