Yan P., Chowell G. Quantitative Methods for Investigating Infectious Disease Outbreaks

Cham: Springer, 2019. — 361 p.This book provides a systematic treatment of the mathematical underpinnings of work in the theory of outbreak dynamics and their control, covering balanced perspectives between theory and practice including new material on contemporary topics in the field of infectious disease modeling. Specifically, it presents a unified mathematical framework linked to the distribution theory of non-negative random variables; the many examples used in the text, are introduced and discussed in light of theoretical perspectives.
The book is organized into 9 chapters: The first motivates the presentation of the material on subsequent chapters; Chapter 2-3 provides a review of basic concepts of probability and statistical models for the distributions of continuous lifetime data and the distributions of random counts and counting processes, which are linked to phenomenological models. Chapters 4 focuses on dynamic behaviors of a disease outbreak during the initial phase while Chapters 5-6 broadly cover compartment models to investigate the consequences of epidemics as the outbreak moves beyond the initial phase. Chapter 7 provides a transition between mostly theoretical topics in earlier chapters and Chapters 8 and 9 where the focus is on the data generating processes and statistical issues of fitting models to data as well as specific mathematical epidemic modeling applications, respectively.
This book is aimed at a wide audience ranging from graduate students to established scientists from quantitatively-oriented fields of epidemiology, mathematics and statistics. The numerous examples and illustrations make understanding of the mathematics of disease transmission and control accessible. Furthermore, the examples and exercises, make the book suitable for motivated students in applied mathematics, either through a lecture course, or through self-study. This text could be used in graduate schools or special summer schools covering research problems in mathematical biology.The Motivation
Structure of the Book with Brief Summary
Shapes of Hazard Functions and Lifetime Distributions
The Hazard Function, the Distribution Functions, and Some Commonly Used Summary Measures
The Shapes of Hazard Functions
The Constant Hazard Function and the Exponential Distribution
Monotonic Hazard Functions Without Upper Limit
The Gamma Distribution
The Inverse-Gaussian Distribution: Non-monotone and Converge to a Positive Constant as x→∞
The Log-Normal Distribution as a Model for the Incubation Period
The Log-Logistic Distribution
The Residual Life Distribution as Uniquely Determined by the Hazard Function
The Pareto Distributions
The Laplace Transform for Life Distributions
Laplace Transform of the Sum of Two Independent Random Variables
Laplace Transform as a Survival Function
Comparing Magnitudes
A General Description of Variability Is Based on ``Majorization''
Laplace Transforms of the Gamma and Inverse-Gaussian Distributions
Mixture of Distributions and Frailty Models
Frailty and Dampened Hazard Functions
Frailty Models with Gamma Distributed u(θ)
Problems and Supplements
Random Counts and Counting Processes
The Probability Functions and Related Quantities
The Power Series Distributions
The Power-Law Distributions
Random Count Distributions as Generated by Stochastic Disease Transmission Models
Mixture of Poisson Distributions and Processes
The Negative Binomial Distribution as a Mixed-Poisson Distribution
Other Mixed-Poisson Distributions
Highly Skewed Data: Proneness, Contagion, or Spells?
General Formulation of a Counting Process
The Time-Homogeneous Poisson Process Given by Definition
Counting Processes with the Negative Binomial Distribution as the Marginal Distribution for Count Numbers
Variance and Covariance Processes for Zero-Mean Martingales
Problems and Supplements
The Branching Process Approximation
The Galton-Watson Branching Process
The Probability Generation Function for N
The Crump-Mode-Jagers (CMJ) Branching Process
Extensions of the CMJ Branching Process
Extinction and the Invasion Probability
The Effects of Variability of N on the Invasion Probability -δ and Generations Toward Extinction
Summary Statements on Invasion Probability and Generations to Extinction
The Geometric Distribution for N
The Poisson Distribution for N
The Negative Binomial Distribution for N
Final Size Distributions for Small Outbreaks
The Distribution of Zs
Some Special Cases when N is Distributed Within the Negative Binomial Family
Examples
Estimation for R Based on the Galton-Watson Branching Process
Modified Harris Estimator Based on Surveillance Data
From Theory to Practice
The Exponential Growth by Generation
The Exponential Growth Derived as the Expected Value of a Linear Pure Birth Markov Process
The Euler-Lotka Equations
When There Is No Latent Period
In the Presence of a Latent Period
Relationships Between r and R for Gamma Distributed TE and TI
Relationships Between r and R When the Exact Distributions for TE and TI Are Unknown
The Initial Phase
The Constant Growth Rate and Asymptotics
Equilibrium Conditions
Periodic Resonance Around a Predominant Exponential Growth
The Sub-exponential Growth
The Sub-exponential Growth as Approximation for the Convex Increasing Part of the Logistic Growth
Frailty Interpretation
Discussion on the Sub-exponential Growth Cd(t)=i(+rvt)v,
A Comprehensive Demonstration of Curve Fitting Using Nonlinear Phenomenological Models to Outbreak Data from the Zika Epidemic in Antioquia, Colombia
Exploratory Analysis
Likelihood Analysis, Estimation and Predictions for the Sub-exponential Model
Least Square Estimation and Predictions for the Sub-exponential Model
Fitting the Logistic Growth Model to the First Epidemic Days of Data
The Logistic Model: Likelihood Based Analyses
The Logistic Model: Least Square Analysis
Fitting Generalized Logistic Models to the First -Day Data with Discussions on Over-parameterization
Data by Day
Least Square Estimates for Other Generalized Logistic Models
Lessons Learned
The Effective Reproduction Number, Rt, with Quantified Uncertainty
Example Based on the Epidemic of Yellow Fever in Two Areas of Angola: Luanda and Huambo
Problems and Supplements
Mechanistic Models with Spatial Structures and Reactive Behavior Change
Metapopulation Spatial Models
Individual-Based Network Models
An Individual-Level Network Model with Household-Community Structure
Capture Dynamic Reactive Behavior Changes Through a Generalized-Growth SEIR Model
Case Study: Modeling the Effectiveness of Contact Tracing During Ebola Epidemics
Model : Homogenous-Mixing SEIR TransmissionModel
Model : Spatially Structured Ebola Transmission Model
Stochastic Simulations
Problems and Supplements