Mathematical Modeling in the Social and Life Sciences

The goal of this book is to encourage the teaching and learning of mathematical model building relatively early in the undergraduate program. The text introduces the student to a number of important mathematical topics and to a variety of models in the social sciences, life sciences, and humanities.

Preface viii

Acknowledgements xiii

1 Mathematical Models 1

I. Mathematical Systems and Models 1

II. An Example: Modeling Free Fall 4

III. Discrete Examples: Credit Cards and Populations 10

IV. Classification of Mathematical Models 16

V. Uses and Limitations of Mathematical Models 18

Exercises 19

Suggested Projects 21

2 Stable and Unstable Arms Races 23

I. The Real-World Setting 23

II. Constructing a Deterministic Model 25

III. A Simple Model for an Arms Race 25

IV. The Richardson Model 28

V. Interpreting and Testing the Richardson Model 45

VI. Obtaining an Exact Solution 53

Exercises 59

Suggested Projects 63

3 Ecological Models: Single Species 65

I. Introduction 65

II. The Pure Birth Process 65

III. Exponential Decay 71

IV. Logistic Population Growth 72

V. The Discrete Model of Logistic Growth and Chaos 80

VI. The Allee Effect 87

VII. Historical and Biographical Notes 89

Exercises 100

Suggested Projects 104

Biographical References 105

4 Ecological Models: Interacting Species 106

I. Introduction 106

II. Two Real-World Situations 106

III. Autonomous Systems 108

IV. The Competitive Hunters Model 116

V. The Predator-Prey Model 123

VI. Concluding Remarks on Simple Models in Population Dynamics 131

VII. Biographical Sketches 133

Exercises 137

Suggested Projects 139

5 Tumor Growth Models 141

I. Introduction 141

II. A General Tumor Growth Model 142

III. The Gompertz Model 145

IV. Modeling Colorectal Cancer 155

V. Historical and Biographical Notes 167

Exercises 176

Suggested Projects 177

6 Social Choice and Voting Procedures 179

I. Three Voting Situations 179

II. Two Voting Mechanisms 180

III. An Axiomatic Approach 185

IV. Arrow’s Impossibility Theorem 187

V. The Liberal Paradox and the Theorem of the Gloomy Alternatives 191

VI. Instant Runoff Voting 197

VII. Approval Voting 203

VIII. Topological Social Choice 207

IX. Historical and Biographical Notes 212

Exercises 224

Suggested Projects 229

7 Foundations of Measurement Theory 232

I. The Registrar’s Problem 232

II. What Is Measurement? 233

III. Simple Measures on Finite Sets 238

IV. Perception of Differences 240

V. An Alternative Approach 242

VI. Some Historical Notes 245

Exercises 245

Suggested Projects 247

8 Introduction to Utility Theory 249

I. Introduction 249

II. Gambles 250

III. Axioms of Utility Theory 251

IV. Existence and Uniqueness of Utility 254

V. Classification of Scales 257

VI. Interpersonal Comparison of Utility 259

VII. Historical and Biographical Notes 261

Exercises 265

Suggested Projects 266

9 Equilibrium in an Exchange Economy 268

I. Introduction 268

II. A Two-Person Economy with Two Commodities 268

III. An m-Person Economy 276

IV. Existence of Economic Equilibrium 283

V. Some Remaining Questions 293

VI. Historical and Biographical Notes 294

Exercises 298

Suggested Projects 301

VII. Additional Historical and Biographical Notes 302

10 Elementary Probability 303

I. The Need for Probability Models 303

II. What Is Probability? 304

III. A Probabilistic Model 322

IV. Stochastic Processes 325

Exercises 331

Suggested Projects 335

11 Markov Processes 336

I. Markov Chains 336

II. Matrix Operations and Markov Chains 341

III. Regular Markov Chains 347

IV. Absorbing Markov Chains 357

V. Historical and Biographical Notes 369

Exercises 371

Suggested Projects 374

12 Two Models of Cultural Stability 375

I. Introduction 375

II. The Gadaa System 375

III. A Deterministic Model 378

IV. A Probabilistic Model 381

V. Criticisms of the Models 383

VI. Hans Hoffmann 384

Exercises 386

Suggested Projects 387

13 Paired-Associate Learning 388

I. The Learning Problem 388

II. The Model 389

III. Testing the Model 397

IV. Historical and Biographical Notes 401

Exercises 404

Suggested Projects 406

14 Epidemics 407

I. Introduction 407

II. Deterministic Models 411

III. A Probabilistic Approach 449

IV. Historical and Biographical Notes 455

Exercises 459

Suggested Projects 463

15 Roulette Wheels and Hospital Beds: A Computer Simulation of Operating and Recovery Room Usage 464

I. Introduction 464

II. The Problems of Interest 468

III. Projecting the Number of Surgical Procedures 468

IV. Estimating Operating Room Demands 469

V. The Simulation Model 474

VI. Other Examples of Simulation 480

VII. Historical and Biographical Notes 484

Exercises 487

Suggested Projects 488

16 Game Theory 490

I. Two Difficult Decisions 490

II. Game Theory Basics 492

III. The Binding of Isaac 502

IV. Tosca and the Prisoners’ Dilemma 507

V. Nash Equilibrium 511

VI. Dynamic Solutions 515

VII. Historical and Biographical Notes 519

Exercises 522

Suggested Projects 526

Appendices

Appendix I: Sets 613

Appendix II: Matrices 617

Appendix III: Solving Systems of Equations 631

Appendix IV: Functions of Two Variables 645

Appendix V: Differential Equations 648

Index 657

Michael Olinick is the author of Mathematical Modeling in the Social and Life Sciences, published by Wiley.