Contents
1.1 Descriptive versus Mechanistic Models
1.4 Many Bathtubs: Compartment Models
1.5 Physics Models: Running and Hopping
1.8 Theoretical versus Practical Models
2 Matrix Models and Structured Population Dynamics
2.1 The Population Balance Law
2.2.2 Warning: Prebreeding versus Postbreeding Models
2.3 Matrix Models Based on Stage Classes
2.4 Matrices and Matrix Operations
2.4.1 Review of Matrix Operations
2.4.2 Solution of the Matrix Model
2.5 Eigenvalues and a Second Solution of the Model
2.6 Some Applications of Matrix Models
2.6.2 Elasticity Analysis and Conservation Biology
2.6.3 How Much Should We Trust These Models?
2.7 Generalizing the Matrix Model
2.7.1 Stochastic Matrix Models
2.7.2 Density-Dependent Matrix Models
2.7.3 Continuous Size Distributions
2.9.1 Existence and Number of Eigenvalues
3 Membrane Channels and Action Potentials
3.1.1 Channel Gating and Conformational States
3.2.3 The Neuromuscular Junction
3.4 Membranes as Electrical Circuits
3.6 Appendix: The Central Limit Theorem
4 Cellular Dynamics: Pathways of Gene Expression
4.2 A Gene Network That Acts as a Clock
4.3 Networks That Act as a Switch
4.4.1 Complex versus Simple Models
5.1 Geometry of a Single Differential Equation
5.2 Mathematical Foundations: A Fundamental Theorem
5.3 Linearization and Linear Systems
5.3.2 Linearization at Equilibria
5.3.3 Solving Linear Systems of Differential Equations
5.5 An Example: The Morris-Lecar Model
6 Differential Equation Models for Infectious Disease
6.1 Sir Ronald Ross and the Epidemic Curve
6.3 Endemic Diseases and Oscillations
6.3.1 Analysis of the SIR Model with Births
6.4 Gonorrhea Dynamics and Control
6.4.1 A Simple Model and a Paradox
6.4.3 Implications for Control
6.6 Within-Host Dynamics of HIV
7.3 Pattern Selection: Steady Patterns
7.4 Moving Patterns: Chemical Waves and Heartbeats
8 Agent-Based and Other Computational Models for Complex Systems
8.1 Individual-Based Models in Ecology
8.1.1 Size-Dependent Predation
8.1.3 Individual-Based Modeling of Extinction Risk
8.3 The Immune System and the Flu
8.4 What Can We Learn from Agent-Based Models?
8.6 Simplifying Computational Models
8.6.1 Separation of Time Scales
8.6.2 Simplifying Spatial Models
8.6.3 Improving the Mean Field Approximation
8.8 Appendix: Derivation of Pair Approximation
9.2.1 Conceptual Model and Diagram
9.3 Developing Equations for Process Rates
9.3.1 Linear Rates: When and Why?
9.3.2 Nonlinear Rates from “First Principles”
9.3.3 Nonlinear Rates from Data: Fitting Parametric Models
9.3.4 Nonlinear Rates from Data: Selecting a Parametric Model
9.4 Nonlinear Rates from Data: Nonparametric Models
9.4.1 Multivariate Rate Equations
9.5.1 Individual-Level Stochasticity
9.5.2 Parameter Drift and Exogenous Shocks
9.6 Fitting Rate Equations by Calibration