arXiv subjects

Jean-François Delmas

Publications and source records attributed to Jean-François Delmas.

At least 37 records · Page 2Linked to original sources

arXiv2022

Optimal vaccinations: Cordons sanitaires, reducible population and optimal rays

We consider the bi-objective problem of allocating doses of a (perfect) vaccine to an infinite-dimensional metapopulation in order to minimize simultaneously the vaccination cost and the effective reproduction number $R_e$, which is defined as the spectral radius of the effective next-generation operator. In this general framework, we prove that a cordon sanitaire, that is, a strategy that effectively disconnects the non-vaccinated population, might not be optimal, but it is still better than the "worst" vaccination strategies. Inspired by graph theory, we also compute the minimal cost which ensures that no infection occurs using independent sets. Using Frobenius decomposition of the whole population into irreducible sub-populations, we give some explicit formulae for optimal ("best" and "worst") vaccinations strategies. Eventually, we provide some sufficient conditions for a scaling of an optimal strategy to still be optimal.

math.OC↗

arXiv2022

The effective reproduction number: convexity, concavity and invariance

Motivated by the question of optimal vaccine allocation strategies in heterogeneous population for epidemic models, we study various properties of the \emph{effective reproduction number}. In the simplest case, given a fixed, non-negative matrix $K$, this corresponds mathematically to the study of the spectral radius $R_e(η)$ of the matrix product $\mathrm{Diag}(η)K$, as a function of $η\in\mathbb{R}_+^n$. The matrix $K$ and the vector $η$ can be interpreted as a next-generation operator and a vaccination strategy. This can be generalized in an infinite dimensional case where the matrix $K$ is replaced by a positive integral compact operator, which is composed with a multiplication by a non-negative function $η$. We give sufficient conditions for the function $R_e$ to be convex or a concave. Eventually, we provide equivalence properties on models which ensure that the function $R_e$ is unchanged.

math.OC↗

arXiv2022

Vaccinating according to the maximal endemic equilibrium achieves herd immunity

We consider the simple epidemiological SIS model for a general heterogeneous population introduced by Lajmanovich and Yorke (1976) in finite dimension, and its infinite dimensional generalization we introduced in previous works. In this model the basic reproducing number $R_0$ is given by the spectral radius of an integral operator. If $R_0>1$, then there exists a maximal endemic equilibrium. In this very general heterogeneous SIS model, we prove that vaccinating according to the profile of this maximal endemic equilibrium ensures herd immunity. Moreover, this vaccination strategy is critical: the resulting effective reproduction number is exactly equal to one. As an application, we estimate that if $R_0 = 2$ in an age-structured community with mixing rates fitted to social activity, applying this strategy would require approximately 29% less vaccine doses than the strategy which consists in vaccinating uniformly a proportion $1 - 1/R_0$ of the population. From a dynamical systems point of view, we prove that the non-maximality of an equilibrium $g$ is equivalent to its linear instability in the original dynamics, and to the linear instability of the disease-free state in the modified dynamics where we vaccinate according to $g$.

math.DS↗

arXiv2022

Conditioning (sub)critical L{é}vy trees by their maximal degree: Decomposition and local limit

We study the maximal degree of (sub)critical L{é}vy trees which arise as the scaling limits of Bienaym{é}-Galton-Watson trees. We determine the genealogical structure of large nodes and establish a Poissonian decomposition of the tree along those nodes. Furthermore, we make sense of the distribution of the L{é}vy tree conditioned to have a fixed maximal degree. In the case where the L{é}vy measure is diffuse, we show that the maximal degree is realized by a unique node whose height is exponentially distributed and we also prove that the conditioned L{é}vy tree can be obtained by grafting a L{é}vy forest on an independent size-biased L{é}vy tree with a degree constraint at a uniformly chosen leaf. Finally, we show that the L{é}vy tree conditioned on having large maximal degree converges locally to an immortal tree (which is the continuous analogue of the Kesten tree) in the critical case and to a condensation tree in the subcritical case. Our results are formulated in terms of the exploration process which allows to drop the Grey condition.

math.PR↗

arXiv2021

Optimal vaccination: Various (counter) intuitive examples

In previous articles, we formalized the problem of optimal allocation strategies for a (perfect) vaccine in an infinite-dimensional metapopulation model. The aim of the current paper is to illustrate this theoretical framework with multiple examples where one can derive the analytic expression of the optimal strategies. We discuss in particular the following points: whether or not it is possible to vaccinate optimally when the vaccine doses are given one at a time (greedy vaccination strategies); the effect of assortativity (that is, the tendency to have more contacts with similar individuals) on the shape of optimal vaccination strategies; the particular case where everybody has the same number of neighbors.

math.OC↗

arXiv2021

Effective reproduction number: Convexity, invariance and cordons sanitaires

We consider the problem of optimal allocation strategies for a (perfect) vaccine in an infinite-metapopulation model (including SIS, SIR, SEIR,. . .), when the loss function is given by the effective reproduction number Re, which is defined as the spectral radius of the effective next generation matrix (in finite dimension) or more generally of the effective next generation operator (in infinite dimension). We give sufficient conditions for Re to be a convex or a concave function of the vaccination strategy. Then, following a previous work, we consider the bi-objective problem of minimizing simultaneously the cost and the loss of the vaccination strategies. In particular, we prove that a cordon sanitaire might not be optimal, but it is still better than the "worst" vaccination strategies. Inspired by the graph theory, we compute the minimal cost which ensures that no infection occurs using independent sets. Using Frobenius decomposition of the whole population into irreducible sub-populations, we give some explicit formulae for optimal ("best" and "worst") vaccinations strategies. Eventually, we provide equivalence properties on models which ensure that the function Re is unchanged.

math.OC↗

arXiv2021

Targeted Vaccination Strategies for an Infinite-dimensional SIS Model

We formalize and study the problem of optimal allocation strategies for a (perfect) vaccine in the infinite-dimensional SIS model. The question may be viewed as a bi-objective minimization problem, where one tries to minimize simultaneously the cost of the vaccination, and a loss that may be either the effective reproduction number, or the proportion of the infected population in the endemic state. We prove the existence of Pareto optimal strategies, describe the corresponding Pareto frontier in both cases, and study its convexity and stability properties. We also show that vaccinating according to the profile of the endemic state is a critical allocation, in the sense that, if the initial reproduction number is larger than 1, then this vaccination strategy yields an effective reproduction number equal to 1.

math.PR↗

arXiv2021

Central limit theorem for kernel estimator of invariant density in bifurcating Markov chains models

Bifurcating Markov chains (BMC) are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. Motivated by the functional estimation of the density of the invariant probability measure which appears as the asymptotic distribution of the trait, we prove the consistence and the Gaussian fluctuations for a kernel estimator of this density based on late generations. In this setting, it is interesting to note that the distinction of the three regimes on the ergodic rate identified in a previous work (for fluctuations of average over large generations) disappears. This result is a first step to go beyond the threshold condition on the ergodic rate given in previous statistical papers on functional estimation.

math.ST↗

arXiv2021

Central limit theorem for bifurcating Markov chains under $L^{2}$-ergodic conditions

Bifurcating Markov chains (BMC) are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. We provide a central limit theorem for additive functionals of BMC under $L^2$-ergodic conditions with three different regimes. This completes the pointwise approach developed in a previous work. As application, we study the elementary case of symmetric bifurcating autoregressive process, which justify the non-trivial hypothesis considered on the kernel transition of the BMC. We illustrate in this example the phase transition observed in the fluctuations.

math.PR↗

arXiv2021

Central limit theorem for bifurcating Markov chains under point-wise ergodic conditions

Bifurcating Markov chains (BMC) are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. We provide a central limit theorem for general additive functionals of BMC, and prove the existence of three regimes. This corresponds to a competition between the reproducing rate (each individual has two children) and the ergodicity rate for the evolution of the trait. This is in contrast with the work of Guyon (2007), where the considered additive functionals are sums of martingale increments, and only one regime appears. Our result can be seen as a discrete time version, but with general trait evolution, of results in the time continuous setting of branching particle system from Adamczak and Miłoś (2015), where the evolution of the trait is given by an Ornstein-Uhlenbeck process.

math.PR↗

arXiv2020

Global Regime for General Additive Functionals of Conditioned Bienaym{é}-Galton-Watson Trees

We give an invariance principle for very general additive functionals of conditioned Bienaym{é}-Galton-Watson trees in the global regime when the offspring distribution lies in the domain of attraction of a stable distribution, the limit being an additive functional of a stable L{é}vy tree. This includes the case when the offspring distribution has finite variance (the L{é}vy tree being then the Brownian tree). We also describe, using an integral test, a phase transition for toll functions depending on the size and height.

math.PR↗

arXiv2020

An Infinite-Dimensional SIS Model

In this article, we introduce an infinite-dimensional deterministic SIS model which takes into account the heterogeneity of the infections and the social network among a large population. We study the long-time behavior of the dynamic. We identify the basic reproduction number $R_0$ which determines whether there exists a stable endemic steady state (super-critical case: $R_0>1$) or if the only equilibrium is disease-free (critical and sub-critical case: $R_0\leq1$). As an application of this general study, we prove that the so-called ``leaky'' and ``all-or-nothing'' vaccination mechanism have the same effect on $R_0$. This framework is also very natural and intuitive to model lockdown policies and study their impact.

math.DS↗

arXiv2020

Some properties of stationary continuous state branching processes

We consider the genealogical tree of a stationary continuous state branching process with immigration. For a sub-critical stable branching mechanism, we consider the genealogical tree of the extant population at some fixed time and prove that, up to a deterministic time-change, it is distributed as a continuous-time Galton-Watson process with immigration. We obtain similar results for a critical stable branching mechanism when only looking at immigrants arriving in some fixed time-interval. For a general sub-critical branching mechanism, we consider the number of individuals that give descendants in the extant population. The associated processes (forward or backward in time) are pure-death or pure-birth Markov processes, for which we compute the transition rates.

math.PR↗

arXiv2020

Exact simulation of the genealogical tree for a stationary branching population and application to the asymptotics of its total length

We consider a model of stationary population with random size given by a continuous state branching process with immigration with a quadratic branching mechanism. We give an exact elementary simulation procedure of the genealogical tree of $n$ individuals randomly chosen among the extant population at a given time. Then, we prove the convergence of the renormalized total length of this genealogical tree as $n$ goes to infinity, see also Pfaffelhuber, Wakolbinger and Weisshaupt (2011) in the context of a constant size population. The limit appears already in Bi and Delmas (2016) but with a different approximation of the full genealogical tree. The proof is based on the ancestral process of the extant population at a fixed time which was defined by Aldous and Popovic (2005) in the critical case.

math.PR↗

arXiv2018

Asymptotic for the cumulative distribution function of the degrees and homomorphism densities for random graphs sampled from a graphon

We give asymptotics for the cumulative distribution function (CDF) for degrees of large dense random graphs sampled from a graphon. The proof is based on precise asymptotics for binomial random variables. Replacing the indicator function in the empirical CDF by a smoother function, we get general asymptotic results for functionals of homomorphism densities for partially labeled graphs with smoother functions. This general setting allows to recover recent results on asymptotics for homomorphism densities of sampled graphon.

math.PR↗

arXiv2017

Asymptotic properties of expansive Galton-Watson trees

We consider a super-critical Galton-Watson tree whose non-degenerate offspring distribution has finite mean. We consider the random trees $τ$n distributed as $τ$ conditioned on the n-th generation, Zn, to be of size an $\in$ N. We identify the possible local limits of $τ$n as n goes to infinity according to the growth rate of an. In the low regime, the local limit $τ$ 0 is the Kesten tree, in the moderate regime the family of local limits, $τ$ $θ$ for $θ$ $\in$ (0, +$\infty$), is distributed as $τ$ conditionally on {W = $θ$}, where W is the (non-trivial) limit of the renormalization of Zn. In the high regime, we prove the local convergence towards $τ$ $\infty$ in the Harris case (finite support of the offspring distribution) and we give a conjecture for the possible limit when the offspring distribution has some exponential moments. When the offspring distribution has a fat tail, the problem is open. The proof relies on the strong ratio theorem for Galton-Watson processes. Those latter results are new in the low regime and high regime, and they can be used to complete the description of the (space-time) Martin boundary of Galton-Watson processes. Eventually, we consider the continuity in distribution of the local limits ($τ$ $θ$ , $θ$ $\in$ [0, $\infty$]).

math.PR↗

arXiv2017

Very fat geometric galton-watson trees

Let $τ$n be a random tree distributed as a Galton-Watson tree with geometric offspring distribution conditioned on {Zn = an} where Zn is the size of the n-th generation and (an, n $\in$ N *) is a deterministic positive sequence. We study the local limit of these trees $τ$n as n $\rightarrow$ $\infty$ and observe three distinct regimes: if (an, n $\in$ N *) grows slowly, the limit consists in an infinite spine decorated with finite trees (which corresponds to the size-biased tree for critical or subcritical offspring distributions), in an intermediate regime, the limiting tree is composed of an infinite skeleton (that does not satisfy the branching property) still decorated with finite trees and, if the sequence (an, n $\in$ N *) increases rapidly, a condensation phenomenon appears and the root of the limiting tree has an infinite number of offspring.

math.PR↗

arXiv2016

Critical Multi-Type Galton-Watson Trees Conditioned to be Large

Under minimal condition, we prove the local convergence of a critical multi-type Galton-Watson tree conditioned on having a large total progeny by types towards a multi-type Kesten's tree. We obtain the result by generalizing Neveu's strong ratio limit theorem for aperiodic random walks on Z^d .

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