Mathematical Modeling in the Social and Life Sciences

590 Pages·2014·18.473 MB·English

by  Michael Olinick

Mathematical Modeling in the
Social and Life Sciences
Michael Olinick
Middlebury College
DEDICATION
To Judy who, without resort to mathematics, is a model wife, parent, and citizen
Publisher: LaurieRosatone
AcquisitionsEditor: DavidDietz
ContentEditor: JacquelineSinacori
EditorialAssistant: MichaelO’Neal
ProductDesigner: TomKulesa
CoverDesigner: KenjiNgieng
MarketingManager: MelanieKurkjian
AssociateProductionManager: JoycePoh
SeniorProductionEditor: JoleneLing
ThisbookwassetbyMPSLimited.
Foundedin1807,JohnWiley&Sons,Inc.hasbeenavaluedsourceofknowledgeandunderstandingformore
than200years,helpingpeoplearoundtheworldmeettheirneedsandfulfilltheiraspirations.Ourcompanyisbuilt
onafoundationofprinciplesthatincluderesponsibilitytothecommunitiesweserveandwhereweliveandwork.
In2008,welaunchedaCorporateCitizenshipInitiative,aglobalefforttoaddresstheenvironmental,social,
economic,andethicalchallengeswefaceinourbusiness.Amongtheissuesweareaddressingarecarbonimpact,
paperspecificationsandprocurement,ethicalconductwithinourbusinessandamongourvendors,andcommunity
andcharitablesupport.Formoreinformation,pleasevisitourwebsite:www.wiley.com/go/citizenship.
Copyright©2014JohnWiley&Sons,Inc.Allrightsreserved.Nopartofthispublicationmaybereproduced,
storedinaretrievalsystemortransmittedinanyformorbyanymeans,electronic,mechanical,photocopying,
recording,scanningorotherwise,exceptaspermittedunderSections107or108ofthe1976UnitedStates
CopyrightAct,withouteitherthepriorwrittenpermissionofthePublisher,orauthorizationthroughpayment
oftheappropriateper-copyfeetotheCopyrightClearanceCenter,Inc.222RosewoodDrive,Danvers,
MA01923,websitewww.copyright.com.RequeststothePublisherforpermissionshouldbeaddressedtothe
PermissionsDepartment,JohnWiley&Sons,Inc.,111RiverStreet,Hoboken,NJ07030-5774,[phone],
fax[phone],websitehttp://www.wiley.com/go/permissions.
Evaluationcopiesareprovidedtoqualifiedacademicsandprofessionalsforreviewpurposesonly,foruseintheir
coursesduringthenextacademicyear.Thesecopiesarelicensedandmaynotbesoldortransferredtoathird
party.Uponcompletionofthereviewperiod,pleasereturntheevaluationcopytoWiley.Returninstructionsanda
freeofchargereturnmailinglabelareavailableatwww.wiley.com/go/returnlabel.Ifyouhavechosentoadoptthis
textbookforuseinyourcourse,pleaseacceptthisbookasyourcomplimentarydeskcopy.OutsideoftheUnited
States,pleasecontactyourlocalsalesrepresentative.
LibraryofCongressCataloging-in-PublicationData
Olinick,Michael.
Mathematicalmodelinginthesocialandlifesciences/MichaelOlinick,MiddleburyCollege.
pages cm
Includesindex.
ISBN978-1-118-64269-6(pbk.)
1. Socialsciences—Mathematicalmodels. 2. Lifesciences—Mathematicalmodels. I. Title.
H61.O483 2014
300.1'51—dc23
2013044538
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
Contents
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
iii
iv CONTENTS
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
Contents v
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
vi CONTENTS
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
Contents vii
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, 527
Appendix II: Matrices, 531
Appendix III: Solving Systems of Equations, 545
Appendix IV: Functions of Two Variables, 559
Appendix V: Differential Equations, 562
Index 571
Online Chapters (www.wiley.com/college/olinick)
17
Recidivism in the Criminal Justice System
18
Evolutionary Game Theory
19
Agent Based Simulation
Preface
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
numberofimportantmathematicaltopicsandtoavarietyofmodelsinthesocialsciences,life
sciences, and humanities. Students with some mathematical maturity and a strong secondary
school background will find many chapters quite accessible. A standard first year calculus
course is sufficient background for the remaining chapters. While many of the models use
differential equations or some elementary linear algebra, no previous experience with these
topics is assumed. The text material will help students gain the necessary knowledge.
Appendices on sets, matrices, systems of linear equations, and functions of two variables
provide additional background material.
Particularproblemsinpoliticalscience,ecology,biology,evolution,medicine,psychol-
ogy,sociology,economics,finance,anthropology,criminaljustice,epidemiology,philosophy,
religion,opera,andhospitalplanningprovidethemotivationforthedevelopmentoftoolsand
techniquesemployedthroughoutappliedmathematics.Theseinclude
DifferentialEquations DiscreteDynamicalSystems,
Axiomatics ProbabilityTheory
RegularMarkovChains AbsorbingMarkovChain
MatrixAlgebra LeastSquaresFittingOfData
Simulation TheoryOfGames
The curricula in many social science and life science disciplines are becoming increas-
ingly infused with the development and analysis of formal models. Students in such majors
(particularly integrated biology/mathematics and economics/mathematics programs) need an
introductiontothemathematicalideasandtechniquesjustlisted.Thistextprovidesonewayof
gainingthisexposureinasinglecourse.
Iselectedmodelsprimarilyfromthebehavioralsciencesforanumberofreasons:
1. Theyshowtherichvarietyofdisciplinestowhichmathematicsismakingimportant
contributions;
2. Because such models require less technical background knowledge than more tra-
ditional models in physics, chemistry, and engineering, students can examine in
depth many different applications in a one-semester course;
viii