Chowell / Castillo-Chavez Mathematical and Statistical Estimation Approaches in Epidemiology

1. Auflage 2009
ISBN: 978-90-481-2313-1
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 363 Seiten

Mathematical and Statistical Estimation Approaches in Epidemiology compiles t- oretical and practical contributions of experts in the analysis of infectious disease epidemics in a single volume. Recent collections have focused in the analyses and simulation of deterministic and stochastic models whose aim is to identify and rank epidemiological and social mechanisms responsible for disease transmission. The contributions in this volume focus on the connections between models and disease data with emphasis on the application of mathematical and statistical approaches that quantify model and data uncertainty. The book is aimed at public health experts, applied mathematicians and sci- tists in the life and social sciences, particularly graduate or advanced undergraduate students, who are interested not only in building and connecting models to data but also in applying and developing methods that quantify uncertainty in the context of infectious diseases. Chowell and Brauer open this volume with an overview of the classical disease transmission models of Kermack-McKendrick including extensions that account for increased levels of epidemiological heterogeneity. Their theoretical tour is followed by the introduction of a simple methodology for the estimation of, the basic reproduction number,R . The use of this methodology 0 is illustrated, using regional data for 1918–1919 and 1968 in uenza pandemics.

Autoren/Hrsg.

Chowell, Gerardo

Castillo-Chavez, Carlos

Inhaltsverzeichnis

1;Preface;5
2;Contents;8
3;Contributors;10
4;The Basic Reproduction Number of Infectious Diseases: Computation and Estimation Using Compartmental Epidemic Models;13
4.1;1 Thresholds in Disease Transmission Models;13
4.2;2 The Simple Kermack-McKendrick Epidemic Model;14
4.3;3 More Elaborate Epidemic Models;17
4.4;4 SI R Models with Demographics;20
4.5;5 The SIS Model;23
4.6;6 Backward Bifurcations;24
4.6.1;6.1 Endemic Equilibria;27
4.7;7 Calculation of Reproduction Numbers;29
4.8;8 Estimating R0 Using a Compartmental Epidemic Model;31
4.8.1;8.1 Parameter Estimation;32
4.8.2;8.2 Bootstrap Confidence Intervals;33
4.8.3;8.3 Example: The Transmissibility of the 1918 Influenza Pandemic in Winnipeg, Canada;34
4.9;9 Estimation of the Reproduction Number Using the Intrinsic Growth Rate r;35
4.9.1;9.1 Example: The Transmissibility of the 1968 Influenza Pandemic in US Cities;37
4.10;References;39
5;Stochastic Epidemic Modeling;43
5.1;1 Introduction;43
5.2;2 History;44
5.3;3 Stochastic Compartmental Models;45
5.4;4 Distribution of the Final Epidemic Size;51
5.5;5 Stochastic Sustained Oscillations;57
5.6;6 Effects of Varying Infectiousness;58
5.7;7 Stochastic and Deterministic Dynamics are Complementary;60
5.8;References;62
6;Two Critical Issues in Quantitative Modeling of Communicable Diseases: Inference of Unobservables and Dependent Happening;65
6.1;1 Introduction;65
6.2;2 Incubation Period and Serial Interval;66
6.2.1;2.1 Incubation Period;67
6.2.2;2.2 Serial Interval;70
6.3;3 Backcalculation and Estimation of the Generation Time;73
6.3.1;3.1 Backcalculation;73
6.3.2;3.2 Generation Time;74
6.4;4 Dependent Happening;80
6.4.1;4.1 What Would Matter Due to Dependence?;80
6.4.2;4.2 Herd Immunity and the Concept of Effectiveness;83
6.5;5 Addressing Dependent Happening;88
6.5.1;5.1 Household Secondary Attack Rate;88
6.5.2;5.2 The Impact of Reductions in Susceptibility and Infectiousness on the Transmission Dynamics;91
6.6;6 Conclusion;94
6.7;References;95
7;The Chain of Infection, Contacts, and Model Parametrization;100
7.1;1 Modeling Infection;100
7.2;2 The Chain of Infection;105
7.3;3 Contact and Transmission Rates;107
7.4;4 Conclusions;111
7.5;References;112
8;The Effective Reproduction Number as a Prelude to Statistical Estimation of Time-Dependent Epidemic Trends;114
8.1;1 Introduction;114
8.2;2 Renewal Equation Offers the Conceptual Understanding of R(t);115
8.2.1;2.1 Infection-Age Structured Model;115
8.2.2;2.2 Deriving the Estimator of the Effective Reproduction Number;119
8.3;3 Applying Theory to the Data;122
8.3.1;3.1 A Simple Example;122
8.3.2;3.2 What to do with the Coarsely Reported Data?;127
8.4;4 Incidence-to-Prevalence Ratio and the Actual Reproduction Number;128
8.5;5 Conclusion;130
8.6;References;130
9;Sensitivity of Model-Based Epidemiological Parameter Estimation to Model Assumptions;133
9.1;1 Introduction;133
9.2;2 The Basic Reproductive Number and Its Estimation Using the Simple SIR Model;134
9.3;3 More Complex Compartmental Models;136
9.3.1;3.1 Inclusion of Latency;136
9.3.2;3.2 More General Compartmental Models: Gamma Distributed Latent and Infectious Periods;138
9.4;4 A General Formulation;140
9.5;5 Comparing R0 Estimates Obtained Using Different Models;143
9.6;6 Sensitivity Analysis;147
9.7;7 Discussion;148
9.8;References;150
10;An Ensemble Trajectory Method for Real-Time Modeling and Prediction of Unfolding Epidemics: Analysis of the 2005 Marburg Fever Outbreak in Angola;152
10.1;1 Introduction;152
10.2;2 Uncertainty Quantification and Model Parameter Estimation;153
10.3;3 Real Time Analysis of Outbreak of Marburg Fever in Angola;157
10.3.1;3.1 Brief Anatomy of the Outbreak;157
10.3.2;3.2 Homogeneously Mixing SEIR Population Model;159
10.3.3;3.3 Parameter Estimation and Outbreak Prediction;159
10.4;4 Discussion and Conclusions;169
10.5;References;169
11;Statistical Challenges in BioSurveillance;171
11.1;1 Introduction;171
11.2;2 Background;173
11.3;3 Public Health Outcome Surveillance;173
11.3.1;3.1 Adjusting for Covariates;174
11.3.2;3.2 Maximally Selected Measures of Evidence;174
11.3.3;3.3 Other Statistical Issues;180
11.4;4 Syndromic Surveillance;181
11.4.1;4.1 Inconsistent Seasonal Effects;182
11.4.2;4.2 Reporting Delays;184
11.4.3;4.3 System Population Coverage;185
11.4.4;4.4 Lack of Effective Training Data;185
11.4.5;4.5 Data Confidentiality;186
11.4.6;4.6 Two SS Systems;187
11.5;5 Discussion;190
11.5.1;5.1 Data Quality;190
11.5.2;5.2 Background Assessment;191
11.5.3;5.3 Complications Arising From Monitoring Multiple Data Sources;191
11.5.4;5.4 Creating Synthetic Outbreak Data;192
11.6;6 Open Challenges;192
11.7;7 Summary;193
11.8;References;193
12;Death Records from Historical Archives: A Valuable Source of Epidemiological Information;196
12.1;1 Introduction;196
12.2;2 The Nature of Historical Death Records;197
12.3;3 Uses of Historical Data;198
12.4;References;200
13;Sensitivity Analysis for Uncertainty Quantification in Mathematical Models;202
13.1;1 Introduction and Overview;202
13.1.1;1.1 Sensitivity Analysis: Forward and Adjoint Sensitivity;202
13.1.2;1.2 Parameter Estimation;204
13.2;2 Sensitivity Analysis;205
13.2.1;2.1 Normalized Sensitivity Index;205
13.2.2;2.2 Motivation for Sensitivity Analysis;206
13.3;3 Linear System of Equations and Eigenvalue Problem;207
13.3.1;3.1 Linear System of Equations: Symbiotic Population;207
13.3.2;3.2 Stability of the Equilibrium Solution: The Eigenvalue Problem;210
13.4;4 Dimensionality Reduction;215
13.4.1;4.1 Principal Component Analysis;216
13.4.2;4.2 Singular Value Decomposition (SVD);216
13.4.3;4.3 Sensitivity of SVD;217
13.5;5 Initial Value Problem;221
13.5.1;5.1 Forward Sensitivity of the IVP;222
13.5.2;5.2 Adjoint Sensitivity Analysis of the IVP;224
13.6;6 Principal Component Analysis of the IVP;226
13.7;7 Algorithmic Differentiation;227
13.7.1;7.1 Sensitivity of the Reproductive Number R0;228
13.7.2;7.2 Forward Sensitivity/Mode;230
13.7.3;7.3 Adjoint/Reverse Mode;235
13.8;8 Optimization Problems;237
13.8.1;8.1 Linear Programming Problem: BVD Disease;238
13.8.2;8.2 Quadratic Programming Problem: Wheat Selection;243
13.8.3;8.3 Adjoint Operator, Problem, and Sensitivity;246
13.9;9 Examples;249
13.9.1;9.1 Sensitivity of the Doubling Time;249
13.9.2;9.2 Sensitivity of a Critical Point;250
13.9.3;9.3 Sensitivity of Periodic Solutions to Parameters;252
13.10;References;253
14;An Inverse Problem Statistical Methodology Summary;255
14.1;1 Introduction;255
14.2;2 Parameter Estimation: MLE, OLS, and GLS;256
14.2.1;2.1 The Underlying Mathematical and Statistical Models;256
14.2.2;2.2 Known Error Processes: Normally Distributed Error;258
14.2.3;2.3 Unspecified Error Distributions and Asymptotic Theory;260
14.3;3 Computation of n, Standard Errors and Confidence Intervals;268
14.4;4 Investigation of Statistical Assumptions;272
14.4.1;4.1 Residual Plots;272
14.4.2;4.2 Example Using Residual Plots;274
14.5;5 Pneumococcal Disease Dynamics Model;279
14.5.1;5.1 Statistical Models of Case Notification Data;280
14.5.2;5.2 Inverse Problem Results: Simulated Data;281
14.5.3;5.3 Inverse Problem Results: Australian Surveillance Data;288
14.6;6 Sensitivity Functions;290
14.6.1;6.1 Traditional Sensitivity Functions;291
14.6.2;6.2 Generalized Sensitivity Functions;292
14.6.3;6.3 TSF and GSF for the Logistic Equation;294
14.7;7 Statistically Based Model Comparison Techniques;297
14.7.1;7.1 RSS Based Statistical Tests;298
14.7.2;7.2 Revisiting the Cat-Brain Problem;300
14.8;8 Epi Model Comparison;301
14.8.1;8.1 Surveillance Data;302
14.8.2;8.2 Test Statistic;303
14.8.3;8.3 Inverse Problem Results;304
14.8.4;8.4 Model Comparison;304
14.9;9 Concluding Remarks;306
14.10;References;307
15;The Epidemiological Impact of Rotavirus Vaccination Programs in the United States and Mexico;309
15.1;1 Introduction;310
15.2;2 Method;312
15.2.1;2.1 Age-Structured Model for Rotavirus Transmission and Its Vaccination;312
15.2.2;2.2 Parameterization;317
15.3;3 Results;320
15.4;4 Conclusions;324
15.5;References;326
15.6;Appendix;327
16;Spatial and Temporal Dynamics of Rubella in Peru, 1997–2006: Geographic Patterns, Age at Infection and Estimation of Transmissibility;330
16.1;1 Introduction;330
16.2;2 Materials and Methods;331
16.2.1;2.1 Demographic and Geographic Data;331
16.2.2;2.2 Rubella Epidemic Data;332
16.2.3;2.4 Estimation of the Basic Reproduction Number, R0;333
16.2.4;2.5 Estimation of the Reproduction Number, R;333
16.2.5;2.6 Critical Community Size;334
16.2.6;2.7 Scaling Laws in the Distributions of Attack Rates and Duration of Epidemics;334
16.2.7;2.8 Spatial Heterogeneity of Epidemics;335
16.3;3 Results;335
16.3.1;3.1 Estimates of the Basic Reproduction Number, R0;336
16.3.2;3.2 Estimates of the Reproduction Number, R, for Individual Rubella Outbreaks;338
16.3.3;3.3 Critical Community Size;338
16.3.4;3.4 Scaling Laws in the Distribution of Attack Rates and Duration of Epidemics;339
16.3.5;3.5 Spatial Heterogeneity;339
16.4;4 Discussion;340
16.5;References;344
17;The Role of Nonlinear Relapse on Contagion Amongst Drinking Communities;347
17.1;1 Introduction;348
17.1.1;1.1 Social Dynamics, Disease Transmission, and Social Structure;349
17.2;2 A Deterministic Contagion Model in Well-Mixed Drinking Communities;350
17.3;3 A Stochastic Contagion Model;353
17.4;4 Drinking Dynamics in Small-World Communities with High Relapse Rates;355
17.5;5 Discussion;360
17.6;Appendix;361
17.7;References;362
18;Index;365