arXiv subjects

Félix Foutel-Rodier

Publications and source records attributed to Félix Foutel-Rodier.

13 recordsLinked to original sources

arXiv2026

Convergence of spatial branching processes to $α$-stable CSBPs: Genealogy of semi-pushed fronts

We consider an inhomogeneous branching diffusion on an unbounded domain of $\mathbb{R}^d$ and propose a simple condition under which we expect the size process (i.e., the number of particles) and the genealogy of the system to converge to those of an $α$-stable continuous-state branching process, with $α\in(1,2)$. This condition can be seen as the spatial analogue of the classical assumption that the tail of the offspring distribution of a Galton--Watson process is regularly varying. We make a first step towards establishing this result by providing a set of sufficient conditions under which the branching diffusion, seen as a random marked metric measure space that captures both the positions and the genealogical structure of the population, converges to an $α$-stable genealogy. These conditions are based on the convergence of the moments of the process, which can be efficiently computed via recursive formulas. We apply this framework to a one-dimensional branching Brownian motion with inhomogeneous branching rate and negative drift. This model was introduced by Tourniaire as a toy model to investigate the internal dynamics of fluctuating pushed fronts. By using our general set of conditions we prove convergence of the genealogy of the process in the semipushed regime, which was conjectured to hold by Birzu, Hallatschek, and Korolev.

math.PR↗

arXiv2026

Moments of density-dependent branching processes and their genealogy

A density-dependent branching process is a particle system in which individuals reproduce independently, but in a way that depends on the current population size. This feature can model a wide range of ecological interactions at the cost of breaking the branching property. We propose a general approach for studying the genealogy of these models based on moments. Building on a recent work of Bansaye, we show how to compute recursively these moments in a similar spirit to the many-to-few formula in the theory of branching processes. These formulas enable one to deduce the convergence of the genealogy by studying the population density, for which stochastic calculus techniques are available. As a first application of these ideas, we consider a density-dependent branching process started close to a stable equilibrium of the ecological dynamics. We show that, under a finite second moment assumption, its genealogy converges to Kingman's coalescent when the carrying capacity of the population goes to infinity.

math.PR↗

arXiv2025

Non-linear branching processes and Crump-Mode-Jagers processes with interaction

We consider a class of Crump-Mode-Jagers processes with interaction, constructed by removing a newly born offspring with a probability that depends on the age structure of the population at its birth time. We prove a law of large numbers for the tree structure of the process in a local topology, and show how this result condenses several other limit theorems (convergence of the empirical age distribution, of ancestral lineages). Beyond this specific example, our work illustrates a more general principle that we formalise. As in standard propagation of chaos, the trees generated by typical individuals become independent as the number of individuals goes to infinity. This allows us to express the distribution of the local tree structure around a typical individual in terms of a time-inhomogeneous branching process, which we call a non-linear branching process.

math.PR↗

arXiv2025

The genealogy of nearly critical branching processes in varying environment

Building on the spinal decomposition technique in Foutel-Rodier and Schertzer (2022) we prove a Yaglom limit law for the rescaled size of a nearly critical branching process in varying environment conditional on survival. In addition, our spinal approach allows us toprove convergence of the genealogical structure of the population at a fixed time horizon -- when the sequence of trees are envisioned as a sequence of metric spaces -- in the Gromov--Hausdorff--Prohorov (GHP) topology. We characterize the limiting metric space as a time-changed version of the Brownian coalescent point process Popovic (2004). Beyond our specific model, we derive several general results allowing one to go from spinal decompositions to convergence of random trees in the GHP topology. As a direct application, we show how this type of convergence naturally condenses the limit of several interesting genealogical quantities: the population size, the time to the most-recent common ancestor, the reduced tree, and the tree generated by $k$ uniformly sampled individuals. As in a recent article by the authors (Foutel-Rodier and Schertzer 2022), we hope that our specific example illustrates a general methodology that could be applied to more complex branching processes.

math.PR↗

arXiv2024

Vague convergence and method of moments for random metric measure spaces

We introduce a notion of vague convergence for random marked metric measure spaces. Our main result shows that convergence of the moments of order $k \ge 1$ of a random marked metric measure space is sufficient to obtain its vague convergence in the Gromov-weak topology. This result improves on previous methods of moments that also require convergence of the moment of order $k=0$, which in applications to critical branching processes amounts to estimating a survival probability. We also derive two useful companion results, namely a continuous mapping theorem and an approximation theorem for vague convergence of random marked metric measure spaces.

math.PR↗

arXiv2024

A moment approach for the convergence of spatial branching processes to the Continuum Random Tree

We consider a general class of branching processes in discrete time, where particles have types belonging to a Polish space and reproduce independently according to their type. If the process is critical and the mean distribution of types converges for large times, we prove that the tree structure of the process converges to the Brownian Continuum Random Tree, under a moment assumption. We provide a general approach to prove similar invariance principles for branching processes, which relies on deducing the convergence of the genealogy from computing its moments. These are obtained using a new many-to-few formula, which provides an expression for the moments of order $k$ of a branching process in terms of a Markov chain indexed by a uniform tree with $k$ leaves.

math.PR↗

arXiv2024

Optimal Vaccination Policy to Prevent Endemicity: A Stochastic Model

We examine here the effects of recurrent vaccination and waning immunity on the establishment of an endemic equilibrium in a population. An individual-based model that incorporates memory effects for transmission rate during infection and subsequent immunity is introduced, considering stochasticity at the individual level. By letting the population size going to infinity, we derive a set of equations describing the large scale behavior of the epidemic. The analysis of the model's equilibria reveals a criterion for the existence of an endemic equilibrium, which depends on the rate of immunity loss and the distribution of time between booster doses. The outcome of a vaccination policy in this context is influenced by the efficiency of the vaccine in blocking transmissions and the distribution pattern of booster doses within the population. Strategies with evenly spaced booster shots at the individual level prove to be more effective in preventing disease spread compared to irregularly spaced boosters, as longer intervals without vaccination increase susceptibility and facilitate more efficient disease transmission. We provide an expression for the critical fraction of the population required to adhere to the vaccination policy in order to eradicate the disease, that resembles a well-known threshold for preventing an outbreak with an imperfect vaccine. We also investigate the consequences of unequal vaccine access in a population and prove that, under reasonable assumptions, fair vaccine allocation is the optimal strategy to prevent endemicity.

q-bio.PE↗

arXiv2023

Convergence of genealogies through spinal decomposition with an application to population genetics

Consider a branching Markov process with values in some general type space. Conditional on survival up to generation $N$, the genealogy of the extant population defines a random marked metric measure space, where individuals are marked by their type and pairwise distances are measured by the time to the most recent common ancestor. In the present manuscript, we devise a general method of moments to prove convergence of such genealogies in the Gromov-weak topology when $N \to \infty$. Informally, the moment of order $k$ of the population is obtained by observing the genealogy of $k$ individuals chosen uniformly at random after size-biasing the population at time $N$ by its $k$-th factorial moment. We show that the sampled genealogy can be expressed in terms of a $k$-spine decomposition of the original branching process, and that convergence reduces to the convergence of the underlying $k$-spines. As an illustration of our framework, we analyse the large-time behavior of a branching approximation of the biparental Wright-Fisher model with recombination. The model exhibits some interesting mathematical features. It starts in a supercritical state but is naturally driven to criticality. We show that the limiting behavior exhibits both critical and supercritical characteristics.

math.PR↗

arXiv2023

General epidemiological models: Law of large numbers and contact tracing

We study a class of individual-based, fixed-population size epidemic models under general assumptions, e.g., heterogeneous contact rates encapsulating changes in behavior and/or enforcement of control measures. We show that the large-population dynamics are deterministic and relate to the Kermack-McKendrick PDE. Our assumptions are minimalistic in the sense that the only important requirement is that the basic reproduction number of the epidemic $R_0$ be finite, and allow us to tackle both Markovian and non-Markovian dynamics. The novelty of our approach is to study the "infection graph" of the population. We show local convergence of this random graph to a Poisson (Galton-Watson) marked tree, recovering Markovian backward-in-time dynamics in the limit as we trace back the transmission chain leading to a focal infection. This effectively models the process of contact tracing in a large population. It is expressed in terms of the Doob $h$-transform of a certain renewal process encoding the time of infection along the chain. Our results provide a mathematical formulation relating a fundamental epidemiological quantity, the generation time distribution, to the successive time of infections along this transmission chain.

math.PR↗

arXiv2022

From individual-based epidemic models to McKendrick-von Foerster PDEs: A guide to modeling and inferring COVID-19 dynamics

We present a unifying, tractable approach for studying the spread of viruses causing complex diseases requiring to be modeled using a large number of types (e.g., infective stage, clinical state, risk factor class). We show that recording each infected individual's infection age, i.e., the time elapsed since infection, has three benefits. First, regardless of the number of types, the age distribution of the population can be described by means of a first-order, one-dimensional partial differential equation (PDE) known as the McKendrick-von Foerster equation. The frequency of type $i$ is simply obtained by integrating the probability of being in state $i$ at a given age against the age distribution. This representation induces a simple methodology based on the additional assumption of Poisson sampling to infer and forecast the epidemic. We illustrate this technique using French data from the COVID-19 epidemic. Second, our approach generalizes and simplifies standard compartmental models using high-dimensional systems of ordinary differential equations (ODEs) to account for disease complexity. We show that such models can always be rewritten in our framework, thus, providing a low-dimensional yet equivalent representation of these complex models. Third, beyond the simplicity of the approach, we show that our population model naturally appears as a universal scaling limit of a large class of fully stochastic individual-based epidemic models, where the initial condition of the PDE emerges as the limiting age structure of an exponentially growing population starting from a single individual.

q-bio.PE↗

arXiv2020

The Moran forest

Starting from any graph on $\{1, \ldots, n\}$, consider the Markov chain where at each time-step a uniformly chosen vertex is disconnected from all of its neighbors and reconnected to another uniformly chosen vertex. This Markov chain has a stationary distribution whose support is the set of non-empty forests on $\{1, \ldots, n\}$. The random forest corresponding to this stationary distribution has interesting connections with the uniform rooted labeled tree and the uniform attachment tree. We fully characterize its degree distribution, the distribution of its number of trees, and the limit distribution of the size of a tree sampled uniformly. We also show that the size of the largest tree is asymptotically $α\log n$, where $α= (1 - \log(e - 1))^{-1} \approx 2.18$, and that the degree of the most connected vertex is asymptotically $\log n / \log\log n$.

math.PR↗

arXiv2019

Exchangeable coalescents, ultrametric spaces, nested interval-partitions: A unifying approach

Kingman (1978)'s representation theorem states that any exchangeable partition of $\mathbb{N}$ can be represented as a paintbox based on a random mass-partition. Similarly, any exchangeable composition (i.e. ordered partition of $\mathbb{N}$) can be represented as a paintbox based on an interval-partition (Gnedin 1997). Our first main result is that any exchangeable coalescent process (not necessarily Markovian) can be represented as a paintbox based on a random non-decreasing process valued in interval-partitions, called nested interval-partition, generalizing the notion of comb metric space introduced by Lambert & Uribe Bravo (2017) to represent compact ultrametric spaces. As a special case, we show that any $Λ$-coalescent can be obtained from a paintbox based on a unique random nested interval partition called $Λ$-comb, which is Markovian with explicit transitions. This nested interval-partition directly relates to the flow of bridges of Bertoin & Le Gall (2003). We also display a particularly simple description of the so-called evolving coalescent (Pfaffelhuber & Wakolbinger 2006) by a comb-valued Markov process. Next, we prove that any measured ultrametric space $U$, under mild measure-theoretic assumptions on $U$, is the leaf set of a tree composed of a separable subtree called the backbone, on which are grafted additional subtrees, which act as star-trees from the standpoint of sampling. Displaying this so-called weak isometry requires us to extend the Gromov-weak topology of Greven et al (2006), that was initially designed for separable metric spaces, to non-separable ultrametric spaces. It allows us to show that for any such ultrametric space $U$, there is a nested interval-partition which is 1) indistinguishable from $U$ in the Gromov-weak topology; 2) weakly isometric to $U$ if $U$ has complete backbone; 3) isometric to $U$ if $U$ is complete and separable.

math.PR↗

arXiv2019

Kingman's coalescent with erosion

Consider the Markov process taking values in the partitions of N such that each pair of blocks merges at rate one, and each integer is eroded, i.e., becomes a singleton block, at rate d. This is a special case of exchangeable fragmentation-coalescence process called Kingman's coalescent with erosion. We provide a new construction of the stationary distribution of this process as a sample from a standard flow of bridges. This allows us to give a representation of the asymptotic frequencies of this stationary distribution in terms of a sequence of hierarchically independent diffusions. Moreover, we introduce a new process called Kingman's coalescent with immigration, where pairs of blocks coalesce at rate one, and new blocks of size one immigrate at rate d. By coupling Kingman's coalescents with erosion and with immigration, we are able to show that the size of a block chosen uniformly at random from the stationary distribution of the restriction of Kingman's coalescent with erosion to {1,...,n} converges to the total progeny of a critical binary branching process.

math.PR↗