125 research outputs found

### 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

### Diffusion limit for many particles in a periodic stochastic acceleration field

The one-dimensional motion of any number \cN of particles in the field of
many independent waves (with strong spatial correlation) is formulated as a
second-order system of stochastic differential equations, driven by two Wiener
processes. In the limit of vanishing particle mass ${\mathfrak{m}} \to 0$, or
equivalently of large noise intensity, we show that the momenta of all $N$
particles converge weakly to $N$ independent Brownian motions, and this
convergence holds even if the noise is periodic. This justifies the usual
application of the diffusion equation to a family of particles in a unique
stochastic force field. The proof rests on the ergodic properties of the
relative velocity of two particles in the scaling limit.Comment: 20 page

### 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.Comment: Part I of "Stochastic Epidemic Models with Inference", T. Britton &
E. Pardoux eds., Lecture Notes in Mathematics 2255, Springer 201

### 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.Comment: 23 pages, 1 figure. This version will appear in ALEA. Compared to v1,
it contains amendmends mainly in Sec. 2 and in the proof of Proposition 4.

### 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.Comment: arXiv admin note: text overlap with arXiv:1304.071

### 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

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