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Publication# The Contact Process

Résumé

This work is a study of three interacting particle systems that are modified versions of the contact process. The contact process is a spin system defined on a graph and is commonly taken as a model for the spread of an infection in a population; transmission of the infection happens by proximity (contact). The first two models we consider – the grass-bushes-trees model and the multitype contact process – are models for competition between species in Ecology. The third model – annealed approximation to boolean networks – approximately describes the transmission of information among genes in a cell. We consider the grass-bushes-trees model on the set of integers, ℤ. Each point of ℤ is a region of space. In the continuous-time dynamics, at each instant each region can be either empty (state 0) or occupied by an individual of one of two existing species (states 1 and 2). Occupants of both species die at rate 1, leaving their regions empty, and send descendents to neighboring regions at rate λ. An individuals of type 1 may be born on a region previously occupied by an individual of type 2, but the converse is forbidden. We take the "heaviside" initial configuration in which all sites to the left of the origin are occupied by type 1 individuals and all sites to the right of the origin are occupied by type 2 individuals. If the birth of new individuals is allowed to occur at sites that are not adjacent to the parent, and if the rate λ is supercritical for the usual contact process on ℤ, we see the formation of an interface region in which both types coexist. Addressing a conjecture of Cox and Durrett (1995), we prove that the size of this region is stochastically tight. The multitype contact process on ℤ is a process identical to the grass-bushes-trees model in every respect except that no births can occur at previously occupied sites; in particular, the model is symmetric for both species. We again start the process from the heaviside configuration and prove that the size of the interface region is tight. In addition, we prove that the position of the interface, when properly rescaled, converges to Brownian motion. Finally, we give necessary and sufficient conditions on the initial configuration so that one of the two species becomes extinct with probability one and also so that both species are present at all times with positive probability. Lastly, we consider a model proposed by Derrida and Pomeau (1986) and recently studied by Chatterjee and Durrett (2009); it is defined as an approximation to S. Kauffman's boolean networks (1969). The model starts with the choice of a random directed graph on n vertices; each node has r input nodes pointing at it. A discrete time threshold contact process is then considered on this graph: at each instant, each site has probability q of choosing to receive input; if it does, and if at least one of its inputs were occupied by a 1 at the previous instant, then it is labeled with a 1; in all other cases, it is labeled with a 0. r and q are kept fixed and n is taken to infinity. Improving a result of Chatterjee and Durrett, we show that if qr > 1, then the time of persistence of the dynamics is exponential in n.

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In this paper we study the speed of infection spread and the survival of the contact process in the random geometric graph G = G (n, r(n), f) of n nodes independently distributed in S = -1/2, 1/2 according to a certain density f(.). In the first part of the paper we assume that infection spreads from one node to another at unit rate and that infected nodes stay in the same state forever. We provide an explicit lower bound on the speed of infection spread and prove that infection spreads in G with speed at least D(1)nr(n)(2). In the second part of the paper we consider the contact process xi(t) on G where infection spreads at rate lambda > 0 from one node to another and each node independently recovers at unit rate. We prove that, for every lambda > 0, with high probability, the contact process on G survives for an exponentially long time; there exist positive constants c(1) and c(2) such that, with probability at least 1 - c(1)/n(4), the contact process xi(1)(t) starting with all nodes infected survives up to time t(n) = exp(c(2)n/log n) for all n.

Thomas Mountford, Daniel Rodrigues Valesin

We consider a model recently proposed by Chatterjee and Durrett [1] as an "annealed approximation" of boolean networks, which are a class of cellular automata on a random graph, as defined by S. Kauffman [5]. The starting point is a random directed graph on n vertices; each vertex has r input vertices pointing to it. For the model of [1], a discrete time threshold contact process is then considered on this graph: at each instant, each vertex has probability q of choosing to receive input; if it does, and if at least one of its input vertices were in state 1 at the previous instant, then it is labelled with a 1; in all other cases, it is labelled with a 0. r and q are kept fixed and n is taken to infinity. Improving one of the results of [1], we show that if qr > 1, then the time of persistence of activity of the dynamics is exponential in n

In order to be able to bear the risk they are taking, insurance companies have to set aside a certain amount of cushion that can guarantee the payment of liabilities, up to a dened probability, and thus to remain solvent in case of bad events. This amount is named capital. The calculation of capital is a complex problem. To be sustainable, capital must consider all possible risk sources that may lead to losses among assets and liabilities of the insurance company, and it must account for the likelihood and the eect of these bad (and usually extreme) events that could occur to the risk sources. Insurance companies build models and tools in order to perform this capital calculation. For that, they have to collect data, build statistical evidence, build mathematical models and tools in order to eciently and accurately derive capital. The papers exposed in this thesis deal with three major diculties. First, the uncertainty behind the choice of a specic model and the quantication of this uncertainty in terms of additional capital. The use of external scenarios, i.e. opinions on the likelihood of some events happening, allows to build a coherent methodology that make the cushion more robust against wrong model specication. Second, the computational complexity in using these models in an industrialized environment, and numerical methods available for increasing their computational eciency. Most of these models cannot provide an analytical formula of capital. Consequently, one has to approximate it via simulation methods. Considering the high number of risk sources and the complexity of insurance contracts, these methods can be slow to run before providing a reasonable accuracy. This often makes these models unusable in practical cases. Enhancements of classical simulation methods are presented in the aim of making these approximations faster to run for the same level of accuracy. Third, the lack of reliable data and the high complexity of problems with long time horizons, and statistical methods for identifying and building reliable proxies in such cases. A typical example is life-insurance contracts that imply being exposed to multiple risks sources over a long horizon. Such contracts can in fact be approximated wisely by proxies that can capture the risk over time.