Title: Epidemics with two levels of mixing

Standard deterministic models of epidemics implicitly assume that the population among which the disease is spreading is locally as well as globally large, in the sense that if the population is partitioned into groups, e.g. by age, sex and/or geographical location, then each of these groups, and not just the total population, is large. The same assumption is made when analysing the large-population behaviour of many stochastic epidemic models. However, this assumption is unrealistic for many human epidemics, since such populations contain small social groups, such as households, school classes and workplaces, in which transmission is likely to be enhanced. Thus, there has been a growing interest in models for epidemics among populations whose structure remains locally finite as the population becomes large.

This talk is concerned with a general class of structured-population epidemic models, in which individuals mix at two levels: global and local. A stochastic model for SIR (susceptible - infected - removed) epidemics among a closed finite population is described, in which during its infectious period a typical infective makes both local and global contacts. Each local contact of a given infective is with an individual chosen independently according to a contact distribution "centred" on that infective and each global contact is with an individual chosen independently and uniformly from the whole population. The threshold behaviour of the model is determined, as is the asymptotic final outcome in the event of an epidemic taking off. The theory is specialised to (i) the households model, in which the population is partitioned into households and local contacts are chosen uniformly within an infective's household; (ii) the overlapping groups model, in which the population is partitioned in several ways, with local uniform mixing within the elements of the partitions; (iii) the great circle model, in which individuals are equally spaced on a circle and local contacts are nearest-neighbour; and (iv) a network model with casual contacts.

The talk ends with a brief discussion of a stochastic SIS (susceptible - infected - susceptible) model for epidemics among a population of households, covering threshold behaviour, endemic level and time to extinction. Much of the talk is based on work done jointly with Peter Neal (University of Manchester).

Title: Identifying separated time scales in stochastic models of reaction networks

Reaction rates and chemical species numbers may vary over several orders of magnitude. Combined these large variations can lead to subnetworks operating on very different time scales. Separation of time scales has been exploited in many contexts as a basis for reducing the complexity of dynamic models, but the interaction of the rate constants and the species numbers makes identifying the appropriate time scales tricky at best. Some systematic approaches to this identification will be discussed and illustrated by application to a simple model of viral infection of a cell and a model of the heat shock response in E. Coli.

Title: Stochastic model and statistical inference for the Cuban HIV epidemic with contact-tracing and unobserved infectious population

This study is motivated by the Cuban HIV-AIDS epidemic, for which a database containing almost 9,000 observations is available. We first explain how the well-known SIR model can be generalized to describe the detection system in Cuba, which includes a contact-tracing program. Precisely, individuals identified as infected may contribute to finding other infectious individuals by providing information related to persons with whom they had possibly infectious contacts. The detected (removed) population is modeled by an individual-based component which takes into account the time since detection. The evolution of the whole population is described by a SDE driven by a Poisson point measure. For adequate scalings, the link with PDEs that are often introduced in epidemiology is done. This sheds a new light on parameter estimation for these PDEs. Second, this approximation result may serve as a key tool for exploring the asymptotic properties of standard inference methods such as maximum likelihood estimation, when the observations are complete. As often when dealing with epidemiological data, the infectious population is unobserved. We use a Bayesian method, called Approximate Bayesian Computation (ABC) for estimating the parameters. ABC, an alternative to data imputation methods such as MCMC integration, is likelihood-free and relies only on simulations of the model.

This is work in collaboration with MG. Blum, S. Clemencon, H. De Arazoza.

Title: Toward testable predictions of the fixation probability of rare beneficial mutations

The fixation probability is a cornerstone of population genetics and has a rich mathematical history. Analytical predictions of the fixation probability were first made almost 80 years ago, and have stood untested since that time. Recently, experimental protocols involving the adaptation of microorganisms, in large replicate studies, have brought experimental tests of the fixation probability within reach. Analytical predictions, however, are quite sensitive to the organism's life history, the population dynamics imposed by the experimental protocol, and the mechanism of the selective advantage. We have developed organism- and protocol-specific predictions of the fixation probability, with the goal of creating testable predictions for laboratory settings. Our approach involves the time evolution of a probability generating function, typically described by a partial differential equation with a delay term.