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2018 | OriginalPaper | Buchkapitel

7. Large Deviations for Infectious Diseases Models

verfasst von : Peter Kratz, Etienne Pardoux

Erschienen in: Séminaire de Probabilités XLIX

Verlag: Springer International Publishing

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Abstract

We study large deviations of a Poisson driven system of stochastic differential equations modeling the propagation of an infectious disease in a large population, considered as a small random perturbation of its law of large numbers ODE limit. Since some of the rates vanish on the boundary of the domain where the solution takes its values, thus making the action functional possibly explode, our system does not obey assumptions which are usually made in the literature. We present the whole theory of large deviations for systems which include the infectious disease models, and apply our results to the study of the time taken for an endemic equilibrium to cease, due to random effects.

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Vorheriges Kapitel Bismut-Elworthy-Li Formulae for Bessel Processes

Nächstes Kapitel The Girsanov Theorem Without (So Much) Stochastic Analysis

Nur mit Berechtigung zugänglich

The reasoning behind this is the following. Assume that an infectious individual meets on average α > 0 other individuals in unit time. If each contact of a susceptible and an infectious individual yields a new infection with probability p, the average number of new infections per unit time is βS(t)I(t)∕N, where β = pα since all individuals are contacted with the same probability (hence S(t)∕N is the probability that a contacted individual is susceptible).

This implies that for z ∈ A ^N with β _j(z) > 0, we have \(\beta _j(z) \geq \underline {\beta }(\lambda _0/N)\).

Here (and later) B(x, r) denotes the open ball around x with radius r.

We do not necessarily have \(\mu ^i\in V_{x,v_i}\) for all x ∈ B _i; this might not be the case if x ∈ ∂A. In such a case \(V_{x,v_i}=\emptyset \) is possible, cf. the discussion about x = (1, 0)^⊤ for the SIRS model below.

The constant − 1 can be replaced by any other constant − C (C > 0). Note that C ₆ then depends on C with C ₆ increasing in C.

Note that here, we also include those j with β _j(x) > 0 and μ _j = 0 for all μ ∈ V _x,y.

Note that this result is not used for the proof of Theorem 7.4.

Note that B ^x depends on x only through β(x).

This theorem says that if \(F:X\times Y\to \mathbb {R}\), where X and Y are convex, one the two being compact, F being quasi-concave and u.s.c. with respect to its first variable, quasi-convex and l.s.c. with respect to the second, then sup_xinf_yF(x, y) =inf_ysup_xF(x, y).

Note that the Assumption (D5) is required here.

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Titel: Large Deviations for Infectious Diseases Models
verfasst von: Peter Kratz
Etienne Pardoux
Verlag: Springer International Publishing
Buch: Séminaire de Probabilités XLIX
Print ISBN: 978-3-319-92419-9

Electronic ISBN: 978-3-319-92420-5

Copyright-Jahr: 2018
DOI: https://doi.org/10.1007/978-3-319-92420-5_7