Models for Life: An Introduction to Discrete Mathematical Modeling with Microsoft Office Excel


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Features an authentic and engaging approach to mathematical modeling driven by real-world applications

With a focus on mathematical models based on real and current data, Models for Life: An Introduction to Discrete Mathematical Modeling with Microsoft Office Excel guides readers in the solution of relevant, practical problems by introducing both mathematical and Excel techniques.

The book begins with a step-by-step introduction to discrete dynamical systems, which are mathematical models that describe how a quantity changes from one point in time to the next. Readers are taken through the process, language, and notation required for the construction of such models as well as their implementation in Excel. The book examines single-compartment models in contexts such as population growth, personal finance, and body weight and provides an introduction to more advanced, multi-compartment models via applications in many areas, including military combat, infectious disease epidemics, and ranking methods. Models for Life: An Introduction to Discrete Mathematical Modeling with Microsoft Office Excel also features:

  • A modular organization that, after the first chapter, allows readers to explore chapters in any order
  • Numerous practical examples and exercises that enable readers to personalize the presented models by using their own data
  • Carefully selected real-world applications that motivate the mathematical material such as predicting blood alcohol concentration, ranking sports teams, and tracking credit card debt
  • References throughout the book to disciplinary research on which the presented models and model parameters are based in order to provide authenticity and resources for further study
  • Relevant Excel concepts with step-by-step guidance, including screenshots to help readers better understand the presented material
  • Both mathematical and graphical techniques for understanding concepts such as equilibrium values, fixed points, disease endemicity, maximum sustainable yield, and a drug’s therapeutic window
  • A companion website that includes the referenced Excel spreadsheets, select solutions to homework problems, and an instructor’s manual with solutions to all homework problems, project ideas, and a test bank

The book is ideal for undergraduate non-mathematics majors enrolled in mathematics or quantitative reasoning courses such as introductory mathematical modeling, applications of mathematics, survey of mathematics, discrete mathematical modeling, and mathematics for liberal arts. The book is also an appropriate supplement and project source for honors and/or independent study courses in mathematical modeling and mathematical biology.

Jeffrey T. Barton, PhD, is Professor of Mathematics in the Mathematics Department at Birmingham-Southern College. A member of the American Mathematical Society and Mathematical Association of America, his mathematical interests include approximation theory, analytic number theory, mathematical biology, mathematical modeling, and the history of mathematics.


Preface xiii

Acknowledgments xvii

1 Density-Independent Population Models 1

1.1 Exponential Growth, 1

1.2 Exponential Growth with Stocking or Harvesting, 22

1.3 Two Fundamental Excel Techniques, 32

1.4 Explicit Formulas, 40

1.5 Equilibrium Values and Stability, 50

2 Personal Finance 59

2.1 Compound Interest and Savings, 60

2.2 Borrowing for Major Purchases, 77

2.3 Credit Cards, 92

2.4 The Time Value of Money: Present Value, 104

2.5 Car Leases, 112

3 Combat Models 119

3.1 Lanchester Combat Model, 120

3.2 Phase Plane Graphs, 140

3.3 The Lanchester Model with Reinforcements, 146

3.4 Hughes Aimed Fire Salvo Model, 153

3.5 Armstrong Salvo Model with Area Fire, 169

4 The Spread of Infectious Diseases 183

4.1 The S–I–R Model, 184

4.2 S–I–R with Vital Dynamics, 203

4.3 Determining Parameters from Real Data, 216

4.4 S–I–R with Vital Dynamics and Routine Vaccinations, 226

5 Density-Dependent Population Models 235

5.1 The Discrete Logistic Model, 235

5.2 Logistic Growth with Allee Effects, 248

5.3 Logistic Growth with Harvesting, 254

5.4 The Discrete Logistic Model and Chaos, 263

5.5 The Ricker Model, 266

6 Blood Alcohol Concentration and Pharmacokinetics 273

6.1 Blood Alcohol Concentration, 273

6.2 The Widmark Model, 280

6.3 The Wagner Model, 283

6.4 Alcohol Consumption Patterns, 289

6.5 More General Drug Elimination, 301

6.6 The Volume of Distribution, 319

6.7 Common Drugs, 321

7 Ranking Methods 329

7.1 Introduction to Markov Models, 329

7.2 Ranking Sports Teams, 342

7.3 Google PageRank, 361

8 Body Weight and Body Composition 381

8.1 Constant Calorie Expenditure, 382

8.2 Variable Calorie Expenditure, 385

8.3 Health Metrics, 394

8.4 Body Composition, 397

8.5 The Body Composition Model for Body Weight, 406

8.6 Points-based Systems: The Weight Watchers Model, 419

Appendix A: The Geometric Series Formula 431

Appendix B: Lanchester’s Square Law and the Fractional Exchange Ratio 433

Appendix C: Derivation of the FER = 1 Line for the Hughes Salvo Model 439

Appendix D: The Waiting Time Principle 441

Appendix E: Creating Cobweb Diagrams in Excel 445

Appendix F: Proportion of Total Credit Distributed Does Not Exceed 1 449

Bibliography 451

Index 459


E.1 Introduction to Excel 5

E.2 Absolute Addressing 12

E.3 Multiple Formulas 15

E.4 Graphing in Excel 32

E.5 Goal Seek 35

E.6 Working with Multiple Columns 53

E.7 Improving Spreadsheet Readability 66

E.8 The SUM Command and Working with Dates 99

E.9 The AVERAGE and COUNT Commands 101

E.10 The MAX Command 102

E.11 IF Statements 123

E.12 Phase Plane Graphs 141

E.13 Phase Planes with Multiple Trajectories 143

E.14 Nested IF Statements 155

E.15 Date Format and Plotting Data Points 193

E.16 Graphing Functions 243

E.17 Exponential and Logarithmic Functions 268

E.18 Data Validation 275

E.19 Setting the Error Tolerance for Goal Seek 295

E.20 Exponents 303

E.21 The MOD Command 311

E.22 Rounding 421

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