An Introduction to Undergraduate Research in Computational and Mathematical Biology: From Birdsongs to Viscosities (Foundations for Undergraduate Research in Mathematics)

Series Preface
Introduction
Contents
Building New Models: Rethinking and Revising ODE Model Assumptions
	1 Introduction
	2 Techniques for Analyzing ODE Models
		2.1 ODEs as Mean Field Models
		2.2 Equilibrium Stability Analysis
		2.3 Bifurcation Analysis
		2.4 A Few Comments on Approximation
		2.5 Fast–Slow Analysis of Systems with Multiple Time Scales
		2.6 Computing Numerical Solutions to ODEs
			2.6.1 Euler\'s Method
			2.6.2 Numerical Solutions in R
			2.6.3 Keeping Numerical Solutions Positive: The Log-Transform Trick
		2.7 ODEs as Statistical Models
			2.7.1 A Brief Overview of Key Statistical Concepts
			2.7.2 Likelihood Based Parameter Estimation
			2.7.3 Likelihood Framework for ODEs
			2.7.4 Parameter Estimation as an Optimization Problem
			2.7.5 Recognizing Identifiability (Estimability) Problems
			2.7.6 Practical Identifiability Analysis
			2.7.7 Structural Identifiability Analysis
			2.7.8 Statistical Analyses Beyond Parameter Estimation
			2.7.9 Alternative Approaches to Parameter Estimation and Uncertainty Quantification
			2.7.10 Closing Remarks on Fitting ODE Models to Data
	3 Identifying and Modifying Model Assumptions
		3.1 Autonomous ODEs to Non-autonomous ODEs
		3.2 Deriving Deterministic Discrete-Time Models
		3.3 Deriving Stochastic Models
			3.3.1 Continuous-Time, Discrete-State Stochastic Models
		3.4 Stochastic Differential Equations (SDEs)
			3.4.1 Numerical Solutions to SDEs
		3.5 Distributed Delay Equations
			3.5.1 Integral and Integro-Differential Equations
			3.5.2 Linear Chain Trick
			3.5.3 Mathematical Foundations of the Linear Chain Trick
			3.5.4 Delay Differential Equations
		3.6 Individual Heterogeneity
		3.7 Spatially Explicit Models
	4 Choosing a Research Project
		4.1 Additional Project Topics
	5 The Importance of Publishing
	Appendix
		Getting Started Writing in LaTeX and Programming in R
		Installing and Using LaTeX
		Installing and Using R
	References
A Tour of the Basic Reproductive Number and the Next Generation of Researchers
	1 Introduction
		1.1 Overview
		1.2 What Is the Basic Reproductive Number and Why Is It Important?
			1.2.1 Definitions of R0
			1.2.2 Methods of Finding R0
		1.3 Additional Reading
	2 An Introductory Example: Student–Teacher Ratio
		2.1 The Model
		2.2 Finding R0 with the Next Generation Matrix
			2.2.1 The Intuition Behind the NGM Strategy
			2.2.2 The Mathematical Approach
		2.3 Interpreting R0
		2.4 Sensitivity Analysis
		2.5 Other Full Examples
	3 Challenges in the of the Next Generation Matrix
		3.1 Defining the Infected Class: Infected but Not Infectious
		3.2 Defining the Infected Class: Multiple Levels of Infection
		3.3 Unclear Which Eigenvalue Is Largest: Co-infection
	4 Challenges in Interpreting R0
		4.1 Multiple Infection Pathways: The Case of SARS
		4.2 The Maximum of Two Reproductive Numbers: Co-infection
		4.3 Two Thresholds and Backwards Bifurcations
		4.4 Geometric Series: Returning to the Infectious Class
		4.5 Geometric Mean: Indirect Transmission
		4.6 Not Quite a Geometric Mean: Generational Models
	5 Ways to Complicate Your Sensitivity Analysis
	6 Conclusions for Students
	7 Generating Possible Research Questions
	8 Notes for Mentors
	References
The Effect of External Perturbations on Ecological Oscillators
	1 Introduction
		1.1 Further Reading
	2 Oscillators
		2.1 Predator–Prey Models
			2.1.1 Lotka–Volterra Predator–Prey Model
			2.1.2 Rosenzweig–MacArthur Model
	3 The Phase of an Oscillator
		3.1 Isochrons
		3.2 The Effects of Perturbations on Phase
	4 Suggested Research Projects
	Appendix
	Appendix 1: MATLAB Code
	Appendix 2: XPP Code
	Appendix 3: Desmos Graphs
	Appendix 4: Other Oscillators
	Appendix 5: Non-dimensionalization and Relaxation Oscillators*
	References
Exploring Modeling by Programming: Insights from Numerical Experimentation
	1 Introduction
		1.1 Differential Equations Refresher
		1.2 Mathematical Modeling Refresher
		1.3 Programming Refresher
	2 Predator–Prey
		2.1 Analytical Results
		2.2 Numerical Results
	3 Toxoplasma gondii Transmission in Cats
		3.1 Role of Technology
		3.2 Dynamics
		3.3 Stability Analysis
		3.4 Implications
	4 Circadian Rhythms and Alcohol Dependence
		4.1 Role of Technology
		4.2 Model Dynamics
		4.3 Bifurcation
	5 Influenza Dynamics, Healthcare Options, and Associated Costs
		5.1 Role of Technology
		5.2 Model Dynamics
		5.3 Stability Analysis
	6 Stochastic Models
		6.1 Simple Birth–Death Process
		6.2 Gillespie Algorithm
			6.2.1 A Sample Calculation
			6.2.2 A Computer Implementation
		6.3 Stochastic Extension of Predator–Prey Model
		6.4 Stochastic Extension of Influenza Dynamics Model
	7 Conclusion
	Appendix 1: Notes on Programming
	Appendix 2: Circadian Rhythm Code
	Appendix 3: Influenza Code
	References
Simulating Bacterial Growth, Competition, and Resistance with Agent-Based Models and Laboratory Experiments
	1 Introduction
		1.1 Introduction to Antibiotic Resistance
			1.1.1 Genetic Basis of Antibiotic Resistance
			1.1.2 Mechanistic Basis of Antibiotic Resistance
		1.2 Spread and Severity of Antibiotic-Resistant Infections
		1.3 The Economic, Social, and Civic Impacts of Antibiotic Resistance
			1.3.1 Economic Burden of Antibiotic Resistance
			1.3.2 Increased Impact on Subpopulations
			1.3.3 Impact on the Food Supply and Agriculture
		1.4 Introduction to Agent-Based Models of Bacteria
	2 Bacteria Growth
		2.1 Bacterial Growth on Agar Plates
		2.2 Effects of Energy Source Availability on Bacterial Growth on an Agar Plate
		2.3 Effects of Energy Source Availability on Bacterial Growth in a Flask
	3 Competition Between Bacteria Strains
	4 Genetic Mutations
	5 Antibiotic Intervention and Resistance
	6 Spread of Antibiotic-Resistant Bacteria in Humans
	Appendix
		Model 1.2.0
		Model 2.2
		Model 3.0.0
			Model 4.3
		Model 5.0
		Model 6.2
	References
Agent-Based Modeling in Mathematical Biology: A Few Examples
	1 Preface
	2 Introduction
	3 Background for Students and Faculty
	4 Introduction to ABM
		4.1 Spatial Transmission
		4.2 Random Walks
		4.3 Reproduction
		4.4 Movement in Graphs
	5 Research Projects
		5.1 Modeling Subcutaneous Infections
		5.2 Research Projects for ABM in Biology
	6 Afterword
		6.1 Lindsey Chludzinski
		6.2 Alexandra Ballow
		6.3 Lindesy and Alex
	References
Network Structure and Dynamics of Biological Systems
	1 Introduction
		1.1 Further Reading
	2 Biological Systems Represented as Dynamic Models on Networks
	3 Fundamentals of Network Theory
		3.1 Mathematics of Undirected Networks
		3.2 Mathematics of Directed Networks
		3.3 Network Metrics
		3.4 Centrality Measures
	4 Network Models
		4.1 Special Types of Graphs
		4.2 Random Graphs
			4.2.1 Erdős–Rényi Random Graph
			4.2.2 Small-World Networks
			4.2.3 Scale-Free Networks
	5 Processes on Networks
		5.1 Percolation
		5.2 Epidemics on Networks
			5.2.1 SI Network Model
			5.2.2 SIR Network Model
			5.2.3 SIS Network Model
		5.3 Neuronal Activity
		5.4 Adaptive Networks
	6 Suggested Research Projects
	Appendix
		R Code for Figures
		Basics of Stochastic Processes
		R Code for Selected Stochastic Processes
	References
What Are the Chances?—Hidden Markov Models
	1 Introduction
		1.1 Case in Point
	2 Example Model
	3 Probability Equalities
		3.1 Joint Probability
		3.2 Marginal Probability
		3.3 Bayes\' Theorem
	4 Probability of a State Given an Observation
	5 Probability of a Sequence of States Generating a Sequence of Observations
	6 Probability of a State Sequence and an Observation Sequence
	7 Probability of a Sequence of Observations: The Forward Algorithm
		7.1 Obvious Solution
		7.2 Forward Algorithm
		7.3 Forward Algorithm Initialization
		7.4  Forward Algorithm Step 2
		7.5 Forward Algorithm Completion
	8 Probability of a Genomics Sequence
		8.1 Biology Background
		8.2 Locations of Genes
	9 Parallel Forward Algorithm (Optional)
		9.1 Communication
		9.2 Implementation of the Parallel Forward Algorithm
	10 Decoding Problem
		10.1 Obvious Solution
		10.2 Viterbi Algorithm
		10.3 Parallel Viterbi Algorithm (Optional)
	11 Detecting CpG Islands
	Exercises
	Projects
	Answers to Quick Review Questions
	Further Reading
	References
Using Neural Networks to Identify Bird Species from BirdsongSamples
	1 Introduction
	2 Creating Scalograms
		2.1 Time Signals, the WAV Audio Format, and Scalograms
		2.2 Wavelets
		2.3 The Continuous Wavelet Transform
		2.4 Transforms on Discrete Signals
		2.5 Creating Scalograms
	3 Neural Networks
		3.1 Densely Connected Neural Networks
		3.2 Training Network Parameters
		3.3 Other Types of Layers (Convolutional, Pooling, and Reshaping)
		3.4 Convolutional Neural Networks
	4 Data Description
	5 Pre-processing and Snippet Selection
		5.1 Making Scalograms
		5.2 Max-Pooling
		5.3 Training–Validation–Testing Split
		5.4 Selecting Snippets from Scalograms
		5.5 Denoising
	6 Results
	7 Suggested Future Research Projects
	References
Using Regularized Singularities to Model Stokes Flow: A Study of Fluid Dynamics Induced by Metachronal Ciliary Waves
	1 Biological Introduction
	2 Mathematical Model
		2.1 Fluid Dynamics
		2.2 Method of Regularized Stokeslets
		2.3 Cilium Beat Form
		2.4 Patches of Cilia
	3 Results
		3.1 No Metachronal Wave
		3.2 Various Metachronal Waves
	4 Conclusion
	5 Suggested Projects
	References