arXiv subjects

Giuseppe Toscani

Publications and source records attributed to Giuseppe Toscani.

At least 19 recordsLinked to original sources

arXiv2026

From Individual-Based Models to Macroscopic Dynamics of Antimicrobial Resistance

We introduce and discuss a kinetic framework describing the time evolution of the statistical distributions of a population divided into the compartments of susceptible, infectious, recovered, and resistant in the presence of a microbial infection driven by susceptible infectious interactions. Our main objective is to quantify the impact of excessive and inappropriate antimicrobial use, which accelerates the spread of resistance by enabling a fraction of infectious individuals to transition into the resistant compartment. The model consists of a system of Boltzmann type equations capturing binary interactions between susceptible and infectious individuals, complemented by linear redistribution operators that represent recovery, the development of resistance, and reinfection processes. In the grazing collision limit, we show that this Boltzmann system is well approximated by a system of coupled Fokker Planck equations. This limiting description allows for a more tractable analysis of the dynamics, including the characterization of the long-time behavior of the population densities. Our analysis highlights how interaction terms drive the system toward a stable equilibrium and quantifies the effects of inappropriate antimicrobial use on the distribution of resistant individuals. Overall, the results offer a multiscale perspective that bridges kinetic theory with classical epidemic modeling.

q-bio.PE↗

arXiv2026

Large-time behaviour for coupled systems of Lotka-Volterra-type Fokker-Planck equations

We study a system of Fokker-Planck equations recently introduced to describe the temporal evolution of statistical distributions of population densities with predator-prey interactions. At the macroscopic level, the system recovers a Lotka-Volterra model and defines an explicit family of equilibrium densities that depend on the form of the diffusion coefficient. By introducing Energy-type distances, we rigorously establish exponential convergence to equilibrium in appropriate homogeneous Sobolev spaces, with a rate explicitly determined by the dissipative contribution of the interaction term. The analysis highlights the intrinsic energy dissipation mechanism governing the dynamics and clarifies how the evolution of expected quantities determines the emergence of a stable equilibrium configuration. This approach provides a new perspective on the convergence to equilibrium for problems with time-dependent coefficients.

math.AP↗

arXiv2025

Individual-Based Foundation of SIR-Type Epidemic Models: mean-field limit and large time behaviour

We introduce a kinetic framework for modeling the time evolution of the statistical distributions of the population densities in the three compartments of susceptible, infectious, and recovered individuals, under epidemic spreading driven by susceptible-infectious interactions. The model is based on a system of Boltzmann-type equations describing binary interactions between susceptible and infectious individuals, supplemented with linear redistribution operators that account for recovery and reinfection dynamics. The mean values of the kinetic system recover a SIR-type model with reinfection, where the macroscopic parameters are explicitly derived from the underlying microscopic interaction rules. In the grazing collision regime, the Boltzmann system can be approximated by a system of coupled Fokker-Planck equations. This limit allows for a more tractable analysis of the dynamics, including the large-time behavior of the population densities. In this context, we rigorously prove the convergence to equilibrium of the resulting mean-field system in a suitable Sobolev space by means of the so-called energy distance. The analysis reveals the dissipative structure of the dynamics and the role of the interaction terms in driving the system toward a stable equilibrium configuration. These results provide a multi-scale perspective connecting kinetic theory with classical epidemic models.

math.AP↗

arXiv2025

On Rank Graduation Metrics for High Dimensional Ordinal Data

Evaluating the reliability of machine learning classifications remains a fundamental challenge in Artificial Intelligence (AI), particularly when the target variable is multidimensional. Classification variables can be expressed by means of a categorical scale which, at best, is ordinal. Because ordinal data lack a natural metric structure in their underlying space, most conventional distance measures aimed at assessing the accuracy of machine learning classifications cannot be directly or meaningfully applied. In this paper, we develop a mathematical framework for comparing ordinal data based on a family of Rank Graduation $(\mathrm{RGX}_p)$ \emph{metrics}. We demonstrate that these metrics can quantify the proportion of variability of the response explained by the predictions, in a similar manner as the predictive $R^2$ for continuous response variables. After establishing theoretical connections between the $\mathrm{RGX}_p$ family and other prominent metrics in AI, we conduct extensive experiments across diverse datasets and learning tasks to evaluate their empirical performance. The results underscore the versatility, interpretability, and robustness of the $\mathrm{RGX}_p$ metrics as a principled foundation for developing trustworthy and SAFE AI systems.

math.OC↗

arXiv2025

Lotka-Volterra-type kinetic equations for interacting species

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.

math.AP↗

arXiv2025

Energy distance and evolution problems: a promising tool for kinetic equations

We study the rate of convergence to equilibrium of the solutions to Fokker-Planck type equations with linear drift by means of Cramér and Energy distances, which have been recently widely used in problems related to AI, in particular for tasks related to machine learning. In all cases in which the Fokker-Planck type equations can be treated through these distances, it is shown that the rate of decay is improved with respect to known results which are based on the decay of relative entropy.

math.AP↗

arXiv2025

From Kinetic Theory to AI: a Rediscovery of High-Dimensional Divergences and Their Properties

Selecting an appropriate divergence measure is a critical aspect of machine learning, as it directly impacts model performance. Among the most widely used, we find the Kullback-Leibler (KL) divergence, originally introduced in kinetic theory as a measure of relative entropy between probability distributions. Just as in machine learning, the ability to quantify the proximity of probability distributions plays a central role in kinetic theory. In this paper, we present a comparative review of divergence measures rooted in kinetic theory, highlighting their theoretical foundations and exploring their potential applications in machine learning and artificial intelligence.

math-ph↗

arXiv2025

Measuring inequality in society-oriented Lotka--Volterra-type kinetic equations

We present a possible approach to measuring inequality in a system of coupled Fokker-Planck-type equations that describe the evolution of distribution densities for two populations interacting pairwise due to social and/or economic factors. The macroscopic dynamics of their mean values follow a Lotka-Volterra system of ordinary differential equations. Unlike classical models of wealth and opinion formation, which tend to converge toward a steady-state profile, the oscillatory behavior of these densities only leads to the formation of local equilibria within the Fokker-Planck system. This makes tracking the evolution of most inequality measures challenging. However, an insightful perspective on the problem is obtained by using the coefficient of variation, a simple inequality measure closely linked to the Gini index. Numerical experiments confirm that, despite the system's oscillatory nature, inequality initially tends to decrease.

q-fin.GN↗

arXiv2025

Supercritical Fokker-Planck equations for consensus dynamics: large-time behaviour and weighted Nash-type inequalities

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 main feature of this Fokker-Planck equation is the presence of a variable diffusion coefficient, a nonlinear drift and boundaries, which introduce new challenging mathematical problems in the study of its long-time behavior. In particular, propagation of regularity is shown as a consequence of new weighted Nash and Gagliardo-Nirenberg inequalities.

math.AP↗

arXiv2024

How to measure multidimensional variation?

The coefficient of variation, which measures the variability of a distribution from its mean, is not uniquely defined in the multidimensional case, and so is the multidimensional Gini index, which measures the inequality of a distribution in terms of the mean differences among its observations. In this paper, we connect these two notions of sparsity, and propose a multidimensional coefficient of variation based on a multidimensional Gini index. We demonstrate that the proposed coefficient possesses the properties of the univariate coefficient of variation. We also show its connection with the Voinov-Nikulin coefficient of variation, and compare it with the other multivariate coefficients available in the literature.

math.ST↗

arXiv2024

Multivariate Gini-type discrepancies

Measuring distances in a multidimensional setting is a challenging problem, which appears in many fields of science and engineering. In this paper, to measure the distance between two multivariate distributions, we introduce a new measure of discrepancy which is scale invariant and which, in the case of two independent copies of the same distribution, and after normalization, coincides with the scaling invariant multidimensional version of the Gini index recently proposed in [34]. A byproduct of the analysis is an easy-to-handle discrepancy metric, obtained by application of the theory to a pair of Gaussian multidimensional densities. The obtained metric does improve the standard metrics, based on the mean squared error, as it is scale invariant. The importance of this theoretical finding is illustrated by means of a real problem that concerns measuring the importance of Environmental, Social and Governance factors for the growth of small and medium enterprises.

stat.ME↗

arXiv2024

Extending the Gini Index to Higher Dimensions via Whitening Processes

Measuring the degree of inequality expressed by a multivariate statistical distribution is a challenging problem, which appears in many fields of science and engineering. In this paper, we propose to extend the well known univariate Gini coefficient to multivariate distributions, by maintaining most of its properties. Our extension is based on the application of whitening processes that possess the property of scale stability.

math.ST↗

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↗

arXiv2024

New Trends on the Systems Approach to Modeling SARS-CoV-2 Pandemics in a Globally Connected Planet

This paper presents a critical analysis of the literature and perspective research ideas for modeling the epidemics caused by the SARS-CoV-2 virus. It goes beyond deterministic population dynamics to consider several key complexity features of the system under consideration. In particular, the multiscale features of the dynamics from contagion to the subsequent dynamics of competition between the immune system and the proliferating virus. Other topics addressed in this work include the propagation of epidemics in a territory, taking into account local transportation networks, the heterogeneity of the population, and the study of social and economic problems in populations involved in the spread of epidemics. The overall content aims to show how new mathematical tools can be developed to address the above topics and how mathematical models and simulations can contribute to the decision making of crisis managers.

q-bio.PE↗

arXiv2024

Modelling contagious viral dynamics: a kinetic approach based on mutual utility

The temporal evolution of a contagious viral disease is modelled as the dynamic progression of different classes of population with individuals interacting pairwise. This interaction follows a binary mechanism typical of kinetic theory, wherein agents aim to improve their condition with respect to a mutual utility target. To this end, we introduce kinetic equations of Boltzmann-type to describe the time evolution of the probability distributions of the multi-agent system. The interactions between agents are defined using principles from price theory, specifically employing Cobb-Douglas utility functions for binary exchange and the Edgeworth box to depict the common exchange area where utility increases for both agents. Several numerical experiments presented in the paper highlight the significance of this mechanism in driving the phenomenon toward endemicity.

q-bio.PE↗

arXiv2024

Measuring multidimensional inequality: a new proposal based on the Fourier transform

Inequality measures are quantitative measures that take values in the unit interval, with a zero value characterizing perfect equality. Although originally proposed to measure economic inequalities, they can be applied to several other situations, in which one is interested in the mutual variability between a set of observations, rather than in their deviations from the mean. While unidimensional measures of inequality, such as the Gini index, are widely known and employed, multidimensional measures, such as Lorenz Zonoids, are difficult to interpret and computationally expensive and, for these reasons, are not much well known. To overcome the problem, in this paper we propose a new scaling invariant multidimensional inequality index, based on the Fourier transform, which exhibits a number of interesting properties, and whose application to the multidimensional case is rather straightforward to calculate and interpret.

physics.soc-ph↗

arXiv2023

On a kinetic description of Lotka-Volterra dynamics

Owing to the analogies between the problem of wealth redistribution with taxation in a multi-agent society, we introduce and discuss a kinetic model describing the statistical distributions in time of the sizes of groups of biological systems with prey-predator dynamic. While the evolution of the mean values is shown to be driven by a classical Lotka-Volterra system of differential equations, it is shown that the time evolution of the probability distributions of the size of groups of the two interacting species is heavily dependent both on a kinetic redistribution operator and the degree of randomness present in the system. Numerical experiments are given to clarify the time-behavior of the distributions of groups of the species.

q-bio.PE↗

arXiv2023

Measuring multidimensional heterogeneity in emergent social phenomena

Measuring inequalities in a multidimensional framework is a challenging problem which is common to most field of science and engineering. Nevertheless, despite the enormous amount of researches illustrating the fields of application of inequality indices, and of the Gini index in particular, very few consider the case of a multidimensional variable. In this paper, we consider in some details a new inequality index, based on the Fourier transform, that can be fruitfully applied to measure the degree of inhomogeneity of multivariate probability distributions. This index exhibits a number of interesting properties that make it very promising in quantifying the degree of inequality in data sets of complex and multifaceted social phenomena.

physics.soc-ph↗