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A general approach to the integral functionals of epidemic processes

Part of: Markov processes General field theory

Published online by Cambridge University Press:  26 July 2018

Claude Lefèvre  and
Philippe Picard
Claude Lefèvre*
Affiliation:
Université Libre de Bruxelles
Philippe Picard*
Affiliation:
Université Lyon 1
*
* Postal address: Université Libre de Bruxelles, Département de Mathématique, Campus de la Plaine C.P. 210, B-1050 Bruxelles, Belgium. Email address: clefevre@ulb.ac.be
** Postal address: Université Lyon 1, ISFA, LSAF EA2429, 50 Avenue Tony Garnier, F-69007 Lyon, France. Email address: philippe.picard69@free.fr
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Abstract

In this paper we consider the integral functionals of the general epidemic model up to its extinction. We develop a new approach to determine the exact Laplace transform of such integrals. In particular, we obtain the Laplace transform of the duration of the epidemic T, the final susceptible size ST, the area under the trajectory of the infectives AT, and the area under the trajectory of the susceptibles BT. The method relies on the construction of a family of martingales and allows us to solve simple recursive relations for the involved parameters. The Laplace transforms are then expanded in terms of a special class of polynomials. The analysis is generalized in part to Markovian epidemic processes with arbitrary state-dependent rates.

Keywords

General epidemicMarkovian epidemic processfinal susceptible sizetotal area under the trajectory of the infectives and the susceptiblesmartingale methodspecial class of polynomials

MSC classification

Primary: 60J28: Applications of continuous-time Markov processes on discrete state spaces 92D30: Epidemiology
Secondary: 12E10: Special polynomials
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 2 , June 2018 , pp. 593 - 609
Copyright
Copyright © Applied Probability Trust 2018 

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