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

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

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

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