

Program > Titles and abstractsNatalia Carbona Tobón, GeorgAugust 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 degreedependent 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 BienayméGaltonWatson 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  Emmanuel Jacob, École Normale Supérieure de Lyon, France Targeted immunisation thresholds for the contact process on powerlaw trees and scalefree networks.
Abstract: We consider the contact process on a GaltonWatson tree with powerlaw offspring distribution, which arises naturally as a local limit of some standard scalefree network models. These trees have enough highdegree 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), AixMarseille 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.  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.  Daniel Valesin, University of Warwick, United Kingdom The interchangeandcontact process We introduce a process called the interchangeandcontact 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 1p. 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.  Sonia Velasco, Université ParisCité, 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 1p, 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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