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Nicolas Saintier

Publications and source records attributed to Nicolas Saintier.

At least 19 recordsLinked to original sources
arXiv2026

Kinetic SIS opinion-driven models with asymmetric awareness feedback: macroscopic limit and polarization

We study a kinetic multi-agent framework coupling opinion dynamics with epidemic spreading, where individual social behaviour both affects and is affected by disease transmission. Each agent is characterised by an epidemiological state and a continuous opinion variable measuring compliance with non-pharmaceutical interventions. The key mechanism of the model is an asymmetric opinion update driven by epidemic encounters: infection events induce more cautious attitudes, while failed transmissions push individuals toward more extreme opinions. We focus on a prototypical SIS setting, for which we derive a macroscopic kinetic description and, in a fast social-interaction regime, a reduced system of differential equations capturing the feedback between epidemic prevalence and opinion evolution. Convergence of the reduced model is rigorously quantified through a modified Wasserstein distance. Numerical simulations highlight how infection-induced awareness and non-infection-driven extremization jointly shape collective epidemic-opinion dynamics.

physics.soc-ph↗
arXiv2025

Fractional Lane-Emden Hamiltonian systems

In this work, our interest lies in proving the existence of solutions to the following Fractional Lane-Emden Hamiltonian system: $$ \begin{cases} (-\Delta)^s u = H_v(x,u,v) & \text{in }\Omega,\\ (-\Delta)^s v = H_u(x,u,v) & \text{in }\Omega,\\ u=v=0 & \text{in } \R^n\setminus\Omega. \end{cases} $$ The method, that can be traced back to the work of De Figueiredo and Felmer \cite{DF-F}, is flexible enough to deal with more general nonlocal operators and make use of a combination of fractional order Sobolev spaces together with functional calculus for self-adjoint operators.

math.AP↗
arXiv2024

Analytical insights from a model of opinion formation based on Persuasive Argument Theory

In recent years, numerous mathematical models of opinion formation have been developed, incorporating diverse interaction mechanisms such as imitation and majority rule. However, limited attention has been given to models grounded in persuasive arguments theory (PAT), which describes how individuals may alter their opinions through the exchange of arguments during discussions. Moreover, analytical investigations of PAT-based models remain sparse. In this study, we propose an analytical model rooted in PAT, demonstrating that a group of agents can exhibit two distinct collective dynamics: quasi-consensus and bipolarization. Specifically, we explore various scenarios characterized by the number of arguments and the degree of homophily, revealing that bipolarization arises within this framework only in the presence of homophily.

physics.soc-ph↗
arXiv2024

Effect of the geometry of butt-joint implant-supported restorations on the fatigue life of prosthetic screws

Statement of problem. Dental implant geometry affects the mechanical performance and fatigue behavior of butt-joint implant-supported restorations. However, failure of the implant component has been generally studied by ignoring the prosthetic screw, which is frequently the critical restoration component Purpose. Evaluate the effect of 3 main implant geometric parameters: the implant body diameter, the platform diameter, and the implant-abutment connection type (external versus internal butt-joint) on the fatigue life of the prosthetic screw. The experimental values were further compared with the theoretical ones obtained by using a previously published methodology M&M. 4 different designs of direct-to-implant dental restorations from the manufacturer BTI were tested. Forty-eight fatigue tests were performed in an axial fatigue testing machine according to ISO 14801. Linear regression models, 95% interval confidence bands for the linear regression, and 95% prediction intervals of the fatigue load-life results were obtained and compared through an analysis of covariance to determine the influence of the 3 parameters under study on the fatigue behavior Results. Linear regression models showed a statistical difference when the implant body diameter was increased by 1 mm; an average 3.5-fold increase in fatigue life was observed. Increasing the implant abutment connection diameter by 1.4 mm also showed a significant difference, leading to 7-fold longer fatigue life on average. No significant statistical evidence was found to demonstrate a difference in fatigue life between internal and external connections Conclusions. Increasing the implant platform and body diameter significantly improved the fatigue life of the screw, whereas external and internal connections provided similar results. In addition, experimental results proved the accuracy of the fatigue life prediction methodology

physics.med-ph↗
arXiv2023

High Resolution Finite Difference Schemes for a Size Structured Coagulation-Fragmentation Model in the Space of Radon Measures

In this paper we develop explicit and semi-implicit second-order high-resolution finite difference schemes for a structured coagulation-fragmentation model formulated on the space of Radon measures. We prove the convergence of each of the two schemes to the unique weak solution of the model. We perform numerical simulations to demonstrate that the second order accuracy is achieved by both schemes.

math.NA↗
arXiv2022

Non-local equations and optimal Sobolev inequalities on compact manifolds

This paper deals with fractional Sobolev spaces on a compact Riemannian manifold. We prove a Sobolev inequality in the critical range with an optimal constant for these fractional Sobolev spaces. We use this result to study the existence of a non-trivial solution for equations driven by a non-local integro-differential operator $\mathcal{L}_{\mathcal{K}}$ with critical non-linearity.

math.AP↗
arXiv2022

Opinion formation process in a hierarchical society

In this work we study the formation of consensus in a hierarchical population. We derive the corresponding kinetic equations, and analyze the long time behaviour of their solutions for the case of finite number of hierarchical obtaining explicit formula for the consensus opinion.

math.AP↗
arXiv2021

Analytical formulation for multidimensional continuous opinion models

Usually, opinion formation models assume that individuals have an opinion about a given topic which can change due to interactions with others. However, individuals can have different opinions in different topics and therefore n-dimensional models are best suited to deal with these cases. While there have been many efforts to develop analytical models for one dimensional opinion models, less attention has been paid to multidimensional ones. In this work, we develop an analytical approach for multidimensional models of continuous opinions where dimensions can be correlated or uncorrelated. We show that for any generic reciprocal interactions between agents, the mean value of initial opinion distribution is conserved. Moreover, for positive social influence interaction mechanisms, the variance of opinion distributions decreases with time and the system converges to a delta distributed function. In particular, we calculate the convergence time when agents get closer in a discrete quantity after interacting, showing a clear difference between correlated and uncorrelated cases.

physics.soc-ph↗
arXiv2020

Interacting particles systems with delay and random delay differential equations

In this work we study a kinetic model of active particles with delayed dynamics, and its limit when the number of particles goes to infinity. This limit turns out to be related to delayed differential equations with random initial conditions. We analyze two different dynamics, one based on the full knowledge of the individual trajectories of each particle, and another one based only on the trace of the particle cloud, loosing track of the individual trajectories. Notice that in the first dynamic the state of a particles is its path, whereas it is simply a point in $\R^d$ in the second case. We analyse in both cases the corresponding mean-field dynamic obtaining an equation for the time evolution of the distribution of the particles states. Well-posedness of the equation is proved by a fixed-point argument. We conclude the paper with some possible future research directions and modelling applications.

math.AP↗
arXiv2020

Evolutionary game theory in mixed strategies: from microscopic interactions to kinetic equations

In this work we propose a kinetic formulation for evolutionary game theory for zero sum games when the agents use mixed strategies. We start with a simple adaptive rule, where after an encounter each agent increases the probability of play the successful pure strategy used in the match. We derive the Boltzmann equation which describes the macroscopic effects of this microscopical rule, and we obtain a first order, nonlocal, partial differential equation as the limit when the probability change goes to zero. We study the relationship between this equation and the well known replicator equations, showing the equivalence between the concepts of Nash equilibria, stationary solutions of the partial differential equation, and the equilibria of the replicator equations. Finally, we relate the long time behavior of solutions to the partial differential equation and the stability of the replicator equations.

math.AP↗
arXiv2020

A model for the competition between political mono-polarization and bi-polarization

We investigate the phenomena of political bi-polarization in a population of interacting agents by means of a generalized version of the model introduced in PRE E 101, 012101 (2020) for the dynamics of voting intention. Each agent has a propensity $p$ in $[0,1]$ to vote for one of two political candidates. In an iteration step, two agents $i$ and $j$ with respective propensities $p_i$ and $p_j$ interact, and then $p_i$ either increases by an amount $h>0$ with a probability that is a nonlinear function of $p_i$ and $p_j$ or decreases by $h$ with the complementary probability. We study the behavior of the system under variations of a parameter $q \ge 0$ that measures the nonlinearity of the propensity update rule. We focus on the stability properties of the two distinct stationary states: mono-polarization in which all agents share the same extreme propensity ($0$ or $1$), and bi-polarization where the population is divided into two groups with opposite and extreme propensities. We find that the bi-polarized state is stable for $q q_c$, where $q_c$ is a transition value that decreases as $h$ decreases. We develop a rate equation approach whose stability analysis reveals that $q_c$ vanishes when $h$ becomes infinitesimally small. This result is supported by the analysis of a transport equation derived in the continuum $h \to 0$ limit. We also show by Monte Carlo simulations that the mean time $τ$ to reach mono-polarization in a system of size $N$ scales as $τ\sim N^α$ at $q_c$ , where $α(h)$ is a non-universal exponent.

physics.soc-ph↗
arXiv2019

The role of voting intention in public opinion polarization

We introduce and study a simple model for the dynamics of voting intention in a population of agents that have to choose between two candidates. The level of indecision of a given agent is modeled by its propensity to vote for one of the two alternatives, represented by a variable $p \in [0,1]$. When an agent $i$ interacts with another agent $j$ with propensity $p_j$, then $i$ either increases its propensity $p_i$ by $h$ with probability $P_{ij}=ωp_i+(1-ω)p_j$, or decreases $p_i$ by $h$ with probability $1-P_{ij}$, where $h$ is a fixed step. We analyze the system by a rate equation approach and contrast the results with Monte Carlo simulations. We found that the dynamics of propensities depends on the weight $ω$ that an agent assigns to its own propensity. When all the weight is assigned to the interacting partner ($ω=0$), agents' propensities are quickly driven to one of the extreme values $p=0$ or $p=1$, until an extremist absorbing consensus is achieved. However, for $ω>0$ the system first reaches a quasi-stationary state of symmetric polarization where the distribution of propensities has the shape of an inverted Gaussian with a minimum at the center $p=1/2$ and two maxima at the extreme values $p=0,1$, until the symmetry is broken and the system is driven to an extremist consensus. A linear stability analysis shows that the lifetime of the polarized state, estimated by the mean consensus time $τ$, diverges as $τ\sim (1-ω)^{-2} \ln N$ when $ω$ approaches $1$, where $N$ is the system size. Finally, a continuous approximation allows to derive a transport equation whose convection term is compatible with a drift of particles from the center towards the extremes.

physics.soc-ph↗
arXiv2018

Opinion formation models with heterogeneous persuasion and zealotry

In this work an opinion formation model with heterogeneous agents is proposed. Each agent is supposed to have different power of persuasion, and besides its own level of zealotry, that is, an individual willingness to being convinced by other agent. In addition, our model includes zealots or stubborn agents, agents that never change opinions. We derive a Bolzmann-like equation for the distribution of agents on the space of opinions, which is approximated by a transport equation with a nonlocal drift term. We study the long-time asymptotic behavior of solutions, characterizing the limit distribution of agents, which consists of the distribution of stubborn agents, plus a delta function at the mean of their opinions, weighted by they power of persuasion. Moreover, explicit bounds on the rate of convergence are given, and the time to convergence is shown to decrease when the number of stubborn agents increases. This is a remarkable fact observed in agent based simulations in different works.

math.AP↗
arXiv2018

Nonlinear elliptic equations with measure valued absorption potential

We study the semilinear elliptic equation --$Δ$u + g(u)$σ$ = $μ$ with Dirichlet boundary condition in a smooth bounded domain where $σ$ is a nonnegative Radon measure, $μ$ a Radon measure and g is an absorbing nonlinearity. We show that the problem is well posed if we assume that $σ$ belongs to some Morrey class. Under this condition we give a general existence result for any bounded measure provided g satisfies a subcritical integral assumption. We study also the supercritical case when g(r) = |r| ^{q--1} r, with q > 1 and $μ$ satisfies an absolute continuity condition expressed in terms of some capacities involving $σ$. 2010 Mathematics Subject Classification. 35 J 61; 31 B 15; 28 C 05 .

math.AP↗
arXiv2017

A game theoretic model of wealth distribution

In this work we consider an agent based model in order to study the wealth distribution problem where the interchange is determined with a symmetric zero sum game. Simultaneously, the agents update their way of play trying to learn the optimal one. Here, the agents use mixed strategies. We study this model using both simulations and theoretical tools. We derive the equations for the learning mechanism, and we show that the mean strategy of the population satisfies an equation close to the classical replicator equation. Concerning the wealth distribution, there are two interesting situations depending on the equilibrium of the game. If the equilibrium is a pure strategy, the wealth distribution is fixed after some transient time, and those players which are close to optimal strategy are richer. When the game has an equilibrium in mixed strategies, the stationary wealth distribution is close to a Gamma distribution with variance depending on the coefficients of the game matrix. We compute theoretically their second moment in this case.

physics.soc-ph↗
arXiv2015

Local existence conditions for an equations involving the $p(x)$-Laplacian with critical exponent in $\mathbb{R}^N$

The purpose of this paper is to formulate sufficient existence conditions for a critical equation involving the $p(x)$-Laplacian posed in $\mathbb{R}^N$. This equation is critical in the sense that the source term has the form $K(x)|u|^{q(x)-2}u$ with an exponent $q$ that can be equal to the critical exponent $p^*$ at some points of $\mathbb{R}^N$ including at infinity. The sufficient existence conditions we find are local in the sense that they depend only on the behaviour of the exponents $p$ and $q$ near these points. We stress that we do not assume any symmetry or periodicity of the coefficients of the equation and that $K$ is not required to vanish in some sense at infinity like in most existing results. The proof of these local existence conditions is based on a notion of localized best Sobolev constant at infinity and a refined concentration-compactness at infinity.

math.AP↗