Bayesian Analysis of Infectious Diseases: COVID-19 and Beyond

by Lyle D. Broemeling

Overview

Bayesian Analysis of Infectious Diseases -COVID-19 and Beyond shows how the Bayesian approach can be used to analyze the evolutionary behavior of infectious diseases, including the coronavirus pandemic. The book describes the foundation of Bayesian statistics while explicating the biology and evolutionary behavior of infectious diseases, including viral and bacterial manifestations of the contagion. The book discusses the application of Markov Chains to contagious diseases, previews data analysis models, the epidemic threshold theorem, and basic properties of the infection process. Also described are the chain binomial model for the evolution of epidemics.

Features:

• Represents the first book on infectious disease from a Bayesian perspective.
• Employs WinBUGS and R to generate observations that follow the course of contagious maladies.
• Includes discussion of the coronavirus pandemic as well as many examples from the past, including the flu epidemic of 1918-1919.
• Compares standard non-Bayesian and Bayesian inferences.
• Offers the R and WinBUGS code on at www.routledge.com/9780367633868

Product Details

ISBN-13:	9780367647247
Publisher:	CRC Press
Publication date:	08/29/2022
Series:	Chapman & Hall/CRC Biostatistics Series
Pages:	342
Product dimensions:	6.12(w) x 9.19(h) x (d)

About the Author

Lyle D. Broemeling, Ph.D., is Director of Broemeling and Associates Inc., and is a consulting biostatistician. He has been involved with academic health science centers for about 20 years and has taught and been a consultant at the University of Texas Medical Branch in Galveston, the University of Texas MD Anderson Cancer Center and the University of Texas School of Public Health. His main interest is in developing Bayesian methods for use in medical and biological problems and in authoring textbooks in statistics. His previous books are Bayesian Biostatistics and Diagnostic Medicine, and Bayesian Methods for Agreement.

Table of Contents

Author x

1 Introduction to Bayesian Inferences for Infectious Diseases 1

1.1 Introduction 1

1.2 A Preview of the Book 1

1.3 Some Key References for Infectious Diseases and their Analysis 3

The Following Articles are Quite Appropriate for the Coronavirus Pandemic 4

Links to Infectious Disease Sources 4

Comments 4

2 Bayesian Analysis 5

2.1 Introduction 5

2.2 Bayes Theorem 6

2.3 Prior Information 8

2.3.1 The Binomial Distribution 8

2.3.2 The Normal Distribution 12

2.4 Posterior Information 13

2.4.1 The Binomial Distribution 13

2.4.2 The Normal Distribution 13

2.4.3 The Poisson Distribution 17

2.5 Inference 19

2.5.1 Introduction 19

2.5.2 Estimation 19

2.5.3 Testing Hypotheses 21

2.6 Predictive Inference 26

2.6.1 Introduction 26

2.6.2 The Binomial Population 26

2.6.3 Forecasting from a Normal Population 28

2.7 An Example of Bayesian Inference in a Stochastic Epidemic 31

2.7.1 Deterministic Model 31

2.7.2 The Stochastic Epidemic 32

2.8 Comments 35

2.9 Exercises 35

References 37

3 Infectious Diseases 39

3.1 Introduction 39

3.2 Antibody Production 40

3.3 How Drugs Fight Disease 41

3.4 How Certain Drugs Attack Infections 41

3.5 Drug Resistance 42

3.6 Vaccine and Hormone Therapy 42

3.7 Infectious Diseases 43

3.8 Contagious Diseases 43

3.9 Emerging Infections 48

3.10 Tuberculosis 55

3.11 Malaria 55

3.12 Coronavirus 55

3.13 Foodborne Pathogens 55

3.14 Animal Virus Threats to Humans 56

References 56

4 Bayesian Inference for Discrete Markov Chains: Their Relevance to Infectious Diseases 59

4.1 Introduction 59

4.2 Examples of Markov Chains with Biased Coins 61

4.3 Fundamental Computations 63

4.4 Limiting Distributions 69

4.5 Stationary Distributions 79

4.6 Where is that Particular State? 83

4.6.1 Introduction 83

4.6.2 Irreducible Chains 85

4.6.3 Bayesian Analysis of Transient and Recurrent States 87

4.7 Period of a Markov Chain 91

4.8 Ergodic Chains and Time Reversibility 94

4.9 Stochastic Epidemic 101

4.10 Tracking the Coronavirus 103

4.11 Comments and Conclusions 104

4.12 Exercises 107

References 111

5 Biological Examples Modeled by Discrete Markov Chains 113

5.1 Introduction 113

5.2 Birth and Death Process 113

5.3 The Logistic Growth Process 121

5.4 Epidemic Processes 125

5.4.1 Introduction 125

5.4.2 Deterministic Model 126

5.4.3 Stochastic Model 127

5.4.4 Chain Binomial Epidemic Models 130

5.5 Duration and Size 136

5.6 Example of an Epidemic with the Greenwood and Reed-Frost Models 138

5.7 The Covid-19 Pandemic. Statistical Concepts and Perspectives 138

5.8 Comments and Conclusions 139

5.9 Exercises 141

References 146

6 Inferences for Markov Chains in Continuous Time 149

6.1 Introduction 149

6.2 The Poisson Process 149

Definition (a) 150

Definition (b) 150

Definition (c) 151

6.3 Bayesian Inferences for λ 152

6.4 Thinning and Superposition 152

6.5 Spatial Poisson Process 155

6.6 Concomitant Poisson Processes 162

6.7 Nonhomogeneous Processes 162

6.7.1 The Intensity Function 162

6.7.2 Choosing the Intensity Function 164

6.8 General Continuous Time Markov Chains 182

6.9 Why Are More Coronavirus Tests Needed Than We Thought Were Needed? 187

6.10 Summary 189

6.11 Exercises 190

References 192

7 Bayesian Inference: Biological Processes that Follow a Continuous Time Markov Chain 195

7.1 Introduction 195

7.2 The Foundation of Continuous Time Markov Chains 195

7.2.1 The Markov Property and Transition Function 196

7.2.2 Transition Rates, Holding Times, and Transition Probabilities 197

7.2.3 The Kolmogorov Forward and Backward Equations and the Matrix Exponential 199

7.2.4 Computing the Transition Function with R 200

7.3 Limiting and Stationary Distributions 201

Basic Limit Theorem 202

7.4 Mean Time to Absorption with R 204

7.5 Time Reversibility 206

7.6 Time Reversibility 207

7.6.1 DNA Evolution 207

7.6.2 Birth and Death Processes 221

7.6.3 A Random Walk 229

7.6.4 The Yule Process 231

7.6.5 Birth and Death With Immigration 232

7.6.6 SI Epidemic Models 234

7.6.7 Stochastic SIS Epidemic Model 236

7.7 The SIR Model 238

7.8 Summary and Conclusions 247

7.9 Exercises 249

References 252

8 Additional Information about Infectious Diseases 253

8.1 Introduction 253

8.2 Contagious Diseases Today 253

8.3 A Preview of Data Analysis and Models 254

8.4 The Epidemic Threshold Theorem 256

8.5 Basic Characteristics of the Infectious Process 257

8.6 Chain Binomial Representations 257

8.7 The Size of the Outbreak is Compared 258

8.8 The Evolution of an Epidemic: Epidemic Chains 260

8.9 A Chain Binomial Model 260

8.10 What is the Size of an Epidemic? 263

8.11 Chain Data for the Common Cold 266

8.12 Generalized Linear Models for the Analysis of Binomial Chain Information 267

8.13 Models for the Common Cold 269

8.14 Random Infectiousness Models 272

8.15 Latent and Infectiousness Periods 275

8.16 Observable Infectious Period 275

8.17 Households of Size Two 277

8.18 Example of Households of Size Two for a Measles Epidemic 280

8.19 Bayesian Analysis of Measles in Households of Size Two 281

8.20 The Exponential Growth of Epidemics 282

8.21 The Coronavirus 283

8.22 Do we Need More Tests for the Virus? 288

8.23 Group Testing 289

8.24 CD4 in HIV Patients 291

8.25 An Analysis of Smallpox Data 295

8.26 Respiratory Disease Data 300

8.27 Bayesian Analysis of Respiratory Disease Information 308

8.28 Comments and Conclusions 310

8.29 Exercises 311

References 312

Index 315

Editorial Reviews

"This book will be useful for both masters and undergraduate students in biostatistics, who are planning to pursue research in Bayesian approaches towards epidemics applications"

- Chitaranjan Mahapatra, International Society for Clinical Biostatistics, 72, 2021

"Since most available textbooks for infectious disease modeling present the subject from differential equations,
mathematical modeling perspective, this text is an important first step towards filling the gap from the statistical
perspective."

Marie V. Ozanne, Mount Holyoke College USA, Biometrics: A Journal of the International Biometric Society, December 2021.

From the Publisher