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Mathematical Epidemiology and More
A Course on Compartmental Models
Meredith L. Greer
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\(\newcommand{\R}{\mathbb R} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)
Front Matter
Colophon
Acknowledgements
Preface
To Students: How to Get the Most from This Text
I
Core Modeling Concepts
1
Prelude: Diseases in Popular News
1.1
Possible Discussion Themes
1.2
Follow Up By Seeking More Precise Data
1.3
For Further Thought
2
Zombies: Can Math Help?
2.1
We Now Have Data: Qualitative Musings
2.2
A First Model
2.3
For Further Thought
3
Visualization: Graphing Model Results and Data
3.1
“Solving” Differential Equations Using Graphs
3.2
Estimating
\(\beta\)
and Building Visualization Skills
3.3
For Further Thought
4
Compartmental Diagrams
4.1
Compartmental Diagrams: Vocabulary and Comments
4.2
From Zombie Game Variation to Compartmental Diagram
4.3
Estimating Gamma and Other Rates
4.4
For Further Thought
5
Model Analysis and the Basic Reproduction Number
\(\mathcal{R}_0\)
5.1
Using Calculus to Analyze Models
5.2
The Basic Reproduction Number
\(\mathcal{R}_0\)
5.3
For Further Thought
6
Vaccines
6.1
Determining a Threshold Vaccine Level
6.2
How Vaccination Level Affects Long Term Outcomes
6.3
Vaccine Efficacy and Effectiveness
6.4
For Further Thought
7
Incubation Periods
7.1
Including an Incubation Period: The SEIR Model
7.2
Practice with an Incubation Model
7.3
For Further Thought
8
Isolation
8.1
Introduction
8.2
Complete Isolation: Models and Parameter Estimation
8.3
Incomplete Isolation, and Quarantine vs. Isolation
8.4
Practice with an Isolation Model
8.5
For Further Thought
9
Preparing Influenza Data to Compare With a Model
9.1
Incidence and Prevalence
9.2
Translating a Typical Data Set Into Numbers We Can Compare With a Model
9.3
For Further Thought
10
Residual Sum of Squares: Improving a Model’s Fit to Data
10.1
Residual Sum of Squares: An Introductory Example
10.2
The Underlying Math of RSS
10.3
Residual Sum of Squares: Quickly Comparing Multiple Parameter Values
10.4
For Further Thought
11
* Akaike Information Criterion: Comparing Different Models
11.1
12
* Epidemic vs. Endemic Illness
12.1
Comparing Endemic Illnesses with Epidemics
13
* Modeling the Spread of an Idea
13.1
II
Bonus Topics
14
The Next Generation Method
14.1
Mathematical Background for the Next Generation Method
14.2
The Next Generation Method: Steps to Follow
14.3
Python Code for the Next Generation Method
14.4
For Further Thought
15
* Case Studies: Vaccination and Isolation
15.1
A Varicella Example with Vaccination
15.2
H1N1 with Testing and Strict Isolation
15.3
A Measles Example with a Vaccine Campaign
16
* Articles Showing Further Examples
16.1
Compartmental Models for Diseases
16.1.1
Legionnaires’ Disease
16.2
Compartmental Models for Applications Other Than Diseases
16.2.1
Elections
16.2.2
Bumblebees and Pollination
17
* Tips on Reading a Scientific Paper
17.1
What are Scientific Papers, and Why Read Them?
18
Greek Letters
18.1
A.
Glossary of Mathematical Epidemiology Terms
Colophon
Colophon
©2020–present Meredith L. Greer
This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, CC BY-NC-SA. To view a copy of this license, visit
CreativeCommons.org.
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