arXiv subjects

Aurélien Velleret

Publications and source records attributed to Aurélien Velleret.

12 recordsLinked to original sources

arXiv2026

Stochastic heterogeneous SIR model with infection-age dependent infectivity on large random graphs

We study an individual-based stochastic SIR epidemic model with infection-age dependent infectivity on a large random graph, capturing individual heterogeneity and non-homogeneous connectivity. Each individual is associated with particular characteristics (for example, spatial location and age structure), which may not be i.i.d., and is represented by a particular node. The connectivities among the individuals are given by a non-homogeneous random graph, whose connecting probabilities may depend on the individual characteristics of the edge. To each individual is associated a random infectivity function of its infection age, which is allowed to depend upon the individual characteristics. We use measure-valued processes to describe the epidemic evolution dynamics, tracking the infection age of all individuals, and their associated characteristics. We consider the epidemic dynamics as the population size grows to infinity under a specific scaling of the connectivity graph related to the convergence to a graphon. In the limit, we obtain a system of measure-valued equations, which can be also represented as a PDE model on graphon, and reflects the heterogeneities in individual characteristics and social connectivity.

math.PR↗

arXiv2026

LAN property for the parameter of the jump rate in mean field interacting systems of neurons

In the context of a large system of $N$ neurons interacting through spike events in a mean-field regime as $N\to \infty$, we characterize the estimation of a multidimensional parameter in the spiking rate, when the neural states are observed over a fixed time horizon. We first prove the local asymptotic normality (LAN) property and leverage classical theory to establish the asymptotic efficiency of the maximum likelihood estimator. While the theory of Ibragimov and Hasminskii yields strong results, up to global asymptotic minimax bound, its applicability appears currently limited to models without state resets at spike times. Following then Höpfner's classical approach, we nevertheless derive, in a general setting including neuron reset, the consistency, asymptotic normality and local asymptotic minimax optimality of the estimator. Keywords: Local Asymptotic Normality (LAN); Mean-field regime; Interacting particle system; Multidimensional parameter estimation; Jump rate estimation; Maximum likelihood estimator (MLE); Asymptotic minimax optimality

math.ST↗

arXiv2026

Quasi-ergodic theorems for Feynman-Kac semigroups and large deviation for additive functionals

We study the long-time behavior of an additive functional that takes into account the jumps of a symmetric Markov process. This process is assumed to be observed through a biased observation scheme that includes the survival to events of extinction and the Feynman-Kac weight by another similar additive functional. Under conditioning for the convergence to a quasi-stationary distribution and for two-sided estimates of the Feynmac-Kac semigroup to be obtained, we shall discuss general assumptions on the symmetric Markov process. For the law of additive functionals, we will prove a quasi-ergodic theorem, namely a conditional version of the ergodic theorem and a conditional functional weak law of large numbers. As an application, we also establish a large deviation principle for the mean ratio of additive functionals.

math.PR↗

arXiv2025

Inter-city infections and the role of size heterogeneity in containment strategies

This study examines the effectiveness of regional lockdown strategies in mitigating pathogen spread across regional units, termed cities hereinafter. We develop simplified models to analyze infection spread across cities within a country during an epidemic wave. Isolation of a city is initiated when infection numbers within the city surpass defined thresholds. We compare two strategies: strategy (P) consists in prescribing thresholds proportionally to city sizes, while the same threshold is used for all cities under strategy (U). Given the heavy-tailed distribution of city sizes, strategy (P) may result in more secondary infections from larger cities than strategy (U). Random graph models are constructed to represent infection spread as a percolation process. In particular, we consider a model in which mobility between cities only depends on city sizes. We assess the relative efficiency of the two strategies by comparing the ratios of the number of individuals under isolation to the total number of infections by the end of the epidemic wave under strategy (P) and (U). Additionally, we derive analytical formulas for disease prevalence and basic reproduction numbers. Our models are calibrated using mobility data from France, Poland and Japan, validated through simulation. The findings indicate that mobility between cities in France and Poland is mainly determined by city sizes. However, a poor fit was observed with Japanese data, highlighting the importance to include other factors like e.g. geography for some countries in modeling. Our analysis suggest similar effectiveness for both strategies in France and Japan, while strategy (U) demonstrates distinct merits in Poland.

physics.soc-ph↗

arXiv2024

Individual based SIS models on (not so) dense large random networks

Starting from a stochastic individual-based description of an SIS epidemic spreading on a random network, we study the dynamics when the size $n$ of the network tends to infinity. We recover in the limit an infinite-dimensional integro-differential equation studied by Delmas, Dronnier and Zitt (2022) for an SIS epidemic propagating on a graphon. Our work covers the case of dense and sparse graphs, provided that the number of edges grows faster than $n$, but not the case of very sparse graphs with $O(n)$ edges. In order to establish our limit theorem, we have to deal with both the convergence of the random graphs to the graphon and the convergence of the stochastic process spreading on top of these random structures: in particular, we propose a coupling between the process of interest and an epidemic that spreads on the complete graph but with a modified infection rate. Keywords: Random graph, mathematical models of epidemics, measure-valued process, large network limit, limit theorem, graphon.

math.PR↗

arXiv2024

Metastability between the clicks of Muller's ratchet

We prove the existence and uniqueness of a quasi-stationary distribution for three stochastic processes derived from the model of Muller's ratchet. This model was invented with the aim of evaluating the limitations of an asexual reproduction mode in preventing the accumulation of deleterious mutations through natural selection alone. The main considered model is non-classical, as it is a stochastic diffusion evolving on an irregular set of infinite dimension with hard killing on an hyperplane. We are nonetheless able to prove exponential convergence in total variation to the quasi-stationary distribution even in this case. The parameters in this last convergence result are directly related to the core parameters of Muller's ratchet. The speed of convergence to the quasi-stationary distribution is deduced both for the infinite dimensional model and for approximations with a large yet finite number of potential mutations. Likewise, we give uniform moment estimates of the empirical distribution of mutations in the population under quasi-stationarity.

math.PR↗

arXiv2023

Exponential quasi-ergodicity for processes with discontinuous trajectories

This paper tackles the issue of establishing a lower-bound on the asymptotic ratio of survival probabilities between two different initial conditions, asymptotically in time for a given Markov process with extinction. Such a comparison is a crucial step in recent techniques for proving exponential convergence to a quasi-stationary distribution. We introduce a weak form of the Harnack inequality as the essential ingredient for such a comparison. This property is actually a consequence of the convergence property we intend to prove. Its complexity appears as the price to pay for the level of flexibility required by our applications. We show in our illustrations how simply and efficiently it can be used nonetheless. As illustrations, we consider two continuous-time processes on R^d that do not satisfy the classical Harnack inequalities, even in a local version. The first one is a piecewise deterministic process while the second is a pure jump process with restrictions on the directions of its jumps.

math.PR↗

arXiv2021

Adaptation of a population to a changing environment under the light of quasi-stationarity

We analyze the long-term stability of a stochastic model designed to illustrate the adaptation of a population to variation in its environment. A piecewise-deterministic process modeling adaptation is coupled to a Feller logistic diffusion modeling population size. As the individual features in the population become further away from the optimal ones, the growth rate declines, making population extinction more likely. Assuming that the environment changes deterministically and steadily in a constant direction, we obtain the existence and uniqueness of the quasi-stationary distribution, the associated survival capacity and the Q-process. Our approach also provides several exponential convergence results (in total variation for the measures). From this synthetic information, we can characterize the efficiency of internal adaptation (i.e. population turnover from mutant invasions). When the latter is lacking, there is still stability, but because of the high level of population extinction. Therefore, such a characterization must be based on specific features of this quasi-ergodic regime.

math.PR↗

arXiv2021

Unique Quasi-Stationary Distribution, with a possibly stabilizing extinction

We establish sufficient conditions for exponential convergence to a unique quasi-stationary distribution in the total variation norm. These conditions also ensure the existence and exponential ergodicity of the Q-process, the process conditionned upon never being absorbed. The technique relies on a coupling procedure that is related to Harris recurrence (for Markov Chains). It applies to general continuous-time and continuous-space Markov processes. The main novelty is that we modulate each coupling step depending both on a final horizon of time (for survival) and on the initial distribution. By this way, we could notably include in the convergence a dependency on the initial condition. As an illustration, we consider a continuous-time birth-death process with catastrophes and a diffusion process describing a (localized) population adapting to its environment.

math.PR↗

arXiv2020

Two level natural selection with a quasi-stationarity approach

In a view for a simple model where natural selection at the individual level is confronted to selection effects at the group level, we consider some individual-based models of some large population subdivided into a large number of groups. We then obtain the convergence to the law of a stochastic process with some Feynman-Kac penalization. To analyze the limiting behavior of this law, we exploit a recent approach, designed for the convergence to quasi-stationary distributions. We are able to deal with the fixation of the stochastic process and relate the convergence to equilibrium to the one where fixation implies extinction. We notably establish different regimes of convergence. Besides the case of an exponential rate (the rate being uniform over the initial condition), critical regimes with convergence in 1/t are also to notice. We finally address the relevance of such limiting behaviors to predict the long-time behavior of the individual-based model and describe more specifically the cases of weak selection. Consequences in term of evolutive dynamics are also derived, where such competition is assumed to occur repeatedly at each de novo mutation.

math.PR↗

arXiv2020

Large deviations for surviving trajectories of general Markov processes

The purpose of this paper is to ensure the conditions of Gärtner-Ellis Theorem for evaluations of the empirical measure. We show that up-to-date conditions for ensuring the convergence to a quasi-stationary distribution can be applied efficiently. By this mean, we are able to prove Large Deviation results even with a conditioning that the process is not extinct at the end of the evaluation. The domain on which these Large Deviation results apply is implicitly given by the range of penalization for which one can prove the above-mentioned results of quasi-stationarity. We propose a way to relate the range of controlled deviations to the range of admissible penalization. Central Limit Theorems are deduced from these results of Large Deviations. As an application, we consider the empirical measure of position and jumps of a continuous-time process on aunbounded domain of $R^d$. This model is inspired by the adaptation of a population to a changing environment. Jumps makes it possible for the process to face a deterministic dynamics leading to high extinction areas.

math.PR↗

arXiv2020

Individual-based models under various time-scales

This article is a presentation of specific recent results describing scaling limits of individual-based models. Thanks to them, we wish to relate the time-scales typical of demographic dynamics and natural selection to the parameters of the individual-based models. Although these results are by no means exhaustive, both on the mathematical and the biological level, they complement each other. Indeed, they provide a viewpoint for many classical time-scales. Namely, they encompass the timescale typical of the life-expectancy of a single individual, the longer one wherein a population can be characterized through its demographic dynamics, and at least four interconnected ones wherein selection occurs. The limiting behavior is generally deterministic. Yet, since there are selective effects on randomness in the history of lineages, probability theory is shown to be a key factor in understanding the results. Besides, randomness can be maintained in the limiting dynamics, for instance to model rare mutations fixing in the population.

math.PR↗