Mathematical Modeling of Random and Deterministic Phenomena E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 404

Veröffentlichungsjahr: 2020

Beschreibung

This book highlights mathematical research interests that appear in real life, such as the study and modeling of random and deterministic phenomena. As such, it provides current research in mathematics, with applications in biological and environmental sciences, ecology, epidemiology and social perspectives. The chapters can be read independently of each other, with dedicated references specific to each chapter. The book is organized in two main parts. The first is devoted to some advanced mathematical problems regarding epidemic models; predictions of biomass; space-time modeling of extreme rainfall; modeling with the piecewise deterministic Markov process; optimal control problems; evolution equations in a periodic environment; and the analysis of the heat equation. The second is devoted to a modelization with interdisciplinarity in ecological, socio-economic, epistemological, demographic and social problems. Mathematical Modeling of Random and Deterministic Phenomena is aimed at expert readers, young researchers, plus graduate and advanced undergraduate students who are interested in probability, statistics, modeling and mathematical analysis.

Leseprobe

Table of Contents

Cover

Preface

Acknowledgments

Introduction

PART 1: Advances in Mathematical Modeling

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

1.1. Introduction

1.2. The three models

1.3. The stochastic model, LLN, CLT and LD

1.4. Moderate deviations

1.5. References

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

2.1. Introduction

2.2. Regression model and predictor

2.3. Large sample properties

2.4. Application to demersal coastal fish off Senegal

2.5. Conclusion

2.6. References

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

3.1. Why?

3.2. How?

3.3. Outlook

3.4. References

4 Change-point Detection for Piecewise Deterministic Markov Processes

4.1. A quick introduction to stochastic control and change-point detection

4.2. Model and problem setting

4.3. Numerical approximation of the value functions

4.4. Simulation study

4.5. Conclusion

4.6. References

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

5.1. Introduction

5.2. Statement of the problem

5.3. Optimal control for the NTB problem with an unknown source

5.4. Characterization of the low-regret control for the NTB system

5.5. Concluding remarks

5.6. References

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

6.1. Introduction

6.2. Preliminaries

6.3. A stochastic integro-differential equation of fractional order

6.4. An illustrative example

6.5. References

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

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

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”

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

8.4. The numerical example of the periodic step function gives way to a physical interpretation

8.5. To invoke arbitrary functions leads to an interpretation of orthogonality relations

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

8.8. Conclusion

8.9. References

PART 2: Some Topics on Mayotte and Its Region

9 Towards a Methodology for Interdisciplinary Modeling of Complex Systems Using Hypergraphs

9.1. Introduction

9.2. Systemic and lexicometric analyses of questionnaires

9.3. Hypergraphic analyses of diagrams

9.4. Discussion and perspectives

9.5. Appendix

9.6. References

10 Modeling of Post-forestry Transitions in Madagascar and the Indian Ocean: Setting Up a Dialogue Between Mathematics, Computer Science and Environmental Sciences

10.1. Introduction

10.2. Interdisciplinary exploration of agrarian transitions

10.3. Community management of resources, looking for consensus

10.4. Discussion and conclusion

10.5. References

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

11.1. Introduction

11.2. Origin of the data

11.3. Methodologies and results

11.4. Discussion

11.5. Conclusion

11.6. References

12 Reflections Upon the Mathematization of Mayotte’s Economy

12.1. Introduction

12.2. Justifying the mathematization of economics

12.3. For a reasonable mathematization of economics: the case of Mayotte

12.4. Concluding remark

12.5. References

List of Authors

Index

End User License Agreement

List of Tables

Chapter 2

Table 2.1. Mean Square Errors of prediction for when the covariate is the sali...

Table 2.2. Mean Square Errors of prediction for when the covariate is the temp...

Chapter 3

Table 3.1. Rainfall field parameters

Chapter 4

Table 4.1. Average cost of the strategies for MA, KF and the approach proposed i...

Table 4.2. Average cost of the strategies for MA, KF and the approach proposed i...

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

Chapter 11

Table 11.1. Number of times the different months went up (in red) or down (in bl...

Table 11.2. The “Observed births” column presents the number of births registere...

Table 11.3. Relationship between two consecutive months. Field: Births in Mayott...

Guide

Cover

Table of Contents

Begin Reading

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Series EditorNikolaos 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 Ltd27-37 St George’s RoadLondon SW19 4EUUK

www.iste.co.uk

John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA

www.wiley.com

© 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 DataA CIP record for this book is available from the British LibraryISBN 978-1-78630-454-4

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 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 MANOU-ABISophie DABO-NIANGJean-Jacques SALONENovember 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 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 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 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) 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.

Introduction written by Solym Mawaki MANOU-ABI, Sophie DABO-NIANG and Jean-Jacques SALONE.

PART 1Advances in Mathematical Modeling