Modèles aléatoires en écologie,
génétique et évolution
Rencontre du 30 janvier 2013
à Paris 13
Les exposés auront lieux dans l'amphithéâtre Fermat, qui se trouve en
aile G au premier étage de l'Institut Galilée. Voir le plan
(L'Institut Galilée est indiqué par le chiffre 2)
Programme et Présentations
Pour se rendre au campus de Villetaneuse de l'université Paris 13, des
Il y a un plan googlemap
pour indiquer le cheminement à pied (bleu) ou
en bus (rouge) depuis la gare d'Epinay-Villetaneuse.
- Marc Lelarge : Diffusions and cascades in random networks
The spread of new ideas, behaviors or technologies has been
extensively studied using epidemic models. Here we consider a model of
diffusion where the individuals' behavior is the result of a strategic
choice. We study a simple coordination game with binary choice and give
a condition for a new action to become widespread in a random network.
Our results differ strongly from the one derived with epidemic models
and show that connectivity plays an ambiguous role: while it allows the
diffusion to spread, when the network is highly connected, the diffusion
is also limited by high-degree nodes which are very stable.
In a second part (joint work with E.Coupechoux), we study a model of
random networks that has both a given degree distribution and a tunable
clustering coefficient. We analyze the impact of clustering on the
cascades (size and frequency).
- Bernard Cazelles & Anton Camacho : Accounting for immunodynamics in epidemiological models
Epidemiological models of influenza transmission usually assume that
recovered individuals instantly develop a fully protective immunity
against the infecting strain. However, recent studies have highlighted
host heterogeneity in the development of this immune response,
characterized by delay and even absence of protection that could lead to
homologous reinfection. Here, we investigate how these immunological
mechanisms at the individual level shape the epidemiological dynamics at
the population level. In particular we assess their roles in driving
multiple-wave influenza outbreaks. We develop a mechanistic model
accounting for host heterogeneity in the immune response. Immunological
parameters are inferred by confronting our dynamical model to a two-wave
influenza epidemic that occurred on the remote island of Tristan da
Cunha in 1971.
- Romain Guy : Statistical inference for epidemic models approximated by diffusion and Gaussian processes.
- Pieter Trapman : Long-range percolation on the hierarchical lattice
The hierarchical lattice of order N, may be seen as the leaves of an
infinite regular N-tree, in which the distance between two vertices is
the distance to their most recent common ancestor in the tree. We create
a random graph by independent long-range percolation on the hierarchical
lattice of order N: The probability that a pair of vertices at
(hierarchical) distance R share an edge depends only on R and is
exponentially decaying in R, furthermore the presence or absence of
different edges are independent. We give criteria for percolation (the
presence of an infinite cluster) and we show that in the supercritical
parameter domain, the infinite component is unique. Furthermore, we show
the percolation probability (the density of the infinite cluster) is
continuous in the model parameters. In particular, there is no
percolation at criticality. Joint work with Slavik Koval and Ronald Meester
Peter Windridge : Law of large numbers for the SIR epidemic on a random graph with given vertex degrees
We consider the limiting behaviour of the SIR epidemic model on a random graph chosen uniformly subject to having given vertex degrees.
In the model, each vertex is either susceptible, infective or recovered. Infective vertices infect their susceptible neighbours, and recover, at a constant rate. Initially there is only one infective vertex.
The infection and recovery rates, together with the limiting vertex degree distribution, determine a 'growth' parameter for the disease. If this parameter is below a (specified) critical threshold then, with high probability, only a small number of vertices ever get infected.
Above the threshold there is a positive probability that many vertices get infected.
In that case, the evolution of the epidemic is almost deterministic in the sense that key quantities, such as the fraction of infective vertices, are concentrated around the solutions to certain ordinary differential equations.
I'll explain these results in detail and sketch our new proof. Compared to existing approaches, ours surmounts a technical assumption on the degree sequence and is shorter.
This is joint work with Malwina Luczak and Svante Janson.
Last modified: 25 Jan 2013