Mathematical Modelling E-Book

Seyed M. Moghadas

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

Seitenzahl: 193

Veröffentlichungsjahr: 2018

Beschreibung

An important resource that provides an overview of mathematical modelling Mathematical Modelling offers a comprehensive guide to both analytical and computational aspects of mathematical modelling that encompasses a wide range of subjects. The authors provide an overview of the basic concepts of mathematical modelling and review the relevant topics from differential equations and linear algebra. The text explores the various types of mathematical models, and includes a range of examples that help to describe a variety of techniques from dynamical systems theory. The book's analytical techniques examine compartmental modelling, stability, bifurcation, discretization, and fixed-point analysis. The theoretical analyses involve systems of ordinary differential equations for deterministic models. The text also contains information on concepts of probability and random variables as the requirements of stochastic processes. In addition, the authors describe algorithms for computer simulation of both deterministic and stochastic models, and review a number of well-known models that illustrate their application in different fields of study. This important resource: * Includes a broad spectrum of models that fall under deterministic and stochastic classes and discusses them in both continuous and discrete forms * Demonstrates the wide spectrum of problems that can be addressed through mathematical modelling based on fundamental tools and techniques in applied mathematics and statistics * Contains an appendix that reveals the overall approach that can be taken to solve exercises in different chapters * Offers many exercises to help better understand the modelling process Written for graduate students in applied mathematics, instructors, and professionals using mathematical modelling for research and training purposes, Mathematical Modelling: A Graduate Textbook covers a broad range of analytical and computational aspects of mathematical modelling.

Leseprobe

Table of Contents

Cover

Preface

1 Basic Concepts and Quick Review

1.1 Modelling Types

1.2 Quick Review

Exercises

2 Compartmental Modelling

2.1 Cascades of Compartments

2.2 Parameter Units

Exercises

3 Analysis Tools

3.1 Stability Analysis

3.2 Phase‐Plane Behavior

3.3 Direction Field

3.4 Routh–Hurwitz Criterion

Exercises

4 Bifurcation

4.1 Transcritical Bifurcation

4.2 Saddle‐Node Bifurcation

4.3 Pitchfork Bifurcation

4.4 Hopf Bifurcation

4.5 Solution Types

Exercises

5 Discretization and Fixed‐Point Analysis

5.1 Discretization

5.2 Fixed‐Point Analysis

Exercises

6 Probability and Random Variables

6.1 Basic Concepts

6.2 Conditional Probabilities

6.3 Random Variables

Exercises

7 Stochastic Modelling

7.1 Stochastic Processes

7.2 Probability Generating Function

7.3 Markov Chains

7.4 Random Walks

Exercises

8 Computer Simulations

8.1 Deterministic Structure

8.2 Stochastic Structure

8.3 Monte Carlo Methods

Exercises

9 Examples of Mathematical Modelling

9.1 Traffic Model

9.2 Michaelis–Menten Kinetics

9.3 The Brusselator System

9.4 Generalized Richards Model

9.5 Spruce Budworm Model

9.6 FitzHugh–Nagumo Model

9.7 Decay Model

9.8 The Gambler's Ruin

Exercises

References

Index

End User License Agreement

List of Tables

Chapter 09

Table 9.1 Parameter values for the Morris–Lecar model.

List of Illustrations

Chapter 01

Figure 1.1 The process of model development, analysis, and validation.

Figure 1.2 Representation of a conical hourglass.

Figure 1.3 Representation of spring motion in a sinusoidal form.

Figure 1.4 Types of deterministic and stochastic models.

Chapter 02

Figure 2.1 Illustration of the balance law in modelling a process.

Figure 2.2 Illustration of radioactive material emitting particles.

Figure 2.3 Compartmental representation of drug absorption by the GI tract and diffusion into the bloodstream.

Figure 2.4 Linear (a) and branching (b) types of cascades in compartmental modelling.

Figure 2.5 Linear cascade for compartmental modelling of population growth.

Figure 2.6 Population size over time in exponential growth and logistic models with

and

.

Figure 2.7 Linear cascade of compartmental modelling for the dynamics of disease spread in a population.

Figure 2.8 Linear cascade of compartmental modelling for the dynamics of predator–prey interactions.

Figure 2.9 Branching cascade of compartmental modelling for the dynamics of influenza spread in a population.

Chapter 03

Figure 3.1 Representation of the stable node

in a neighborhood.

Figure 3.2 Representation of the solutions for system (3.2) as circles around the origin.

Figure 3.3 Representation of the solutions of (3.3) approaching the stable node

as

goes to

.

Figure 3.4 Representation of the local dynamics of a saddle node in the

plane.

Figure 3.5 Representation of the local dynamics of (a) a stable node (sink); and (b) an unstable node, in the

plane.

Figure 3.6 Representation of the local dynamics of (a) a stable spiral node; and (b) an unstable spiral node, in the

plane.

Figure 3.7 Representation of the local dynamics of a center in the

plane.

Figure 3.8 Representation of the dynamical behavior of system (3.8) around its critical points based on the constant values of

and

.

Figure 3.9 Representation of the phase plane for system (3.9).

Figure 3.10 Solutions of the system (3.10) approaching

when

. Parameter values are

,

,

,

, and

.

Figure 3.11 Solutions of the system (3.10) starting in the (

) region that approach

when

. Parameter values are

,

,

,

, and

.

Figure 3.12 Direction field and phase plane for the system (3.12).

Figure 3.13 Direction field and isoclines for the equation

with

(solid curve)

(dashed curve), and

(dot‐dashed curve).

Figure 3.14 Nullcline (bold curve) for the equation

in Example 3.10. Other curves show the same isoclines illustrated in Figure 3.13.

Chapter 04

Figure 4.1 Schematic diagram of transcritical bifurcation with (a)

and (b)

. Solid and dashed lines represent the presence of stable and unstable critical points, respectively.

Figure 4.2 Schematic diagram of the saddle‐node bifurcation in equation (4.2).

Figure 4.3 Schematic diagram of the saddle‐node bifurcation in equation (4.3).

Figure 4.4 Schematic diagram of the pitchfork bifurcation in equation (4.4).

Figure 4.5 Schematic diagram of the pitchfork bifurcation in equation (4.5).

Figure 4.6 Phase plane of system (4.7) for the presence of a period solution when the system undergoes a Hopf bifurcation with

. The period solution is stable and the Hopf bifurcation is supercritical.

Figure 4.7 Oscillatory dynamics of prey (

) and predator (

) for model (4.8) over time with

,

,

,

, and

.

Figure 4.8 Phase‐plane behavior of solutions with

and (a)

(stable critical point); (b)

(stable critical point); (c)

(stable periodic solution and unstable critical point); (d)

(stable critical point).

Figure 4.9 Representation of (a) stable and (b) unstable limit cycles.

Figure 4.10 Representation of semi‐stable limit cycles: (a) stable from outside and unstable from inside; (b) stable from inside and unstable from outside.

Figure 4.11 Representation of a homoclinic orbit.

Figure 4.12 Representation of a heteroclinic orbit.

Figure 4.13 The stable limit cycle for system (4.12) with

.

Chapter 05

Figure 5.1 Discretization of the interval

with a fixed time‐step

.

Figure 5.2 Approximations using the Euler method for the solution of (5.3) with

.

Figure 5.3 Approximations using the nonstandard method for the solution of (5.3) with

.

Figure 5.4 Time profile of the fraction of the population infected. with (a)

, (b)

, and (c)

. Other parameter values are

and

. Initial population sizes at time

are

,

, and

.

Chapter 06

Figure 6.1 Representation of sample spaces for the conditional probability of

.

Figure 6.2 Representation of a random variable defined on the sample space

.

Figure 6.3 Representation of a dartboard.

Figure 6.4 Probability of hitting the gray area on the dartboard in Figure 6.3 exactly once as a function of the number of throws.

Figure 6.5 Representation of a probability density function.

Chapter 07

Figure 7.1 Monthly exchange rate of the Canadian dollar against the US dollar over the period 1990–2017.

Figure 7.2 Monitoring mouse movements in a maze.

Figure 7.3 Transition diagram for the movements of the mouse in Example 7.3.

Figure 7.4 Representation of the areas for movements of a zebra.

Figure 7.5 Transition diagram for the movements of the zebra in Example 7.5.

Figure 7.6 Possible pathways for the zebra moving from the dry area (

) to water (

) in exactly two transitions (Figure 7.5). The probabilities of moves between states are indicated on the connecting arrows.

Figure 7.7 Transition diagram for a Markov chain process.

Figure 7.8 Transition diagram for a Markov chain process.

Figure 7.9 Transition diagram for a Markov chain process.

Chapter 08

Listing 8.1 : Matlab

©

code for simulating the predator–prey system (8.1). To run the simulations, parameter values, initial conditions, time interval, and time-step must be given as input.

Figure 8.1 Time profiles of the prey (solid line) and predator (dashed line) populations simulated with the Euler method using parameter values

,

,

, and

in the time interval

with fixed time‐step

.

Figure 8.2 Phase plane for the predator–prey system (8.1) simulated with the Euler method using parameter values

,

,

, and

in the time interval

with fixed time‐step

.

Listing 8.2 : Matlab

©

code for the SIR epidemic model, part I. Module for updating the state variables.

Figure 8.3 Simulation logic diagram for model (8.3).

Listing 8.3 : Matlab

©

code for the SIR-epidemic model, part II. Module for stochastic iterations.

Listing 8.4 : Matlab

©

code for the SIR epidemic model, part III. Module for simulations.

Listing 8.5 : Matlab

©

code for the SIR epidemic model, part I. Module for updating the state variables.

Figure 8.4 Stochastic simulations of the SIR epidemic model with

and

in a population with initial conditions

,

, and

. Gray curves show 200 independent realizations, and the black curve represents the average of these realizations. Simulations were performed using Listings 8.2–8.5.

Listing 8.6 : Matlab

©

code for the drunkard’s random walk.

Figure 8.5 Stochastic simulations of the random walk for the drunkard, using Listing 8.6.

Listing 8.7 : Matlab

©

code sampling a random variable with exponential distribution.

Figure 8.6 The inverse transform sampling method for the exponential distribution with mean

. The random number was sampled 100 times.

Chapter 09

Figure 9.1 Representation of traffic in a single lane.

Figure 9.2 Simulations for the Brusselator system with (a):

and

(for which the solutions approach the stable critical point

); and (b)

and

(for which the solutions approach the stable limit cycle).

Figure 9.3 The solution curves of the Richards model with

and

for different values of

: solid curve (

); dashed curve (

); and dot‐dashed curve (

).

Listing 9.1 : Matlab

©

code for simulating the generalized Richards model.

Figure 9.4 Representation of the critical points for model (9.21). Dashed lines correspond to different values of

.

Listing 9.2 : Matlab

©

code for simulating the FitzHugh–Nagumo model.

Figure 9.5 Simulations of the FitzHugh–Nagumo model with

,

,

, and

. The solid curve shows the stable limit cycle generated by the Hopf bifurcation.

Figure 9.6 Probability that 50% of the nuclei are still in the compartment at time

as a function of the initial number of nuclei.

Listing 9.3 : Matlab

©

code for simulating the decay model.

Figure 9.7 Representation of the gambler's ruin model.

Figure 9.8 Equivalent circuit for the Morris–Lecar model.

Figure 9.9 Hopf trajectories of the Morris–Lecar model with (a)

; (b)

; (c) 

; (d)

.

Guide

Cover

Table of Contents

Begin Reading

Pages

iii

iv

v

ix

x

xi

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

77

78

79

80

81

82

83

84

85

86

87

88

89

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

Mathematical Modelling

A Graduate Textbook

Seyed M. Moghadas

York University, Canada

Majid Jaberi-Douraki

York University, Canada

This edition first published 2019© 2019 John Wiley & Sons, Inc.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.

The right of Seyed M. Moghadas and Majid Jaberi-Douraki be identified as the authors of the material in this work has been asserted in accordance with law.

Registered Office(s)John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA

Editorial Office111 River Street, Hoboken, NJ 07030, USA

For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com.

Wiley also publishes its books in a variety of electronic formats and by print‐on‐demand. Some content that appears in standard print versions of this book may not be available in other formats.

Limit of Liability/Disclaimer of WarrantyIn view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of experimental reagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

Library of Congress Cataloging‐in‐Publication DataNames: Moghadas, Seyed M., author. | Jaberi-Douraki, Majid, author.Title: Mathematical modelling : a graduate textbook / by Seyed M. Moghadas, Majid Jaberi-Douraki.Description: 1st edition. | Hoboken, NJ : John Wiley & Sons, 2018. | Includes bibliographical references and index. |Identifiers: LCCN 2018008846 (print) | LCCN 2018013540 (ebook) | ISBN 9781119484028 (pdf) | ISBN 9781119483991 (epub) | ISBN 9781119483953 (cloth)Subjects: LCSH: Mathematical models–Textbooks.Classification: LCC TA342 (ebook) | LCC TA342 .M636 2018 (print) | DDC 511/.8–dc23LC record available at https://lccn.loc.gov/2018008846

Cover Design: WileyCover Image: © Oli Kellett/Gettyimages

To our parents

Preface

Mathematical modelling has evolved to become an important tool for understanding the underlying mechanisms of real‐life problems, most notably in biological and medical sciences. With advances in computational methods and computing power, the last two decades have witnessed the emergence of a new generation of models that have opened up novel vistas for mathematics and statistics to play an ever more significant role in many disciplines, enhancing research efforts for new discoveries, from basic sciences to more applied and practical settings. Tracing back to its historical roots, the importance of mathematical modelling was eloquently characterized by Daniel Bernoulli (1760) in his work to predict the gain in life expectancy that would result from controlling smallpox, a baneful disease of humans at the time. This characterization was highlighted in his statement, “I simply wish that, in a matter which so closely concerns the well‐being of mankind, no decision shall be made without all the knowledge which a little analysis and calculation can provide.” More than 250 years later, this quote still remains a compelling argument for the use of mathematical models to generate new knowledge in various fields, including biology, physiology, ecology, finance, and medicine.

Due to the widening scope and application of mathematical models in various disciplines, there has been a surge of interest in training highly qualified personnel with professional skills and expertise in mathematical modelling. This training often starts with a course in this subject. The purpose of this book is to provide essential materials for an enriched training that is suitable for both undergraduate and graduate levels. While we strive to provide the necessary background and theoretical foundation for topics covered in this book, readers and researchers may consult the references for other advanced textbooks for more details on related topics. Our aim here is to demonstrate a wide spectrum of problems that can be addressed through mathematical modelling with the use of fundamental tools and techniques in applied mathematics and statistics. However, it should be emphasized that many of the theoretical concepts utilized throughout the book require sufficient knowledge of applied mathematics, and the audience would be advised to acquire the relevant materials from topic‐specific textbooks, especially in differential equations and linear algebra.

This textbook contains nine chapters, starting with a chapter that provides the basic concepts of mathematical modelling, and a brief review of the relevant topics from differential equations and linear algebra. Chapters 2–5 elaborate on different types of mathematical models, and describe various techniques from dynamical systems theory for their analysis with a wide variety of examples. The analytical techniques are concerned with compartmental modelling, stability, bifurcation, discretization, and fixed‐point analysis. The theoretical analyses in these chapters primarily involve systems of ordinary differential equations for deterministic models. Chapter 6 briefly reviews the concepts of probability and random variables, which are required for the stochastic processes and Markov chains presented in Chapter 7. In Chapter 8 we describe algorithms for computer simulations of both deterministic and stochastic models, and provide examples of Matlab© codes for simulating models. Finally, in Chapter 9, a number of well‐known models are detailed to illustrate their application in different fields of study. Each chapter contains a number of exercises to help students better understand the modelling process, and develop enquiring and creative minds in this important subject.

This textbook is suitable for a modelling course in applied sciences at both undergraduate and graduate levels. It is also accessible to more theoretically oriented bioscientists who have some knowledge of linear algebra, systems of differential equations, and probability. In contextualizing this textbook, we have made a special effort to develop an interdisciplinary content for fundamentals and applications of mathematical modelling. The contents have been taught by the authors in one‐semester mathematical modelling courses at the graduate level, and therefore benefited from continual improvements and feedback provided by students over a number of years. We hope that this textbook will be a resource that eliminates the need for a collection of multiple books by instructors and students in order to cover a number of important topics in a mathematical modelling course.

November 2017

Seyed M. MoghadasYork University, Canada

Majid Jaberi‐DourakiKansas State University, USA

About the Companion Website

www.wiley.com/go/Moghadas/Mathematicalmodelling

The Companion Website provides the solution manual, presenting a guideline for solutions to the exercises within each chapter of the textbook. All solutions are based on the methods and techniques discussed in the textbook.

1Basic Concepts and Quick Review

A standard scientific practice is to formulate an explanation for an observed phenomenon and then test this formulation by projecting the outcomes of various experiments under pertinent conditions. Projections are generally compared with experimental data. If there is agreement, the explanation can be accepted as a valid theory, whereas discrepancies point to a need for reformulation of the explanation. A model that describes the main features of the phenomenon, often represented mathematically, can be iteratively improved in the process of reformulation to resolve its discrepancies with observations or experimental data. This iterative process is known as the modelling cycle (Figure 1.1).

In simple terms, mathematical modelling is a process by which we derive a model to describe a phenomenon that may or may not be observable. For example, the movement of a pendulum is an observable phenomenon, but the transmission of a disease in the population may not be observable. In the latter case, the outcomes of infection and illness indicate that the epidemic phenomenon may be taking place and the disease is being transmitted among individuals. The process of modelling consists of several important steps. In general, the model represents a framework that includes simplification, assumptions, and approximation to describe the phenomenon under investigation. This framework can be expressed by mathematical equations and analyzed using the theory of dynamical systems and computational tools for model validation and comparison with available data (Figure 1.1).

Figure 1.1 The process of model development, analysis, and validation.

Before proceeding further, let us present an example of developing a simple mathematical model. In this example, we wish to calculate the volume of sand that falls from the top half to the bottom half of a conical hourglass within a period of time (Figure 1.2). Suppose that the sand flows at the rate of per second from the top half to bottom half of the hourglass. We remember from calculus that the volume of a cone with height and radius is given by . Here, we will first find the volume of sand in the bottom half of the conical hourglass. From the dimensions given in Figure 1.2, this volume is given by:

Using the property of similar triangles, we can write