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Etienne Pardoux

Publications and source records attributed to Etienne Pardoux.

At least 37 records · Page 2Linked to original sources

arXiv2021

Epidemic models with varying infectivity

We introduce an epidemic model with varying infectivity and general exposed and infectious periods, where the infectivity of each individual is a random function of the elapsed time since infection, those function being i.i.d. for the various individuals in the population. This approach models infection-age dependent infectivity, and extends the classical SIR and SEIR models. We focus on the infectivity process (total force of infection at each time), and prove a functional law of large number (FLLN). In the deterministic limit of this LLN, the infectivity process and the susceptible process are determined by a two-dimensional deterministic integral equation. From its solutions, we then derive the exposed, infectious and recovered processes, again using integral equations. For the early phase, we study the stochastic model directly by using an approximate (non--Markovian) branching process, and show that the epidemic grows at an exponential rate on the event of non-extinction, which matches the rate of growth derived from the deterministic linearized equations. We also use these equations to derive the basic reproduction number $R_0$ during the early stage of an epidemic, in terms of the average individual infectivity function and the exponential rate of growth of the epidemic.

math.PR↗

arXiv2020

The height process of a continuous state branching process with interaction

For a generalized continuous state branching process with non-vanishing diffusion part, finite expectation and a directed ("left-to-right") interaction, we construct the height process of its forest of genealogical trees. The connection between this height process and the population size process is given by an extension of the second Ray--Knight theorem. This paper generalizes earlier work of the two last authors which was restricted to the case of continuous branching mechanisms. Our approach is different from that of Berestycki et al. There the diffusion part of the population process was allowed to vanish, but the class of interactions was more restricted.

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arXiv2020

A Spatial Stochastic Epidemic Model: Law of Large Numbers and Central Limit Theorem

We consider an individual-based SIR stochastic epidemic model in continuous space. The evolution of the epidemic involves the rates of infection and cure of individuals. We assume that individuals move randomly on the two-dimensional torus according to independent Brownian motions. We define the empirical measures $μ^{S,N}$, $μ^{I,N}$ and $μ^{R,N}$ which describe the evolution of the position of the susceptible, infected and removed individuals. We prove the convergence in propbability, as $N\rightarrow \infty$, of the sequence $(μ^{S,N},μ^{I,N})$ towards $(μ^{S},μ^{I})$ solution of a system of parabolic PDEs. We show that the sequence $(U^{N}=\sqrt{N}(μ^{S,N}-μ^{S}),V^{N}=\sqrt{N}(μ^{I,N}-μ^{I}))$ converges in law, as $N\rightarrow\infty$, towards a Gaussian distribution valued process, solution of a system of linear PDEs with highly singular Gaussian driving processes. In the case where the individuals do not move we also define and study the law of large numbers and central limit theorem of the sequence $(μ^{S,N},μ^{I,N})$.

math.PR↗

arXiv2020

Stochastic epidemics in a homogeneous community

These notes describe stochastic epidemics in a homogenous community. Our main concern is stochastic compartmental models (i.e. models where each individual belongs to a compartment, which stands for its status regarding the epidemic under study : S for susceptible, E for exposed, I for infectious, R for recovered) for the spread of an infectious disease. In the present notes we restrict ourselves to homogeneously mixed communities. We present our general model and study the early stage of the epidemic in chapter 1. Chapter 2 studies the particular case of Markov models, especially in the asymptotic of a large population, which leads to a law of large numbers and a central limit theorem. Chapter 3 considers the case of a closed population, and describes the final size of the epidemic (i.e. the total number of individuals who ever get infected). Chapter 4 considers models with a constant influx of susceptibles (either by birth, immigration of loss of immunity of recovered individuals), and exploits the CLT and Large Deviations to study how long it takes for the stochastic disturbances to stop an endemic situation which is stable for the deterministic epidemic model. The document ends with an Appendix which presents several mathematical notions which are used in these notes, as well as solutions to many of the exercises which are proposed in the various chapters.

math.PR↗

arXiv2019

Approximation of the Height process of a CSBP with interaction

In this work, we first show that the properly rescaled height process of the genealogical tree of a continuous time branching process converges to the height process of the genealogy of a (possibly discontinuous) continuous state branching process. We then prove the same type of result for generalized branching processes with interaction.

math.PR↗

arXiv2019

Moderate deviations and extinction of an epidemic

Consider an epidemic model with a constant flux of susceptibles, in a situation where the corresponding deterministic epidemic model has a unique stable endemic equilibrium. For the associated stochastic model, whose law of large numbers limit is the deterministic model, the disease free equilibrium is an absorbing state, which is reached soon or later by the process. However, for a large population size, i.e. when the stochastic model is close to its deterministic limit, the time needed for the stochastic perturbations to stop the epidemic may be enormous. In this paper, we discuss how the Central Limit Theorem, Moderate and Large Deviations allow us to give estimates of the extinction time of the epidemic, depending upon the size of the population.

math.PR↗

arXiv2018

Large Deviations of the Exit Measure through a Characteristic Boundary for a Poisson driven SDE

Let O the basin of attraction of the unique stable equilibrium of a dynamical system, which is the law of large numbers limit of a Poissonian SDE. We consider the law of the exit point from O of that Poissonian SDE. We adapt the approach of M. Day (1990) for the same problem for a Brownian SDE. For that purpose, we will use the Large deviation for the Poissonian SDE reflected at the boundary of O studied in our recent work Pardoux and Samegni (2018).

math.PR↗

arXiv2018

Large Deviation Principle for Reflected Poisson driven SDEs in Epidemic Models

We establish a large deviation principle for a reflected Poisson driven SDE. Our motivation is to study in a forthcoming paper the problem of exit of such a process from the basin of attraction of a locally stable equilibrium associated with its law of large numbers. Two examples are described in which we verify the assumptions that we make to establish the large deviation principle.

math.PR↗

arXiv2017

Convergence to a Continuous State Branching Process with jumps and Height Process

In this work, we study asymptotics of the genealogy of Galton-Watson processes. Thus we consider a offspring distribution such that the rescaled Galton-Watson processes converges to a continuous state branching process (CSBP) with jumps. After we show that the rescaled height (or exploration) process of the corresponding Galton-Watson family tree, converges in a functional sense, to the continuous height process that Le Gall and Le Jan introduced in 1998 on their paper "branching processes in Lévy processes : The exploration process".

math.PR↗

arXiv2016

Large deviation principle for Poisson driven SDEs in epidemic models

We consider a general class of epidemic models obtained by applying a random time change to a collection of Poisson processes and we show the large deviation principle for such models. We generalize to a more general situation the approach followed by Dolgoaschinnykh [3] in the case of the SIR epidemic model. Thanks to an assumption which is satisfied in many examples, we simplify the recent work by P. Kratz and E. Pardoux [8].

math.PR↗

arXiv2016

Large deviations for infectious diseases models

We establish large deviation estimates for Poisson driven models of infectious disease, and apply those estimates to the time of exit from the basin of attraction of an endemic equilibrium. We apply our results to the classical SIRS model.

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arXiv2016

Non-binary branching process and Non-Markovian exploration process

We study the exploration (or height) process of a continuous time non-binary Galton-Watson random tree, in the subcritical, critical and supercritical cases. Thus we consider the branching process in continuous time (Z_{t})_{t\geq 0}, which describes the number of offspring alive at time t. We then renormalize our branching process and exploration process, and take the weak limit as the size of the population tends to infinity. Finally we deduce a Ray-Knight representation.

math.PR↗

arXiv2016

Continuity of the Feynman-Kac formula for a generalized parabolic equation

It is well-known since the work of Pardoux and Peng [12] that Backward Stochastic Differential Equations provide probabilistic formulae for the solution of (systems of) second order elliptic and parabolic equations, thus providing an extension of the Feynman-Kac formula to semilinear PDEs, see also Pardoux and Rascanu [14]. This method was applied to the class of PDEs with a nonlinear Neumann boundary condition first by Pardoux and Zhang [15]. However, the proof of continuity of the extended Feynman-Kac formula with respect to x (resp. to (t,x)) is not correct in that paper. Here we consider a more general situation, where both the equation and the boundary condition involve the (possibly multivalued) gradient of a convex function. We prove the required continuity. The result for the class of equations studied in [15] is a Corollary of our main results.

math.PR↗

arXiv2016

On the large time behaviour of the solution of an SDE driven by a Poisson Point Process

We study a stochastic differential equation driven by a Poisson point process, which models continuous changes in a population's environment, as well as the stochastic fixation of beneficial mutations that might compensate for this change. The fixation probability of a given mutation increases as the phenotypic lag X_t between the population and the optimum grows larger, and successful mutations are assumed to fix instantaneously (leading to an adaptive jump). Our main result is that the process is transient (i.e., continued adaptation is impossible) if the rate of environmental change v exceeds a parameter m, which can be interpreted as the rate of adaptation in case every beneficial mutation gets fixed with probability 1. If v < m, the process is positive recurrent, while in the limiting case m=v, null recurrence or transience depends upon additional technical conditions. We show how our results can be extended to the case of a time varying rate of environmental change.

math.PR↗

arXiv2015

Stochastic Variational Inequalities on Non-Convex Domains

The objective of this work is to prove, in a first step, the existence and the uniqueness of a solution of the following multivalued deterministic differential equation: $dx(t)+\partial ^-φ(x(t))(dt)\ni dm(t),\ t>0$, $x(0)=x_0$, where $m:\mathbb{R}_+\rightarrow\mathbb{R}^d$ is a continuous function and $\partial^-φ$ is the Fréchet subdifferential of a semiconvex function $φ$; the domain of $φ$ can be non-convex, but some regularities of the boundary are required. The continuity of the map $m\mapsto x:C([0,T];\mathbb{R}^{d})\rightarrow C([0,T] ;\mathbb{R}^{d})$, which associate the input function $m$ with the solution $x$ of the above equation, as well as tightness criteria allow to pass from the above deterministic case to the following stochastic variational inequality driven by a multi-dimensional Brownian motion: $X_t+K_t = ξ+\int_0^t F(s,X_{s})ds + \int_0^t G(s,X_s) dB_s,\; t\geq0$, $\;$ with $dK_{t}(ω)\in\partial^-φ( X_t (ω))(dt)$.

math.DS↗

arXiv2015

A mixing tree-valued process arising under neutral evolution with recombination

The genealogy at a single locus of a constant size $N$ population in equilibrium is given by the well-known Kingman's coalescent. When considering multiple loci under recombination, the ancestral recombination graph encodes the genealogies at all loci in one graph. For a continuous genome $G$, we study the tree-valued process $(T^N_u)_{u\in G}$ of genealogies along the genome in the limit $N\to\infty$. Encoding trees as metric measure spaces, we show convergence to a tree-valued process with cadlag paths. In addition, we study mixing properties of the resulting process for loci which are far apart.

math.PR↗

arXiv2015

A path-valued Markov process indexed by the ancestral mass

A family of Feller branching diffusions $Z^x$, $x \ge 0$, with nonlinear drift and initial value $x$ can, with a suitable coupling over the {\em ancestral masses} $x$, be viewed as a path-valued process indexed by $x$. For a coupling due to Dawson and Li, which in case of a linear drift describes the corresponding Feller branching diffusion, and in our case makes the path-valued process Markovian, we find an SDE solved by $Z$, which is driven by a random point measure on excursion space. In this way we are able to identify the infinitesimal generator of the path-valued process. We also establish path properties of $x\mapsto Z^x$ using various couplings of $Z$ with classical Feller branching diffusions.

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arXiv2015

Numerical methods in the context of compartmental models in epidemiology

We consider compartmental models in epidemiology. For the study of the divergence of the stochastic model from its corresponding deterministic limit (i.e., the solution of an ODE) for long time horizon, a large deviations principle suggests a thorough numerical analysis of the two models. The aim of this paper is to present three such motivated numerical works. We first compute the solution of the ODE model by means of a non-standard finite difference scheme. Next we solve a constraint optimization problem via discrete-time dynamic programming: this enables us to compute the leading term in the large deviations principle of the time of extinction of a given disease. Finally, we apply the τ-leaping algorithm to the stochastic model in order to simulate its solution efficiently. We illustrate these numerical methods by applying them to two examples.

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