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.

Dedication

Preface

Chapter 1 Basic Concepts and Quick Review

1.1 Modelling Types

1.2 Quick Review

1.2.1 First–order differential equations

1.2.2 Second order differential equations

1.2.3 Linear algebra

1.2.4 Scaling

Exercises

Chapter 2 Compartmental Modelling

2.1 Cascades of Compartments

2.2 Parameter Units .

Exercises

Chapter 3 Analysis Tools

3.1 Stability Analysis

3.2 Phase–Plane Behaviour

3.3 Direction Field

3.4 Routh–Hurwitz Criterion

Exercises

Chapter 4 Bifurcation

4.1 Transcritical Bifurcation

4.2 Saddle–Node Bifurcation

4.3 Pitchfork Bifurcation

4.4 Hopf Bifurcation

4.5 Solution Types

Exercises

Chapter 5 Discretization and Fixed Point Analysis

5.1 Discretization

5.1.1 Method of Euler

5.1.2 Non–standard methods

5.2 Fixed Point Analysis

Exercises

Chapter 6 Probability and Random Variables

6.1 Basic Concepts

6.2 Conditional Probabilities

6.3 Random Variables

6.3.1 Cumulative distribution function

6.3.2 Discrete random variables

6.3.3 Continuous random variables

6.3.4 Waiting time

Exercises

Chapter 7 Stochastic Modelling

7.1 Stochastic Processes

7.2 Probability Generating Function

7.3 Markov Chain

7.4 Random Walks

Exercises

Chapter 8 Computer Simulations

8.1 Deterministic Structure

8.2 Stochastic Structure

8.3 Monte–Carlo Methods

Exercises

Chapter 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 The Spruce Budworm Model

9.6 The FitzHugh–Nagumo Model

9.7 The Decay Model

9.8 The Gambler s Ruin

Exercises

Bibliography

Index

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.