Program > Titles and abstracts

Natalia Carbona Tobón, Georg-August Universität Göttingen, Germany

The contact process on dynamical random trees with degree dependence

In this talk we investigate the contact process in the case when the underlying structure evolves dynamically as a degree-dependent dynamical percolation model. Starting with a connected locally finite base graph we initially declare edges independently open with a probability that is allowed to depend on the degree of the adjacent vertices and closed otherwise. Edges are independently updated with a rate depending on the degrees and then are again declared open and closed with the same probabilities. We are interested in the contact process, where infections are only allowed to spread via open edges. Our aim is to analyse the impact of the update speed and the probability for edges to be open on the existence of a phase transition. For a general connected locally finite graph, our first result gives sufficient conditions for the critical value for survival to be strictly positive. Furthermore, in the setting of Bienaymé-Galton-Watson trees, we show that the process survives strongly with positive probability for any infection rate if the offspring distribution has a stretched exponential tail with an exponent depending on the percolation probability and the update speed. In particular, if the offspring distribution follows a power law and the connection probability is given by a product kernel and the update speed exhibits polynomial behaviour, we provide a complete characterisation of the phase transition.

This talk is based on join work with Marcel Ortgiese (University of Bath), Marco Seiler (University of Frankfurt) and Anja Sturm (University of Göttingen).

Henk Don, Radboud University Nijmegen, The Nederlands

The contact process on finite graphs

The contact process on a finite graph dies out with probability 1. Nevertheless, we will discuss how one can still identify a phase transition between quick extinction and long survival. In the long survival phase, the process exhibits metastable behavior. In this talk we will review some results on the phase transition, the extinction time and the metastable distribution of the contact process on finite graphs. In particular we will discuss the complete graph and the Erdös-Rényi graph.

Emmanuel Jacob, École Normale Supérieure de Lyon, France

Targeted immunisation thresholds for the contact process on power-law trees and scale-free networks.

Abstract: We consider the contact process on a Galton-Watson tree with power-law offspring distribution, which arises naturally as a local limit of some standard scale-free network models. These trees have enough high-degree vertices to allow propagation and survival of the contact process even with an arbitrarily small infection rate. We then investigate the effect of immunisation of all vertices with degree above a threshold, which is allowed to depend on the infection rate. Depending on the value of this threshold, we prove that the survival probability of the contact process after immunisation is essentially unchanged, or severely reduced, or equal to zero. This is joint work with John Fernley.

Júlia Komjáthy, Delft University of Technology, The Nederlands

Degree dependent contact process on Galton Watson trees and the configuration model

In this talk we look at degree dependent contact processes. This is a variant of the usual contact process, where transmissions through edges depend on the degrees of the transmitter and the receiver vertex. The dependence is such that in a unit time, any transmitter vertex infects in expectation a sublinear but increasing number of its neighbors. We unfold the phase transitions of this new process with respect to the small lambda behavior on Galton Watson trees with degree distributions of unbounded support. More precisely, we show that there is a phase transition at the square root function: when in expectation, a vertex infects more than square root of its neighbors, then the process behaves qualitatively similar to the classical process; while when it infects less on average, then new phases occur in the dynamics.

Régine Marchand, Institut Élie Cartan de Lorraine (IECL), Nancy, France

The asymptotic shape theorem for some versions of the contact process

The first aim of the talk is the introduce our guest star: the contact process. Then I will focus on the case of the supercritical contact process on Zd,where, conditionaly on survival, the growth of the contact process starting from a finite configuration is governed by a shape theorem. I will try to explain the main ingredients of the proof, possible extensions to some random environments and still open questions. In the last part of the talk, Il will try to present current questions about the contact process and its variations, beyond the Zd case.

Bruno Schapira, Institut de Mathématiques de Marseille (I2M), Aix-Marseille Université, France

Contact process on a dynamic random regular graph

We consider the contact process on a dynamic random regular graph. We show that there exists a critical value for the infection parameter, below which the contact process dies out in a time which is logarithmic in the size of the graph. This completes an earlier result of da Silva, Oliveira and Valesin, showing that above this critical value, the process survives a time exponential in the size of the graph.
Joint work with Daniel Valesin.

Marco Seiler, Frankfurt Institute for Advanced Studies (FIAS) and Goethe University Frankfurt, Germany

Asymptotic behaviour of the contact process in an evolving random environment

We study a contact process in an evolving (edge) random environment on (infinite) connected and transitive graphs. We assume that the evolving random environment is described by an autonomous ergodic spin systems with finite range, for example by dynamical percolation. This background process determines which edges are open or closed for infections.
In particular, we discuss the phase transition of survival and the dependence of the associated critical infection rate on the random environment and on the initial configuration of the system. For the latter, we state sufficient conditions such that the initial configuration of the system has no influence on the phase transition between extinction and survival. We show that this phase transition coincides with the phase transition between ergodicity and non-ergodicity and we discuss a complete convergences result of the process.
At the end of the talk we present some partial results regarding the expansion speed and asymptotic shape of the infection process conditioned on survival on a d-dimensional integer lattice.
This talk is based on joint work with Anja Sturm and on going work with Noemi Kurt and Michel Reitmeier.

Daniel Valesin, University of Warwick, United Kingdom

The interchange-and-contact process

We introduce a process called the interchange-and-contact process, which is defined on an arbitrary graph as follows. At any point in time, vertices are in one of three states: empty, occupied by a healthy individual, or occupied by an infected individual. Infected individuals recover with rate 1 and infect healthy individuals in neighboring vertices with rate lambda. Additionally, each edge has a clock with rate v, and when this clock rings, the states of the two vertices of the edge are exchanged. This means that particles perform an interchange process with rate v, moving around and, when infected, carrying the infection with them. We study this process on Z^d, with an initial configuration where there is an infected particle at the origin, and every other vertex contains a healthy particle with probability p and is empty with probability 1-p. We define lambda_c(v, p) as the infimum of the values of lambda for which the process survives with positive probability. We prove results about the asymptotic behavior of \lambda_c when p is fixed and v is taken to zero and to infinity.
Joint work with Daniel Ungaretti, Marcelo Hilário and Maria Eulalia Vares.

Sonia Velasco, Université Paris-Cité, France

Extinction and survival in inherited sterility

We introduce an interacting particle system which models the inherited sterility method. Individuals evolve on Z^d according to a contact process with parameter lambda > 0. With probability p in [0,1] an offspring is fertile and can give birth to other individuals at rate lambda. 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 lambda, 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.

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