Mathematical Modelling A Graduate Textbook

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.

Seyed M. Moghadas, PhD, is Associate Professor of Applied Mathematics and Computational Epidemiology, and Director of the Agent-Based Modelling Laboratory at York University in Toronto, Ontario, Canada. Dr. Moghadas is an Associate Editor of Infectious Diseases in the Scientific Reports, Nature Publishing Group.

Majid Jaberi-Douraki, PhD, is Assistant Professor of Biomathematics at Kansas State University, Manhattan, Kansas, USA. His research involves modelling dynamical systems and optimal control theory in a wide range of real-world problems.

Preface ix

About the CompanionWebsite xi

1 Basic Concepts and Quick Review 1

1.1 Modelling Types 4

1.2 Quick Review 5

1.2.1 First-order Differential Equations 5

1.2.2 Second-order Differential Equations 7

1.2.3 Linear Algebra 9

1.2.4 Scaling 12

Exercises 14

2 Compartmental Modelling 15

2.1 Cascades of Compartments 18

2.2 Parameter Units 23

Exercises 25

3 Analysis Tools 29

3.1 Stability Analysis 32

3.2 Phase-Plane Behavior 37

3.3 Direction Field 47

3.4 Routh–Hurwitz Criterion 49

Exercises 54

4 Bifurcation 59

4.1 Transcritical Bifurcation 60

4.2 Saddle-Node Bifurcation 61

4.3 Pitchfork Bifurcation 62

4.4 Hopf Bifurcation 64

4.5 Solution Types 69

Exercises 73

5 Discretization and Fixed-Point Analysis 77

5.1 Discretization 78

5.1.1 Euler Method 78

5.1.2 Nonstandard Methods 79

5.2 Fixed-Point Analysis 83

Exercises 88

6 Probability and Random Variables 91

6.1 Basic Concepts 91

6.2 Conditional Probabilities 93

6.3 Random Variables 95

6.3.1 Cumulative Distribution Function 95

6.3.2 Discrete Random Variables 96

6.3.3 Continuous Random Variables 98

6.3.4 Waiting Time 101

Exercises 101

7 Stochastic Modelling 103

7.1 Stochastic Processes 103

7.2 Probability Generating Function 108

7.3 Markov Chains 110

7.4 RandomWalks 118

Exercises 119

8 Computer Simulations 125

8.1 Deterministic Structure 125

8.2 Stochastic Structure 128

8.3 Monte Carlo Methods 136

Exercises 138

9 Examples of Mathematical Modelling 143

9.1 Traffic Model 143

9.2 Michaelis–Menten Kinetics 146

9.3 The Brusselator System 148

9.4 Generalized Richards Model 149

9.5 Spruce Budworm Model 152

9.6 FitzHugh–Nagumo Model 153

9.7 Decay Model 155

9.8 The Gambler’s Ruin 158

Exercises 160

References 165

Index 169

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