Large deviation principle for epidemic models

Abstract

International audienceWe consider a general class of epidemic models obtained by applying the random time changes of Ethier and Kurtz (2005) to a collection of Poisson processes and we show the large deviation principle for such models. We generalise the approach followed by Dolgoarshinnykh (2009) in the case of the SIR epidemic model. Thanks to an additional assumption which is satisfied in many examples, we simplify the recent work of Kratz and Pardoux (2017)

Large deviation principle for reflected Poisson driven stochastic differential equations in epidemic models

“…This section is summarized by the following result whose proof is essentially the same as that of Theorem 2.4 in section 2 of [13]. It mainly uses a Girsanov change of probability for doubly stochastic Poisson processes as well as the law of large numbers established in section 3.…”

“…In [13], the upper bound was established as a consequence of a result in [4], which does not apply here. This is why we need to detail the proof of the upper bound.…”

“…with b(z) = k j=1 β j (z)h j . We have established in [13] a large deviation principle for the solution of such an SDE.…”

A stochastic SIR model on a graph with epidemiological and population dynamics occurring over the same time scale

“…The following result provides bounds for population fluctuations over very large time intervals for such initial conditions, which will prove useful to describe the early phase of the epidemic in the next section. Its proof is based on a Freidlin-Wentzelltype results on large deviations from a deterministic approximation given in [39] (see also [29,12]) and can be found in the Appendix. The constant α 0 above is the exit cost from B 2 (z * , ε) starting from z * for the dynamical system y = Ay + B and the Poisson perturbation considered, that is, the minimal value of the quasipotential for this system and the perturbation with respect to z * on the boundary of this ball (see Chapter V of [21], see [39], and see the proof of Proposition 3 for an expression of the quasipotential).…”

Moderate deviations and extinction of an epidemic

“…The Large Deviations results are close to those presented in Shwartz and Weiss [9], [10], although their assumptions are not quite satisfied in our models. Derivations adapted to our setup can be found in Kratz and Pardoux [5], Pardoux and Samegni-Kepgnou [6], and Britton and Pardoux [1]. The results concerning moderate deviations are new and constitute the core of this paper.…”