arXiv subjects

Sonia Velasco

Publications and source records attributed to Sonia Velasco.

4 recordsLinked to original sources

arXiv2025

Extinction and survival in inherited sterility

We introduce an interacting particle system which models the inherited sterility method. Individuals evolve on $\mathbb{Z}^d$ according to a contact process with parameter $λ>0$. With probability $p \in [0,1]$ an offspring is fertile and can give birth to other individuals at rate $λ$. With probability $1-p$, an offspring is sterile and blocks the site it sits on until it dies. The goal is to prove that at fixed $λ$, the system survives for large enough $p$ and dies out for small enough $p$. The model is not attractive, since an increase of fertile individuals potentially causes that of sterile ones. However, thanks to a comparison argument with attractive models, we are able to answer our question.

math.PR↗

arXiv2025

Duality for some models of epidemic spreading

We examine the role of boundaries and the structure of nontrivial duality functions for three non conservative interacting particle systems in one dimension that model epidemic spreading: (i) the diffusive contact process (DCP), (ii) a model that we introduce and call generalized diffusive contact process (GDCP), both in finite volume in contact with boundary reservoirs, i.e., with open boundaries, and (iii) the susceptible-infectious-recovered (SIR) model on $\Z$. We establish duality relations for each system through an analytical approach. It turns out that with open boundaries self-duality breaks down and qualitatively different properties compared to closed boundaries (i.e., finite volume without reservoirs) arise: both the DCP and GDCP are ergodic but no longer absorbing, while the respective dual processes are absorbing but not ergodic. We provide expressions for the stationary correlation functions in terms of the dual absorption probabilities. We perform explicit computations for a small sized DCP, and for arbitrary size in a particular setting of the GDCP. The duality function is factorized for the DCP and GDCP, contrary to the SIR model for which the duality relation is nonlocal and yields an explicit expression of the time evolution of some specific correlation functions, describing the time decay of the sizes of clusters of susceptible individuals.

math.PR↗

arXiv2023

Hydrodynamic and hydrostatic limit for a generalized contact process with mixed boundary conditions

We consider an interacting particle system which models the sterile insect technique. It is the superposition of a generalized contact process with exchanges of particles on a finite cylinder with open boundaries (see Kuoch et al., 2017). We show that when the system is in contact with reservoirs at different slowdown rates, the hydrodynamic limit is a set of coupled non linear reaction-diffusion equations with mixed boundary conditions. We also prove the hydrostatic limit when the macroscopic equations exhibit a unique attractor.

math.PR↗

arXiv2023

Dynamic and static large deviations of a one dimensional SSEP in weak contact with reservoirs

We derive a formula for the quasi-potential of one-dimensional symmetric exclusion process in weak contact with reservoirs. The interaction with the boundary is so weak that, in the diffusive scale, the density profile evolves as the one of the exclusion process with reflecting boundary conditions. In order to observe an evolution of the total mass, the process has to be observed in a longer time-scale, in which the density profile becomes immediately constant.

math.PR↗