Mathematical Modeling of Random and Deterministic Phenomena (Mathematics and Statistics) [1 ed.] 1786304546, 9781786304544

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Mathematical Modeling of Random and Deterministic Phenomena

Series Editor Nikolaos Limnios

Mathematical Modeling of Random and Deterministic Phenomena

Edited by

Solym Mawaki Manou-Abi Sophie Dabo-Niang Jean-Jacques Salone

First published 2020 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK

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© ISTE Ltd 2020 The rights of Solym Mawaki Manou-Abi, Sophie Dabo-Niang and Jean-Jacques Salone to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2019952987 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-454-4

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments

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xi xiii

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Solym Mawaki M ANOU -A BI, Sophie DABO -N IANG and Jean-Jacques S ALONE Part 1. Advances in Mathematical Modeling . . . . . . . . . . . . . . . .

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Chapter 1. Deviations From the Law of Large Numbers and Extinction of an Endemic Disease . . . . . . . . . . . . . . . . . . . . . . Étienne PARDOUX

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1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . 1.2. The three models . . . . . . . . . . . . . . . . . . . . 1.2.1. The SIS model . . . . . . . . . . . . . . . . . . . 1.2.2. The SIRS model . . . . . . . . . . . . . . . . . . 1.2.3. The SIR model with demography . . . . . . . . 1.3. The stochastic model, LLN, CLT and LD . . . . . . 1.3.1. The stochastic model . . . . . . . . . . . . . . . 1.3.2. Law of large numbers . . . . . . . . . . . . . . . 1.3.3. Central limit theorem . . . . . . . . . . . . . . . 1.3.4. Large deviations and extinction of an epidemic 1.4. Moderate deviations . . . . . . . . . . . . . . . . . . 1.4.1. CLT and extinction of an endemic disease . . . 1.4.2. Moderate deviations . . . . . . . . . . . . . . . . 1.5. References . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 2. Nonparametric Prediction for Spatial Dependent Functional Data: Application to Demersal Coastal Fish off Senegal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mamadou N’ DIAYE, Sophie DABO -N IANG, Papa N GOM, Ndiaga T HIAM, Massal FALL and Patrice B REHMER 2.1. Introduction . . . . . . . . . . . . . . . . . . . . 2.2. Regression model and predictor . . . . . . . . . 2.3. Large sample properties . . . . . . . . . . . . . . 2.4. Application to demersal coastal fish off Senegal 2.4.1. Procedure of prediction . . . . . . . . . . . . 2.4.2. Demersal coastal fish off Senegal data set . . 2.4.3. Measuring prediction performance . . . . . 2.5. Conclusion . . . . . . . . . . . . . . . . . . . . . 2.6. References . . . . . . . . . . . . . . . . . . . . .

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Chapter 3. Space–Time Simulations of Extreme Rainfall: Why and How? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gwladys T OULEMONDE, Julie C ARREAU and Vincent G UINOT

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3.1. Why? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Rainfall-induced urban floods . . . . . . . . . . . . . 3.1.2. Sample hydraulic simulation of a rainfall-induced urban flood . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. How? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Spatial stochastic rainfall generator . . . . . . . . . . 3.2.2. Modeling extreme events . . . . . . . . . . . . . . . . 3.2.3. Stochastic rainfall generator geared towards extreme events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. References . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 4. Change-point Detection for Piecewise Deterministic Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alice C LEYNEN and Benoîte DE S APORTA

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4.1. A quick introduction to stochastic control and change-point detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Model and problem setting . . . . . . . . . . . . . . . . . . . 4.2.1. Continuous-time PDMP model . . . . . . . . . . . . . . . 4.2.2. Optimal stopping problem under partial observations . . 4.2.3. Fully observed optimal stopping problem . . . . . . . . . 4.3. Numerical approximation of the value functions . . . . . . . 4.3.1. Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Discretizations . . . . . . . . . . . . . . . . . . . . . . . .

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73 76 77 78 80 82 83 84

Contents

4.3.3. Construction of a stopping strategy 4.4. Simulation study . . . . . . . . . . . . . 4.4.1. Linear model . . . . . . . . . . . . . 4.4.2. Nonlinear model . . . . . . . . . . . 4.5. Conclusion . . . . . . . . . . . . . . . . 4.6. References . . . . . . . . . . . . . . . .

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Chapter 5. Optimal Control of Advection–Diffusion Problems for Cropping Systems with an Unknown Nutrient Service Plant Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loïc L OUISON and Abdennebi O MRANE 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Statement of the problem . . . . . . . . . . . . . . . . . . . . . 5.2.1. Existence of a solution to the NTB uptake system . . . . . 5.3. Optimal control for the NTB problem with an unknown source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1. Existence of a solution to the adjoint problem of NTB uptake system with an unknown source . . . . . . . . . . . . . . 5.4. Characterization of the low-regret control for the NTB system 5.5. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . 5.6. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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103 107 110 111

Chapter 6. Existence of an Asymptotically Periodic Solution for a Stochastic Fractional Integro-differential Equation . . . . . . . 113 Solym Mawaki M ANOU -A BI, William D IMBOUR and Mamadou Moustapha M BAYE 6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1. Asymptotically periodic process and periodic limit processes 6.2.2. Sectorial operators . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. A stochastic integro-differential equation of fractional order . . . 6.4. An illustrative example . . . . . . . . . . . . . . . . . . . . . . . . 6.5. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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113 115 115 117 118 137 138

Chapter 7. Bounded Solutions for Impulsive Semilinear Evolution Equations with Non-local Conditions . . . . . . . . . . . . . . . . . . . . 141 Toka D IAGANA and Hugo L EIVA 7.1. Introduction . . . . . . . . . . . . . . . . 7.2. Preliminaries . . . . . . . . . . . . . . . . 7.3. Main theorems . . . . . . . . . . . . . . . 7.4. The smoothness of the bounded solution 7.5. Application to the Burgers equation . . . 7.6. References . . . . . . . . . . . . . . . . .

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Chapter 8. The History of a Mathematical Model and Some of Its Criticisms up to Today: The Diffusion of Heat That Started with a Fourier “Thought Experiment” . . . . . . . . . . . . . . 161 Jean D HOMBRES 8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. A physical invention is translated into mathematics thanks to the heat flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. The proper story of proper modes . . . . . . . . . . . . . . . . . . . . . 8.3.1. Mathematical position of the lamina problem . . . . . . . . . . . . 8.3.2. Simple modes are naturally involved . . . . . . . . . . . . . . . . . 8.3.3. A remarkable switch to proper modes . . . . . . . . . . . . . . . . . 8.4. The numerical example of the periodic step function gives way to a physical interpretation . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1. A calculation that a priori imposes an extension to the function f at the base of the lamina . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2. A crazy calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3. Fourier is happily confronted with the task of finding an explanation for the simplicity of the result about coefficients . . . . . . . 8.4.4. Criticisms of the modeling . . . . . . . . . . . . . . . . . . . . . . . 8.5. To invoke arbitrary functions leads to an interpretation of orthogonality relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1. Function is a leitmotiv in Fourier’s intellectual career . . . . . . . . 8.6. The modeling of the temperature of the Earth and the greenhouse effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7. Axiomatic shaping by Hilbert spaces provides a good account for another dictionary part in Fourier’s theory, and also to its limits, so that his representation finally had to be modified to achieve efficient numerical purposes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1. Another dictionary: the Fourier transform for tempered distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.2. Heisenberg inequalities may just be deduced from the existence of a scalar product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.3. Orthogonality and a quick look to wavelets . . . . . . . . . . . . . . 8.8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

161 163 164 165 166 167 169 169 170 174 175 177 180 181

184 184 185 187 187 189

Part 2. Some Topics on Mayotte and Its Region . . . . . . . . . . . . . 191 Chapter 9. Towards a Methodology for Interdisciplinary Modeling of Complex Systems Using Hypergraphs . . . . . . . . . . . . . . . . . 193 Jean-Jacques S ALONE 9.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 9.1.1. The ARESMA project . . . . . . . . . . . . . . . . . . . . . . . . . 193

Contents

9.1.2. Towards a methodology of interdisciplinary modeling 9.2. Systemic and lexicometric analyses of questionnaires . . . 9.2.1. Complex systems . . . . . . . . . . . . . . . . . . . . . 9.2.2. Methodology . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.4. Conclusion of the section . . . . . . . . . . . . . . . . . 9.3. Hypergraphic analyses of diagrams . . . . . . . . . . . . . 9.3.1. Hypergraphs and modeling of a complex system . . . . 9.3.2. Methodology . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.4. Conclusion of the section . . . . . . . . . . . . . . . . . 9.4. Discussion and perspectives . . . . . . . . . . . . . . . . . 9.5. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1. Other properties of a connected hypergraph . . . . . . 9.5.2. Metric over an FHT . . . . . . . . . . . . . . . . . . . . 9.6. References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 10. Modeling of Post-forestry Transitions in Madagascar and the Indian Ocean: Setting Up a Dialogue Between Mathematics, Computer Science and Environmental Sciences . . . 221 Dominique H ERVÉ 10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. Interdisciplinary exploration of agrarian transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1. Exploration of post-forestry transitions in rainforests of Madagascar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2. Applications to dry forests in southwestern Madagascar . . 10.2.3. Viability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3. Community management of resources, looking for consensus 10.3.1. Degradation, violation, sanction . . . . . . . . . . . . . . . 10.3.2. Local farmers’ maps and conceptual graphs . . . . . . . . . 10.4. Discussion and conclusion . . . . . . . . . . . . . . . . . . . . 10.5. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 11. Structural and Predictive Analysis of the Birth Curve in Mayotte from 2011 to 2017 . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Julien BALICCHI and Anne BARBAIL 11.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1. Motivation . . . . . . . . . . . . . . . . . . . . . . 11.1.2. Context . . . . . . . . . . . . . . . . . . . . . . . . 11.1.3. About the literature on the birth curve in Mayotte 11.1.4. Objective of ARS OI . . . . . . . . . . . . . . . . . 11.2. Origin of the data . . . . . . . . . . . . . . . . . . . . .

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11.3. Methodologies and results . . . . . . . . . . . . . . . . . 11.3.1. Methodological approach . . . . . . . . . . . . . . . 11.3.2. Annual trend . . . . . . . . . . . . . . . . . . . . . . 11.3.3. Monthly trend . . . . . . . . . . . . . . . . . . . . . 11.3.4. Characterization of the explosion risk of the number of births . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.5. Autocorrelation . . . . . . . . . . . . . . . . . . . . . 11.3.6. Modeling by an ARIMA process (p, d, q) . . . . . . 11.3.7. Predictions for the year 2018 . . . . . . . . . . . . . 11.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 11.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 11.6. References . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 12. Reflections Upon the Mathematization of Mayotte’s Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Victor B IANCHINI and Antoine H OCHET 12.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2. Justifying the mathematization of economics . . . . . . . . . . 12.2.1. The ontological and linguistic arguments . . . . . . . . . . 12.2.2. Towards a naturalization of modeling in economics . . . . 12.2.3. A number of caveats . . . . . . . . . . . . . . . . . . . . . . 12.3. For a reasonable mathematization of economics: the case of Mayotte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1. The trend towards the mathematization of the economics of Mayotte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2. From Mayotte’s formal economy to its informal one . . . . 12.3.3. When the formal economy interacts with the informal one: some issues for the modelization of complex systems . . . . . . . 12.4. Concluding remark . . . . . . . . . . . . . . . . . . . . . . . . . 12.5. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Index

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Preface

In order to identify mathematical modeling and interdisciplinary research issues in evolutionary biology, epidemiology, epistemology, environmental and social sciences encountered by researchers in Mayotte, the first international conference on mathematical modeling (CIMOM’18) was held in Dembéni, Mayotte, from November 15 to 17, 2018, at the Centre Universitaire de Formation et de Recherche. The objective was to focus on mathematical research with interdisciplinarity. This book aims to highlight some of the mathematical research interests that appear in real life, for example the study of random and deterministic phenomena. It also aims to contribute to the future emergence of mathematical modeling tools that can provide answers to some of the specific research questions encountered in Mayotte. In Mayotte and its region, including the coastal zone of Africa, climate change has impacted ecological, biological, epidemiological, environmental, social and natural systems. There is an urgent need to use mathematical tools to understand what is happening and what may happen and to help decision-makers. The modeling of such complex systems has therefore become a necessity, in particular, to preserve the ecological, environmental, economic, social and natural environments of Mayotte. Mayotte is, in fact, a research laboratory, where the scientific fields converge. The CIMOM’18 conference was an effective opportunity to present not only recent advances in mathematical modeling, with an emphasis on epidemiology, ecology, the environment, evolution biology and socio-economic issues, but also new interdisciplinary research questions. Most of the documents presented in this book have been collected from a variety of sources, including communication documents at the CIMOM’18. It contains not only chapters related to the research questions above-mentioned, but also potential mathematical modeling tools for some important research questions. After the CIMOM’18, we invited the original authors (or speakers) to write journal articles to provide contributions on these questions, with a common structure

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for each chapter, in terms of pointing out mathematical models, illustrative examples and applications on advanced topics, with a view to publishing this Wiley Mathematics and Statistics series book. Each chapter has been reviewed by one or two independent reviewers and the book publishers. Some chapters have undergone major revisions based on the reviews before being definitively accepted. We hope that this book will promote mathematical modeling tools in real applications and inspire more researchers in Mayotte and other regions to further explore emerging research issues and impacts. Solym Mawaki M ANOU -A BI Sophie DABO -N IANG Jean-Jacques S ALONE November 2019

Acknowledgments

This book was made possible through the collaboration of many people and institutions whom we would like to thank. The idea for its drafting was born from the organization of the international conference on mathematical modeling in Mayotte (CIMOM’18). Very quickly it became clear to us that it was necessary to write articles in the form of a collective book that could serve as a basis for the development of mathematical tools for the modeling of complex systems. Mathematics is the foundation of science, and it is essential for the economic development of a region or a country. Mayotte can, and must, participate more in access to mathematic and scientific research. We would like to thank the Centre Universitaire de Mayotte and its Scientific Commission, the University of Montpellier and the Vice-Rectorate of Mayotte for their scientific, financial and logistical support. We would like to thank all the authors, speakers, guest speakers and people who have contributed to this beautiful project, namely: Etienne Pardoux, Benoîte De Saporta, Jean Dhombres, Abdennebi Omrane, Loïc Louison, William Dimbour, Gwladys Toulemonde, Dominique Hervé, Angelo Raherinirina, Sylvain Dotti, Éloïse Comte, André Mas, Christian Delhommé, Jean Diatta, Bertrand Cloez, Jean-Michel Marin, Aurélien Siri, Elliott Sucré, Abal-Kassim Cheik Ahamed, Laurent Souchard and all the students involved. We also thank Nikolaos Limnios, who was the capable editor for this book. In addition to being very familiar with the subject of mathematical modeling, he was able to help us during the various stages of the book’s production. Many renowned anonymous researchers helped to review the chapters of this book and we would also like to thank them a lot. A special thanks to Cédric Villani and Charles Torossian for their exceptional lectures at the CIMOM’18 and for supporting this project.

Introduction

This book, entitled “Mathematical Modeling of Random and Deterministic Phenomena”, was written to provide details on current research in applied mathematics that can help to answer many of the modeling questions encountered in Mayotte. It is aimed at expert readers, young researchers, beginning graduate and advanced undergraduate students, who are interested in statistics, probability, mathematical analysis and modeling. The basic background for the understanding of the material presented is timely provided throughout the chapters. This book was written after the international conference on mathematical modeling in Mayotte, where a call for chapters of the book was made. They were written in the form of journal articles, with new results extending the talks given during the conference and were reviewed by independent reviewers and book publishers. This book discusses key aspects of recent developments in applied mathematical analysis and modeling. It also highlights a wide range of applications in the fields of biological and environmental sciences, epidemiology and social perspectives. Each chapter examines selected research problems and presents a balanced mix of theory and applications on some selected topics. Particular emphasis is placed on presenting the fundamental developments in mathematical analysis and modeling and highlighting the latest developments in different fields of probability and statistics. The chapters are presented independently and contain enough references to allow the reader to explore the various topics presented. The book is primarily intended for graduate students, researchers and educators; and is useful to readers interested in some recent developments on mathematical analysis, modeling and applications. The book is organized into two main parts. The first part is devoted to the analysis of some advanced mathematical modeling problems with a particular focus on Introduction written by Solym Mawaki M ANOU -A BI, Sophie DABO -N IANG and Jean-Jacques S ALONE.

xvi

Mathematical Modeling of Random and Deterministic Phenomena

epidemiology, environmental ecology, biology and epistemology. The second part is devoted to a mathematical modelization with interdisciplinarity in ecological, socio-economic, epistemological, natural and social problems. In Chapter 1, we present large population approximations for several deterministic and stochastic epidemic models. The hypothesis of constant population of susceptibles is explained through some realistic situations. After recalling the definition of SIS, SIRS and SIR models, a law of large numbers (LLN) is presented as well as a central limit theorem (CLT) to estimate the time of extinction of an epidemic and a principle of great deviation to estimate the error. This chapter then describes the principle of moderate deviations. These results are then used to deduce the critical population sizes for launching an epidemic. It explains how it can be used to predict the time taken for an epidemic to cease. Chapter 2 is devoted to the study of non-parametric prediction of biomass of demersal fish in a coastal area, with a case study in Senegal. The inputs of the regression model are spatio-functional, i.e. the temperature and salinity of the water are depth curves recorded at different fishing locations. The prediction is done through a dual kernel estimator accounting the proximity between the temperature or salinity observations and locations. The originality of the approach lies in the functional nature of the exogeneous variables. Some theoretical asymptotic results on the predictor are provided. Chapter 3 is concerned with the study of urban flood risk in urban areas caused by heavy rainfall, that may trigger considerable damage. The simulated water depths are very sensitive to the temporal and spatial distribution of rainfall. Besides, rainfall, owing in particular to its intermittency, is one of the most complex meteorological processes. Its simulation requires an accurate characterization of the spatio-temporal variability and intensity from available data. Classical stochastic approaches are not designed explicitly to deal with extreme events. To this end, spatial and spatio-temporal processes are proposed in the sound asymptotic framework provided by extreme value theory. Realistic simulation of extreme events raises a number of issues such as the ability to reproduce flexible dependence structure and the simulation of such processes. In Chapter 4, we consider a problem of change-point detection for a continuous-time stochastic process in the family of piecewise deterministic Markov processes. The process is observed in discrete-time and through noise, and the aim is to propose a numerical method to accurately detect both the date of the change of dynamics and the new regime after the change. To do so, we state the problem as an optimal stopping problem for a partially observed discrete-time Markov decision process, taking values in a continuous state space, and provide a discretization of the state space based on quantization to approximate the value function and build a tractable stopping policy. We provide error bounds for the approximation of the value

Introduction

xvii

function and numerical simulations to assess the performance of our candidate policy. An application concerns treatment optimization for cancer patients. The change point then corresponds to a sudden deterioration of the health of the patient. It must be detected early, so that the treatment can be adapted. The context of Chapter 5 is the nutrient transfer mechanism in croplands. The authors study the case of an additional nutrient which comes from a “service plant” (meaning a natural input), as a control function. The Nye-Tinker-Barber model is introduced with a perturbation as an unknown source of nutrient. An optimal control formulation of this problem is studied and adapted for the incomplete data case. A characterization of the low-regret optimal control is provided In Chapter 6, basic stochastic evolution equations in long-time periodic environment are developed. Periodicity often appears in implicit ways in various phenomena. For instance, this is the case when we study the effects of fluctuating environments on population dynamics. Some classical books gave a nice presentation of various extensions of the concepts of periodicity, such as almost periodicity, asymptotically periodicity, almost automorphy, as well as pertinent results in this area. Recently, there has been an increasing interest in extending certain results to stochastic differential equations in separable Hilbert space. This is due to the fact that almost all problems in a real life situation, to which mathematical models are applicable, are basically stochastic rather than deterministic. In this chapter, we deal with a stochastic fractional integro-differential equation, for which a result of existence and uniqueness of an asymptotically periodic solution is given. In Chapter 7, we study the existence of solutions in semilinear evolution equations with impulse, where the differential operator generates a strongly compact semi-group. The chapter generalizes a recent published work by one of the co-authors to the non-local initial condition case. In the previous work, the existence, stability and smoothness of bounded solutions for impulsive semilinear parabolic equations with Dirichlet boundary conditions, are obtained using the Banach fixed point theorem, under the classical Lipschitz assumptions. In Chapter 8, we discuss the history and criticisms of a mathematical model, namely the diffusion of heat. The starting point is a “thought experiment” on the diffusion of heat through an infinite rectangular flat lamina. This is the path along which Fourier invented the representation of functions that bears his name; and we mainly treat the typical example of the periodic step function. Fourier thus invented the notion of proper modes, also known today as eigen modes, and found the orthogonality relations. Following Fourier, we then consider an example, the diffusion of heat in a sphere like the Earth, and come up with the required adaptation that, for the first time, allowed us to investigate the greenhouse effect. We then examine some of the criticisms related to Fourier’s representation until functional analysis was created in the 20th Century, answering various questions. Still, an

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Mathematical Modeling of Random and Deterministic Phenomena

interesting creation came with a critique from quantum mechanics in the 1930s, perhaps not understood as such, but which led to wavelets as developed in the 21st Century, and a remarkable new tool that can be adapted to various situations. The text, in a story form, aims to combine mathematics, physics and also epistemology in a history that is rigorous with respect for original texts; it also tries to understand the meaning of a scientific posterity for the construction of science, as well as how a thought experiment has been transformed into a realistic modeling. The second part is dedicated to the development of interdisciplinary modeling with mathematical approaches. In Chapter 9, we present a methodology for interdisciplinary modeling of complex systems using hypergraphs. This project begins by setting out the research stakes related to the sustainable management of mangrove forests in Mayotte: Mangroves are coastal ecosystems that have undergone global upheavals while facing a number of issues regarding biodiversity, pressures for natural hazards and attractiveness for the socio-economic development of territories. The mangroves of Mayotte thus present high stakes of preservation and management. This sustainable management is conceived in a participatory framework where, “it seems necessary for the users of the mangrove and those involved in the management of these wetlands, to exchange their experiences and knowledge further”. The author proposes an interdisciplinary system approach in ecology, geography, literature and modeling that aims at “the identification of variables” and “interactions in order to co-construct conceptual models combining societal and ecological dimensions” and “the identification of key variables to guide reflection on the sustainable management of these mangroves. The author aims to contribute to the implementing of integrated management of Mayotte’s mangroves in order to preserve them and ensure the maintenance of their ecosystem services”. In Chapter 10, we discuss modeling of post-forestry transitions in Madagascar and the Indian Ocean by setting up a dialogue between mathematics, computer science and environmental sciences. We discuss mathematical tools, implemented to model and analyze the dynamics of complex socio-ecological systems, made up of cultivated and inhabited areas after deforestation in Madagascar. In Chapter 11, the authors propose a descriptive analysis and a modelization of the evolution of the birth rate in Mayotte. Finally, in Chapter 12, we develop the idea that excessive mathematical modeling of the Mahoran economy would be ineffective to really take into account the weight of informal economy sectors, even though a systemic modeling seems to be an interesting perspective. The argument is based, in a historical and epistemological approach, on the critical discussion of two classic arguments for mathematization economy: the ontological argument that the economy is based on numbers (and laws)

Introduction

xix

and is therefore arithmetic-algebraic in nature, and the linguistic argument that considers mathematical language as a bearer minima of universality, logic and rigor. Examples of economic situations encountered in Mayotte support this argument, showing the complex links that exist between the formal and informal economy, between modern society and traditional practices. The statistician drift is denounced. The diversity and multiplicity of stakeholders and economic factors also appear as obstacles to mathematical modeling. Last but not least, we are grateful to our families for their continued support, encouragement and especially for supporting us during all the long hours we spent away from them while working on this book.

PART 1

Advances in Mathematical Modeling

Mathematical Modeling of Random and Deterministic Phenomena, First Edition. Edited by Solym Mawaki Manou-Abi, Sophie Dabo-Niang and Jean-Jacques Salone. © ISTE Ltd 2020. Published by ISTE Ltd and John Wiley & Sons, Inc.

1 Deviations From the Law of Large Numbers and Extinction of an Endemic Disease

1.1. Introduction We consider epidemic models with a constant flux of susceptibles, either because an infected individual becomes susceptible immediately after healing, or after some time when the individual becomes immune to the illness, or because there is a constant flux of newborn or immigrant susceptibles. In the above-mentioned three cases, for certain values of the parameters, there is an endemic equilibrium, which is a stable equilibrium of the associated deterministic epidemic model. The deterministic model can be considered as the law of large numbers limit (as the size of the population tends to ∞) of a stochastic model, where infections, healings, births and deaths happen according to Poisson processes, whose rates depend upon the numbers of individuals in each compartment. Since the disease-free states are absorbing, it follows from an irreducibility property, which is clearly valid in our models, that the epidemic will stop sooner or later in the more realistic stochastic model. However, the time which the stochastic perturbances will need to stop the epidemic may be enormous when the size N of the population is large. The aim of this chapter is to describe, based on the central limit theorem (CLT), large and moderate deviations (LD, MD), the time it takes for the epidemic to stop in the stochastic model. The chapter is organized as follows. In section 1.2, we describe the three deterministic and stochastic models which we have in mind, namely, the SIS, SIRS Chapter written by Étienne PARDOUX.

4

Mathematical Modeling of Random and Deterministic Phenomena

and SIR model with demography. In section 1.3, we give the general formulation of the stochastic models, and recall the law of large numbers, the central limit theorem and the theory of large deviations, and their application to the time of extinction of an epidemic. Finally, in section 1.4, we present the moderate deviations result for the SIS model (which is the simplest of our three models), and explain how it can be used to predict the time taken for an epidemic to cease. Those results will be proved in more generality, with full details of the proofs in Pardoux (2019). The results concerning the law of large numbers and the large deviations can be found in Kratz and Pardoux (2018), Pardoux and Samegni-Kepgnou (2017), and Britton and Pardoux (2019b), where the central limit theorem is also established. Note that the three above-mentioned references present different approaches to the large deviations results. The moderate deviations results will appear in Pardoux (2019). We conclude this introduction with a short history and a few references to books and lecture notes which describe models of infectious diseases and epidemics. Mathematical modeling of infectious diseases has a long history of being useful. The first such mathematical model was probably the one proposed by Bernoulli in Bernoulli (1760), with a model of smallpox. A little more than one hundred years ago, Sir Ronald Ross, a British medical doctor and Nobel laureate, who contributed to the understanding of malaria wrote: As a matter of fact all epidemiology, concerned as it is with variation of disease from time to time and from place to place, must be considered mathematically (...) and the mathematical method of treatment is really nothing but the application of careful reasoning to the problems at hand. As a matter of fact, Ross deduced, from mathematical arguments, conclusions concerning malaria, which his physician colleagues found hard to accept. One of the first books devoted to mathematical modeling of infectious diseases is Bailey (1975). A book which has had huge impact is Anderson and May (1991), which deals exclusively with deterministic models. Since then, there has been steady production of new research monographs, for example Andersson and Britton (2000) also looking at inference methodology, Daley and Gani (1999) focusing mainly on stochastic models, Keeling and Rohani (2008) dealing also with animal populations, and Diekmann, Heesterbeek, and Britton (2013) covering both deterministic and stochastic modeling. Finally, Britton and Pardoux (2019a) will soon present the broadest treatment of stochastic epidemic models ever published in one volume, covering both classical and new results and methods, from mathematical models to statistical procedures.

Deviations From the Law of Large Numbers and Extinction of an Endemic Disease

5

1.2. The three models 1.2.1. The SIS model The deterministic SIS model is the following. Let s(t) (respectively i(t)) denote the proportion of susceptible (respectively infectious) individuals in the population. Given an infection parameter λ and a recovery parameter γ, the deterministic SIS model can be written as   s (t) = −λs(t)i(t) + γi(t), i (t) = λs(t)i(t) − γi(t). Since clearly s(t) + i(t) ≡ 1, the system can be reduced to a one-dimensional ordinary differential equation. If we let z(t) = i(t), we have s(t) = 1 − z(t), and we obtain the ordinary differential equation z  (t) = λz(t)(1 − z(t)) − γz(t) . It is easy to verify that this ordinary differential equation has a so-called “diseasefree equilibrium”, which is z(t) = 0. If λ > γ, this equilibrium is unstable, and there is a stable endemic equilibrium z(t) = 1 − γ/λ. The corresponding stochastic model is as follows. Let StN (respectively ItN ) denote the proportion of susceptible (respectively infectious) individuals in a population of total size N .     ⎧ ˆ t ˆ t 1 1 ⎪ N N N N N ⎪ Sr Ir dr + Prec γN Ir dr , ⎪ ⎨ St = S0 − N Pinf λN N 0 0     ˆ t ˆ t ⎪ ⎪ 1 1 N N N ⎪ ⎩ ItN = I0N + Pinf λN Sr Ir dr − Prec γN Ir dr . N N 0 0 Here Pinf (t) and Prec (t) are two mutually independent standard (i.e. rate 1) Poisson processes. Let us give some explanations, first concerning the modeling, then concerning the mathematical formulation. Let StN (respectively ItN ) denote the number of susceptible (respectively infectious) individuals in the population. The equations for those quantities are the above equations, multiplied by N . The argument of Pinf (t) can be written as ˆ

t

λ 0

SrN N I dr . N r

The justification for such a rate of infections in the total population is as follows. Each infectious individual meets other individuals in the population at some rate β.

6

Mathematical Modeling of Random and Deterministic Phenomena

The encounter results in a new infection with probability p if the partner of the encounter is susceptible, which happens with probability StN /N , since we assume that each individual in the population has the same probability of being that partner, and with probability 0 if the partner is an infectious individual. Letting λ = βp and summing over the infectious at time t gives the above rate. Concerning recovery, it is assumed that each infectious recovers at rate γ, independently of the others. R EMARK 1.1.– Let us comment about the fact that we write our stochastic models in terms of Poisson processes. The fact that the infection events happen according to a Poisson process is a rather natural assumption. However, concerning the recovery from infection, our model assumes that the duration of the infectious period follows an exponential distribution. This is not realistic. We are forced to make such an assumption if we want to have a Markov model. We must confess that this assumption is done for mathematical convenience. However, we expect to extend our results to non-Markovian models in forthcoming publications. Note that there is an equivalent, but slightly more complicated way of writing the Poisson terms, which we now present. Let Minf and Mrec denote two mutually independent Poisson random measures on (0, +∞)2 , with mean measure the Lebesgue measure.  Pinf

ˆ

t

λN 0

 SrN IrN dr

ˆ tˆ

∞

can be rewritten as 0

0

1u≤λN SrN IrN dr Minf (dr, du)

and   ˆ t ˆ tˆ Prec γN IrN dr can be rewritten as 0

0

∞

0

1u≤γN IrN dr Mrec (dr, du) .

Again we have StN + ItN = 1, and ZtN = ItN satisfies ZtN

=

Z0N

1 + Pinf N



ˆ λN 0

t

 (1 −

ZrN )ZrN dr

  ˆ t 1 N − Prec γN Zr dr . N 0

1.2.2. The SIRS model In the SIRS model, contrary to the SIS model, an infectious who heals is first immune to the illness, he is “recovered”, and only after some time does he lose his immunity and turn susceptible. The deterministic SIRS model can be written as ⎧  s (t) = −λs(t)i(t) + ρr(t), ⎪ ⎪ ⎨ i (t) = λs(t)i(t) − γi(t), ⎪ ⎪ ⎩  r (t) = γi(t) − ρr(t),

Deviations From the Law of Large Numbers and Extinction of an Endemic Disease

7

while the stochastic SIRS model can be written as     ˆ t ˆ t ⎧ 1 1 N N N N N ⎪ ⎪ St = S0 − Pinf λN Sr Ir dr + Ploim ρN Rr dr , ⎪ ⎪ N N ⎪ 0 0 ⎪ ⎪ ⎪     ˆ t ˆ t ⎨ 1 1 N N N N N Sr Ir dr − Prec γN Ir dr It = I0 + Pinf λN ⎪ N N ⎪ 0 0 ⎪ ⎪     ⎪ ˆ t ˆ t ⎪ ⎪ 1 ⎪ N N ⎩ RtN = R0N + 1 Prec γN Ir dr − Ploim ρN Rr dr . N N 0 0 These two models could be reduced to two-dimensional models for z(t) = (i(t), s(t)) (respectively ZtN = (ItN , StN )). 1.2.3. The SIR model with demography In this model, recovered individuals remain immune forever, but there is a flux of susceptibles by births at rate μN , while individuals from each of the three compartments die at rate μ. Thus, the deterministic model ⎧  s (t) = μ − λs(t)i(t) − μs(t) ⎪ ⎪ ⎨ i (t) = λs(t)i(t) − γi(t) − μi(t) ⎪ ⎪ ⎩  r (t) = γi(t) − μr(t), whose stochastic variant can be written as ⎧  ˆ t    ˆ t 1 1 1 ⎪ N N N N N ⎪ ⎪ St = S0 − Pinf λN Sr Ir dr + Pbirth(ρN t)− Pds μN Sr dr , ⎪ ⎪ N N N ⎪ 0 0 ⎪ ⎪     ⎪ ˆ t ˆ t ⎪ ⎪ 1 1 ⎪ N N N N N ⎪ Sr Ir dr − Prec γN Ir dr ⎪ ⎨ It = I0 + N Pinf λN N 0 0   ˆ t ⎪ ⎪ 1 N ⎪ ⎪ Ir dr , − Pdi μN ⎪ ⎪ N ⎪ 0 ⎪ ⎪ ⎪     ˆ t ˆ t ⎪ ⎪ 1 1 ⎪ N N N N ⎪ Ir dr − Pdr μN Rr dr . ⎩ Rt = R0 + Prec γN N N 0 0 R EMARK 1.2.– We may think that it would be more natural to decide that births happen at rate μ times the total population. Then the total population process would be a critical branching process, which would go extinct in finite time a.s., which we do not want. Next it might seem more natural to replace, in the infection rate, the ratio StN /N by StN /(StN + ItN + RN t ), which is the actual ratio of susceptibles in the population at time t. It is easy to show that StN + ItN + RN t is close to N , so we choose the simplest formulation.

8

Mathematical Modeling of Random and Deterministic Phenomena

Again, we can reduce these models to two-dimensional models for z(t) = (i(t), s(t)) (respectively ZtN = (ItN , StN )), by deleting the r (respectively RN ) component. 1.3. The stochastic model, LLN, CLT and LD 1.3.1. The stochastic model The three above-mentioned stochastic models are of the following form. ZtN

 ˆ t  k 1 N = zN + hj P j N βj (Zs )ds N j=1 0 ˆ = zN +

0

t

b(ZsN )ds

 ˆ t  k 1 + hj Mj N βj (ZsN )ds , N j=1 0

[1.1]

where {Pj (t), t ≥ 0}0≤j≤k are mutually independent standard Poisson processes,

k Mj (t) = Pj (t) − t, and b(z) = j=1 βj (z)hj . ZtN takes its values in Rd . In the case of the SIS model, d = 1, k = 2, h1 = 1, β1 (z) = λz(1 − z), h2 = −1 and β2 (z) = γz.   1 , β1 (z) = λz1 z2 , In the case of the SIRS model, d = 2, k = 3, h1 = −1     −1 0 h2 = , β2 (z) = γz1 and h3 = , β3 (z) = ρ(1 − z1 − z2 ). 0 1 In the case of the SIR ourselves to d = 2,  model  with demography, we can  restrict  1 −1 while k = 4, h1 = , β1 (z) = λz1 z2 , h2 = , β2 (z) = (γ + μ)z1 , −1 0     0 0 h3 = , β3 (z) = μ, h4 = , β4 (z) = μz2 . 1 −1 While the above formulation has the advantage of being concise, for certain purposes it is more convenient to rewrite [1.1] using the equivalent formulation already described in the case of the SIS model. Let {Mj , 1 ≤ j ≤ k} be mutually independent Poisson random measures on R2+ with mean measure the Lebesgue

Deviations From the Law of Large Numbers and Extinction of an Endemic Disease

9

measure, and let Mj (ds, du) = Mj (ds, du) − ds du, 1 ≤ j ≤ k. We can rewrite [1.1] in the form ZtN

ˆ t ˆ N βj (ZsN ) k 1 = zN + hj Mj (ds, du) N j=1 0 0 ˆ = zN +

t

0

b(ZsN )ds

ˆ t ˆ N βj (ZsN ) k 1 + hj Mj (ds, du), N j=1 0 0

[1.2]

in the sense that the joint law of {Z N , N ≥ 1} is the same law of a sequence of random elements of the Skorohod space D([0, T ]; Rd ), whether we use [1.1] or [1.2] for its definition. We will now state a few results, without specifying particular assumptions. Those results are valid at least in the case of the three above examples. See Britton and Pardoux (2019b) for details of the proofs, and precise assumptions under which those results hold true. Concerning the initial condition, we assume that for some z ∈ [0, 1]d , zN = [N z]/N , where [N z] ∈ Zd+ is the vector whose i-th component is the integer part of the real number N z i . 1.3.2. Law of large numbers We have a law of large numbers T HEOREM 1.1.– Let ZtN denote the solution of the stochastic differential equation [1.1]. Assume that the βj are locally bounded, b is locally Lipschitz and the unique solution of equation [1.3] does not explode in finite time. Then ZtN → zt a.s. locally uniformly in t, where {zt , t ≥ 0} is the unique solution of the ordinary differential equation dzt = b(zt ), dt

z0 = x.

[1.3]

The main argument in the proof of the above theorem is the fact that, locally uniformly in t, P (N t) →t N

a.s. as N → ∞.

10

Mathematical Modeling of Random and Deterministic Phenomena

1.3.3. Central Limit Theorem We also have a Central Limit Theorem. Let UtN :=

√

N (ZtN − z(t)).

T HEOREM 1.2.– Assume in addition to the hypotheses of Theorem 1.1 that b is of class C 1 . Then, as N → ∞, {UtN , t ≥ 0} ⇒ {Ut , t ≥ 0} for the topology of locally uniform convergence, where {Ut , t ≥ 0} is a Gaussian process of the form ˆ Ut =

t 0

∇x b(zs )Us ds +

k

ˆ t hj βj (zs )dBj (s), t ≥ 0 ,

j=1

[1.4]

0

where {(B1 (t), B2 (t), . . . , Bk (t)), t ≥ 0} are mutually independent standard Brownian motions. 1.3.4. Large deviations and extinction of an epidemic We denote by AC T,d the set of absolutely continuous functions from [0, T ] into Rd . For any φ ∈ AC T,d , let Ak (φ) denote the (possibly empty) set of functions c ∈ L1 (0, T ; Rk+ ) such that cj (t) = 0 a.e. on the set {t, βj (φt ) = 0} and dφt cj (t)hj , = dt j=1 k

t a.e.

We define the rate function  inf c∈Ak (φ) IT (φ|c), IT (φ) := ∞,

if φ ∈ AC T,A ; otherwise,

where as usual the infimum over an empty set is +∞, and ˆ IT (φ|c) =

T 0

k

g(cj (t), βj (φt ))dt

j=1

with g(ν, ω) = ν log(ν/ω) − ν + ω. We assume in the definition of g(ν, ω) that for all ν > 0, log(ν/0) = ∞ and 0 log(0/0) = 0 log(0) = 0. It is not hard to verify that IT (φ) = 0 if and only if φ solves the ordinary differential equation [1.3]. IT (φ) can be interpreted as an energy needed for letting φ deviate from being a solution of [1.3]. The collection Z N obeys a large deviations principle, in the sense that

Deviations From the Law of Large Numbers and Extinction of an Endemic Disease

11

T HEOREM 1.3.– For any open subset O ⊂ D([0, T ]; Rd ), lim inf N →∞

1 log IP Z N,zN ∈ O ≥ −IT,z (O). N

For any closed subset F ⊂ D([0, T ]; Rd ), lim sup N →∞

1 log IP(Z N,zN ∈ F ) ≤ −IT,z (F ) , N

where for any z ∈ Rd , A ⊂ D([0, T ]; Rd ), IT,z (A) :=

inf

φ∈A,φ(0)=z

IT (φ) .

A slight reinforcement of this theorem allows us to conclude a Wentzell–Freidlin type of result. Wentzell and Freidlin have studied small random perturbations of an ordinary differential equation like [1.3] (see Freidlin and Wentzell (2012)). One of their main results is to compute, asymptotically, the time needed for a small random perturbation of such an equation to drive the solution outside of the basin of attraction of a stable equilibrium. The theory has been originally developed for Brownian perturbations. Here we give a statement of the same type, for a Poissonian perturbation. In what follows, we assume that the first component of ZtN (respectively z(t)) is ItN (respectively i(t)). Assume that the deterministic ordinary differential equation [1.3] has a unique stable equilibrium z ∗ whose first component satisfies z1∗ > 0. We define V := inf

inf

T >0 φ∈AC T ,d ,φ(0)=z ∗ ,φ1 (T )=0

IT (φ).

Let now N,z TExt = inf{t > 0, Z1N (t) = 0, if Z N (0) = zN }.

We have the T HEOREM 1.4.– Given any η > 0, for any z with z1 > 0,

N,z lim P exp{N (V − η)} < TExt < exp{N (V + η)} = 1.

N →∞

Moreover, for all η > 0 and N large enough, N,z ) ≤ exp{N (V + η)}. exp{N (V − η)} ≤ E(TExt

12

Mathematical Modeling of Random and Deterministic Phenomena

It is important to evaluate the quantity V . Note that it is the value function of an optimal control problem. In case of the SIS model, which is one-dimensional, we can solve this control problem explicitly with the help of Pontryagin’s maximum principle1, see Pontryagin et al. (1962) or for a concise introduction adapted to this application section A.6 in Britton and Pardoux (2019b), and deduce in that case that V = log λγ − 1 + λγ . For other models, we can compute numerically the value of V for each given value of the parameters. 1.4. Moderate deviations 1.4.1. CLT and extinction of an endemic disease Consider the SIR with demography. i (t) = λi(t)s(t) − γi(t) − μi(t), s (t) = −λi(t)s(t) + μ − μs(t). We assume that λ > γ + μ, in which case there is a unique stable endemic μ equilibrium, namely, z ∗ = (i∗ , s∗ ) = ( γ+μ − μλ , γ+μ λ ). Following section 4.1 in Britton and Pardoux (2019b), we can study the extinction of an epidemic in the above model using the CLT. We note that the basic reproduction number R0 (the expected number of infectious contacts by one infectious at the start of the epidemic, i.e. when s(t) 1) and the expected relative time of a life an individual is infected, ε, are given by R0 =

λ , γ+μ

ε=

1/(γ + μ) μ = . 1/μ γ+μ

[1.5]

The rate of recovery γ is much larger than the death rate μ (52 compared to 1/75 for a one week infectious period and 75 year life length) so for all practical purposes, the two expressions can be approximated by R0 ≈ λ/γ and ε ≈ μ/γ. Denote again by ItN the fraction of the population which is infectious in a population of size N . The law of large tells us that for N and t large, ItN is close to i∗ . The CLT tell √ numbers N ∗ us that N (It − i ) converges to a Gaussian process, whose asymptotic variance can be shown to be well approximated by R0−1 − R0−2 ∼ R0−1 . This suggests that for large t, the number of infectious in  the population is approximately Gaussian, with mean N i∗ and standard deviation N/R0 . Since we expect a Gaussian process with  marginal N (0, 1) to hit −2 fairly quickly, we expect that if N i∗ −2 N/R0 ≤ 0, then 1 Pontryagin’s maximum principle states sufficient conditions for a control to be optimal. In the case of the SIS model, the corresponding control problem is one-dimensional, and Pontryagin’s conditions allow us to compute explicitly, the optimal trajectory.

Deviations From the Law of Large Numbers and Extinction of an Endemic Disease

13

 the epidemic will stop rather quickly, while if N i∗ − 4 N/R0 ≥ 0, it is not clear that the time of extinction will be of order 1 (as a function of N ). This gives a critical population size roughly of the order of Nc =

9 9 = 2 . (i∗ )2 R0  (1 − R0−1 )2 R0

Note that the factor 9 is rather arbitrary. This Nc is rather large since i∗ is relatively small. Clearly, even if everybody in the population gets ill at some point, being ill one week in a lifetime of 75 years on average gives a small fraction of infectious in the population. Consider measles prior to vaccination. If we assume that R0 ≈ 15 and the infectious period is 1 week (1/52 years) and life duration 75 years, implying that 1/75 ε ≈ 1/(1/52)+1/75 ≈ 1/3750, we arrive at Nc ≈ 9(3750)2 /15 ≈ 8 · 106 . Therefore, if the population is at most a couple of million, we expect that the disease will go extinct quickly, whereas the disease will become endemic (for a rather long time) in a population larger than, for example, 20 million people. This confirms the empirical observation prior to vaccination that measles was continuously endemic in the UK, whereas it died out quickly in Iceland (and was later reintroduced by infectious people visiting the country), see Anderson and May (1991). 1.4.2. Moderate deviations If the CLT allows us to predict extinction of an endemic disease for population sizes under a given threshold Nc , and large deviations gives predictions for arbitrarily large population sizes, it is fair to look at moderate deviations, which describes ranges of fluctuations between those of the CLT and those of the LD. We shall present the moderate deviations approach in the specific case of the SIS model. In other words, our model from now on is one-dimensional, which can be written in the form ˆ t N N Zt = Z 0 + b(ZsN )ds + YtN , where 0

b(z) = [λ(1 − z) − γ]z, and N N N ˆ ˆ ˆ ˆ 1 t λN Zs (1−Zs ) 1 t γN Zs N Yt = M1 (ds, du) − M2 (ds, du). N 0 0 N 0 0 We consider the case λ > γ and recall that the unique stable equilibrium of the ∗ deterministic model then is z ∗ = 1 − λγ . We assume that Z0N = zN := [N z ∗ ]/N . We have ˆ t ∗ ZtN − z ∗ = zN − z∗ − λ ZsN (ZsN − z ∗ )ds + YtN . 0

14

Mathematical Modeling of Random and Deterministic Phenomena

It follows that ∗ − z ∗ )e−λ ZtN − z ∗ = (zN

´t 0

ZsN ds

ˆ + YtN − λ

t

0

ZsN e−λ

´t s

ZrN dr

YsN ds.

Consequently, ∗ |ZtN − z ∗ | ≤ |zN − z ∗ | + 2 sup |YsN |.

[1.6]

0≤s≤t

We can also rewrite the above stochastic differential equation in the form ∗ − z ∗ − (λ − γ) ZtN − z ∗ = zN

YtN = YtN − λ

ˆ

t

0

(ZsN

ˆ

t 0

(ZsN − z ∗ )ds + YtN , where [1.7]

∗ 2

− z ) ds.

A combination of [1.6] and [1.7] yields the existence of a constant C such that     ˆ T N ∗ 2 ∗ ∗ N N 2 (Zt − z ) dt ≤ |zN − z | + C sup |Yt | ∧ sup |Yt | . 0≤t≤T

0

0≤t≤T

[1.8] For the bound by sup0≤t≤T |YtN |2 , we first take the square in [1.6]. We now define, for 0 < α < 1/2, YtN,α = N α [YtN − λ

ˆ 0

t

(ZsN − z ∗ )2 ds].

and deduce from [1.7] ∗ N α (ZtN − z ∗ ) = N α (zN − z ∗ ) − (λ − γ)

ˆ

t 0

N α (ZsN − z ∗ )ds + YtN,α . [1.9]

It follows from [1.9] that the map Y N,α → N α (Z N − z ∗ ) is continuous from D([0, T ]) into itself. Here we equip D([0, T ]) with the sup norm topology, which makes it a Hausdorff topologic vector space (equipped with the Skorohod topology, D([0, T ]) is not a topologic vector space).

Deviations From the Law of Large Numbers and Extinction of an Endemic Disease

15

We are interested in the large deviations of N α (Z N − z ∗ ), which means moderate deviations of Z N − z ∗ . Note that the deviations of N α (Z N − z ∗ ) in case α = 1/2 are analyzed by the central limit theorem, and in case α = 0 by the large deviations. So with 0 < α < 1/2, we are clearly in a regime here which is intermediate between the CLT and LD, which is called the regime of moderate deviations. We first note that the LD of N α (Z N − z ∗ ) will be deduced from those of Y N,α thanks to the contraction principle, see for example, Theorem 4.2.1 in Dembo and Zeitouni (1998). So we essentially have to analyze the LD of Y N,α . In fact, we will proceed in three steps. In the first step, we shall analyze the large deviations of N,α

Yt

:= N α−1

ˆ tˆ 0

λN z ∗ (1−z ∗ )

0

ˆ tˆ

M1 (ds, du) − N α−1

γN z ∗

0

0

M2 (ds, du),

at the speed N 2α−1 , or in other words, the moderate deviations of Y

N t

1 := N

ˆ tˆ 0

λN z ∗ (1−z ∗ )

0

1 M1 (ds, du) − N

ˆ tˆ 0

γN z ∗

0

M2 (ds, du).

N,α The second step will consist of showing that Y N,α and Y have the same behavior as regards large deviations. Finally, the third step will consist of applying the contraction principle, in order to deduce the LD of N α (Z N − z ∗ ).

1.4.2.1. Step 1: moderate deviations of Y

N

We shall use the notations aN = N 2α−1 and Y measure ν on [0, T ], we write

N,α

N

:= N α Y . Given a signed

ˆ   N,α N,α ΛN (ν) = log E eν(Y ) , where ν(Y )=

N,α

[0,T ]

for the logarithmic moment generating function of Y

N,α

Yt

ν(dt)

at ν.

The crucial step of our derivation is the P ROPOSITION 1.1.– For any signed measure ν on [0, T ], as N → ∞, aN ΛN (a−1 N ν) → Λ(ν) :=

 1  E ν(Y )2 , 2

where Yt :=

ˆ tˆ 0

λz ∗ (1−z ∗ ) 0

M1 (ds, du) −

ˆ tˆ 0

0

γz ∗

M2 (ds, du).

16

Mathematical Modeling of Random and Deterministic Phenomena

P ROOF.– We first rewrite a−1 N Y N,α

a−1 N Yt

= N −α

N,α

1 N

in the form

(−1)j Mj, (βj (z ∗ )t) ,

j=0 =1

here β0 (z) = λz(1 − z) and β1 (z) = γz and the processes {Mj, , j = 0, 1; 1 ≤ ≤ N } are i.i.d. compensated standard Poisson processes. We now have   −1 N,α aN ΛN (a−1 ν) = a log E exp ν(a Y ) N N N ⎡ ⎛ ⎞⎤ 1 = N aN log E exp ⎣N −α ν ⎝ (−1)j Mj,1 (βj (z ∗ )·)⎠⎦ j=0

⎛ ⎞ 1 = N 2α log E 1 + N −α ν ⎝ (−1)j Mj,1 (βj (z ∗ )·)⎠ 

j=0

⎞2 ⎛  1

N −2α ⎝ (−1)j Mj,1 (βj (z ∗ )·)⎠ + O N −3α + ν 2 j=0

⎡ ⎛ ⎞2 ⎤ 1 1 ⎢ ⎥ → E ⎣ν ⎝ (−1)j Mj,1 (βj (z ∗ )·)⎠ ⎦ 2 j=0 =

σ2 2

ˆ t ∧ s ν(dt)ν(ds)

[0,T ]2

as N → ∞, where we have used the notation σ 2 := 2 λγ (λ − γ). The next step consists of establishing exponential tightness of the laws of Y in the sense that P ROPOSITION 1.2.– For any R > 0, there exists a compact set KR D([0, T ]; Rd ), such that

 N,α

,

⊂⊂

lim sup aN log IP(Y N,α ∈ (KR )c ) ≤ −R . N

The proof of this Proposition essentially follows the lines of the proof of exponential tightness in section 4.2.4 of Britton and Pardoux (2019b).

Deviations From the Law of Large Numbers and Extinction of an Endemic Disease

17

We now define the Fenchel–Legendre transform of Λ. Recall that we equip D([0, T ]) with the supnorm topology. For each φ ∈ D([0, T ]; Rd ), Λ∗ (φ) =

sup

ν∈(D([0,T ]))∗

{ν(φ) − Λ(ν)} .

From Proposition 1.1 and Proposition 1.2 combined with an approximation of N,α Y by a piecewise linear continuous process (see Pardoux (2019) for the details), we deduce from Corollary 4.6.14 from Dembo and Zeitouni (1998) the following. N,α

T HEOREM 1.5.– The sequence {Y , N ≥ 1} satisfies the Large Deviation Principle in D([0, T ]; Rd ) with the convex, good rate function Λ∗ and with speed aN , in the sense that for any Borel subset Γ ⊂ D([0, T ]; Rd ), N,α

− inf Λ∗ (φ) ≤ lim inf aN log IP(Y φ∈˚ Γ

N

≤ lim sup aN log IP(Y

N,α

N

∈ Γ) ∈ Γ) ≤ − inf Λ∗ (φ) . φ∈Γ

Let us compute Λ∗ . With the notation s ∧ t := inf(s, t), ˆ σ2 s ∧ t ν(ds)ν(dt) . Λ(ν) = 2 [0,T ]2 It is easily seen that Λ∗ (φ) = +∞ if φ(0) = 0. Let now φ ∈ C 2 ([0, T ]) such that φ(0) = 0. The gradient of the map ν → ν(φ) − Λ(ν) can be written as ˆ s ∧ t ν(ds). φ(t) − σ 2 [0,T ]

We look for ν ∗ such that this gradient equals 0. This implies that φ (t) = σ 2 ν ∗ ((t, T ]), hence φ (T ) = σ 2 ν ∗ ({T }),

φ (T ) −

ˆ

T

φ (s)ds = σ 2 ν ∗ ((t, T ]) ,

t

1 1 ν ∗ (dt) = − 2 φ (t)dt + 2 φ (T )δT (dt) . σ σ From those identities, combined with φ(0) = 0, we deduce that 1 Λ (φ) = 2σ 2 ∗

ˆ 0

T

|φ (t)|2 dt .

18

Mathematical Modeling of Random and Deterministic Phenomena

1.4.2.2. Step 2: moderate deviations of Y N We want to show in this step that Y N,α satisfies exactly the same large deviations N,α result as Y . This will follow if we prove that Y N,α satisfies Proposition 1.1 (with the same expression in the limit) and Proposition 1.2. Let us state a property that allows us to conclude that Y N,α satisfies Proposition 1.1 with the correct limit. P ROPOSITION 1.3.– For any C > 0, as N → ∞,    N,α − Y N,α ) → 0, aN log E exp Ca−1 λ( Y N   N,α  N,α ) → 0 . aN log E exp Ca−1 λ(Y − Y N

[1.10]

We first prove C OROLLARY 1.1.– Given Proposition 1.1, if Proposition 1.3 holds true, then for any signed measure ν on [0, T ], as N → ∞, ˆ  −1 N,α  σ2 aN log E eaN ν(Y ) → t ∧ s ν(dt)ν(ds) . 2 [0,T ]2 P ROOF.– For any δ > 0, we deduce from Hölder’s inequality "# !  N,α aN log E exp ν a−1 Y N "# ! ! ""#  ! N,α N,α = aN log E exp ν a−1 exp {ν a−1 Y N,α − Y N Y N ! N,α "# aN ≤ log E exp (1 + δ)a−1 N ν Y 1+δ $ "% aN δ 1 + δ !  N,α N,α , ν Y −Y + log E exp 1+δ δaN so that, if we combine Proposition 1.1 and Proposition 1.3, we deduce that  N,α )} ≤ lim sup aN log E exp{ν(a−1 N Y N

(1 + δ)σ 2 2

ˆ [0,T ]2

t ∧ s ν(dt)ν(ds),

and letting δ → 0, we conclude that  N,α )} ≤ lim sup aN log E exp{ν(a−1 N Y N

σ2 2

ˆ [0,T ]2

t ∧ s ν(dt)ν(ds).

Deviations From the Law of Large Numbers and Extinction of an Endemic Disease

19

For the inequality in the other direction, we note that, by similar arguments, $ % 1 aN N,α  N,α )} ) ≤ aN log E exp ν(Y log E exp{a−1 N ν(Y aN (1 + δ) 1+δ +

aN δ N,α − Y N,α )}, log E exp{(δaN )−1 ν(Y 1+δ

so that lim inf N

σ2 ≥ 2(1 + δ)

 N,α )} aN log E exp{a−1 N ν(Y

ˆ [0,T ]2

t ∧ s ν(dt)ν(ds),

hence, letting δ → 0 we conclude that N,α )} ≥ lim inf aN log E exp{a−1 N ν(Y N

σ2 2

ˆ [0,T ]2

t ∧ s ν(dt)ν(ds). 

Before we prove Proposition 1.3, we first need to establish a technical Lemma. L EMMA 1.1.– Let M be a standard Poisson random measure on R2+ , and M(dt, du) = M(dt, du) − dt du the associated compensated measure. If ϕ is an ´T R+ -valued predictable process, such that 0 ϕt dt has exponential moments of any order, and a ∈ R, then there exists a constant C such that for any 0 ≤ t ≤ T , & $ ˆ s ˆ ϕr %' E sup exp a M(dr, du) 0≤s≤t

0

0



$

≤ C E exp (e

2a

ˆ − 1 − 2a)

t

0

%1/2 ϕs ds .

P ROOF.– Consider with b ≥ 0 the process Xt = a

ˆ tˆ 0

ϕs 0

ˆ M(ds, du) − b

0

t

ϕs ds .

It follows from Itô’s formula that ˆ tˆ ˆ t Xt Xs e ϕs ds + a e =1−b 0

+ (e − 1 − a) a

0

ˆ tˆ 0

ϕs 0

ϕs 0

eXs− M(ds, du)

eXs− M(ds, du) .

[1.11]

20

Mathematical Modeling of Random and Deterministic Phenomena

´t´ϕ It follows from Lemma 1.2 below that Mt = 0 0 s eXs− M(ds, du) is a martingale. Hence, eX is a martingale, if b = (ea − 1 − a), a submartingale if we replace = by . Hence if b ≥ (ea −1−a), EeXt ≤ 1. Now, first using Doob’s L2 inequality for submartingales, and later Cauchy’s inequality, we have & $ ˆ s ˆ ϕr %' E sup exp a M(dr, du) 0≤s≤t

$ ˆ tˆ  E exp a  ≤

$

0

E exp 2a

0

0

ϕs 0

% M(ds, du)

ˆ tˆ 0

0

ϕs

ˆ M(ds, du) − 2b

 $ ˆ t %1/2 × E exp 2b ϕs ds

%1/2

t 0

ϕs ds

0

If 2b = e2a − 1 − 2a, the first factor on the last right-hand side equals 1.



In order to complete the proof of Lemma 1.1, we still need to establish L EMMA 1.2.– The process ϕ satisfying ´ t ´ ϕ the same assumptions as in Lemma 1.1, and Xt being given by [1.11], Mt = 0 0 s eXs− M(ds, du) is a martingale. P ROOF.– It is plain that Mt is a local martingale, whose predictable quadratic variation is given as ˆ < M >t =

0

t

⎧ #´ ´ ´ t ⎨≤ exp 2a t ϕs M(ds, du) ϕs ds, 0 0 #´ 0 e2Xs ϕs ds ´t t ⎩≤ exp −2(a + b) ϕs ds ϕ ds, 0 0 s

if a > 0 ; if a ≤ 0 .

All we need to show is that the above quantity is integrable. It is clearly a consequence of the assumption in case a < 0. In case a > 0, the second factor of the right-hand side has finite exponential moments, so is square integrable, and all we need to show is that $ E exp 4a

ˆ tˆ 0

ϕs 0

% M(ds, du)

< ∞.

[1.12]

Deviations From the Law of Large Numbers and Extinction of an Endemic Disease

21

Using Itô’s formula we have $ ˆ t ˆ ϕs % ˆ t M(ds, du) − (e8a − 1) ϕs ds Yt = exp 8a 0

0

= 1 + (e8a − 1)

0

ˆ tˆ 0

ϕs

0

Ys− M(ds, du).

It is easy to conclude that EYt ≤ 1. It follows from Cauchy–Schwartz that $ E exp 4a

ˆ tˆ 0

ϕs 0

% M(ds, du)

≤



( EYt

$ % ˆ t E exp (e8a − 1) ϕs ds , 0



and the result follows from our assumption on ϕ. We now turn to the P ROOF OF P ROPOSITION 1.3 We note that N,α

Yt

− YtN,α = Y t

N,α

ˆ − YtN,α + λ

t

0

(ZsN − z ∗ )2 ds.

Proposition 1.3 will follow from the fact that for any C > 0, as N → ∞,   N,α N,α aN log E exp Ca−1 ν(Y − Y ) → 0, N   N,α aN log E exp Ca−1 − Y N,α ) → 0, N ν(Y & ˆ · ' α N ∗ 2 aN log E exp Ca−1 N ν (Z − z ) ds → 0. s N

[1.13] [1.14] [1.15]

0

We shall prove [1.13] and [1.15]. The proof of [1.14] is quite similar to that of [1.13]. S TEP 1: PROOF OF [1.13] It suffices to consider one of the terms in the sum over j, and we suppress the index j for simplicity. We note that N,α a−1 N (Yt

−Y

N,α ) t

=N

−α

−N

ˆ tˆ

N β(z ∗ )

0 −α

N [β(ZsN )∨β(z ∗ )]

ˆ tˆ 0

M(ds, du)

N [β(ZsN )∨β(z ∗ )]

N β(ZsN )

M(ds, du).

22

Mathematical Modeling of Random and Deterministic Phenomena

It is not hard to see that we can treat each of the two terms on the right separately, and treat only the first term, with the treatment of the second one being quite similar. We note that there exists a compensated Poisson process on R+ M such that this first term can be rewritten as VtN := N −α M



ˆ N 0

t

(β(ZsN ) − β(z ∗ ))+ ds

 .

N We need to estimate E exp[Ca−1 N ν(V )]. If we decompose the signed measure ν as the difference of two measures as follows ν = ν+ − ν− , we again have two terms, and it suffices to treat one of them, say ν+ . Of course, it suffices to treat the case where ν+ = 0. Since the positive constant C is arbitrary, without loss of generality, we can assume that ν+ is a probability measure on [0, T ]. It is then clear that

 exp

Ca−1 N

ˆ

T

0

 VtN ν+ (dt)

& ≤ exp

Ca−1 N

' sup

0≤t≤T

VtN

.

We choose a new parameter 0 <  < α, and we split the expression whose expectation needs to be estimated in two terms. $ % $ % exp CN −α sup VtN = exp CN −α sup VtN 1sup0≤t≤T |ZtN −z∗ |≤N − 0≤t≤T

0≤t≤T

$

+ exp CN −α sup VtN 0≤t≤T

% 1sup0≤t≤T |ZtN −z∗ |>N − [1.16]

We now estimate the first term on the right-hand side of [1.16]. In this case, we define the stopping time σN = inf{0 ≤ t ≤ T ; |ZtN − z ∗ | > N − } and note that $  ˆ t % exp CN −α sup M N (β(ZsN ) − β(z ∗ ))+ ds 1sup0≤t≤T |ZtN −z∗ |≤N − 0≤t≤T

0

$  ˆ ≤ exp CN −α sup M N 0≤t≤T

0

t∧σN

(β(ZsN ) − β(z ∗ ))+ ds

% .

Deviations From the Law of Large Numbers and Extinction of an Endemic Disease

23

Consequently, the expectation of the first term on the right of [1.16] is bounded from above by $  ˆ t∧σN % E exp CN −α sup M N (β(ZsN ) − β(z ∗ ))+ ds 0≤t≤T

0



2CN −α

≤ E exp (e

− 1 − 2CN

−α

ˆ )N

+ * ≤ exp cN 1−2α− ,

T ∧σN

0

) (β(ZtN )

∗

+

− β(z )) dt

where the first inequality follows from Lemma 1.1, and the second one exploits the Lipschitz property of β. Consider now the second term on the right-hand side of [1.16].  $  ˆ t −α E exp N sup M N (β(ZsN ) 0≤t≤T

∗

+

0

%

−β(z )) ds



1sup0≤t≤T |ZtN −z∗ |>N −

 $  ˆ t %1/2 ≤ E exp 2N −α sup M N (β(ZsN ) − β(z ∗ ))+ ds 0≤t≤T

0

1/2  × IP sup |ZtN − z ∗ | > N −

0≤t≤T

* + ≤ exp cN 1−2α IP

,  ˆ t 1/2 , , , β(ZsN )ds ,, > N 1−

, sup ,,M N 0≤t≤T



0

where the second inequality follows from Lemma 1.1 and the boundedness of β. For the second factor in the last expression, we need to consider   ˆ t   and β(ZsN )ds > N 1−

IP sup M N 0≤t≤T

 IP



sup

 −M

0≤t≤T

0

ˆ

t

N 0

 β(ZsN )ds

>N

1−

 .

Both probabilities are estimated in a similar way. By an exponential estimate,  IP

 sup M

0≤s≤t

ˆ N 0

s

 β(ZrN )dr

> N 1−



 exp{−(16ct)−1 N 1−2 },

[1.17]

24

Mathematical Modeling of Random and Deterministic Phenomena

for N large enough. Finally, the expectation of the second term of the right-hand side of [1.16] is bounded by exp{c1 N 1−2α−c2 N 1−2 }, with c1 , c2 > 0, and $  ˆ t % E exp N −α sup M N (β(ZsN ) − β(z ∗ ))+ ds 0≤s≤t

≤ ecN

1−2α−

0

+ ec1 N

1−2α

−c2 N 1−2

.

From the inequality log(a + b) ≤ log(2) + log(sup(a, b)), for N large enough, $  ˆ t % aN log E exp N −α sup M N (β(ZsN ) − β(z ∗ ))+ ds 0≤s≤t

0

≤ aN log(2) + cN − , which establishes [1.13]. S TEP 2:

PROOF OF



[1.15] We, in fact, must prove that for any C > 0, as N → ∞,  ) ˆ

α aN log E exp a−1 N CN

T

0

(ZtN − z ∗ )2 dt

→ 0.

In this proof, C will denote a constant whose value may change from line to line. We now introduce a new process, where β¯ = sup0≤z≤1 β(z), XtN := the event AN b

1 N

ˆ tˆ 0

N β¯

0

$ :=

sup |Y

0≤t≤T

M(ds, du),

N t |

%

* + ¯ , ≤ b ∩ XtN ≤ βT

and the stopping time N ¯ }, τ¯b := inf{t > 0, |Y t | > b} ∧ inf{t, XtN > βT ∗ | ≤ N −1 where the constant b will be chosen below. From [1.8], the fact that |z ∗ − zN and Cauchy–Schwartz,   ) ˆ α aN log E exp a−1 N CN

T

0

(ZtN − z ∗ )2 dt

%' & $ α N N [1.18] CN sup |Y |1  N α−1 + aN log E exp a−1 c (A ) t N b

0≤t≤T

%' $ α N 2 N , + aN log E exp a−1 CN sup |Y | 1 A s N &

0≤t≤T

b

[1.19]

Deviations From the Law of Large Numbers and Extinction of an Endemic Disease

25

we take the limit successively in the two terms of the above right-hand side. S TEP 2a: estimate of [1.18] We have %' & $ −1 α N E exp aN CN sup |Ys |1(AN )c b

0≤t≤T

,) ,ˆ ˆ  , , t N β(ZsN ) , , −α ≤ E exp CN sup , M(ds, du), 1(AN )c + 1. b , 0≤t≤T , 0 0 



It remains to be noted that ,) ,ˆ ˆ   , , t N β(ZsN ) 1−2α , , −α  eCN E exp CN sup , M(ds, du), , , , 0≤t≤T 0 0 and

c ≤ IP IP( AN b )



   ¯  e−CN , sup |YtN | > b + IP sup XtN > βT

0≤t≤T

0≤t≤T

for some positive constant C, so that finally there exist two positive constants C1 and C2 such that, for N large enough, &

$

E exp

α a−1 N CN

%' sup

0≤t≤T

|YsN |1(AN )c b

≤ 1 + exp{C1 N 1−2α − C2 N } ≤ 2.

S TEP 2b: estimate of [1.19] Since Y N is a martingale, it is clear that the process $ exp

C α N 2 a−1 N |Yt | N

%

2

is a submartingale. Consequently, from Doob’s L2 submartingale inequality, & $ %'  * + α N 2 α N 2 N E exp a−1 ≤ 4E exp a−1 CN sup |Y | 1 A t τb | N N CN |YT ∧¯ 0≤t≤T

b

-  "# ! N ≤ E exp 2CN 1−α |YTN∧¯τb |2 − |Y T ∧¯τb |2 -  # N × E exp 2CN 1−α |Y T ∧¯τb |2 .

[1.20]

26

Mathematical Modeling of Random and Deterministic Phenomena

Consider first the first factor on the right-hand side of [1.20]. We have "! " ! N N N |YTN∧¯τb |2 − |Y T ∧¯τb |2 = YTN∧¯τb − Y T ∧¯τb YTN∧¯τb + Y T ∧¯τb , , ¯ ) ,,Y N − Y N ,, , ≤ (b + βT T ∧¯ τb T ∧¯ τb and the result follows from [1.13] and [1.14]. We finally consider the second term on the right-hand side of [1.20]. We have , , , , , N ,2 , N ,2 ,Y T ∧¯τb , ≤ ,Y T , 1{|YTN |≤b} + (b + N −1 )2 1{¯τb 0, ZtN,α ≤ −c}. For any δ > 0,

N,α lim IP exp{a−1 < exp{a−1 N (V c − δ)} < Tc N (V c + δ)} = 1.

N →∞

Moreover, lim aN log E(TcN,α ) = V c .

N →∞

1−2α Recall that a−1 . In the CLT regime, α = 1/2, a−1 N =N N = 1, while in the LD −1 regime, α = 0, aN = N . 1−2α

Vc Let us now compute the corresponding critical population size. eN is the N order of magnitude of the time needed for Zt − zt to make a deviation of size cN −α . This is sufficient to extinguish an epidemic, provided i∗ is of the same order, so that the corresponding critical size is Nα ∼ (1/i∗ )1/α , that is roughly the CLT critical population size raised to the power 1/2α. In the case of the SIR model with demography for measles, the CLT critical population size is of the order of a few million; thus, for example, with α = 1/3, we go from 106 to 109 , i.e. a few billion, which is the order of magnitude of the biggest countries, i.e. China and India.

Deviations From the Law of Large Numbers and Extinction of an Endemic Disease

29

1.5. References Anderson, R.M. and May, R.M. (1991). Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford. Andersson, H. and Britton, T. (2000). Stochastic Epidemic Models and Their Statistical Analysis. Springer Lecture Notes in Statistics. Springer Verlag, New York. Bailey, N. (1975). The Mathematical Theory of Infectious Diseases and its Applications. Griffin, London. Bernoulli, D. (1760). Essai d’une nouvelle analyse de la mortalité causée par la petite vérole et des avantages de l’inoculation pour la prévenir. Mém. Math. Phys. Acad. Roy. Sci., Paris, 1–45. Britton, T. and Pardoux, E. (eds) (2019a). Stochastic Epidemic Models with Inference. Springer Verlag, New York. Britton, T. and Pardoux, E. (2019b). Stochastic epidemics in a homogeneous community. In Stochastic Epidemic Models with Inference, Britton, T. and Pardoux, E. (eds). Springer Verlag, New York. Daley, D. and Gani, J. (1999). Epidemic Modeling: An Introduction. Cambridge University Press, Cambridge. Dembo, A. and Zeitouni, O. (1998). Large Deviations, Techniques and Applications, 2nd edition. Applications of Mathematics, 38, Springer, New York. Diekmann, O., Heesterbeek, J., and Britton, T. (2013). Mathematical Tools for Understanding Infectious Disease Dynamics. Princeton University Press, Princeton. Freidlin, M. and Wentzell, A. (2012). Random Perturbations of Dynamical Systems. Springer, New York. Keeling, M. and Rohani, P. (2008). Modeling Infectious Diseases in Humans and Animals. Princeton University Press, Princeton. Kratz, P. and Pardoux, E. (2018). Large deviations for infectious diseases models. Séminaire de Probabilités XLIX, 221–327. Pardoux, E. (2019). Moderate deviations and extinction of an endemic disease. arXiv:1905.08986.

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Mathematical Modeling of Random and Deterministic Phenomena

Pardoux, E. and Samegni-Kepgnou, B. (2017). Large deviation principle for epidemic models. Journal of Applied Probability, 54, 905–920. Pontryagin, L., Boltyanskii, V., Gramkrelidze, R., and Mishchenko, E. (1962). The Mathematical Theory of Optimal Processes. Translated by Trirogoff, K.N., Neustadt, L.W. (eds), John Wiley & Sons, New York.

2 Nonparametric Prediction for Spatial Dependent Functional Data: Application to Demersal Coastal Fish off Senegal

2.1. Introduction The description of a fine-scale demersal habitat is essential to better understand the functioning of the ecosystems that host abundant fish resources. Understanding how this type of complex system works is therefore of particular interest to decision-makers and stakeholder managers. Fisheries research shows, in an ecosystem approach, how the environment or ecological parameters affect the variability of density or biomass of one species or a group of species of wildlife by studying data at different capture locations over a long period of time (Planque and Buffaz 2008, Young and Carr 2015, Luan et al. 2018). They prove that environmental conditions influence the spatio-temporal distribution of species. In such situations, studying the link between environmental variables for the purpose of prediction, using massive and complex fish data with a space and/or time dynamic component is of great importance for better monitoring of fishery resources in marine ecosystems. Spatial or space–time data abound in many fields, particularly in the description of oceanological systems, where the study of relationships between variables consisting of complex and high-dimensional spatio-temporal vectors and/or functional components is considered for understanding the functioning of the natural systems. Such data are often based on dense sampling schemes of observations over space, time, and other continuum measures. The growth of high-quality, informative and massive Chapter written by Mamadou N’ DIAYE, Sophie DABO -N IANG, Papa N GOM, Ndiaga T HIAM, Massal FALL and Patrice B REHMER.

Mathematical Modeling of Random and Deterministic Phenomena, First Edition. Edited by Solym Mawaki Manou-Abi, Sophie Dabo-Niang and Jean-Jacques Salone. © ISTE Ltd 2020. Published by ISTE Ltd and John Wiley & Sons, Inc.

32

Mathematical Modeling of Random and Deterministic Phenomena

data implies a rising scientific interest in new statistical methods able to handle and analyze open problems. Classical multivariate statistical techniques such as multivariate spatial parametric prediction (Kriging) models (Cressie 2015, Rivoirard et al. 2000) are still applied to evaluate and predict fish abundance. This kind of model is linear and requires Gaussian distribution and parametric covariance assumptions. Failure of these hypotheses may have a significant inference effect. Alternative multivariate methods such as K-function (Ripley 1987, Heppell et al. 1999, Gardner et al. 2008, Lefort et al. 2011) and SDMs using conventional statistical methods as generalized linear models and generalized additive models (Young and Carr 2015, Luan et al. 2018, Pollock et al. 2014) in a nonlinear context have also been used in marine biology. Note that when the sample of interest is a massive data set, dimension reduction techniques are common approaches. In general, to solve the dimension problem, several multivariate regression methods using a huge number of predictors consider dimension as a nuisance parameter and use low-density conditions, primarily for convenience. They do not capture the additional information coming from the underlying generating process that underpins the data. An alternative to multivariate models, when handling massive spatial data on a continuum, may be spatial functional data analysis, a recent research area combining the well-developed branches of functional statistics and spatial statistics, showing the ability to analyze such complex multivariate data. Functional data analysis (FDA) transforms large data into functional data, i.e. data objects, such as curves, shapes, images or more complex mathematical objects. Rather than using correlation models between predictors, a data set from multidimensional variables for several dependent spatial units can be analyzed by FDA methods using the high dimensionality of the predictors. Basically, a FDA object corresponds to the realization of a stochastic process, generally assumed to be smooth on a continuum and valued in a metric, semi-metric, Hilbert or Banach space of eventually infinite dimension. In this chapter, we are studying ways to model and predict high-dimensional oceanological data by FDA for a better management of fishery resources. More precisely, we aim to predict the quantity of fish biomass at a given location, taking into account an eventual spatial heterogeneity and environmental impact, in a functional setting. For that we use the available massive spatio-functional environmental and fish abundance data and consider a non-parametric prediction functional approach, an alternative to Kriging and other multivariate models cited above. The proposed non-parametric method is adapted to high-dimensional spatial data, without any parametric assumption on the law underlying the generating process.

Nonparametric Prediction for Spatial Dependent Functional Data

33

Non-parametric regression and prediction problems in the spatial setting have been widely studied in the literature when variables are of finite dimensions compared to the functional case where some of the covariates are of functional nature (curves, shapes, etc.) as in many areas, such as soil sciences, oceanology, epidemiology, climatology and fisheries among others. The first results in a non-parametric spatial prediction setting are those of Biau and Cadre (2004) based on a kernel prediction of a strictly stationary random field indexed in (N∗ )N . Later, Dabo-Niang and Yao (2007) contributed to Biau and Cadre (2004)’s investigation and proposed a kernel regression estimation and prediction of continuously indexed strictly stationary random fields. Menezes et al. (2010) proposed a spatial non-parametric kernel predictor under a stochastic sampling design (spatial locations are random). Wang et al. (2012) proposed a local linear spatio-temporal prediction model, using a kernel weight function, taking into account the distances between sites. Their prediction procedure is based on the assumption that the error term of the model is auto-correlated and the underlying process is strictly stationary. In Dabo-Niang et al. (2016), nonparametric prediction in a spatial multivariate setting for locally stationary processes is proposed. Their predictor depends on two kernels: one controls the distance between observations, while the other controls the geographic proximity of the locations. This idea was presented in Dabo-Niang et al. (2014) in the context of density estimation and in Ternynck (2014) to deal with a regression problem for strictly stationary functional processes. It has also been extended to the nearest neighbors method (K-NN) by Ahmed et al. (2018). The idea of explicitly incorporating a spatial correlation structure into a non-parametric regression estimator, assuming that the error term is a second-order stationary process with a parametric correlation model, has been used in Francisco-Fernández and Opsomer (2005) and Francisco-Fernández et al. (2012). These authors used a local linear regression estimator and a generalized cross-validation criterion to account for the spatial correlation. The present work goes along the direction of a spatial proximity structure into a kernel predictor and takes advantage of these previous works (Francisco-Fernández and Opsomer 2005, Francisco-Fernández et al. 2012, Dabo-Niang et al. 2014, Ternynck 2014, DaboNiang et al. 2016). The originality of the considered predictor (Dabo-Niang et al. 2016) is that the covariate is real-valued, no parametric correlation model on the error term is assumed and the observations are assumed to be, locally identically distributed compared to Ternynck (2014), who considered a functional covariate and strictly stationary processes. The outline of this chapter is as follows. In section 2.2, we introduce the regression model, allowing us to define the predictor. Section 2.3 is dedicated to properties of large samples. Section 2.4 gives the application of demersal coastal fish off Senegal to spatial prediction. Section 2.5 is devoted to some conclusions.

34

Mathematical Modeling of Random and Deterministic Phenomena

2.2. Regression model and predictor Denote the integer lattice points in the N -dimensional Euclidean space by ZN , N ≥ 1. We consider a spatial process {Zi = (Xi , Yi ), i ∈ ZN } defined over the probability space (Ω, F, P). In geostatistics, we will generally consider the case N = 2. A point in bold i = (i1 , . . . , iN ) ∈ ZN will be referred to as a site. Suppose Xi takes values in an infinite-dimensional separable Banach space (E, d(·, ·)) (i.e. Xi is a functional random variable and d a semi-metric) and Yi is scalar. In the following,  ·  will denote any norm in Rd or RN (there will be no ambiguity since the vectors of RN are in bold), C will indicate some arbitrary positive constant that may vary from line to line. Moreover, we write un = O(vn ) means that ∃C such that |un /vn | ≤ C as vn → ∞, where n ∈ NN . As it is classically assumed in the literature, the process under study {Zi , i ∈ ZN } is observable over the rectangular domain. In = {i = (i1 , . . . , iN ), 1 ≤ ik ≤ nk , k = 1, . . . , N }. Let n = (n1 , . . . , nN ) and  = n1 × . . . × nN be the cardinal of In . From now on, we assume for simplicity n that n1 = n2 = . . . = nN = n (e.g. El Machkouri (2007), El Machkouri and Stoica (2010) and El Machkouri (2011)), but the following results can be extended to a more general framework. We write n → ∞ if n → ∞. The main goal is to predict Yi0 at an unobserved site i0 under the information that can be drawn on Xi0 and observations (Xi , Yi )i∈On , where On is the observed spatial set of finite cardinal tending to ∞ as n → +∞ and contained in In , with i0 ∈ / On . Let (Xi0 , Yi0 ) be of same distribution as (X, Y ) ∈ E × R. We do not suppose strict stationarity. We assume that the variables (Xi , Yi )i∈On are locally (Dabo-Niang et al. 2016) identically distributed (see, for instance, Klemelä (2008) who considered density estimation for locally identically time-series data): a sufficient number of (Xi , Yi ) have a distribution close to that of (X, Y ). Let E|Yi | < ∞, and assume that the distributions of (X, Y ) and (Xi , Yi ), i ∈ On , are unknown. We may imagine that if there are enough sites i close to i0 , then the sequence (Xi , Yi )i∈On may be used to predict Yi0 , under the condition Xi0 = xi0 , denoted as x in the following with the abuse of notation.  Let kn = kn,i0 = i∈On 1[i−i0 ≤dn ] denote the number of neighbors i for which the distance to i0 is less than or equal to dn > 0 where dn → ∞ as n → ∞. This last assumption states that the proximity between locations (eventually) increases as the sample size increases. We suppose that the spatial process of interest satisfies the following non-parametric regression model: Yi := r(Xi ) + i ,

[2.1]

Nonparametric Prediction for Spatial Dependent Functional Data

35

where r(.) = E(Yi |Xi = .), is assumed to be independent of i, the noise i is centered, α-mixing and independent of Xi . The main task in non-parametric regression is to build an estimator rn of r in equation [2.1]. The prediction is based on the following regression estimate, where we assume that the observed region On is the rectangular In :

rn (x) =

⎧ gn (x) ⎪ ⎪ , if fn (x) = 0 ⎪ ⎪ fn (x) ⎨ ⎪ 1  ⎪ ⎪ Yi otherwise, ⎪ ⎩n 

[2.2]

i∈On

where the functions fn and gn are defined by

d(x, Xi ) 1  fn (x) = K2ρn (i0 − i) K1 an bn i∈On

d(x, Xi ) 1  K2ρn (i0 − i) gn (x) = Yi K1 an bn i∈On

 d(x, Xi ) , where K2ρn (i0 − i) = K2ρn (i0 − i) E K1 bn i∈O

n

  i0 −i i1 i2 iN i = K K2 (i0 −i)/n 2 ρn nρn n = ( n , n , ... n ) , bn and ρn are bandwidths with an =



tending to zero; K1 and K2 are kernels defined from R+ to R+ such that: • ∃C11 , C12 , 0 < C11 < C12 < ∞, C11 I[0,1] (t) ≤ K1 (t) ≤ C12 I[0,1] (t), • ∃C21 , C22 , 0 < C21 ≤ C22 < ∞, C21 I[0,1] (s) ≤ K2 (s) ≤ C22 I[0,1] (s).

[2.3]

β Hereinafter, we assume that dn = nρn , kn = CN dN n + O(dn ) as dn → +∞, 0 < β < N and CN is a constant that depends on N . This assumption is verified when taking the Euclidean distance and N = 2 (square grid). In fact, in this case we have, kn ≤ 4d2n − 4dn + 4 which leads to kn = O(d2n ) = O( nρ2n ), see Dabo-Niang et al. (2016) and Ternynck (2014) for more details.

36

Mathematical Modeling of Random and Deterministic Phenomena

To build the predictor of Yi0 , we use the regression estimator rn (x) in equation [2.2] over the sample sites On = In \ {i0 }, with optimal bandwidths bn and ρn detailed in the prediction procedure of section 2.4.1. In fact, the predictor is defined as:

  d(Xi0 ,Xi ) K2ρn (i0 − i) Yi K 1  i∈O n bn

 Yi0 =  , [2.4] d(Xi0 ,Xi ) K  (i0 − i)  i∈On K1 2ρ n b n

if the denominator is not null, otherwise it is equal to the empirical mean. This proposed predictor will be compared with the one that does not take into account the spatial structure and is defined as:

 d(Xi0 ,Xi ) Yi K 1  b n

 , Yi0 =  d(Xi0 ,Xi ) K 1 i∈On b 

i∈On

[2.5]

n

with bn an optimal bandwidth detailed in section 2.4.1. Remark that equation [2.5] is based on the classical non-parametric regression estimate Dabo-Niang et al. (2011) without the second kernel on the locations:

 d(x,Xi ) Y K i 1 i∈On bn

 . rncl (x) =  d(x,Xi ) K 1 i∈On bn 

[2.6]

2.3. Large sample properties First we introduce some mixing assumptions. In fact, to take into account the spatial dependency, we assume that the process {Zi = (Xi , Yi ), i ∈ ZN } satisfies a mixing condition defined in Carbon et al. (1997) as follows: there exists a function χ(t) 0 as t → ∞, such that α(σ(S), σ(S  )) = sup{|P(A ∩ B) − P(A)P(B)|, A ∈ σ(S), B ∈ σ(S  )}, ≤ ψ(Card(S), Card(S  ))χ(dist(S, S  )), where dist(S, S  ) is the Euclidean distance between two finite sets of sites S ⊂ ZN and S  ⊂ ZN (the smallest Euclidean distance between two elements of S and S  ) and

Nonparametric Prediction for Spatial Dependent Functional Data

37

σ(S) (respectively σ(S  )) denotes the σ-fields generated by {Zi , i ∈ S} (respectively {Zi , i ∈ S  }) and ψ(·) is a positive symmetric function non-decreasing in each variable. We recall that the process is said to be strongly mixing if ψ ≡ 1. As usual, we will assume that one of the following conditions on χ(i): is defined by χ(i) ≤ Ci−θ , for some θ > 0,

[2.7]

i.e. that χ(i) tends to zero at a polynomial rate, or χ(i) ≤ C exp(−si), for some s > 0, i.e. that χ(i) tends to zero at an exponential rate. Concerning the function χ(·), for the sake of simplicity, the large sample properties given below concern only the case where χ(·) tends to zero at a polynomial rate. However, similar asymptotic results could be obtained with χ(·) tending to zero at an exponential rate (which implies the polynomial case). Throughout the chapter, it will be assumed that ψ satisfies either ∀n, m ∈ N,

ψ(n, m) ≤ C min(n, m),

[2.8]

or ψ(n, m) ≤ C(n + m + 1)κ

[2.9]

for some C > 0, and some κ ≥ 1. Such assumptions on ψ(n, m) can be found, for instance, in Tran (1990), Carbon et al. (1997), Hallin et al. (2004), Biau and Cadre (2004), Dabo-Niang and Yao (2013). Let B(x, h) be a ball of center x ∈ E and radius h. The small ball probabilities are denoted by ϕi,x (h) = P[Xi ∈ B(x, h)], ϕi,x (h) goes to zero when h goes to zero (see, for example, Ferraty and Vieu (2006) for more details). For any random real-valued variable Z and p ∈ N∗ , let Zp = (E [|Z|p ])

1/p

.

The mean square and almost sure consistency results of rn are obtained under non-parametric assumptions on r, on the kernel, the bandwidth and local dependence conditions, see Ternynck (2014) and N’diaye et al. (2019) for more details on

38

Mathematical Modeling of Random and Deterministic Phenomena

all required assumptions (only some of them are given here). More precisely, we have 

 rn (x) − r(x)2 = O bn +

1  ρN n ϕ n x (bn )

 ,

[2.10]

where, for all i = j ∈ NN , the joint probability distribution pi,j of Xi and Xj satisfies ∃ε ∈ (0, 1], pi,j (B(x, bn ) × B(x, bn )) ≤ C4 (ϕi,x (bn )ϕj,x (bn ))

1+ε 2

,

and for all i and x, there exist positive constants C1 and C2 and a function ϕx (h) tending to zero as h goes to zero such that 0 < C1 ϕx (h) ≤ ϕi,x (h) ≤ C2 ϕx (h). In order to give vnalmost sure convergence of rn , we consider a set D ⊂ E such that D ⊂ k=1 Bk , where Bk = B(xk , n ), vn > 0 is some integer, xk ∈ E, k = 1, ..., vn , and n > 0. Note that such set can always be built (Ferraty and Vieu 2006, Dabo-Niang et al. 2011). We assume that there exists a non-increasing positive function Γ such that, i) limn→∞ Γ(bn ) = 0 and 0 < C1 Γ(bn ) ≤ ϕi,x (bn ) ≤ C2 Γ(bn ), for all i, x ∈ D, where C1 and C2 are some positive constants. ii) limn→∞

 nρN n Γ(bn ) → ∞. log  n

Note that the mean-square convergence result [2.10] is proved first in Ternynck (2014) when the underlying process is strictly stationary. It has been extended by N’diaye et al. (2019) in the context of a local stationary setting considered in this contribution. In addition, a uniform almost sure consistency result of rn is established in N’diaye et al. (2019) with the above local stationary assumption: 



sup |rn (x) − r(x)| = O bn +

x∈D

 log n  ρN n Γ(b n) n

 almost surely.

From these results, we can derive an almost sure consistency of the predictor (the proof is given in N’diaye et al. (2019))     Yi0 − Yi0  −→ 0 almost surely, n→∞

[2.11]

Nonparametric Prediction for Spatial Dependent Functional Data

39

where

 d(Xi0 ,Xi ) K2ρn (i0 − i) Yi K 1 bn

 Yi0 =  , d(Xi0 ,Xi ) K2ρn (i0 − i) i∈On K1 bn 

i∈On

[2.12]

is the predictor of Yi0 at a location i0 . 2.4. Application to demersal coastal fish off Senegal In this section, we study the performance of the proposed predictor through some numerical experiments which highlight its importance. The proposed predictor is compared with that from the classical kernel method and does not take into account the spatial interaction (proximity between locations) (Dabo-Niang and Yao 2007, Biau and Cadre 2004). Let us first describe the prediction procedure. It allows us to compute optimal bandwidths using cross-validation based on the regression model of equation [2.1]. 2.4.1. Procedure of prediction The bandwidth choice (even in a finite or infinite dimensional setting) is a crucial question in non-parametric estimation. We propose to choose the optimal bandwidths by cross-validation. 2.4.1.1. Step 1 Specify sets bandwidths S1 and S2 for, respectively, K1 and K2 . 2.4.1.2. Step 2 For each bn ∈ S1 , ρn ∈ S2 and i0 ∈ In , compute equation [2.2] 2.4.1.3. Step 3 Compute optimal bandwidths bn and ρn by applying a cross-validation procedure over S1 and S2 . More precisely, consider the following minimization problem, i.e. ˆ sites determine bn and ρn that minimize the mean square error over the n min

bn ∈S1 ,ρn ∈S2

1  (rn (Xi0 ) − Yi0 )2 ˆ n i0 ∈In

and denote them bn and ρn .

[2.13]

40

Mathematical Modeling of Random and Deterministic Phenomena

The same procedure is applied to equation [2.6] for computing bn by replacing rn (.) with rncl (.) in equation [2.13] and minimize with respect to S1 . 2.4.1.4. Step 4 For each site i0 , predict Yi0 by: – the proposed predictor Yi0 computed with bn and ρn (see equation [2.4]); – comparing it with the one that does not take into account the spatial proximity: Yi∗0 using bn (see equation [2.5]). 2.4.2. Demersal coastal fish off Senegal data set Data were obtained from coastal demersal surveys off Senegal performed by the scientific team of CRODT (-Oceanographic Research Center of Dakar Centre de Recherche Océographique Dakar-Thiaroye):, the oceanography research center of ISRA, in both hot and cold seasons, in the north, center and south areas, from 2001 to 2015. The fishing gear used is a standard fish trawl, with the lengh of 31.82 m, with a 33.9 m long bead, 24.5 m for the back rope and 45 mm for the size of the meshes stretched at the level of the pocket. Fishing stations were visited from sunrise to sunset (diurnal strokes) at the rate of 1/2 hour per station, according to the usual working methodology of CRODT. They were essentially fired by stratified sampling, following double stratification by area (north, center and south) and bathymetry (0–50, 50–100, 100–150 and 150–200 m depth). In practice, during the survey, the sampling design is fixed. This allows us to consider that data are collected following deterministic spatial sampling schemes as the framework of this study. The database includes 495 stations described, among others, by identifying variables (campaign, number of station or trawl), temporal information (date, year, season, starting and ending trawl times, duration time and time stratum), spatial coordinates (starting and ending latitudes and longitudes, area, starting and ending depths, average depth and bathymetric stratum), biological parameters (catch per unit effort (CPUE), species/or group of species, family, zoological group and specific status) and environmental variables (temperature and salinity) at different depths. Figure 2.1 shows the spatial distribution of fishing stations off Senegal over the continental shelf. Since Senegalese and Mauritanian upwellings affect the spatial and seasonal distribution of coastal demersal fish, a spatial analysis of these distributions, taking into account environmental conditions, is of great importance. Among the 495 trawl sites during 2015, environmental variables, salinity and temperature, at different consecutive depths are available at only 154 sites. The shift of the variation is 1 m per depth, so we have 200 depth measurements for each environmental parameter. These measurements may then be considered as observations of functional variables. The data set used in the following is composed of the CPUE

Nonparametric Prediction for Spatial Dependent Functional Data

41

and environmental variables (salinity and temperature) observed in 2015 at 154 locations.

Figure 2.1. Map of 495 fishing stations off Senegal over the continental shelf

2.4.3. Measuring prediction performance In this section, we are interested in predicting, at the 154 sample locations, the CPUE (catch per unit effort), which is, here, the quantity of fish biomass, considered

42

Mathematical Modeling of Random and Deterministic Phenomena

as the response variable Y using an environmental functional covariate X(t), t ∈ [0, 200]. Here t is the depth in meters. Two prediction models will be used, each with a functional covariate (salinity or temperature). An overview of these environmental characteristics of the studied area (off Senegal with coastlines 10 m by depth) is given in Figure 2.2. The top left panel of this figure shows the sea bottom salinity (SBS) off Senegal. That is the salinity at depth t = 200 m. The top right panel of Figure 2.2 gives the sea surface salinity (SSS); the salinity at depth t = 0. The bottom left panel of Figure 2.2 shows the sea bottom temperature (SBT) (t = 200) while the bottom right panel gives the sea surface temperature (SST) at the surface, namely, at depth t = 0. Figure 2.2 is only a description of the spatial variation of salinity and temperature in the bottom and at the surface. Overall, we can say that there is some spatial heterogeneity of temperature and salinity, as shown by the various ranges of values. Figure 2.3 gives the CPUE across the fishing stations. The latter is the variable of interest which we will be interested in predicting according to an environmental variable such as salinity or temperature. In theory, the CPUE is the weight average per 30 mn. We consider, in this work, that the CPUE expresses the quantity of biomass in a given station. A pre-processing process has been applied on the environmental variables at the 154 sample locations using R rainbow package (Shang et al. 2019) to transform the discrete temperature and salinity at the 200 consecutive depth measurements as functional objects and to detect eventual outliers, see Figures 2.4 and 2.5. They give the original (left panel) and reconstructed/smoothed (right panel) temperature and salinity curves. Note that these smoothed curves, considered here as observations of a functional co-variable, are the profiles of environmental parameters (salinity and temperature) in the bathymetry (depth). In the pre-processing, we detect outlying curves. After removing them, the sample sizes of salinity and temperature curves are  s = 113 and n  t = 119. respectively n After removing the outliers, the remaining environmental curves are plotted in Figure 2.6 according to the bathymetric profile (different depths). The top panels give the smoothed curves, whereas the bottom panels give the means of the smoothed curves. Panel (a) shows heterogeneity of salinity between locations whereas panel (c) shows important variation of salinity according to the depth (bathymetry). This can also be observed for the temperature in panel (b). Let us now focus on the CPUE prediction. We use the functional data set without outliers (see panels (a) and (b) of  Figure 2.6) and the corresponding CPUE. Let In be the set of sample locations and n  s or n  t ) the sample size. We wish to predict the CPUE at a given location (equal to n i0 in In where we supposed that Yi0 (CPUE at site i0 ) is not measured, using the data  − 1 locations in On = In \ {i0 }. Let Xi (.) be the functional covariate, such set at n as salinity or temperature, then the sample is (Yi , Xi )i∈On . The proposed prediction of the CPUE at location i0 is Yi0 defined in equation [2.4] (see section 2.2). In order to highlight the performance of this predictor, we compare it, by the criterion of Mean

Nonparametric Prediction for Spatial Dependent Functional Data

43

Square Error (MSE), with the classical kernel predictor Yi0 (see equation [2.5]) that does not take into account the spatial proximity between sites. We compute Mean Square Error M SE  and M SE  for, respectively, Yi0 and Yi0 . The Mean Square Error is denoted as M SE (+) : M SE (+) =

1  + (Yi0 − Yi0 )2 with Yi+ = Yi0 or Yi0 . 0  n i0 ∈O

Figure 2.2. Spatial variation of salinity (top) and temperature (bottom) off Senegal at the bottom (left) and surface (right). For a color version of this figure, see www.iste.co.uk/manouabi/modeling.zip

[2.14]

44

Mathematical Modeling of Random and Deterministic Phenomena

Figure 2.3. CPUE over fishing stations off Senegal. For a color version of this figure, see www.iste.co.uk/manouabi/modeling.zip

Nonparametric Prediction for Spatial Dependent Functional Data

45

Figure 2.4. Salinity at the 154 locations (original data in the left panel; the smoothed (reconstructed) functional data in the right panel). For a color version of this figure, see www.iste.co.uk/manouabi/modeling.zip

Figure 2.5. Temperature at the 154 locations (original data in the left panel; the smoothed (reconstructed) functional data in the right panel). For a color version of this figure, see www.iste.co.uk/manouabi/modeling.zip

Equation [2.14] is computed for several kernels, and the results are given in Tables 2.1 and 2.2. The first two columns give several used kernels. The last two columns give M SE  and M SE  respectively. The last row gives the p-value of a Wilcoxon test in order to determinate if M SE  is significantly less than M SE  .

46

Mathematical Modeling of Random and Deterministic Phenomena

(b): Smoothed temperatures

20

Temperature

35.6 35.2

15

35.4

Salinity

35.8

25

36.0

30

(a): Smoothed salinities

50

100

150

200

0

100

150

Depth(m)

(c): Mean of smoothed salinities

(d): Mean of smoothed temperatures

35.68

200

18

35.60

19

20

Temperature

21

35.66 35.64 35.62

Salinity

50

Depth(m)

22

0

0

50

100 Depth(m)

150

200

0

50

100

150

200

Depth(m)

Figure 2.6. Smoothed environmental curves without outliers according to the bathymetric profile. All curves are smoothed using B-spline basis. For a color version of this figure, see www.iste.co.uk/manouabi/modeling.zip

Generally, the proposed predictor outperforms. For the two models considered (see Tables 2.1 and 2.2 ), M SE  is significantly smaller than M SE  , with very low p value. More precisely, for the model with salinity as co-variate (see Table 2.1), the smallest M SE  is equal to 1.2845134 with the K1 Triangular kernel whereas the smallest M SE  equals 1.228367 with the K2 Tukey kernel. In addition, note also that the smallest Average Mean Square Error for Yi0 is equal to 1.35, while that of Yi0 is equal to 1.46. For the model with environmental temperature as co-variate (see Table 2.2), the proposed predictor also performs (M SE  > M SE  ) with the smallest

Nonparametric Prediction for Spatial Dependent Functional Data

47

M SE  = 1.1825773 for the Tukey K2 kernel compared to M SE  = 1.3601111. The average Mean Square Error for Yi0 is equal to 1.38, while that of Yi0 is equal to 1.47. The results show that the proposed predictor is a good alternative to the basic kernel method in this particular spatially heterogeneous Senegalese fish biomass data set. When comparing the two prediction models, namely, with salinity and temperature covariates, we see that the model with salinity produces less error with respect to average Mean Square Error, while the model with temperature as an explanatory variable is more performant in terms of the Mean Square Error. Kernel1

Kernel2

M SE 

M SE 

Epanechnikov 1.2426774 1.2845134

Triangle

Biweight

1.2467063 1.2845134

Triweight

1.2424457 1.2845134

Gaussian

1.2425748 1.2845134

Parzen

1.2353714 1.2845134

Quad

1.2426774 1.2845134

Silverman

1.2437404 1.2845134

Tukey

1.2447179 1.2845134

Epanechnikov 1.4477781 1.4679125

Epanechnikov

Biweight

1.3818974 1.4679125

Triweight

1.3370865 1.4679125

Gaussian

1.4484400 1.4679125

Parzen

1.4167447 1.4679125

Quad

1.4477781 1.4679125

Silverman

1.4484608 1.4679125

Tukey

1.4446595 1.4679125

Epanechnikov 1.623466

Parzen

p.value

1.627702

Biweight

1.626234

1.627702

Triweight

1.579487

1.627702

Gaussian

1.618070

1.627702

Parzen

1.566932

1.627702

Quad

1.623466

1.627702

Silverman

1.617559

1.627702

Tukey

1.228367

3.702191 × 10

1.627702 −21

Table 2.1. Mean Square Errors of prediction for Yi0 and Yi0 when the covariate is the salinity

48

Mathematical Modeling of Random and Deterministic Phenomena

Kernel1

Kernel2 Triangle

M SE 

M SE 

1.4833015 1.6148783

Epanechnikov 1.517042 1.6148783 Biweight

Biweight

1.4620797 1.6148783

Triweight

1.561286 1.6148783

Gaussian

1.529699 1.6148783

Quad

1.517042 1.6148783

Silverman

1.534799 1.6148783

Tukey

1.4597212 1.6148783

Triangle

1.2663868 1.3601111

Epanechnikov 1.3038538 1.3601111 Gaussian

Biweight

1.3262059 1.3601111

Triweight

1.3392725 1.3601111

Gaussian

1.2340943 1.3601111

Quad

1.3038538 1.3601111

Silverman

1.2548942 1.3601111

Tukey

1.1825773 1.3601111

Triangle

1.3567657 1.4435814

Epanechnikov 1.3885122 1.4435814

Parzen

p.value

Biweight

1.4135013 1.4435814

Triweight

1.2471777 1.4435814

Gaussian

1.4313367 1.4435814

Quad

1.3885122 1.4435814

Silverman

1.4072615 1.4435814

Tukey

1.3599626 1.4435814

2.177332 × 10−20

Table 2.2. Mean Square Errors of prediction for Yi0 and Yi0 when the covariate is the temperature

2.5. Conclusion In this work, we propose a non-parametric spatial predictor of the catch per unit of effort of Senegalese coastal demersal fish species. The originality of the proposed method is taking the spatial distribution of demersal species into account. Indeed, the method takes into account both the distances between sites and that between the observations in a functional setting. This contribution gives an extension and application of the predictor of Dabo-Niang et al. (2016) to a functional covariate context. The numerical results show that the proposed method outperforms the

Nonparametric Prediction for Spatial Dependent Functional Data

49

classical kernel predictor. They show the importance of taking the spatial structure and environmental effects into account in predicting the quantity of fish biomass. To go further in this marine application, a prediction model including, simultaneously, the two different environmental parameters could improve the results. This may be useful to identify the areas of potential concentration of coastal species during hot and cold seasons in Senegal. This methodology may permit us to improve the understanding of Senegalese coastal demersal fish distributions and thus propose stock management models to make fishery exploitation more sustainable. In this contribution, we only use catch per unit of effort of Senegalese coastal demersal fish data observed in 2015, working with all the available data during the period 2001–2015 has to be considered to account for some spatio-temporal variability and study the impact of climate change on biomass. In fact, the problems that will arise come from missing data at some locations and the irregularity in time and space. Such problems may be handled by new FDA progress on sparse data in time and space. 2.6. References Ahmed, M.S., N’diaye, M., Attouch, M. et al. (2018). K-nearest neighbors prediction and classification for spatial data. arXiv Preprint. Biau, G. and Cadre, B. (2004). Nonparametric spatial prediction. Statistical Inference for Stochastic Processes, 7(3), 327–349. Carbon, M., Tran, L.T., and Wu, B. (1997). Kernel density estimation for random fields. Statistics & Probability Letters, 36(2), 115–125. Cressie, N.A.C. (2015). Statistics for Spatial Data, Revised Edition. WileyInterscience, New York. Dabo-Niang, S. and Yao, A.-F. (2007). Kernel regression estimation for continuous spatial processes. Mathematical Methods of Statistics, 16(4), 298–317. Dabo-Niang, S. and Yao, A.-F. (2013). Kernel spatial density estimation in infinite dimension space. Metrika, 76(1), 19–52. Dabo-Niang, S., Rachdi, M., and Yao, A.-F. (2011). Kernel regression estimation for spatial functional random variables. Far East Journal of Theoretical Statistics, 37(2), 77–113. Dabo-Niang, S., Hamdad, L., and Ternynck, C. et al. (2014). A kernel spatial density estimation allowing for the analysis of spatial clustering: Application to Monsoon Asia Drought Atlas data. Stoch. Environ. Res. Risk Assess., 28(8), 2075–2099. Dabo-Niang, S., Ternynck, C., and Yao, A.-F. (2016). Nonparametric prediction of spatial multivariate data. Nonparametric Statistics, 2, 428–458.

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El Machkouri, M. (2007). Nonparametric regression estimation for random fields in a fixed-design. Stat. Inference Stoch. Process., 10(1), 29–47. El Machkouri, M. (2011). Asymptotic normality of the Parzen–Rosenblatt density estimator for strongly mixing random fields. Statistical Inference for Stochastic Processes, 14(1), 73–84. El Machkouri, M. and Stoica, R. (2010). Asymptotic normality of kernel estimates in a regression model for random fields. J. Nonparametr. Stat., 22(8), 955–971. Ferraty, F. and Vieu, P. (2006). Nonparametric Functional Data Analysis: Theory and Practice. Springer, Berlin. Francisco-Fernández, M., and Opsomer, J.D. (2005). Smoothing parameter selection methods for nonparametric regression with spatially correlated errors. Canad. J. Statist., 33(2), 279–295. Francisco-Fernández, M., Quintela-del Río, A., and Fernández-Casal, R. (2012). Nonparametric methods for spatial regression. An application to seismic events. Environmetrics, 23(1), 85–93. Gardner, B., Sullivan, P.J., Morreale, S.J. et al. (2008). Spatial and temporal statistical analysis of bycatch data: Patterns of sea turtle bycatch in the North Atlantic. Canadian Journal of Fisheries and Aquatic Sciences, 65(11), 2461–2470. Hallin, M., Lu, Z., and Tran, L.T. (2004). Local linear spatial regression. The Annals of Statistics, 32(6), 2469–2500. Heppell, S.S., Crowder, L.B., and Menzel, T.R. (1999). Life table analysis of long-lived marine species with implications for conservation and management. In American Fisheries Society Symposium, 23, 137–148. Klemelä, J. (2008). Density estimation with locally identically distributed data and with locally stationary data. J. Time Ser. Anal., 29(1), 125–141. Lefort, R., Fablet, R., Berger, L. et al. (2011). Spatial statistics of objects in 3-D sonar images: Application to fisheries acoustics. IEEE Geoscience and Remote Sensing Letters, 9(1), 56–59. Luan, J., Zhang, C., Xu, B. et al. (2018). Modeling the spatial distribution of three Portunidae crabs in Haizhou Bay, China. PloS ONE, 13(11), e0207457. Menezes, R., García-Soidán, P., and Ferreira, C. (2010). Nonparametric spatial prediction under stochastic sampling design. Journal of Nonparametric Statistics, 22(3), 363–377. N’diaye, M., Dabo-Niang, S., Ngom, P. et al. (2019). Nonparametric prediction and supervised classification for spatial dependent functional data under fixed sampling design. Forthcoming.

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Planque, B. and Buffaz, L. (2008). Quantile regression models for fish recruitmentenvironment relationships: Four case studies. Marine Ecology Progress Series, 357, 213–223. Pollock, L.J., Tingley, R., Morris, W.K. et al. (2014). Understanding co-occurrence by modeling species simultaneously with a joint species distribution model (jsdm). Methods in Ecology and Evolution, 5(5), 397–406. Ripley, B. (1987). Spatial point pattern analysis in ecology. In Developments in Numerical Ecology, Legendre, P. and Legendre, L. (eds), 407–429. Springer, Berlin. Rivoirard, J., Simmonds, J., Foote, K. et al. (2000). Geostatistics for Estimating Fish Abundance. Wiley-Blackwell, Hoboken. Shang, H.L., Hyndman, R.J., and Shang, M.H.L. (2019). Package ‘rainbow’. Package. Available at: https://cran.opencpu.org/web/packages/rainbow/rainbow.pdf. Ternynck, C. (2014). Spatial regression estimation for functional data with spatial dependency. Journal de la Société Française de Statistique, 155(2). Tran, L.T. (1990). Kernel density estimation on random fields. Journal of Multivariate Analysis, 34(1), 37–53. Wang, H., Wang, J., and Huang, B. (2012). Prediction for spatio-temporal models with autoregression in errors. Journal of Nonparametric Statistics, 24(1), 217–244. Young, M. and Carr, M.H. (2015). Application of species distribution models to explain and predict the distribution, abundance and assemblage structure of nearshore temperate reef fishes. Diversity and Distributions, 21(12), 1428–1440.

3 Space–Time Simulations of Extreme Rainfall: Why and How?

3.1. Why? 3.1.1. Rainfall-induced urban floods Floods are regarded as the most widespread and globally costly natural disaster. Their human and economic impact is obviously the largest in the most densely populated areas. Since the fraction of the world’s population living in urban areas increases steadily, the impact and cost of urban floods can be expected to rise in the future. In Mediterranean areas, heavy rainfall events occur mostly in autumn owing to temperature differences between the moisture coming from the sea and the surface of the land. Orographic factors may also play a significant role. These rainfall events which could be either very localized, high intensity and last a few hours, or could be long-lasting events with moderate intensity affecting large areas - might lead to floods. Not only does urbanization lead to an increase in population density (and with it the density of stakes), but it also leads to dramatic changes in land use. The fraction of impervious and/or low permeability areas increases. This contributes to reducing the infiltration capacity of soils. Intense to moderate rainfall, that would otherwise infiltrate, are retained on the ground and generate large runoff volumes. Roads, parking lots, etc., being significantly smoother than natural grounds, contribute to speeding up of the propagation of the runoff signal. Urban drainage networks may also contribute to enhancing the dynamics of urban catchments. Chapter written by Gwladys T OULEMONDE, Julie C ARREAU and Vincent G UINOT.

Mathematical Modeling of Random and Deterministic Phenomena, First Edition. Edited by Solym Mawaki Manou-Abi, Sophie Dabo-Niang and Jean-Jacques Salone. © ISTE Ltd 2020. Published by ISTE Ltd and John Wiley & Sons, Inc.

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Mathematical Modeling of Random and Deterministic Phenomena

Urban flood crisis management gathers a wide variety of stakeholders. These include local authorities, urban planning divisions, rescue and civil protection services, weather forecasting and warning services, insurance and reassurance companies, etc. Not all stakeholders have the same needs. Urban planning divisions and insurance companies are mostly concerned with long-term measures for disaster mitigation and vulnerability reduction. Warning services, local authorities and civil protection play a key role during the crisis, by managing communications, rescue actions, and establishing priorities for stake protection. Ideally, decision-makers would like to be supplied with real-time knowledge or a forecast of the rainfall event (i.e. how is the rainfall field likely to vary in space and time). The knowledge of the space–time behavior of the rainfall field would allow rainfall-runoff models to be operated to forecast the consequences of the rainfall event, in terms of flood-induced risk and damages on the scale of the conurbation. An “ideal” urban flood forecasting and warning chain should include meteorological monitoring and forecasting modules, one or several rainfall-runoff and/or free surface flow models and a decision support system, prioritizing and synthesizing information for the crisis management group. The design of flood crisis management systems is beyond the scope of the present chapter. Only rainfall generating and free surface modeling aspects are covered hereafter. The following section presents hydraulic simulations of a rainfall-induced urban flood. These simulations illustrate the need for space–time simulations of extreme rainfall highlighting the effect of the localization and the extension of the rainfall field on the flooding pattern. The second section deals with “how to perform space–time simulations of extreme rainfall?”. After discussing the spatial stochastic rainfall generators, the emphasis is put on extreme events modeling to understand the main associated issues. The key challenges for the construction of a stochastic rainfall generator geared towards extreme events are then presented. Finally, the third section is devoted to outlooks from an operational point of view, and a possible framework for an integrated rainfall-induced urban flood crisis management system is proposed. 3.1.2. Sample hydraulic simulation of a rainfall-induced urban flood The present section illustrates the sensitivity of the flooding pattern to the localization and extension of the rainfall field. A two-dimensional model of a part of the Ecusson district (Montpellier city, France) is built. Synthetic rainfall fields are used as inputs to the model in the form of a source term in the two-dimensional shallow water equations (Guinot & Soares-Frazão 2006).

Space–Time Simulations of Extreme Rainfall

55

The rainfall fields are assumed time-independent and follow a radial distribution in the form   (x − x0 )2 + (y − y0 )2 ∀t [3.1] Pθ (x, y, t) = Pmax × exp − R2 where (x, y, t) are the space and time coordinates and θ = (x0 , y0 , R, Pmax ) with (x0 , y0 ) the coordinates of the center of the field, R a scaling radius and Pmax the maximum precipitation. Four such fields are generated with the parameters in Table 3.1. It is acknowledged that the assumption of a time-independent and static rainfall field undermines the realistic character of the simulations. However, the purpose here is only to illustrate the sensitivity analysis of the computed runoff to the location of the center of the rainfall field. Figure 3.1 shows the normalized rainfall θ fields PPmax (x, y, t) for the four simulations. It should be noted that these four rainfall fields have almost the same average value when averaged on the scale of a 1 km × 1 km radar pixel, centered on the modeled area. Although unusual, intensities of 125 mm/day and 500 mm/day are commensurate with values observed in the South of France during extreme rainfall events (Delrieu et al. 2005, Brunet et al. 2018). Simulation Pmax (mm/d) R (m) x0 (m) y0 (m) 1

125

200

300

500

2

125

200

700

300

3

500

100

300

500

4

500

100

700

300

Table 3.1. Rainfall field parameters

Figure 3.2 shows the maximum water depth fields obtained by forcing a software package solving the two-dimensional shallow water equations (Guinot & SoaresFrazão 2006) with the aforementioned rainfall fields. The maximum water depth field hmax (x, y) is computed from the simulated water depth field hθ (x, y, t) as hmax (x, y) = max hθ (x, y, t) 0≤t≤T

[3.2]

where T is the simulated period. In the present simulations, T = 15 minutes. This corresponds to the time needed for the hydraulic fields to reach the asymptotic steady state under a steady-state rainfall forcing. The maximum water depth is selected because it has been identified as a danger indicator, in particular for pedestrians, and as a damage indicator for buildings (Blanco-Vogt & Schanze 2014, Merz et al. 2010, Wagenaar et al. 2016).

56

Mathematical Modeling of Random and Deterministic Phenomena


 








 

Figure 3.1. Normalized rainfall fields Pθ (x, y, t)/Pmax . x− and y− grid spacing: 100 m. For a color version of this figure, see www.iste.co.uk/manouabi/modeling.zip

Figure 3.2 allows the following conclusions to be drawn: – The maximum rainfall intensity does not exert a significant influence on the range of hmax . While Pmax varies by a factor 4 between Simulations 1 and 3, it yields very similar hmax maps (see Figure 3.2, left). The same holds for Simulations 2 and 4 (see Figure 3.2, right). In contrast, Pmax appears to be a controlling factor for the extension of the flooded area. – Comparing Simulations 1 and 2 (Figure 3.2, top) with 3 and 4 (Figure 3.2, bottom) shows that shifting the center (x0 , y0 ) of the rainfall field from (300 m, 500 m) to (700 m, 300 m) also induces significant differences in the mapped hmax (x, y). The contrast is all the more significant as the scaling radius R is smaller.

Space–Time Simulations of Extreme Rainfall


 








 

57

Figure 3.2. Maximum simulated water depths hmax (x, y). x− and y− grid spacing: 100 m. The corresponding rainfall fields are shown in Figure 3.1. For a color version of this figure, see www.iste.co.uk/manouabi/modeling.zip

In conclusion, a 1 km × 1 km rainfall field resolution is too coarse for an accurate mapping of the flood hazard in urban areas. A 100 m × 100 m resolution appears more appropriate. Note that only static rainfall simulations are used in the present example, while rainfall fields are known to exhibit strong space–time dependencies (Cox & Isham 1988, Kleiber et al. 2012, Baxevani & Lennartsson 2015) especially at high spatio-temporal resolution, see for instance Benoit et al. (2018). Since free surface flows are governed by wave propagation phenomena, it is most likely that the need for highly spatially resolved rainfall fields also comes with a similar need for a high temporal resolution. This latter aspect is not covered in the present chapter for the sake of conciseness. Therefore, any (deterministic or stochastic) rainfall field generator should be developed and used with the following questions in mind: (i) what is the minimum

58

Mathematical Modeling of Random and Deterministic Phenomena

required spatial and temporal resolution to provide a sound assessment of the free surface flow variables? (ii) how can the temporal and spatial distribution of rainfall over a given area be reproduced? Stochastic rainfall generators have the potential to characterize several aspects of the space–time dependence structure of rainfall fields. These are described in the next section, with an emphasis on extreme event modeling. 3.2. How? 3.2.1. Spatial stochastic rainfall generator Since the outputs of meteorological models are uncertain in essence, an alternative to providing (inevitably biased and inaccurate) deterministic rainfall fields consists of generating ensemble rainfall scenarios, the characteristics of which are controlled in a statistical fashion. These rainfall scenarios can supply boundary conditions for hydraulic models. The latter can simulate water depths, flow velocities and other hydraulic variables that are in turn used to assess flood risk and to map damages. To generate realistic ensemble rainfall scenarios, including dry sequences, stochastic rainfall generators may be employed (Ailliot et al. 2015). Spatial generators, that are capable of simulating continuous rainfall fields, are complex probabilistic models that combine several stochastic mechanisms in order to reproduce various features of the rainfall fields. To estimate the parameters that tune the stochastic mechanisms of the generators, statistical inference schemes are designed to draw information from rainfall observation series, whether from rain-gauged stations or from other types of data, such as radar data. Thanks to these inference schemes, simulations of the generators are expected to be as close to reality as possible. One of the foremost features that stochastic rainfall generators seek to reproduce is the alternation of periods with and without rainfall. The alternation of rain and no rain can be thought of as basic weather types which are usually modeled with a Markov chain, hidden or not (Ailliot et al. 2015). These weather types can be further refined by decomposing the rainfall type into several sub-types, such as drizzle, moderate intensity and heavy rainfall. The weather-type decomposition is designed to account for the non-stationarity in time and possibly in space if the weather-type description includes spatial information (Garavaglia et al. 2010, Leblois 2012). Further spatial non-stationarity due to orographic effects can be modeled by resorting to landscape variables (Arnaud et al. 2006). A related feature is the so-called intermittency of rainfall, i.e. for a given time step, the alternation of areas with and without rainfall. Intermittency may be modeled

Space–Time Simulations of Extreme Rainfall

59

by considering a binary random variable that indicates whether it is raining or not (Kleiber et al. 2012, Leblois & Creutin 2013). Alternatively, the random variable that models positive rainfall may be truncated below zero, thereby enforcing, by construction, consistency in the rainfall amounts at the boundaries between wet and dry regions (Allard & Bourotte 2014, Baxevani & Lennartsson 2015). Another core feature is the spatio-temporal dependence of the rainfall fields. Within rainy periods (that may be related to different weather types), the generator must simulate positive rainfall values, i.e. rainfall intensities, with spatio-temporal patterns similar to the ones observed. Rodriguez-Iturbe et al. (1987) and Cox & Isham (1988) propose spatio-temporal rainfall models considering that storm events induce a cluster of rain cells, which are represented as cylinders in space–time. Gaussian processes with spatio-temporal covariance functions can also be employed (Kleiber et al. 2012, Baxevani & Lennartsson 2015). However, the marginal distributions of rainfall intensities are well known to be non-Gaussian. In particular, they are asymmetrical and, in some areas such as in the South of France, they are heavy tailed (Carreau et al. 2017). This non-Gaussianity of the marginals can be handled by a transformation to suitable univariate non-Gaussian distributions. Nevertheless, this significantly increases the complexity of the inference scheme for spatio-temporal Gaussian processes (Kleiber et al. 2012, Baxevani & Lennartsson 2015). In addition, although anisotropy can be accounted for (Baxevani & Lennartsson 2015), the dependence structure of Gaussian processes may be inadequate owing to its symmetry (Carreau & Bouvier 2016) or to its properties regarding extreme events (in particular, the asymptotic independence property, see section 3.2.2). With the exception of Evin et al. (2018) in the multivariate case, no spatial stochastic rainfall generator has stochastic mechanisms specially dedicated to account for the spatio-temporal behavior of extreme events (Ailliot et al. 2015). This is a crucial feature in the context of urban flood risk studies. The next section presents an overview of existing approaches to modeling extreme events, both in terms of intensity and spatial and spatio-temporal patterns. 3.2.2. Modeling extreme events The key result of Extreme Value Theory (EVT) is due to Fisher & Tippett (1928) and has found many applications in domains such as finance and insurance (Embrechts et al. 1997) along with environmental sciences (Katz 2002). Interest in extreme value analyses in climate science is relatively recent (Kharin et al. 2007, Goubanova & Li 2006, Salvadori & Rosso 2007). The result of Fisher & Tippett (1928) shows that the behavior of the maximum, i.e. the largest value of a sample, correctly centered and standardized, converges in distribution to the Generalized Extreme Value (GEV) distribution. The GEV distribution encompasses the three extreme-value distributions that share the max-stability property, i.e. the maximum of two independent copies

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of random variables from a given extreme-value distribution belongs to the same extreme-value distribution. In practice, modeling is performed on block maxima where each block represents a given time interval. Obviously this approach is debatable, since it may include rather low values for blocks within which no large events occurred; and it leaves unexploited information contained in other large values of the sample, for example, when several large events occurred within the same block. An alternative approach consists of considering the largest values of the sample defined, as excesses above a high threshold. The distribution of these strictly positive excesses can then be approximated by a Generalized Pareto Distribution (GPD) (Pickands III 1975). In both cases (working on block maxima or excesses), we obtain an approximation of the upper tail of the distribution of the variable of interest. This allows us to extrapolate beyond the largest observed values. In particular, probabilities of exceeding high thresholds or high quantiles, i.e. levels that are exceeded with a very low probability, can be estimated thanks to the upper tail approximation by the GEV distribution or by the GPD. Similar approaches allow a lower tail approximation to be obtained. In a spatial context, max-stable processes are the straightforward extension of the maxima approach, defined previously, since they appear as the natural limits for spatial maxima taken site by site (de Haan 1984). By definition, a max-stable process fulfills the max-stability property which implies, in particular, that its margins follow a GEV distribution. More precisely, a stochastic process Z is max-stable if for each n ≥ 1 there exists normalizing functions an > 0 and bn ∈ R such that maxi=1,...,n Zi (x)−bn d = Z with Z1 , . . . , Zn a sequence of independent copies of the an stochastic process Z. Max-stable processes have led to a very rich literature with various proposed parametric models (Brown & Resnick 1977, Smith 1990, Schlather 2002, Kabluchko et al. 2009, Davison & Gholamrezaee 2012, Opitz 2013) and were widely used as models for spatial extreme events, especially for environmental phenomena. A common representation of a max-stable process is the one due to Schlather (2002) in which the process Z(x), a stationary max-stable process on Rp with unit Fréchet marginal distributions, is written as maxj Sj max{0, Wj (x)}, + −2 where {Sj }∞ ds and j=1 are the points of a Poisson process on R with intensity s p {Wj (x)}∞ are independent replicates of a stationary process W (x) on R satisfying j=1 E max(0, Wj (o)) = 1 with o denoting the origin. As we aim to simulate fields of extreme phenomena, we focus in this chapter on unconditional stochastic simulations. As a first example of climate extreme modeling, Blanchet & Davison (2011) proposed complex models based on max-stable processes, designed for heavy snow events. One of their results consists of a risk analysis by computing joint survival probabilities of groupwise annual maxima. Max-stable models can also help to characterize significant heights of extreme waves in the Gulf of

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Lions (Chailan 2014) and to establish different long-term scenarios of littoral erosion that depend on different inputs like wind. Max-stable processes in space were extended to the space–time context in Davis et al. (2013a). Moreover, Davis et al. (2013b) proposed statistical inference for such models based on pairwise likelihood. Huser & Davison (2014) obtained consistent estimation thanks to a pairwise censored likelihood to model extreme values of space–time rainfall data. Max-stable processes being constructed from the pointwise maxima of underlying processes, Dombry et al. (2016) presented an algorithm for the exact simulation of a max-stable process at a finite number of locations, by focusing on the processes that effectively contribute to the maxima. An extension with a reduced computational cost, but restricted to Brown–Resnick processes, is introduced in Oesting et al. (2018) when the number of locations is large. The physical interpretation of spatial max-stable processes is not straightforward since they represent maxima of spatial fields. In general, one simulation of a spatial max-stable process does not represent a real event, since pointwise maxima are likely to occur at different times. It is even more blatant in a space–time framework where their use and interpretation for simulation purposes become extremely difficult. As previously explained, in the univariate context, an alternative to maxima models are threshold exceedance models. The generalization of the GPD to spatial processes yields the so-called Pareto processes (Ferreira & de Haan 2014, Opitz et al. 2015, Dombry & Ribatet 2015, Thibaud & Opitz 2015, de Fondeville & Davison 2018). Constructively, the Pareto process corresponds to the product of a scale component and a component on the spatial structure, called spectral process. There is no unique definition of a spatial extreme event. Dombry & Ribatet (2015) defined the notion of -Pareto processes by considering general exceedances, introducing a homogeneous cost functional denoted by . In practice, the choice of  must mainly depend on the nature of the considered phenomenon. Possible examples are functions of the maximum, minimum, or mean. For rainfall, the spatio-temporal mean is a good option combining duration, spatial extent and magnitude of the event. Clear advantages of thresholding techniques are their potential to exploit information from more data and to explicitly model the original event. This last point is essential for the stochastic spatial generator construction purposes. So far, Pareto processes were mostly used in a parametric framework, thereby using assumptions on the choice of the underlying dependence structure that may be too strong. In this context, Thibaud & Opitz (2015) are interested in -Pareto processes and propose an exact simulation procedure for the limiting processes of threshold exceedances of all asymptotically dependent elliptical processes. It is also

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possible to relax the parametric assumption for the spectral processes by relying on non-parametric estimates, deduced from observed spectral processes. Building on this non-parametric idea and stemming from original works of Caires et al. (2011) and Ferreira & de Haan (2014), Chailan et al. (2017) developed a semi-parametric approach to generate extreme spatio-temporal fields of waves in the Gulf of Lions (South of France). The first step consists of selecting space–time extreme events called storms, among a 52-year hindcast of wave features over the north-western Mediterranean sea. In the second step, the selected storms are uplifted after proper standardization. As a result, extreme storms of greater intensity than ones observed are generated and are illustrated in a case-study concerning the quantification of the long-shore mass flux of energy in a coastal area. Palacios-Rodríguez et al. (2018) extended this semi-parametric approach by setting up a sound space–time framework, thanks to links with Pareto processes. A key benefit of the proposed method is the possibility of generating an unlimited number of realizations of extreme storms. Another extension concerns the selection of extreme episodes with a general space–time cost functional  quantifying how extreme episodes are, in other words the extremeness of episodes. The aforementioned simulation approaches for spatial extremes, in the max-stable framework, rely on the hypothesis of asymptotic dependence. This means that dependence is assumed to remain constant, regardless of the extreme level under consideration. These approaches are not suitable when the dependence strength decreases at high levels and may ultimately vanish. This behavior, called asymptotic independence (AI), is very difficult to detect in practice. However, analyses of hourly precipitation in the South of France (Bacro et al. 2019) suggest AI behavior (see also Davison et al. 2013, Thibaud et al. 2013, Le et al. 2018). Models allowing for AI behavior requires the development of specific approaches. Stationary Gaussian processes are examples of AI processes because, except in the case of perfect correlation, bivariate Gaussian variables are AI (Sibuya 1960). AI processes can also be obtained by inverting max-stable processes (Wadsworth & Tawn 2012) or as pointwise maxima of samples from a ratio of Gaussian processes, with common correlation function (Padoan 2013). Models with flexible dependence such as max-mixture models (Wadsworth & Tawn 2012, Bacro et al. 2016) for maxima and other processes for threshold exceedances, such as Gaussian scales mixture processes or related works (Opitz 2016, Huser et al. 2017, Huser & Wadsworth 2018) constitute inspiring and promising works. Following an idea of Wadsworth & Tawn (2012), Bacro et al. (2016) exploited a max-mixture approach to propose a general spatial model which is capable of dealing with extremal dependence at small distances, possible independence at large distances and AI at intermediate ones.

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These models, allowing AI behavior, commonly assume temporal independence for inference purposes. However, developing flexible space–time modeling for extremes is crucial to characterize the temporal dynamics and the persistence of extreme events, spanning several time steps. Bacro et al. (2019) proposed a two-stage model for spatio-temporal exceedances that remains physically interpretable in an AI context. Following Bortot & Gaetan (2014), they use the representation of the GPD as a Gamma mixture of an exponential distribution to formulate a hierarchical model, integrating space–time dependence thanks to a latent space–time Gamma process (Wolpert & Ickstadt 1998). This Gamma process relies on an elliptical cylinder allowing for an efficient physical interpretation in terms of storms. Statistical inference of model parameters is performed thanks to a pairwise log-likelihood for the observed censored excesses. The interest of this model was exemplified on a real dataset of rainfall in the South of France, and was validated by computing empirical estimates of various multivariate conditional probabilities involving spatio-temporal aspects. This hierarchical model is related to the temporal trawl processes (Barndorff-Nielsen et al. 2014, Noven et al. 2015) of which Opitz (2017) proposed spatial extensions. 3.2.3. Stochastic rainfall generator geared towards extreme events As argued in section 3.2.1, to be useful for urban flood risk studies, a spatial stochastic rainfall generator should be capable of mimicking the spatio-temporal patterns of extreme events. Therefore, the stochastic mechanisms associated with extreme events should draw from the modeling techniques described in section 3.2.2. We identify the following three key challenges. As extreme events modeling is not adapted for regular events, a natural strategy would be to define a special weather type dedicated to extreme events. To our knowledge, in existing approaches, although extreme weather type may be identified a posteriori, no particular technique has been proposed for their identification (Leblois 2012). Both intensity and spatio-temporal pattern information should enter in to the definition of this extremal weather type. To this end, non-parametric descriptors of the extremal dependence structure, such as the madogram, could be useful (Cooley et al. 2006, Vannitsem & Naveau 2007, Erhardt & Smith 2012). Within the weather type dedicated to extreme events, regular and extreme events are likely to be present in different areas and in different time periods. Transitions between regular and extreme events must be modeled both in the univariate marginal distributions and in the spatio-temporal dependence structure. Therefore, a second issue to be considered is the need to rely on mixed univariate marginal distributions that can characterize both regular and extreme events. To this end, Carreau & Bengio (2009) stitched together a Gaussian distribution for the lower part and a GPD for the upper tail. This hybrid Pareto distribution was used in a mixture model. It was shown to be able to adapt to various complex heavy tailed distributions. One drawback of

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this approach is the presence of a potentially non-negligible lower tail which may be inadequate to model phenomena such as precipitation. A recent alternative to this hybrid was proposed in Naveau et al. (2016) who take benefit from EVT for both large and low values (excluding zeros). Their statistical model ensures a smooth transition between the upper and the lower tails, with a reasonable number of parameters. The last and probably most challenging issue concerns the need to design spatio-temporal processes with transitions in the dependence structure between extreme and ordinary events. As far as we know, there is no proposition along these lines. Nevertheless, one possibility to simulate spatial fields that contain both regular and extreme events is to rely on the approach in Thibaud et al. (2013) in which a single dependence structure, inherited from either a max-stable or an inverted max-stable process, is employed. A related problem is the non-stationarity of the dependence structure in space and time where few proposals exist. For instance, Fox & Dunson (2015) developed dynamic latent factor models in the Gaussian framework. In the max-stable framework, Huser & Genton (2016) integrated covariates in the dependence structure. In both cases, the dependence structure is non-stationary but is not able to change distribution family, as would be required to make a transition between ordinary and extreme rainfall events. 3.3. Outlook As far as an operational use is concerned, using the fields from a stochastic rainfall generator as inputs for a free surface flow model, such as that presented in section 3.1, raises a number of issues. In each of the 15 minute shallow water simulations presented in this section, 2 CPU seconds are needed to simulate 1 second using a standard PC. This is because the urban geometry is complex. Refined computational grids are needed to capture hydraulic singularities accurately. The typical computational cell width in a two-dimensional urban shallow water model is 1 m. This precludes refined shallow water models to be used on the scale of the district (let alone the entire conurbation) for real-time purposes. Besides, a stochastic rainfall generator implies that many realizations of the rainfall field are to be used as inputs for as many shallow water simulations, so as to obtain a statistical description of danger/damage over the area of concern. This increases the computational effort even more. An alternative consists of upscaling the free surface flow model. Fast-running free surface flow models can be obtained by averaging the shallow water equations over large domains (the size of a house block, for example). The properties of the buildings and other singularities that influence the flow are described in a statistical fashion. A key parameter that emerges from averaging is the porosity of the urban medium, i.e. the plan view fraction of the urban area available for the storage and transport of water. A wide variety of porosity models (Guinot & Soares-Frazão 2006, Sanders

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et al. 2008, Guinot 2012, Özgen et al. 2016, Guinot et al. 2017, 2018, Viero 2019) have been proposed and the field is in rapid development. Porosity models have been reported to run two to three orders of magnitude faster that their classical shallow water counterparts (Sanders et al. 2008, Guinot et al. 2018). Such computational rapidity is compatible with the stochastic simulation of multiple rainfall scenarios. A possible framework for an integrated rainfall-induced urban flood crisis management system can be as follows: 1) generation of stochastic rainfall fields on the scale of the conurbation, with the temporal and spatial resolution required by the free surface flow model (i.e. 100 m, see section 3.1); 2) use of the multiple realizations of the rainfall fields to run fast, porosity-based shallow water simulations. This allows the urban areas at higher risk to be identified on a coarse scale; 3) selection of those areas where the flood risk has been identified as higher, and where a detailed mapping of the risk on the metric scale is deemed necessary. For these, run refined simulations covering the (limited) spatial extension where the detailed mapping is needed. To do so, the initial and boundary conditions for the refined flow model must be interpolated from those of the porosity model. The same goes with the rainfall field. Although simple in its principle, the above sequence is not straightforward to implement. Concerning step 1, with a few exceptions (Benoit et al. 2018), observation series are not available over a sufficiently dense network of sites and over long enough observation periods at the desired resolution. In these cases, downscaling techniques can be employed by making use of auxiliary information such as rainfall radar data (Delrieu et al. 2014). Few downscaling techniques were proposed in the extreme value framework. Very recently, Engelke et al. (2019) developed a method based on a theoretical link between the extremal distribution of the aggregated data and the corresponding underlying process. Concerning step 3, very little is known about the constraints attached to initial and boundary condition scale transfer from the porosity model to the local, refined, shallow water model. This specific issue is currently under study in the Inria Lemon research team. As far as urban floods are concerned, the generation of rainfall scenarios is not the only possible application field of Extreme Value Theory (EVT). In coastal areas, urban flooding may also result from storm surges. Applying the EVT to wave forcing so as to obtain probabilistic assessments of the marine submersion risk, is also a path for research.

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3.4. References Ailliot, P., Allard, D., Monbet, V., and Naveau, P. (2015). Stochastic weather generators: an overview of weather type models. Journal de la Société Française de Statistique, 156(1), 101–113. Allard, D. and Bourotte, M. (2014). Disaggregating daily precipitations into hourly values with a transformed censored latent Gaussian process. Stochastic Environmental Research and Risk Assessment, 29(2), 453–462. Arnaud, P., Lavabre, J., Sol, B., and Desouches, C. (2006). Cartographie de l’aléa pluviographique de la France. La Houille Blanche, 5, 102–111. Bacro, J.N., Gaetan, C., and Toulemonde, G. (2016). A flexible dependence model for spatial extremes. Journal of Statistical Planning and Inference, 172, 36–52. Bacro, J.N., Gaetan, C., Opitz, T., and Toulemonde, G. (2019). Hierarchical spacetime modeling of asymptotically independent exceedances with an application to precipitation data. Journal of the American Statistical Association, 0(0), 1–26. Barndorff-Nielsen, O.E., Lunde, A., Shepard, N., and Veraat, A.E.D. (2014). Integervalued trawl processes: A class of stationary infinitively divisible processes. Scandinavian Journal of Statistics, 41, 693–724. Baxevani, A. and Lennartsson, J. (2015). A spatiotemporal precipitation generator based on a censored latent Gaussian field. Water Resources Research, 51(6), 4338–4358. Benoit, L., Allard, D., and Mariethoz, G. (2018). Stochastic rainfall modeling at sub-kilometer scale. Water Resources Research, 54(6), 4108–4130. Blanchet, J. and Davison, A.C. (2011). Spatial modeling of extreme snow depth. The Annals of Applied Statistics, 5(3), 1699–1725. Blanco-Vogt, A. and Schanze, J. (2014). Assessment of the physical flood susceptibility of buildings on a large scale: Conceptual and methodological frameworks. Natural Hazards and Earth System Sciences, 14(8), 2105–2117. Bortot, P. and Gaetan, C. (2014). A latent process model for temporal extremes. Scandinavian Journal of Statistics, 41, 606–621. Brown, B. and Resnick, S. (1977). Extreme values of independent stochastic processes. Journal of Applied Probability, 14, 732–739. Brunet, P., Bouvier, C., and Neppel, L. (2018). Retour d’expérience sur les crues des 6 et 7 octobre 2014 à Montpellier-Grabels (Hérault, France) : Caractéristiques hydro-météorologiques et contexte historique de l’épisode. Géographie physique et environnement, 12, 43–59.

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Caires, S., de Haan, L., and Smith, R.L. (2011). On the determination of the temporal and spatial evolution of extreme events. Technical Report, Rijkswaterstaat, Centre for Water Management. Carreau, J. and Bengio, Y. (2009). A hybrid Pareto model for asymmetric fat-tailed data: The univariate case. Extremes, 12(1), 53–76. Carreau, J. and Bouvier, C. (2016). Multivariate density model comparison for multisite flood-risk rainfall in the French Mediterranean area. Stochastic Environmental Research Risk Assessement, 30, 1591–1612. Carreau, J., Naveau, P., and Neppel, L. (2017). Partitioning into hazard subregions for regional peaks-over-threshold modeling of heavy precipitation. Water Resources Research, 53(5), 4407–4426. Chailan, R., Toulemonde, G., and Bacro, J.N. (2017). A semiparametric method to simulate bivariate space-time extremes. The Annals of Applied Statistics, 11(3), 1403–1428. Chailan, R., Toulemonde, G., Bouchette, F., Laurent, A., Sevault, F., and Michaud, H. (2014). Spatial assessment of extreme significant waves heights in the Gulf of Lions. Coastal Engineering Proceedings, 34. Cooley, D., Naveau, P., and Poncet, P. (2006). Variograms for spatial max-stable random fields. In Bertail, P., Doukhan, P., and Soulier, P. (eds). Dependence in Probability and Statistics, Springer. Cox, D.R. and Isham, V. (1988). A simple spatial-temporal model of rainfall. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 415, 317–328. Davis, R.A., Klüppelberg, C., and Steinkohl, C. (2013a). Max-stable processes for modeling extremes observed in space and time. Journal of the Korean Statistical Society, 42, 399–414. Davis, R.A., Klüppelberg, C., and Steinkohl, C. (2013b). Statistical inference for maxstable processes in space and time. Journal of the Royal Statistical Society, 75, 791–819. Davison, A.C., and Gholamrezaee, M.M. (2012). Geostatistics of extremes. Proceedings of the Royal Society A, 468, 581–608. Davison, A.C., Huser, R., and Thibaud, E. (2013). Geostatistics of dependent and asymptotically independent extremes. Journal of Mathematical Geosciences, 45, 511–529.

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Delrieu, G., Nicol, J., Yates, E., Kirstetter, P.-E., Creutin, J.-D., Anquetin, S., Obled, C., Saulnier, G.-M., Ducrocq, V., Gaume, E., Payrastre, O., Andrieu, H., Ayral, P.-A., Bouvier, C., Neppel, L., Livet, M., Lang, M., du Châtelet, J.P., Walpersdorf, A., and Wobrock, W. (2005). The catastrophic flash-flood event of 8–9 September 2002 in the gard region, France: A first case study for the Cévennes-Vivarais Mediterranean Hydrometeorological Observatory. Journal of Hydrometeorology, 6(1), 34–52. Delrieu, G., Wijbrans, A., Boudevillain, B., Faure, D., Bonnifait, L., and Kirstetter, P.-E. (2014). Geostatistical radar–raingauge merging: A novel method for the quantification of rain estimation accuracy. Advances in Water Resources, 71, 110–124. Dombry, C. and Ribatet, M. (2015). Functional regular variations, Pareto processes and peaks over threshold. Statistics and Its Interface, 8(1), 9–17. Dombry, C., Engelke, S., and Oesting, M. (2016). Exact simulation of max-stable processes. Biometrika, 103(2), 303–317. Embrechts, P., Klüppelberg, C., and Mikosch, T. (1997). Modeling Extremal Events: For Insurance and Finance. Corrected edition. Springer. Engelke, Sebastian, De Fondeville, Raphaël, and Oesting, Marco. (2019). Extremal behaviour of aggregated data with an application to downscaling. Biometrika, 106(1), 127–144. Erhardt, R.J. and Smith, R.L. (2012). Approximate Bayesian computing for spatial extremes. Computational Statistics and Data Analysis, 56(6), 1468–1481. Evin, G., Favre, A.-C., and Hingray, B. (2018). Stochastic generation of multisite daily precipitation focusing on extreme events. Hydrology and Earth System Sciences, 22(1), 655–672. Ferreira, A. and de Haan, L. (2014). The generalized Pareto process; with a view towards application and simulation. Bernoulli, 20, 1717–1737. Fisher, R.A. and Tippett, L.H.C. (1928). Limiting forms of the frequency of the largest or smallest member of a sample. Math. Proc. Cambridge Philos. Soc., 24, 180–190. de Fondeville, R. and Davison, A.C. (2018). High-dimensional peaks-over-threshold inference. Biometrika, 105(3), 575–592. Fox, E.B., and Dunson, D.B. (2015). Bayesian nonparametric covariance regression. J. Mach. Learn. Res., 16, 2501–2542. Garavaglia, F., Gailhard, J., Paquet, E., Lang, M., Garçon, R., Bernardara, P. (2010). Introducing a rainfall compound distribution model based on weather patterns sub-sampling. Hydrology and Earth System Sciences Discussions, 14.

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Goubanova, K. and Li, L. (2006). Extremes in temperature and precipitation around the Mediterranean basin in an ensemble of future climate scenario simulations. Global and Planetary Change, 57, 27–42. Guinot, V. (2012). Multiple porosity shallow water models for macroscopic modeling of urban floods. Advances in Water Resources, 37(Mar), 40–72. Guinot, V. and Soares-Frazão, S. (2006). Flux and source term discretization in two-dimensional shallow water models with porosity on unstructured grids. International Journal for Numerical Methods in Fluids, 50(3), 309–345. Guinot, V., Delenne, C., Rousseau, A. and Boutron, O. (2017). Flux closures and source term models for shallow water models with depth-dependent integral porosity. Advances in Water Resources, 122, 1–26. Guinot, V., Sanders, B.F., and Schubert, J.E. (2018). Dual integral porosity shallow water model for urban flood modeling. Advances in Water Resources, 103, 16–31. de Haan, L. (1984). A spectral representation for max-stable processes. Annals of Probability, 12, 1194–1204. Huser, R. and Davison, A.C. (2014). Space-time modeling of extreme events. Journal of the Royal Statistical Society: Series B, 76, 439–461. Huser, R. and Genton, M. (2016). Non-stationary dependence structures for spatial extremes. Journal of Agricultural, Biological, and Environmental Statistics, 21(3), 470–491. Huser, R. and Wadsworth, J.L. (2018). Modeling spatial processes with unknown extremal dependence class. Journal of the American Statistical Association, 1–11. Huser, R., Opitz, T., and Thibaud, E. (2017). Bridging asymptotic independence and dependence in spatial extremes using Gaussian scale mixtures. Spatial Statistics, 21, 166–186. Kabluchko, Z., Schlather, M., and de Haan, L. (2009). Stationary max-stable fields associated to negative definite functions. The Annals of Probability, 37, 2042–2065. Katz, R.W., Parlange, M.B., and Naveau, P. (2002). Statistics of extremes in hydrology. Advances in Water Resources, 25, 1287–1304. Kharin, V., Zwiers, F., Zhang, X., and Hegerl, G. (2007). Changes in temperature and precipitation extremes in the IPCC ensemble of global coupled model simulations. Journal of Climate, 20, 1419–1444. Kleiber, W., Katz, R.W., and Rajagopalan, B. (2012). Daily spatiotemporal precipitation simulation using latent and transformed Gaussian processes. Water Resources Research, 48(1).

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Le, P.D., Davison, A.C., Engelke, S., Leonard, M., and Westra, S. (2018). Dependence properties of spatial rainfall extremes and areal reduction factors. Journal of Hydrology, 565, 711–719. Leblois, E. (2012). Le bassin versant, système spatialement structuré et soumis au climat. HDR, University of Grenoble. Leblois, E. and Creutin, J.-D. (2013). Space-time simulation of intermittent rainfall with prescribed advection field: Adaptation of the turning band method. Water Resources Research, 49(6), 3375–3387. Merz, B., Kreibich, H., Schwarze, R., and Thieken, A. (2010). Review article “Assessment of economic flood damage”. Natural Hazards and Earth System Science, 10(8), 1697–1724. Naveau, P., Huser, R., Ribereau, P., and Hannart, A. (2016). Modeling jointly low, moderate, and heavy rainfall intensities without a threshold selection. Water Resources Research, 52(4), 2753–2769. Noven, R.C., Veraart, A.E.D., and Gandy, A. (2015). A latent trawl process model for extreme values. arXiv preprint arXiv:1511.08190. Oesting, M., Schlather, M., and Zhou, C. (2018). Exact and fast simulation of maxstable processes on a compact set using the normalized spectral representation. Bernoulli, 24(2), 1497–1530. Opitz, T. (2013). Extremal t processes: Elliptical domain of attraction and a spectral representation. Journal of Multivariate Analysis, 122, 409–413. Opitz, T. (2016). Modeling asymptotically independent spatial extremes based on Laplace random fields. Spatial Statistics, 16, 1–18. Opitz, T. (2017). Spatial random field models based on Lévy indicator convolutions. arXiv preprint arXiv:1710.06826. Opitz, T., Bacro, J.N., and Ribereau, P. (2015). The spectrogram: A threshold-based inferential tool for extremes of stochastic processes. Electronic Journal of Statistics, 9, 842–868. Özgen, I., Zhao, J., Liang, D., and Hinkelmann, R. (2016). Urban flood modeling using shallow water equations with depth-dependent anisotropic porosity. Journal of Hydrology, 541, 1165–1184. Padoan, S.A. (2013). Extreme dependence models based on event magnitude. J. Multivariate Anal., 122, 1–19. Palacios-Rodríguez, F., Toulemonde, G., Carreau, J., and Opitz, T. (2018). Space-time extreme processes simulation for flash floods in Mediterranean France. METMA 2018 - 9th Workshop on spatio-temporal modeling.

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4 Change-point Detection for Piecewise Deterministic Markov Processes

4.1. A quick introduction to stochastic control and change-point detection Stochastic control problems are the class of dynamic decision-making problems under uncertainty. The controller acts on a time-dependent random process to change its dynamics. Such actions can be applied continuously at all times or punctually at some controller-chosen dates. The former is called continuous control, the latter impulse control. Actions are selected by the controller in order to fulfill some objectives that typically correspond to minimizing or maximizing some criterion. For instance, the driver of a car acts continuously on the accelerator pedal and changes gear punctually in order to drive as fast as authorized despite uncertainty on the random behavior of other cars, traffic lights. The questions of interest regarding stochastic optimal control are many. They include, on the one hand, the study of the value function which is the best possible performance: how regular is it, can it be characterized as the unique solution of some equations, can it be approximated numerically, among others. On the other hand, we are interested in optimal strategies that can reach the optimum: do they exist, in which form, how do they depend on the problem parameters, how to compute them numerically and so on. The interested reader may consult, for example, Fleming and Soner (2006), Fleming and Rishel (1975) for a precise formulation of such problems and questions. Regarding applications of optimal control, they are also many and varied. We present here only two examples that correspond to recently defended theses and are Chapter written by Alice C LEYNEN and Benoîte DE S APORTA. Mathematical Modeling of Random and Deterministic Phenomena, First Edition. Edited by Solym Mawaki Manou-Abi, Sophie Dabo-Niang and Jean-Jacques Salone. © ISTE Ltd 2020. Published by ISTE Ltd and John Wiley & Sons, Inc.

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closely related to the present work. The first example was developed in the PhD thesis of Chloé Pasin (2018). It concerns a population of HIV patients treated with interleukin (IL) injections. The controller chooses the dates of injection, the number of injections and the dose injected in order to minimize the expected time spent by the patient below a prescribed threshold for the CD4+ T lymphocytes count. The sources of randomness in this context include a random response to injections and individual variability between patients. The second example was developed in the PhD thesis of Alizée Geeraert (2017). It concerns a maintenance optimization problem for a multi-component equipment subject to random deteriorations or failures. The controller chooses the dates of intervention to perform maintenance operations as well as the operations to be executed (change or repair) for each component. The objective is to minimize the expected unavailability and maintenance costs. Both these examples fall into the category of impulse control problems, and both PhD theses were focusing on a numerical approximation of the value function and/or optimal strategies. The underlying stochastic processes in both works are Piecewise Deterministic Markov Processes (PDMPs). PDMPs are a general class of non-diffusion processes introduced by M. Davis in the 1980s (Davis 1984) covering a wide range of applications from workshop optimization, queuing theory (Davis 1993), Internet networks (Bardet et al. 2013), reliability (de Saporta, Dufour and Zhang 2016), insurance and finance (Bäuerle and Rieder 2011) or biology (Doumic et al. 2015, Riedler and Thieullen 2015, Riedler et al. 2012) for instance. PDMPs are continuous time hybrid processes with a discrete component called the mode or regime and a Euclidean component. The process follows a deterministic trajectory punctuated by random jumps. In the special case where the Euclidean component is continuous, the jumps correspond to a change of regime. In the HIV example, the regime contains all the discrete parameters of the treatment, including the number of cycles and chosen dose and the Euclidean component is the CD4+ T lymphocytes count that jumps upwards at each injection and then decays exponentially fast. In the maintenance example, the regime is the state (nominal, degraded, failed) of each component and the Euclidean variable is simply the functioning time of each component. For many real-life applications, the regime and jump times are not observed and the Euclidean variable is measured in discrete-time and corrupted by noise. It may be, for example, a degradation or failure of some components of a system, see Baysse et al. (2014) where the Euclidean component is some cool-down time that increases with the degradation of the system, or the cancer cell load of remission patients monitored through proxy tumor markers at regular follow-up blood tests to detect relapse (Abbott and Michor 2006). Although the optimal control of PDMPs has attracted a lot of attention since the 1980s, see, for example, (Cohen et al. 2012, Costa and Dufour 2013, Davis 1993, Dempster and Ye 1995, Ga¸tarek 1992, Lenhart 1989), very few works consider such models under partial observations, which is not surprising as the non-observation of the jump times leads to highly

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technical difficulties. To the authors’ best knowledge, only two papers address his question. In (Brandejsky, de Saporta, and Dufour 2013), the authors consider an optimal stopping problem for PDMPs where the jump times are perfectly observed and the observation of the post-jump locations are corrupted by noise. They derive the dynamic programming equations of the problem, as well as a numerical approximation of the value function and a computable -optimal stopping time. In (Bäuerle and Lange 2017), the authors consider a general continuous control problem where the jump times are also observed and the observation of the post-jump locations are corrupted by noise. They reduce the problem to a discrete-time Markov Decision Process (MDP) and prove the existence of optimal policies, but provide no numerical approximation of the value function or optimal strategies. MDPs correspond to sequential decision-making problems in discrete time, see, for example, (Bäuerle and Rieder 2011, Hernández-Lerma and Lasserre 1996). The aim of the present work is to address the impulse control problem for PDMPs under partial observation. As the general problem is very difficult to tackle, we only deal with the easiest special case, namely, change-point detection, when only one change of regime can occur. The problem can thus be formulated as an optimal stopping problem for PDMPs under partial observations. However, unlike Bäuerle and Lange (2017), and Brandejsky, de Saporta, and Dufour (2013), we do not suppose that the observations are made at the jump times. Instead, we suppose that the observation times are deterministic and on a regular grid of step size δ. We thus have a discrete time collection of noisy observations of the continuous Euclidean variable and want to detect the change in regime as early as possible, as well as detect the new regime after the jump. The reference approach for change-point detection when the data are independent and identically distributed, with a different mean before and after the jump, is the moving average. The controller chooses a window length over which to compute the average of past observations and a threshold above which we can consider that the mean has changed, see, for example, (Basseville and Nikiforov 1993). This method is model-free and very easy to implement but it is not applicable in our context as the points of the underlying PDMP do not share the same distribution and may have very different means over time. A second approach found in the literature is segmentation, see, for example, (Cleynen 2013, section 1.2.2) for a review. Instead of making online decisions, we observe the whole sequence of data until the horizon and detect the change points a posteriori. This is relevant for many applications, such as locating genes on a DNA sequence, but it is not applicable in our context of on-line decision-making when the controller wants to detect the change as fast as possible in order to possibly act on the process.

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A third approach is filtering. At each observation time, we can compute the probability of being in each regime given the observations. In the special framework of linear dynamics, the Kalman filter is especially efficient as it can be updated recursively, see, for example, (Balakrishnan 1984) for an introduction. The controller then selects a threshold to detect the change and estimate the most probable new regime. If the model is not (approximately) linear or if the observation points are too far apart, the Kalman filter approach does not perform well, as will be illustrated in our simulation study in section 4.4. For a general PDMP, the filter is not computable, but can only be approximated, see section 4.3. To solve our generic problem of change-point detection for PDMPs, we choose a different approach. We formulate the problem as an optimal stopping problem, the simplest form of impulse control, for the discrete-time Markov chain of the sample points and their noisy observations. The model thus falls into the class of partially observed MDPs. The equivalent fully observed MDP for the filter process is still in discrete-time but on an infinite state space. We then propose a two-step discretization of this MDP, following an idea from de Saporta, Dufour and Nivot (2016). The first step is a time-dependent discretization of the state space of the original PDMP. The second step is a joint discretization of the thus obtained approximate filter, together with an approximation of the observation process. The main material of this chapter was originally published in (Cleynen 2018), where the reader will find the detailed statements and proofs of the main results. Compared to (Cleynen 2018), the structure of the present chapter is different. We tried to avoid technicalities as much as possible and to make the main results accessible to readers unfamiliar with optimal control. In addition, the present introduction to stochastic control and change-point detection, the examples mentioned therein and the simulation study are new. The chapter is organized as follows. In section 4.2, we introduce our continuoustime PDMP model as well as the observation model. We define the change-point detection problem as an optimal stopping problem under partial observations and give the equivalent fully observed dynamic programming equations for the filter process. In section 4.3, we propose a two-step discretization approach by quantization to numerically solve the optimization problem and build a tractable strategy. In section 4.4, we investigate the performance of our candidate strategy and compare our approach to moving average and Kalman filtering when possible. A conclusion and perspectives for further work are given in section 4.5. 4.2. Model and problem setting In this section, we present the special class of PDMPs we will focus on, define the observation process and state the change-point detection problem as

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an optimal stopping problem under partial observation. We then derive the filter recursive equation and state the equivalent fully observed optimal stopping problem as well as the corresponding dynamic programming equations on which we base our discretization. 4.2.1. Continuous-time PDMP model We consider the problem of detecting a change-point in the dynamic of a special class of PDMPs which is observed with noise at discrete observation times. The process Xt = (mt , xt , ut ) is defined on a state space E = M × K × R+ , where M = {0, 1, . . . , d} is the finite set of regimes or modes, K is a compact subset of R and ut is the time since the last jump. We will further denote X = M × K. For each mode m, the local characteristics of the PDMP are: – a flow Ψm : K × R2+ → K × R+ of the special form Ψm ((x, u), t) = (Φm (x, t), u + t); – a jump intensity λm : K × R+ → R+ such that λ0 (x, u) = λ(u) and λi (x, u) = 0, for all positive i in M; – a (sub)Markovian jump kernel Qm : (B(E), K × R+ ) → E such that Qm ({i} × A × {0}|x, u) = πi 1{0} (m)1A (x) with πi > 0 for all positive i ∈ M d and i=1 πi = 1. In other words, the PDMP has a single jump at some random time T and evolves deterministically before and after the jump. Before the jump, the trajectory follows the flow Φ0 (0, t) that corresponds to the solution at time t of an ordinary differential equation starting from x at time 0. After the jump, the trajectory randomly selects a new regime m ∈ {1, . . . , d} and follows a new flow Φm starting from position Φ0 (0, T ), as shown in Figure 4.1. The distribution of the jump time does not depend on the position xt but only on the running time: P(0,x,0) (T > t) = e−

´t 0

λ(s)ds

.

At the jump time, the location x remains unchanged, the time since the last jump u is set to 0 and a new mode is selected according to the distribution π. The third component of Xt , namely, the running time since the last jump, only intervenes in the jump time distribution. It is necessary to obtain a strong Markov process. As we will see in the sequel, solving the change-point detection problem is not straightforward, even for such simple dynamics.

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Figure 4.1. Typical trajectory of the PDMP (blue solid line) and observation process (red dots). For a color version of this figure, see www.iste.co.uk/manouabi/modeling.zip

We suppose that the observation times (tn )n∈N are deterministic and on a regular grid of fixed step size δ until a finite horizon N δ, and that a noisy observation of xtn is available at each time tn : Yn = F (Xtn ) + εn = F (xtn ) + εn ,

[4.1]

where F is a deterministic link function, (εn ) are i.i.d. real-valued random variables with density f with respect to the Lebesgue measure on R and independent of the process (Xt ). We further assume that Y takes its values in Y, which is a subset of R. 4.2.2. Optimal stopping problem under partial observations We are interested in detecting the jump-time T and the mode after the jump, based on the observations Yn . We choose to formulate this problem as an optimal stopping problem for a discrete-time Markov chain. However, in our framework, it is important to note that the underlying process is time-continuous, and in particular that the jump-time T may occur between observation dates. In this chapter, we will only allow detections at the observation times. Allowing detection between observation times is still an open problem. In the sequel, we will simply denote Xn = (mtn , xtn ). As our PDMP has only one jump, we can explicitly write the kernels Pn of the time-inhomogeneous discrete-time

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Markov chain (Xn ). For any Borel subsets A ⊂ M, B ⊂ K, any (m, x) ∈ X and n ≥ 0, we have Pn (A × B|m, x) = P(Xn+1 ∈ A × B|Xn = (m, x)) = 1{0} (m)1A (0)1B (Φ0 (x, δ))e− +

d 

ˆ πi 1{0} (m)1A (i)

i=1 ´ − 0s λ(nδ+z)dz

e +

d 

δ

´δ 0

λ(nδ+s)ds

λ(nδ + s)

[4.2]

0

1B (Φi (Φ0 (x, s), δ − s))ds

1{i} (m)1A (i)1B (Φi (x, δ)).

i=1

For any y  ∈ Y, let fFy be the function from X onto R defined by fFy : (m, x) → f (y  − F (x)). Thus, the kernels Rn of the Markov chain (Xn , Yn ) are, for any Borel subsets A ⊂ X, C ⊂ Y, any (m, x, y) ∈ X × Y and n ≥ 0 Rn (A × C|m, x, y) = P((Xn+1 , Yn+1 ) ∈ A × C|(Xn , Yn ) = (m, x, y)) ˆ Pn (fFy 1A )(m, x)dy  , [4.3] = C

Note that the Rn kernels do not depend on y. We can now state our change-point detection problem. For 0 ≤ n ≤ N , set Fn = σ(Xk , Yk , 0 ≤ k ≤ n) the σ-field generated by the Markov chain (Xn , Yn ) up to time n, and FnY = σ(Yk , 0 ≤ k ≤ n) the σ-field generated by the observations up to time n. Let T Y be the set of F Y -stopping times. We do not restrict ourselves to the stopping times bounded by N because it may be optimal not to stop at all until the horizon N is reached. A decision taken at the stopping-time τ ∈ T Y is a FτY measurable random variable A taking values in M+ = {1, 2, . . . , d} if τ ≤ N , equal to 0 if τ > N . Decision A = i corresponds to deciding Xτ is in mode i. Let AY τ be the set of admissible decisions at the stopping time τ . Until the stopping-time τ , the cost-per-stage function is denoted by c and the terminal cost (at stopping time τ ) when taking decision a is C, where c(i, x, y) = βi δ, C(m, x, y, 0) = c(m, x, y), C(m, x, y, a) = α1(m=0) + γ1(m=a;m>0) ,

for a ∈ M+ ,

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with β0 = 0 and for positive i, βi = β > 0. Thus, β represents the penalty for late detection, α the false alarm penalty and γ the cost of selecting a wrong mode. The cost of an admissible strategy (τ, Aτ ) ∈ T Y × AY τ for starting the point ξ ∈ X × Y is (τ −1)∧N   c(Xn , Yn )+1(τ ≤N ) C(Xτ ∧N , Yτ ∧N , Aτ ) , J(τ, Aτ , ξ) = Eξ n=0

and the value function of the problem is V (ξ) =

inf

(τ,Aτ )∈T Y ×AY τ

J(τ, Aτ , ξ).

The optimal (possibly not achievable) cost is 0 when the jump is detected at the first observation after its occurrence and the right post-jump mode is selected. The aim of this chapter is to derive a numerically tractable approximation of the value function V as well as propose a computable strategy close to optimality. 4.2.3. Fully observed optimal stopping problem The classical approach to deal with partial observations is to introduce the filter process and the corresponding completely observed optimal stopping problem for filtered trajectories. For any starting point ξ = (0, x, y) ∈ X×Y, set Θ0 = θ0 = δ(0,x) and for 1 ≤ n ≤ N , and any Borel subset A of X set Θn (A) = Pξ (Xn ∈ A|FnY ), the filter for the unobserved part of the process. The filter is recursively obtained as follows. P ROPOSITION 4.1.– For any n ≥ 0, conditionally on (Θn = θ, Yn+1 = y  ), we have Θn+1 = Ψn (θ, y  ) with ´ 

Pn (fFy 1A )(m, x)dθ(m, x) , P (fFy )(m, x)dθ(m, x) X n

Ψn (θ, y )(A) = X´

[4.4]

for any Borel subset A of X. The proof of this proposition is quite classical and therefore omitted. It relies on the standard prediction-correction approach. Similar computations can be found, for example, in (Bäuerle and Rieder 2011), in the framework of MDPs, with the notable difference that in our context we do not assume that the kernels have a density with respect to any fixed measure, or in (Brandejsky, de Saporta, and Dufour 2013), for a different class of PDMPs.

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Denote P(X) the set of probability measures on X. Thus, (Θn , Yn ) is a Markov chain on P(X) × Y, with transition kernels defined, for any Borel subsets P ⊂ P(X), C ⊂ Y, and any (θ, y) ∈ P(X) × Y, by R n (P × C|θ, y) = P((Θn+1 , Yn+1 ) ∈ P × C|(Θn , Yn ) = (θ, y)) ˆ 1P (Ψn (θ, y  )) Pn (fFy )(m, x)dθ(m, x)dy  . = X×C

Again, this kernel does not depend on y. The partially observed optimal stopping problem defined in section 4.2.2 is equivalent to a fully observed optimal stopping problem using the filtered trajectories introduced above. The fully observed state space is thus P(X) × Y, the initial point is ξ  = (δ(0,x) , y) for some (x, y) ∈ K × Y. In this framework, the cost of an admissible strategy (τ, Aτ ) ∈ T Y × AY τ starting from the point ξ ∈ X × Y is  (τ −1)∧N       J (τ, Aτ , ξ ) = Eξ c (Θn , Yn )+1(τ ≤N ) C (Θτ ∧N , Yτ ∧N , Aτ ) , n=0

where, for ´g from X × Y onto R g  is the function from P(X) × Y onto R such that g  (θ, y) = X g(m, x, y)dθ(m, x), here for g = c or g = C. The value function of the problem is V  (ξ  ) =

inf

(τ,Aτ )∈T Y ×AY τ

J  (τ, Aτ , ξ  ).

[4.5]

The value function is then solution of the dynamic programming equations.  (θ, y) = mina∈M C  (θ, y, a) and for 0 ≤ n ≤ N − 1 T HEOREM 4.1.– Set vN

vn (θ, y) = min



  min C  (θ, y, a); c (θ, y) + Rn vn+1 (θ, y) .

a∈M+

Let ξ0 = (δ(0,x) , y) ∈ P(X) × Y. Then we have v0 (ξ0 ) = V  (ξ0 ) = V (0, x, y). Note that none of the above functions actually depend on y. Again, the proof of this statement relies on standard arguments and is omitted. The proper framework for the proof is that of Partially Observed MDPs (POMDPs). We first define the equivalent POMDP to the optimal stopping problem under partial observation and then prove the equivalence with the fully observed MDP corresponding to the fully observed optimal stopping problem. The dynamic programming is then straightforward. Similar derivations can be found, for instance, in (Bäuerle and Rieder 2011, de Saporta, Dufour and Nivot 2016).

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The dynamic programming recursion cannot be exactly computed because integration against the kernels Rn is numerically intractable and involves the Bayes operators for the filter Ψn , which cannot be computed either. Instead, we build a numerically tractable approximated value function based on the dynamic programming recursion and some discretization of operators Rn and Ψn . 4.3. Numerical approximation of the value functions We now describe our construction of a numerically tractable approximation of the optimal value function V  defined in equation [4.5] and a corresponding candidate optimal strategy. The main difficulties are, first, that the filter Θn is measure-valued and thus infinite-dimensional and, second, that this filter cannot be simulated as the Bayes operators Ψn involve continuous integration. To build our approximation, we start from the dynamic programming equations from Theorem 4.1 and propose a two-step discretization of the operators Rn , 0 ≤ n ≤ N − 1. Our global approach and the relationships between the different Markov chains we introduce, together with their state space and kernels, are summarized in Figure 4.2. The left column corresponds to the construction presented in section 4.2 from the original continuous-time PDMP to the fully observed dynamic programming equations for the observations and filter process. The first step in the middle column corresponds to a time-dependent discretization of the state space of the Markov chain (Xn ). We obtain a finite state space Markov ¯ n ) that we plug into the observation equation [4.1] and filter operator to chain (X ¯ n , Y¯n ) with kernels R ¯ n and (Θ ¯ n , Y¯n ) with kernels R ¯ . obtain Markov chains (X n   ¯ Finally, we replace Rn with Rn in the dynamic programming equations to obtain the first sequence of approximate value functions. Note that by doing so, Y¯n does not ¯ n is not the filter of X ¯n correspond to a discretization of the observations Yn and Θ given the observations Yn . By this procedure, we start from a finite state space Markov ¯ n ) and obtain a simulatable filter Θ ¯ n that is still measure-valued but can be chain (X identified to finite-dimensional vectors. One more approximation is still required to obtain a finite state space Markov chain. The second step in the right column consists of the joint discretization of the ¯ n , Y¯n ). We obtain a finite state space Markov chain (Θ ˆ n , Yˆn ) with Markov chain (Θ ˆ  . Again, we plug this new kernel into the dynamic programming equations. kernel R n ¯ n , Y¯n ) has a finite state space, integrating with respect to R ˆ As the Markov chain (Θ n simply corresponds to computing weighted sums. Hence, the dynamic programming equations are now fully numerically solvable. This leads both to a numerically tractable approximation of the original value function and to a candidate -optimal strategy.

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Figure 4.2. Two-step approximation of the value functions. For a color version of this figure, see www.iste.co.uk/manouabi/modeling.zip

Both steps are based on discretization by optimal quantization of the vector-valued Markov chains. In this section, we briefly recall the optimal quantization procedure and its main properties, construct the two discretizations and state the convergence result of the approximate value functions to the original one. The details of the construction and the proofs can be found in (Cleynen 2018). 4.3.1. Quantization The main discretization tool we use is optimal quantization. The quantization of a  of X on random variable X consists of finding a finite grid such that the projection X  There exists extensive this grid minimizes some Lp norm of the difference X − X. literature on quantization methods for random variables and processes. The interested reader may, for instance, consult (Gray and Neuhoff 1998, Pagès et al. 2004b) and the references therein. Quantization methods have been developed recently in numerical probability or optimal stochastic control with applications in finance, see, for example (Bally and Pagès 2003, Bally, Pagès, and Printems 2005, Pagès 1998, Pagès et al. 2004b).

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We recall that for an Rq -valued random variable Z such that E[|Z|2 ] < ∞ and  a fixed integer, the optimal L2 -quantization of the random variable Z consists of finding the best possible L2 -approximation of Z by a random variable Z taking at most  values in Rq , which will be denoted by Γ = {z 1 , . . . , z  }. The asymptotic properties of the L2 -quantization are given by Zador’s theorem (see, for example, Bally, Pagès, and Printems (2005), Theorem 4.3), recalled below, which uses the notation pΓ (z) for the closest neighbor projection of z ∈ Rq on a grid Γ ⊂ Rq . T HEOREM 4.2.– Let Z be an Rq -valued random variable, and suppose that for some  > 0 we have E[|Z|2+ ] < + ∞. Then, as  tends to infinity, we have min E[|Z − pΓ (Z)|2 ] = O(−2/q ).

|Γ|≤

There exist algorithms that can numerically find, for a fixed , the quantization of Z (or, equivalently, the grid {z1 , . . . , z } attaining the minimum in Theorem 4.2 above and its distribution) as soon as Z is simulatable (Pagès 1998, Pagès et al. 2004b). Roughly speaking, such a grid will have more points in the areas of high density of Z and fewer points in the areas of low density of X. Replacing Z by Z turns integrals into finite sums and makes numerical computations possible, with easy derivation of error bounds for Lipschitz functionals of the random variable thanks to Theorem 4.2. Optimal quantization can also readily be extended to (discrete-time) Markov chains (Pagès et al. 2004a). We thus retrieve a quantization grid at each time step and the transition matrices between two consecutive grids. 4.3.2. Discretizations We propose a time-dependent discretization of the state space X based on the optimal quadratic quantization of the discrete-time Markov chain (Xn )n≥0 . Let Ωn , n ≥ 0 be a sequence of optimal quantization grids for (Xn )n≥0 . The cardinality of ¯n Ωn is denoted by NΩ and does not depend on n, and Ωn = {ωn1 , . . . , ωnNΩ }. Let X ¯ n = pΩ (Xn ), and set be the nearest-neighbor projection of Xn onto Ωn : X n j ¯ n+1 = ω j |X ¯ n = ωni ). P¯n (ωk+1 |ωni ) = p¯n,i,j = P(X n+1

As the mode component is already discrete, we will assume in the sequel that the ¯ n = (m, x projection preserves the mode, i.e. if Xn = (m, x), then X ¯). To define an approximation for the kernels Rn , we replace, in the definition of Rn , ¯ n ). Namely, we the quantities related to the Markov chain (Xn ) by those related to (X define:

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¯ n from Ωn × Y onto Ωn+1 × Y: – a family of Markov kernels R ¯ n (ω j , dy  |ωni ) ¯ n (ω j , dy  |ωni , y) = R R n+1 n+1 j = p¯n,i,j f (y  − F (ωn+1 ))dy  ,

¯ n from P(Ωn ) × R onto P(Ωn+1 ): – a family of operators Ψ NΩ j ¯k,i,j fFy (ωk+1 )θ(ωki ) j i=1 p  ¯ Ψn (θ, y )(ωn+1 ) = NΩ NΩ . j ¯k,i,j  fFy (ωk+1 )θ(ωki ) j  =1 i=1 p ¯ n From these two ingredients, we construct a new family of Markov kernels R from P(Ωn ) × Y onto P(Ωn+1 ) × Y by setting, for all Borel subsets P¯ ⊂ P(Ωn ) and C¯ ⊂ Y, ¯ y¯) = R ¯ ¯ θ, ¯ n (P¯ × C| ¯ θ) ¯ n (P¯ × C| R ˆ  j ¯ y¯ ))¯ ¯ i ). ¯ n (θ, 1P¯ (Ψ pn,i,j f (y  − F (ωn+1 ))dy  θ(ω = n ¯ 1≤i≤NΩC 1≤j≤NΩ

¯ n ), we can construct a Markov chain From the family of Markov kernels (R ¯ n , Y¯n )n≥0 by setting (Θ ¯ 0 = δ(0,x ) , Y¯0 = y¯) = 1 P(Θ 0 ¯ Y¯n = y¯) = R ¯ ¯ n = θ, ¯ n (P¯ × C| ¯ θ). ¯ n+1 , Y¯n+1 ) ∈ P¯ × C¯ |Θ P((Θ Note that – (Y¯n ) does not have the same dynamics as the original observations (Yn ), nor does it correspond to a function of these observations; ¯ n (ω i ) = P(X ¯ n = ω i |Y¯0 , . . . Y¯n ), which thus corresponds to the filter – we have Θ n n ¯ ¯ ¯ n ) given the original of (Xn ) given the (Yn ), but does not correspond to the filter of (X observations (Yn ), or to a function of the original filter (Θn ) nor of observations (Yn ); ¯ n is numerically – unlike the recursion for the filter Θn , the recursion for Θ ¯ n ) is simulatable. tractable, thus (Θ ¯ n is characterized by the random weights Θ ¯i = Θ ¯ n (ω i ), for The random filter Θ n n i = 1, . . . , NΩ and can be identified with a finite dimensional random vector valued in the NΩ -simplex in RNΩ of dimension NΩ − 1. This identification will be in force throughout this chapter.

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Finally, we define the main quantities of interest for this section, namely, the approximate value functions v¯n from P(Ωn ) × Y onto R as  ¯  ¯ y  , a) (θ, y ) = min C  (θ, v¯N a∈M

¯ y ) v¯n (θ,



= min

   ¯     ¯ ¯ ¯ min C (θ, y , a); c (θ, y ) + Rn v¯n+1 (θ, y ) . 

a∈M+

The value functions v¯n are not directly numerically computable as they involve ¯  on the continuous space P(Ωn ) × Y. A second integration by operators R n discretization is thus needed. To do so, we use optimal quantization again. Following ¯ n , Y¯n ). the generic direction given in (Pham et al. 2005), we discretize jointly (Θ ¯ ¯ ¯ n only Note that (Θn , Yn ) is easy to simulate because the recursive construction of Θ involves finite weighted sums. This approach would not have been possible on the chain (Θn , Yn ) as Θn cannot be simulated exactly. Our second discretization step thus now consists of replacing the Markov chain ˆ n , Yˆn ). By construction, ¯ n , Y¯n ) by its optimal quantization approximation (Θ (Θ ˆ n , Yˆn ) takes a finite number of values on a grid Γn of size NΓ : Γn = (Θ {γni }1≤i≤NΓ = {(πn1 , yn1 ), . . . , (πnNΓ , ynNΓ )}. We now set ˆ n+1 , Yˆn+1 ) = γ j |(Θ ˆ n , Yˆn ) = γni ). ˆ n (γ j |γni ) = P((Θ R n+1 n+1 With these new transition kernels, we define the approximate value functions vˆn from Γn onto R as  (γ) = min C  (γ, a), vˆN a∈M

vˆn (γ) = min



  ˆ n vˆn+1 min C  (γ, a); c (γ) + R (γ) .

a∈M+

These functions can be numerically computed on the grids Γn . T HEOREM 4.3.– Under technical assumptions, for any starting point (0, x0 , y0 ), we have −1/NΩ

|v0 (δ(0,x0 ) , y0 ) − vˆ0 (δ(0,x0 ) , y0 )| = O(NΩ−1 + NΓ

).

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This rate of convergence is very slow, which is not surprising given that we have to discretize infinite dimension measure-valued random variables. This is the well-known curse of dimensionality we are faced with when dealing with partial observations. 4.3.3. Construction of a stopping strategy We can now construct a computable stopping strategy using the fully discretized value function. Suppose that the process starts from point ξ0 = (0, x0 , y0 ) and observations y0 , . . . , yn are available at time n. We cannot compute the filter Pξ0 (Xn ∈ ·|(Y0 , . . . , Yn ) = (y0 , . . . , yn )) because of the continuous integrals in the definition of the Bayes operator from Proposition 4.1. However, we can recursively compute an approximate filter as follows: θ¯0 = δ(0,x0 ) ,

¯ k−1 (θ¯k−1 , yk ), 1 ≤ k ≤ n. θ¯k = Ψ

By construction, θ¯k belongs to P(Ωk ) for all k. This approximate filter can then be projected onto the quantization grids (Γk )0≤k≤n : (θˆk , yˆk ) = pΓk (θ¯k , yk ), for all 1 ≤ k ≤ n. Finally, the values of vˆk (θˆk , yˆk ) are available for all 0 ≤ k ≤ n. Now we define two sequences of function (rn )0≤n≤N and (an )0≤n≤N as – for 0 ≤ n ≤ N − 1, rn : P(Ωn ) × Y → {0, 1} and an : P(Ωn ) × Y → M+ are such that ¯ y¯) = 1 rn (θ,

¯ y ),a)0) . (rn (θ,¯ n a∈M+

– rN : P(ΩN ) × Y → {0, 1} and aN : P(ΩN ) × Y → M are such that ¯ y¯) = 1 rN (θ, (arg mina∈M C  (pΓ

N

¯ y ),a)>0) , (θ,¯

¯ y¯) = arg min C  (pΓ (θ, ¯ y¯), a). aN (θ, N a∈M

Thus, rn is a stopping indicator depending on which term won the minimization in the dynamic programming, and an corresponds to the mode to be selected after the jump.

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Our candidate stopping strategy is the following, as shown in Figure 4.3: Compute r0 (δ(0,x0 ) , y0 ) – if r0 (δ(0,x0 ) , y0 ) = 1, stop at time 0 and select decision a0 (δ(0,x0 ) , y0 ) ¯ 0 (δ(0,x ) , y1 ) and r1 (θ¯1 , y1 ) – otherwise compute θ¯1 = Ψ 0 - if r1 (θ¯1 , y1 ) = 1, stop at time 1 and select decision a1 (θ¯1 , y1 ) ¯ 1 (θ¯1 , y2 ) and r2 (θ¯2 , y2 ) and so on until time N - otherwise compute θ¯2 = Ψ ¯ N −1 (θ¯N −1 , yN ) - if the process was not stopped before N , compute θ¯N = Ψ and rN (θ¯N , yN ) - if rN (θ¯N , yN ) = 1, stop at time N and select decision aN (θ¯N , yN ) - otherwise select decision a = 0.

Figure 4.3. Computable stopping strategy

Note that all quantities can be computed numerically and that this strategy is nonanticipative. However, the sequence (θ¯n , yn ) is not a realization of the Markov chain ¯ n , Y¯n ), nor of (Θn , Yn ). Therefore, theoretically assessing the performance of this (Θ strategy is an open question that will be the subject of future works. Its numerical performance is assessed in section 4.4.

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4.4. Simulation study We assessed the performance of our candidate strategy on simulations for two different families of models: a linear model where the moving average and Kalman filter approaches are expected to perform well, and a more challenging non one-toone flow where none of the other approaches described in the introduction can be used. Recall that the best (possibly unfeasible) performance is 0. In all examples, the jump intensity is of the form λ(u) = u so that the probability to jump gets higher as time increases. The Markov kernel Q is the uniform distribution on the possible post-jump modes. The distribution of the noise is centered Gaussian with variance σ 2 truncated at [−s, s] for some s ∈ R. We investigate several forms for the flow. The link function between the process and the observations will be either F (x) = x or F (x) = x−1 . The performance of the approaches is evaluated through the average cost of the strategies over 1000 Monte Carlo simulations, for NΩ = 10 and NΓ = 30. We consider different combinations of values for the costs of stopping too early, stopping too late, or choosing the wrong mode, namely, α ∈ {3, 4, 5, 6} β ∈ {0.5, 1, 1.5, 2} γ ∈ {0.5, 1, 1.5, 2}. 4.4.1. Linear model In order to compare our method with other state of the moving average and Kalman filtering approaches, we first study exponential or linear trajectories. The process starts at X0 = (0, 1, 0). The flows Φm are defined as  Φ0 (x, t) = x, Φi (x, t) = evi t x, vi = 0; 1 ≤ i ≤ 3, so that Xn+1 = evi δ Xn and Yn+1 = Xn+1 + n+1 to fit the Kalman linear model. We choose the following parameters: v1 = 0.1, v2 = 0.5, v3 = 1, sample step δ = 1/6, and the starting point is x = 1. Sample trajectories for different levels of noise are illustrated in Figure 4.4. Clearly, the detection problem is more and more difficult as the level of noise increases. Table 4.1 presents the results for the link function F (x) = x for β = 1 (cost of stopping too late), γ = 1.5 (cost of selecting the wrong mode) and the four values of α (cost of an early detection), for the moving average (MA), Kalman filter (KA) and our approach. For MA, only the results for a threshold value of 2 are presented, as they are always better.

Mathematical Modeling of Random and Deterministic Phenomena

0

1

2

3

4

5

4 3 2 obs

6

1 0 −1

−1

−1

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2 4

0 4

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obs 3

−1 3

6

1 2

4

4 3 2 obs 1 0

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t

−1

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−1 0

6

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3

4 2 obs 1 0 −1 1

2

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4 3 2 obs 1 0 −1 0

1

t

0

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2

t

3

4

5

6

0

1

2

t

3 t

Figure 4.4. Sample trajectories for the linear model (solid blue line) and observations (red crosses) for and increasing level of noise from left to right (0.1, 0.5 and 1). The vertical lines correspond to the (unobserved) jump time to be detected. For a color version of this figure, see www.iste.co.uk/manouabi/modeling.zip α σ2 0.1 3 0.5 1 0.1 4 0.5 1 0.1 5 0.5 1 0.1 6 0.5 1

MA, s = 2 window 2 3 4 5 0.41 0.40 0.40 0.41 0.92 0.81 0.76 0.72 1.70 1.45 1.27 1.18 0.41 0.40 0.40 0.41 0.93 0.81 0.76 0.72 2.02 1.61 1.35 1.23 0.41 0.40 0.40 0.41 0.94 0.81 0.77 0.73 2.34 1.78 1.44 1.29 0.41 0.40 0.40 0.41 0.95 0.82 0.77 0.73 2.67 1.94 1.52 1.35

0.5 2.30 1.45 1.22 3.00 1.78 1.42 3.70 2.11 1.62 4.41 2.44 1.82

KF threshold 0.75 0.9 0.59 0.42 0.56 0.49 0.58 0.65 0.66 0.42 0.59 0.49 0.59 0.65 0.74 0.42 0.62 0.49 0.60 0.65 0.82 0.42 0.65 0.49 0.61 0.65

New approach cal 0.41 0.48 0.66 0.41 0.48 0.66 0.41 0.48 0.66 0.41 0.48 0.66

0.70 0.79 0.99 0.68 0.75 0.96 0.67 0.70 0.96 0.66 0.67 0.93

Table 4.1. Average cost of the strategies for MA, KF and the approach proposed in this work for different parameter values. Best scores (i.e. minimal strategy cost) by method are indicated in bold, while the overall best score is indicated in blue. For a color version of this table, see www.iste.co.uk/manouabi/modeling.zip

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When the level of noise is low, the moving average, which is also the simplest method, has the best performance. However, the global performance deteriorates dramatically when the noise increases. With higher levels of noise, the Kalman filtering approach performs best, which is not surprising, as the process follows a linear model. However, if we deviate from the linear model, for instance, by taking the inverse link function F (x) = 1/x for the observations, our approach has the best performance. The same result is observed for the linear link function when the observation points are far apart from each other, see Table 4.2. This is mainly due to the fact the underlying process is continuous and the jump may occur in between observations. Our approach takes this into account, but the Kalman model does not. link function MA KF New Approach F (x) = x 1.42 1.60

1.00

F (x) = 1/x 2.17 1.81

1.17

Table 4.2. Average cost of the strategies for MA, KF and the approach proposed in this work for α = 4, β = 1.5, γ = 0.5, σ 2 = 0.5 and δ = 1/3

4.4.2. Nonlinear model We study the more challenging example of non one-to-one flows, trying to detect a sudden change in the frequency of a sinusoidal trajectory. The process is initiated at X0 = (0, 0, 0, 0) and d = 1. The flows Φm are defined as 

Φ0 (x, u, t) = (sin(3π(u + t)), u + t), Φ1 (x, u, t) = (sin(5π(u + t), u + t).

A sample trajectory is given in Figure 4.5 with an observation step δ = 1/6 and noise σ 2 = 1. To the best of our knowledge, no other algorithm is adapted to change-point detection for such models. Table 4.3 shows the influence of the observation time steps δ and of the variance σ 2 in terms of time to jump-detection. While increasing δ significantly increases the amount of time required to detect the jump, it slightly decreases the number of observations needed after the jump for its detection. The number of early detection is more influenced by the cost parameters than by the time steps, with a strong tendency to increase with the observation noise. This, in fact, leads to strategy costs very close to zero for small time steps and variance values, with a tendency to increase with δ, σ 2 and to decrease with α.

3 2 1

obs

−2

−1

0

1 −2

−1

0

obs

2

3

4

Mathematical Modeling of Random and Deterministic Phenomena

4

92

0

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5

6

0

1

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3

time

4

5

6

time

Figure 4.5. Sample trajectories for the sine model. Left: unobserved trajectory (solid gray line), true sample points (yellow dots) and observations (blue crosses). Right: only the observation points for the same trajectory. For a color version of this figure, see www.iste.co.uk/manouabi/modeling.zip

α = 3, β = 2 δ

σ 2 ΔT

α = 6, β = 0.5

sd Nb Obs Nb early ΔT

sd Nb Obs Nb early

0.1 0.35 0.06

4

0

0.39 0.06

4

0

1/10 0.5 0.48 0.17

5

16

0.59 0.19

6

10

1.0 0.44 0.44

4

103

0.67 0.46

7

60

0.1 0.44 0.09

3

0

0.54 0.12

3

0

1/6 0.5 0.59 0.23

4

20

0.71 0.26

4

12

1.0 0.48 0.51

3

124

0.91 0.54

5

38

0.1 0.71 0.48

3

10

1.06 0.95

4

0

1/4 0.5 0.67 0.54

3

81

1.27 0.78

5

3

1.0 0.57 0.62

2

152

1.28 0.73

5

22

Table 4.3. Time to jump detection for different δ values. Average real-time and observation-time to detection for the sine model

4.5. Conclusion We have proposed a numerically feasible numerical scheme to approximate the value function of a change-point detection problem for a simple class of PDMPs. We obtain error bounds for this approximation explicitly depending on the parameters of the problem. We have also proposed a feasible stopping strategy that performs well compared to state-of-the-art methods when such methods are applicable, despite the very slow convergence rate. We believe that this is a promising start for the more general study of impulse control problems for general PDMPs when there are no observations of the jump times. The easiest extension is certainly going from scalar-valued PDMPs to multivariate ones. We believe that our proofs would hold in

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this context. Allowing more than one jump should be more challenging as we would not be able to write the explicit form of the kernels Pn . However, the underlying POMDP framework is suitable for several interventions, so it should be possible to extend our results in this direction, although probably technically involved. Another interesting research avenue is considering that the next observation date is also a decision to be taken by the controller. This would be highly relevant for medical applications. Finally, the important open question concerns the optimality of our candidate strategy. It cannot be directly linked to our various operators, but we are hopeful that further work will enable us to prove theoretically that it is close to optimality. 4.6. References Abbott, L.H. and Michor, M. (2006). Mathematical models of targeted cancer therapy. British Journal of Cancer, 95, 1136–1141. Balakrishnan, A.V. (1984). Kalman Filtering Theory. University Series in Modern Engineering. Optimization Software, Inc., Publications Division, New York. Bally, V. and Pagès, G. (2003). A quantization algorithm for solving multidimensional discrete-time optimal stopping problems. Bernoulli, 9(6), 1003–1049. Bally, V., Pagès, G., and Printems, J. (2005). A quantization tree method for pricing and hedging multidimensional American options. Math. Finance, 15(1), 119–168. Bardet, J.B., Christen, A., Guillin, A., Malrieu, F., and Zitt, P.A. (2013). Total variation estimates for the TCP process. Electron. J. Probab., 18(10), 21. Basseville, M. and Nikiforov, I.V. (1993). Detection of abrupt changes: Theory and application. Prentice Hall, Englewood Cliffs. Bäuerle, N. and Lange, D. (2017). Optimal control of partially observable piecewise deterministic Markov processes. SIAM Journal on Control and Optimization, 56(2), 1441–1462. Bäuerle, N. and Rieder, U. (2011). Markov Decision Processes with Applications to Finance. Springer, Heidelberg. Baysse, C., Bihannic, D., Gégout-Petit, A., Prenat, M., and Saracco, J. (2014). Hidden Markov model for the detection of a degraded operating mode of optronic equipment. J. SFdS, 155(3), 48–61. Brandejsky, A., de Saporta, B., and Dufour, F. (2013). Optimal stopping for partially observed piecewise-deterministic Markov processes. Stochastic Process. Appl., 123(8), 3201–3238.

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Cleynen, A. (2013). Approches statistiques en segmentation : Application à la ré-annotation de génome. PhD thesis, Paris 11. Cleynen, A. and de Saporta, B. (2018). Change-point detection for piecewise deterministic Markov processes. Automatica J. IFAC, 97, 234–247. Cohen, S., Madan, D., Siu, T., and Yang, H. (eds) (2012). Stochastic Processes, Finance and Control, A Fest schrift in honor of Robert J. Elliott, World Scientific Publishing Company Ltd., Hackensack. Costa, O. and Dufour, F. (2013). Continuous Average Control of Piecewise Deterministic Markov Processes. Springer, New York. Davis, M. (1984). Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models. J. Roy. Statist. Soc. Ser. B, 46(3), 353–388. Davis, M. (1993). Markov Models and Optimization. Chapman & Hall, London. Dempster, M. and Ye, J. (1995). Impulse control of piecewise deterministic Markov processes. Ann. Appl. Probab., 5(2), 399–423. Doumic, M., Hoffmann, M., Krell, N., and Robert, L. (2015). Statistical estimation of a growth-fragmentation model observed on a genealogical tree. Bernoulli. 21(3), 1760–1799. Available at: http://dx.doi.org/10.3150/14-BEJ623. Fleming, W. and Rishel, R. (1975). Deterministic and Stochastic Optimal Control. Springer-Verlag, New York. Fleming, W. and Soner, H. (2006). Controlled Markov Processes and Viscosity Solutions, 2nd edition. Springer, New York. Ga¸tarek, D. (1992). Optimality conditions for impulsive control of piecewisedeterministic processes. Math. Control Signals Systems, 5(2), 217–232. Geeraert, A. (2017). Contrôle optimal stochastique des processus de Markov déterministes par morceaux et application à l’optimisation de maintenance. PhD thesis, Université de Bordeaux. Gray, R. and Neuhoff, D. (1998). Quantization. IEEE Trans. Inform. Theory, 44(6), 2325–2383. Hernández-Lerma, O. and Lasserre, J. (1996). Discrete-time Markov Control Processes. Springer-Verlag, New York. Lenhart, S. (1989). Viscosity solutions associated with impulse control problems for piecewise-deterministic processes. Internat. J. Math. Math. Sci., 12(1), 145–157. Pagès, G. (1998). A space quantization method for numerical integration. J. Comput. Appl. Math., 89(1), 1–38.

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Pagès, G., Pham, H., and Printems, J. (2004a). An optimal Markovian quantization algorithm for multi-dimensional stochastic control problems. Stoch. Dyn.. 4(4), 501–545. Pagès, G., Pham, H., and Printems, J. (2004b). Optimal quantization methods and applications to numerical problems in finance. In Handbook of Computational and Numerical Methods in Finance, Rachev, S.T. (ed.), 253–297, Birkhäuser Boston. Pasin, C. (2018). Modélisation et optimisation de la réponse à des vaccins et à des interventions immunothérapeutiques : Application au virus Ebola et au VIH. PhD thesis, Université de Bordeaux. Pham, H., Runggaldier, W., and Sellami, A. (2005). Approximation by quantization of the filter process and applications to optimal stopping problems under partial observation. Monte Carlo Methods Appl., 11(1), 57–81. Riedler, M. and Thieullen, M. (2015). Spatio-temporal hybrid (PDMP) models: Central limit theorem and Langevin approximation for global fluctuations. Application to electrophysiology. Bernoulli, 21(2), 647–696. Available at: http://dx.doi.org/10.3150/13-BEJ583. Riedler, M., Thieullen, M., and Wainrib, G. (2012). Limit theorems for infinitedimensional piecewise deterministic Markov processes. Applications to stochastic excitable membrane models. Electron. J. Probab., 17,(55), 48. Available at: http://dx.doi.org/10.1214/EJP.v17-1946. de Saporta, B., Dufour, F., and Nivot, C. (2016). Partially observed optimal stopping problem for discrete-time Markov processes. 4OR. Available at: https://doi.org/10.1007/s10288-016-0337-8. de Saporta, B., Dufour, F., and Zhang, H. (2016). Numerical Methods for Simulation and Optimization of Piecewise Deterministic Markov Processes. ISTE Ltd, and John Wiley & Sons, New York.

5 Optimal Control of Advection–Diffusion Problems for Cropping Systems with an Unknown Nutrient Service Plant Source

5.1. Introduction In this chapter, we present a mathematical analysis and optimal control for the Nye–Tinker–Barber (NTB) model, governed by an advection–diffusion system. The model describes the nutrient transfer mechanism in cropping systems. Here, we study the case of additional nutrient quantity by the mean of a control function which represents the nutrients coming from a “service plant”. The term service plant is given for plant species implanted close to principal plants, with the aim of providing a service to the cropping system, as they contribute to the nutrition without using chemicals. Farmers have been using chemicals to feed plants for a long time but new cropping techniques are used in order to replace chemical nutrients using natural processes, as soils are suffering from chemicals, pesticides and insecticides as well. As an example, we refer to chlordecone (CLD), which is an organochlorine insecticide used against the banana weevil. Many reports highlighted the contamination of freshwater and terrestrial fauna by CLD (see, for example Kermarrec 1980 or Snegaroff 1977). Several years after the spread of CLD began the soils in many banana fields in Caribbean islands have a CLD content, higher than average (see (Cabidoche et al. 2009). As a result, service plants are used. Nutrients are then produced naturally, and are present in the soil at various levels of concentration. This is the case for nitrogen fixation and transfer in mixed cropping systems (see Jalonen 2009). For example, the Chapter written by Loïc L OUISON and Abdennebi O MRANE. Mathematical Modeling of Random and Deterministic Phenomena, First Edition. Edited by Solym Mawaki Manou-Abi, Sophie Dabo-Niang and Jean-Jacques Salone. © ISTE Ltd 2020. Published by ISTE Ltd and John Wiley & Sons, Inc.

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canavalia service plant implanted which is a legume, is, close to the banana (principal) plant so as to provide nitrogen. It is well-known that the banana is not able to fix the mineral nitrogen. Plant growth is strongly linked to the amount of soil nutrients absorbed from its roots. Modeling is required to understand the nutrient dynamics and transfer, and to test scenarios for the optimization of crop nutrition. A lot of plant nutrition models emerged in the 1960s. Among them, we refer to the work of Nye (1969) or Itoh (1983), who suggested a general framework of the model of nutrient uptake using the term, ‘source-sink’, modeling either the increase or the decrease in solute concentration w.r.t. time and space. The root nutrient uptake, as well as the solute movement in the soil, are well explained by Tinker (2000). They describe the nutrient motion processes from a biological and chemical point of view, using a partial differential equation (PDE) known as the Nye–Tinker–Barber (NTB) model. Recently, Roose (2001) used the NTB system in order to reflect in a more accurate way the morphology of the root system (modeling of root growth, root hair, mycorrhizae, etc.) and the spatio-temporal dynamics of the solute in the soil. We also mention the work of Ptashnyk (2010), where the author studied a process of nutrient uptake by a single root branch using the asymptotic expansion method. More recently, Louison (2015) studied the optimal control for the NTB model to determine the optimal amount of required nutrients for plant growth. In this chapter, we focus on the optimal control of the NTB system, where we introduce a perturbation as an unknown source of nutrient, present in the rhizosphere. We then obtain a problem of incomplete data that we study from the optimal control point of view. Here, the results presented generalize the work in Louison (2015) to the low-regret control. We show that the so-called low-regret (or least-regret) control by Lions (1992) (see also Lions 1994) fits on the control of this problem. The notion of low-regret optimal control of Lions is applied to PDE systems where there are controls and unknown perturbations. We then look for the control not making things worse with respect to a nominal control w0 (or to doing nothing, w0 = 0 in our case), independently of the perturbations which may be of infinite number. Lions used the notion of the low-regret control in application to the control of systems with perturbations or having missing data. Here we extend the notion to nutrient uptake problems of advection–diffusion type, where the nutrient concentration present in the rhizosphere (interaction zone close to the roots) is not well known.

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We give the precise optimality system for the low-regret control for the NTB problem using quadratic perturbations, Nakoulima (2000) (see also Nakoulima 2003). In each instance, we give the characterization of the low-regret control by means of singular optimality systems. The chapter is organized as follows. We first present the model of nutrient uptake and provide a proof of the existence and positivity of a solution. Then we study the optimal control question (existence, uniqueness, etc.) and adapt it to our incomplete data case. We finally give a characterization of the low-regret optimal control by a singular optimality system (SOS). 5.2. Statement of the problem Let Ω ⊂ Rd , 1 ≤ d ≤ 3 be the part of the soil close to the root surface called the rhizosphere. We suppose that Ω is of regular boundary Γ = Γ1 ∪ Γ2 of class C ∞ with Γ1 ∩ Γ2 = ∅. Here, Γ1 represents the root surface and Γ2 plays the role of the rhizosphere frontier close to another plant root system or the frontier with the rest of the soil. During a time t ∈ [0, T [, the transport-diffusion and uptake of nutrients by roots is governed by the following Nye–Tinker–Barber (NTB) system: ⎧ ∂c ⎪ ⎪ α + q · ∇c − DΔc = ⎪ ⎪ ⎪ ∂t ⎪ ⎪ div q = ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(D∇c − 12 qc).n (D∇c − 12 qc).n c(0, x)

 c

0 Ic = K = 0 = c0 (x)

in Q :=]0, T [×Ω, in Q, on Σ1 :=]0, T [×Γ1 ,

[5.1]

on Σ2 :=]0, T [×Γ2 , in Ω,

where c = c(t, x) is the concentration of nutrient density at time t in the position x, the function  c(t, x) represents the unknown source of nutrients in the rhizosphere, which is absorbed by the main plant and which is already present in the soil. This source may also be provided by humans or by secondary companion as fungi in other situations. The coefficient α = b + θ is a constant, b represents the buffer power and θ is the liquid saturation. The vector q = q(t, x) represents the Darcy flux, and D is the diffusion coefficient (a positive constant). The function h(c) = Ic/K is the Michaelis–Menten uptake function, it represents the inflow nutrient density at the root surface. We suppose that there is no nutrient on the second rhizosphere frontier Γ2 , or entering from it. Here, I and K are respectively the maximum uptake and Michaelis– Menten constants. R EMARK 5.1.– The mathematical analysis is understood in the classical infinite dimensional spaces: Lp spaces (L∞ , L2 ..) and Sobolev spaces as H 1 or the spaces of continuous functions C, endowed with their related norms.

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R EMARK 5.2.– The Michaelis–Menten uptake function h(c) = Ic/K has a nonlinear Ic version h(c) = . The nonlinear case has been studied by Louison (2015). K +c 5.2.1. Existence of a solution to the NTB uptake system This part is devoted to the weak formulation to problem [5.1] and to the existence of a unique non-negative solution c(t, x). It is well known that there is not a regular solution in general for such problems. We then have to consider weak solutions. We will consider the function t → c(t; .) ∈ H 1 (Ω). Finally, we suppose that  c= c(t, x) ∈ L2 ((0, T )×Ω) (= L2 (0, T ; L2 (Ω))). Then, it is easy to see that we have the following result: L EMMA 5.1.– The weak formulation of the problem [5.1] is as follows: given c0 ∈ L2 (Ω), find c : t ∈ [0, T ] → c(t; .) ∈ H 1 (Ω) such that: ⎧ ⎨

ˆ ˆ d c(t) ψ dx + a(t; c, ψ)=  c(t; .)ψ dx dt Ω Ω ⎩ c(0, x) = c (x), 0 α

[5.2]

a.e. t ∈]0, T [ and for all ψ ∈ H 1 (Ω), where 1 a(t; c, ψ) = 2

ˆ

ˆ Ω

q. (ψ∇c − c∇ψ) dx + D

Ω

ˆ ∇c ∇ψ dx −

Γ1

Ic ψ dx. [5.3] K

P ROPOSITION 5.1 (Existence).– Suppose that the flux q is uniformly bounded q ∈ (L∞ ∩ C 1 (Q))d . Then, there exists a unique solution c ∈ L2 (]0, T [; H 1 (Ω)) ∩ C(]0, T [; L2 (Ω)) to the problem [5.2]–[5.3]. P ROOF.– First, we show that the bilinear form a is continuous. We have by the Cauchy–Schwarz inequality: |a(t; c, ψ)| ≤ (q∞ + D)cH 1 (Ω) ψH 1 (Ω) + cL2 (Γ1 ) ψL2 (Γ1 ) where q∞ = max qi L∞ (Q) . Then, thanks to the trace theorem on H 1 (Ω), there 1≤i≤d

is a constant β = β(Ω) such that ϕL2 (Γ1 ) ≤ βϕH 1 (Ω) , ∀ϕ ∈ H 1 (Ω). Hence, |a(t; c, ψ)| ≤ CcH 1 (Ω) ψH 1 (Ω) where C = q∞ + D + β 2 , which implies that a is continuous.

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Now we show that a is semi-coercive. We have: ˆ ˆ I 2 a(t; c, c) = D |∇c|2 dx − |c| dx ≥ D∇c2L2 (Ω) − c2L2 (Γ1 ) Ω Γ1 K ≥ D∇c2L2 (Ω) − β 2 c2H 1 (Ω) . Introducing the term Dc2L2 (Ω) on the right-hand side of the above inequality, we obtain the Gårding’s inequality:   a(t; c, c) ≥ D − β 2 c2H 1 (Ω) − Dc2L2 (Ω) ,

∀ c ∈ H 1 (Ω).

[5.4]

Hence, the bilinear form a satisfies the conditions of Lions’s Theorem, when D > β 2 . Consequently, [5.1] has a unique weak solution c ∈ L2 (]0, T [; H 1 (Ω)) ∩ C(]0, T [; L2 (Ω)). Moreover, we have   ∂c ∈ L2 (0, T ; H 1 (Ω) ). ∂t The Lions theorem is well known. We can also see Brézis (1984) or Lions and Magenes (1970).

R EMARK 5.3.– The assumption on the flux q is a classical mathematical hypothesis and not a physical one. It allows us to avoid the difficulty in the estimation of the convective term. L EMMA 5.2 (Positivity).– Let c be the solution to the NTB system. Suppose that c0 ≥ 0 and c|Σ ≥ 0 and that  c ≥ 0, then c is non-negative and we have: 1

c(T, .) ≥ 0, ∀ T > 0. P ROOF.– As usual, we decompose the solution as c = c+ − c− , where c+ and c− are the classical non-negative parts of c. We will show that c− = 0. We multiply the NTB system by c− and we integrate by parts over Q. We have for each integral: ˆ −α

T

0

ˆ

∂c− − α )c dxdt = − ( 2 Ω ∂t

ˆ Ω

|c− (T )|2 − |c− (0)|2 dx ,

and, div q = 0 in Ω, ˆ − Q

(q.∇c− )c− dxdt = −

ˆ 0

T

1 2

ˆ Γ

(q.c− )c− .n dx dt,

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and,

ˆ D



 Δc− c− dxdt = D

Q

ˆ ˆ  

 − 2 ∇c− c− .n dxdt − D ∇c  dxdt. Σ

Q

We sum up the three integrals, we find: ˆ

ˆ I α − 2 − 2 − |c (T )| − |c (0)| dx + |c− |2 .n dxdt 2 Ω Σ1 K ˆ =D |∇c− |2 dxdt ≥ 0. Q

Now since c− |Σ = 0, we obtain 1

c− (T, .)L2 (Ω) ≤ c− (0, .)L2 (Ω) = 0, ∀ T ≥ 0.

5.3. Optimal control for the NTB problem with an unknown source Here we study the optimal control for the NTB system [5.1]. Nutrients enter by exudates through Γ2 . It is there where we put a control function. We rewrite the system [5.1] using simpler notations: ⎧ ∂c ⎪ ⎪ c in Q :=]0, T [×Ω, ⎪ α + q · ∇c − DΔc =  ⎪ ∂t ⎪ ⎪ ⎪ div q = 0 in Q, ⎪ ⎨ 1 I [5.5] (D∇c − qc).n = c on Σ1 =]0, T [×Γ1 , ⎪ 2 K ⎪ ⎪ 1 ⎪ ⎪ (D∇c − qc).n = −v on Σ2 =]0, T [×Γ2 , ⎪ ⎪ ⎪ 2 ⎩ c(0, x) = 0 in Ω, where we choose c0 (x) = 0 for simplicity, and where v is a positive control function depending on t and x. It corresponds to the addition of nutrients in the soil through the frontier Γ2 . The rhizosphere contains an unknown source  c= c(t, x). The solution is then denoted by: c := (t, x; v,  c) = c(v,  c). The goal of this section is to give a characterization of the control which minimizes the cost function: J(v,  c) = c(v,  c) − z¯2L2 (Σ1 ) + N v2L2 (Σ2 ) , where z ∈ L2 (Σ1 ) is the observation, and where N > 0 is a positive constant.

[5.6]

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103

We study the optimal control to the NTB system with an unknown nutrient source [5.1] or equivalently [5.5]. The natural minimization problem is: inf 2



v∈L (Σ2 )

sup J(v,  c)



c∈L2 (Q)

c) = +∞ (this can be easily sup J(v,  c∈L2 (Q) verified as we have vectorial spaces). We then use the low-regret concept of Lions, which is applied to systems where there are controls and unknown perturbations. The notion of low-regret control was introduced for PDE’s by Lions (1992).

but this problem has no solution since

The low regret control is to solve the minmax problem: inf 2

v∈L (Σ2 )



sup

c∈L2 (Q)



J(v,  c) − J(0,  c) − γ c2L2 (Q) .

[5.7]

We then look for the control(s) not making things worse with respect to a nominal control v = 0, which corresponds to the case where no control acts on the system, but not too bad, up to a small real positive term γ c2L2 (Q) that we can choose with γ β 2 . Since c(v, 0) ∈ H 1 (Ω) and thanks to the trace theorem, the linear form L is continuous on H 1 (Ω). Consequently, [5.8] has a unique weak solution ξ ∈ L2 (]0, T [; H 1 (Ω)) ∩ C(]0, T [; L2 (Ω)). P ROPOSITION 5.3.– Let c(0,  c) be the nutrient concentration when v = 0 and c(v, 0) the concentration when  c = 0. Then we have: J(v,  c) − J(0,  c) = J(v, 0) − J(0, 0) + 2ξ(v),  cL2 (Q) ,

[5.12]

where ξ = ξ(v) is the solution to the adjoint problem [5.8] and where < ., . > is the habitual inner product in the L2 space. P ROOF.– First, from the linearity of problem [5.5], we obtain: J(v,  c) − J(0,  c) = J(v, 0) − J(0, 0) + 2c(v, 0), c(0,  c)L2 (Σ1 ) .

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In particular, notice that we have the relation c(v, c˜) = c(v, 0) + c(0, c˜). We now have to estimate c(v, 0), c(0,  c)L2 (Σ1 ) . We multiply the first equation in [5.8] by c(0,  c), solution of the problem [5.5] (with v = 0), and we integrate by parts on Q. We then have: ˆ

−α

Q

∂ξ

(v) c(0,  c) dxdt = ∂t

ˆ

∂c α (0,  c) ξ(v) dxdt + 0 ∂t Q

as ξ(t = T ) = c(t = 0) = 0. We also have: ˆ ˆ

 

q.∇ξ(v) c(0,  c) dxdt = − div q ξ(v) c(0,  c) dxdt + 0, − Q

Q

thanks to the Gauss condition. So that from

 

  div q ξ(v) c(0,  c) = div q ξ(v) c(0,  c) + qξ(v) .∇c(0,  c), we obtain ˆ ˆ



− q.∇ξ(v) c(0,  c) dxdt = ξ(v) q.∇c(0,  c) dxdt Q

Q

ˆ

− qξ(v)c(0,  c) .n dσdt. Σ

Now the Green formula for the Laplacian gives: ˆ ˆ



DΔξ(v) c(0,  c) dxdt = − D∇ξ(v) c(0,  c).n dσdt − Σ

Q

ˆ ˆ



D∇c(0,  c) ξ(v).n dσdt − DΔc(0,  c) ξ(v) dxdt. + Σ

Q

We sum up using the boundary conditions in [5.5] and [5.8], we obtain:

ˆ ∂ξ −α 0= − q.∇ξ − DΔξ c(0,  c) dxdt ∂t Q

ˆ ∂ ξ(v) α + q · ∇ − DΔ c(0,  c) dxdt = ∂t Q

[5.13]

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ˆ

1 D∇ξ(v) + qξ(v) c(0,  c).n dσdt 2 Σ ˆ

1 D∇c(0,  c) − qc(0,  c) ξ(v).n dσdt + 2 Σ ˆ ˆ

I c(v, 0) + ξ(v) c(0,  ξ(v)  c dxdt − c).n dσ1 dt = K Q Σ1 ˆ

I + c(0,  c) ξ(v).n dσ1 dt Σ1 K ˆ ˆ = ξ(v)  c dxdt − c(v, 0)c(0,  c).n dσ1 dt. −

Σ1

Q

That is: cL2 (Q) . c(v, 0), c(0,  c)L2 (Σ1 ) = ξ(v),  For more details, see the work by ?. T HEOREM 5.1.– The problem [5.7] is equivalent to the classical optimal control problem: inf

v∈L2 (Σ2 )

J γ (v)

[5.14]

where J γ (v) = J(v, 0) − J(0, 0) +

1 ξ(v)2L2 (Q) . γ

[5.15]

Moreover, the minimization problem [5.14]–[5.15] admits a unique solution uγ called the low-regret control. P ROOF.– Indeed, we have from above: J(v,  c) − J(0,  c) = J(v, 0) − J(0, 0) + 2ξ(v),  cL2 (Q) .

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Hence, sup

c∈L2 (Q) +



J(v,  c) − J(0,  c) − γ c2L2 (Q) = J(v, 0) − J(0, 0)

sup

c∈L2 (Q)



2ξ(v),  cL2 (Q) − γ c2L2 (Q) .

But, using the conjugate property, we have:

1 2ξ(v),  cL2 (Q) − γ c2L2 (Q) = ξ(v)2L2 (Q) . γ c∈L2 (Q) sup

Hence, the low-regret control – if it exists – satisfies to the classical optimal control problem:  

2 inf sup J(v,  c) − J(0,  c) − γ cL2 (Q) = inf J γ (v) v∈L2 (Σ2 ) v∈L2 (Σ2 ) 2 c∈L (Q) where J γ (v) is given by [5.15]. Now we prove the existence and uniqueness for a low-regret optimal control for the NTB system, with an unknown nutrient source term. The cost function J γ (v) satisfies J γ (v) ≥ −J(0, 0), for any v ∈ L2 (Σ2 ). Therefore, it exists kγ = inf J γ (v). 2 v∈L (Σ2 )

We consider a minimizing sequence {vn (γ)} = {vn }. Then, it converges to kγ (which is independent of n). We obtain −J(0, 0) ≤ J γ (vn ) ≤ kγ + 1, so that we have  vn L2 (Σ2 ) ≤

kγ + 1 + ˜ c2L2 (Σ N

1)

= Cγ .

Hence, there is a subsequence that we can still denote by {vn }, which converges weakly to a low-regret control function uγ ∈ L2 (Σ2 ). The uniqueness of the low-regret control uγ is obvious and thanks to the strict convexity of the cost function J γ .

5.4. Characterization of the low-regret control for the NTB system In this section, we give a characterization of the low-regret optimal control by a Singular Optimality System (SOS) (the problem is singular due to the missing terms).

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T HEOREM 5.2.– The low-regret control uγ solution to [5.14]–[5.15] is characterized by the unique quadruplet {cγ , ργ , ξγ , pγ } solution to the optimality system: ⎧ ∂cγ ⎪ α + q · ∇cγ − DΔcγ = 0, ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎪ I ⎪ cγ , (D∇cγ − 12 qcγ ).n = K ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (D∇cγ − 12 qcγ ).n = −uγ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ cγ (0) = 0, and ⎪ ⎪ 1 ∂ρ ⎪ ⎪ α γ + q · ∇ργ − DΔργ = ξγ , ⎪ ⎪ ∂t γ ⎪ ⎪ ⎪ ⎪ I ⎪ ργ , (D∇ργ − 12 qργ ).n = K ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (D∇ργ − 12 qργ ).n = 0, ⎪ ⎪ ⎩ ργ (0) = 0,

∂ξγ − q.∇ξγ − DΔξγ = 0, ∂t 1 I (−D∇ξγ − qξγ ).n = −cγ − ξγ , 2 K 1 (−D∇ξγ − qξγ ).n = 0, 2 ξγ (T ) = 0,

−α

∂pγ − q.∇pγ − DΔpγ = 0 ∂t 1 I (−D∇pγ − qpγ ).n = cγ − z + ργ − pγ 2 K 1 (−D∇pγ − qpγ ).n = 0 2 pγ (T ) = 0 −α

in Q, on Σ1 , on Σ2 , in Ω, in Q, on Σ1 , on Σ2 , in Ω,

with div q = 0 and with the adjoint equation: pγ + N uγ = 0

in

L2 (Σ2 ),

[5.16]

where cγ = c(uγ , 0), ξγ = ξ(uγ , 0), ργ = ρ(uγ , 0) and pγ = p(uγ , 0). P ROOF.– Indeed, the low-regret control uγ satisfies to the Euler–Lagrange formula: lim

λ→0

J γ (uγ + λw) − J γ (uγ ) λ

= 0,

∀ w ∈ L2 (Σ2 ).

Then we have: J γ (uγ + λw) − J γ (uγ ) = J(uγ + λw, 0) − J(uγ , 0) − +

1 ξ(uγ )2L2 (Q) γ

1 ξ(uγ + λw)2L2 (Q) γ

= 2λc(uγ , 0) − z, c(w, 0)L2 (Σ1 ) + λ2 c(w, 0)2L2 (Σ1 ) + 2λN uγ , wL2 (Σ2 ) + λ2 N w2L2 (Σ2 ) +

λ2 ξ(w)2L2 (Q) γ

λ + 2 ξ(uγ ), ξ(w)L2 (Q) . γ

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109

When λ → 0, we obtain 1 c(uγ , 0) − z, c(w, 0)L2 (Σ1 ) + N uγ , wL2 (Σ2 ) +  ξ(uγ ), ξ(w)L2 (Q) = 0. [5.17] γ We develop the term  γ1 ξ(uγ ), ξ(w)L2 (Q) as the following. We define ργ = ρ(uγ , 0) solution of the problem: ⎧ ∂ργ ⎪ ⎪ α + q · ∇ργ − DΔργ ⎪ ⎨ ∂t (D∇ργ − 12 qργ ).n 1 ⎪ ⎪ ⎪ (D∇ργ − 2 qργ ).n ⎩ ργ (0)

= = = =

1 γ ξγ I K ργ

0 0

in Q, on Σ1 , on Σ2 , in Ω,

[5.18]

where ξγ = ξ(uγ , 0). Then, ∂ργ + q · ∇ργ − DΔργ , ξ(w)L2 (Q) ∂t ˆ

∂ α ργ + q.∇ργ − DΔργ ξ(w) dxdt = ∂t Q ˆ

∂ α ξ(w) + q.∇ξ(w) + DΔξ(w) ργ dxdt =− ∂t Q ˆ

1 D∇ργ − qργ ξ(w).n dσ1 dt − 2 Σ1 ˆ

1 + D∇ξ(w) + qξ(w) ργ .n dσ1 dt 2 Σ1

α

= c(w, 0), ργ L2 (Σ1 ) . ∂ργ 1 + q · ∇ργ − DΔργ , ξ(w)L2 (Q) =  ξ(uγ ), ξ(w)L2 (Q) = ∂t γ c(w, 0), ργ L2 (Σ1 ) , so that [5.17] reduces to: Finally, α

cγ − z + ργ , c(w, 0)L2 (Σ1 ) + N uγ , wL2 (Σ2 ) = 0.

[5.19]

Then, we introduce the adjoint state p(uγ , 0) = pγ solution to: ⎧ ∂p −α ∂tγ − q.∇pγ − DΔpγ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎨ (−D∇pγ − qpγ ).n 2 1 ⎪ ⎪ ⎪ (−D∇pγ − qpγ ).n ⎪ ⎪ 2 ⎩ pγ (T )

=

0

= cγ − z + ργ − =

0

=

0

in Q, I K pγ

on Σ1 , on Σ2 , in

Ω.

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Mathematical Modeling of Random and Deterministic Phenomena

We then have, by using the Green formula: ∂pγ 0 = −α − q.∇pγ − DΔpγ , c(w, 0)L2 (Q) ∂t ˆ

∂ α pγ + q.∇pγ + DΔpγ c(w, 0) dxdt =− ∂t Q ˆ

∂ α c(w, 0) + q.∇c(w, 0) − DΔc(w, 0) pγ dσdt = ∂t Q ˆ ˆ + (cγ − z + ργ ) c(w, 0).n dσ1 dt − wpγ .n dσ2 dt Σ1

Σ2

= cγ − z + ργ , c(w, 0)L2 (Σ1 ) − pγ , wL2 (Σ2 ) . Hence, cγ − z + ργ , c(w, 0)L2 (Σ1 ) = pγ , wL2 (Σ2 ) .

[5.20]

Summing [5.19] and [5.20], we finally obtain the adjoint state equality: pγ + N uγ , wL2 (Σ2 ) = 0,

∀ w ∈ L2 (Σ2 ).

5.5. Concluding remarks We presented the mathematical analysis for the Nye–Tinker–Barber (NTB) model, which describes the nutrient uptake density for plants. The model is governed by an advection–diffusion system. We then studied the optimal control of the NTB system, where we introduced a perturbation as an unknown source of nutrient, present naturally in the rhizosphere. We obtained a problem of incomplete data that we treated from the optimal control point of view using the techniques of Lions (the low-regret control) as a generalization for the NTB-like model. An interesting program is to work on numerical simulations in the goal to compare the results with experiments, and then confirm these theoretical results. We also have to take into account the pollution caused by CLD and by other kinds of pollution already existing in the soil.

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111

5.6. References Allaire, G. (2007). Numerical Analysis and Optimization. An Introduction to Mathematical Modeling and Numerical Simulation. Oxford Science Publications. Brézis, H. (1984). Analyse Fonctionnelle. Masson, Paris. Cabidoche, Y.-M., Achard, R., Cattan, P., Clermont-Dauphin, C., Massat, F., Sansoulet, J. (2009). Long-term pollution by chlordecone of tropical volcanic soils in the French West Indies: A simple leaching model accounts for current residue. Environmental Pollution, 157, 1697–1705. Itoh, S., Barber, S.A. (1983). A numerical solution of whole plant nutrient uptake for soil-root systems with root hairs. Plant and Soil, 70, 403–413. Jacob, B., Omrane, A. (2010). Optimal control for age-structured population dynamics of incomplete data. Journal of Mathematical Analysis and Applications, 370, 42–48. Jalonen, R., Nygren, P., Sierra, J. (2009). Transfer of nitrogen from a tropical legume tree to an associated fodder grass via root exudation and common mycelial networks. Plant, Cell & Environment, 32(10), 1366–1376. Kermarrec, A. (1980). Niveau actuel de la contamination des chaînes biologiques en Guadeloupe : Pesticides et métaux lourds 1979–1980. NRA Antilles-Guyane. PetitBourg. Guadeloupe. France. Lions, J.-L., Magenes, E. (1970). Problèmes aux limites non homogènes et applications, 1. Lions, J.-L. (1992). Contrôle à moindres regrets des systèmes distribués. C.R. Acad. Sci. Paris Ser. I Math., 315, 1253–1257. Lions, J.-L., Diaz, J. (1994). No-regret and Low Regret Control. Environment, Economics and their Mathematical Models, Masson, Paris. Louison, L., Omrane, A., Ozier-lafontaine, H., Picart, D. (2015). Modeling plant nutrient uptake: Mathematical analysis and optimal control. Evolution Equations and Control Theory, 4(2), 193–203. Nakoulima, O., Omrane, A., Velin, J. (2000). Perturbations à moindres regrets dans les systèmes distribués à données manquantes, C.R. Acad. Sc. de Paris. 330, 801–806. Nakoulima, O., Omrane, A., Velin, J. (2003). On the Pareto control and no-regret control for distributed systems with incomplete data. SIAM J. Control Optim., 42, 1167–1184.

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Nye, P.H., Marriott, F.H.C. (1969). A theoretical study of the distribution of substances around roots resulting from simultaneous diffusion and mass flow. Plant and Soil, 3, 459–472. Ptashnyk, M. (2010). Derivation of a macroscopic model for nutrient uptake by hairyroots. Nonlinear Analysis: Real World Application, 11, 4586–4596. Roose, T., Fowler, A.C., Darrah, P.R. (2001), A mathematical model of plant nutrient uptake. J. Math. Biology, 42, 347–360. Snegaroff, J. (1977). Les résidus d’insecticides organochlorés dans les sols et les rivières de la région bananière de Guadeloupe. Phytiatrie-Phytopharmacie, 26, 251–268. Tinker, P.B., Nye, P.H. (2000). Solute Movement in the Rhizosphere. Oxford University Press, Oxford.

6 Existence of an Asymptotically Periodic Solution for a Stochastic Fractional Integro-differential Equation

6.1. Introduction Periodicity often appears in implicit ways in various deterministic or random phenomena. For instance, this is the case when we study the effects of fluctuating environments on population dynamics (predator–prey systems, for example). However, the dynamics observed in some phenomena in the real world are not always periodic, and therefore, may be almost periodic or asymptotically periodic. The theory of almost periodicity or asymptotic periodicity has been developed in connection with problems related to differential equations, dynamical systems, and other areas of mathematics. Recently, there has been an increasing interest in extending certain classical results to stochastic differential equations in separable Hilbert spaces. This is because almost all problems in a real-life situation to which mathematical models are applicable are basically stochastic rather than deterministic. The impetus of this chapter comes from two main sources. The first source is our recent paper (Manou-Abi and Dimbour 2019), in which the concept of the periodic limit process was introduced to study the existence and uniqueness of asymptotically periodic solutions to specific stochastic differential equations. The second source is a Chapter written by Solym Mawaki M ANOU -A BI, William D IMBOUR and Mamadou Moustapha M BAYE.

Mathematical Modeling of Random and Deterministic Phenomena, First Edition. Edited by Solym Mawaki Manou-Abi, Sophie Dabo-Niang and Jean-Jacques Salone. © ISTE Ltd 2020. Published by ISTE Ltd and John Wiley & Sons, Inc.

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Mathematical Modeling of Random and Deterministic Phenomena

paper by Cuevas and de Souza (2010) or Xia Xia (2014), in which the following class of semilinear integro-differential equations of fractional order was studied: 

´t dx(t) = 0 x(0) = x0 .

(t−s)α−2 Γ(α−1) Ax(s)dsdt

+ f (t, x(t))dt

t≥0

It is natural to allow a random noise effect like a Brownian motion, so we consider and study in this chapter the existence and uniqueness of asymptotically periodic solutions in a Hilbert space H of the following class of stochastic fractional integro-differential equations driven by a Brownian motion: 

´t dX(t) = 0 X(0) = c0 ,

(t−s)α−2 Γ(α−1) AX(s)dsdt

+ f (t, X(t))dt + g(t, X(t))dB(t),

t≥0

[6.1]

where 1 < α < 2, c0 ∈ L2 (P, H) and (A, D(A)) is a linear densely defined operator and the functions f : R+ × L2 (P, H) → L2 (P, H), g : R+ × L2 (P, H) → L2 (P, H) are Lipschitz continuous and bounded. The process B = (B(t))t∈R is a two-sided standard one-dimensional Brownian motion, which is defined on the filtered complete probability space (Ω, F, Ft , P) with values in the separable Hilbert space H, where Ft = σ{B(u)−B(v)/u, v ≤ t}. Notice that the convolution integral in (6.4) is known as the Riemann–Liouville fractional integral (see Cuestas (2007), Podlubny (1999)). The study of fractional differential equations has been gaining importance in recent years, due to their applications in various fields of science such as mathematical biology and the fact the fractional order models are capable to describe the realistic situation of various phenomena. For instance, we refer to the monographs of Bazhlekova (2001), Miller and Ross (1993), Podlubny (1999). The recurrence of dynamics for stochastic processes produced by many different kinds of stochastic equations is one of the most important topics in the qualitative theory of stochastic processes, due its applications in many scientific fields. The concept of periodicity and various extensions such as almost automorphy, almost periodicity and asymptotic periodicity was studied for dynamics of stochastic processes. (see (Bezandry and Diagana 2011), (Manou-Abi and Dimbour 2017)). This chapter is organized as follows: in section 6.2, we preliminarily introduce the space of the square mean ω-periodic limit process and its qualitative properties.

Existence of an Asymptotically Periodic Solution

115

Based on the results presented in section 6.2 and some suitable conditions, we prove in section 6.3 the existence, as well as the uniqueness of the square mean asymptotically ω-periodic solution for the above fractional stochastic integro-differential equations driven by a Brownian motion. Finally, we provide an example to illustrate our results. 6.2. Preliminaries In this section, we recall some notations, definitions, lemmas and results which are used in what follows. 6.2.1. Asymptotically periodic process and periodic limit processes Let us consider a real separable Hilbert space (H, ||.||) and a probability space (Ω, F, P) equipped with a filtration (Ft )t . Denote by L2 (P, H) the space of all strongly ´ measurable square-integrable H-valued random variables such that E||X||2 = Ω ||X(ω)||2 dP(ω) < ∞. For X ∈ L2 (P, H), let ||X||2 = (E||X||2 )1/2 . Then it is routine to check that L2 (P, H) is a Hilbert space equipped with the norm ||.||2 . D EFINITION 6.1.– A stochastic process X : R+ → L2 (P, H) is said to be continuous whenever lim E||X(t) − X(s)||2 = 0.

t→s

The process X is said to be bounded if there exists a constant K > 0 such that E||X(t)||2 ≤ K

∀t ≥ 0.

D EFINITION 6.2.– (Manou-Abi and Dimbour 2019) A continuous and bounded stochastic process X : R+ → L2 (P, H) is said to be square mean ω-periodic limit if there exists ω > 0 such that 2 ˜ =0 lim E||X(t + nω) − X(t)||

n→+∞

˜ : R+ → is well-defined for each t ≥ 0 when n ∈ N for some stochastic process X L2 (P, H). The  collection of such square mean ω-periodic limit process is denoted by Pω L R+ , L2 (P, H) .   T HEOREM 6.1.– (Manou-Abi and Dimbour 2019) The space Pω L R+ , L2 (P, H)  1/2 = is a Banach space equipped with the norm ||X||∞ = supt≥0 E||X(t)||2 supt≥0 ||X(t)||2 .

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Mathematical Modeling of Random and Deterministic Phenomena

D EFINITION 6.3.– A continuous and bounded process X : R+ → L2 (P, H) is said to be square mean asymptotically ω-periodic if X = Y + Z where Y and Z are continuous bounded processes so that E||Y (t + ω) − Y (t)||2 = 0

and

lim E||Z(t)||2 = 0.

t→∞

    We write Y ∈ Pω R+ , L2 (P, H) , Z ∈ C0 R+ , L2 (P, H) and we denote the space of all square ω-periodic stochastic process X : R+ →   mean asymptotically L2 (P, H) by APω R+ , L2 (P, H) .   2 T HEOREM 6.2.– (Manou-Abi  and2 Dimbour  2019) The space APω R+ , L (P, H) is a closed subspace of Pω L R+ , L (P, H) . Thus, the space   APω R+ , L2 (P, H) is a Banach space equipped with the norm ||.||∞ . The following result provides some interesting properties. T HEOREM 6.3.– (Manou-Abi and Dimbour 2019) Let X be a continuous and bounded stochastic process and ω > 0. Then the following statements are equivalent   i) X ∈ APω R+ , L2 (P, H) ii) We have 2

lim E ||X(t + nω) − Y (t)|| = 0

n→∞

uniformly on t ∈ R+ for some stochastic process Y : R+ → L2 (P, H). iii) We have 2

lim E ||X(t + nω) − Y (t)|| = 0

n→∞

uniformly on compact subsets of R+ for some stochastic process Y : R+ → L2 (P, H). iv) We also have 2

lim E ||X(t + nω) − Y (t)|| = 0

n→∞

uniformly on [0, ω] for some stochastic process Y : R+ → L2 (P, H). We have the following composition result:

Existence of an Asymptotically Periodic Solution

117

T HEOREM 6.4.– (Manou-Abi and Dimbour 2019) Assume that f : R+ × L2 (P, H) → L2 (P, H) is a square mean ω-periodic limit process uniformly for Y ∈ L2 (P, H) in bounded sets of L2 (P, H) and satisfies the Lipschitz condition, i.e. there exists a constant L > 0 so that E||f (t, Y ) − f (t, Z)||2 ≤ L E||Y − Z||2

∀t ≥ 0, ∀ Y, Z ∈ L2 (P, H).

Let X : R+ → L2 (P, H) be a square mean ω-periodic limit process. Then the process F (t) = (f (t, X(t)))t≥0 is a square mean ω-periodic limit process. Now, let us end this part with the following property of a Brownian motion. P ROPOSITION 6.1 (Weak Markov Property).– Let B = (B(s))s≥0 be a two-sided standard one-dimensional Brownian motion and set for h ∈ R, ˜ h (u) = B(u + h) − B(h), u ∈ R. B ˜ h is a two-sided Brownian motion, independent of {B(s) : s ≤ Then the process B ˜ h (u) + B(h). h}. In others words, B(u + h) has the same law as B 6.2.2. Sectorial operators We recall some definitions about sectorial operators which have been studied in the past decades. Let X be a Banach space. D EFINITION 6.4 (Lunardi (1995)).– A closed and dense defined linear operator A : D(A) ⊆ X → X with domain D(A) in the Banach space X is called sectorial operator of type μ if there exists 0 < θ < π/2, M > 0 and μ ∈ R such that its resolvent ρ(A) exists outside the sector μ + Sθ = {μ + λ : λ ∈ C, |arg(−λ)| < θ} and ||(λI − A)−1 || ≤

M when λ does not belong to μ + Sθ , |λ − μ|

D EFINITION 6.5 (Bazhlekova (2001)).– Let A be a closed and linear operator with domain D(A) defined on a Banach space X. We call A the generator of a solution operator if there exist μ ∈ R and a strong continuous function Sα : R+ → L(X, X) so that for x ∈ X, {λα : Re(λ) > μ} ⊂ ρ(A) and λα−1 (λα − A)−1 x ˆ ∞ = e−λt Sα (t)dt, Re(λ) > μ. 0

118

Mathematical Modeling of Random and Deterministic Phenomena

In this case, Sα (.) is called the solution operator generated by A. In particular, Sα (0) = I. If A is sectorial of type μ with 0 < θ < π(1 − solution operator given by Sα (t) =

1 2πi

ˆ γ

α 2 ),

then A is the generator of a

eλt λα−1 (λα − A)−1 dλ,

where γ is a suitable path lying outside the sector μ + Sθ (see Cuestas (2007), Bazhlekova (2001)). Eduardo Cuesta (2007) proved that if A is a sectorial operator of type μ < 0, for some M > 0 and 0 < θ < π(1 − α/2) then there exists a constant C > 0 such that ||Sα (t)|| ≤

CM , 1 + |μ|tα

t ≥ 0.

[6.2]

Furthermore, if 1 < α < 2, then ˆ

+∞ 0

1 |μ|−1/α π dt = , α 1 + |μ|t α sin(π/α)

[6.3]

and therefore Sα (t) is integrable on [0, ∞[. 6.3. A stochastic integro-differential equation of fractional order In this section, we investigate the existence of the square mean asymptotically ω-periodic solution to the following fractional stochastic integro-differential equation driven by a Brownian motion: 

´t dX(t) = 0 X(0) = c0 ,

(t−s)α−2 Γ(α−1) AX(s)ds

+ f (t, X(t))dt + g(t, X(t))dB(t),

t≥0

[6.4]

where 1 < α < 2, c0 ∈ L2 (P, H) and (A, D(A)) is a linear densely defined operator of sectorial type and the functions f : R+ × L2 (P, H) → L2 (P, H), g : R+ → L2 (P, H) are Lipschitz continuous and bounded, and B(t) is a two-sided standard one-dimensional Brownian motion, which is defined on the filtered complete

Existence of an Asymptotically Periodic Solution

119

probability space (Ω, F, Ft , P) with values in the separable Hilbert space H and Ft adapted. Recall that Ft = σ{B(u) − B(v)/u, v ≤ t}. In order to establish our main result, we impose the following Assumption. (H1): A is a sectorial operator of type μ < 0 for some M > 0, and angle θ with 0 ≤ θ ≤ π(1 − α/2). Now, let us first consider the following linear stochastic integro-differential equation of fractional order: 

´t dX(t) = 0 X(0) = c0 ,

(t−s)α−2 Γ(α−1) AX(s)dsdt

+ F (t)dt + G(t)dB(t)

t≥0

Now, let A be a sectorial operator of type μ with 0 < θ < π(1 − denote the solution of (6.5) under this hypothesis as X.

α 2 ),

[6.5] and let us

Recall that the Laplace transform of the abstract function G is defined by ˆ G(λ) =

ˆ

∞

e−λt G(t)dB(t).

0

Since equation (6.5) can be written by means of the Laplace transform as ˆ ˆ ˆ + λFˆ (λ) + G(λ)dB(λ), λX(λ) = c0 + λ1−α AX(λ)

λα ∈ μ + Sθ ,

ˆ Fˆ and G ˆ stand for the Laplace transform of X, F and G respectively, the where X, variation of constants formula allows us to write the solution of (6.5) as ˆ X(t) = Sα (t)c0 +

0

ˆ

t

Sα (t − s)F (s)ds +

0

t

Sα (t − s)G(s)dB(s).

The above consideration motivates the following definition: D EFINITION 6.6.– Assume that A generates a solution operator Sα (t). A stochastic process, {X(t), t ≥ 0}, is said to be a mild solution to (6.4) if X(t) is Ft -adapted and satisfies the following equation: ˆ t ˆ t X(t) = Sα (t)c0 + Sα (t − s)f (s, X(s))ds + Sα (t − s)g(s, X(s))dB(s). 0

0

We refer to Xia (2014) or Cuevas and de Souza (2009) when g = 0. Now, we will establish some technical results.

120

Mathematical Modeling of Random and Deterministic Phenomena

L EMMA 6.1.– Let F be a square mean ω-periodic limit process in L2 (P, H). Under Assumption (H1), the sequence of stochastic processes (Xn (t))n≥1 , t ≥ 0, defined by ˆ Xn (t) =

nω

Sα (t + s)F (nω − s)ds

0

is a Cauchy sequence in L2 (P, H) for all t ≥ 0. We shall denote by U = (U (t))t≥0 the limit process of (Xn (t))n≥1 , t ≥ 0, in L2 (P, H). P ROOF.– We have E||Xn+p (t) − Xn (t)||2  ˆ  (n+p)ω  = E  Sα (t + s)F ((n + p)ω − s)ds  0 ˆ − ≤ 2E

nω

0

ˆ

+ 2E

 2  Sα (t + s)F (nω − s)ds

(n+p)ω

nω

ˆ

0

nω

||Sα (t + s)F ((n + p)ω − s)|| ds

2

   2 Sα (t + s) F ((n + p)ω − s) − F (nω − s)  ds

= I1 (t, n, p) + I2 (t, n, p)

where I1 (t, n, p) = 2E I2 (t, n, p) = 2E

ˆ

(n+p)ω

nω nω

ˆ

0

||Sα (t + s)F ((n + p)ω − s)|| ds

2

   2 Sα (t + s) F ((n + p)ω − s) − F (nω − s)  ds .

We have I1 (t, n, p) = 2E

ˆ

(n+p)ω

nω

||Sα (t + s)F ((n + p)ω − s)|| ds

2

Existence of an Asymptotically Periodic Solution

≤ 2E

ˆ

(n+p)ω

nω

ˆ

||Sα (t + s)|| ||F ((n + p)ω − s)||ds

121

2

 2 CM ||F ((n + p)ω − s)||ds 1 + |μ|(t + s)α nω ˆ (n+p)ω  2 ˆ (n+p)ω CM ≤2 ds E||F ((n + p)ω − s)||2 ds 1 + |μ|(t + s)α nω nω ˆ (n+p)ω C 2M 2 ≤ 2Kpω ds |μ|2 (s)2α nω C 2M 2  1  (nω)−2α+1 . = 2Kpω |μ|2 2α − 1 ≤ 2E

(n+p)ω



Now we consider the integers N1 and N2 such that C 2M 2  1  (nω)−2α+1 <

|μ|2 2α − 1

2Kpω

16KC 2 M 2
0.

n→∞

However,

ˆ I4 (t, n, p) ≤ 4E

nω

Nω nω

≤ 4E

ˆ

= 4E

Nω nω

ˆ

Nω

2 CM ||F ((n + p)ω − s) − F (nω − s)||ds 1 + |μ|(t + s)α 2 CM ||F ((n + p)ω − s) − F (nω − s)||ds |μ|sα 2 CM 1 ||F ((n + p)ω − s) − F (nω − s)||ds . |μ|sα/2 sα/2

Again, using Cauchy–Schwarz inequality, it follows that ˆ nω 2 2 ˆ nω C M 1 ds E||F ((n + p)ω − s) − F (nω − s)||2 ds I4 (t, n, p) ≤ 4 2 α α N ω |μ| s Nω s ˆ 16KC 2 M 2  +∞ 1 2 ds ≤ α |μ|2 Nω s =

16KC 2 M 2 |μ|2 (α − 1)2 (N 2 ω 2 )α−1

and hence I4 (t, n, p) ≤ . This show that (Xn (t))n≥1 , t ≥ 0, is a Cauchy sequence in L2 (P, H). L EMMA 6.2.– Let F be a square mean ω-periodic limit process in L2 (P, H) so that lim E||F (t + nω) − F˜ (t)||2 = 0

n→+∞

for all t ≥ 0. Define V (t) =

´t 0

Sα (t − s)F (s)ds. Under Assumption (H1), we have

lim E||V (t + nω) − V ∗ (t)||2 = 0

n→+∞

124

Mathematical Modeling of Random and Deterministic Phenomena

uniformly on t ≥ 0 where ˆ

V ∗ (t) = U (t) +

t 0

Sα (t − s)F˜ (s)ds.

P ROOF.– Let us rewrite ˆ ˆ t+nω Sα (t + nω − s)F (s)ds = V (t + nω) = 0

ˆ =

−nω

ˆ =

−nω

ˆ

0

Sα (t − s)F (s + nω)ds + ˆ

nω

Sα (t + s)F (s + nω)ds +

0

t

Sα (t − s)F (s + nω)ds

t

0

Sα (t − s)F (s + nω)ds

t

Sα (t − s)F (s + nω)ds

0

= Xn (t, n) + I(t, n). We have E||V (t + nω) − V ∗ (t)||2 = E ||Xn (t) + I(t, n) − U (t)  2 ˆ t  ˜ − Sα (t − s)F (s)ds 0

2

≤ 2E ||Xn (t) − U (t)||   2 ˆ t   + 2E I(t, n) − Sα (t − s)F˜ (s)ds . 0

Using Lemma 6.1, it follows that 2

E ||Xn (t) − U (t)|| → 0 for all t ≥ 0. Now, for mω ≤ t < (m + 1)ω; m ∈ N, we have ˆ t  2    Sα (t − s)F˜ (s)ds EI(t, n) − 0

 ˆ t   2  Sα (t − s) F (s + nω) − F˜ (s) ds = E  0

Existence of an Asymptotically Periodic Solution

≤E

ˆ

2 CM ||F (s + nω) − F˜ (s)||ds α 1 + |μ|(t − s)

t 0

≤ 2E + 2E

ˆ

ˆ

125

2 CM ˜ (s)||ds ||F (s + nω) − F 1 + |μ|(t − s)α 2 CM ˜ (s)||ds . ||F (s + nω) − F 1 + |μ|(t − s)α

mω 0 t

mω

But first, 2E

ˆ

mω

0

2 CM ||F (s + nω) − F˜ (s)||ds α 1 + |μ|(t − s)

≤ 2C 2 M 2 E

m−1 ˆ  k=0

(k+1)ω

kω

1 1 + |μ|(t − (k + 1)ω)α

2 ||F (s + nω) − F˜ (s)||ds m−1 ˆ  ω 1 2 2 ≤ 2C M E α 0 1 + |μ|(t − (k + 1)ω) k=0

||F (s + (n + k)ω) − F˜ (s + kω)||ds 2

≤ 2C M

2

ˆ

ω 0

m−1  k=0

2

1 1 + |μ|(t − (k + 1)ω)α

2

ˆ

ω

ds 0

E||F (s + (n + k)ω)

− F˜ (s + kω)||2 ds 2 ˆ ˆ m−1 ω 1 2C 2 M 2 ω  ≤ ds E||F (s + (n + k)ω) |μ|2 (t − (k + 1)ω)α 0 0 k=0

− F˜ (s + kω)||2 ds. Since lim E||F (s + (n + k)ω) − F˜ (s + kω)||2 = 0

n→+∞

for s ∈ [0, ω] and the fact that E||F (s + (n + k)ω) − F˜ (s + kω)||2 ≤ 2K,

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Mathematical Modeling of Random and Deterministic Phenomena

it follows by Lebesgue’s dominated convergence theorem that: ˆ lim

n→+∞

ω

0

E||F (s + (n + k)ω) − F˜ (s + kω)||2 = 0.

Note that m−1  k=0

+∞ 1 1 1  ≤ < ∞, α α (t − (k + 1)ω) ω (k + 1)α k=0

hence m−1  k=0

1 (t − (k + 1)ω)α

2 < ∞,

uniformly in t,

therefore ˆ lim 2E

n→+∞

However, ˆ 2E

mω

0

CM ||F (s + nω) − F˜ (s)||ds 1 + |μ|(t − s)α

2 = 0.

2 CM ||F (s + nω) − F˜ (s)||ds α mω 1 + |μ|(t − s)  ˆ (m+1)ω 2 2 2 ≤ 2C M E ||F (s + nω) − F˜ (s)||ds t

≤ 2C 2 M 2

mω ω

ˆ 0

E||F (s + (n + m)ω) − F˜ (s + mω)||ds

2

.

But E||F (s + (n + m)ω) − F˜ (s + mω)||2 ≤ 2E||F (s + (n + m)ω) − F˜ (s)||2 + 2E||F˜ (s + mω) − F˜ (s)||2 so that lim E||F (s + (n + m)ω) − F˜ (s + mω)||2 = 0.

n→+∞

Again, using Lebesgue’s dominated convergence theorem, we obtain ˆ lim

n→+∞

0

ω

E||F (s + (n + m)ω) − F˜ (s + mω)||2 ds = 0,

Existence of an Asymptotically Periodic Solution

127

and hence lim 2E

ˆ

n→+∞

t

mω

2 CM ˜ (s)||ds = 0. ||F (s + nω) − F 1 + |μ|(t − s)α

Thus, ˆ t   2   Sα (t − s)F˜ (s)ds = 0 lim EI(t, n) −

n→+∞

0

for all t ≥ 0. Therefore, lim E||V (t + nω) − V ∗ (t)||2 = 0

n→+∞

uniformly on t ≥ 0 for some stochastic process V ∗ (t) : R+ → L2 (P, H). L EMMA 6.3.– Let G be a square mean ω-periodic limit process in L2 (P, H) and B(t) a two-sided standard one-dimensional Brownian motion. Under Assumption (H1), the sequence of stochastic process (Yn (t))n≥1 , t ≥ 0, defined by ˆ Yn (t) =

0

−nω

Sα (t − s)G(s + nω)dB(s)

is a Cauchy sequence in L2 (P, H) for all t ≥ 0. We will denote by U ∗ = (U ∗ (t))t≥0 the limit process of (Yn (t))n≥1 , t ≥ 0, in L (P, H). 2

P ROOF.– We have,

ˆ  0  S (t − s)G((n + p)ω + s)dB(s) E||Yn+p (t) − Yn (t)|| = E   −(n+p)ω α 2

ˆ −

0

−nω

 2  Sα (t − s)G(nω + s)dB(s)

2 ˆ   −nω   Sα (t − s)G((n + p)ω + s)dB(s) ≤ 2E    −(n+p)ω  ˆ  + 2E 

0

−nω

 2    Sα (t − s) G((n + p)ω + s) − G(nω + s) dB(s)

128

Mathematical Modeling of Random and Deterministic Phenomena

≤ 2E

ˆ

−nω −(n+p)ω

+ 2E

ˆ

ˆ ≤ 2E

−nω

||Sα (t − s)||2 ||G((n + p)ω + s)||2 ds

−(n+p)ω

ˆ

ˆ

0

||Sα (t − s)||2 ||G((n + p)ω + s) − G(nω + s)||2 ds

−nω

(n+p)ω

nω

ˆ

+ 2E

nω

0

2

2 ||Sα (t − s)|| ||G((n + p)ω + s) − G(nω + s)||dB(s)

−nω

+ 2E ≤ 2E

0

||Sα (t − s)|| ||G((n + p)ω + s)||dB(s)

||Sα (t + s)||2 ||G((n + p)ω − s)||2 ds

||Sα (t + s)||2 ||G((n + p)ω − s) − G(nω − s)||2 ds

≤ J1 (t, n, p) + J2 (t, n, p) where ˆ J1 (t, n, p) = 2E

(n+p)ω

nω nω

ˆ J2 (t, n, p) = 2E

0

||Sα (t + s)||2 ||G((n + p)ω − s)||2 ds

||Sα (t + s)||2 ||G((n + p)ω − s) − G(nω − s)||2 ds.

Estimates of J1 (t, n, p). ˆ J1 (t, n, p) = 2E ˆ ≤2

(n+p)ω

nω (n+p)ω



||Sα (t + s)||2 ||G((n + p)ω − s)||2 ds. C 2M 2

2 E||G((n s)α

1 + |μ|(t + ˆ C 2 M 2 K (n+p)ω −2α ≤2 s ds μ2 nω ˆ C 2 M 2 K +∞ −2α s ds ≤2 μ2 nω =2

nω

C 2M 2K . μ2 (2α − 1)(nω)2α−1

+ p)ω − s)||2 ds

Existence of an Asymptotically Periodic Solution

129

Now we consider the integers N1 and N2 such that 2

C 2M 2K ≤ ∀n ≥ N1 μ2 (2α − 1)(nω)2α−1

4

C 2M 2K ≤ ∀n ≥ N2 μ2 (2α − 1)(nω)2α−1

and set N = max (N1 , N2 ). For n ≥ N , we have: ˆ nω ||Sα (t + s)||2 ||G((n + p)ω − s) − G(nω − s)||2 ds J2 (t, n, p) = 2E 0

ˆ ≤2

nω



0

ˆ ≤2

Nω 0

ˆ +2

nω

C 2M 2 1 + |μ|(t + s)α

2 2 E||G((n + p)ω − s) − G(nω − s)|| ds

C 2M 2 2  2 E||G((n + p)ω − s) − G(nω − s)|| ds 1 + |μ|(t + s)α C 2M 2 2  2 E||G((n + p)ω − s) − G(nω − s)|| ds 1 + |μ|(t + s)α

Nω

≤ J3 (t, n, p) + J4 (t, n, p) where ˆ J3 (t, n, p) = 2

Nω 0

ˆ J4 (t, n, p) = 2

C 2M 2 2  2 E||G((n + p)ω − s) − G(nω − s)|| ds α 1 + |μ|(t + s)

nω Nω



C 2M 2 1 + |μ|(t + s)α

2 2 E||G((n + p)ω − s) − G(nω − s)|| ds.

Since G is a square mean ω periodic limit process, then 2 ˜ = 0, lim E||G(t + kω) − G(t)||

k→+∞

∀t ≥ 0

˜ : R+ → is well-defined for each t ≥ 0 when k ∈ N for some stochastic process G 2 L (P, H). Now, ˆ J3 (t, n, p) = 2

0

Nω



C 2M 2 1 + |μ|(t + s)α

2 2 E||G((n + p)ω − s) − G(nω − s)|| ds

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Mathematical Modeling of Random and Deterministic Phenomena

ˆ =2

Nω



0

C 2M 2 1 + |μ|(t − s + N ω)α

2 E||G((n − N + p)ω + s)

− G((n − N )ω + s)||2 ds ˆ Nω E||G((n − N + p)ω + s) − G((n − N )ω + s)||2 ds ≤ 2C 2 M 2 ≤ 4C 2 M 2

ˆ

0

Nω 0

+ 4C 2 M 2

ˆ

2 ˜ E||G((n − N + p)ω + s) − G(s)|| ds

Nω

0

2 ˜ E||G((n − N + p)ω + s) − G(s)|| ds

Since 2 ˜ lim E||G((n − N + p)ω + s) − G(s)|| =0

n→∞

2 ˜ =0 lim E||G((n − N )ω + s) − G(s)||

n→∞

2 ˜ and using the fact that max{E||G((n − N + p)ω + s) − G(s)|| , E||G((n − N )ω + 2 ˜ s) − G(s)|| } ≤ 2K, it follows by Lebesgue’s dominated convergence theorem that

lim J3 (t, n, p) = 0

n→∞

for all t ≥ 0. However,

ˆ

nω

C 2M 2

2 2 E||G((n + p)ω − s) − G(nω − s)|| ds. 1 + |μ|(t + s)α ˆ 4C 2 M 2 K +∞ −2α ≤ s ds μ2 Nω

J4 (t, n, p) = 2

≤ ≤

Nω



4C 2 M 2 K μ2 (2α − 1)(N ω)2α−1 μ2 (2α

4C 2 M 2 K , − 1)(N2 ω)2α−1

so that J4 (t, n, p) ≤ . This proves that (Yn (t))n≥1 , t ≥ 0, is a Cauchy sequence in L2 (P, H)

Existence of an Asymptotically Periodic Solution

131

L EMMA 6.4.– Let G be square mean ω-periodic limit in L2 (P, H) such that 2 ˜ lim E||G(t + nω) − G(t)|| =0

n→+∞

for all t ≥ 0. Define ˆ H(t) =

0

t

Sα (t − s)G(s)dB(s)

Under assumption (H1), we have lim E||H(t + nω) − H ∗ (t)||2 = 0

n→+∞

uniformly on t ≥ 0 where ∗

ˆ

∗

H (t) = U (t) +

t

0

˜ Sα (t − s)G(s)dB(s).

P ROOF.– Let us rewrite, ˆ t+nω H(t + nω) = Sα (t + nω − s)G(s)dB(s) ˆ =

0

t

−nω

Sα (t − s)G(s + nω)dB(s + nω).

˜ ˜ Let B(s) = B(s + nω) − B(nω) for each s ∈ R. The weak Markov property B is also a two-sided Brownian motion and has the same distribution as B. Moreover, ˜ {B(s), s ∈ R} is a two-sided Brownian motion independent of B(nω). Thus, ˆ H(t + nω) = ˆ =

t

−nω 0 −nω

ˆ

+

0

t

˜ Sα (t − s)G(s + nω)dB(s) ˜ Sα (t − s)G(s + nω)dB(s) ˜ Sα (t − s)G(s + nω)dB(s)

= Yn (t) + J(t, n)

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Mathematical Modeling of Random and Deterministic Phenomena

where ˆ J(t, n) =

t

0

˜ Sα (t − s)G(s + nω)dB(s) =

ˆ 0

t

Sα (t − s)G(s + nω)dB(s).

We have E||H(t + nω) − H ∗ (t)||2  2 ˆ t   ∗ ˜   = E Yn (t) + J(t, n) − U (t) − Sα (t − s)G(s)ds 0

2

∗

≤ E ||Yn (t) − U (t)||   2 ˆ t   ˜  + E J(t, n) − Sα (t − s)G(s)ds  0

Using Lemma 6.3, it follows that 2

E ||Yn (t) − U ∗ (t)|| → 0 for all t ≥ 0, when n → +∞. For mω ≤ t < (m + 1)ω; m ∈ N, we have ˆ t   2   ˜ EJ(t, n) − Sα (t − s)G(s)dB(s)  0

 ˆ t  2     ˜ = E  Sα (t − s) G(s + nω) − G(s) dB(s) ≤E

ˆ t

0

ˆ ≤ ˆ

mω



t

mω

Now ˆ mω 

2 CM ˜ ||G(s + nω) − G(s)||dB(s) 1 + |μ|(t − s)α

C 2M 2 2 ˜ ds  2 E||G(s + nω) − G(s)|| 1 + |μ|(t − s)α

0

+

0

t 0

ˆ =

0



C 2M 2 1 + |μ|(t − s)α C 2M 2

1 + |μ|(t −

2 ˜ ds 2 E||G(s + nω) − G(s)||

2 s)α

C 2M 2 1 + |μ|(t − s)α

2 ˜ E||G(s + nω) − G(s)|| ds

2 ˜ ds 2 E||G(s + nω) − G(s)||

Existence of an Asymptotically Periodic Solution

≤ C 2M 2

m−1  ˆ (k+1)ω k=0

kω



1 1 + |μ|(t − (k + 1)ω)α

133

2

2 ˜ E||G(s + nω) − G(s)|| ds

= C 2M 2

m−1 ˆ ω k=0

0

1  2 1 + |μ|(t − (k + 1)ω)α

˜ + kω)||2 ds E||G(s + (n + k)ω) − G(s  ˆ ω m−1  1 2 2 ≤C M   α 2 0 k=0 1 + |μ|(t − (k + 1)ω) ˜ + kω)||2 ds E||G(s + (n + k)ω) − G(s  ˆ m−1 1 C 2M 2 ω  ≤ μ2 (t − (k + 1)ω)2α 0 k=0

˜ + kω)||2 ds. E||G(s + (n + k)ω) − G(s Let us point out that m−1  k=0

+∞ 1 1 1  ≤ < ∞, (t − (k + 1)ω)2α ω 2α (k + 1)2α k=0

and using the fact that ˜ + kω)||2 = 0, lim E||G(s + (n + k)ω) − G(s

n→+∞

2 ˜ for s ∈ [0, ω] for all k ≥ 0, and E||G(s + (n + k)ω) − G(s)|| ≤ 2K, it follows by Lebesgue’s dominated convergence theorem that:

ˆ lim

n→+∞

0

mω



C 2M 2 1 + |μ|(t +

2 s)α

2 ˜ E||G(s + nω) − G(s)|| ds = 0,

uniformly on t ≥ 0. However, ˆ

t

mω

C 2M 2 2 ˜ ds  2 E||G(s + nω) − G(s)|| α 1 + |μ|(t − s)

134

Mathematical Modeling of Random and Deterministic Phenomena

ˆ ≤

(m+1)ω

mω

ˆ =

ω

C 2M 2 2 ˜ ds  2 E||G(s + nω) − G(s)|| 1 + |μ|(t − s)α C 2M 2



2 mω)α

˜ + mω)||2 ds E||G(s + (n + m)ω) − G(s

1 + |μ|(t − s − ˆ ω 2 2 ˜ + mω)||2 ds. E||G(s + (n + m)ω) − G(s ≤C M 0

0

Since ˜ + mω)||2 = 0, lim E||G(s + (n + m)ω) − G(s

n→+∞

for s ∈ [0, ω], m ≥ 0 and 2 ˜ ≤ 2K, E||G(s + (n + m)ω) − G(s)||

we obtain, by the Lebesgue dominated convergence theorem that ˆ lim

n→+∞

t

mω



C 2M 2 1 + |μ|(t − s)α

2 ˜ ds = 0. 2 E||G(s + nω) − G(s)||

In view of the above, it follows that ˆ t   2   ˜ Sα (t − s)G(s)dB(s) lim EJ(t, n) −  = 0

n→+∞

0

uniformly on t ≥ 0. Therefore, lim E||H(t + nω) − H ∗ (t)||2 = 0

n→+∞

uniformly in t ≥ 0 for the stochastic process H ∗ (t) : R+ → L2 (P, H) defined as above. Now we can establish the main result of this section. T HEOREM 6.5.– Let f, g : R+ × L2 (P, H) → L2 (P, H) be square mean ω-periodic limit processes in t ≥ 0 uniformly in X for bounded subsets of L2 (P, H). Assume that f, g satisfies a local Lipschitz condition, uniformly in t ≥ 0: i.e. there exists constant Lf > 0 and Lg > 0 so that E||f (t, X) − f (t, Y )||2 ≤ Lf E||X − Y ||2

∀t ≥ 0, ∀X, Y ∈ L2 (P, H)

E||g(t, X) − g(t, Y )||2 ≤ Lg E||X − Y ||2

∀t ≥ 0, ∀X, Y ∈ L2 (P, H).

Existence of an Asymptotically Periodic Solution

135

Under (H1) if  |μ|−2/α π 2 |μ|−1/α π  0. Clearly, both the f and g satisfy the Lipschitz conditions with Lf = β 2 ||ψ||2∞ and Lg = β 2 ||Φ||2∞ . For instance, ψ(t) and Φ(t) can be chosen to be equal to the following 2-periodic limit (function) deterministic process (see (Xie and Zhang 2015)) given by ⎧ t = 2n − 1, n ∈ N ⎨ 1, t ∈ {0, 2} ∪ {2n + 1 − kn } ∪ {2n + 1 + kn } a{kn } (t) = 0, ⎩ linear, in between. where {kn } ⊂]0, 1[ such that kn > kn+1 , kn → 0 as n → +∞. Note that if we define b : R+ → R by  b(t) =

1, t = 2n − 1, n ∈ N 0, otherwise.

then we have b(t) = limm→+∞ a{kn } (t+2m) so a{kn } is a 2-periodic limit (function) deterministic process. Denote H = L2 [0, 1]. In order to write the system [6.6] on the abstract form [6.4], we consider the linear operator A : D(A) ⊂ L2 [0, 1] → L2 [0, 1], given by Au = u − νu,

where

ν > 0,

with domain D(A) = {u ∈ L2 [0, 1], u ∈ L2 [0, 1],

u(0) = u(1) = 0}.

138

Mathematical Modeling of Random and Deterministic Phenomena

It is well-known that A is a sectorial operator of type μ = −ν < 0 (see Lunardi (1995)). Then the system (6.6) takes the following abstract form 

´t du(t) = 0 u(0) = c0 .

(t−s)α−2 Γ(α−1) Au(s)dsdt

+ f (t, u(t))dt + g(t, u(t))dB(t),

t≥0

where u(t) = u(t, .) and c0 = h(x). Therefore, by Theorem 6.5, equation 6.6 has a unique square mean asymptotically ω-periodic mild solution R+ whenever β is small enough. 6.5. References Bezandry, P. and Diagana, T. (2007). Existence of almost periodic solutions to some stochastic differential equations. Applicable Analysis, 86(7), 819–827. Bezandry, P. and Diagana, T. (2011). Almost Periodic Stochastic Processes. Springer, New York. Bazhlekova, E. (2001). Fractional Evolution Equation in Banach Spaces. PhD thesis, Eindhoven University of Technology. Cuevas, C. and de Souza, J.C. (2009). S-asymptotically ω-periodic solutions of semilinear fractional integro-differential equations. Applied Mathematics Letters, 22, 865–870. Cuevas, C. and de Souza J.C. (2010). Existence of s-asymptotically ω-periodic solutions for fractional order functional integro-differential equations with infinite delay. Nonlinear Analysis, 72(3), 1683–1689. Cuesta, E. (2007). Asymptotic behaviour of the solutions of fractional integrodifferential equations and some time discretizations. Discrete Contin. Dyn. Syst., 277–285. Diop, M.A., Ezzinbi, and Mbaye, M.M. (2015). Existence and global attractiveness of a pseudo almost periodic solution in p-th mean sense for stochastic evolution equation driven by a fractional brownian motion, Stochastic, 87(6), 1061–1093. Available at: http://dx.doi.org/10.1080/17442508.2015.1026345 Lunardi, A. (1995). Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel. Manou-Abi, S.M. and Dimbour, W. (2019). Asymptotically periodic solution of a stochastic differential equation. Bull. Malays. Math. Sci. Soc., 1(1), 1–29.

Existence of an Asymptotically Periodic Solution

139

Manou-Abi, S.M. and Dimbour, W. (2017). On the p-th mean s-asymptotically omega periodic solution for some stochastic evolution equation driven by Q-brownian motion. Adv. Sci. Technol. Eng. Syst. J., Proceedings of ICAM’17, Taza, Morocco, 2(5), 124–133. Miller, K.S. and Ross, B. (1993). Theory and Applications of Fractional Differential Equations. Wiley, New York. Podlubny, I. (1999). Fractional Differential Equations. Academic Press, New York. Xia, Z. (2014). Asymptotically periodic solutions of semilinear fractional integrodifferential equations. Advances in Difference Equations, 2014(9), 19. Xie, R. and Zhang, L. (2015). Space of ω-periodic limit functions and its applications to an abstract cauchy problem. J. Funct. Spaces, 2015(10).

7 Bounded Solutions for Impulsive Semilinear Evolution Equations with Non-local Conditions

7.1. Introduction For semilinear evolution equations in function spaces, difficulties arise when the nonlinear term consists of a composition operator, commonly called Nemytskiis operator, which almost never maps a function space into itself unless the generator function is affine. For example, the composition of two functions of Lp (Ω) does not necessarily belong to Lp (Ω). Examples of this type include the Burgers equation and the Benjamin–Bona–Mahony (BBM) equation with both impulses and non-local conditions, which involve a nonlinear term with spatial derivatives, which greatly complicates the problem when we try to study the approximate controllability of this equation on a fixed interval [0, τ ]. Indeed, for each control, we need to have a corresponding solution defined on the same fixed interval of time. To address these particular examples, we must use the fact that the operator Δ generates an analytic semi-group, which is compact; and then make use of fractional power spaces to write the problem as an abstract ordinary differential equation in L2 (Ω). The fundamental problem is that the composition operator associated with the nonlinear term is well defined only from adequate fractional power spaces to the L2 (Ω) space. Therefore, the novelty of this work lies in the fact that we allow nonlinear terms involving spatial derivative, the use of fractional power spaces and the Banach Fixed Point Theorem. Moreover, our technique can be applied to other equations such as Navier–Stokes equations. Chapter written by Toka D IAGANA and Hugo L EIVA.

Mathematical Modeling of Random and Deterministic Phenomena, First Edition. Edited by Solym Mawaki Manou-Abi, Sophie Dabo-Niang and Jean-Jacques Salone. © ISTE Ltd 2020. Published by ISTE Ltd and John Wiley & Sons, Inc.

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Mathematical Modeling of Random and Deterministic Phenomena

For that, we study the existence of bounded solutions for the following semilinear evolution equation with impulses and non-local condition ⎧  z ∈ Z, t > t0 , t = tk , ⎨ z = −Az + F (t, z), z(t0 ) + g(z) = z0 , ⎩ + z(tk ) = z(t− k = 1, 2, 3, . . . , p, k ) + Jk (z(tk )),

[7.1]

where A : D(A) ⊂ Z → Z is a sectorial operator on Z (Z equipped with  · Z =  ·  being a Banach space) with −A being the infinitesimal generator of a strongly continuous compact semi-group {T (t)}t≥0 , g : P C(R, Z α ) → Z α and Jk : Z α → Z α , and F : R × Z α → Z with Z α = D(Aα ). Among other things, the results obtained in this chapter generalize those of Leiva and Sivoli (2018) on the existence of bounded solutions for the system of parabolic equations with only impulses. Recently, in Diagana (2017) the existence of solutions for some damped elastic system and also the existence of bounded solutions for broader elastic systems, was studied, which also motivates this work. However, in Leiva and Sequera (2003) the existence and the stability of bounded solutions for a semilinear system of parabolic equations were proved. In Leiva and Sivoli (2003), the existence, stability and smoothness of bounded solutions for nonlinear time-varying thermoelastic plate equations were studied. In Leiva (1999), the existence of bounded solutions of a second-order system with dissipation was studied. Similarly, in Leiva (2001), the existence of bounded solutions of second-order evolution equations was studied and more recently in Leiva and Sivoli (2018) the existence, stability and the smoothness of the bounded solution of a system of parabolic equations with impulses were obtained. In this chapter, we generalize the above-mentioned results to the case of semilinear impulsive evolution equations with non-local conditions and nonlinear term involving spatial derivative such as the well-known Burges equation. Also, under some condition, we prove that these bounded solutions are periodic in some cases and almost periodic in other cases. 7.2. Preliminaries We begin this section with the definition of a sectorial operator and some preliminary results on compact analytic semi-groups. D EFINITION 7.1.– (See Henry (1981)) We call a linear operator A in a Banach space Z a sectorial operator if it is a closed, densely defined operator such that, for some Φ ∈ (0, π2 ), M ≥ 1 and a real number a, the sector Sa,Φ = {λ ∈ C : Φ ≤ | arg(λ − a)| ≤ π,

λ = a},

Bounded Solutions for Impulsive Semilinear Evolution Equations

143

is in the resolvent set of A and (λ − A)−1  ≤

M , |(λ − a)−1 |

∀λ ∈ Sa,Φ .

Throughout this chapter, the operator A : D(A) ⊂ Z → Z is sectorial and −A is the infinitesimal generator of a compact analytic semi-group of uniformly bounded linear operators {T (t)}t≥0 ⊂ Z, with 0 ∈ (A) ((A) being the resolvent of A). Therefore, fractional power operators Aα , 0 < α ≤ 1, are well-defined. Moreover, since Aα is a closed operator, its domain D(Aα ) is a Banach space endowed with the graph norm zα = Aα z, z ∈ D(Aα ). The previous Banach space will be denoted by Z α = D(Aα ) and is dense in Z. Moreover, for 0 < β < α ≤ 1, the embedding Z α → Z β is compact whenever the resolvent operator of A is compact. For the previous semi-group, the following properties will be used: there are constants, η > 0, Mα ≥ 0 and C1−α , so that T (t)z ≤ e−ηt z, ∀z ∈ Z,

t ≥ 0,

Mα −ηt e , t > 0, tα Aα T (t)z = T (t)Aα z, ∀z ∈ Z α ,

Aα T (t) ≤

(T (t) − I)z ≤

C1−α α α t A z, t > 0, α

[7.2] [7.3] [7.4]

∀z ∈ Z α .

[7.5]

For more on sectorial operators and strongly continuous semi-groups and related issues, we refer the reader to the landmark book by Henry (1981). In this work, we will make extensive use of the following spaces, − − Zbα = {z : R → Z α , z ∈ Cb (R , Z α ), ∃z(t+ k ), z(tk ), and z(tk ) = z(tk )} ,

where R = R\{t1 , t2 , ..., tp } and Cb (R , Z α ) the space of bounded continuous functions on R . Then Zbα is a Banach space when it is equipped with sup-norm defined by zb = sup {z(t)α : t ∈ R} ,

∀z ∈ Z α .

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Mathematical Modeling of Random and Deterministic Phenomena

A ball centered at zero with radius ρ > 0 in Zbα is given by Bρb = {z ∈ Zbα : z(t)b ≤ ρ, t ∈ R}. The space of piecewise continuous functions P C(R) = P C(R, Z α ) is defined by, P C(R) = P C(R, Z α ) = {z : R → Z α , z ∈ C(R , Z α ), − − ∃z(t+ k ), z(tk ), and z(tk ) = z(tk )} .

7.3. Main theorems In this section, we shall prove the main result of this work, i.e. the existence of bounded mild solutions to [7.1]. Given that the functions F , g and Jk are smooth enough, the initial value problem [7.1] admits a unique mild solution (see Fan and Li (2010), Jain and Dhakne (2013), Leiva and Sundar (2017), Leiva (2018)) given by the following formula: ˆ z(t) = T (t − t0 )[z0 − g(z)] + +



t

t0

T (t − s)F (s, z(s))ds

[7.6]

T (t − tk )Jk (z(tk )), t ∈ [t0 , τ ],

0 0 such that for all a ∈ R, g(z)α
0 such that F (t, z1 ) − F (s, z2 ) ≤ K[|t − s| + z1 − z2 α ], z1 , z2 ∈ Brα (0),

t, s ∈ [a, b].

Also, there exists a constant LF > 0 such that F (t, 0) ≤ LF ,

∀t ∈ R.

Bounded Solutions for Impulsive Semilinear Evolution Equations

(H3) There exist constants Sk , Ljk > 0,

k = 1, 2, 3, . . . , p, so that

Jk (t, z1 ) − Jk (z2 )α ≤ Sk z1 − z2 α , Jk (t, 0)α ≤ Ljk ,

145

∀z1 , z2 ∈ Z α ,

k = 1, 2, 3, . . . , p,

k = 1, 2, 3, . . . , p, t ∈ R.

(H4) The function F is globally Lipschitz in (t, z), i.e. there exists a constant L > 0 such that F (t, z1 ) − F (s, z2 ) ≤ L[|t − s| + z1 − z2 α ], ∀z1 , z2 ∈ Z α ,

∀t, s ∈ R.

Now, we are ready to formulate the main results of this chapter. But, before that, we shall prove the following lemma. L EMMA 7.1.– A function z ∈ Zbα is a mild solution of [7.1] if, and only if, z is a solution of the following integral equation: ˆ z(t) =

t

T (t−s)F (s, z(s))ds+

−∞



T (t−tk )Jk (tk , z(tk )), t ∈ R, t = tk .[7.7]

0 5%). The application of the Box–Jenkins method makes it possible to determine an optimal modeling of the time series. First, the choice of the number of integrations d to be made to make the process stationary is set according to the threshold at which the p-value associated with the Ljung– Box test statistic becomes significant, i.e. a single derivation (Figure 11.7) (p ≤ 5%). Since the process is non-stationary for a parameter d = 1, the number of degrees to be considered for the auto-regressive part can be determined. Based on the autocorrelation graph (Figure 11.8), which gives a visualization of the offsets correlated with the original series, the values for the parameter p = {1, 2, 8, 9, 12, 13} are possible solutions, corresponding to the peaks for which the autocorrelations are significant. By convention, we will take the minimum value and set AR(2) as a slope of direction determined between Xt−2 , Xt−1 and Xt . The issue is to determine the moving-average part. This time the partial autocorrelations are used, which gives information on the offsets correlated with the series of origin independently of the previous correlations, and we observe that for q = {1, 5, 6, 7, 8, 10, 13}, the peaks are significant. We choose M A(5) as the predicted value that will be smoothed over the last five months. The model is selected by choosing ARIM A(2, 1, 5). The model obtained is of the form: – auto-regressive part: 1 − 1, 732B ∗∗ (1) + 0, 998B ∗∗ (2) – moving-average part: 1 − 2, 209B ∗∗ (1) + 1, 757B ∗∗ (2) − 0, 375B ∗∗ (3) − 0, 080B ∗∗ (4) + 0, 007B ∗∗ (5) – performance: AIC = 926

254

Mathematical Modeling of Random and Deterministic Phenomena

Figure 11.6. Auto-correlog rams on the 2011, 2012, 2013, 2014, 2015, 2016, 2017 series. Field: Births in Mayotte. Source: CHM. Exploitation: ARS OI

Figure 11.7. Time series from 2011 to 2017 complete data. Field: Births in Mayotte. Source: CHM Exploitation: ARS OI. For a color version of this figure, see www.iste.co.uk/manouabi/modeling.zip

Structural and Predictive Analysis of the Birth Curve in Mayotte from 2011 to 2017

Figure 11.8. Time series obtained after a diversion. Field: Births in Mayotte. Source: CHM. Exploitation: ARS OI. For a color version of this figure, see www.iste.co.uk/manouabi/modeling.zip

Figure 11.9. Autocorrelation Graph. Source: CHM. Exploitation: ARS OI. For a color version of this figure, see www.iste.co.uk/manouabi/modeling.zip

Figure 11.10. Partial autocorrelation graph. Source: CHM. Exploitation: ARS OI. For a color version of this figure, see www.iste.co.uk/manouabi/modeling.zip

255

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It remains to optimize the model by looking at the significance of the coefficients. It can thus be observed that the coefficients associated with the terms of degree 4 and 5 of the moving-average part are particularly close to 0 and potentially zero. The nullity test of the coefficient helps to conclude that degree 5 is statistically no different from 0(p > 5%); by suppressing it, we observe a decrease of the AIC from 926 to 924, which is synonymous with a performance gain. An update of the tests shows that the coefficient associated with degree 4 of the moving-average part is also close to 0 (p > 5%), which allows us to remove it from the predictive model and observe a new performance gain (AIC = 922). After this step, all the coefficients reject the hypothesis of nullity. The final predictive model is then: auto-regressive part :

1 − 1, 732B ∗∗ (1) + 0, 998B ∗∗ (2)

moving-average part :

1 − 2, 246B ∗∗ (1) + 1, 885B ∗∗ (2) − 0, 535B ∗∗ (3)

performance :

AIC = 922

11.3.7. Predictions for the year 2018 Based on the constructed model and taking into account the probabilities of the activities of the number of births, predictions for the year 2018 can be constructed. The central hypothesis, based on the model predictions, is then used when the empirical probability of observing an over- or under-activity is less than 75%. When it is above 75%, for months of high activity, the strong assumption (upper bound of the 95% interval) is chosen. On the other hand, when it is above threshold for months of low activity, it is the weak hypothesis (lower limit of the 95% interval). The prediction curve shows the characteristics shown with an increase in the number of births from January to May, followed by a drop from June to August and a stabilization from September to December. According to the model: 9, 684 births are to be expected, of which 44% during March–June. The number of births observed is 9,441, of which 40% during the period of high activity. The product model has an error rate of 2.6% compared to actual data with an average difference (in absolute value) of 131 births per month. Compared to the unadjusted model (systematic choice of the central hypothesis) and the probability of an explosion in the number of births, 9,873 births were predicted, i.e. a higher error rate (4.6%).

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Figure 11.11. Change in the number of births for the years 2011 to 2017 and predictions for the year 2017 Field: Births in Mayotte Source: CHM Exploitation: ARS OI

11.4. Discussion Statistical analysis allowed to characterize the birth curve. Thus, from 2011 to 2014, the trend is towards the stabilization of the number of births with significant growth in 2015. The number of births in 2016 is similar to that in 2015, 2018 included (p > 5%). Among the plausible and identifiable factors that explain the significant increase in 2015, the migratory flows that have particularly intensified over the last five years and the fertility rate that has increased for both foreigners and Mayotte can be put forward. Nevertheless, these factors do not justify the stabilization of the curve over the last three years. The increase or decrease of the number of births from one month to another is gradual and not abrupt with stalls every three months. It takes place from January to June, followed by a drop from July to September and a stabilization from October to December. The months that explode in terms of number of births are March, May and June. On the other hand, the months with the lowest number of births are January, September, October and November. Finally, the probability of an explosion

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of the number of births for a particular month is based on an empirical approach; the evolution of this concept towards a theoretical distribution would provide a strategic estimate of risk interest.
H

-





J











t 645 709 767 805 810 778 717 641 568 516 498 517

s*

s t 762 837 906 954 966 940 883 810 740 692 679 704

s*** t 878 965 1,046 1,103 1,122 1,101 1,048 979 912 869 861 892

645 837 1,046 954 1,122 1,101 883 810 568 516 498 704

664 669* 908 1,012 979 839 851 676* 695 714 759 676

Table 11.2. The “Observed births” column presents the number of births registered for each month considered. ∗ indicates that the number of births observed is close to that predicted in the low hypothesis, ∗∗ indicates that it is close to that predicted in the central hypothesis, ∗∗∗ indicates that it is close to that predicted in the high hypothesis. Field: Births in Mayotte. Source: Mamoudzou Hospital Center Exploitation: ARS OI Studies and Statistics Department

Despite the good performance of the model built, we can see that for some months the prediction is clearly far from the data observed. Thus, the months of September, August and November have about 527 births on average, which seems to be absurd. This characteristic of the model can be explained by one of the results observed, namely the more the number of births increases for one year, the greater the variability is important. By observing the peak of this activity, it represents on average 39% of births over the years 2011–2017 (40% for 2018), while the model provides 44%. Similarly, for those with low activity, on average, 22% of the total births for the period of September, October and November (23% in 2018) against 16% for the model built. In addition to this first finding related to the defect of the model, others seem to be inherent in the specifics of the year 2018. Thus, the forecasts for the month of February are particularly bad, overestimating by 25% the number of actual births. For this month, it is important to know that the average growth observed over the years 2011–2017 is +12% (with the exception of the year 2016 for which it was only 3%), whereas for the year 2018, a stabilization can be observed (+1% compared to January). The increase in March caught up with this delay in a unique way, since the average increase was 18.3% while for 2018, growth has doubled (record observed from 2011 to 2018). The month of August also shows an unexpected decrease in 2018, −21% of birth compared to July, while on average, the expected rate is −9%. This is

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followed by a stabilization or a slight increase of the number of births during the period of September, October and November, whereas for the previous years, the number of births fell until September (−9% on average) and oscillated until to December. For December 2018, the number of births fell once again exceptionally (−11% against an average of +1% over the 2011–2017 period).

Table 11.3. Relationship between two consecutive months. Field: Births in Mayotte. Source: Mamoudzou Hospital Center. Exploitation: ARS OI - Studies and Statistics Department

11.5. Conclusion The results presented may be of great interest strategic for the staff of the Mayotte Hospital Center. It is best to target the recruitment of missionaries concentrated on the periods of three to four months over the period from March to June. The understanding of the phenomenon remains important in order to guide prevention actions more effectively. The possibility of crossing the birth curve to various factors remains complex. Regarding the first data, until 2017, the census took place in its five-year format. For the second, it is data collected by field survey that do not allow such a cross on the short term. In addition, the indicators tend to go upward despite an “ARS OIC campaign”, which aims to encourage the population to have no more than three children. While waiting to find the right prevention lever adapted to the specificities of the Mayotte territory, further work to improve the monthly predictions of the model must be performed in order to be able to support the staff of the hospital as effectively as possible. 11.6. References Bai, J. and Perron, P. (1998). Estimating and testing linear models with multiple structural changes. Econometrica, 66(1), 47–78. Balicchi, J., Al. (2014). Mayotte, département le plus jeune de france. Insee Première No.1488. 06/02/2014.

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Chaussy, C., Merceron, S., and Genay, V. (2019). A Mayotte, près d’un habitant sur deux est de nationalité étrangère. Insee Première No.1737.07/02/2019. Genay, V. and Merceron, S. (2017). 256 500 habitants à Mayotte en 2017: la population augmente plus rapidement qu’avant. Insee Analyses No.15.14/12/2017. Marie, C.-V. and Al. (2017). Migrations, natalité et solidarités familiales : La société de Mayotte en pleine mutation. Insee Analyses No.12.10/03/2017. Parenton, F. (2018). Enquête nationale périnatale 2016 et extension à Mayotte. ARS OI. Santé Publique France. Parenton, F. (2018). Panel 2016 des indicateurs de santé périnatale à Mayotte. ARS OI 30/11/2018. S ERVICE E TUDES ET S TATISTIQUES A RS Données au 1er janvier 2018.

OI

(2018). Fiches nos îles, notre santé.

Zeileis, A., Kleiber, C., Krämer, W., and Hornik, K. (2003). Testing and dating of structural changes in practice, Computational Statistics & Data Analysis, 44(1), 109–123. Special Issue in Honour of Stan Azen: A Birthday Celebration. Available at: http://www.sciencedirect.com/science/article/ pii/S0167947303000306.

12 Reflections Upon the Mathematization of Mayotte’s Economy

12.1. Introduction From a European point of view, Mayotte could be seen as an island of which we know little more than the name. So why reflect upon the relations between mathematics and Mayotte’s economy? First, because the teaching of and research in economics are sometimes criticized for being excessively mathematized1, knowing that such criticisms are not really known in Mayotte. The question of knowing whether or not it is suitable to model economics mathematically is obviously not new. Since the 19th Century, there have been arguments claiming or putting into perspective the idea that economics is quantitatively ontological, or that mathematics should be the proper language of economics, that have led to some sort of naturalization of modelization in economics since the end of World War II. Second, thinking through the relations between mathematics and Mayotte’s economy is valuable at a time when the island is in a process of statistization, especially since it has become a Chapter written by Victor B IANCHINI and Antoine H OCHET. 1 For national teaching movements, see UQAM’s student strike in economics at Montreal in 1978; “Le mouvement des étudiants pour la réforme de l’enseignement de l’économie”, alias “Autisme-économie” in 2000, and “Pour un Enseignement Pluraliste dans le Supérieur en Economie” (PEPS-économie, 2013), in France. Internationally, we can refer to the International Student Initiative for Pluralism in Economics in 2013. Concerning the critics in research, we can refer to the controversy between the collective of the French association for political economy (2015) and Cahuc and Zylberberg (2016). Of course, the question of the relation between mathematics and economics is not the sole matter of controversy.

Mathematical Modeling of Random and Deterministic Phenomena, First Edition. Edited by Solym Mawaki Manou-Abi, Sophie Dabo-Niang and Jean-Jacques Salone. © ISTE Ltd 2020. Published by ISTE Ltd and John Wiley & Sons, Inc.

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French department and a European outermost region. Let us therefore encourage the development of any statistical or econometric studies on Mayotte. Nevertheless, data collection is rather difficult in a territory where the State apparatus and its census organs are very recent, so that measuring Mayotte’s economy is already challenging, and is all the more so when economic indicators say nothing of the interactions between formal activities and informal ones, with the latter being very widespread in Mayotte. Drawing upon economic, historical and anthropological approaches, in this chapter, we argue in favor of what we call a “reasonable mathematization” of Mayotte’s economy, a mathematization that does not ignore the cultural dimensions of the social organization in which the economy is embedded, and which forms the “plural wealth” of the island. The point is the following. After having shown that economics is not ontologically mathematical (in the traditional sense) and is embedded in various cultural dimensions, we claim that the mathematization process in which Mayotte’s economy has found itself since becoming a French department and a European overseas region should not hide the various economic specificities of the island, constantly in interaction with informal sectors. This chapter will bring to the fore some instances of such interactions, which cannot be properly understood without interdisciplinary research between anthropology and economics. Finally, we envisage that Mayotte’s economy could be viewed as a significant issue for the modelization of complex systems in the systemic paradigm – a paradigm which has come into being since the second half of the 20th Century, but has not really penetrated economics yet. Before tackling the central questions, a definition of the term mathematization is needed. Following Mongin (2001, 2003) to a certain extent, we consider that mathematization is a generic term2 within which it is necessary 2 Symbolization consists of replacing some words from our “initial” language - written and spoken – with some borrowed or created signs derived from an “artificial” one. Such is the case today when an economist symbolizes consumer pleasure by a functional notation or, further back in time, when Marx called constant capital C. When symbolization becomes autonomous, i.e. when we perform operations on the “artificial” signs by putting aside their significations, then we can talk about formalization. In the case of standard consumer theory, a well-known instance of formalization is when we study by derivations the necessary conditions for the existence of a maxima in the consumer utility function, or, in the past, to a degree, when Marx began to perform algebraic operations in the third book of Das Kapital. Quantification refers to the fact of transforming a quality into a quantity. To do so, it is necessary to adopt a standard and to build an instrument. Measuring is usually carried out downstream from quantification. It is the empirical evaluation of an object by an instrument.

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to distinguish many specific ones, such as that of symbolization, formalization, quantification, axiomatization3, modelization4 and statistization5. 12.2. Justifying the mathematization of economics Despite the long debate on the so-called mathematization of economics since the 19th Century, advanced knowledge in mathematics is nowadays a necessity for any postgraduate economist. Mathematics is not just any language, certain economists say; it gives knowledge about economic phenomena themselves. In Irving Fisher’s terms, it is “the lantern” of economists which allows them to see such phenomena better and further6. From this point of view, mathematics is not only the proper language of economics, but also its ontological foundation (section 12.1). These linguistic and ontological arguments contribute to opening the path to a naturalization of modeling in economics (section 12.2). If some examples of models were to be found at the end of the 18th Century, the modelization of economics became widespread after World War II. Finally, some caveats about excessive mathematization of economics will be provided (section 12.3).

3 As for the delicate term of axiomatization, it can be understood in an old way – in the way of Euclid – or in a modern one – in the way of Hilbert in geometry, and Peano in arithmetic. Briefly, the former conceived axioms as “fundamentally true”, and their function was to convey their “truth” deductively within the other propositions. For the latter, the axioms are viewed as “neutral” – not necessarily “true” – but they are easy to formulate, consistent and not too numerous, with the internal rigor of the reasoning through demonstration being what really matters. Whereas the old conception of axioms is not necessarily related to symbolization and formalization, such is not the case for the modern one. Modern axiomatization – and therefore formalization also – opens the path to a formal system. It will take time for the modern conception of axiomatization to penetrate economics. The first axiomatization in the field is to be observed with von Neumann and Morgenstein. 4 Let us now turn to modelization. There are, of course, many varied definitions of what a model is, either conceived as a prototype or as a type. Focusing on the standard meaning of a model in economics, we define it as a simplified and usually mathematical representation of one or some aspects of an economic system, which can be derived from theoretical principles. In most cases, models are not axiomatized, but they are expressed in a formal way, though formalization is sometimes not fully finalized. In this chapter, we do not deal with the question of whether there is a difference of nature Mongin (2003) rather than a difference of degree Armatte (2003) between axiomatization and modelization. For an interesting study of the old and new meanings of the notion of model in social sciences, see Armatte (2005). 5 Finally, statistization is the process through which statistical indicators emerge for facilitating decision-making related to various concerns of “general interest”. 6 (Fisher 1892, p. 119).

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12.2.1. The ontological and linguistic arguments Though there are several arguments for justifying the mathematization of economics, it is sufficient in this chapter to group them into two categories, which are linked: the ontological argument and the linguistic one. Such arguments are not necessarily recent in the history of ideas. William Stanley Jevons, for instance, who used mathematical functions through marginalist procedures as a formal way to present economic ideas, like others before him or during his time7, claimed that “our science must be mathematical, simply because it deals with quantities (...). Wherever things treated are capable of being greater or less, there the laws and relations must be mathematical in nature. The ordinary laws of supply and demand treat entirely of quantities of commodities demanded or supplied, and express the manner in which the quantities vary in connection with price. In consequence of this fact, the laws are mathematical” (Jevons 1888, pp. 3–5). This quote is interesting in that it emphasizes both the ontological and linguistic arguments. But for now, let us focus on the former. The fact that political economy has successively and historically moved away from the moral philosophy from which it emerged has opened the path to the idea according to which the ontological foundation of political economy is analogous to natural philosophy. As observed by (Mirowski 2012, p. 120), this analogy rests upon the following syllogism. Science has shown that nature is mathematical; economy is a phenomenon of nature; therefore, economics is mathematical. The relevance of this syllogism should, of course, be nuanced according to the different ways in which the fundamental nature of economics can be conceived. This being so, the idea that economics deals with quantities, and therefore with mathematics, still seems present at times in economists’ mentalities to this day, as evidenced by the words of a former Nobel laureate in economics: “I came to the position that mathematical analysis is not one of the many ways of doing economic theory: it is the only way”, wrote (Lucas 2001, p. 9) in his professional memoir. “Economic theory is mathematical analysis. Everything else is just pictures and talk.” It was by reading the famous Foundations of Economic Analysis during his economics training that Lucas found his faith in mathematical economics. The book was written in 1947 by Paul Samuelson, another but older Nobel laureate in economics. Apart from the fact that Samuelson has written on many economic matters, he is also known for having provided the contemporary idea that mathematics is the proper language of economics (Samuelson 1952, p. 56): proper language, in the sense that it is more efficient and transparent than the verbal one8; more efficient, because it expresses quantities (or aggregations) and their relations more easily than words do; more transparent, since contrary to 7 For example, Johann von Thünen, Herman Gossen, Augustin Cournot, Leon Walras and Vilfredo Pareto. Nevertheless, note that these authors did not necessarily follow or express Jevons’ arguments in the same way. 8 Note that Samuelson’s arguments are not so far from what can be called “mathematical formalism”, from which a dialogue between the Bourbaki school and economists emerged after World War II.

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verbal language, it ensures a “logical rigor” in the reasoning that is not subject to interpretations, so that the chain of reasoning from assumption to conclusion is clear, without erroneous deductive inference. Though Samuelson did not seem to claim that mathematics is a substitute for genuine insight or knowledge of the real world, he warned young economists that they had best be prepared for psychological problems if they ignored THE language: “without mathematics you [younger economists] run grave psychological risks. As you grow older, you are sure to resent the method increasingly. Either you will get an inferiority complex and retire from the field of theory or you will get an inferiority complex and become aggressive about your dislike of it” (Ibid., p. 65; see also Munir 1994). It is interesting to compare this idea in the light of different studies about how students in economics perceive mathematics in their training. If certain studies in the field are to be believed, among graduate students in six leading universities in the United States, while a substantial majority considered excellence in mathematics as fundamental for professional success in the 1980s (Colander and Klamer 1987), more recent data point to a slight change in the trend (Colander 2005). Nowadays, despite some criticisms concerning an excessive mathematization of economics, the learning of mathematics still plays a considerable role in the training of economists.9 12.2.2. Towards a naturalization of modeling in economics Despite the fact that it is an anachronism to talk about model before the mid-20th Century, it is useful to refer to the 18th and 19th Centuries to understand the historical development of modeling in economics. As Morgan (2012, pp. 6–14)10 wrote, it is possible to identify three moments of time in this field. The first ran from the late 18th Century to the early 19th Century. At this time, which can be called “prehistory”, there were only a few isolated examples of models. François Quesnay’s Tableau Economique published in 1758, for instance, can be viewed as a physiocratic model showing that agricultural surpluses, by flowing through the economy in the form of rent, wages and purchases, were the real economic powers. Another instance is Johann von Thünen’s spatial diagram of farm prices in relation to distance from towns (Von Thünen 1966). Another is David Ricardo’s table of farm accounts (Ricardo 1821), which might be understood as a classical model explaining how income is distributed in the agricultural economy. The case of David Ricardo deserves special attention. Like other authors, he used mathematics in a formal manner through quantified, simple examples. During a controversy on the determination of the value of goods with Thomas Robert Malthus, Ricardo made mathematical examples in order to see whether value is independent of distribution. The result of such examples was against his intuition, so elementary mathematics can help someone to test the validity of their idea. This is an instance of the idea that mathematics can constitute 9 For the case of France, see PEPS-économie (2013). 10 To whom we are indebted in this chapter.

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an “impartial spectator” in the analysis of economic problems. After about 1870 came the second moment in the historical development of modeling in economics. The use of formal equations, algebra, calculus, geometry and other basic mathematical tools was common by this time. As outlined by Mirowski (1991), the incursion into political economy by scientists trained in physics, all familiar with the concept of equilibrium in a field of force, led them to use the mathematical tools of physics in economics. As stated previously, we will give three instances of attempts at modeling, which are generally still studied in the training of economists. In 1892, Irving Fisher provided a modelization of a mini-economy, characterized by three goods and three consumers, through a hydraulic mechanism. Some years before, in 1881, Francis Ysidro Edgeworth (1881) highlighted a diagram representing the range of possible contracts between two agents on an island; how Robinson Crusoe might gain Friday’s help in cultivating their island economy. Just before, in 1879, and in order to explain more clearly how two countries trade with each other, Marshall (1879) drew diagrams in which the curves represent the offers of German iron for English cloth and vice versa as relative prices change. The case of Marshall is rather amusing in light of the previous instances. Marshall’s diagrams often appeared in footnotes, and he considered them useful only if they could be illustrated with real examples from economic life. If not, he thought that they should be burnt! It was in the 1930s that the contemporary idea of the term model became explicit in economics. The “scientific conception of the world” outlined in the Vienna Circle manifesto, relying upon empirical and positive knowledge based on immediate data and application of the logical analysis method, was in tune with the times. The desire to unify science and scientific language was resurfacing, and the figure of the “social scientist” was becoming increasingly widespread. In addition, two international organizations linked to Bourbaki’s formalist school, whose aim was to promote the relation between economic theory, mathematics and statistics, saw the light of day: the Econometric Society in 1930, and the Cowles Commission for Research in Economics in 1932. Also of note were two economists who won the first Nobel Prize in Economics in 1969 and who are important names to associate with model-based research: the Norwegian Ragnar Frisch and the Dutchman Jan Timbergen. The former developed one of the first mathematical models of the business cycle during the Great Depression, in 1933. Then, his “macro-dynamic system” formed the basis for the first econometric model of a whole economy in the history of economics, the model embedded statistical information from the Dutch economy in the parameters of the business cycles’ equations, (Tinbergen 1937) in order to see how to pull the Netherlands out of the Depression. From this moment on, and particularly after World War II, the use of mathematics has intensified in economics. If we believe Debreu (1991), in 1940, fewer than 3% of the pages of the American Economic Review (Volume 30, for instance) included mathematical expressions, while in 1990, 40% of the pages (Volume 80, for example) included sophisticated mathematics. The naturalization of modeling in economics peaked in 1980, when Robert Lucas, Nobel Prize winner in 1995, wrote the following

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words: “I believe to be the more essential truth, stressed in my introduction, that progress in economic thinking means getting better and better abstract, analogue economic models, not better verbal observations about the world” (Lucas 1980, p. 700). 12.2.3. A number of caveats The point in this chapter is not to criticize mathematical economics “freely”. Nonetheless, the previous arguments we have given to justify the mathematization of economics, i.e. the ontological argument and the language argument, may easily lead to Lucas’ reductionist conception of progress in economic thinking, putting aside “verbal observations about the world”. We hold that such a chain of ideas gives rise to what we call “an excessive mathematization of economics”. We will now point out a number of caveats in this regard. Regarding the language argument, mathematics may indeed be useful for economics, mainly for its logical rigor. This helps the cumulative process of knowledge, making it easier for research studies to build on previous findings. When Lucas published his paper on the neutrality of money in 1972, his 20 pages of mathematical economics were not really open to interpretation. For economists, his model is rigorous, and likewise the transparency of himself and his results, so that the choice for the reader is relatively easy. The point is less to know what Lucas really meant in his paper than to know whether we agree with him or not. In contrast, the case of John Maynard Keynes’ The General Theory of Employment Interest More is interesting. This major book in the history of economic thought, despite Keynes’ attempt to formalize a piece of his theory, is today still being debated among scholars11. This analogy between Keynes and Lucas already appeared in De Vroey (2002). Still, even if mathematics allows hypothesis and the results of an economic theory to be explicited, and gives it logical rigor, it is not always transparent, in that formalisms can be interpreted in different ways. As outlined by Boumans (2005, p. 14), the manner and the context in which authors such as Frisch, Tinbergen and Michal Kalecki claimed that mixed differencedifferential equations are the most suitable formalism for business-cycle models is a good illustration of the caveat regarding the viewing of mathematics as perfectly transparent. After all, in some respect, mathematics may be viewed as a product of thought, and as such can belong to Karl Popper’s third world12. Regarding the ontological argument, it is not because economics deals with quantifiable notions, such as price and money, that it is basically mathematical. Though prices in modern markets conform to specific algebraic structures, we do not know whether most economic entities have historically presented the algebraic structure and invariance property required to apply mathematical processing to them. Quite the contrary, as Mirowski (2012, p. 122) observed, there is no evidence that prices, convenience units and money have always been conceived as numbers, in a purely ontological sense: they depended, and still depend, on too many other social practices 11 This analogy between Keynes and Lucas already appeared in De Vroey (2002) 12 Popper (1968)

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(Kua 1986 and Hadden 1993). Prices are not the a priori products of nature; rather, they are provisional invariances imposed upon the motley variety of human perception by various conventions and social structures. Both economics and mathematics depend on cultural and institutional elements, so that it is difficult for the former to be a matter of the latter. Moreover, the mathematician and economist Georgescu-Roegen (1966, p. 124), praised by Samuelson, acknowledged that “the difficulty of the subject of economics does not lie in the mathematics it needs, but in the fact that the subject itself is too involved to be fully accessible to mathematics. And what makes the subject (economics) not fully amenable to mathematics is the role that cultural propensities play in the economic process”. To a certain extent, this echoes Boulding’s criticisms of Samuelson, drawing on Poincaré (1952), according to which economic variables are always in relation to the whole and are therefore internally heterogeneous, which makes it difficult to capture them with numerical averages (Boulding 1948, p. 188; Munir 1994, pp. 254–255). More generally, many thinkers, such as that of the Historical School or Karl Polanyi, have viewed economics as embedded in complex social environments and history and thus put into perspective any attempts to naturalize mathematical modeling in economics. In brief, despite its utility, an excessive mathematization of economics has a price: it puts aside realities of the world which are not mathematizable but expressible by verbal language. Let us therefore conclude this part by quoting the case of John Hicks, Nobel Prize winner in 1972, who is widely known for his role in the development of famed modeling in economics such as the IS/LM model Hicks (1937). When Hicks was asked whether or not Britain should enter the European Monetary System, he answered: “That is the sort of question about which economists should have something to say... A lot of these mathematical models, including some of my own, are really terribly much in the air. They lost their feet off the ground” (Klamer 1989, p. 180). 12.3. For a reasonable mathematization of economics: the case of Mayotte Even an island such as Mayotte, located in the Mozambique Channel and the Indian Ocean, cannot resist the increasing process of what Israel (1986) calls “the mathematization of the real”. Mayotte’s, economy is no exception to the rule, especially since becoming a French department territory (section 12.1). Nonetheless, this is a difficult endeavor in light of the formal and informal characteristics of Mayotte’s economy. After briefly presenting a macroeconomic panorama of Mayotte’s economy, we will see that the formal indicators easily enable us to deduce the existence of significant informal activities (section 12.2). Yet knowing that there are such activities tells us nothing about how to understand them, especially when institutions prioritize measuring Mayotte’s formal economy, which is already a difficult task for France’s youngest department. We will use examples to show the remarkable fact that Mayotte’s formal economy cannot be strictly separated from

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informal activities and that their interactions, difficult to mathematize traditionally, might be viewed as issues for the modelization of complex systems (section 12.3). 12.3.1. The trend towards the mathematization of the economics of Mayotte In a world which tends to be mathematized, figures or numbers tend to impact public opinion. Statistics are used in many dimensions of human activities to study various phenomena, so that no one should ignore this “data science”. When economics contributes to making history and takes on an increasingly large place in current societies, we observe some sort of “economic statistization” of daily life, illustrated by the development of economic indicators (gross domestic product, unemployment rate, inflation rate, etc.). Every day, the media informs us about their evolution. Mayotte now has to deal with such indicators. On 29 March 2009, 95.2% of the electorate voted in favor of the transformation of Mayotte into a French Overseas Department and Region. Almost two years later, on March 31, 2011, the departmental community of Mayotte became the 101st French Department and the 5th Overseas Department and Region. Finally, on July 11, 2012, the European Parliament declared it was favorable to Mayotte becoming a European Outermost Region, which has been effective since January 1, 2014. Since then, Mayotte has been obliged to provide the European Commission with statistics in order to supply economic indicators. Though “the measure of Mayotte’s formal economy is already difficult”13 in a territory in which the implementation of census tools is very recent, it is now important to collect data properly. As such, any statistical and econometric study of Mayotte’s economy is obviously welcome. 12.3.2. From Mayotte’s formal economy to its informal one “The island of records”: this is one definition of Mayotte that comes to mind when we consider its characteristics in terms of (sub)development in France Mayotte Hebdo (2016). Though the growth rate of Mayotte’s domestic product in volume was about 10.5% in 2017 (this is the first record for the economic annual growth of a French department nowadays), this growth is still more exclusive than inclusive. In short, if we focus on Mayotte’s per capita growth, the island is the poorest French department and one of the poorest European region. Mayotte is a “champion” in terms of “migratory pressure”, whereas paradoxically, the migratory balance is negative. In other words, on the one hand, many migrants come to Mayotte to attempt to ensure their subsistence; on the other hand, many other individuals (usually graduates or those who can aspire to a French mainland education) leave Mayotte in order to bolster their existence. Besides the “migratory pressure”, there is a high birth rate, 13 As admitted by an economist from the Issuing Institute of the Overseas Departments.

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meaning that Mayotte is characterized by the highest population growth in France. Mayotte is also a “champion” regarding the difficulties of access to healthcare and education: its ranking rises to 100th worldwide in terms of the human development index (Goujon and Hermet 2012). This is also the case in respect of the unemployment rate, which stands at 25.9% (according to the ILO definition), knowing that there are also many jobless natives of Mayotte who are desirous to find one but who are not considered as “unemployed” in the ILO sense. In total, the unemployed and the inactive who wish to work make up at least 35% of the population (IEDOM 2018, p. 42). Having drawn the broad lines of the macroeconomic panorama, we can use other indicators to indirectly deduce the existence of a significant informal economy in Mayotte, which we define in a broad sense as any kind of economic activities beyond the legal and statistical norms (Charmes 1994). For instance, besides the “halo” dimension regarding the measure of its unemployment rate, bank accounts per capita stand at only 0.67, whereas there is strong issuance and evolution of cash (INSEE 2019). These formal signs of informal activities are completed by some brief reports on the phenomena, such as that of the Association for the Right to Economic Initiative (ADIE) published in 2008 and that of France’s National Institute of Statistics and Economic Studies (INSEE) published in 2018. According to the latter report, broadly speaking, whereas informal firms14 accounted for two-thirds of all merchant companies in 2015, they provide only 9% of the added value of all companies, representing 54 million euros. They are usually small (1.3 workers per business on average). Half of the leaders of these units are women; compared to the rest of Mayotte’s population, they have a lower level of education and are more often natives of the Comoros, so that they invest little and their productivity is low. As useful as such quantitative information may be, and regardless of its pertinence, it does not really highlight the complexities of the phenomena of informal activities in Mayotte. The concept of informal economy is already complicated in secondary literature (Portes 2003). In this chapter, we broadly define the phenomena as Hart (1973) did, that is undeclared income-generating activities in urban administration. One of these complexities is that they are not independent of formal activities. Some illustrations from our paper (Bianchini and Hochet 2019) are presented below and give some food for thought for the modelization of complex systems. 12.3.3. When the formal economy interacts with the informal one: some issues for the modelization of complex systems Let us first make some observations about the delicate notion of complex systems. The idea according to which “God” has written “the truth” in a mathematical language revealing “natural” laws seems to be put into perspective by what could be called the “complexity approach”, which nowadays is usually associated with 14 That are defined as companies with or without a legal existence evidenced by the listing in the Sirene directory, and which are not known to the tax authorities in terms of tax declarations.

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the Sante Fe Institute, an institution of teaching and multidisciplinary graduate research (Waldrop 1992, Arthur et al. 1997, Brock 2000). Although it is evident that life is a complex matter and that we all interact in a complex world, the definition of complexity is far from being a simple matter, which could explain why there is no unified “science of complexity”. As observed by Talbott (2001, p. 16), “(t)here is no consensus definition of complexity studies . . . Indeed the subject matter is often taken to be scarcely distinguishable from ‘everything’, which is perhaps why the disciplines at issue have so far yielded a richer harvest in vague hunches than concrete results”. Nonetheless, many scholars try to define a framework for complex systems via the numerous and various properties which characterize them which could be, according to Hooker’s classification (Hooker 2011, pp. 20–21): nonlinearity, irreversibility; constraints (holonomic and nonholonomic); equilibria and stabilities (static and dynamic); amplification; sensitivity to initial conditions; finite deterministic unpredictability; symmetry breaking; bifurcations; self-organization; emergence; constraints (enabling and coordinated); intrinsically global coherence; order; organization; modularity; hierarchy; path dependence and historicity; constraint duality; super-system formation; coordinated spatial and temporal differentiation with functional organization; multi-scale and multi-order functional organization; autonomy; adaptation; adaptiveness; learning; model specificity/model plurality; model centeredness; condition-dependent laws. Despite the different meanings which might sometimes be viewed as too “formal” (Israel 2005), it appears that a complex system necessarily involves the idea that “the whole is greater than its sum”. In a general way, therefore, it is possible to draw on this idea, which, as stated by Israel (2005, p. 492) “can be traced as far back as Aristotle”, is as old as it is new, for talking about complexity. Today, many authors, such as Edgar Morin, are taking up the old adage, according to which it is not reasonable to know the parts without trying to know the whole, nor to know the whole without knowing the parts, because both are inseparable. In other words, complexity is closely tied to interaction, and it is in this general way that we understand the notion in this chapter. Let us now try to see in what way Mayotte’s economy is a complex matter, in that its formal activities can interact with informal ones15. One of the most common and wellknown interactions is outsourcing, which consists of employing undeclared employees on construction sites, small businesses or sweatshops. This type of configuration, in which legal construction companies respond to public procurement and employ undeclared labor, is observed in both the most developed countries and Mayotte. Addon activities in addition to a declared job are also widespread informal practices. In Mayotte, they sometimes take on original forms, such as those mainly observed in 15 This paper suggests that Mayotte’s economy could be studied through what we today call “systemic economy”. On such matter, we can refer to Lesourne (1978), Fuerxer (1992), Le Moigne (1998) or Farmer and Foley (2009). This being, it is not the aim of this chapter to review the too various literature existing on the subject, or to define a complex system which could properly model Mayotte’s economy. These two interesting stakes deserve special and considerable attention, so that they should be the objects of other papers.

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female employees, for example, the Mahoran official who for several years has been organizing traveling trips in the Arabian Peninsula, China, India and South-East Asia during her vacations. Her purchases in these regions will thus be sent to Mayotte in containers filled with fabrics, furniture and other manufactured products of lesser value intended for family, villagers or resale in the shop run by her husband. Of course, there are methods, both direct (questionnaires) and indirect (macro-modelization), for estimating these kinds of informal activities, but they do not really highlight the interactions between informal practices and formal ones. The inter-penetrations between these sectors appear more, for instance, in socio-anthropological studies on corrupt practices considered as illegal practices but tolerated to different degrees. These studies show the preponderance of the social, political and cultural context to apprehend the different degrees of tolerance of the population with regard to a range of “practical norms” that do not comply with legal norms, be it in the sphere of tax optimization of small businesses or white-collar insider trading (Heidenheimer and Johnston 2011; Blundo and Olivier de Sardan 2007). We will finish with the land question. As a DIECTTE director has remarked, “(t)he first problem for economic activity in Mayotte is the land question, because it compromises the establishment and development of businesses. But this does not only concern companies: it is also necessary to understand that today 50% of private constructions are made without planning permission” (Interview of May 5, 2017 with M. L., DIECCTE). Indeed, the land question is an issue in Mayotte, as well as an interesting field study highlighting the interactions between formal and informal activities, which might be understood as different social logics of reciprocity specific to a moral economy. In the context of the regularization of ownership titles initiated by the public authorities at the end of the 1990s, land regularization was intended to facilitate the consolidation of larger exploitable areas in order to stimulate development policies to support intensive crops. But the cost added to the length of the process of demarcation and registration of a nominative title paradoxically led to the consequences of reducing titled areas, and encouraging the use of informal work to exploit and secure the vacant plots of land inherited according to customary rules. Therefore, instead of encouraging the population to embark on the legal procedures for registering their properties on the land register, the application of positive law had the unexpected effect, especially in rural areas, of increasing the use of illegal labor. And, by doing so, the effect was to multiply informal transactions privately or by witness according to the traditional rules of the local moral economy. Remarkably, the application of French common law has therefore not only maintained, and even strengthened, the permanence of customary land governance directly related to the informal economy, particularly in agriculture, but also generated formal land uncertainty, which is presented by the authorities as a general brake on investment in all economic sectors and for spatial planning (Court of Auditors, 2016). Last but not least, the transition towards French positive law has been accompanied by changes in the symbolic and monetary value of the land. On the one hand, land has become a means of producing a new monetary surplus thanks to the perception of a rental income; on the other hand, land remains

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a “prestigious good” allowing the strengthening of its community memberships by a territorial anchorage. These transformations can therefore contribute to explaining the rise of the construction sector, one of the island’s main economic sectors, and in which the use of informal labor appears widespread. Despite the application of French common law, which is tending to transform land values in Mayotte, informal activities remain and seem to be a resilient phenomenon. The resilience of Mayotte’s informal economy thus appears to be an interesting issue for studies on complexity16. 12.4. Concluding remark So, what is an excessive mathematization of Mayotte’s economy about? A conception of an economy whose language would correspond to its so-called mathematical nature, and whose progress would be independent of verbal observations. This would thus put aside the cultural and historical dimensions in which Mayotte’s economy is embedded. Reducing Mayotte’s economy to numbers or symbols might then be equivalent to dispossessing the island of its qualities, rather like claiming that quantification of the world conflicts with the personification of behaviors Musil (1956). Despite this caveat, Mayotte is now a French department and, as such, should collect data and robust economic indicators. A considerable issue might then be to strike a balance between the inevitable mathematization of Mayotte’s economy and its plural diversities. Though systemic thinking has not really penetrated economics yet, let us hope that it could be sensitive to finding the right balance. 12.5. References ADIE (2008). Le travail indépendant informel à Mayotte. Study report. Armatte, M. (2003). L’axiomatisation et les théories économiques : un commentaire. Revue économique, 54(1), 99–138. Armatte, M. (2005). La notion de modèle dans les sciences sociales: Anciennes et nouvelles significations. Mathématiques et Sciences Humaines. Online since 22 April 2006. Arthur, W., Durlauf, S., and Lane, D. (eds) (1997). The Economy as an Evolving Complex System II. Perseus, Cambridge. Bianchini, V. and Hochet, A. (2019). Comprendre l’économie informelle à Mayotte : Fonctions, paradoxes, et dynamique des valeurs foncières. Working paper. 16 A well-known, though delicate, property of many complex systems is indeed the ability of the system to react to perturbations, internal failures and environmental events by absorbing the disturbance and/or reorganizing to maintain its functions: this is resilience. For a review on the state of the art on the resilience of complex systems, see Fraccascia et al. (2018).

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Fuerxer, J. (1992). Économie et systémique. In Systémique, Théorie et applications, Le Gallou, F. and Bouchon-Menier, B. (eds). Gesta, 248–258. Georgescu-Roegen, N. (1966). Analytical Economics: Issues and Problems. Harvard University Press. Goujon, M. and Hermet, F. (2012). L’indice de développement humain: une évaluation pour Mayotte. Région et Développement, LEAD. Université du Sud, Toulon. Hadden, R. (1993). On the Shoulders of Merchants : Exchange and the Mathematical Conception of Nature. SUNY Press, Albany. Hart, K. (1973). Informal income opportunities and urban employment in Ghana. The Journal of Modern African Studies, 11, 61–89. Heidenheimer, A. and Johnston, M. (2011). Political Corruption: Concepts and Contexts. Transaction Publishers, New Brunswick. Hicks, J. (1937). Mr Keynes and the classics: Econometrica, 5(2), 147–159.

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Hooker, C. (ed.) (2011). Philosophy of Complex Systems, Handbook of the Philosophy of Science, 10, Elsevier. IEDOM (2018). Mayotte. Annual report 2017. INSEE (2018). Enquête sur les entreprise mahoraises en 2015. Analyse no.16, February. INSEE (2019). Un taux de chômage de 35%. Flash info no. 82, February. Israel, G. (1986). La mathématisation du réel. Essai sur la modélisation du réel. Le Seuil, Paris. Israel, G. (2005). The science of complexity: Epistemological problems, and perspectives. Science in Context, 18(3), 479–509. Jevons, W.S. (1888). Theory of Political Economy, 3rd edition. MacMillan and Co, London. Klamer, A. (1989). An accountant among economists: Conversations with sir John R. Hicks. Journal of Economic Perspectives, Fall 1989, 167–180. Kua, W. (1986). Measures and Men. Princeton University Press, Princeton. Le Moigne, J.L. (1998). Modéliser socioéconomiques. Economica, Paris.

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List of Authors

Julien BALICCHI ARS Mayotte France

Sophie DABO-NIANG University of Lille France

Anne BARBAIL ARS Mayotte France

Benoîte DE SAPORTA University of Montpellier France

Victor BIANCHINI CUFR de Mayotte and University of Nîmes France

Toka DIAGANA University of Alabama in Huntsville USA and King Fahd University of Petroleum and Minerals Saudi Arabia

Patrice BREHMER IRD Centre de Bretagne France Julie CARREAU IRD University of Montpellier CNRS France Alice CLEYNEN University of Montpellier France

William DIMBOUR University of French Guiana France Jean DHOMBRES Centre Alexandre Koyré France Massal FALL Senegalese Institute for Agricultural Research CRODT-ISRA Senegal

Mathematical Modeling of Random and Deterministic Phenomena, First Edition. Edited by Solym Mawaki Manou-Abi, Sophie Dabo-Niang and Jean-Jacques Salone. © ISTE Ltd 2020. Published by ISTE Ltd and John Wiley & Sons, Inc.

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Vincent GUINOT University of Montpellier CNRS IRD INRIA France Dominique HERVÉ IRD Madagascar Antoine HOCHET CUFR de Mayotte and GRED/IRD Montpellier France Hugo LEIVA University Yachay Tech Ecuador Loïc LOUISON University of French Guiana France Solym Mawaki MANOU-ABI IMAG de Montpellier and CUFR de Mayotte France Mamadou Moustapha MBAYE University of Dakar Senegal

Mamadou N’DIAYE University of Dakar Senegal Papa NGOM University of Dakar Senegal Abdennebi OMRANE University of French Guiana France Étienne PARDOUX Aix-Marseille University France Jean-Jacques SALONE IMAG de Montpellier and CUFR de Mayotte France Ndiaga THIAM Oceanographic Research Center of Dakar-Thiaroye CRODT-ISRA Senegal Gwladys TOULEMONDE University of Montpellier CNRS INRIA France

Index

A, B, C analytic semi-group, 141–143, 157 ARESMA, 193–196, 199, 201, 203, 204, 213 ARIMA model, 249, 253 Banach fixed point theorem, 136, 141 birth curve, 245, 247, 257, 259 bounded solution, 141, 142, 147, 150–155, 158 Burgers equation, 141, 156, 158 central limit theorem, 3, 4, 10, 15 complex system, 193, 195–199, 205, 208, 209, 262, 269–271 conceptual model, 194, 202 convergence theorem, 123, 126, 130, 133, 134 D, E, F dry forests, 222, 223, 228 dynamic systems, 229, 237 ecosystems, 221, 237, 238 epidemic model, 3, 4 extreme rainfall, 53–55, 64 fish biomass prediction, 32, 42, 47, 49 fixed design, 40 Fourier series, 161, 174, 176, 178–180, 186, 187

fractional integro-differential equation, 113, 114 power spaces, 141 functional spatial dependent data, 31 H, I, L heat equation, 137 flux, 163, 164 partial differential equation, 187 Hilbert space, 179, 183, 184 hypergraphs, 193, 197, 205–207, 212 impulsive semilinear evolution equation, 141 large deviations, 4, 10, 13, 15, 17, 18, 27 law of large numbers, 3, 4, 9, 12 low-regret control, 98, 99, 103, 106–108, 110 M, N, O mangrove, 193–199, 201–205, 208–210, 212, 213 Markov chain, 223 mathematization of economics, 263–265, 267, 268 model, 261, 263, 265–268, 271

Mathematical Modeling of Random and Deterministic Phenomena, First Edition. Edited by Solym Mawaki Manou-Abi, Sophie Dabo-Niang and Jean-Jacques Salone. © ISTE Ltd 2020. Published by ISTE Ltd and John Wiley & Sons, Inc.

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moderate deviations, 3, 4, 12, 13, 15, 18, 27 non-local condition, 141, 142, 153, 156, 158 nonparametric prediction, 31 nutrient uptake, 98, 99, 103, 110 Nye-Tinker-Barber model, 97, 98, 110 optimal stopping problem, 75, 76, 78, 80, 81 orthogonality relation, 162, 174, 177, 187 P, Q, S periodic function, 161, 167, 169, 173, 177–180, 183 solution, 113, 114, 115, 118 piecewise deterministic Markov processes (PDMPs), 73, 74 post-forestry, 221–223 prediction, 245, 249, 256–259

public health, 248 quantization, 76, 83, 84, 86, 87 smoothness, 142, 151 space-time modeling, 63 spatial stochastic generator, 54, 57–59, 61, 63–65 stability, 142, 150, 151 statistic, 266, 269, 270 statistical analysis, 252, 257 step function, 169, 174, 175, 183, 187 stochastic, 3–9, 14, 27 control, 73, 76, 83 process, 114–116, 119, 120, 122, 127, 129, 134 T, U, W thought experiment, 161, 162, 175, 187 urban flood risk, 59, 63 wave, 164, 167, 174–176, 186–188 wavelet, 162, 163, 186–188

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