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