May 15, 2019

Preview: Medallion lecturer, Étienne Pardoux

Étienne Pardoux received his PhD in 1975 from the Université Paris-Sud Orsay, under the joint supervision of Alain Bensoussan and Roger Temam. He held a position at CNRS, before joining the Université de Provence at Marseille (now Aix Marseille Université) in 1979, where he has worked ever since and, since 2017, is professor emeritus. Étienne’s research interests include stochastic partial differential equations, nonlinear filtering, anticipating stochastic calculus, backward stochastic differential equations, homogenization of PDEs with periodic and random coefficients, and, more recently, probabilistic models in evolutionary biology and epidemics. He received the Monthyon Prize from the French Academy of Sciences in 1993. Étienne’s Medallion Lecture will be delivered at the Stochastic Processes and their Applications (SPA) meeting, 8-12 July 2019, in Evanston, IL, USA: https://sites.math.northwestern.edu/SPA2019/.

Fluctuations around a law of large numbers, and extinction of an endemic disease

We consider epidemic models where there is a constant flux of susceptible individuals, either because the infected individuals, when they recover, don’t gain any immunity, or they lose their immunity after some time, or because of demography (birth or immigration of susceptible individuals). Under certain conditions on the parameters, the associated deterministic epidemic model, which is an ODE, has a stable endemic equilibrium. This ODE is a large population law of large numbers limit of a system of stochastic Poisson driven SDEs. The stochastic model has a disease free absorbing state, which by irreducibility, is reached soon or later by the process. It might however be that the time it takes for this to happen, i.e. for the random fluctuations to drive the system out of the basin of attraction of the endemic equilibrium of the deterministic limiting ODE is enormous, and does not give any encouraging information concerning the epidemic.

It is therefore of interest to try to predict the time it takes for the random fluctuations inherent in the model to drive the system to the disease-free absorbing state. This can be done using the central limit theorem, moderate and large deviations. The relevance of each approach will depend upon the size of the population.

Most results are given for a homogeneous model (i.e. where each infectious individual is likely to infect with equal likelihood each susceptible individual in the population). However, there are extensions of those results for a population distributed over space. Another model of interest is the so-called “household model,” where there are both local infections in each household, and global infections between households. In that model, the law of large numbers limit is given by a type of “propagation of chaos” result.

This is joint work with R. Forien, P. Kratz, B. Samegni-Kepgnou and T. Yeo.

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