Statistics for Biology and Health. Series Editors: M. Gail K. Krickeberg J. Sarnet A. Tsiatis W. Wong

Transcription

1 Statistics for Biology and Health Series Editors: M. Gail K. Krickeberg J. Sarnet A. Tsiatis W. Wong

2 Statistics for Biology and Health Aalen/Borgan/Gjessing: Survival and Event History Analysis: A Process Point of View Bacchieri/Cioppa: Fundamentals of Clinical Research Borchers/Buckland/Zucchini: Estimating Animal Abundance: Closed Populations Burzykowski/Molenberghs/Buyse: The Evaluation of Surrogate Endpoints Duchateau/Janssen: The Frailty Model Everitt/Rabe-Hesketh: Analyzing Medical Data Using S-PLUS Ewens/Grant: Statistical Methods in Bioinformatics: An Introduction, 2 nd ed. Gentleman/Carey/Huber/Irizarry/Dudoit: Bioinformatics and Computational Biology Solutions Using R and Bioconductor Hougaard: Analysis of Multivariate Survival Data Keyfitz/Caswell: Applied Mathematical Demography, 3 rd ed. Klein/Moeschberger: Survival Analysis: Techniques for Censored and Truncated Data, 2 nd ed. Kleinbaum/Klein: Survival Analysis: A Self-Learning Text, 2 nd ed. Kleinbaum/Klein: Logistic Regression: A Self-Learning Text, 2 nd ed. Lange: Mathematical and Statistical Methods for Genetic Analysis, 2 nd ed. Manton/Singer/Suzman: Forecasting the Health of Elderly Populations Martinussen/Scheike: Dynamic Regression Models for Survival Data Moyé: Multiple Analyses in Clinical Trials: Fundamentals for Investigators Nielsen: Statistical Methods in Molecular Evolution O Quigley: Proportional Hazards Regression Parmigiani/Garrett/Irizarry/Zeger: The Analysis of Gene Expression Data: Methods and Software Proschan/LanWittes: Statistical Monitoring of Clinical Trials: A Unified Approach Schlattmann: Medical Applications of Finite Mixture Models Siegmund/Yakir: The Statistics of Gene Mapping Simon/Korn/McShane/Radmacher/Wright/Zhao: Design and Analysis of DNA Microarray Investigations Sorensen/Gianola: Likelihood, Bayesian, and MCMC Methods in Quantitative Genetics Stallard/Manton/Cohen: Forecasting Product Liability Claims: Epidemiology and Modeling in the Manville Asbestos Case Sun: The Statistical Analysis of Interval-censored Failure Time Data Therneau/Grambsch: Modeling Survival Data: Extending the Cox Model Ting: Dose Finding in Drug Development Vittinghoff/Glidden/Shiboski/McCulloch: Regression Methods in Biostatistics: Linear, Logistic, Survival, and Repeated Measures Models Wu/Ma/Casella: Statistical Genetics of Quantitative Traits: Linkage, Maps, and QTL Zhang/Singer: Recursive Partitioning in the Health Sciences Zuur/Ieno/Smith: Analysing Ecological Data For further volumes:

3 Niel Hens Ziv Shkedy Marc Aerts Christel Faes Pierre Van Damme Philippe Beutels Modeling Infectious Disease Parameters Based on Serological and Social Contact Data A Modern Statistical Perspective 123

4 Niel Hens Center for Statistics, Interuniversity Institute for Biostatistics and statistical Bioinformatics Hasselt University Diepenbeek, Belgium Centre for Health Economic Research and Modelling Infectious Diseases, Vaccine and Infectious Disease Institute University of Antwerp Universiteitsplein Antwerpen, Belgium Marc Aerts Center for Statistics, Interuniversity Institute for Biostatistics and statistical Bioinformatics Hasselt University Diepenbeek, Belgium Pierre Van Damme Centre for the Evaluation of Vaccination, Vaccine and Infectious Disease Institute University of Antwerp Antwerpen, Belgium Ziv Shkedy Center for Statistics, Interuniversity Institute for Biostatistics and statistical Bioinformatics Hasselt University Diepenbeek, Belgium Christel Faes Center for Statistics, Interuniversity Institute for Biostatistics and statistical Bioinformatics Hasselt University Diepenbeek, Belgium Philippe Beutels Centre for Health Economic Research and Modelling Infectious Diseases, Vaccine and Infectious Disease Institute University of Antwerp Antwerpen, Belgium Please note that additional material for this book can be downloaded from ISSN ISBN ISBN (ebook) DOI / Springer New York Heidelberg Dordrecht London Library of Congress Control Number: Springer Science+Business Media New York 2012 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 To Kelly, Lena and Louisa To my family To Els, Anouk and Lauren To Sven, Hanne and Marte To Myrjam, Bruno, Tom and Eline To Conny and Robin

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7 Preface Mathematical epidemiology of infectious diseases usually involves describing the flow of individuals between mutually exclusive infection states. One of the key parameters describing the transition from the susceptible to the infected class is the hazard of infection, often referred to as the force of infection. The force of infection reflects the degree of contact with potential for transmission between infected and susceptible individuals. The mathematical relation between the force of infection and effective contact patterns is generally assumed to be subjected to the mass action principle, which yields the necessary information to estimate the basic reproduction number, another key parameter in infectious disease epidemiology. It is within this context that the Center for Statistics (CenStat, Interuniversity Institute for Biostatistics and statistical Bioinformatics, Hasselt University) and the Centre for the Evaluation of Vaccination and the Centre for Health Economic Research and Modelling Infectious Diseases (CEV, CHERMID, Vaccine and Infectious Disease Institute, University of Antwerp) have collaborated over the past 15 years. This book demonstrates the past and current research activities of these institutes and can be considered to be a milestone in this collaboration. This book is focused on the application of modern statistical methods and models to estimate infectious disease parameters. We want to provide the readers with software guidance, such as R packages, and with data, as far as they can be made publicly available. Please visit for data and code pertaining to this book. Many persons have contributed either directly or indirectly to this book for which we are very Hasselt University: Steven Abrams, Girma Minalu Ayele, Kaatje Bollaerts, Emanuele Del Fava, Nele Goeyvaerts, Geert Molenberghs, Harriet Namata, and Kim Van University of Antwerp: Mathieu Andraud, Joke Bilcke, Olivier Lejeune, Elke Leuridan, Benson Ogunjimi, and Heidi Theeten, and elsewhere: Kari Auranen, Benoit Dervaux, John Edmunds, Paddy Farrington, Nigel Gay, Janneke Heijne, Daniel Hlubinka, Mark Jit, Peter Kung U Kimani, Mira Kojouhorova, Andrea Kvitkovicova, Marco Massari, Rafael Mikolajczyk, Joël Mossong, Magdalena Rosinka, Malgorzata Sadkowska-Todys, Stefania Salmaso, Gianpaolo Scalia Tomba, Thierry Van Effelterre, Jacco Wallinga, Andreas Wienke, vii

8 viii Preface and James Wood. We also wish to thank the editorial team at Springer, more particularly John Kimmel, Marc Strauss, and Hannah Bracken for their guidance throughout the writing process and the reviewers Piero Manfredi, James Wood, Jacco Wallinga, and Gianpaolo Scalia Tomba whose identity was revealed to us and whose suggestions improved the book considerably. The authors would appreciate being informed of errors and may be contacted by electronic mail: Hasselt and Antwerp Niel Hens Ziv Shkedy Marc Aerts Christel Faes Pierre Van Damme Philippe Beutels

9 Acknowledgements This book would not have been possible without the strategic basic research project SIMID funded by the agency for the Promotion of Innovation by Science and Technology in Flanders (IWT project number ). We further gratefully acknowledge the chair in evidence-based vaccinology, which was sponsored by a gift from Pfizer. Some of the material in this book was developed based on earlier work funded by the Flemish fund for scientific research (FWO project number G039304), EC s sixth framework project POLYMOD, and the IAP research network nr P6/03 of the Belgian Government (Belgian Science Policy). We thank the production team at Springer for their support at the final stage of the publication process. ix

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11 Contents Part I Introducing the Concept of the Book 1 Why This Book? An Introduction Terms for Germs Models of Infectious Diseases Where Does This Book Fit in the Field? A Road Map for This Book... 6 Part II Mathematical Models for Infectious Diseases: An Introduction 2 A Priori and A Posteriori Models for Infectious Diseases The Theory of Happenings An Example: A Basic Model for HIV/AIDS A Mathematical Model for HIV/AIDS A Statistical Model for the Initial HIV/AIDS Outbreak The Mechanism that Generates the Data The Basic Epidemiological Parameters Discussion The SIR Model Introduction The SIR Model The Basic Model Dynamics The Basic Model in R Vaccination in the Basic Model The Basic SIR Model with Vaccination in R The Critical Vaccination Coverage The SIR Model in Endemic Equilibrium Compartments in the Time Homogeneous SIR Model xi

12 xii Contents The SIR Model with Constant Force of Infection at Endemic State in R The Critical Proportion of Vaccination in the Time Homogeneous SIR Model The Time Homogeneous SIR Model and Serological Data Models with Maternal Antibodies and Latent Periods Transmission Within Multiple Subpopulations An SIR Model with Interacting Subpopulations Transmission Over Age and Time Estimating the WAIFW Matrix Discussion Part III Data Sources 4 Data Sources for Modeling Infectious Diseases Serological Data Hepatitis A Hepatitis B Hepatitis C Mumps Parvovirus B Rubella Tuberculosis Varicella Multisera Data Hepatitis B Incidence Data Belgian Contact Survey Part IV Estimating the Force of Infection 5 Estimating the Force of Infection from Incidence and Prevalence Serological Data Age-Dependent Force of Infection Modeling Issues Incidence Data Parametric Approaches to Model the Prevalence and Force of Infection Modeling the Force of Infection: A Historical Perspective Polynomial Models Nonlinear Models Discussion Fractional Polynomial Models Motivating Example Model Selection

13 Contents xiii Constrained Fractional Polynomials Selection of Powers and Back to Nonlinear Models Application to the Data Influence of the Link Function From Fractional Polynomials Back to Nonlinear Models Discussion Nonparametric Approaches to Model the Prevalence and Force of Infection Nonparametric Approaches The First Nonparametric Approaches Local Estimation by Polynomials Application to UK Mumps Data Concluding Remarks Semiparametric Approaches to Model the Prevalence and Force of Infection Semiparametric Approaches Penalized Splines Penalized Likelihood Framework Generalized Linear Mixed Model Framework Covariate Effects Adaptive Spline Smoothing Synthesis Non-, Semi- and Parametric Methods to Estimate the Prevalence and Force of Infection: A Case Study The Constraint of Monotonicity Introduction Piecewise Constant Forces of Infection Isotonic Regression The Pool Adjacent Violator Algorithm Keiding (1991) Smooth Then Constrain P-spline Regression with Shape Constraints Concluding Remarks Hierarchical Bayesian Models for the Force of Infection Introduction Exploratory Data Analysis Hierarchical Bayesian Models for the Force of Infection Nonlinear Hierarchical Model Fitting Farrington s Model in Winbugs Hierarchical Log-Logistic Model Fitting the Log-Logistic Model in Winbugs Model Selection Application to the Data

14 xiv Contents 10.4 Hierarchical Nonparametric Model Hierarchical Beta/Binomial Model Fitting the Model in Winbugs Application to the Data Discussion Modeling the Prevalence and the Force of Infection Directly from Antibody Levels Serological and Current Status Data The Threshold Approach A Direct Approach Application to VZV Data Fitting a Mixture Fitting the Mean Antibody Levels as a Function of Age Combining the Mixture with the Least Squares Regression Fit Modeling the Force of Infection Directly from Antibody Titers Using Hierarchical Mixture Models Model Formulation Age-Dependent Mixture Probability Application to the Data Determining the Current Infection Status Implementation in Winbugs Concluding Remarks Modeling Multivariate Serological Data Introduction Marginal and Conditional Models The Bivariate Dale Model Applied to Airborne Infections The Marginal, Conditional and Joint Force of Infection Individual Heterogeneity Concluding Remarks Estimating Age-Time Dependent Prevalence and Force of Infection from Serial Prevalence Data Introduction Proportional Hazards Model Semiparametric Model Non-parametric Model Example I: Tuberculosis Non-proportional Hazards Model The Tuberculosis Data Revisited Example II: Hepatitis A Monotonicity Concluding Remarks

15 Contents xv Part V Estimating Mixing Patterns and the Basic Reproduction Number 14 Who Acquires Infection from Whom? The Traditional Approach Who Acquires Infection from Whom? Estimation from Serological Data The Discretized Mass Action Principle Imposing Mixing Patterns: The Traditional Approach of Anderson and May (1991) Exploiting an Underlying Continuous Mixing Surface Topics in Estimating WAIFW Matrices Model Selection Lifelong Immunity Concluding Remarks Informing WAIFW with Data on Social Contacts Estimation from Serological Data and Data on Social Contacts The Social Contact Hypothesis Refinements to the Social Contact Data Approach The Bootstrap and Multi-model Inference The Nonparametric Bootstrap Multi-model Inference Concluding Remarks Part VI Integration of Statistical Models into the Mathematical Model Framework 16 Integrating Estimated Parameters in a Basic SIR Model Introduction Application Model Comparison Uncertainty Vaccination Discussion A A Note on the R Environment A.1 Data Structures in R A.2 Reading and Writing Data A.3 Graphical Procedures A.4 Statistical Models in R A.5 Packages A.6 References B Statistical Inference B.1 Maximum Likelihood Estimation and Likelihood Inference

16 xvi Contents B.2 Generalized Linear Models B.3 Profile Likelihood and Other Likelihoods B.3.1 Profile Likelihood Estimation B.3.2 Profile Likelihood Confidence Intervals B.4 The Bootstrap B.4.1 Bootstrap Confidence Intervals B.4.2 Nonparametric Bootstrap B.4.3 Parametric Bootstrap B.5 Bayesian Methodology References Index