Part I Mathematical Modeling.. 1 Modeling in Biology.. 1.1 Working with Parameters.. 1.2 Mathematics in Biology.. 1.3 Quantifying Randomness in Data.. 1.4 Basic Concepts of Modeling.. 1.5 Case Study: An Agent-Based Epidemic Model.. 1.6 Projects.. References.. 2 Empirical Modeling.. 2.1 The Basic Linear Least Squares Method (y = mx).. 2.2 Fitting Linear and Linearized Models to Data.. 2.3 Fitting Semilinear Models to Data.. 2.4 Model Selection.. 2.5 Case Study: Michaelis–Menten Kinetics.. 2.6 Project.. References.. 3 Mechanistic Modeling.. 3.1 Transition Processes.. 3.3 Compartment Analysis—The SEIR Epidemic Model.. 3.4 SEIR Model Analysis.. 3.5 Case Study: Two Scenarios from the COVID-19 Pandemic.. 3.6 Equivalent Forms.. 3.7 Case Study: Lead Poisoning.. 3.8 Case Study: Enzyme Kinetics.. 3.9 Case Study: Adding Demographics to Make an Endemic.. 3.10 Projects.. References.. Part II Dynamical Systems.. 4 Dynamics of Single Populations.. 4.1 Discrete Population Models.. 4.2 Cobweb Analysis.. 4.3 Continuous Dynamics.. 4.4 Linearized Stability Analysis.. 4.5 Case Study: A Mathematical Model of Resource Conservation.. 4.6 Projects.. References.. 5 Discrete Linear Systems.. 5.1 Discrete Linear Systems.. 5.2 Case Study: Peregrine Falcons.. 5.3 A Matrix Algebra Primer.. 5.4 Long-Term Behavior of Linear Models.. 5.5 Case Study: Loggerhead Turtles.. 5.6 Case Study: Phylogenetic Distance.. 6 Nonlinear Dynamical Systems.. 6.2 Linearized Stability Analysis Using Eigenvalues.. 6.3 Stability Analysis with the Routh–Hurwitz Conditions.. 6.4 Case Study: Onchocerciasis.. 6.5 Discrete Nonlinear Systems.. 6.6 Projects.. References.. Appendix A: Using MATLAB and Octave.. Appendix B: Derivatives and Differentiation.. Appendix C: Nonlinear Optimization.. Appendix D: A Runge–Kutta Method for Numerical Solution of Differential Equations.. Appendix E: Scales and Dimensionless Parameters.. Appendix F: Approximating a Nonlinear System at an Equilibrium Point.. Appendix G: Best Practices in the Use of Algebra.. Hints and Answers to Selected Problems.. Index

Mathematical Modeling for Epidemiology and Ecology provides readers with the mathematical tools needed to understand and use mathematical models and read advanced mathematical biology books. It presents mathematics in biological contexts, focusing on the central mathematical ideas and the biological implications, with detailed explanations. The author assumes no mathematics background beyond elementary differential calculus. An introductory chapter on basic principles of mathematical modeling is followed by chapters on empirical modeling and mechanistic modeling. These chapters contain a thorough treatment of key ideas and techniques that are often neglected in mathematics books, such as the Akaike Information Criterion. The second half of the book focuses on analysis of dynamical systems, emphasizing tools to simplify analysis, such as the Routh-Hurwitz conditions and asymptotic analysis. Courses can be focused on either half of the book or thematically chosen material from both halves, such as a course on mathematical epidemiology. The biological content is self-contained and includes many topics in epidemiology and ecology. Some of this material appears in case studies that focus on a single detailed example, and some is based on recent research by the author on vaccination modeling and scenarios from the COVID-19 pandemic. The problem sets feature linked problems where one biological setting appears in multi-step problems that are sorted into the appropriate section, allowing readers to gradually develop complete investigations of topics such as HIV immunology and harvesting of natural resources. Some problems use programs written by the author for Matlab or Octave; these combine with more traditional mathematical exercises to give students a full set of tools for model analysis. Each chapter contains additional case studies in the form of projects with detailed directions. New appendices contain mathematical details on optimization, numerical solution of differential equations, scaling, linearization, and sophisticated use of elementary algebra to simplify problems. Inglés