Conditional limit theorems for multitype branching processes and illustration in epidemiological risk analysis [Elektronische Ressource] / eingereicht von Sophie Pénisson

Sophie Pénisson

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• Mathematics

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• Publié par: universitat_potsdam
• Publié le: 01 janvier 2010
• Nombre de lectures: 13
• Langue: English
• Poids de l'ouvrage: 1 Mo

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Institut für Mathematik der Universität Potsdam
Lehrstuhl für Wahrscheinlichkeitstheorie
Ecole Doctorale Santé Publique Paris-Sud 11 Paris Descartes
CONDITIONAL LIMIT THEOREMS
FOR MULTITYPE BRANCHING PROCESSES
AND ILLUSTRATION IN EPIDEMIOLOGICAL RISK ANALYSIS
Dissertation / Thèse de Doctorat
eingereicht von / présentée par
Sophie Pénisson
zur Erlangung des akademischen Grades
Doctor rerum naturalium
- Dr. rer. nat. -
1in der Wissenschaftsdisziplin Stochastik
pour obtenir le grade de
Docteur
2dans le domaine Mathématiques Appliquées
am 16. Juli 2010
an der Mathematisch-Naturwissenschaftlichen Fakultät
der Universität Potsdam.
Prüfungskommission / Membres du jury
Prof. Dr. Gerold Alsmeyer G utachter / Rapporteur
Dr. Habil. Michel Chavance
Prof. Dr. Jana Eccard
Prof. Dr. Elisabeth Gassiat
Dr. Habil. Christine Jacob (Betreuerin / Directrice de thèse)
Prof. Dr. Peter Jagers G utachter / Rapporteur
Prof. DrH.e nning Läuter V orsitz / Président
Prof. Dr. Sylvie Roelly (Betreuerin / Directrtihèces ede) G utachter / Rapporteur
1 Probabilités et Statistique
2 Angewandte MathematikThis work is licensed under a Creative Commons License:
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Published online at the
Institutional Repository of the University of Potsdam:
URL http://opus.kobv.de/ubp/volltexte/2010/4530/
URN urn:nbn:de:kobv:517-opus-45307
http://nbn-resolving.org/urn:nbn:de:kobv:517-opus-45307 Acknowledgments
I would like to express my gratitude to my supervisors, Prof. Sylvie R lly and Dr. Christine
Jacob, who were abundantly helpful, patient and devoted. They both o ered me invaluable
support and guidance throughout these three years.
I must also acknowledge Prof. Jagers and Prof. Alsmeyer who kindly accepted to serve as my
external reviewers, as well as all the members of the committee.
I am very grateful to Prof. R lly who encouraged me to complete my thesis under a joint
supervision, which has been made possible thanks to the Franco-German University. This thesis
bene ted in addition from the nancial assistance of the graduate program IRTG Stochastic Mo-
dels of Complex Processes, to which I am very thankful for the high-quality scienti c training
that it o ered to me. A very special thanks goes to the Unite MIA from the French National
Institute for Agricultural Research in Jouy-en-Josas, who kindly hosted me for several months.
I would like to acknowledge more generally the di erent structures or institutions involved in
the development of my thesis: Institut fur Mathematik der Universit at Potsdam, Institut fur
Mathematik der Technische Universit at Berlin, Ecole Doctorale Sante Publique Paris-Sud 11, and
College Doctoral Franco-Allemand Paris Ouest Nanterre / Potsdam.
Lastly, I would like to thank my friends and family for their support, and express my gratitude
to those who patiently proofread this thesis.
34Summary
This thesis is concerned with the issue of extinction of populations composed of di erent
types of individuals, and their behavior before and in case of a very late extinction.
We approach this question rstly from a strictly probabilistic viewpoint, and secondly from the
standpoint of risk analysis related to the extinction of a particular model of population dynamics.
In this context we propose several statistical tools.
The population size is modeled by a branching process, which is either a continuous-time
multitype Bienayme-Galton-Watson process (BGWc), or its continuous-state counterpart, the
multitype Feller di usion process. We are interested in di erent kinds of conditioning on non-
extinction, and in the associated equilibrium states. These ways of conditioning have been widely
studied in the monotype case. However the literature on multitype processes is much less extensive,
and there is no systematic work establishing connections between the results for BGWc processes
and those for Feller di usion processes.
In the rst part of this thesis, we investigate the behavior of the population before its extinction
by conditioning the associated branching processX on non-extinction (X = 0), or more generallyt t
on non-extinction in a near future 06 <1 (X = 0), and by letting t tend to in nity. Wet+
prove the result, new in the multitype framework and for > 0, that this limit exists and is non-
degenerate. This re ects a stationary behavior for the dynamics of the population conditioned
on non-extinction, and provides a generalization of the so-called Yaglom limit, corresponding to
the case = 0. In a second step we study the behavior of the population in case of a very late
extinction, obtained as the limit when tends to in nity of the process conditioned by X = 0.t+
The resulting conditioned process is a known object in the monotype case (sometimes referred
to as Q-process), and has also been studied when X is a multitype Feller di usion process.t
We investigate the not yet considered case where X is a multitype BGWc process and provet
the existence of the associated Q-process. In addition, we examine its properties, including the
asymptotic ones, and propose several interpretations of the process. Finally, we are interested in
interchanging the limits in t and, as well as in the not yet studied commutativity of these limits
with respect to the high-density-type relationship between BGWc processes and Feller processes.
We prove an original and exhaustive list of all possible exchanges of limit (long-time limit in t,
increasing delay of extinction , di usion limit).
The second part of this work is devoted to the risk analysis related both to the extinction of
a population and to its very late extinction. We consider a branching population model (arising
notably in the epidemiological context) for which a parameter related to the rst moments of
the o spring distribution is unknown. We build several estimators adapted to di erent stages
of evolution of the population (phase growth, decay phase, and decay phase when extinction is
expected very late), and prove moreover their asymptotic properties (consistency, normality). In
particular, we build a least squares estimator adapted to theQ-process, allowing a prediction of the
population development in the case of a very late extinction. This would correspond to the best or
to the worst-case scenario, depending on whether the population is threatened or invasive. These
tools enable us to study the extinction phase of the Bovine Spongiform Encephalopathy epidemic
in Great Britain, for which we estimate the infection parameter corresponding to a possible source
of horizontal infection persisting after the removal in 1988 of the major route of infection (meat
and bone meal). This allows us to predict the evolution of the spread of the disease, including the
year of extinction, the number of future cases and the number of infected animals. In particular,
we produce a very ne analysis of the evolution of the epidemic in the unlikely event of a very late
extinction.
5
6666Zusammenfassung
Diese Arbeit befasst sich mit der Frage des Aussterbens von Populationen verschiedener Typen
von Individuen. Uns interessiert das Verhalten vor dem Aussterben sowie insbesondere im Falle
eines sehr sp aten Aussterbens. Wir untersuchen diese Fragestellung zum einen von einer rein
wahrscheinlichkeitstheoretischen Sicht und zum anderen vom Standpunkt der Risikoanalyse aus,
welche im Zusammenhang mit dem Aussterben eines bestimmten Modells der Populationsdynamik
steht. In diesem Kontext schlagen wir mehrere statistische Werkzeuge vor.
Die Populationsgr o e wird entweder durch einen zeitkontinuierlichen mehrtyp-Bienayme-Gal-
ton-Watson Verzweigungsprozess (BGWc) oder durch sein Analogon mit kontinuierlichem Zu-
standsraum, den Feller Di usionsprozess, modelliert. Wir interessieren uns fur die unterschiedli-
chen Arten auf Uberleben zu bedingen sowie fur die hierbei auftretenden Gleichgewichtszust ande.
Diese Bedingungen wurden bereits weitreichend im Falle eines einzelnen Typen studiert. Im
Kontext von mehrtyp-Verzweigungsprozessen hingegen ist die Literatur weniger umfangreich und
es gibt keine systematischen Arbeiten, welche die Ergebnisse von BGWc Prozessen mit denen der
Feller Di usionsprozesse verbinden. Wir versuchen hiermit diese Luc ke zu schliessen.
Im ersten Teil dieser Arbeit untersuchen wir das Verhalten von Populationen vor ihrem Ausster-
ben, indem wir das zeitasymptotysche Verhalten des auf Uberleben bedingten zugeh origen Verzwei-
gungsprozesses (XjX = 0) betrachten (oder allgemeiner auf Uberleben in naher Zukunft 06<t t t
1, (XjX = 0) ). Wir beweisen das Ergebnis, neuartig im mehrtypen Rahmen und fur > 0,t t+ t
dass dieser Grenzwert existiert und nicht-degeneriert ist. Dies spiegelt ein station ares Verhalten
fur auf Uberleben bedingte Bev olkerungsdynamiken wider und liefert eine Verallgemeinerung des
sogenannten Yaglom Grenzwertes (welcher dem