6 research outputs found
Time reversal dualities for some random forests
We consider a random forest , defined as a sequence of i.i.d.
birth-death (BD) trees, each started at time 0 from a single ancestor, stopped
at the first tree having survived up to a fixed time . We denote by
the population size process associated
to this forest, and we prove that if the BD trees are supercritical, then the
time-reversed process , has the same
distribution as , the
corresponding population size process of an equally defined forest
, but where the underlying BD trees are subcritical,
obtained by swapping birth and death rates or equivalently, conditioning on
ultimate extinction.
We generalize this result to splitting trees (i.e. life durations of
individuals are not necessarily exponential), provided that the i.i.d.
lifetimes of the ancestors have a specific explicit distribution, different
from that of their descendants. The results are based on an identity between
the contour of these random forests truncated up to and the duality
property of L\'evy processes. This identity allows us to also derive other
useful properties such as the distribution of the population size process
conditional on the reconstructed tree of individuals alive at , which has
potential applications in epidemiology.Comment: 28 pages, 3 figure
Décompositions trajectorielles de processus de Lévy: application à la modélisation de dynamiques épidémiologiques
This dissertation is devoted to the study of some pathwise decompositions of spectrally positive LĂ©vy processes, and duality relationships for certain (possibly non-Markovian) branching processes, driven by the use of the latter as probabilistic models of epidemiological dynamics. More precisely, we model the transmission tree of a disease as a splitting tree, i.e. individuals evolve independently from one another, have i.i.d. lifetimes (periods of infectiousness) that are not necessarily exponential, and give birth (secondary infections) at a constant rate during their lifetime. The incidence of the disease under this model is a Crump-Mode-Jagers process (CMJ); the overarching goal of the two first chapters is to characterize the law of this incidence process through time, jointly with the partially observed (inferred from sequence data) transmission tree. In Chapter I we obtain a description, in terms of probability generating functions, of the conditional likelihood of the number of infectious individuals at multiple times, given the transmission network linking individuals that are currently infected. In the second chapter, a more elegant version of this characterization is given, passing by a general result of invariance under time reversal for a class of branching processes. Finally, in Chapter III we are interested in the law of the (sub)critical branching process seen from its extinction time. We obtain a duality result that implies in particular the invariance under time reversal from their extinction time of the (sub)critical CMJ processes and the excursion away from 0 of the critical Feller diffusion (the width process of the continuum random tree).Cette thĂšse est consacrĂ©e Ă l'Ă©tude de dĂ©compositions trajectorielles de processus de LĂ©vy spectralement positifs et des relations de dualitĂ© pour des processus de ramification, motivĂ©e par l'utilisation de ces derniers comme modĂšles probabilistes d'une dynamique Ă©pidĂ©miologique. Nous modĂ©lisons l'arbre de transmission d'une maladie comme un arbre de ramification, oĂč les individus Ă©voluent indĂ©pendamment les uns des autres, ont des durĂ©es de vie i.i.d. (pĂ©riodes d'infectiositĂ©) et donnent naissance (infections secondaires) Ă un taux constant durant leur vie. Le processus d'incidence dans ce modĂšle est un processus de Crump-Mode-Jagers (CMJ) et le but principal des deux premiers chapitres est d'en caractĂ©riser la loi conjointement avec l'arbre de transmission partiellement observĂ©, inferĂ© Ă partir des donnĂ©es de sĂ©quences. Dans le Chapitre I, nous obtenons une description en termes de fonctions gĂ©nĂ©ratrices de la loi du nombre d'individus infectieux, conditionnellement Ă l'arbre de transmission reliant les individus actuellement infectĂ©s. Une version plus Ă©lĂ©gante de cette caractĂ©risation est donnĂ©e dans le Chapitre II, en passant par un rĂ©sultat gĂ©nĂ©ral d'invariance par retournement du temps pour une classe de processus de ramification. Finallement, dans le Chapitre III nous nous intĂ©ressons Ă la loi d'un processus de ramification (sous)critique vu depuis son temps d'extinction. Nous obtenons un rĂ©sultat de dualitĂ© qui implique en particulier l'invariance par retournement du temps depuis leur temps d'extinction des processus CMJ (sous)critiques et de l'excursion hors de 0 de la diffusion de Feller critique (le processus de largeur de l'arbre alĂ©atoire de continuum)
Branching processes seen from their extinction time via path decompositions of reflected LĂ©vy processes
International audienceWe consider a spectrally positive LĂ©vy process X that does not drift to +â, viewed as coding for the genealogical structure of a (sub)critical branching process, in the sense of a contour or exploration process [34, 29]. We denote by I the past infimum process defined for each tâ„0 by It:=inf[0,t]X and we let Îł be the unique time at which the excursion of the reflected process XâI away from 0 attains its supremum. We prove that the pre-Îł and the post-Îł subpaths of this excursion are invariant under space-time reversal, which has several other consequences in terms of duality for excursions of LĂ©vy processes. It implies in particular that the local time process of this excursion is also invariant when seen backward from its height. As a corollary, we obtain that some (sub)critical branching processes such as the binary, homogeneous (sub)critical Crump-Mode-Jagers (CMJ) processes and the excursion away from 0 of the critical Feller diffusion, which is the width process of the continuum random tree, are invariant under time reversal from their extinction time
On Computer-Intensive Simulation and Estimation Methods for Rare Event Analysis in Epidemic Models
International audienceThis article focuses, in the context of epidemic models, on rare events that may possibly correspond to crisis situations from the perspective of Public Health. In general, no close analytic form for their occurrence probabilities is available and crude Monte-Carlo procedures fail. We show how recent intensive computer simulation techniques, such as interacting branching particle methods, can be used for estimation purposes, as well as for generating model paths that correspond to realizations of such events. Applications of these simulation-based methods to several epidemic models are also considered and discussed thoroughly