arXiv subjects

Emmanuel Schertzer

Publications and source records attributed to Emmanuel Schertzer.

At least 37 records · Page 2Linked to original sources

arXiv2021

The stochastic dynamics of early epidemics: probability of establishment, initial growth rate, and infection cluster size at first detection

Emerging epidemics and local infection clusters are initially prone to stochastic effects that can substantially impact the epidemic trajectory. While numerous studies are devoted to the deterministic regime of an established epidemic, mathematical descriptions of the initial phase of epidemic growth are comparatively rarer. Here, we review existing mathematical results on the epidemic size over time, and derive new results to elucidate the early dynamics of an infection cluster started by a single infected individual. We show that the initial growth of epidemics that eventually take off is accelerated by stochasticity. These results are critical to improve early cluster detection and control. As an application, we compute the distribution of the first detection time of an infected individual in an infection cluster depending on the testing effort, and estimate that the SARS-CoV-2 variant of concern Alpha detected in September 2020 first appeared in the United Kingdom early August 2020. We also compute a minimal testing frequency to detect clusters before they exceed a given threshold size. These results improve our theoretical understanding of early epidemics and will be useful for the study and control of local infectious disease clusters.

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arXiv2021

General selection models: Bernstein duality and minimal ancestral structures

$Λ$-Wright--Fisher processes provide a robust framework to describe the type-frequency evolution of an infinite neutral population. We add a polynomial drift to the corresponding stochastic differential equation to incorporate frequency-dependent selection. A decomposition of the drift allows us to approximate the solution of the stochastic differential equation by a sequence of Moran models. The genealogical structure underlying the Moran model leads in the large population limit to a generalisation of the ancestral selection graph of Krone and Neuhauser. Building on this object, we construct a continuous-time Markov chain and relate it to the forward process via a new form of duality, which we call Bernstein duality. We adapt classical methods based on the moment duality to determine the time to absorption and criteria for the accessibility of the boundaries; this extends a recent result by González Casanova and Spanò. An intriguing feature of the construction is that the same forward process is compatible with multiple backward models. In this context we introduce suitable notions for minimality among the ancestral processes and characterise the corresponding parameter sets. In this way we recover classic ancestral structures as minimal ones.

math.PR↗

arXiv2021

Height and contour processes of Crump-Mode-Jagers forests (III): The binary, homogeneous universality class

This paper belongs to a series of papers aiming to investigate scaling limits of Crump-Mode-Jagers (CMJ) trees. In the previous two papers we identified general conditions under which CMJ trees belong to the universality class of Galton-Watson and Bellman-Harris processes. In this paper we identify general conditions for CMJ trees to belong to the universality class of binary, homogeneous CMJ trees. These conditions state that the offspring process should 'look like' a renewal process, and also that it should not accumulate too many atoms near the origin. We show in particular that any renewal process satisfies these conditions.

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arXiv2020

Chromosome Painting: how recombination mixes ancestral colors

We consider a Moran model with recombination in a haploid population of size $N$. At each birth event, with probability $1-ρ_N R$ the offspring copies one parent's chromosome, and with probability $ρ_N R$ she inherits a chromosome that is a mosaic of both parental chromosomes. We assume that at time $0$ each individual has her chromosome painted in a different color and we study the color partition of the chromosome that is asymptotically fixed in a large population, when we look at a portion of the chromosome such that $ρ:= \lim_{N\to \infty} \frac{ρ_N N}{2}\to \infty$. To do so, we follow backwards in time the ancestry of the chromosome of a randomly sampled individual. This yields a Markov process valued in the color partitions of the half-line, that was introduced by Esser et al. (2016), in which blocks can merge and split, called the partitioning process. Its stationary distribution is closely related to the fixed chromosome in our Moran model with recombination. We are able to provide an approximation of this stationary distribution when $ρ\gg 1$ and an error bound. This allows us to show that the distribution of the (renormalised) length of the leftmost block of the partition (i.e. the region of the chromosome that carries the same color as 0) converges to an exponential distribution. In addition, the geometry of this block can be described in terms of a Poisson point process with an explicit intensity measure.

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

Perturbations of supercritical oriented percolation and sticky Brownian webs

Previously, Sarkar and Sun have shown that for supercritical oriented percolation in dimension $1+1$, the set of rightmost infinite open paths converges to the Brownian web after proper centering and scaling. In this note, we show that a pair of sticky Brownian webs arise naturally if one considers the set of right-most infinite open paths for two coupled percolation configurations with distinct (but close) percolation parameters. This leads to a natural conjecture on the convergence of the dynamical supercritical oriented percolation model to the so-called dynamical Brownian web.

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↗

arXiv2018

How does geographical distance translate into genetic distance?

Geographic structure can affect patterns of genetic differentiation and speciation rates. In this article, we investigate the dynamics of genetic distances in a geographically structured metapopulation. We model the metapopulation as a weighted directed graph, with d vertices corresponding to d subpopulations that evolve according to an individual based model. The dynamics of the genetic distances is then controlled by two types of transitions -mutation and migration events. We show that, under a rare mutation - rare migration regime, intra subpopulation diversity can be neglected and our model can be approximated by a population based model. We show that under a large population - large number of loci limit, the genetic distance between two subpopulations converges to a deterministic quantity that can asymptotically be expressed in terms of the hitting time between two random walks in the metapopulation graph. Our result shows that the genetic distance between two subpopulations does not only depend on the direct migration rates between them but on the whole metapopulation structure.

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arXiv2018

Coagulation-transport equations and the nested coalescents

The nested Kingman coalescent describes the dynamics of particles (called genes) contained in larger components (called species), where pairs of species coalesce at constant rate and pairs of genes coalesce at constant rate provided they lie within the same species. We prove that starting from $rn$ species, the empirical distribution of species masses (numbers of genes$/n$) at time $t/n$ converges as $n\to\infty$ to a solution of the deterministic coagulation-transport equation $$ \partial_t d \ = \ \partial_x ( ψd ) \ + \ a(t)\left(d\star d - d \right), $$ where $ψ(x) = cx^2$, $\star$ denotes convolution and $a(t)= 1/(t+δ)$ with $δ=2/r$. The most interesting case when $δ=0$ corresponds to an infinite initial number of species. This equation describes the evolution of the distribution of species of mass $x$, where pairs of species can coalesce and each species' mass evolves like $\dot x = -ψ(x)$. We provide two natural probabilistic solutions of the latter IPDE and address in detail the case when $δ=0$. The first solution is expressed in terms of a branching particle system where particles carry masses behaving as independent continuous-state branching processes. The second one is the law of the solution to the following McKean-Vlasov equation $$ dx_t \ = \ - ψ(x_t) \,dt \ + \ v_t\,ΔJ_t $$ where $J$ is an inhomogeneous Poisson process with rate $1/(t+δ)$ and $(v_t; t\geq0)$ is a sequence of independent rvs such that $ {\mathcal L}(v_t) = {\mathcal L}(x_t)$. We show that there is a unique solution to this equation and we construct this solution with the help of a marked Brownian coalescent point process. When $ψ(x)=x^γ$, we show the existence of a self-similar solution for the PDE which relates when $γ=2$ to the speed of coming down from infinity of the nested Kingman coalescent.

math.PR↗

arXiv2018

The sequential loss of allelic diversity

This paper gives a new flavor of what Peter Jagers and his co-authors call `the path to extinction'. In a neutral population with constant size $N$, we assume that each individual at time $0$ carries a distinct type, or allele. We consider the joint dynamics of these $N$ alleles, for example the dynamics of their respective frequencies and more plainly the nonincreasing process counting the number of alleles remaining by time $t$. We call this process the extinction process. We show that in the Moran model, the extinction process is distributed as the process counting (in backward time) the number of common ancestors to the whole population, also known as the block counting process of the $N$-Kingman coalescent. Stimulated by this result, we investigate: (1) whether it extends to an identity between the frequencies of blocks in the Kingman coalescent and the frequencies of alleles in the extinction process, both evaluated at jump times; (2) whether it extends to the general case of $Λ$-Fleming-Viot processes.

math.PR↗

arXiv2017

The Brownian web, the Brownian net, and their universality

The Brownian web is a collection of one-dimensional coalescing Brownian motions starting from everywhere in space and time, and the Brownian net is a generalization that also allows branching. They appear in the diffusive scaling limits of many one-dimensional interacting particle systems with branching and coalescence. This article gives an introduction to the Brownian web and net, and how they arise in the scaling limits of various one-dimensional models, focusing mainly on coalescing random walks and random walks in i.i.d. space-time random environments. We will also briefly survey models and results connected to the Brownian web and net, including alternative topologies, population genetic models, true self-repelling motion, planar aggregation, drainage networks, oriented percolation, black noise and critical percolation. Some open questions are discussed at the end.

math.PR↗

arXiv2016

Recovering the Brownian Coalescent Point Process from the Kingman Coalescent by Conditional Sampling

We consider a continuous population whose dynamics is described by the standard stationary Fleming-Viot process, so that the genealogy of $n$ uniformly sampled individuals is distributed as the Kingman $n$-coalescent. In this note, we study some genealogical properties of this population when the sample is conditioned to fall entirely into a subpopulation with most recent common ancestor (MRCA) shorter than $\varepsilon$. First, using the comb representation of the total genealogy (Lambert & Uribe Bravo 2016), we show that the genealogy of the descendance of the MRCA of the sample on the timescale $\varepsilon$ converges as $\varepsilon\to 0$. The limit is the so-called Brownian coalescent point process (CPP) stopped at an independent Gamma random variable with parameter $n$, which can be seen as the genealogy at a large time of the total population of a rescaled critical birth-death process, biased by the $n$-th power of its size. Secondly, we show that in this limit the coalescence times of the $n$ sampled individuals are i.i.d. uniform random variables in $(0,1)$. These results provide a coupling between two standard models for the genealogy of a random exchangeable population: the Kingman coalescent and the Brownian CPP.

math.PR↗

arXiv2016

Height and contour processes of Crump-Mode-Jagers forests (I): general distribution and scaling limits in the case of short edges

Crump-Mode-Jagers (CMJ) trees generalize Galton-Watson trees by allowing individuals to live for an arbitrary duration and give birth at arbitrary times during their life-time. In this paper, we are interested in the height and contour processes encoding a general CMJ tree. We show that the one-dimensional distribution of the height process can be expressed in terms of a random transformation of the ladder height process associated with the underlying Lukasiewicz path. As an application of this result, when edges of the tree are "short" we show that, asymptotically, (1) the height process is obtained by stretching by a constant factor the height process of the associated genealogical Galton-Watson tree, (2) the contour process is obtained from the height process by a constant time change and (3) the CMJ trees converge in the sense of finite-dimensional distributions.

math.PR↗

arXiv2015

Renewal Structure of the Brownian Taut String

In a recent paper, M. Lifshits and E. Setterqvist introduced the taut string of a Brownian motion $w$, defined as the function of minimal quadratic energy on $[0,T]$ staying in a tube of fixed width $h>0$ around $w$. The authors showed a Law of Large Number (L.L.N.) for the quadratic energy spent by the string for a large time $T$. In this note, we exhibit a natural renewal structure for the Brownian taut string, which is directly related to the time decomposition of the Brownian motion in terms of its $h$-extrema (as first introduced by Neveu and Pitman). Using this renewal structure, we derive an expression for the constant in the L.L.N. given by M. Lifshits and E. Setterqvist. In addition, we provide a Central Limit Theorem (C.L.T.) for the fluctuations of the energy spent by the Brownian taut string.

math.PR↗

arXiv2014

Trimming a Tree and the Two-Sided Skorohod Reflection

The $h$-trimming of a tree is a natural regularization procedure which consists in pruning the small branches of a tree: given $h\geq0$, it is obtained by only keeping the vertices having at least one leaf above them at a distance greater or equal to $h$. The $h$-cut of a function $f$ is the function of minimal total variation uniformly approximating the increments of $f$ with accuracy $h$, and can be explicitly constructed via the two-sided Skorohod reflection of $f$ on the interval $[0,h]$. In this work, we show that the contour path of the $h$-trimming of a rooted real tree is given by the $h$-cut of its original contour path. We provide two applications of this result. First, we recover a famous result of Neveu and Pitman, which states that the $h$-trimming of a tree coded by a Brownian excursion is distributed as a standard binary tree. In addition, we provide the joint distribution of this Brownian tree and its trimmed version in terms of the local time of the two-sided reflection of its contour path. As a second application, we relate the maximum of a sticky Brownian motion to the local time of its driving process.

math.PR↗

arXiv2013

The Brownian Net with Killing

Motivated by its relevance for the study of perturbations of one-dimensional voter models, including stochastic Potts models at low temperature, we consider diffusively rescaled coalescing random walks with branching and killing. Our main result is convergence to a new continuum process, in which the random space-time paths of the Sun-Swart Brownian net are terminated at a Poisson cloud of killing points. We also prove existence of a percolation transition as the killing rate varies. Key issues for convergence are the relations of the discrete model killing points and their Poisson intensity measure to the continuum counterparts.

math.PR↗

arXiv2013

Stochastic flows in the Brownian web and net

Certain one-dimensional nearest-neighbor random walks in i.i.d. random space-time environments are known to have diffusive scaling limits. In the continuum limit, the random environment is represented by a `stochastic flow of kernels', which is a collection of random kernels that can be loosely interpreted as the transition probabilities of a Markov process in a random environment. The theory of stochastic flows of kernels was introduced by Le Jan and Raimond, who showed that each such flow is characterized by its n-point motions. We focus on a class of stochastic flows of kernels with Brownian n-point motions which, after their inventors, will be called Howitt-Warren flows. We give a graphical construction of general Howitt-Warren flows, where the underlying random environment takes on the form of a suitably marked Brownian web. Alternatively, we show that a special subclass of the Howitt-Warren flows can be constructed as random flows of mass in a Brownian net. Using these constructions, we prove some new results for the Howitt-Warren flows. In particular, we show that the kernels spread with a finite speed and have a locally finite support at deterministic times if and only if the flow is embeddable in a Brownian net. We show that the kernels are always purely atomic at deterministic times, but with the exception of a special subclass known as the erosion flows, exhibit random times when the kernels are purely non-atomic. We moreover prove ergodic statements for a class of measure-valued processes induced by the Howitt-Warren flows. Along the way, we also prove some new results for the Brownian web and net.

math.PR↗

arXiv2010

The Impossible Trio in CDO Modeling

We show that stochastic recovery always leads to counter-intuitive behaviors in the risk measures of a CDO tranche - namely, continuity on default and positive credit spread risk cannot be ensured simultaneously. We then propose a simple recovery variance regularization method to control the magnitude of negative credit spread risk while preserving the continuity on default.

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