Crowds in Equations: An Introduction to the Microscopic Modeling of Crowds

201 Pages·2019·15.044 MB·English

by  Bertrand Maury ,  Sylvain Faure

Crowds
in
Equations
An Introduction to the Microscopic
Modeling of Crowds
Q0163hc_9781786345516_tp.indd 1 14/5/18 9:14 AM
Advanced Textbooks in Mathematics
Print ISSN: 2059-769X
Online ISSN: 2059-7703

The Advanced Textbooks in Mathematics explores important topics for post-
graduate students in pure and applied mathematics. Subjects covered within this
textbook series cover key fields which appear on MSc, MRes, PhD and other
multidisciplinary postgraduate courses which involve mathematics.
Written by senior academics and lecturers recognised for their teaching skills,
these textbooks offer a precise, introductory approach to advanced mathematical
theories and concepts, including probability theory, statistics and computational
methods.
Published
Crowds in Equations: An Introduction to the Microscopic Modeling
of Crowds
by Bertrand Maury and Sylvain Faure
The Wigner Transform
by Maurice de Gosson
Periods and Special Functions in Transcendence
by Paula B Tretkoff
Mathematics of Planet Earth: A Primer
by Jochen Bröcker, Ben Calderhead, Davoud Cheraghi, Colin Cotter,
Darryl Holm, Tobias Kuna, Beatrice Pelloni, Ted Shepherd and Hilary Weller
edited by Dan Crisan
Forthcoming
Conformal Maps and Geometry
by Dmitry Beliaev
Vishnu Mohan - Q0163 - Crowds in Equations.indd 1 05-07-18 3:50:24 PM
Advanced Textbooks in Mathematics
Crowds
in
Equations
An Introduction to the Microscopic
Modeling of Crowds
Jochen Bröcker • Ben Calderhead • Davoud Cheraghi
Bertrand Maury
Colin Cotter • Darryl Holm • Tobias Kuna
Université Paris-Sud, France Beatrice Pelloni • Ted Shepherd • Hilary Weller
Sylvain Faure
Université Paris-Sud & CNRS, France
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
Q0163hc_9781786345516_tp.indd 2 14/5/18 9:14 AM
Published by
World Scientific Publishing Europe Ltd.
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Head office: 5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
Library of Congress Cataloging-in-Publication Data
Names: Maury, Bertrand, author. | Faure, Sylvain, 1976– author.
Title: Crowds in equations : an introduction to the microscopic modeling of crowds / by
Bertrand Maury (Université Paris-Sud, France),
Sylvain Faure (Université Paris-Sud & CNRS, France).
Description: New Jersey : World Scientific, 2018. | Series: Advanced textbooks in mathematics |
Includes bibliographical references and index.
Identifiers: LCCN 2018013165 | ISBN 9781786345516 (hc : alk. paper)
Subjects: LCSH: Mathematical analysis. | Mathematical models. | Communication in mathematics.
Classification: LCC QA401 .M448 2018 | DDC 511/.8--dc23
LC record available at https://lccn.loc.gov/2018013165
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2019 by World Scientific Publishing Europe Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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For any available supplementary material, please visit
http://www.worldscientific.com/worldscibooks/10.1142/Q0163#t=suppl
Desk Editors: V. Vishnu Mohan/Jennifer Brough/Koe Shi Ying
Typeset by Stallion Press
Email: [email protected]
Printed in Singapore
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June29,2018 10:20 CrowdsinEquations 9inx6in b3216-fm pagev
Foreword
Building bridges, remarking similarities, crossing methods are essential
driving forces of the scientific activity. Since ancient times, physics and
mathematics have been interwoven.Today, althoughthe amountof knowl-
edgemakesitessentiallyimpossibletohaveaglobalviewontheseso-called
hard sciences, interactions remain fruitful. On the other hand, social sci-
ences are — at first sight — unconcerned with this thinking system: they
areinterestedinamultitude ofbehaviourswhichseemtoescapethe deter-
ministic laws of physics and any mathematical formulation.
A small revolution of the last decade has been to remove this barrier
to allow a new research field to emerge. If one observes typical behaviours
(at the scale of many individuals) and that one can identify parameters
influencingthesebehaviours,thenthereisnoreasonthatthemathematical
languagecannotdescribethem!Ofcourse,theconnectionisnotsoeasyand
this new field is still in its infancy, not always considered seriously by its
elders.
In this book, intended to graduate students and researchers in
mathematics, Sylvain Faure and Bertrand Maury invite us to discover the
challengesandthe firstsuccessesofmathematics appliedto socialsciences.
As a preamble, they clearly explain the difficulties of the exercise, due, in
particular,tothefreedomofindividualsandtothedecisionprocesseswhich
are neither symmetric, nor interchangeable.
The book continues with the rigorous analysis of some models, essen-
tiallyatthemicroscopicscale,whichserveasmathematicalprototypesand
exhibit interesting phenomenologies.
v
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vi Foreword
Starting from this picture, the authors propose eventually to extract
the minimal elements which should be contained in a mathematical model
in order to reproduce some typical and sometimes paradoxical properties
of crowdmotions: “Faster-is-Slower”effect, “Stop-and-Go” waves,and flu-
idizing effects of an obstacle. This is fascinating! A quick and easy read,
which makes me want to learn more.
Laure Saint-Raymond
E´cole Normale Sup´erieure de Lyon &
Acad´emie des Sciences, France
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Contents
Foreword v
1. Introduction 1
1.1 From Passive to Active Entities . . . . . . . . . . . . . . . 1
1.2 Basics on Crowd Motion Modeling . . . . . . . . . . . . . 3
1.3 The Mathematical Standpoint . . . . . . . . . . . . . . . . 5
1.4 How to Use this Book? . . . . . . . . . . . . . . . . . . . . 12
2. One-Dimensional Microscopic Models 13
2.1 Follow-the-Leader Model . . . . . . . . . . . . . . . . . . . 14
2.2 Accounting for Inertia/Delays . . . . . . . . . . . . . . . . 30
3. Social Force Model, Native and Overdamped Forms 37
3.1 Inertial Social Force Model . . . . . . . . . . . . . . . . . 37
3.2 Overdamped Social Force Model . . . . . . . . . . . . . . 49
3.3 Alternative Approaches . . . . . . . . . . . . . . . . . . . 56
4. Granular Models 59
4.1 One-Dimensional Model . . . . . . . . . . . . . . . . . . . 59
4.2 Two-Dimensional Model . . . . . . . . . . . . . . . . . . . 61
4.3 Numerical Scheme . . . . . . . . . . . . . . . . . . . . . . 64
4.4 Numerical Experiments . . . . . . . . . . . . . . . . . . . 66
4.5 Mathematical Issues . . . . . . . . . . . . . . . . . . . . . 68
4.6 Critical Discussion . . . . . . . . . . . . . . . . . . . . . . 76
vii
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viii Contents
5. Cellular Automata 83
5.1 Cellular Automata: General Principles . . . . . . . . . . . 84
5.2 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.3 Variations, Extensions . . . . . . . . . . . . . . . . . . . . 92
5.4 Cellular Automata, Mathematical Issues . . . . . . . . . . 93
6. Compartment Models 97
6.1 Compartment Models: Toy Versions
and General Setting . . . . . . . . . . . . . . . . . . . . . 97
6.2 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . 101
6.3 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.4 Numerical Illustration . . . . . . . . . . . . . . . . . . . . 104
6.5 Mathematical Framework: A Cascade
of Gradient Flows . . . . . . . . . . . . . . . . . . . . . . . 104
7. Toward Macroscopic Models 111
7.1 One-Dimensional Macroscopic Traffic Model . . . . . . . . 112
7.2 Two-Dimensional Models . . . . . . . . . . . . . . . . . . 115
7.3 Granular Models: Hard Congestion . . . . . . . . . . . . . 117
7.4 Micro–Macro Issues . . . . . . . . . . . . . . . . . . . . . . 123
7.5 Alternative Macroscopic Models . . . . . . . . . . . . . . . 125
8. Computing Distances and Desired Velocities 127
8.1 Shortest Path Problem on a Graph . . . . . . . . . . . . . 130
8.2 Shortest Path on a Domain: The Eikonal Equation . . . . 132
8.3 Non-homogenous Domains, Various Extensions . . . . . . 135
8.4 Shortest Paths in a Dynamic Environment . . . . . . . . . 139
8.5 Alternative Way to Compute Desired Velocities . . . . . . 142
8.6 Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . 143
9. Data, Observable Phenomena 145
9.1 Diameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
9.2 Proxemics, Interpersonal Distances, Density . . . . . . . . 146
9.3 Cone of Vision . . . . . . . . . . . . . . . . . . . . . . . . 148
9.4 Pedestrian Speed, Fundamental Diagram . . . . . . . . . . 148
9.5 Door Capacity . . . . . . . . . . . . . . . . . . . . . . . . 151
9.6 Capacity Drop Phenomenon . . . . . . . . . . . . . . . . . 151
9.7 Faster-is-SlowerEffect . . . . . . . . . . . . . . . . . . . . 152
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Contents ix
9.8 Influence of an Obstacle . . . . . . . . . . . . . . . . . . . 154
9.9 Stop-and-Go Waves . . . . . . . . . . . . . . . . . . . . . . 157
9.10 Further Considerations on Human Behavior . . . . . . . . 158
10. A Wider Look on Characteristic Phenomena in Crowds 161
10.1 Faster-is-SlowerEffect . . . . . . . . . . . . . . . . . . . . 161
10.2 Fluidizing Effect of an Obstacle . . . . . . . . . . . . . . . 168
10.3 Damping, Propagation,and Stop-and-Go Waves. . . . . . 170
Appendix 177
A.1 Ordinary Differential Equations . . . . . . . . . . . . . . . 177
A.2 Constrained Optimization . . . . . . . . . . . . . . . . . . 180
Bibliography 183
Index 189