Mathematical and Statistical Estimation Approaches in Epidemiology (eBook)

Gerardo Chowell is an associate professor and a Second Century Initiative Scholar (2CI) in the School of Public Health at Georgia State University in Atlanta. His research program includes the development and application of quantitative approaches for understanding the transmission dynamics and control of infectious diseases including influenza, Ebola, and dengue fever. His work has appeared in high-impact journals including The New England Journal of Medicine, PLOS Medicine, and BMC Medicine, and has been cited by major media outlets including the Washington Post and TIME magazine.James (Mac) Hyman has developed and analyzed mathematical models for the transmission of HIV/AIDs, influenza, malaria, dengue fever, chikungunya, and infections.  His current focus is to identify approaches where these models can help public health workers be more effective in mitigating the impact of emerging diseases.  He was a research scientist at Los Alamos National Laboratory for over thirty years, is a past president of the Society for Industrial and Applied Mathematics (SIAM),  and now holds the Phillips Distinguished Chair in Mathematics at Tulane University.

Preface 5
Contents 8
Contributors 10
The Basic Reproduction Number of Infectious Diseases: Computation and Estimation Using Compartmental Epidemic Models 13
1 Thresholds in Disease Transmission Models 13
2 The Simple Kermack-McKendrick Epidemic Model 14
3 More Elaborate Epidemic Models 17
4 SI R Models with Demographics 20
5 The SIS Model 23
6 Backward Bifurcations 24
6.1 Endemic Equilibria 27
7 Calculation of Reproduction Numbers 29
8 Estimating R0 Using a Compartmental Epidemic Model 31
8.1 Parameter Estimation 32
8.2 Bootstrap Confidence Intervals 33
8.3 Example: The Transmissibility of the 1918 Influenza Pandemic in Winnipeg, Canada 34
9 Estimation of the Reproduction Number Using the Intrinsic Growth Rate r 35
9.1 Example: The Transmissibility of the 1968 Influenza Pandemic in US Cities 37
References 39
Stochastic Epidemic Modeling 43
1 Introduction 43
2 History 44
3 Stochastic Compartmental Models 45
4 Distribution of the Final Epidemic Size 51
5 Stochastic Sustained Oscillations 57
6 Effects of Varying Infectiousness 58
7 Stochastic and Deterministic Dynamics are Complementary 60
References 62
Two Critical Issues in Quantitative Modeling of Communicable Diseases: Inference of Unobservables and Dependent Happening 65
1 Introduction 65
2 Incubation Period and Serial Interval 66
2.1 Incubation Period 67
2.2 Serial Interval 70
3 Backcalculation and Estimation of the Generation Time 73
3.1 Backcalculation 73
3.2 Generation Time 74
4 Dependent Happening 80
4.1 What Would Matter Due to Dependence? 80
4.2 Herd Immunity and the Concept of Effectiveness 83
5 Addressing Dependent Happening 88
5.1 Household Secondary Attack Rate 88
5.2 The Impact of Reductions in Susceptibility and Infectiousness on the Transmission Dynamics 91
6 Conclusion 94
References 95
The Chain of Infection, Contacts, and Model Parametrization 100
1 Modeling Infection 100
2 The Chain of Infection 105
3 Contact and Transmission Rates 107
4 Conclusions 111
References 112
The Effective Reproduction Number as a Prelude to Statistical Estimation of Time-Dependent Epidemic Trends 114
1 Introduction 114
2 Renewal Equation Offers the Conceptual Understanding of R(t) 115
2.1 Infection-Age Structured Model 115
2.2 Deriving the Estimator of the Effective Reproduction Number 119
3 Applying Theory to the Data 122
3.1 A Simple Example 122
3.2 What to do with the Coarsely Reported Data? 127
4 Incidence-to-Prevalence Ratio and the Actual Reproduction Number 128
5 Conclusion 130
References 130
Sensitivity of Model-Based Epidemiological Parameter Estimation to Model Assumptions 133
1 Introduction 133
2 The Basic Reproductive Number and Its Estimation Using the Simple SIR Model 134
3 More Complex Compartmental Models 136
3.1 Inclusion of Latency 136
3.2 More General Compartmental Models: Gamma Distributed Latent and Infectious Periods 138
4 A General Formulation 140
5 Comparing R0 Estimates Obtained Using Different Models 143
6 Sensitivity Analysis 147
7 Discussion 148
References 150
An Ensemble Trajectory Method for Real-Time Modeling and Prediction of Unfolding Epidemics: Analysis of the 2005 Marburg Fever Outbreak in Angola 152
1 Introduction 152
2 Uncertainty Quantification and Model Parameter Estimation 153
3 Real Time Analysis of Outbreak of Marburg Fever in Angola 157
3.1 Brief Anatomy of the Outbreak 157
3.2 Homogeneously Mixing SEIR Population Model 159
3.3 Parameter Estimation and Outbreak Prediction 159
4 Discussion and Conclusions 169
References 169
Statistical Challenges in BioSurveillance 171
1 Introduction 171
2 Background 173
3 Public Health Outcome Surveillance 173
3.1 Adjusting for Covariates 174
3.2 Maximally Selected Measures of Evidence 174
3.3 Other Statistical Issues 180
4 Syndromic Surveillance 181
4.1 Inconsistent Seasonal Effects 182
4.2 Reporting Delays 184
4.3 System Population Coverage 185
4.4 Lack of Effective Training Data 185
4.5 Data Confidentiality 186
4.6 Two SS Systems 187
5 Discussion 190
5.1 Data Quality 190
5.2 Background Assessment 191
5.3 Complications Arising From Monitoring Multiple Data Sources 191
5.4 Creating Synthetic Outbreak Data 192
6 Open Challenges 192
7 Summary 193
References 193
Death Records from Historical Archives: A Valuable Source of Epidemiological Information 196
1 Introduction 196
2 The Nature of Historical Death Records 197
3 Uses of Historical Data 198
References 200
Sensitivity Analysis for Uncertainty Quantification in Mathematical Models 202
1 Introduction and Overview 202
1.1 Sensitivity Analysis: Forward and Adjoint Sensitivity 202
1.2 Parameter Estimation 204
2 Sensitivity Analysis 205
2.1 Normalized Sensitivity Index 205
2.2 Motivation for Sensitivity Analysis 206
3 Linear System of Equations and Eigenvalue Problem 207
3.1 Linear System of Equations: Symbiotic Population 207
3.2 Stability of the Equilibrium Solution: The Eigenvalue Problem 210
4 Dimensionality Reduction 215
4.1 Principal Component Analysis 216
4.2 Singular Value Decomposition (SVD) 216
4.3 Sensitivity of SVD 217
5 Initial Value Problem 221
5.1 Forward Sensitivity of the IVP 222
5.2 Adjoint Sensitivity Analysis of the IVP 224
6 Principal Component Analysis of the IVP 226
7 Algorithmic Differentiation 227
7.1 Sensitivity of the Reproductive Number R0 228
7.2 Forward Sensitivity/Mode 230
7.3 Adjoint/Reverse Mode 235
8 Optimization Problems 237
8.1 Linear Programming Problem: BVD Disease 238
8.2 Quadratic Programming Problem: Wheat Selection 243
8.3 Adjoint Operator, Problem, and Sensitivity 246
9 Examples 249
9.1 Sensitivity of the Doubling Time 249
9.2 Sensitivity of a Critical Point 250
9.3 Sensitivity of Periodic Solutions to Parameters 252
References 253
An Inverse Problem Statistical Methodology Summary 255
1 Introduction 255
2 Parameter Estimation: MLE, OLS, and GLS 256
2.1 The Underlying Mathematical and Statistical Models 256
2.2 Known Error Processes: Normally Distributed Error 258
2.3 Unspecified Error Distributions and Asymptotic Theory 260
3 Computation of n, Standard Errors and Confidence Intervals 268
4 Investigation of Statistical Assumptions 272
4.1 Residual Plots 272
4.2 Example Using Residual Plots 274
5 Pneumococcal Disease Dynamics Model 279
5.1 Statistical Models of Case Notification Data 280
5.2 Inverse Problem Results: Simulated Data 281
5.3 Inverse Problem Results: Australian Surveillance Data 288
6 Sensitivity Functions 290
6.1 Traditional Sensitivity Functions 291
6.2 Generalized Sensitivity Functions 292
6.3 TSF and GSF for the Logistic Equation 294
7 Statistically Based Model Comparison Techniques 297
7.1 RSS Based Statistical Tests 298
7.2 Revisiting the Cat-Brain Problem 300
8 Epi Model Comparison 301
8.1 Surveillance Data 302
8.2 Test Statistic 303
8.3 Inverse Problem Results 304
8.4 Model Comparison 304
9 Concluding Remarks 306
References 307
The Epidemiological Impact of Rotavirus Vaccination Programs in the United States and Mexico 309
1 Introduction 310
2 Method 312
2.1 Age-Structured Model for Rotavirus Transmission and Its Vaccination 312
2.2 Parameterization 317
3 Results 320
4 Conclusions 324
References 326
Appendix 327
Spatial and Temporal Dynamics of Rubella in Peru, 1997–2006: Geographic Patterns, Age at Infection and Estimation of Transmissibility 330
1 Introduction 330
2 Materials and Methods 331
2.1 Demographic and Geographic Data 331
2.2 Rubella Epidemic Data 332
2.4 Estimation of the Basic Reproduction Number, R0 333
2.5 Estimation of the Reproduction Number, R 333
2.6 Critical Community Size 334
2.7 Scaling Laws in the Distributions of Attack Rates and Duration of Epidemics 334
2.8 Spatial Heterogeneity of Epidemics 335
3 Results 335
3.1 Estimates of the Basic Reproduction Number, R0 336
3.2 Estimates of the Reproduction Number, R, for Individual Rubella Outbreaks 338
3.3 Critical Community Size 338
3.4 Scaling Laws in the Distribution of Attack Rates and Duration of Epidemics 339
3.5 Spatial Heterogeneity 339
4 Discussion 340
References 344
The Role of Nonlinear Relapse on Contagion Amongst Drinking Communities 347
1 Introduction 348
1.1 Social Dynamics, Disease Transmission, and Social Structure 349
2 A Deterministic Contagion Model in Well-Mixed Drinking Communities 350
3 A Stochastic Contagion Model 353
4 Drinking Dynamics in Small-World Communities with High Relapse Rates 355
5 Discussion 360
Appendix 361
References 362
Index 365