Mathematical Modeling for Epidemiology and Ecology

Preface
Changes from the First Edition
A Focus on Modeling
Mathematical Epidemiology
Scientific Computation
Description of Contents
Ways to Use This Book
A Textbook for a Course
A Supplementary Text for a Course
A Reference for Models and Research Techniques
Models and Problem Sets
Contents
Modeling in Biology
1.1 Working with Parameters
1.1.1 Scaling Parameters
1.1.2 Nonlinear Parameters
1.1.3 Bifurcations
1.2 Mathematics in Biology
1.2.1 Biological Data
1.2.2 Deterministic Patterns in a Random World
1.3 Quantifying Randomness in Data
1.3.1 Probability Distributions
1.3.2 Probability Distributions of Sample Means
1.4 Basic Concepts of Modeling
1.4.1 Mechanistic and Empirical Modeling
1.4.2 Aims of Mathematical Modeling
1.4.3 The Narrow and Broad Views of Mathematical Models
1.4.4 Accuracy, Precision, and Interpretation of Results
1.5 Case Study: An Agent-Based Epidemic Model
1.5.1 Model Description and Physical Simulation
1.5.2 Matlab Implementation
1.6 Projects
References
Empirical Modeling
2.1 The Basic Linear Least Squares Method (y=mx)
2.1.1 Overview of the Method
2.1.2 Development of the Method
2.1.3 Implied Assumption of Least Squares
2.2 Fitting Linear and Linearized Models to Data
2.2.1 Adapting the Method for y=mx to the General Linear Model
2.2.2 Fitting the Exponential Model by Linear Least Squares
2.2.3 Fitting the Power Function Model y=Axp by Linear Least Squares
2.3 Fitting Semilinear Models to Data
2.3.1 Finding the Best A for Given p
2.3.2 Finding the Best p
2.3.3 The Semilinear Least Squares Method
2.3.4 To Linearize or Not?
2.4 Model Selection
2.4.1 Quantitative Accuracy
2.4.2 Complexity
2.4.3 The Akaike Information Criterion
2.4.4 Choosing Among Models
2.4.5 Some Recommendations
2.5 Case Study: Michaelis–Menten Kinetics
2.5.1 The Michaelis–Menten Model and its Linearizations
2.5.2 Comparison of Methods
2.5.3 Conclusion
2.6 Project
References
Mechanistic Modeling
3.1 Transition Processes
3.1.1 Dimensional Analysis
3.1.2 Spontaneous Transition
3.1.3 ``Let the Buyer Beware''
3.1.4 A Model for Vaccination
3.1.5 Multi-Phase Transitions
3.2 Interaction Processes
3.2.1 Person-to-Person Disease Transmission
3.2.2 Models for Consumption and Predation
3.3 Compartment Analysis—The SEIR Epidemic Model
3.3.1 Classification of Epidemiological Models
3.3.2 Compartment Analysis
3.3.3 Model Behavior
3.3.4 Parameterization from Data
3.4 SEIR Model Analysis
3.4.1 The Basic Reproduction Number
3.4.2 Goals of the Analysis
3.4.3 Early-Phase Exponential Growth
3.4.4 The End State
3.5 Case Study: Two Scenarios from the COVID-19 Pandemic
3.5.1 March 2020
3.5.2 January 2021
3.6 Equivalent Forms
3.6.1 Notation
3.6.2 Algebraic Equivalence
3.6.3 Different Parameters
3.6.4 Visualizing Models with Graphs
3.6.5 Dimensionless Variables
3.6.6 Dimensionless Forms
3.6.7 Scaling of Differential Equation Models
3.7 Case Study: Lead Poisoning
3.7.1 A Simplified Model
3.7.2 The Dimensionless Model
3.8 Case Study: Enzyme Kinetics
3.8.1 Scaling
3.8.2 Simulation
3.8.3 Asymptotic Approximation
3.9 Case Study: Adding Demographics to Make an Endemic Disease Model
3.9.1 A Generic SIR Model with Demographics
3.9.2 Several Approaches to a Variable Population Version
3.9.3 Scaling
3.9.4 Simulations
3.9.5 Rescaling
3.10 Projects
References
Dynamics of Single Populations
4.1 Discrete Population Models
4.1.1 A General Seasonal Population Model
4.1.2 Discrete Exponential Growth
4.1.3 The Discrete Logistic Model
4.1.4 Simulations
4.1.5 Fixed Points
4.2 Cobweb Analysis
4.2.1 Cobweb Plots
4.2.2 Stability Analysis
4.3 Continuous Dynamics
4.3.1 Exponential Growth
4.3.2 Logistic Growth
4.3.3 Dynamical Systems
4.3.4 Equilibrium Points and Stability
4.3.5 The Phase Line
4.4 Linearized Stability Analysis
4.4.1 Stability Analysis for Discrete Models: A Motivating Example
4.4.2 Stability Analysis for Discrete Models: The General Case
4.4.3 Stability Analysis for Continuous Models
4.4.4 Comparison of Discrete and Continuous Dynamics
4.5 Case Study: A Mathematical Model of Resource Conservation
4.5.1 Growth and Harvesting Functions
4.5.2 Scaling
4.5.3 Plan for Analysis
4.5.4 A Structured Approach to Phase Line Analysis
4.5.5 A Reconstructed History of Whale Populations
4.5.6 Bifurcation Analysis
4.6 Projects
References
Discrete Linear Systems
5.1 Discrete Linear Systems
5.1.1 Simple Structured Models
5.1.2 Finding the Growth Rate and Stable Stage Distribution
5.1.3 General Properties of Discrete Linear Models
5.2 Case Study: Peregrine Falcons
5.2.1 Mathematical Analysis
5.2.2 General Analysis Questions
5.3 A Matrix Algebra Primer
5.3.1 Matrices and Vectors
5.3.2 Population Models in Matrix Notation
5.3.3 The Central Problem of Matrix Algebra
5.3.4 The Determinant
5.3.5 The Equation Ax=0
5.4 Long-Term Behavior of Linear Models
5.4.1 Eigenvalues and Eigenvectors
5.4.2 Eigenvalue Decoupling
5.4.3 Long-Term Behavior
5.5 Case Study: Loggerhead Turtles
5.5.1 Status Quo for South Carolina Loggerheads in 1994
5.5.2 A Model that Accounts for Trawler Mortality
5.5.3 A Simple Experiment to Test the Value of Turtle Excluder Devices
5.6 Case Study: Phylogenetic Distance
5.6.1 Some Scientific Background
5.6.2 A Model for DNA Change
5.6.3 Equilibrium Analysis of Markov Chain Models
5.6.4 Analysis of the DNA Change Model
References
Nonlinear Dynamical Systems
6.1 Phase Plane Analysis
6.1.1 Solution Curves in the Phase Plane
6.1.2 Nullclines and Equilibria
6.1.3 Nullcline Analysis
6.1.4 Nullcline Analysis in General
6.2 Linearized Stability Analysis Using Eigenvalues
6.2.1 Two-Component Linear Systems
6.2.2 Eigenvalues and Stability
6.2.3 The Jacobian Matrix and Stability
6.3 Stability Analysis with the Routh–Hurwitz Conditions
6.3.1 The Routh–Hurwitz Conditions for Two-Component Systems
6.3.2 The Routh–Hurwitz Conditions for Three-Component Systems
6.4 Case Study: Onchocerciasis
6.4.1 Model Development
6.4.2 Preparation for Analysis
6.4.3 Analysis of the Three-Component System
6.4.4 The Endemic Disease Equilibrium
6.4.5 Analysis of the Two-Component System
6.4.6 Simulation
6.5 Discrete Nonlinear Systems
6.5.1 Linearization for Discrete Nonlinear Systems
6.5.2 A Structured Population Model with One Nonlinearity
6.5.3 Choosing a Discrete or Continuous Model
6.6 Projects
References
A Using MATLAB and Octave
C.1 A Guess and Check Method
C.2 The Bisection/Quintsection Methods
D.1 Accuracy of Runge–Kutta Methods
D.2 The Runge–Kutta rk4 Method
Index