An Introduction to Mathematical Modelling

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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .