Etienne Pardoux: Large Deviations: applied to the time of extinction of an endemic disease

Abstract:
Consider an epidemic compartment model, of type SIRS or SIR with demography, or any other type of a compartment model which possesses a locally stable equilibrium. Our point of view is to consider the ODE model as a Law of Large Numbers limit or a stochastic model which counts each event of infection and removal. This stochastic equation can be considered as a small random perturbation of the ODE limit. The Freidlin-Wentzell theory of "small perturbations of dynamical systems" predicts that after a large time, the small perturbations will eventually produce a Large Deviation of the stochastic equation from the ODE. In the case of our epidemic model, this large deviation can consist in the solution of the stochastic equation leaving the basin of attraction of an endemic equilibrium, resulting in the extinction of the endemic disease. The Freidlin-Wentzell theory tells us how long it will take, asymptotically as the size of the population tends to infinity, for such a large deviation to happen. This talk will describe such results, and is based upon recent joint work with Peter Kratz and Brice Samegni-Kepgnou.

The lecture is part of the PhD course on Stochastic epidemic models, given at Stockholm University, but is open for anyone interested in the subject.

M. Freidlin, A. Wentzell : Small random perturbations of dynamical
systems, 3d ed. Springer 2012
P. Kratz, E. Pardoux : Large deviations for infection diseases models,
arXiv 1602.02803
E. Pardoux, B. Samegni-Kepgnou : Large deviations for Poisson driven
SDEs in epidemics models, arXiv 1606.01619