## An Introduction to Mathematical Modelling

This is one of the fundamental differences between classical mechanics and relatively theory. Application of the .... As we shall see later spatial heterogeneity can often qualitatively change model behaviour. Î»p model ..... Each of these could be plotted against a chosen variable to test for homogeneity in performance. It is.
An Introduction to Mathematical Modelling Glenn Marion, Bioinformatics and Statistics Scotland Given 2008 by Daniel Lawson and Glenn Marion 2008

Contents 1 Introduction

1

1.1

What is mathematical modelling? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

What objectives can modelling achieve? . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.3

Classifications of models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.4

Stages of modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

2 Building models

4

2.1

Getting started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.2

Systems analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.2.1

Making assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.2.2

Flow diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

Choosing mathematical equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.3.1

Equations from the literature . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.3.2

Analogies from physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.3.3

Data exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

Solving equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.4.1

Analytically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.4.2

Numerically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.3

2.4

3 Studying models

12

3.1

Dimensionless form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

3.2

Asymptotic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

3.3

Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

3.4

Modelling model output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

4 Testing models

18

4.1

Testing the assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

4.2

Model structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

i

4.3

Prediction of previously unused data . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

4.3.1

Reasons for prediction errors . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

4.4

Estimating model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

4.5

Comparing two models for the same system . . . . . . . . . . . . . . . . . . . . . . . .

21

5 Using models

23

5.1

Predictions with estimates of precision . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

5.2

Decision support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .