Epidemic models and inference for the transmission of hospital ... - Core

Epidemic models and inference for the transmission of hospital pathogens

Marie Forrester Bachelor of Commerce, Bachelor of Arts University of Queensland

A thesis submitted for the degree of

Doctor of Philosophy July 2006

Principal Supervisor: Prof Tony Pettitt Queensland University of Technology School of Mathematical Sciences Faculty of Science Brisbane, Queensland, 4001, Australia

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Thesis examination page

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Abstract

The primary objective of this dissertation is to utilise, adapt and extend current stochastic models and statistical inference techniques to describe the transmission of nosocomial pathogens, i.e. hospital-acquired pathogens, and multiply-resistant organisms within the hospital setting. The emergence of higher levels of antibiotic resistance is threatening the long term viability of current treatment options and placing greater emphasis on the use of infection control procedures. The relative importance and value of various infection control practices is often debated and there is a lack of quantitative evidence concerning their effectiveness. The methods developed in this dissertation are applied to data of methicillin-resistant Staphylococcus aureus occurrence in intensive care units to quantify the effectiveness of infection control procedures. Analysis of infectious disease or carriage data is complicated by dependencies within the data and partial observation of the transmission process. Dependencies within the data are inherent because the risk of colonisation depends on the number of other colonised individuals. The colonisation times, chain and duration are often not visible to the human eye making only partial observation of the transmission process possible. Within a hospital setting, routine surveillance monitoring permits knowledge of interval-censored colonisation times. However, consideration needs to be given to the possibility of false negative outcomes when relying on observations from routine surveillance monitoring. SI (Susceptible, Infected) models are commonly used to describe community epidemic processes and allow for any inherent dependencies. Statistical inference techniques, such as the expectation-maximisation (EM) algorithm and Markov chain Monte Carlo (MCMC) can be used to estimate the model parameters when only partial observation of the epidemic process is possible. These methods appear well suited for the analysis of hospital infectious disease data but need to be adapted for short patient stays through migration. This thesis focuses on the use of Bayesian statistics to explore the posterior distributions of the unknown parameters. MCMC techniques are introduced to overcome analytical intractability caused by partial observation of the epidemic process. Statistical issues such as model adequacy and MCMC convergence assessment are discussed throughout the thesis. The new methodology allows the quantification of the relative importance of differ-

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ent transmission routes and the benefits of hospital practices, in terms of changed transmission rates. Evidence-based decisions can therefore be made on the impact of infection control procedures which is otherwise difficult on the basis of clinical studies alone. The methods are applied to data describing the occurrence of methicillin-resistant Staphylococcus aureus within intensive care units in hospitals in Brisbane and London

Keywords

Bayesian inference, Markov chain Monte Carlo, reversible jump, transdimensional, stochastic epidemic model, susceptible-infected model, SI model, generalised linear model, hospital epidemiology, infectious diseases, infection control, nosocomial infection, hospital-acquired infection, multiply-resistant organisms, antibioticresistant bacteria, Staphylococcus aureus, methicillin-resistant Staphylococcus aureus, sensitivity, detectability

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Contents

Contents

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List of Tables

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List of Figures

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List of Abbreviations

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List of Notation 1 Overview 1.1 Introduction . . . . . 1.2 Scope . . . . . . . . . 1.3 Outline of thesis . . . 1.4 Contribution of thesis

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2 Review of literature 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Epidemic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Deterministic models . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Stochastic models . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Comparison of deterministic and stochastic models . . . . . . . 2.3 Statistical inference techniques . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Maximum likelihood (ML-) estimation . . . . . . . . . . . . . . . 2.3.2 Martingale techniques . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Bayesian inference using Markov chain Monte Carlo techniques (MCMC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Statistical inference techniques applied to stochastic epidemic models 2.4.1 Chain binomial and other independent household models . . . 2.4.2 Generalised linear models . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Stochastic epidemic models . . . . . . . . . . . . . . . . . . . . . 2.4.4 Non-transmission models . . . . . . . . . . . . . . . . . . . . . . 2.5 Case study: methicillin-resistant Staphylococcus aureus (MRSA) . . . . 2.5.1 Staphylococcus aureus and MRSA . . . . . . . . . . . . . . . . . . 2.5.2 Transmission dynamics . . . . . . . . . . . . . . . . . . . . . . .

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2.5.3 Infection control procedures 2.5.4 Epidemic models . . . . . . . 2.5.5 Statistical inference . . . . . . 2.6 Discussion . . . . . . . . . . . . . . .

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3 Case studies of methicillin resistant Staphylococcus aureus 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Princess Alexandra Hospital (PAH) intensive care unit (ICU) . 3.2.1 Infection control procedures . . . . . . . . . . . . . . . 3.2.2 Population and occurrence of MRSA . . . . . . . . . . . 3.3 ICUs within two London (LON) hospitals . . . . . . . . . . . . 3.3.1 Infection control procedures . . . . . . . . . . . . . . . 3.3.2 Analysis of the patient population and extent of MRSA 3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Mechanistic description of the transmission process 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . 4.2 Model definition and assumptions . . . . . . . . 4.3 Deterministic epidemic model . . . . . . . . . . 4.4 Stochastic epidemic model . . . . . . . . . . . . . 4.5 Statistical inference . . . . . . . . . . . . . . . . . 4.5.1 Data and notation . . . . . . . . . . . . . . 4.6 Discussion . . . . . . . . . . . . . . . . . . . . . .

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5 Generalised linear model (GLM) and inference 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 5.2 Model and methodology . . . . . . . . . . . . . . . . 5.3 Statistical inference . . . . . . . . . . . . . . . . . . . 5.3.1 Data and notation . . . . . . . . . . . . . . . . 5.3.2 Maximum likelihood estimation . . . . . . . 5.3.3 Bayesian inference using MCMC techniques 5.3.4 Model adequacy . . . . . . . . . . . . . . . . . 5.3.5 Model comparison . . . . . . . . . . . . . . . 5.4 Case study: PAH ICU data . . . . . . . . . . . . . . . 5.4.1 Results . . . . . . . . . . . . . . . . . . . . . . 5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . .

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6 Stochastic epidemic model (SEM) and inference 6.1 Introduction . . . . . . . . . . . . . . . . . . 6.2 Model and methodology . . . . . . . . . . . 6.3 Data and notation . . . . . . . . . . . . . . . 6.4 Joint likelihood . . . . . . . . . . . . . . . . 6.5 Maximum likelihood estimation . . . . . . . 6.6 Bayesian inference . . . . . . . . . . . . . . 6.6.1 MCMC algorithm . . . . . . . . . . . 6.7 Simulated data based on the PAH data . . . 6.7.1 Maximum likelihood estimation . . 6.7.2 Bayesian inference . . . . . . . . . . 6.8 Case study: PAH ICU . . . . . . . . . . . . . 6.8.1 Bayesian inference . . . . . . . . . . 6.9 Discussion . . . . . . . . . . . . . . . . . . .

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7 Stochastic epidemic model extended for imperfect sensitivity (ESEM) and inference 101 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 7.2 Model and methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 7.3 Data and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.4 Joint likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.5 ML- estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 7.6 Bayesian inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 7.6.1 MCMC algorithm and convergence assessment . . . . . . . . . . 105 7.6.2 Model adequacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.7 Simulated data based on the PAH data . . . . . . . . . . . . . . . . . . . 111 7.7.1 Maximum likelihood estimation . . . . . . . . . . . . . . . . . . 112 7.7.2 Bayesian inference . . . . . . . . . . . . . . . . . . . . . . . . . . 114 7.8 Case study: PAH ICU data . . . . . . . . . . . . . . . . . . . . . . . . . . 116 7.8.1 Model assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 7.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 8 Stochastic epidemic model for an intervention study and inference 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Model and methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Data and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Bayesian inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 MCMC algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Model adequacy . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Simulated data based on the London data and the hazard-phaseeffects (HPE) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Maximum likelihood estimation . . . . . . . . . . . . . . . . . 8.5.2 Bayesian inference . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Case study: hazard-phase-effects model, London ICU . . . . . . . . . 8.6.1 Model assessment . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Simulated data based on the London data and the detection-phaseeffects (DPE) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Conclusions and future work 9.1 Summary of methodology 9.2 Comparison of results . . . 9.3 Future work . . . . . . . . 9.4 Concluding remarks . . . .

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A General epidemic model for exponentially distributed infectious periods 167 B Data of the Princess Alexandra Hospital ICU B.1 Sources of data . . . . . . . . . . . . . . . . . B.1.1 Patient data . . . . . . . . . . . . . . B.1.2 Positive MRSA swab data . . . . . . . B.1.3 MRSA notification data . . . . . . . . B.2 Data formatting . . . . . . . . . . . . . . . . B.2.1 Discrepancies between data sources C Antibiotic usage at the London ICU

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C.1 C.2 C.3 C.4 C.5

Antibiotic usage . . . . . . . . . . . . . . . . . . . Anti-Staphylococcal properties . . . . . . . . . . Gram-negative properties . . . . . . . . . . . . . Amenogycide, cephalosporin and quinolone use Quantity of antibiotics used . . . . . . . . . . . .

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D Source model

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E Stochastic simulation of the epidemic model

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F Derivation of the data required for the generalised linear model

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G Program code for inference of the generalised linear model 201 G.1 ML-estimation using standard statistical software . . . . . . . . . . . . 201 G.2 Bayesian inference using WinBUGS . . . . . . . . . . . . . . . . . . . . . 203 H Maximum likelihood estimation for the stochastic epidemic model with(out) imperfect sensitivity 205 H.1 Unconstrained parameter values . . . . . . . . . . . . . . . . . . . . . . 206 H.1.1 Score function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 H.1.2 Hessian matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 H.2 Constrained parameter values . . . . . . . . . . . . . . . . . . . . . . . . 210 H.2.1 Score function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 H.2.2 Hessian matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 I

Proposal distribution for patient colonisation times in the presence of imperfect sensitivity 217

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Convergence diagnostics J.1 Stochastic epidemic model . . . . . . . . . . . . . . . . . J.1.1 Simulated data . . . . . . . . . . . . . . . . . . . J.1.2 PAH ICU data . . . . . . . . . . . . . . . . . . . . J.2 Stochastic epidemic model with imperfect sensitivity . J.3 Stochastic epidemic model with imperfect sensitivity phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . J.3.1 Hazard-phase-effects model, London ICU . . . . J.3.2 Detection-phase-effects model, simulated data .

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K Bayesian latent residual method of model assessment

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Bibliography

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List of Tables

2. Review of literature 2.1 Gelman and Rubin convergence diagnostic . . . . . . . . . . . . . . . . 18 2.2 Brooks and Guidici convergence diagnostic . . . . . . . . . . . . . . . . 20 3. Case studies of MRSA 3.1 Source of patient admissions to the PAH ICU . . . . . . . . . . . . . . . 3.2 Positive swabs per day in the PAH ICU . . . . . . . . . . . . . . . . . . . 3.3 Details of admissions to the PAH ICU . . . . . . . . . . . . . . . . . . . . 3.4 Time from admission to first swab for detected patients in the PAH ICU 3.5 MRSA patients in the PAH ICU . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Key characteristics of the London ICUs . . . . . . . . . . . . . . . . . . . 3.7 Number of admissions and transfers to the London ICUs . . . . . . . . 3.8 Key characteristics of the 18-bed London ICU . . . . . . . . . . . . . . . 3.9 Screening swab patterns and frequency for admissions to the London ICUs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Properties of antibiotics administered to the London ICU patients . . . 5. Generalised linear model (GLM) 5.1 MLEs of the GLM parameters for MRSA transmission within the PAH ICU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Posterior summaries of the GLM parameters for MRSA transmission within the PAH ICU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Deviance information criterion for the GLM fitted to the PAH data using MCMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Effect of bed occupancy on GLM transmission parameters within the PAH ICU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Posterior summaries for parameters of a GLM, that allows annual variation in transmission, fitted to the PAH ICU data . . . . . . . . . . . .

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6. Stochastic epidemic model (SEM) 6.1 Key characteristics of data simulated according to the stochastic epidemic model (SEM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6.2 SEM MLEs for MRSA transmission based on simulated data . . . . . . . 94 6.3 SEM MCMC sampler efficiency . . . . . . . . . . . . . . . . . . . . . . . 95

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6.4 SEM posterior summaries for MRSA transmission based on simulated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.5 SEM posterior summaries for MRSA transmission within the PAH ICU 96 7. Stochastic epidemic model extended for imperfect sensitivity (ESEM) 7.1 ESEM MLEs for MRSA transmission based on simulated data . . . . . 7.2 Summary statistics of the posterior means of ESEM parameters for simulated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Posterior summaries for the ESEM parameters within the PAH ICU . 7.4 Effect of prior swab sensitivity information on the estimated benefit of isolation, β1 − β2 , in the PAH ICU (based on the ESEM) . . . . . . . . 8. Stochastic epidemic model for an intervention study 8.1 Parameter values used to simulate data according to the hazardphase-effects (HPE) model . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 MLEs for the HPE model parameters based on simulated data . . . . 8.3 Posterior summaries for the HPE model parameters based on simulated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Posterior summaries for the HPE model parameters for the London ICU data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Parameter values to simulate data according to the detection-phaseeffects (DPE) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 MLEs for the DPE model parameters based on simulated data . . . . 8.7 Posterior summaries for the DPE model parameters based on simulated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9. Conclusions and future work 9.1 Comparison of MRSA parameter estimates . . . . . . . . . . . . . . . . 161 F.1

Calculations to derive data for the generalised linear model . . . . . . . 199

G.1 GLM data variables in S-Plus program code . . . . . . . . . . . . . . . . 202 J.13 Starting values for Markov chains for the HPE model parameters within the London ICU . . . . . . . . . . . . . . . . . . . . . . . . . . . J.14 Geweke convergence diagnostics for the HPE parameters within the London ICU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J.15 Raftery and Lewis convergence diagnostic for the HPE model parameters within the London ICU . . . . . . . . . . . . . . . . . . . . . . . . J.16 Heidelberger and Welch convergence diagnostic for the HPE model parameters within the London ICU . . . . . . . . . . . . . . . . . . . . J.17 Geweke convergence diagnostics for the DPE parameters within the London ICU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J.18 Raftery and Lewis convergence diagnostic for the DPE model parameters within the London ICU . . . . . . . . . . . . . . . . . . . . . . . . J.19 Heidelberger and Welch convergence diagnostic for the DPE model parameters within the London ICU . . . . . . . . . . . . . . . . . . . .

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K.1 Notation to describe the Bayesian latent residual method . . . . . . . . 236

List of Figures

2. Review of literature 2.1 Dynamics of infection and disease states . . . . . . . . . . . . . . . . . . 3. Case studies of MRSA 3.1 PAH ICU layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 PAH ICU patient lengths of stay . . . . . . . . . . . . . . . . . . . . . . . 3.3 Bed occupancy, MRSA prevalence and MRSA incidence at the PAH ICU 3.4 Layout of the London ICUs . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Number of patients and positive swabs taken each day in the 18-bed London ICU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4. Mechanistic description of the transmission process 4.1 Transmission dynamics of a nosocomial antibiotic resistant bacterium 58 4.2 Stochastic compartmental model of the infection dynamics of nosocomial antibiotic resistant bacteria . . . . . . . . . . . . . . . . . . . . . 59 4.3 Example clinical time-line for a hospital ward patient colonised with antibiotic resistant bacteria . . . . . . . . . . . . . . . . . . . . . . . . . 59 5. Generalised linear model (GLM) 5.1 Relationships between data of the generalised linear model . . . . . . 5.2 Directed acyclic graph of the generalised linear model . . . . . . . . . 5.3 Marginal posterior densities of the generalised linear model parameters obtained using MCMC within a Bayesian framework. . . . . . . . 5.4 Correlations for posterior densities of the GLM parameters for the PAH ICU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Deviance residuals for the posterior means of the GLM parameters for the PAH ICU data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Stochastic epidemic model (SEM) 6.1 Numbers of detected patients in data simulated according to the SEM 6.2 Prevalence and incidence for data simulated using the SEM . . . . . . 6.3 Posterior distributions for the SEM parameters fitted to the PAH ICU 6.4 Weighted latent residuals for the SEM . . . . . . . . . . . . . . . . . . 7. Stochastic epidemic model extended for imperfect sensitivity (ESEM)

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91 93 97 98

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7.1 Allowable Markov chain move types for updating a patient’s final colonisation status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Scatter plots showing correlation between the ESEM parameters fitted to simulated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Augmented data for the PAH ICU and the ESEM . . . . . . . . . . . . . 7.4 Posterior distributions for the ESEM parameters fitted to the PAH ICU . 7.5 Correlations for posterior densities of ESEM parameters fitted to PAH ICU data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Posterior distributions for the relative risks of transmission of the ESEM in the PAH ICU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Markov chain realisations for the ESEM parameters fitted to the PAH ICU data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Posterior predictive discrepancies for the ESEM fitted to PAH ICU data 7.9 Cross-validation residuals (method 1) for the ESEM fitted to PAH ICU data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10 Cross-validation residuals (method 2) for the ESEM fitted to PAH ICU data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11 Latent residuals for the ESEM fitted to PAH ICU data . . . . . . . . . . . 7.12 Effect of prior swab sensitivity information on the estimated benefit of isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Stochastic epidemic model for an intervention study 8.1 Posterior distributions for the HPE model parameters fitted to simulated data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Posterior distributions for the HPE model parameters fitted to the observed London ICU data . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Markov chain realisations for the HPE model parameters fitted to observed data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Correlations for posterior densities of the HPE model parameters fitted to observed data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Posterior predictive discrepancies for the HPE model fitted to the London ICU data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Cross-validation residuals (method 1) for the HPE model fitted to the London ICU data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Cross-validation residuals (method 2) for the HPE model fitted to the London ICU data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Latent residuals for the HPE model fitted to the London ICU data . . 8.9 Posterior distributions for the DPE model parameters fitted to simulated data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.10 Markov chain realisations for the DPE model parameters fitted to simulated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

106 115 117 118 119 120 121 123 124 125 126 128

. 141 . 143 . 144 . 145 . 147 . 148 . 149 . 150 . 153 . 154

C.1 18-bed London ICU patients (a) on antibiotics and (b) not on antibiotics.174 C.2 Positive 18-bed London ICU admissions (split according to antibiotic usage) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 C.3 18-bed London ICU patients on (a) anti-MRSA, (b) antiStaphylococcal, (c) other and (d) no antibiotics. . . . . . . . . . . . . . . 177 C.4 Positive 18-bed London ICU admissions (split according to antiStaphylococcal antibiotic usage) . . . . . . . . . . . . . . . . . . . . . . 178 C.5 18-bed London ICU patients on (a) gram-negative, (b) gram-positive and (c) no antibiotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

xvii

C.6 Positive 18-bed London ICU admissions (split according to gramnegative antibiotic usage) . . . . . . . . . . . . . . . . . . . . . . . . . C.7 18-bed London ICU patients taking (a) amenogycides, (b) cephalosporins, (c) quinolones and (d) other antibiotics. . . . . . . . . . . . . C.8 Positive 18-bed London ICU admissions (split according to amenogocycide/quinlone/cephalosporin usage) . . . . . . . . . . . . . . . . . . C.9 18-bed London ICU patients administered (a) one, (b) two, (c) three or more and (d) no antibiotics . . . . . . . . . . . . . . . . . . . . . . . .

. 182 . 185 . 186 . 189

I.1

Proposal distribution for patient colonisation times . . . . . . . . . . . 217

J.1

Castelloe and Zimmerman convergence diagnostics for Markov chains of ESEM parameters fitted to PAH ICU data . . . . . . . . . . . . 228 Castelloe and Zimmerman convergence diagnostics for Markov chains of the HPE model parameters fitted to the London data . . . . . 231

J.2

xviii

A list of abbreviations

AIC BIC DAG DIC DPE EM ESEM GLM HCW HPE IACF ICU LON LOS MCMC MLMLE MRSA MSSA NA PAH RJMCMC SEM VRE CPU

Akaike information criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bayesian information criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Directed acyclic graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deviance information criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Detection-phase-effects model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expectation-Maximisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stochastic epidemic model extended for imperfect sensitivity . . . . Generalised linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Health-care worker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hazard-phase-effects model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integrated autocorrelation time function . . . . . . . . . . . . . . . . . . . . . . . . . Intensive care unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . London . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Length of stay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Markov chain Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maximum likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maximum likelihood estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methicillin-resistant Staphylococcus aureus . . . . . . . . . . . . . . . . . . . . . . Methicillin-susceptible Staphylococcus aureus . . . . . . . . . . . . . . . . . . . Not a number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Princess Alexandra Hospital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reversible jump MCMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stochastic epidemic model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vancomycin-resistant enterococci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computer processing unit time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 73 24 134 12 102 68 33 134 21 3 47 33 13 11 11 31 38 226 41 16 85 28 95

xx

A list of symbols and notation

ai a aθ bθ BU BL ci C CIx Cj C(t) di dij D() D() ˆ avg () D D Dd Dobs Dm ej e f () F () g G h(t) H(θ) jm (s⋆ |s) Kx , Kx′ l() L() N (t) ˆC N j

Admission time of patient i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Admission times for all patients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Parameter 1 of the prior distribution for θ . . . . . . . . . . . . . . . . . . . . . . . . 72 Parameter 2 of the prior distribution for θ . . . . . . . . . . . . . . . . . . . . . . . . 72 Upper bound of the uniform distribution . . . . . . . . . . . . . . . . . . . . . . . . 90 Lower bound of the uniform distribution . . . . . . . . . . . . . . . . . . . . . . . . 89 Colonisation time of patient i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Non-isolated (and colonised) patient state . . . . . . . . . . . . . . . . . . . . . . . 57 x% credible interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Number of non-isolated patients in the ward at swabtime tj . . . . . 69 Number of susceptible patients in the ward at time t . . . . . . . . . . . . . 58 Bayesian latent residual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Time from the j th swab, which is positive, taken from patient i and notification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Discrepancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Deviance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Posterior deviance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Complete data, includes observed and augmented data . . . . . . . . . 12 Data emanating from deterministic processes . . . . . . . . . . . . . . . . . . . 64 Observed data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Latent or missing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 j th event time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Event times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Density function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Markov chain iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Total number of MCMC iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Hazard function at time t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Hessian matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 State proposal probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Move probabilities defined for x=1, 4 and 5 . . . . . . . . . . . . . . . . . . . . . . 106 Loglikelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Number of patients in the ward at time t . . . . . . . . . . . . . . . . . . . . . . . . . 58 Expected number of colonisations in the j th interval . . . . . . . . . . . . . 76

xxii

NjC NjQ NA Ne NF N (D) NT P Nsp (s) P (D|θ) P (θ|D) q() qi q Q Q(t) Qj ri r R rDj sd sp ss S Sj S(θ) S(t) tiv−1 ti ts t− j i : ti ∈ ph2 vi v X() yj yj yobs yrep y 1x α() β0 β1 β2 η ηph(t) θ θ⋆ θˆ

Number of colonisation events in the j th interval . . . . . . . . . . . . . . . . Number of isolation events in the j th interval . . . . . . . . . . . . . . . . . . . . Number of admissions during the observation period . . . . . . . . . . . Number of events during the observation period . . . . . . . . . . . . . . . . Number of false negative swabs, which is a function of the data D Number of detected patients, or true positive swabs . . . . . . . . . . . . . Number of patients colonised on admission . . . . . . . . . . . . . . . . . . . . . Likelihood of the data D given the parameters θ . . . . . . . . . . . . . . . . . Posterior of the parameters θ given the data D . . . . . . . . . . . . . . . . . . . Proposal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isolation time of patient i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isolation times for all patients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isolated patient state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Number of isolated patients in the ward at time t . . . . . . . . . . . . . . . . Number of isolated patients in the ward at swabtime tj . . . . . . . . . . Discharge time of patient i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discharge times for all patients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Removed patient state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deviance residual for the j th interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . Final colonisation status of colonised in ward . . . . . . . . . . . . . . . . . . . Final colonisation status of colonised on admission . . . . . . . . . . . . . Final colonisation status of susceptible . . . . . . . . . . . . . . . . . . . . . . . . . . Susceptible patient state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Number of susceptible patients in the ward at time tj . . . . . . . . . . . . Score function of parameter θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Number of susceptible patients at time t . . . . . . . . . . . . . . . . . . . . . . . . . Last negative swab time for patient i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Swabtimes of patient i, comprising {ti1 , ti2 , . . .} . . . . . . . . . . . . . . . . . . . Swabtimes of all patients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time infinitesimally preceding tj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Logical condition that time t is in phase 2 . . . . . . . . . . . . . . . . . . . . . . . . Positive swab time of patient i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Positive swab times for all patients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Survivor function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Number of patients escaping colonisation in the j th interval . . . . . Observed data used to assess a model(cross-validation) . . . . . . . . . Observed data used to assess a model (posterior prediction) . . . . . Replicated data used to assess a model . . . . . . . . . . . . . . . . . . . . . . . . . . Data for model fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Indicator function taking value 1 if x is true and 0 otherwise . . . . . Acceptance probability for the MCMC algorithm . . . . . . . . . . . . . . . . Background transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rate of transmission from non-isolated (colonised) patients . . . . . Rate of transmission from isolated (colonised) patients . . . . . . . . . . Multiplicative effect on the baseline hazard . . . . . . . . . . . . . . . . . . . . . . Indicator function taking the value η if time t occurs in the move phase and one otherwise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unknown model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proposed value of θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Point estimate of θ, i.e. posterior mean or MLE . . . . . . . . . . . . . . . . . .

69 69 103 85 103 89 103 13 13 14 63 63 57 58 69 63 63 57 75 63 63 63 57 69 11 58 86 59 64 62 205 63 63 62 71 23 22 22 24 72 14 61 61 61 134 134 11 14 24

xxiii

π() λj πj ρ σc ϕ

Target distribution or full conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constant hazard rate in the interval [tj−1 , tj ) . . . . . . . . . . . . . . . . . . . . . Conditional probability of colonisation in an interval [tj−1 , tj ) . . . Imperfect sensitivity; probability of a false negative swab . . . . . . . . Standard distribution of the proposal distribution for the colonisation times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Probability that a patient is colonised on admission to the ward .

13 62 68 59 89 58

xxiv

Statement of Original Authorship

The work contained in this thesis has not been previously submitted for a degree or diploma at any other higher education institution. To the best of my knowledge and belief, this thesis contains no material previously published or written by any other person except where due reference is made.

Signed: Date:

xxvi

Acknowledgements

Thank-you to the following people and organisations for support during this degree: Supervisor, Tony Pettitt – expert guidance, time and encouragement. I am especially grateful for the opportunities provided to collaborate with researchers abroad. Family and friends – camping, capoeira and making life away from this thesis fun and interesting. Thanks in particular to my parents for milk provisions and Sonia for assistance to locate glasses, keys, etc. Collaborators and colleagues – to those within the School of Mathematical Sciences, for making staff room visits a worthwhile distraction, and to those who showed an interest in this work and offered suggestions or advice. In particular, Gavin Gibson who assisted with later models of this thesis and Ben Cooper for data and enthusiasm to assist with and discuss this thesis, facilitate my time in London, and to provide information on prospective employers. School of Mathematical Sciences administration and IT support staff – rapid assistance to administrative and technical problems encountered. Princess Alexandra Hospital – financial support, data and information regarding the intensive care unit policies and procedures. Australian Research Council – financial support. Lindt & Sprungli ¨ – rewarding the boring and mundane parts of this thesis.

xxviii

CHAPTER 1

Overview

1.1

Introduction

The primary objective of this study is to develop statistical inference techniques for application to hospital data. The methodology provides a means of quantifying transmission and the estimated benefit, in terms of changed transmission rates, of infection control interventions. Despite a large body of clinical research, there is a growing dissent between practitioners concerning the effectiveness of widely implemented infection control procedures. Clinical studies are generally limited to quasi-experimental designs making causal relationships difficult to establish. Dependencies within the epidemic process make standard statistical tests invalid, making it difficult to rule out stochastic fluctuation as a cause for any observed changes. Systematic reviews (Cooper et al., 2003; Loveday et al., 2006) have found major methodological weaknesses in published research evaluating the effectiveness of infection control procedures designed to curb the spread of nosocomial pathogens and in particular, methicillin-resistant Staphylococcus aureus (MRSA). Epidemic models and statistical inference are well-equipped to tackle any difficulties arising from dependencies within the epidemic process. Furthermore, such methods are able to quantify the estimated effects of infection control interventions, in terms of changed transmission rates, which is otherwise difficult on the basis of clinical studies. Patients colonised with nosocomial pathogens such as MRSA may be asymptomatic so that the transmission process can only be partially observed by routine swabs testing for colonisation. Markov chain Monte Carlo (MCMC) methods are currently popular techniques (e.g. Gibson and Renshaw, 1998, 2001; O’Neill and Roberts, 1999; Streftaris and Gibson, 2004b) used to analyse data of partially observed infectious diseases within the community. The methods appear well suited to routinely collected hospital data but must be adapted to allow for patient turnover. Community

2

Chapter 1. Overview

populations have relatively small turnover and are typically assumed to be closed. The predominantly short patient stays of hospital ward populations requires the removal of this assumption. This thesis investigates the use of epidemic models and statistical inference techniques to evaluate the effectiveness of infection control procedures designed to curb nosocomial transmission of pathogens. Emphasis is placed on the use of MCMC methods within a Bayesian framework. The methods are adapted for application to routine surveillance data and extended to allow for imperfect sensitivity of the surveillance process. The methods are applied to data describing the occurrence of methicillin-resistant Staphylococcus aureus within a Brisbane and London intensive care unit.

1.2 Scope The scope of this thesis is to estimate parameters for stochastic epidemic models describing transmission of a single nosocomial pathogen that can be carried asymptomatically.

Epidemic models and statistical inference Stochastic epidemic models to describe the transmission of infectious diseases within a small number of individuals are described and developed. Theoretical properties of stochastic epidemic models are not investigated. The models are introduced to infer model parameters describing transmission from data assumed to have been generated from the defined model. Techniques to infer parameters of the stochastic epidemic model given only partial observation of the transmission process are developed.

MCMC within a Bayesian framework MCMC techniques, including reversible jump MCMC, are employed to estimate parameters of the stochastic epidemic model. Convergence diagnostics and methods for assessing goodness-of-fit for models fitted to date within a Bayesian framework are applied. Methods to improve the speed of implementation and convergence are not a primary focus of this thesis.

Quantification of hospital practices designed to curb nosocomial transmission Based on the given epidemic model, assumptions and data describing the occurrence of nosocomial pathogens, the rates at which the pathogen is transmitted are estimated. The effect of control practices are quantified, for example, by including separate transmission parameters for pre- and post-intervention transmission. The results of this thesis are not generalisable to different populations, settings or time. Other than reporting the effects of infection control measures, recommendations for infection control practices are not within the scope of this thesis.

1.3 Outline of thesis

1.3

3

Outline of thesis

The remaining chapters of this thesis are organised as follows: Chapter 2 presents an investigation of epidemic models and associated statistical inference techniques that have been described in the literature. The use of epidemic models and statistical inference techniques to evaluate the effectiveness of hospital practices designed to control MRSA and other antibiotic resistant bacteria is discussed, complimented by a brief introduction to the epidemiology of MRSA and the main results and limitations of clinical research designed to investigate the effectiveness of infection control procedures. Details of admissions, the extent of MRSA and infection control procedures employed by the Princess Alexandra Hospital (PAH) and three London (LON) intensive care units (ICUs) are given in Chapter 3. A stochastic epidemic model to describe the transmission of nosocomial pathogens that can be carried asymptomatically, e.g. MRSA, is described in Chapter 4. The model describes the flow of patients through susceptible–colonised–isolated– removed states according to admission, colonisation, isolation and discharge events. A critical examination of factors and assumptions which may affect parameter values is performed. The objectives and data requirements of statistical inference for the stochastic epidemic model are introduced. Problems posed by dependence and partial observation of the transmission process are discussed. In Chapter 5, the stochastic epidemic model is approximated by a generalised linear model and data is aggregated according to discrete intervals of time. By considering the data in discrete form, inference techniques are not affected by partial observation and so standard inference techniques, such as maximum likelihood estimation, can be used. Bayesian inference using Markov chain Monte Carlo techniques is introduced for comparative and precision purposes. The methods are applied to data describing MRSA occurrence within the PAH ICU. Chapter 6 describes a Bayesian framework using MCMC techniques to infer the parameters of the stochastic epidemic model. The use of MCMC avoids any problems associated with partial observation of the epidemic process. In this chapter, the swabs allowing for partial observation of asymptomatic pathogen transmission are assumed to have perfect sensitivity. Knowledge of a patient’s colonisation status on admission is also assumed. The methods are illustrated using the PAH data. In Chapter 7, techniques are introduced to infer parameters of a stochastic epidemic model allowing for imperfect sensitivity. Reversible jump MCMC is used to facilitate inference. A by-product of the methodology is that knowledge of a patient’s colonisation status is no longer required nor assumed. The methodology is used to infer the rates of cross-colonisation, spontaneous acquisition, importation probability and screening swab sensitivity within the PAH ICU. Chapter 8 is motivated by an intervention study which took place at an 18-bed ICU in London. The stochastic epidemic model and techniques for inference are extended to estimate pre- and post-intervention model parameters. An overview and discussion of the methodology and results are given in Chapter 9. Possible extensions to the research presented in this thesis are indicated.

4

Chapter 1. Overview

1.4 Contribution of thesis The main purpose of this thesis is to develop statistical methodology to enable estimation of parameters for stochastic epidemic models that describe transmission of single nosocomial pathogens that can be carried asymptomatically. The techniques are applied to hospital surveillance data. This thesis provides the following: Methodological contributions • adapts methods for quantitative infectious disease epidemiology to hospital epidemiology • provides a tool for estimating parameters that describe the transmission of asymptomatic pathogens in small populations with high turnover • develops reversible jump MCMC methodology to allow for imperfect sensitivity Applied contributions • provides insight into the underlying transmission process for the spread of a single nosocomial pathogen that can be carried asymptomatically • quantifies the relative importance of cross-colonisation verse spontaneous acquisition (as measured by background transmission) • develops a method for analysing routine surveillance data • quantifies the effect of infection control measures Publications • Forrester, M., Pettitt, A., and Gibson, G. (2006). Bayesian inference of hospital transmission and control given imperfect surveillance data. Biostatistics. Accepted for publication. • Forrester, M. and Pettitt, A. N. (2005). Use of stochastic epidemic modeling to quantify transmission rates of colonization with methicillin-resistant Staphylococcus aureus in an intensive care unit. Infect Control Hosp Epidemiol, 26(7): 598–606.

CHAPTER 2

Review of literature

2.1

Introduction

Analysis of infectious disease data provides insight into underlying epidemic processes, a means of assessing proposed interventions aimed at reducing transmission and a summary of the data (Becker, 1989). By summarising the data according to key parameters, the disease can be compared between different areas and times and with other diseases. The task of the analysis is essentially one of statistical inference: given an observation of a process, which is believed to be governed by a known epidemic model, what range of model parameters could plausibly explain the observations (Gibson and Renshaw, 1998). The behaviour of an epidemic model is characterised by the model parameters. Both epidemic modelling and the analysis of infectious disease data are complicated by dependencies within the epidemic process (Becker, 1989; Becker and Britton, 1999; Andersson and Britton, 2000; O’Neill, 2002). Dependencies arise because the risk of acquisition depends on the number of individuals who are infected. If for example, the occurrence of the disease did not require the presence of infected individuals, data could be analysed using survival analysis (Cox and Oakes, 1984; Collett, 1994) techniques. Such techniques require the specification of the hazard function to describe the risk of an individual becoming infected. The hazard function may be specific to each individual and/or contain a random parameter correlated between related individuals, which is known as a frailty model. Cox’s proportional hazard is a well known hazard function. Partial observation of the epidemic process further complicates the task of infectious disease data analysis (Becker and Britton, 1999; Andersson and Britton, 2000; O’Neill, 2002). For example, it is usually unknown who infects whom, the time of infection and the duration of infectiousness. Poor quality data and low and fluctuating reporting rates are often further obstacles to statistical inference (Becker and

6

Chapter 2. Review of literature

Britton, 1999). Becker and Britton (1999) highlight the importance of the data and a need to consider the type and quantity of data required to make objective decisions concerning infectious disease control procedures. The focus of this thesis lies with statistical inference techniques for disease data. Epidemic models form a basis for statistical inference and as such, some general epidemic models (Section 2.2) are described below, followed by a discussion of statistical inference techniques (Section 2.3).

2.2 Epidemic models Epidemic models describe the dynamics of infectiousness as opposed to the dynamics of disease (see Figure 2.1) (Halloran, 1998). The infectious process can be repre-

This figure is not available online. Please consult the hardcopy thesis available from the QUT Library

F IGURE 2.1. Dynamics of infection and disease states. sented by a succession of states or compartments. Such a representation is referred to as a compartmental model (Jacquez, 1996). For example, diseases characterised by Figure 2.1 can be represented by a SEIR compartmental model which consists of four compartments; susceptible (S), exposed (E), infected (I) and removed (R). Individuals are initially considered susceptible. Upon contact with an infectious individual, the individual is exposed. The individual remains exposed for a duration of time, the latent period, until becoming infectious. Upon termination of infectiousness, the individual is considered removed from the infectious cycle. Removal can correspond to immunity or death. There are many well-known variants to the SEIR model (Hethcote, 1994), for example, using the same notation, possible models are SI, SIS, SEI, SEIS, SIR, SIRS, SEIR, and SEIRS. The SEIR model and variants are approximations to the epidemic process; a complete epidemic model would also describe the infectious load in each member of the host population (Dietz and Schenzle, 1985). A detailed model describing the number of parasites in each member of the host population was attempted by Rvachev in 1967 (Dietz and Schenzle, 1985), however most models ignore the number of parasites per host or more generally, the infectious load. The compartmental model can be described by either deterministic or stochastic mathematical equations. In a deterministic model, the number of infections in a short time interval can be assumed proportional to the number of susceptible and

2.2 Epidemic models

7

infectious individuals and to the time interval. In a stochastic model, the probability of one new case in a short interval is equivalently proportional to the same quantity. SIR models involve only a minor variation to predator-prey models such as the deterministic Lotka-Volterra and the stochastic Volterra models applied in ecology (Renshaw, 1999). Mathematical models can be applied to both epidemic and endemic situations. The term “epidemic” refers to outbreaks which can usually be attributed to a point source. Mathematical models are used to describe biological and transmission mechanisms, threshold densities and to predict the course of epidemics (Bailey, 1975). In particular, they are used to predict the initial conditions which lead to an epidemic, the shape of the epidemic curve, the number of cases at the peak of the epidemic, the duration of the total epidemic and the total number of cases (Dietz and Schenzle, 1985). The term “endemic” is used to refer to randomly occurring low numbers of cases punctuated by the intermittent re-introduction of the disease (Thompson, 2004). In endemic situations, mathematical models are important tools for evaluating and comparing control interventions (Bailey, 1975; Dietz and Schenzle, 1985). For example, they can be used to answer questions concerning the control effort required to make the positive equilibrium unstable and zero incidence stable. For viruses, they can estimate the required vaccination coverage and for vector transmitted pathogens, the minimum reduction in vector density required.

2.2.1 Deterministic models Deterministic models are generally based on the mass-action principle which states that the course of an epidemic depends on the number of susceptible individuals and the contact rate between susceptible and infectious individuals. The mass action principle was formulated in discrete time by Hamer in 1906 and then in continuous time by Ross in 1908 (Andersson and Britton, 2000). In a deterministic model, the future state of the epidemic process can be determined from the initial numbers of susceptible and infectious individuals, together with the infection-, recovery-, birthand death-rates (Bailey, 1975). The first mathematical model is generally attributed to Kermack and McKendrick (Kermack and McKendrick, 1927). The set of equations were however first published by Ross and Hudson in 1917 (Diekmann et al., 1995; Dietz and Schenzle, 1985). The mathematical equations describe the dynamics of a SIR disease in continuous time. It is assumed that the disease being modelled occurs in a large, closed, homogenous and uniformly mixing population of equally susceptible individuals, contacts are made according to the law of mass action and infection triggers an autonomous process within the host (Diekmann and Heesterbeek, 2000). These assumptions are summarised by the equation Z ∞ dS ′ ¯ )S ′ (t − τ ) dτ, S (t) = = S(t) A(τ (2.1) dt 0 where S(t) is the spatial density of susceptibles, i.e. the number of susceptibles per unit area, at time t, −S ′ (t) is the incidence, i.e. the number of infection events in ¯ ) is the expected infectivity of an individual that a unit of time, at time t and A(τ became infected τ time units ago (Diekmann et al., 1995).

8


Equation 2.1, referred to as the Kermack and McKendrick model, is usually expressed for the specific case in which infectivity has an exponential distribution. If β is the rate at which an infectious individual has contact (sufficient for transmission) with susceptible members and γ is the removal rate, then the Kermack and McKendrick model with exponential infectivity is S ′ (t) = −βS(t)I(t),

(2.2a)

′

I (t) = βS(t)I(t) − γI(t).

(2.2b) R∞

′ ¯ ) = βe−γτ and I(t) = − 1 ¯ The derivation is obtained by defining A(τ β 0 A(τ )S (t − R 1 ∞ ¯ τ )dτ = − β 0 A(t − τ )S ′ (τ )dτ and differentiating. The proof is included in Appendix A.

The Kermack and McKendrick model is often referred to as the general epidemic model. To quote Bailey (1975), the term “general” is used in the sense that the model is not confined to infection only, i.e. the possibility of removal is also considered. Hethcote (1994) provides a brief outline of possible generalisations. Deterministic models approximate actual state changes by considering that integer valued variables are varying continuously. When numbers of susceptible and infective individuals are both large and mixing is reasonably homogeneous, the deterministic model is likely to be satisfactory as a first approximation (Bailey, 1975). The approximation will not be good when any of the integer-valued variables become small such that the population becomes close to extinction (Mollison et al., 1994; Daley and Gani, 1999; Andersson and Britton, 2000). Deterministic epidemic models are discussed in detail by Bailey (1975), Anderson and May (1991) and Daley and Gani (1999).

2.2.2 Stochastic models In a stochastic model, probability distributions of the numbers of susceptible and infectious individuals occurring at any instant replace the real-values of deterministic treatments. The majority of stochastic models are based on variants of two classical models, namely the general epidemic model and the chain binomial model (Lefèvre, 1988). Both classical models are special cases of a more general SIR model of a closed and homogeneous mixing population in which pairs of individuals make independent contact. The infectious periods are assumed to be independent and identically distributed. Assumptions concerning the distribution of the infectious periods differ between the two classical models. For example, within the general epidemic model the infectious period is assumed to be exponential and in the chain binomial model, the infectious period is assumed to be of a pre-determined fixed length. The general epidemic model is usually described in continuous time dynamics and the chain binomial model in discrete time dynamics. Stochastic epidemic models are discussed in detail by Bailey (1975), Lefèvre (1988), Daley and Gani (1999) and Andersson and Britton (2000). The references provide an overview of analyses of stochastic epidemic models. For example, asymptotic and exact distributions of the final size of the epidemic, the total area under the trajectory of infective individuals, and approximations to the epidemic model are discussed. In addition, generalisations of the stochastic epidemic model to allow for several classes of susceptible and infected individuals are made and the phenomena of recurrence and competition and spatial aspects are considered.

2.2 Epidemic models

9

The general stochastic epidemic model was first studied by McKendrick in 1926 and then ignored until 1949 when analysed by Bartlett. Deterministic general epidemic models assume that the actual number of new infections in a time interval is proportional to the product of the susceptible and infective population sizes and the time interval. Stochastic general epidemic models, on the other hand, assume that the probability of a new infection in a short interval is proportional to this same amount, i.e. the product of the susceptible and infective population sizes and the time interval. The stochastic version of Equation 2.2 expresses the probability of infection and removal, respectively, as Pr{S(t + δt) = s − 1, I(t + δt) = i + 1|S(t) = s, I(t) = i} = βsiδt + o(δt)

(2.3a)

and Pr{S(t + δt) = s, I(t + δt) = i − 1|S(t) = s, I(t) = i} = γiδt + o(δt).

(2.3b)

The force-of-infection is defined as the rate at which an individual is infected, i.e. βI(t). This formulation for the force-of-infection is known as the “pseudo massaction” assumption (de Jong et al., 1995). It is used when the number of effective transmissions is expected to remain the same regardless of the population size. Another approach, “true mass-action” assumes that the probability of contact decreases as population size increases. In this case, β should be divided by the population size. The force-of-infection at a time t is sometimes referred to as the hazard h(t). For the model described by the system of equations (2.3), the hazard at time t is h(t) = βI(t).

(2.4)

The system of equations (2.3) define an infection process in which contacts between uniformly mixing susceptible and infectious individuals occur at times given by points of a homogeneous Poisson process with intensity β. An implicit assumption is that the infectious periods are exponentially distributed with mean γ −1 . It is only necessary to know the times and nature of all events occurring in the course of an epidemic; the individuals to which each event applies do not affect the next event. A further implication is that the conditional distribution of an infection time in given previous infection times i0 , i1 , . . . in−1 , depends only on the most recent infection time in−1 . Infectious or latent periods that are modelled as random variables from an exponential distribution are said to be Markovian. The Selke construction (Selke, 1983) provides a biological interpretation of the general stochastic epidemic model. Each susceptible individual is considered to have a threshold to infection which is exponentially distributed with unit mean. Once the infection pressure reaches this level, the susceptible individual becomes infected. The total infection pressure exerted on a susceptible up to time t is the integral of the hazard from exposure to time t. Stochastic models that do not assume homogenous mixing have received considerable attention. Such models include multitype models (Ball, 1986; Ball et al., 1997; Andersson and Britton, 2000), which divide the population into homogenous subpopulations, and their extensions (Ball and Neal, 2003), social cluster models (Schinazi, 2002) and random network models (Andersson, 1999).

10


Generalised linear models can be used to model epidemics in the community (Becker, 1983, 1986). Equation 2.3a of the general stochastic epidemic model implies that the probability of a given susceptible escaping infection in the interval (t, t + 1), conditional on the individual being susceptible at time t, is Z t+1 exp − βI(s)ds . (2.5) t

If the time unit is chosen so that not many events, or infections, occur in one unit of time then the conditional probability in Equation 2.5 can be approximated by exp{−βI(t)}.

If Ni (t) is the number of individuals infected in time (t, t + 1) and individuals are infected independently during a given time unit, the number of infections is binomially distributed, Ni (t) ∼ Bin S(t), 1 − exp{−βI(t)} . Chain binomial models are suitable for diseases with a short infectious period and a longer constant latent period. Long latent periods form a means of separating disease progression into stages or generations. Specifically, the latent period is used to model the time between generations. With short infectious periods, it follows that infectives remain infectious for one generation only. The number of infections in any generation therefore depends on the state of the epidemic in the previous generation only. The first chain-binomial model was anticipated by En’ko in 1889 (Dietz and Schenzle, 1985). However, the most well-known chain-binomial model was put forward by Reed and Frost in lectures in 1928. Another well-known variant is the Greenwood model which was formulated independently by Greenwood and published in 1931. If α is the probability of no infection by any single infective and α does not vary between generations, the chain-binomial model can be expressed for each of the described variants,  ij k  , 1 − Bin s En’ko,  j  N −1 i P (Sj+1 = sj+1 |Sj = sj , Ij = sj ) = Bin(sj , α j ) (2.6a) Reed-Frost,   Bin(s , α) Greenwood, j

ij = sj−1 − sj .

(2.6b)

The En’ko model assumes a fixed number of contacts k during one time interval, whereas the Reed-Frost model does not specify the distribution of contacts (Dietz and Schenzle, 1985). In the Reed-Frost model, the probability of escaping infection when exposed simultaneously to i infective individuals in one generation is equivalent to the probability of being exposed to a single infective in each of i separate generations. It is applicable to diseases which are primarily transmitted by close contact between individuals (Becker, 1989). The Greenwood model assumes that simultaneous exposure to two or more infectives in one generation is equivalent to being exposed to just one infective. It is applicable if the household environment is “saturated” with infectious material (Becker, 1989), i.e. if the risk of infection of a susceptible depends only on the presence of viruses in the environment, not on the number of infectives. The Greenwood model is therefore applicable to airborne spread of infection (Dietz and Schenzle, 1985).

2.3 Statistical inference techniques

11

2.2.3 Comparison of deterministic and stochastic models The stochastic versus deterministic model debate is often centred around model simplicity and realism (Andersson and Britton, 2000). Deterministic models are simpler to analyse than their stochastic counterparts (Mollison et al., 1994; Bailey, 1975). For a stochastic model to be mathematically manageable, a simpler and not entirely realistic model is required. Although algebraic determination of stochastic model properties is difficult, approximations can provide information about the intrinsic variability of the system (Mollison et al., 1994). Deterministic models are unsuitable for small populations. For larger populations, the mean number of infectives in a stochastic model may not always be approximated satisfactorily by the equivalent deterministic model (Mollison et al., 1994; Daley and Gani, 1999; Andersson and Britton, 2000). According to Renshaw (1999), it should always be assumed that stochastic effects play an important role in any given process unless proven otherwise.

2.3

Statistical inference techniques

The models described in Section 2.2 are used to describe the mechanism by which observed data are generated. Statistical inference is then used to estimate the parameters of the model. Although deterministic models can be fitted to observations, for example, by least-squares minimisation, the subsequent discussion concerns statistical inference for data said to arise from a stochastic model. Methods of inference for the stochastic models described in the preceding section include maximum likelihood estimation, Bayesian inference using Markov chain Monte Carlo and Martingale techniques. An introduction to statistical inference techniques for non-transmission models is given towards the end of the section.

2.3.1 Maximum likelihood (ML-) estimation For data D generated from a stochastic epidemic model defined by parameters θ, a maximum likelihood estimate (MLE) of θ is the value θ that maximises the likelihood L(θ; D), or equivalently, the loglikelihood l(θ; D). If the likelihood is differentiable, unimodal and bounded above (Tanner, 1996), the MLE is unique and found to zero and solving for θ. Numerical apby setting the score function S(θ) = ∂l(θ;D) ∂θ proaches can be used when the maximum of the likelihood can not be determined analytically. The Newton-Raphson and the expectation-maximisation (EM) algorithms are discussed below; other numerical approaches include the Nelder-Mead simplex algorithm, quadratic optimisation and the quasi-Newton method. Newton-Raphson The Newton-Raphson algorithm is based on the Taylor series expansion of f (x) about x[k−1] for scalar x: f (x) = f (x[k−1] ) + f ′ (x[k−1] )(x − x[k−1] ) + ≃ f (x[k−1] ) + f ′ (x[k−1] )(x − x[k−1] ).

f ′′ (x[k−1] )(x − x[k−1] )2 + ... 2! (2.7)

12


The stationary points of the linear approximation in Equation 2.7 are found by setting Equation 2.7 to zero and solving for x. Doing so implies that x = x[k−1] −

f (x[k−1] ) . f ′ (x[k−1] )

(2.8)

Equation 2.8 can be applied to the vector of score functions S(θ) to obtain an iterative solution of the MLE. If f (x) = S(θ) is the score function and f ′ (x) = H(θ) is the 2 L(θ;D) Hessian matrix with ij th element ∂ ∂θ , an updated estimate, θ [k] , of a current esi ∂θj timate, θ[k−1] , is given by θ [k] = θ [k−1] − H −1 (θ [k−1] )S(θ [k−1] ). Given an initial starting value θ [0] , the sequence θ [0] , θ [1] , . . . converges to a root of S(θ). Expectation-Maximisation (EM) algorithm The expectation-maximisation (EM) algorithm, first named so by Dempster et al. (1977), can simplify ML-estimation of the parameter vector θ by considering a more complete and hypothetical data set. The complete data set D is formed by augmenting the observed data Dobs with fictitious data Dm (referred to as latent or missing data). The latent data should be chosen such that the loglikelihood of the complete data is relatively straightforward. The algorithm is an iterative method which consists of two steps: the E-step (expectation step) and the M-step (maximisation step). Once an initial parameter choice θ [0] is chosen, the E-step and M-step are performed repeatedly until convergence occurs, that is until the difference between successive iterates is negligible. E-step. The E-step consists of computing the expected value of the complete data loglikelihood conditional on the observed data and current estimate, EDm |θ[i] ,Dobs l(θ; Dobs Dm ). M-Step. The M-step requires maximising the expectation calculated in the E-step with respect to θ to obtain the next iterate θ [i] . Iterates obtained using the EM algorithm converge to a turning point of the likelihood. Readers are referred to Hastie et al. (2001) for an explanation as to why the EM algorithm works. A numerical example is provided in Tanner (1996).

2.3.2 Martingale techniques Martingale methods can be used to infer parameters of a stochastic epidemic model given partial observation of the epidemic process. A drawback of using martingale techniques for inference is the necessity of the Markov assumption (O’Neill, 2002). The literature on Martingales requires a sound foundation in measure theory, among other topics, and will not be described further. For an introduction to Martingales without the imposition of mathematical rigour, readers are referred to Lan and Lachin (2005).


13

2.3.3 Bayesian inference using Markov chain Monte Carlo techniques (MCMC) Bayesian inference (Gelman et al., 2000; Congdon, 2001) is the process of fitting a model to data and summarising the results by a probability distribution, known as the posterior distribution, on the parameters and unobserved quantities. The treatment of model parameters as random and data as fixed is in contrast to the frequentist framework in which model parameters are treated as fixed and uncertainty is expressed in terms of potential replicates of the data. The posterior distributions obtained within a Bayesian framework provide information about parameter uncertainty and permit the formulation of direct probability statements, appropriate for small size samples, about parameters. Frequentist approaches, on the other hand, estimate only the standard errors rather than full probability distributions. Probability statements within a frequentist approach rely on indirect statements based on confidence intervals and p-values. Calculation of the frequentist confidence levels may require development of appropriate theoretical results and the usual conditions that require asymptotic normality of maximum likelihood estimators are often violated (O’Neill, 2002). A Bayesian framework incorporates prior information on the model parameters θ in the form of a prior distribution P (θ). This prior distribution along with the likelihood of the data D, P (D|θ) = L(θ; D), defines the joint posterior distribution, denoted P (θ|D). The posterior distribution is given by P (θ|D) = R

P (θ)P (D|θ) . θ P (θ)P (D|θ) dθ

(2.9)

The posterior is the distribution of the model parameters conditional on the data. Because the denominator of Equation 2.9 is not a function of θ, and since integration is with respect to θ, the posterior distribution is proportional to the product of the prior and likelihood distributions, P (θ|D) ∝ P (θ)P (D|θ).

(2.10)

When making inference about the parameters, one is usually concerned with point and interval summaries, such as the mean, variance or quantiles, of the posterior distribution. Point and interval summaries are expressed in terms of their expectation of a function of the unknown parameters, R f (θ)P (θ)P (D|θ) dθ E f (θ)|D = θ R . θ P (θ)P (D|θ) dθ

Except in the simplest cases, the integrals cannot be evaluated analytically. Given a realisation of a Markov chain {θ [g] }, g = 1, 2, . . . (obtained using a MCMC algorithm) whose stationary distribution is the joint posterior distribution, the integral E f (θ)|D can be estimated by Monte Carlo integration, G 1 X E f (θ)|D ≃ f (θ[g] ). G g=1

Markov chain realisations can be obtained using MCMC techniques (Geyer, 1992; Gilks et al., 1996b; Brooks, 1998) which iteratively generate samples from some target distribution π(θ) that is known only up to proportionality. Within a Bayesian

14


framework, the target distribution is the joint posterior distribution of the model parameters (see Equations 2.9 and 2.10). The drawn samples form the Markov chain. The process is continued until the chain has converged to its stationary distribution π(θ). Once converged and after discarding initial samples (to remove dependence of the simulated chain on its starting location), the Markov chain can be used to estimate functions of the target distribution (Kass et al., 1998). The transition kernel P(θ [g+1] |θ [g] ) is the probability that the next state of the chain lies within some set, given that that the chain is currently in state θ[g] (Brooks, 1998). If the transition kernel satisfies the detailed balance condition, π(θ [g] )p(θ [g] , θ [g+1] ) = π(θ [g+1] )p(θ [g] , θ [g+1] ), then the Markov chain will have a stationary, distribution π (Brooks, 1998; Tierney, 1994). Markov chains with transition kernels that satisfy the detailed balance equation are said to be reversible (p.230 Robert and Casella, 2004; Chib and Greenberg, 1995). For convergence of the Markov chain to the stationary, regularity conditions of irreducibility, aperiodicity and positive recurrence are required (Roberts, 1996). An irreducible Markov chain will reach any non-empty set with positive probability in some number of iterations. The aperiodic condition prevents the Markov chain from oscillating between different states in a regular periodic fashion. If a Markov chain is positive recurrent and an initial value is sampled from a distribution π(θ) then all subsequent iterations will also be distributed according to π(θ). The Metropolis-Hastings algorithm (Metropolis et al., 1953; Hastings, 1970; Chib and Greenberg, 1995) is a Markov chain simulation method useful for drawing samples from Bayesian posterior distributions (Gelman et al., 2000). The Metropolis sampler (Metropolis et al., 1953) and the Gibbs sampler (Geman and Geman, 1984; Gelfand and Smith, 1990; Tanner and Wong, 1987; Casella and George, 1992) are special cases of the Metropolis-Hastings algorithm. The Metropolis-Hastings algorithm is as follows: • start with an initial value θ [0] • obtain a realisation θ [1] , θ [2] , . . . from a Markov chain by repeating the following steps for g = 0, 1, . . .: 1. sample a point θ ⋆ from some proposal distribution q(θ ⋆ |θ [g] ),

2. evaluate

π(θ ⋆ )q(θ [g] |θ ⋆ ) α(θ [g] , θ ⋆ ) = min 1, , π(θ [g] )q(θ ⋆ |θ [g] )

(2.11)

3. set θ [g+1] equal to θ ⋆ with probability α(θ [g] , θ ⋆ ), otherwise set θ [g+1] to θ [g] . The transition kernel for the Metropolis-Hastings algorithm, ( q(θ [g+1] |θ [g] )α(θ [g] , θ [g+1] ) if θ ⋆ is accepted such that θ [g+1] = θ ⋆ , + R P(θ [g 1] |θ [g] ) = 1 − q(θ ⋆ |θ [g] )α(θ [g] , θ ⋆ )dθ ⋆ if θ ⋆ is rejected such that θ [g+1] = θ [g] ,

satisfies the detailed balance equation,

π(θ [g] )P(θ [g+1] |θ [g] ) = π(θ [g+1] )P(θ [g] |θ [g+1] ),

(2.12)


15

(Roberts, 1996; Robert and Casella, 2004, p.272). This implies that π is the stationary distribution of the Markov chain (Brooks, 1998; Tierney, 1994; Chib and Greenberg, 1995). The irreducible, aperiodic and positive recurrent conditions, which regulate convergence to the stationary distribution, will be satisfied by Markov chains generated using the Metropolis-Hastings algorithm if the proposal distribution provides full support for π (Robert and Casella, 2004, p.272). It should be noted that Markov chains that are irreducible with stationary distribution π will be positive recurrent (Roberts, 1996). It is often more computationally efficient to update the components of θ one by one rather than all at once using a single component Metropolis-Hastings algorithm [g] [g] [g] (Gilks et al., 1996a). For θ [g] = {θ1 , θ2 , . . . , θh } at iteration g of the Markov chain let [g] [g+1] [g+1] [g] [g] , . . . θi−1 , θi+1 , . . . , θh }. In place of steps 1 to 3 above, the algorithm is θ −i = {θ1 to instead, for each component of θ, [g]

[g]

1. sample a point θi⋆ from some proposal distribution q(θi⋆ |θi , θ −i ), [g] [g] 2. evaluate α(θ −i , θi , θi⋆ ) = min 1, [g+1]

3. set θi

[g]

[g]

[g]

π(θi⋆ |θ−i )qi (θi |θi⋆ ,θ−i ) [g]

[g]

[g]

[g]

π(θi |θ−i )qi (θi⋆ |θi ,θ−i )

,

[g+1]

equal to θi⋆ with probability α(θ [g] , θi⋆ ), otherwise set θi

[g]

to θi .

Note that π(θi |θ −i ) is known as the full conditional; it is the distribution of the ith component of θ conditioning on all remaining components, π(θi |θ −i ) = R

π(θ) . π(θ)dθi

If the proposal distribution is symmetric, i.e. if q(θ ⋆ |θ) = q(θ|θ ⋆ ), then a special case of the Metropolis-Hastings algorithm, referred to as the Metropolis algorithm, can be used. Given a symmetric proposal distribution, the candidate value θ ⋆ is accepted as ⋆) the next value with probability α = min 1, π(θ π(θ) . Gibbs sampling is a special case of the single-component Metropolis-Hastings algorithm in which the proposal distribution for updating the ith component of θ at the [g] [g] gth iteration of the Markov chain is same as the full conditional. If q(θi⋆ |θi , θ −i ) = [g]

π(θi⋆ |θ−i ),

α = min 1,

[g]

[g]

π(θi⋆ |θ −i )qi (θi |θi⋆ , θ −i ) [g]

[g]

π(θi |θ −i )qi (θi⋆ |θi , θ −i ) π(θ ⋆ |θ )π(θ |θ ) i −i i −i = min 1, π(θi |θ −i )π(θi⋆ |θ −i )

=1,

such that the proposal candidate is always accepted. A Markov chain defined by the Metropolis-Hastings algorithm will satisfy the irreducible, aperiodic and positive recurrent conditions required for convergence to a stationary distribution as long as the proposal distribution ensures that the support of θ may be fully explored (Robert and Casella, 2004, pp.272-276).

16


A number of frameworks exist that allow parameter subspaces of differing dimensionality (Green, 1995, 2003). These include jump diffusion (Grenander and Miller, 1994; Phillips and Smith, 1996), point processes (Preston, 1977; Ripley, 1977; Geyer and Møller, 1994; Stephens, 2000), product space formulations (Carlin and Chib, 1995; Godsill, 2001, 2003) and reversible jump MCMC (RJMCMC) (Green, 1995). ⋆

When making transdimensional proposals from a state θ [g] of dimension dm to θ of dimension d⋆m , Green (1995, 2003) shows that detailed balance (2.12) will be preserved if the new state is accepted with probability π(θ ⋆ )j (θ ⋆ )g⋆ (u⋆ ) ∂(θ ⋆ , u⋆ ) m m α(θ [g] , θ ⋆ ) = min 1, (2.13) . [g] [g] [g] π(θ )jm (θ )gm (u) ∂(θ , u)

Here jm (θ [g] ) is the probability of choosing a move m which takes θ[g] of dimension dm to θ ⋆ of dimension d⋆m , gm (u) is the probability of generating u of dimension rm , ⋆ and d +r = d⋆ +r ⋆ . Retaining detailed balance ensures that u⋆ is of dimension rm m m m m the target distribution is the stationary distribution provided the chain is irreducible and aperiodic (Dellaportas et al., 2002). Given the current state of the Markov chain θ [g] of dimension dm , the reversible jump MCMC (RJMCMC) algorithm (Green, 1995, 2003) is to repeat the following four steps: 1. propose a move m which takes θ [g] of dimension dm to θ ⋆ of dimension d⋆m with probability jm (θ [g] ), 2. generate u from gm (u), 3. set θ ⋆ = h(θ, u), 4. accept θ ⋆ with probability α(θ [g] , θ ⋆ ) defined in Equation 2.13. The Jacobian appears due to the 1:1 deterministic transformation used in the proposal mechanism and the change of variable in the requirement for the condition of reversibility (Waagepetersen and Sorensen, 2001). Burn-in Early iterations in MCMC simulations are discarded to diminish the effect of the starting value (Brooks, 1998). The discarded iterations are referred to as burn-in. The length of the burn-in period will depend on the starting value and the rate of convergence of the Markov chain to the stationary distribution (Gilks et al., 1996a). The burn-in period can be determined by visual inspection of plots of the Monte Carlo output, analytic determination or convergence diagnostics (discussed below) (Gilks et al., 1996a). Geyer (1992) suggests that calculating the burn-in is unnecessary and that discarding 1 to 2% of the run length should be sufficient if extreme starting values are discarded. A more conservative rule of thumb is used by Gelman et al. (1996, p.295), who generally discard the first half of the iterations. Thin factor Markov chain sequences may be thinned by keeping every kth simulation draw from each sequence and discarding the rest. Thinning is useful in problems with large


17

numbers of parameters where computer storage is a problem (Gelman et al., 1996, p.295). A thinned Markov chain of length G will have less autocorrelation than a full sample of the same length (Geyer, 1992). As the thin factor k goes to infinity, the Markov chain will become almost independent (Geyer, 1992). The amount of thinning must be weighed against the cost of sampling as the full sample will have more information. Choice of the proposal distribution An ideal proposal distribution would reproduce the target distribution (Robert and Casella, 2004, p.293). When direct sampling from the target distribution is not possible, the proposal distribution should be easy to calculate, facilitate computation of the acceptance ratio (2.11), lead to reasonable differences between consecutive values of the Markov chain and optimise the acceptance rate (Gelman et al., 2000, pp.291-292). Random walks constitute a common class of proposal distributions. Typically the normal distribution is used with the mean equal to the current value and the variance tuned to optimise algorithm efficiency (Gelman et al., 2000, pp.305-307). Optimality criteria may include the rate of convergence of the Markov chain, asymptotic efficiency of the Markov chain and the acceptance ratio (Raftery and Lewis, 1996). Theoretical justification (Roberts, 1996, see) has been given to aim for an acceptance ratio in the range of [0.15, 0.50] depending on the dimension of the target distribution. For a one-dimensional target distribution, the optimal proposal distribution will have an acceptance ratio of around 0.44 (Gelman et al., 2000, p.306). The best algorithm will also depend on the computational time for the iteration (Gelman et al., 2000, p.306). Another common class of proposal distribution is the independence sampler, in which the proposal distribution does not depend on the current value, i.e q(θ ⋆ |θ) = q(θ ⋆ |θ). For further details on proposal distribution choice, readers are referred to Robert and Casella (2004, pp.292-301), Chib and Greenberg (1995) and Gelman et al. (2000, pp.305-306). Assessing convergence and behaviour of the chain Diagnostics are required to determine when samples from the Markov chain have converged to the underlying stationary distribution. Some of the more popular diagnostics include those proposed by Gelman and Rubin (1992), Geweke (1992), Raftery and Lewis (1992) and Heidelberger and Welch (1983). Each of these diagnostics can be performed using the CODA software (Medical Research Council Biostatistics Unit, United Kingdom). These methods are explained from a practical point of view by Best et al. (1995) and along with other diagnostics in a comparative review by Cowles and Carlin (1996). See Kass et al. (1998) for a discussion on difficulties associated with convergence monitoring. The use of the score statistic for convergence assessment is detailed by Fan et al. (2006). The diagnostic proposed by Gelman and Rubin (1992) requires that several, say I, Markov chains are run with different starting values (chosen from an overdispersed distribution) for a total of 2T iterations. For each parameter θ, the between-chain

18


(B) and within-chain variances (si ) for each chain i, i = 1 . . . , I, are computed. By discarding the first T observations and denoting the average within-chain variance W , an approximation for the target variance σ 2 is T − 1 I + 1B V = W+ . T I T

Notation and calculations required to determine the within-chain variance W and between chain variance B are provided in Table 2.1. Convergence is said to occur V is approximately unity. A ratio greater when the weighted ratio of the variances W than unity may indicate that the simulated sequences have not yet made a full tour of the target distribution. In this case, further simulation will either increase the within chain variance W or decrease the total variance V .

This table is not available online. Please consult the hardcopy thesis available from the QUT Library

TABLE 2.1. Gelman and Rubin (1992) convergence diagnostic. Calculations. The diagnostic proposed by Geweke (1992) is based on a time series analysis of a given Markov chain. To test for convergence, one performs a hypothesis test with the null H0 and alternative H1 hypotheses: H0: the means of the first x percent and last y percent of the chain are equal, i.e. µx = µy , H1: the means are not equal, i.e. µx 6= µy . The difference between the means in the first and last windows divided by the asymptotic chain standard deviation σ is denoted z, z=

µx − µy . σ

If the chain has converged, z will have a standard normal distribution. The null hypothesis is therefore accepted at the 1 − α significance level if −zα/2 < z < zα/2 or if the p-value P (|Z| > z) >= α. The p-value is the percentage of the times that one would expect to obtain the given, or more extreme, outcome when the null hypothesis holds; it is the smallest level of significance at which the null hypothesis would be rejected when a specified test procedure is used on a given data set. Values of z in the extreme tails of a standard normal distribution suggest that the chain has not converged within the first window of the chain. The Heidelberger and Welch (1983) diagnostic uses Brownian bridge theory and the Cramer von Mises statistic to test the null hypothesis that the sequence of iterates in the given Markov chain(s) has converged. If the null hypothesis is rejected, the first ten percent of the samples are discarded and the test is repeated. This continues until the null hypothesis is accepted or fifty percent of the iterations have been rejected.


19

If stationarity exists within fifty percent or more of the latter original iterations then the half-width test is passed. Otherwise it is recommended that more iterations are performed. Using the Heidelberger and Welch diagnostic, the CODA software determines the number of iterations to keep, the number of iterations to discard and mean and halfwidth of the parameter based on the retained iterates. The halfwidth is equal to 1.96 times the asymptotic standard error of the retained iterates. The Raftery and Lewis (1992) diagnostic relies on Markov chain theory and the binomial variance, both to detect convergence to the stationary distribution and to provide bounds for estimating the quantiles of functions of parameters to some desired level of accuracy. The CODA package may be used to determine the: total iterations the total number of iterations that should be run for each variable, lower bound the minimum number of iterations required to attain a pre-specified probability of estimating the parameter quantiles to the desired level of accuracy using independent samples, burn-in the total number of initial iterations to discard as burn-in, thin factor the thinning interval to be used, dependence factor the increase in the number of iterations needed to reach convergence due to within-chain dependence. The lower bound is determined in advance and will be less than the total iterations required when negative correlations exist between consecutive iterates. Dependence factors greater than five often indicate problems that can be alleviated by model reparameterisation or by changing the starting value or MCMC algorithm (Raftery and Lewis, 1992). The Gelman and Rubin, Geweke, Raftery and Lewis, and Heidelberger and Welch diagnostics are not suitable for detecting convergence within each dimension of the subspace of MCMC samplers across varying dimensions. Brooks and Giudici (1998) extend the Gelman and Rubin (1992) method to monitor convergence of MCMC samplers for varying dimension problems. Like Gelman and Rubin (1992) they propose running parallel chains but additionally split total variation between models. Models are defined according to the dimension of the subspace. Using the notation given in Table 2.2, the variation of a parameter θ is split between (i) within chain variation, Wc (θ) =

C M Rcm r −θ ¯ )2 1 XXX (θcm c , C c=1 m=1 r=1 G−1

(2.14a)

(ii) within model variation, C M Rcm r −θ ¯ )2 1 XXX (θcm m , Wm (θ) = M c=1 m=1 r=1 Rm − 1

(2.14b)

20


(iii) within model within chain variation, Wm Wc (θ) =

C M Rcm r −θ ¯ )2 (θcm 1 XXX cm , M c=1 m=1 r=1 CRcm − 1

(2.14c)

M −θ X ¯ )2 (θ¯m , M −1

(2.14d)

(iv) between model variation, Bm (θ) =

m=1

and (v) between model within chain variation, Bm W c(θ) =

C X M −θ X ¯ )2 (θ¯cm c . C(M − 1) c=1 m=1

(2.14e)

The total variance V is estimated by the sample variance Vˆ (θ) =

C X M R cm X X 1 r (θcm − θ¯ )2 . CG − 1

(2.15)

c=1 m=1 r=1

For a large number of iterations, all chains are said to have converged to the same stationary distribution if the estimators for each of 1. total variance {Vˆ (θ), Wc (θ)}, 2. within-model variance {Wm (θ), Wm Wc (θ)} and 3. between-model variance {Bm (θ), Bm Wc (θ)} ˆ

V are equal. Equality of these estimators is assessed using the statistics W , Wm and c Wm Wc Bm Bm Wc , respectively. Convergence is said to occur when each of these statistics is apˆ

V proximately equal to one. The statistic W monitoring equality of the total variance c estimators approximates the original Gelman and Rubin diagnostic.

This table is not available online. Please consult the hardcopy thesis available from the QUT Library

TABLE 2.2. Brooks and Giudici (1998) convergence diagnostic. Notation. A problem with the Brooks and Guidici approach is that the approximations break down if a rare model is visited by one of the chains. Castelloe and Zimmerman (2002)


21

propose an alteration of the Brooks and Guidici approach to lessen the impact of rare model visits on the convergence diagnostics. In particular, they propose monitoring only the estimators for total and within-model variances. Minor changes to the calculation of the within-model and within-model-within-chain variances are also proposed. Using the notation given in Table 2.2, the variances are calculated as follows: (i) within-model variation Wm (θ) =

C X M R cm X X 1 2 (θ r − θ¯m ) , CG − M c=1 m=1 r=1 cm

(ii) within-model-within-chain variation C X M R cm X X 1 (θ r − θ¯cm )2 . Wm Wc (θ) = C(G − M ) c=1 m=1 r=1 cm

The total and within-chain variances are calculated according to Equations 2.15 and 2.14a, respectively. Efficiency of the MCMC sampler The performance of a Markov chain Monte Carlo sampler is evaluated by the efficiency in estimating the expectation E(θ) of a given parameter θ under the stationary distribution (see for example, Green and Mira (2001) and Neal and Roberts (2005)). Both the running time G needed to obtain a fixed number of sweeps and the integrated autocorrelation time function (IACF) Cθ must be considered. Let θ = θ [0] , θ [1] , . . . , θ [G] denote a sample of the Markov chain under the stationary distribution. An estimator for the integrated autocorrelation time function for θ is PK ˆ Cθ = 1 + 2 k=1 ρˆk (Berg, 2004) where ρˆk is the empirical autocorrelation of θ at lag k and K is the maximum lag at which the normalised autocorrelation is evaluated. The value of K should be chosen such that ρˆK significantly contributes to the serial dependence of the sampled value and ρˆk , k = K + 1, K + 2, . . . are not significant (Kass et al., 1998). The normalised autocorrelation at lag k is G−k G−k G−k 1 X θ [g]θ [g+k] X θ [g] X θ [g+k] ρk = 2 − . G−k G−k σθ g=1 G − k g=1 g=1

¯ = 1 Cθ Var(θ) (Sokal, The sample mean θ¯ for a sample of size G has variance var(θ) G 1996). The number of effective independent samples in a run-length of G is approximately CGθ . An integrated autocorrelation time function value, Cθ , more than one indicates that MCMC sampling is less efficient than i.i.d. sampling. The product of G and Cθ estimates the running time required to obtain the same precision in estimating E(θ) as from G independent draws from the stationary distribution.

22


Model determination Model determination encompasses both model adequacy and model comparison (Gelman et al., 1996). Model adequacy checks the fit of the model to the data whereas model selection entails comparing candidate models. Bayesian posteriors should be robust to changes in the prior (if uninformative), provide reasonable posterior estimates and fit the data. Model adequacy Methods to assess the fit of Bayesian models to data include residual checking (Carlin and Louis, 2000), posterior predictive assessment (Gelman et al., 1996, 2000), cross-validation (Gelfand et al., 1992) and posteriors from the simulation (Dey et al., 1998; Sinha and Dey, 1997). Methods applicable to survival analysis models include the Bayesian latent residual approach (Aslanidou et al., 1998; Ibrahim et al., 2001) and prequential methods (Ibrahim et al., 2001). The residual checking approach is analogous to classical approaches in which models are assessed by analysing residuals (Carlin and Louis, 2000). The approach requires two independent data samples y and z, where y is an independent validation sample and z a sample of data used to fit the model. The Bayesian standardised residual is defined as yj − E(Yj |z) dj = p , j = 1, . . . , n. Var(Yj |z) If two sets of data are not available, some portion of the data may be reserved from the outset for model validation. When data is scarce, other approaches may be more suitable.

The posterior predictive assessment method (Rubin, 1984; Gelman et al., 1996, 2000) compares observed yobs and replicated yrep data sets. If the model fits, replicated data generated under the model should look similar to observed data. The observed and replicated data sets are compared on the basis of test statistics {D(yobs ), D(yrep )} (Rubin, 1984) and discrepancies {D(yobs , θ), D(yrep , θ)} (Gelman et al., 1996). Discrepancies depend on the data and the parameters unlike test statistics which depend only on the data. Example test discrepancies include the Pn

2

yj −E(yj |θ) Var(yj |θ)

Pearson chi-squared discrepancy D(y, θ) = j=1 , (e.g. Gelman et al., 1996, 2000; Stern and Cressie, 2000), p P √ Freeman-Tukey discrepancy D(y, θ) = E(yj |θ))2 , (e.g. Brooks et al., j ( yj − 2000) and the deviance discrepancy D(y, θ) = −2l(θ; y), (e.g. King et al., 2006). The Freeman-Tukey statistic is advocated when cells contain few observations (Brooks et al., 2000). Choice of a suitable discrepancy measure should depend on the problem (Sinharay and Stern, 2003) (for examples, see Glickman and Stern (1998) and Belin and Rubin (1995)). The algorithm for the posterior predictive assessment method is to construct a replicated dataset for each draw from the posterior distribution, evaluate the discrepancy for the drawn parameter values and each of the datasets; observed and replicated, and produce a scatter plot with the discrepancy of the observed data on the x-axis


23

and the discrepancy of the replicated data on the y-axis. The Bayesian p-value is rep the proportion of times D(yj , θ) is greater than or equal to the observed discrepancy D(yjobs , θj ), or the proportion of points in the scatter plot which lie above the line of unit slope. The p-value quantifies the degree of surprise associated with the observed data under the assumed model and the prior (Brooks et al., 2000). Brooks et al. (2000) highlight the need to produce both a graphical summary and Bayesian p-value as the distributions may differ but still provide an optimal p-value of 0.5. The posterior predictive assessment method is criticised for being conservative with non-uniform p-values (Stern and Cressie, 2000; Sinharay and Stern, 2003). An alternative method is given by Bayarri and Berger (2000). The cross-validation technique (Gelfand et al., 1992) uses existing sequential data, rather than hypothetical realisations, to validate the model. It compares the observed responses yj to those expected from the data with the j th response missing. Various checking functions are possible, including the standardised residual

yj − E(yj |y−j ) dj = p , Var(yj |y−j )

Freeman-Tukey residual dj =

√

yj −

q

(2.16)

E(yj |y−j ), and the

(2.17)

tail area probability ( P (Yj ≤ yj |y−j ) of the response, dj = (2.18) P f (Yj |y−j ) ≤ f (yj |y−j )|y−j for a function of the response. Values for the checking functions should be centred around zero, with large absolute values causing concern. Plotting dj against j should reveal patterns of over-/underfitting. In practice, the following approximations can be used to calculate the crossvalidation checking functions, G

E(yj |y−j ) ≈

1 X E(yj |θ [g] ) G

E(yj2 |y−j ) ≈

1 X {Var(yj |θ [g] ) + [E(yj |θ [g] )]2 } G g=1

(2.19)

g=1 G

(2.20)

G

1 X P (Yj ≤ yj |y−j ) ≈ 1 . [g] G g=1 E(yj |θ )≤yj

(2.21)

Based on these approximations the mean is given in Equation 2.19 and the variance is Var(yj |y−j ) = E(yj2 |y−j ) − [E(yj |y−j )]2 ≈

(2.22)

G G 1 X 2 1 X {Var(yj |θ [g] ) + [E(yj |θ [g] )]2 } − E(yj |θ [g] ) . G G g=1

g=1

(2.23)

24


Unless the dataset is small and yj is an extreme outlier, these approximations should be adequate (Carlin and Louis, 2000). The mean and variance can also be approximated using importance weighting and importance re-sampling (Stern and Cressie, 2000). For applied examples of the cross-validation residual technique, see Auranen et al. (2000) and Stern and Cressie (2000). The posteriors from the simulation method (Dey et al., 1998; Sinha and Dey, 1997) compares the posterior distributions of the observed data to the associated posterior distributions of data simulated from the specific model. If the posterior distribution of the observed data for a given parameter fits among the posterior distributions obtained from the replicated dataset then the model fits the data well for that particular discrepancy function. The Bayesian latent residual method (Ibrahim et al., 2001; Aslanidou et al., 1998) is used to assess the adequacy of survival models and in particular the parametric specification of the baseline hazard. If the model is adequate, the latent residuals for each individual i should have a uniform distribution (a proof is given in Appendix K). Following Ibrahim et al. (2001) and Aslanidou et al. (1998), the latent residual di , for an individual i with infection time t, for the stochastic epidemic model is Z t di = exp{− h(u)du}. 0

Prequential methods (Arjas and Gasbarra, 1997) of model assessment are based on the idea that a model is good if it can be used to predict future values that match well with values that are later realised. Prequential model assessment considers sequential predictions and compares them with the actual observed development of the process. The method relies on the theory of continuous time counting processes. Examples of the approach are provided by Arjas and Gasbarra (1997). Prequential model assessment is used by Arjas and Andreev (2000) to assess a Bayesian analysis of acute ear infections. Ibrahim et al. (2001) provides an overview of the approach. Model comparison Model comparison can be made on the basis of model fit and complexity (Gelman et al., 2000, chapter 6). The deviance is often used as a measure of model fit, D(y, θ) = −2l(θ; y). f Within a Bayesian setting, it makes sense to average the deviance over the posterior distribution, G X ˆ avg (y, θ) = 1 D D(y, θ [g] ). (2.24) G g=1 The Deviance Information Criterion (DIC) (Spiegelhalter et al., 2002) is a measure incorporating both model fit and complexity. Model complexity is measured according to the effective number of parameters, represented by the difference between the posterior mean deviance and the deviance at some point estimate, e.g. posterior ˆ of θ, D ˆ The DIC is the sum of the terms for fit, D ˆ avg (y, θ), ˆ avg (y, θ) − D(y, θ). mean, θ, ˆ ˆ avg (y, θ) − D(y, θ), and complexity, D ˆ ˆ avg (y, θ) − D(y, θ). DIC = 2D

(2.25)

2.4 Statistical inference techniques applied to stochastic epidemic models

25

The model with the smallest DIC should best enable the prediction of a replicate dataset with the same structure. Models within 1-2 units of the best DIC deserve consideration and within 3-7 of the best have considerably less support (Spiegelhalter et al., 2002). The DIC can be viewed as a Bayesian alternative to the Akaike Information Criterion (AIC). The AIC (Akaike, 1973) is ˆ y) + 2p, AIC = −2l(θ; where p is the number of parameters. When prior information is negligible, the DIC and AIC are roughly equivalent. However the AIC and DIC will vary when comparing complex hierarchical models when the number of parameters are not clearly defined or, for example, when the use of prior distributions is likely to decrease the effective dimensionality of the model(s) (Spiegelhalter et al., 2002). If one and only one of the competing models are known to be true, Bayes factors (Kass and Raftery, 1995) can be used (Spiegelhalter et al., 2002). The Bayesian Information Criterion (BIC) (Hastie et al., 2001, pp.193-224) is a common approximation. The BIC can be viewed as an estimate of the marginal probability of the model given the data. The BIC tends to favour simpler models than the AIC and DIC. It aims to find the model best supported by the observed data whereas the AIC and DIC aim to find models with good prediction properties. The BIC is ˆ y) + p log n, BIC = −2l(θ; where p is the number of parameters and n the number of data observations.

2.4

Statistical inference techniques applied to stochastic epidemic models

Statistical inference techniques such as ML-estimation and Bayesian inference are used to infer parameters from data believed to be generated from a given stochastic epidemic model. This section describes the use of techniques to infer parameters of models such as the chain binomial, generalised linear and general stochastic epidemic models used to describe observed data. Inference for non-transmission models is touched on briefly towards the end of the section.

2.4.1 Chain binomial and other independent household models Chain binomial models are used to analyse disease incidence data of small groups, such as households, of known size. Two fundamental assumptions exist when making inferences for parameters in a chain binomial model, namely that outbreaks within infected households evolve independently of each other and that disease evolves in generations (Becker and Britton, 1999). Data may describe the epidemic chain or the final outbreak size. Epidemic chain data identifies the generation in which each individual is infected; it describes the number of infectives in each household in each generation of the epidemic. Final outbreak size data describes the total number of cases per household at the end of the epidemic. Final outbreak size data

26


can be determined with greater reliability but leads to a loss of precision in parameter estimation. ML-estimation has been used to infer the probability of an individual escaping infection by measles (Bailey, 1975) and the common cold (Becker, 1989) from both epidemic chain and final outbreak size data. The EM algorithm facilitates ML-estimation for data from larger households (Becker, 1997; Becker and Britton, 1999). Chain binomial data can be analysed using generalised linear models (Becker, 1989). This technique allows the probability of escaping infection to depend on demographic factors such as age, sex, generation, health of either the susceptible or infected individual and the number of infected individuals. Within a Bayesian framework, MCMC techniques can be used to infer parameters of chain binomial models applied to data of measles outbreaks (O’Neill and Roberts, 1999; O’Neill et al., 2000) and meningococcal disease (Ranta et al., 1999). Using a Bayesian framework, O’Neill (2003) uses perfect simulation in place of MCMC, to infer the parameters of a chain binomial model. Perfect simulation, derived from the ‘coupling from the past’ algorithm introduced in Propp and Wilson (1996), allows for exact sampling from the stationary distribution. This means that a burn-in period is not required and is in contrast to MCMC sampling which is an approximate sampling method. A Bayesian hierarchical approach using MCMC methods can be used to fit models which allow transmission to vary between households (Li et al., 2003). Chu et al. (2004) describe a method using MCMC techniques to estimate the transmission probability adjusting for covariates of both the susceptibles and the infectives when multiple infectives are present. Longini and Koopman (1982) allow for the chance of acquiring infection from outside the household while continuing to assume that households act independently of one another. O’Neill et al. (2000) extend the Longini and Koopman (1982) model by using MCMC techniques. Models allowing for dependencies between infected households, (e.g. Ball et al., 1997), which consider within-household and between-household infections, are discussed later. If the latent and infectious periods are not more-or-less constant and/or if the infectious period is not short relative to the latent period, the chain-binomial model is not appropriate; recursive formulae defining the distribution of the final size must be used, (e.g. Ball, 1986; Addy et al., 1991; Baker and Stevens, 1995; O’Neill et al., 2000).

2.4.2 Generalised linear models Generalised linear models (GLMs) allow for inference of continuous data from outbreaks in larger communities by discretising the data and considering the number of infected individuals in successive time units. Time is considered in absolute terms, whereas for inference of chain-binomial models time is defined in terms of generations. GLMs are applicable when the latent and infectious periods are essentially of fixed duration, or the infectious period is indicated by a show of symptoms. MLestimation to infer parameters is often implemented using statistical packages such as GLIM (Becker, 1983, 1986). The method is described and illustrated in Becker (1989, chapter 6) using common cold and smallpox data. Becker (1989) notes that application to data can present difficulties if there are time periods with zero cases. To overcome these difficulties, it is recommended that the GLM be fitted to the data using an identity link function with Poisson error distribution rather than using a log link function and binomial error distribution. The methodology has been applied to incidence data of Cryptosporidium parvum to make inferences concerning


27

transmission from both the environment and from exposure to infected individuals (Brookhart et al., 2002). The model allows for a gamma distributed latent period by decomposing the latent compartment into several exponentially distributed sublatent periods. An over-dispersion parameter is introduced to account for sources of variability not accounted for in the model and thereby allows for the possibility that the data exhibit extra-binomial variability.

2.4.3 Stochastic epidemic models Maximum likelihood estimation (ML-estimation) can be used to make inferences about the parameters of a stochastic epidemic model when the epidemic is fully observable and infection times are known. The likelihood of a stochastic epidemic model can be derived using survival analysis techniques (Ibrahim et al., 2001), counting methods (Andersen et al., 1993), or as a point process (Davison, 2003). The likelihood L(β, γ) of the stochastic epidemic model described in Equation 2.3 is L(β, γ) = β N1 (t) exp

n

−β

Z

0

t

Z t o n o S(u)I(u) du γ N2 (t) exp − γ I(u) du , 0

where N1 (t) and N2 (t) are the number of infection and removal events at time t, respectively (Becker, 1995; Becker and Britton, 1999; Andersson and Britton, 2000). The MLEs for the transmission (β) and removal (γ) rates are

and

N1 (t) βˆ = R t 0 I(u)S(u)du N2 (t) , γˆ = R t 0 I(u)du

respectively (Andersson and Britton, 2000). Rida (1991) shows that the MLEs are consistent and asymptotically normal. ML-estimation is cumbersome when the epidemic process is only partly observed. For example, given a data-set of symptom appearance times, evaluation of the transmission rate requires integration over all possible unknown infection times. The complexity is demonstrated by Bailey and Thomas (1971), who employ ML-estimation to estimate the contact and removal rates, in the general stochastic epidemic model, for a Nigerian smallpox epidemic. An overview is provided in Section 6.83 of Bailey (1975). Interactions between susceptible and infected individuals whose numbers change over time make it difficult to compute the expectation at the E-step of the EM algorithm (Becker, 1997; O’Neill, 2002). An alternative approach is to disregard that the data are interval censored. The general concern about taking interval censoring into account is questioned by Lindsey (1998) in a study of interval censoring in parametric regression models. If interval censoring is ignored then data consisting of prevalence (or incidence) counts can be analysed using a Markov condition to express the probability of observed changes to the prevalence in terms of transmission parameters and ML-estimation to infer parameter values. Pelupessy et al. (2002) use a Markov condition to express the probability of observed changes to the numbers of colonised patients in a hospital

28


ward over time in terms of a spontaneous colonisation rate, transmission rate and decolonisation rate. The methodology was applied to data describing vancomycinresistant enterococci (VRE) and Pseudomonas aeruginosa colonisation to determine the described rates. The model assumes perfect sensitivity of the surveillance procedure used to detect colonised patients and that bed occupancy in the ward is constant. The Markov model was extended to a hidden Markov model (MacDonald and Zucchini, 1997) by Cooper and Lipsitch (2004), in which the numbers of colonised or infected patients in a single hospital ward {ct } at times t are treated as hidden and the numbers of infected patients {ot } at times t are observed. The probability of changes to the observed numbers of colonised or infected patients is expressed using a Markov process (as in the study by Pelupessy et al. (2002) in the preceding paragraph) in terms of the transmission rate, colonised on admission rate and discharge rate parameters. The observed number of infected patients ot , conditional on the underlying hidden number of colonised or infected patients ct , at time t is described by a Poisson distribution parameterised with the infection rate parameter as the mean of the distribution. Bed occupancy in the ward is assumed constant. In contrast to the Markov model (Pelupessy et al., 2002), the hidden Markov model relies on the observed prevalence of infected patients and therefore does not rely on an assumption of perfect sensitivity of swabbing procedures to detect colonisation status. The hidden Markov model was applied to infection counts of MRSA, VRE and third-generation cephalosporin Gram-negative rods by Cooper and Lipsitch (2004). Martingale techniques can be used to make inferences about partially observed epidemics by setting up estimating equations. If D is the epidemic data and θ the parameter of interest, an estimate for θ can be obtained by finding a function f such that E f (D, θ) = 0 and solving for θ. The function f is referred to as the estimating function (Becker, 1995). Martingale methods for the analysis of epidemic data are described in detail by Becker (1989), Becker and Yip (1989), Becker (1993), Becker and Hasofer (1997) and Britton (1998). The use of Martingale techniques for inference is restricted to Markovian models, whereas MCMC techniques are not. MCMC methods for data imputation are relatively straightforward to implement; they are used to explore the joint posterior distribution of model parameters and augmented data. In an epidemic model, the unknown infection times of each individual are included as model parameters in addition to the transmission rate parameters. When the number of infectious individuals is unknown, the MCMC algorithm must allow for evaluation across an unknown and varying number of infection events and therefore model parameters. Gibson and Renshaw (1998) give details on a reversible jump MCMC algorithm to infer transmission rates of an SEIR model given data on births and removals. Building on the work by Gibson and Renshaw (1998) and Renshaw and Gibson (1998) investigate the use of MCMC methods to infer aspects of a stochastic epidemic process relying purely on removal time data. Potential problems are also highlighted. The methodology was employed by O’Neill and Roberts (1999) to estimate the infection and removal rates of the Nigerian smallpox outbreak. The same data set was subsequently analysed by Boys and Giles (2004), who use RJMCMC to infer the infection and removal rates based on a model that allows the removal rate to change an unknown number of times. MCMC data augmentation methods have been adapted for patient populations displaying high patient turnover to infer the transmission rates of VRE (Cooper et al., 2005). An inherent assumption of the general epidemic model is that the population con-


29

sists of uniformly mixing individuals who are homogeneous (in terms of infection). However, in reality, individual characteristics and community structure can lead to variation in the susceptibility and/or infectivity of individuals in the population. Unidentified individual characteristics or community structure can be modelled by modifying the hazard (2.4) of the stochastic epidemic model to allow for random effects. The hazard for an individual i at time t, hi (t) = I(t)αi , incorporates a random effect through the frailty parameter αi ∼ Gamma(u, v). O’Neill and Becker (2001) used a random effects model to estimate the parameters describing susceptibility and the infectious period of an SEIR model fitted to smallpox data. Britton and O’Neill (2002) describe methodology using Bayesian inference for an epidemic model incorporating an unobserved social structure. A Bernoulli random graph was used to describe potential contacts among population members. The probability of infectious contact with each neighbour in the graph was said to occur according to an independent Poisson process. Using two types of data, namely (i) the times of infection and removal of each individual who became infected and (ii) the removal times only, Britton and O’Neill (2002) were able to infer the infection rate, the mean infectious period and the probability that any two individuals had regular social contact. Multitype models, i.e. models which split the total population into homogenous subpopulations, are used to incorporate heterogeneity arising from identifiable individual characteristics or community structure. Hayakawa et al. (2003) propose a multitype model to analyse data from a Bayesian perspective. The model allows the susceptibility of individuals between groups to vary but assumes that any susceptible in a given group is affected equally by any infective individual irrespective of the subpopulation to which the infective belongs. Given data describing the type and removal time of each individual, posterior distributions can be derived for parameters describing the rate of transmission from each subpopulation in addition to the removal rate of the entire population. The model was illustrated using respiratory disease data in which the population was partitioned into three groups according to the individual’s age. Höhle et al. (2005) adapted the multitype model to include spatial effects. The population was divided into subgroups according to spatial location. Rates for transmission within subgroups and between neighbouring units can be estimated using MCMC in a Bayesian framework. Hayakawa et al. (2003) and Höhle et al. (2005) used removal time data to make inferences for the multitype models. Final size data, as opposed to temporal data, can also be used to infer the local and global infection rates using a Bayesian framework (e.g. Demiris and O’Neill, 2005a,b). Demiris and O’Neill (2005a) augment the data with the so-called final severity of the epidemic, which is the sum of the infectious periods for all individuals. Imputing the final severity leads to an approximate analysis based on the independent groups analysis made by Addy et al. (1991). Classical inference for the model and a description of the asymptotic results were previously given by Ball et al. (1997). In a second paper, Demiris and O’Neill (2005b) eliminate the need to use approximation methods by augmenting the data with more detailed information about the epidemic, namely the set of susceptible individuals that each infected individual infects. In an earlier paper, Gibson (1997) fitted a spatio-temporal SI model to infrequent

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temporal data describing the spread of citrus tristeza virus in plants. Here it was assumed that the number of infection events was known. The model allows the infective challenge to increase as the distance between the relative positions of infected and susceptible individuals decreases. An MCMC algorithm is recommended to integrate out uncertainty regarding the order in which events occur and thereby estimate the relationship between the infection rate and relative positions of infected and susceptible plants. Auranen et al. (2000) estimated acquisition and clearance rates of Streptococcus pneumonia in 97 Finnish families. Acquisition from within the family was considered distinct to acquisition from the surrounding community. Both rates were allowed to depend on the ages of the family member. Heterogeneity of susceptibility based on age was introduced by the use of a hazard function hi (t) at time t for individual i, similar to ( I(t)λ1 if individual i was a child, hi (t) = I(t)λ2 if individual i was an adult. Neal and Roberts (2004) estimated the rate of transmission of measles within households, within classrooms and between households from a highly detailed dataset. A related approach was employed by O’Neill and Marks (2005) to investigate infection routes in an outbreak of gastroenteritis. The general stochastic model typically assumes that the infectious period is exponentially distributed. Models deviating from this assumption require knowledge of the explicit history of each individual; the system can no longer be characterised according to the numbers of individuals in each compartment at any time t. O’Neill and Becker (2001) allowed for gamma distributed infectious periods. Streftaris and Gibson (2004a) fitted an SEIR model to foot and mouth disease data using Weibull distributions for the latent and infectious periods. The use of a Weibull distribution to describe the latent and infectious periods is also described by Streftaris and Gibson (2004b). Non-centred and partially non-centred MCMC algorithms (Papaspiliopoulos et al., 2003) have been recommended to assist MCMC mixing with minimal extra computation costs when applied to the general stochastic epidemic model (Neal and Roberts, 2005). Fearnhead and Meligkotsidou (2004) demonstrate the use of a forward-backward algorithm, see for example MacDonald and Zucchini (1997), to simulate from the posterior distribution of infection times for the general epidemic model. If used in conjunction with the Bayesian data augmentation approach in which the data is augmented with unobserved infection times, the algorithm enables all infection times to be updated in a single step rather than individually as in the approach of O’Neill and Roberts (1999). The forward-backward algorithm relies on the memoryless property of the Markovian assumption (Fearnhead and Meligkotsidou, 2004). Consequently, the algorithm may not be amenable for application to models assuming non-Markovian latent or infectious periods. Non-Markovian models, such as those with latent or infectious periods modelled as random variables from a Weibull distribution, require knowledge of the individual to which each event applies (Streftaris and Gibson, 2004b). For a Markovian model, knowledge of the times and nature of all events is sufficient (Streftaris and Gibson, 2004b).

2.5 Case study: methicillin-resistant Staphylococcus aureus (MRSA)

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2.4.4 Non-transmission models Transmission of HIV is often ignored when analysing AIDS data as the transmission parameters are likely to change during the long incubation period of HIV. Furthermore, multiple transmission modes make models complicated and introduce many parameters difficult to estimate from available data. Instead the backprojection method is used. The backprojection method uses AIDS incidence data to reconstruct the HIV infection curve. It is based on the knowledge that infection always precedes disease diagnosis. The reconstructed curve can be used to estimate the total number infected and predict future AIDS incidences. Backprojection and the use of the EM algorithm to analyse data on HIV/AIDS is described in Becker (1997). A Bayesian formulation incorporating reversible jump Markov chain Monte Carlo methods has been described and applied to data for homosexual and bisexual males in England and Wales (de Angelis et al., 1998). Mezzetti and Robertson (1999) applied similar methodology to back-calculate age-specific cancer rates.

2.5

Case study: methicillin-resistant Staphylococcus aureus (MRSA)

The emergence of antibiotic resistant pathogens such as methicillin-resistant Staphylococcus Aureus (MRSA) has emphasised the importance of transmission prevention within hospitals. A particular concern is the widespread use of antibiotics contributing to the emergence of higher levels of antibiotic resistance among pathogenic MRSA (Ayliffe, 1997; Rubin et al., 1999). With antibiotic therapy of Staphylococcus Aureus (S. aureus) infections rendered ineffective, a return to high patient mortality rates of up to eighty-two percent (Oliveira et al., 2002) seen in the pre-antibiotic era is a possibility. Most infections are preceded by colonisation (Muder et al., 1991; Hardy et al., 2004) therefore interventions to eliminate colonisation should reduce nosocomial infections. In accordance with Emori and Gaynes (1993), the following definitions are used to describe colonisation and infection by a given pathogen, infection: clinical disease or the existence of the pathogen in body fluids such as blood, colonisation: the presence of the pathogen, at both normal (e.g. nares (nostrils or nasal openings), throat, perineum, groin, axilla, pharynx and skin (Coello et al., 1994; Wertheim et al., 2005) and non-normal carriage sites, with no adverse effect on the host. MRSA incidence may be higher in ICUs than other wards (Hardy et al., 2004) and the ICU is seen to play a central role in the dissemination of MRSA (HoefnagelsSchuermans et al., 1997; Carlet et al., 2004). The admission and discharge from and to other wards and hospitals contributes to the creation and dissemination of MRSA from the ICU to other wards and hospitals.

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2.5.1 Staphylococcus aureus and MRSA S. aureus is common body commensal of about 30% of the population (Bischoff et al., 2004; Zanelli et al., 2002; Nakamura et al., 2002; Lee et al., 1999) but can be pathogenic where normal defenses are interrupted or there is diminished resistance to infection. Without treatment, individuals can remain colonised for periods ranging from several months to more than three years (Sanford et al., 1994; Scanvic et al., 2001; Vriens et al., 2005). Within a population of healthy adults, approximately 20% (range 12-30%) are persistent carriers, 30% (range 16-70%) are intermittent carriers and 50% (16-69%) are non-carriers (Wertheim et al., 2005). The nares are considered the principle carriage site (Coello et al., 1994; Solberg, 2000). Other carriage sites include the throat, perineum and axillae (Coello et al., 1994). Carriage of S. aureus at multiple sites is common (Coello et al., 1994). Treatment options include mupirocin for nasal colonisation, chlorhexidine for skin colonisation and rifampicin and sodium fusidate for throat colonisation (Cox et al., 1995). MRSA is a heterogeneous group of S. aureus strains resistant to all beta-lactam antibiotics and many other antibiotics. The first strain of MRSA was reported in UK soon after the introduction of methicillin in 1959 (Baird and Hawley, 2000). MRSA strains emerged in Australia as early as 1965 and are now endemic in tertiary care institutions of capital cities of Brisbane, Sydney and Melbourne (Turnidge and Bell, 2000). MRSA carriage rates in the community are still low but seem to be rising rapidly in certain parts of the world (Wertheim et al., 2005; Creech et al., 2006), particularly among young persons (Maltezou and Giamarellou, 2006). In 2001, community prevalence in geographically diverse regions within the US found prevalence rates ranging from 0.6 to 2.8% (Creech et al., 2006; Kuehnert et al., 2006; Nakamura et al., 2002). A 1998 study in Birmingham (Abudu and I. Blair, 2001) found prevalence rates of 1.5%. Later studies have found prevalence rates ranging from 8 to 22% (Fridkin et al., 2005; Creech et al., 2006). MRSA accounted for twenty-four to forty-two percent of S. aureus isolates in Brisbane hospitals during the period 1986 to 1994 (Turnidge et al., 1996). Between 30 to 60% of critically ill patients colonised with MRSA become infected (Hardy et al., 2004). Common types of infections include bloodstream infections, lower respiratory tract infections, surgical site infections and pneumonia (Graffunder and Venezia, 2002; Ibelings and Bruining, 1998; Theaker et al., 2001). Nosocomial infections are associated with increased morbidity, mortality, resource use, lengths of stay, antibiotic costs, surgery, infection control measures and investigations (Cosgrove et al., 2003; Talon et al., 2002; Schulgen et al., 2000; Chaix et al., 1999; Cox et al., 1995; Rubin et al., 1999). Multi-drug resistance raises the possibility of infections against which none of the current antimicrobial agents are effective (Sébille et al., 1997). Vancomycin is the preferred method of treatment for MRSA infections, however the recent appearance of S. aureus with reduced susceptibility to vancomycin (Hiramatsu et al., 1997; Smith et al., 1999; Cassone et al., 2004) and vancomycin-resistant S. aureus (Centers for Disease Control and Prevention, 2002a,b, 2004) is threatening the long-term viability of vancomycin to treat MRSA infections. Newer alternatives are linezolid and quiniupristin/dalfopritit but side effects, costs and resistance may limit their usefulness (Haddadin et al., 2002). The risk of MRSA in critically ill patients is increased by the severity of illness (Ibelings and Bruining, 1998), length of stay (Ibelings and Bruining, 1998; Ismail and Pettitt, 2004), use of intravascular devices (Pujol et al., 1994), intensity of exposure to in-


33

fected patients (Merrer et al., 2000), low frequency of HCWs disinfecting their hands (Albert and Condie, 1981; Doebbeling et al., 1992), understaffing (Haley and Bregman, 1982; Grundmann et al., 2002), overcrowding (Griffiths et al., 2002) and use of antimicrobials (Pujol et al., 1994; Graffunder and Venezia, 2002). MRSA is associated with increased lengths of stay (LOS) and costs (Chaix et al., 1999; Herr et al., 2003; Lepelletier et al., 2004; Cosgrove et al., 2005). The mean cost attributable to individual MRSA infections in an intensive care unit between 1993 and 1997 was 9275USD (median 5885, inter-quartile range 1400 to 16720 USD) (Chaix et al., 1999). Within a Boston hospital, MRSA bacteria in the blood had a median attributable LOS of two days and a median attributable hospital charge of 6916USD between July 1997 and June 2000 (Cosgrove et al., 2005). In a German hospital, between August 1999 and August 2000, one study found avoidable costs of 9261.56 euros per MRSA case.

2.5.2 Transmission dynamics Antibiotic resistance within the ICU is facilitated by patients being colonised on admission to the ICU, selection for resistance in bacteria by antibiotic use and by transmission within the ICU (Marshall et al., 2005; Bonten, 2002). Published rates for the proportion of patients colonised with MRSA on admission include 6.8% for an Australian ICU (Marshall et al., 2003), 6.9% (range 3.7 to 20%) among 14 French ICUs (Lucet et al., 2003) and 10% for an English ICU (Thompson, 2004). More recent studies (Troché et al., 2005; Eveillard et al., 2005) in France detected MRSA in 4.2 to 10.1% of patient admissions. Patients with longer lengths of stay in the hospital prior to admission to the ICU are more likely to be colonised on admission to the ICU (Marshall et al., 2003; Lucet et al., 2003). For directly admitted patients, a history of ICU stay, age and open skin lesions are risk factors (Lucet et al., 2003). With limited use of antibiotics, antibiotic-sensitive bacteria are likely to flourish. With the introduction or increased use of antibiotics, these antibiotic-sensitive bacteria are destroyed and replaced by multi-resistant strains of bacteria (Queensland Government, 2001; Marshall et al., 2004). Antibiotic pressures can cause pre-existing resistant flora to flourish within 24-48 hours (Graffunder and Venezia, 2002). Antibiotic therapy may also destroy the protective flora of the patient potentially increasing the risk of colonisation. A review of more than 20 studies found support for a causal relationship between antimicrobial drug use and MRSA (Graffunder and Venezia, 2002). Antibiotic usage is correlated with MRSA incidence at patient, ward, hospital and national levels (Monnet, 1998; Marshall et al., 2004). Transient hand carriage on health-care worker (HCW) hands is the main mechanism for patient-to-patient transmission (Thompson et al., 1982; Bauer et al., 1990). 17%[CI95 9,25] of contacts between a HCW and MRSA colonised patient results in transmission of MRSA from a patient to the gloves of a HCW. HCWs who dress MRSA infected wounds may carry the organism on their hands for up to three hours, however this can be eradicated by handwashing after patient contact (Cooper et al., 2003). Although HCWs contribute to cross-contamination of patients through lack of handwashing and ineffective use of infection control procedures, they are generally not found to be colonised (Thompson et al., 1982; Jernigan et al., 1996). The prevalence of carriage by HCWs within a hospital was found to be 6.2% in a non-outbreak situa-

34


tion (Eveillard et al., 2004). Similar rates were determined by Cesur and C ¸ okça (2004). Transmission may also occur by contact with equipment (Bures et al., 2000; Bhalla et al., 2004; Griffiths et al., 2002). Airborne transmission is rarely significant however exceptions exist (Cotterill et al., 1996). In one study (Wilson et al., 2004), the presence of airborne MRSA was found to be strongly related to the presence and number of MRSA colonised or infected patients. Colonised and infected patients are major institutional reservoirs of MRSA (Merrer et al., 2000; Thompson et al., 1982; Jernigan et al., 1996).

2.5.3 Infection control procedures New strains of MRSA showing reduced vancomycin susceptibility and resistance means that treatment options are becoming increasingly limited. This in addition to the morbidity and mortality associated with infections means that infection control procedures are important. Colonisation often precedes infection (Muder et al., 1991; Hardy et al., 2004) therefore interventions to eliminate transmission should reduce nosocomial infections. Current infection control practices may include surveillance, isolation or barrier precautions, eradication of carriage, good standards of general hygiene, especially handhygiene and restricted or modified use of antibiotics (Thompson, 2004; Marshall et al., 2005). Isolation or barrier precautions may include placing patients in a single room, wearing gloves and a disposable gown or plastic apron for all contacts with the patient or their potentially contaminated environment, and wearing a mask in the case of respiratory infection or extensive wound and/or skin colonisation. Implementation of infection control procedures is limited by an inability to detect all patients, the time required until detection, unavailability of side-rooms and nurse cohorts, treatment that is only partially effective and lack of compliance (Thompson, 2004). Most medical personnel wash hands less than 50% of the time (Albert and Condie, 1981; Simmons et al., 1990; Doebbeling et al., 1992; Pittet et al., 1999; Boyce et al., 2004). Compliance to glove, gown and mask use has also shown to be low (Boyce et al., 2004). Compliance to infection control practices will depend on staff levels, workload, bed occupancy and turnover and education (Haley and Bregman, 1982; Fridkin et al., 1996; Pittet et al., 1999; Vicca, 1999; Grundmann et al., 2002; Cunningham et al., 2005; Blatnik and Leˇsniˇcar, 2006). As such, improved staff levels and education and awareness of infection control procedures are considered key factors in the management of MRSA (Baird and Hawley, 2000). There have been a number of papers questioning the value of current practices (Teare and Barrett, 1997; Barrett et al., 1998; Farr, 2004). There is a large body of clinical studies giving empirical associations between infection control interventions and MRSA however recommendations are often conflicting. For example, handwashing is often heralded as the most important infection control procedure and associated with reduced nosocomial infections (Doebbeling et al., 1992; Pittet et al., 2000; Johnson et al., 2005) however Simmons et al. (1990) was unable to find evidence of a close relationship between handwashing and cross-infection. Larson (1988) reviewed published literature linking handwashing to infection and concluded that the emphasis of handwashing as a primary infection control measure should continue. Findings from clinical research concerning the effectiveness of barrier precautions are also conflicting (Jernigan et al., 1996). Some studies have found isolation to be as-


35

sociated with reduced MRSA bacteria in the blood (Gastmeier et al., 2004; Pan et al., 2005) and costs (Lucet et al., 2003; Chaix et al., 1999) whilst other studies (Jernigan et al., 1996; Cepeda et al., 2005) have not. Similar disputes exist regarding the role of cohorting of known MRSA patients (McDonald, 1997), clearance by topical or systemic antimicrobial agents (McDonald, 1997) and antibiotic cycling (Bonten, 2002) in the control of MRSA. Many studies have looked at the effect of combined interventions, again with contradictory results. Duckworth et al. (1988) found evidence to support the combination of extended screening, mupirocin for treatment of carriage and use of an isolation ward. Harbarth et al. (2000) found evidence to support an infection control program combining visitor education, single room isolation or cohorting and handwashing signs. Tomic et al. (2004) found that a combined hand hygiene active surveillance cultures and admission to identify MRSA carriers, strict application of barrier precautions for patients with MRSA, eradication of MRSA carriage and continuous education of HCWs led to decrease in MRSA acquisition. A strategy of nurse cohorting, strict isolation, active surveillance to detect carriers and colonisation eradication was successful in the Netherlands, Finland, USA and Rhode Island (Bonten, 2002). Cox et al. (1995) found that the establishment of isolation wards in two of the three hospitals, treatment of all colonised patients and staff to eradicate carriage and screening of all patients upon discharge from wards where MRSA had ever been detected were key parts of a strategy which eventually contained a local outbreak. Further studies in which a multi-faceted approach was associated with successful control of MRSA include Farrington et al. (1998), Cosseron-Zerbib et al. (1998) and Pittet et al. (2000). Cox et al. (1995) found that extensive screening (from the nose, wounds, perineum and clinically indicated locations) of staff and patients and side-room isolation of colonised and isolated patients failed to contain outbreaks, however it is likely that a large number of colonised patients were not detected as throat screening was not performed (Cox et al., 1995). A Brisbane hospital (Faoagali et al., 1992) was unable to control MRSA using a control program combining isolation of infected and colonised patients, surveillance, education and an antibiotic restriction policy. A review of the literature was made by Marshall et al. (2005) to evaluate the evidence regarding the value of widely recommended measures to control endemic MRSA. Most of the literature reviewed were based on observational studies with only a few based on randomised, controlled trials. It was concluded that more well-designed studies are required. Systematic reviews of published research (Cooper et al., 2003; Loveday et al., 2006) have found methodological weaknesses and inadequate reporting in studies evaluating the effectiveness of infection control procedures in the hospital management of MRSA. Consequently, factors other than the infection control procedures being evaluated may explain observed changes to MRSA acquisition rates in many clinical studies. An ideal experimental study to determine the effectiveness of an infection control procedure would compare two groups of randomly allocated patients, with only one group receiving the infection control method. Randomisation is often neither ethical nor practical in hospital settings (Sébille et al., 1997; Harris et al., 2004). For this reason, most clinical studies evaluating the effectiveness of infection control procedures are quasi-experimental (Harris et al., 2004). Quasi-experimental studies evaluate interventions but do not use randomisation.

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The lack of randomisation is a major weakness of quasi-experimental study designs making causal relationships difficult to establish. Alternative explanations for changes to the observed outcome often can not be ruled out due to difficulties in controlling for confounding variables and results being subject to regression to the mean, seasonal effects and trend or maturation effects (Cooper et al., 2003; Harris et al., 2004). Confounding variables may include staffing levels, HCW handwashing practices, staff-patient contact patterns, antibiotic consumption, ward cleaning, handwashing agents, length of stay, bed occupancy, MRSA clearance therapy and strain of MRSA. Some strains spread more quickly in hospitals (Cookson, 1997). Changes to the length of stay may also represent attrition bias as shorter lengths of stay provide less opportunity for detection (Cooper et al., 2003). A trend for decreasing length of stay could give the appearance of a reduction in MRSA (Cooper et al., 2003). Clinical studies should therefore measure confounding variables to ensure no changes occurred. Multiple pre- and post-intervention measurements should be presented so that any trends can be seen. Time series should be long enough to counteract any seasonal effects (Cooper et al., 2003). Statistical analysis is required to ensure that observed changes did not occur by chance alone. Standard tests in non-infectious disease epidemiology include the χ2 tests and logistic regression analyses. These tests assume that patients are affected independently. This is not true for transmitted pathogens. The proportion of other patients being colonised (the colonisation pressure) amplifies the risk for non-colonised patients to be colonised. To show statistical validity, epidemic models, such as those described in Section 2.2, are required. Statistical inference (Section 2.3) can also be used. Pre- and post-intervention transmission rates estimated using statistical inference techniques can be compared to determine the effect of interventions on cross-transmission and spontaneous acquisition (Pelupessy et al., 2002). Epidemic models and statistical inference enable quantification of the magnitude and time scales of estimated effects of infection control procedures, which is otherwise difficult on the basis of clinical studies alone (Lipsitch et al., 2000).

2.5.4 Epidemic models A collaboration between clinical epidemiology and theoretical science, i.e. epidemic models and statistical inference, is recommended to quantify estimations of the dynamics of MRSA and the benefits of infection control (Bonten, 2002). Stochastic epidemic models seem the preferred choice to describe the spread of nosocomial pathogens such as MRSA. Chain binomial models seem less popular, perhaps because the latent periods of nosocomial infections are small relative to the colonisation and infectious periods. In one study (Escolano et al., 2000), six days was found to be the median number of days a patient spent in the ICU before nosocomial infection arose and the median infection time was found to be eighteen days. Several epidemic models have been used to analyse the effect of infection control procedures on transmission of nosocomial pathogens. Most consider stochastic variation by simulating the deterministic model. Due to the small numbers of patients in hospital wards, stochastic variations will be large and should be considered (Cooper et al., 1999; Grundmann et al., 2002; Bonten et al., 2001). Deterministic models assume values for rate parameters. They generally categorise patients and possibly staff according to whether or not they are colonised (or infected) and differential equations are used to describe the rates of change to each


37

of these populations. The rates of change are usually said to depend on parameters such as the rate at which patients are admitted and discharged, the staff-patient contact rate, the probability of handwashing/disinfection, the duration of colonisation or transient hand-carriage by staff and the probability of transmission from patient to staff and vice-versa. Such parameters are usually derived from observations or estimated. The deterministic model of Sébille et al. (1997) considered interactions between sensitive and resistant strains of nosocomial pathogens and the effects of different control measures such as effective handwashing compliance, antibiotic prescribing policies and curtailing admission of colonised patients. Patients were categorised according to whether they were non-colonised, colonised with a resistant strain or colonised with a susceptible strain. Staff were categorised according to whether they were colonised or non-colonised. The model demonstrated that effective handwashing compliance reduced staff member colonisation but only slightly reduced patient colonisation, the emergence of resistance was only slightly related to the ICU antibiotic policy and that although difficult to implement, curtailing the admission of colonised patients is important in eradicating nosocomial pathogens. Likewise, Cooper et al. (1999) used computer simulations to explore the properties of a deterministic compartmental model that described the spread of MRSA within a single hospital. Cooper et al. (1999) found that small changes in transmissibility can result in large changes in observed prevalence and in the total number of colonised patient-days. Increasing the frequency of effective hand-washes and reducing the number of colonised patients admitted to the ward were also found to be effective control measures. Higher levels of handwashing compliance (frequency greater than forty percent) was found to make little difference to prevalence and intensity. Increasing surveillance activities was found to have little effect on the introduction rate but was associated with a linear reduction in prevalence and number of patient days. Conversely, shorter lengths of stay were accompanied by higher introduction rates but small changes to the prevalence and number of patient days. The principle conclusion that was drawn from the study, however, was that stochastic models are essential for the interpretation and analysis of nosocomial infections. Austin et al. (1999a) used a compartmental model to show quantitatively how infection control measures such as handwashing, cohorting of staff and antibiotic restriction affected transmission of VRE within an ICU. Like MRSA, transmission via the hands of HCWs is considered an important determinant of the spread and persistence of VRE in an ICU. The model categorised patients according to whether they were non-colonised or colonised and the HCW population according to whether they were VRE-free or contaminated. The transmission dynamics of nosocomial spread was captured using a set of differential equations to describe changes in the numbers of people in each compartment over time. Monte Carlo simulations of the admission and discharge of non-colonised and colonised patients, of the colonisation of non-colonised patients and of the contamination and decontamination of HCWs were used to determine the patient population rather than assuming a constant patient number. Austin et al. (1999a) found that handwashing and cohorting were the most powerful control measures. It was concluded that staff cohorting or compliance for handwashing much greater than the reported level would be required to prevent nosocomial transmission of VRE in endemic settings. A model by Lipsitch et al. (2000) explained why rapid changes to antibiotic resistance occurred with the introduction of antibiotic cycling; when one antibiotic was

38


replaced by another, the admission of new patients with susceptible flora diluted the prevalence of resistance. The speed that resistance decreased was related to patient turnover within the ward. Grundmann et al. (2002) used Monte Carlo simulations of a compartmental model to explore the transmission dynamics of MRSA within an ICU. It was found that transmission was likely to occur with staff deficits or increased workload. Whilst the authors noted that adherence to hand hygiene policies might have compensated for staff shortage and prevented transmission during periods of overcrowding, shared care and high workload, such adherence would be hard to achieve. Further examples of this approach include Austin et al. (1997), Austin et al. (1999b), D’Agata et al. (2005) and Raboud et al. (2005). Mathematical models to evaluate hospital infection control procedures are reviewed by Grundmann and Hellriegel (2006). Such studies are limited by the lack of knowledge and uncertainty concerning model parameters and subsequent assumptions made about such parameters. Cooper et al. (1999) noted that whilst transmissibility was possibly the most influential parameter there was a lack of direct information. Statistical inference can be used to determine a range of plausible values of the transmission rates. The derived values can also be used as input to simulation studies.

2.5.5 Statistical inference Jernigan et al. (1996) give transmission rate estimates for isolated patients (0.009 transmissions per day) and non-isolated patients (0.140 transmissions per day) for a 7-month outbreak in a neonatal Intensive Care Unit (ICU). The analysis required the correct identification of sources and times of transmissions, which is rarely feasible. Nijssen et al. (2005) used surveillance cultures and genotyping to distinguish between cross-transmission and endogenous (due to selection of pre-existing resistant flora by antibiotic pressure) acquisition of MRSA and methicillin-susceptible S. aureus (MSSA). No evidence for cross-transmission of MRSA or MSSA was found in the 10-week study at a medical ICU. Only 2 of the 158 admitted patients (1.3%) acquired MSSA endogenously. The remaining 53 cases of MSSA and 9 cases of MRSA were imported into the medical ICU. These studies did not consider that different realisations of the same process could have occurred; that is, they did not consider the element of randomness. In small populations, such as those in hospital wards, significant fluctuations in the incidence and prevalence of colonisation and infection will occur and therefore a stochastic analysis should be undertaken (Bonten et al., 2001; Renshaw, 1999). One approach, that allows for a stochastic analysis, is described and implemented by Jackson et al. (2005). The authors (Jackson et al., 2005) used Monte Carlo simulations of a mathematical model to calculate a test statistic across a range of possible parameter values; the generated test statistic was compared to the actual test statistic of S. aureus and Pseudomonas aeruginosa strain-typing data to derive MLEs of the model parameters. With constant bed occupancy and a 0.2 probability of colonisation with S. aureus at admission assumed, the transmission rate per colonised patient per day for S. aureus was estimated to be 0.36[CI95 0.11,2.1] for a surgical ICU, 0.13[CI95 0.006,0.58] for a neurological ICU and 0.0[CI95 0.0, 0.18] for a newborn ICU during the 8-month study. Pelupessy et al. (2002) propose a Markov model to allow for a stochastic analysis


39

of routine hospital surveillance data. The model assumes a sequence of surveillance swabs capable of detecting carriage with certainty. Using maximum likelihood techniques the transmission rates of VRE colonisation were estimated. The model forms the basis of the hidden Markov model proposed by Cooper and Lipsitch (2004). The hidden Markov model allows the rate of colonisation (which can not be observed) or infection to be inferred from the number of infected patients. The underlying Markov model describes the number of patients harbouring the organism. The observed number of infections are assumed to follow a Poisson distribution conditional on the unknown number of patients harbouring the organism and infected. By applying the model to counts of MRSA infections within U.S. hospitals over 40 months, Cooper and Lipsitch (2004) estimated that MRSA transmission was 0.329[CI95 0.233,0.465] per day, the probability of colonisation on admission was 0.009[CI95 0.002,0.035] per day and the infection rate was 0.349[CI95 0.209,0.490] per day. True-mass action dynamics and a constant 10-bed occupancy level were assumed. If pseudo-mass action dynamics were assumed, the transmission rate is equivalently 0.0329[CI95 0.0233,0.0465] per colonised patient per day. There is otherwise a lack of information about such rates (Cooper et al., 1999) possibly because the analysis of routine surveillance data is subject to the difficulties associated with infectious disease data in addition to a population with high turnover. As mentioned earlier, infectious disease data is often highly correlated and often arise from partially observable epidemic processes. For example, MRSA acquisition depends on the number of other patients who are colonised (Merrer et al., 2000) leading to dependencies in the epidemic process. Nosocomial pathogens are typically carried asymptomatically and so the acquisition times are imperfectly observed through infrequent routine swabs. Imperfections in the observation process are confounded by false negative swabs. The ability of the swabbing procedure to detect a colonised patient may depend on number and location of patient sites swabbed, the pathogenic load within the patient and laboratory procedures. For example, patients colonised only in the perineum will not be detected from nasal swabs. The nares is the principle carriage site and cultures taken from the nares have been found to detect between 77 to 93% of MRSA patients (Sewell et al., 1993; Coello et al., 1994; Sanford et al., 1994). The pathogenic load may depend on the type of carrier (intermittent or persistent) (Nouwen et al., 2005) and antibiotic usage (Aly et al., 1970). Antibiotic usage may affect the carriage level of VRE making detection more difficult (Harbarth et al., 2002; Cooper et al., 2005) and it is possible that antibiotic usage has a similar affect on the detectability of MRSA. Laboratory methods using a PCR method which involved preincubation in broth for four hours for the detection of MRSA from the nose, groin and other sites had a 66.7%[CI95 51.9,83.3] sensitivity (Hope et al., 2004). Modifications to the PCR method such as increased broth time and use of selective broth may improve sensitivity. In a separate study, the PCR method was found to have a sensitivity of 81% for detecting S. aureus nasal colonisation and a sensitivity of 87% for detecting S. aureus tracheal colonisation (Keene et al., 2005). Detection can be improved by taking more than one specimen from multiple sites (Coello et al., 1994; Sewell et al., 1993). Sample-handling mistakes and laboratory errors may also effect the sensitivity of the swabbing procedure (Nouwen et al., 2004). Detection of MRSA from the nose, groin and other sites using PCR methods have approximately 99% specificity (Hope et al., 2004; Keene et al., 2005). The EM algorithm and MCMC techniques discussed in Section 2.3 appear well-suited

40


to routine surveillance data but must be adapted to allow for the admission and discharge of patients. Community populations have relatively small turnover and so the populations are typically assumed closed. The predominantly short patient stays of hospital ward populations requires the removal of this assumption. Cooper et al. (2005) applied an MCMC approach to hospital infection data on VRE to infer transmission parameters. Patients detected as colonised were assumed to either have been colonised on admission or via transmission from a colonised patient within the ward. The importance of considering imperfect sensitivity was emphasised but not allowed for in the paper. Swab sensitivity has been inferred from disease data in non-infectious disease analyses (Smith and Vounatsou, 2003; Trotter and Gay, 2003).

2.6 Discussion This chapter provides an overview of epidemic models and techniques to infer parameters of the model assumed to govern an observed epidemic process. Techniques covered to infer model parameters included ML-estimation, estimating equations using Martingales and Bayesian inference using Markov chain Monte Carlo techniques. The use of epidemic models and statistical inference in conjunction with clinical epidemiology is recommended to quantify the dynamics of methicillinresistant Staphylococcus aureus and the benefits of infection control. Methicillinresistant Staphylococcus aureus (MRSA) is a heterogeneous group of Staphylococcus aureus strains resistant to many antibiotics. It is a significant cause of morbidity, mortality and costs in the intensive care unit. Emphasis is placed on infection control procedures to limit the spread of MRSA within hospital wards, however the value of such infection control procedures are often questioned. This thesis will develop statistical inference techniques to quantify parameters of stochastic epidemic models describing the transmission of hospital-acquired pathogens, such as MRSA. Parameter values, and their differences, will provide scientific evidence concerning the value of infection control procedures, which is otherwise difficult on the basis of clinical studies alone.

CHAPTER 3

Case studies of methicillin resistant Staphylococcus aureus

3.1

Introduction

This chapter presents details of admissions and the extent of methicillin-resistant Staphylococcus aureus (MRSA) within intensive care units (ICUs) of the Princess Alexandra Hospital and two London hospitals. The infection control procedures employed to limit the spread of MRSA and other nosocomial pathogens are described.

3.2

Princess Alexandra Hospital (PAH) intensive care unit (ICU)

The PAH is an 800 bed metropolitan adult hospital providing tertiary referral services for trauma, transplantation and the major surgical and medical sub-specialties. The General ICU (CC7) is on the seventh floor and commenced in approximately 1993 with 12 beds: 2 isolation rooms, 2 × 2-bed bays and 2 × 3-bed bays (see Figure 3.1). Details of admissions to the General ICU and routine surveillance data describing MRSA occurrence within the ICU from 1 January 1995 to 28 March 1997 exist in the dataset analysed in the PhD thesis of Ismail (1999). A list of the available data and a description of formatting made to the data is given in Appendix B. At the time of the study MRSA was endemic within the ICU.

42

Chapter 3. Case studies of methicillin resistant Staphylococcus aureus

Fire Escape 5

7

6

4

Nurse’s Station

3 2

8

9 10 11

1

12

Isolation 1 Legend = Bed

Isolation 2

Corridor

= Sink

F IGURE 3.1. PAH ICU layout.

3.2 Princess Alexandra Hospital (PAH) intensive care unit (ICU)

43

3.2.1 Infection control procedures In the ICU, one nurse would look after one patient. There was one float nurse for four patients and an extra nurse in charge per unit. Bedside staff wore disposable gowns, however it was not necessary to wear gowns for all patients. It was unit policy to wash hands before and after all patient contact. Standard procedures associated with barrier precautions, room cleaning and waste disposal were followed. Routine nasal or groin swabs testing for the presence of MRSA were taken on Mondays and Thursdays from all patients in the ICU, with 100% compliance assumed. Swabs were taken from patients on transfer from other hospitals and from patients with signs or symptoms requiring investigation. Routine and non-routine swabs were assumed to take place at eleven am on the date of swabbing, unless otherwise stated. No information is available for non-routine swabs except for those with positive isolates. Upon notification of a patient being colonised with MRSA, patients were placed in isolation. Unless otherwise stated, notification was assumed to take place at eleven am on the date of notification. Where there were three or more patients with MRSA, they were cohorted in a two- or three-bed bay and the isolation rooms were used as needed. In contrast to the general ward, the isolation room had a basin for each patient bed and the handwashing policy was signposted at the entrance to the room. Each sink had dispensers for gloves, soap, antibacterial scrub solution (chlorhexidine or iodophor) and skin moisturiser. All HCWs and visitors to the isolation room were required to wear gowns. Only the allocated nurse or nominated relief staff would contact the isolated or cohorted patient. Eradication procedures, antibiotic use, staff workload, staff levels, culture typing and procedures for removing patients from isolation once cleared are unknown. Compliance to infection control procedures is also unknown.

3.2.2 Population and occurrence of MRSA There were 4 patients in the ICU at the start of the observation period and a further 2584 admissions during the observation period; 818 days from 1 January 1995 to 28 March 1997. The 2588 admissions consisted of 2407 hospital admissions; 116 patients with two ICU admissions, 22 with three and 6 with more than three admissions. Table 3.1 gives the number of admissions from each source of admission. Source Direct admission Emergency room Hospital ward Operating room Other hospital Recovery room

Number 1 535 358 1293 345 56 2588

% Admission source 0.04 20.67 13.83 49.96 13.33 2.16 100.00

No. MRSA 0 10 36 47 25 3 121

% MRSA 0 1.87 10.06 3.63 7.25 5.36

TABLE 3.1. Source of patient admissions to the PAH ICU. Of the 2588 admissions, 121(4.7%) had positive cultures; 13 positive cultures were

44


taken before admission to the ICU. Of the 108 positive cultures taken in the ICU, 41 (38.0%) were from the patients’ first swab in the ICU and 64 (59.3%) were taken on routine swab days (see Table 3.2). Of the 13 admissions with detection, or a positive swab date, prior to admission, notification of detection was not made until after admission, so that no patients were known to carry MRSA on admission to the ICU. Table 3.3 lists the length of stay (LOS) and other admission details. The days until positive swab for positive admissions are given in Table 3.4. No information was available for swabs taken after the first positive swab. The average LOS of patients preceding each day is illustrated in Figure 3.2. The positive result was known after discharge for 43 of the 121 admissions with a positive isolate. Reporting time from positive swab (or admission) to notification was 2.37[CI95 0.1,5] days, range(0,6.2). Notification was made for 78 detected admissions whilst the patient was still in the ICU; these patients remained in the ICU for 8.25[CI95 0.12,27.91] days, range (0.08,61.19) after notification. Day Monday Tuesday Wednesday Thursday Friday Saturday Sunday

Count 40 9 14 24 10 4 7 108

% 37 8.3 13 22.2 9.3 3.7 6.5

TABLE 3.2. Positive swabs per day in the PAH ICU.

% Male Age Acute Physiological Score Length of stay Number of swabs

MRSA Not MRSA All 62.0 61.9 61.9 53.4[CI95 18, 83] 52.2[CI95 18, 82] 52.3[CI95 18, 82] 50.1[CI95 15, 99] 38.6[CI95 9, 104] 39.1[CI95 9, 104] 12.6[CI95 0.8, 36.7] 2.4[CI95 0.3, 12.3] 2.9[CI95 0.3, 17.1] 2.0[CI95 0, 6] 0.7[CI95 0, 4] 0.8[CI95 0, 4]

TABLE 3.3. Details of admissions to the PAH ICU. No. days to 1st swab 0 1 2 3 4

No. neg. 14 17 21 11 4

No. pos. 5 14 6 13 3

% pos. on 1st swab 26.32 45.16 22.22 54.17 42.86

TABLE 3.4. Time from admission to first swab for detected patients in the PAH ICU. For the 108 patients detected in the ICU, 67 were not detected on the first swab and 41 were. The 12 beds in the ICU had an occupancy rate of 9.3[CI95 5,12]. Figure 3.3 shows the number of occupied beds, positive isolates and number of admissions known

3.2 Princess Alexandra Hospital (PAH) intensive care unit (ICU)

45

to have MRSA per day. There was a mean number of 1.3[CI95 0,4] detected patients in the ICU representing a prevalence of 14.0%[CI95 0.0,42.9]. Table 3.5 shows that the ICU was free from MRSA patients on only 229 of the 818 days. Of the 7443.2 patient days of the study, 286.4(3.8%) were taken by detected patients prior to notification of the positive isolate and 643.5(8.6%) were taken by detected patients postnotification of the positive isolate. The time taken by detected patients prior to notification represents the cumulative time that patients remain non-isolated due to the notification delay. No. MRSA patients No. days Total days(%)

0 229 28

1 276 33.74

2 186 22.74

3 90 11

4 33 4.03

5 3 0.37

6 1 0.12

0.04 0.03 0.02 0.0

0.01

Average LOS (days)

0.05

TABLE 3.5. MRSA patients in the PAH ICU.

01-01-95

13-06-95

23-11-95

04-05-96

14-10-96

26-03-97

Date

F IGURE 3.2. PAH ICU patient lengths of stay. For a given date x, the graph shows the average number of days that patients, in the ICU at x, have been in the ICU prior to x.


10 12 8 6 4 2

Occupied beds

46

01-01-95

13-06-95

23-11-95

04-05-96

14-10-96

26-03-97

04-05-96

14-10-96

26-03-97

04-05-96

14-10-96

26-03-97

4 3 2 1 0

Positive isolates cultured

Date

01-01-95

13-06-95

23-11-95

5 6 2 3 4 0 1

Detected patients in ICU

Date

01-01-95

13-06-95

23-11-95

Date

F IGURE 3.3. Bed occupancy, MRSA prevalence and MRSA incidence at the PAH ICU. Each figure illustrates the respective numbers per swab day. Prevalence and incidence are based on the positive swab date. Occupancy is defined as the number of patients in beds at 11am.

3.3 ICUs within two London (LON) hospitals

3.3

47

ICUs within two London (LON) hospitals

This section provides a summary of the 1-year prospective study undertaken by Cepeda et al. (2005) and presents results from further analysis of the data collected for the Cepeda et al. (2005) study. For more detailed information of infection control procedures, screening, eradication and typing procedures, readers are referred to the original source (Cepeda et al., 2005). As part of the 1-year prospective study by Cepeda et al. (2005), patient admissions, swabbing procedures and infection control procedures were observed for three London intensive care units (ICUs); two within hospital A and one within hospital B. In hospital A, one ward is an 18-bed unit and the other is a 4-bed unit. The 18-bed unit has four single beds, one two-bed bay and four three-bed bays. Hospital B has a 10bed unit consisting of four single beds, one four-bed bay and one two-bed bay. The layout of the three ICUs is illustrated in Figure 3.4. The observation period took place between 12 June 2000 and 11 June 2001. At the time of the study, MRSA was endemic in all ICU units.

This figure is not available online. Please consult the hardcopy thesis available from the QUT Library

F IGURE 3.4. Layout of the London ICUs. Source: Cepeda et al. (2005).

3.3.1 Infection control procedures Except for the policy concerning isolation of MRSA patients, infection control procedures were constant throughout the observation period. The observation period can be divided into three phases according to the implemented isolation policy: Phase 1 12 June 2000 to 11 September 2000,

48


Phase 2 12 September 2000 to 10 March 2001, Phase 3 11 March 2001 to 11 June 2001. In phases 1 and 3, upon notification of being MRSA positive (colonised or infected), patients were moved to a single room or cohort-isolated in an open bay with other MRSA patients. Nurse cohorting took place for all patients with MRSA. These phases are referred to as the move phase. In phase 2, MRSA patients were only moved or cohort-isolated if they had other multiresistant or notifiable pathogens or needed protective isolation. Nurse cohorting was only required for MRSA patients with other multiresistant pathogens. Patients colonised with MRSA were not prescribed antibiotic treatment. Infected patients were treated with antibiotics (glycopeptide, linezolid, or combination of rifampicin and trimethoprim). Infection was said to occur if MRSA was present in a clinical isolate. In all phases of the study, standard procedures concerning the use of disposable aprons, gloves for invasive procedures, washing and turning of patients and contact and disposal of body fluids were followed regardless of the patients’ MRSA status. Handwashing (with alcohol handrub) was encouraged through regular education sessions. A 21% handwashing compliance and 99% apron usage rate was observed. Patients were screened for MRSA (using nose or groin swabs) within twenty-four hours of admission, weekly thereafter and upon discharge. Samples such as sputum, wound or blood cultures) were taken when clinically indicated. There was a three day reporting delay from collection until availability of results. In this thesis, only results from swabs taken from the nose are used.

3.3.2 Analysis of the patient population and extent of MRSA Table 3.6 summarises key characteristics of the London ICUs during the observation period. The number of direct admissions and transfers between ICUs are given in Table 3.7. Key characteristics of the 18-bed London ICU for each phase are provided in Table 3.8.


Data No. beds in ICU (mean occupancy) Duration of study period, days No. swab intervals, Nint No. admissions No. patients admitted No. admissions with first swab positive No. positive isolate notifications No. positive isolates taken in ICU No. pre-, during-, post-admission +ive isolates No. non-detected patient days No. detected patient days No. isolated patient days Mean no. swabs per patient Mean no. detected patients in ICU, per day, Mean prevalence of detected patients Mean LOS of detected patients, days Mean LOS of non-detected patients, days Mean no. of detected patients in ICU, per day Mean time from detection to discharge, days

49

18-bed 18(14.01) 365 364 688 1 622 63 65 111 125 4003.37 228.29 875.37 1.83 3.07 22.29 15.8 5.81 3.07 10.21

Unit 4-bed 4(2.9) 365 364 208 2 204 8 6 15 15 1006.48 21.21 28.83 1.41 0.15 4.21 6.38 4.98 0.15 3.74

10-bed 10(8.65) 365 364 336 3 298 71 84 103 103 1838.08 278.13 1029.53 2.19 3.46 39.92 15.76 6.54 3.46 12.84

TABLE 3.6. Key characteristics of the London ICUs. Based on the observation period 12 June 2000 to 11 June 2001. 1 2 3

Includes the 16 patients in the ICU at beginning of observation period Includes the 4 patients in the ICU at beginning of observation period Includes the 5 patients in the ICU at beginning of observation period

Source Direct Admission Transfer from 18-bed unit Transfer from 4-bed unit Transfer from 10-bed unit Total Admissions

18-bed 647 (107) 0 41 (4) 0 688 (111)

Unit 4-bed 194 (12) 14 (3) 0 208 (15)

10-bed 336 (103) 0 0 0 336 (103)

TABLE 3.7. Number of admissions and transfers to the London ICUs. The number of positive isolates from each admission source is in the brackets.


50

Mean bed occupancy Duration of study period, days No. swab intervals, Nint No. admissions No. patients admitted No. admissions with first swab positive No. positive isolates taken in ICU No. pre-, during-, post-admission +ive isolates No. positive isolate notifications No. non-detected patient days No. detected patient days No. isolated patient days Mean no. swabs per patient Mean no. detected patients in ICU, per day, Mean prevalence of detected patients Mean LOS of detected patients, days Mean LOS of non-detected patients, days Mean no. detected patients in ICU, per day Mean time from detection to discharge, days

Phase 2 13.4 180 179 294 2 269 27 49 60 25 1892.95 91.31 429.96 1.99 2.88 21.75 16.11 6.63 2.88 12.17

1 14.12 92 91 243 1 227 24 33 38 17 1052.32 67.41 177.34 1.34 2.82 20.63 11.85 4.32 2.82 10.48

3 15.09 93 92 173 3 155 18 34 42 22 1096.78 75.68 233.26 1.99 3.32 22.36 16.28 6.13 3.32 9.26

TABLE 3.8. Key characteristics of the 18-bed London ICU.

20 15

10 5 0

0

0

5

10

15

Positive other swabs

20

Negative Nasopharyngeal swabs

20 15

10

15 0

5

10

Number of patients

20

3

Includes the 16 patients in the ICU at beginning of observation period Includes the 8 patients in the ICU at beginning of observation period Includes the 18 patients in the ICU at beginning of observation period

5

2

Positive nasopharyngeal swabs

1

12-06-0012-10-0011-02-0111-06-01

12-06-0012-10-0011-02-0111-06-01

12-06-0012-10-0011-02-0111-06-01

12-06-0012-10-0011-02-0111-06-01

Date (a)

Date (b)

Date (c)

Date (d)

F IGURE 3.5. Number of patients and positive swabs taken each day in the 18-bed London ICU.


51

Screening swab patterns Table 3.9 lists the screening swab patterns and the frequency at which these patterns were observed for admissions to the London ICUs. The table is divided into three sub-tables; with patterns of positive swabs not followed by negative swabs in 3.8(a), positive swabs followed by negative swabs in 3.8(b) and positive swabs interrupted by intermittent negative swabs in 3.8(c). These swab patterns can be analysed to investigate the probabilities of MRSA acquisition verse importation, screening swab sensitivity and specificity and carriage persistence. Patterns in 3.8(c) with one negative swab interrupting a sequence of positive swabs indicate that the likelihood of imperfect sensitivity; isolation of false negative swabs from persistent carriers. Patterns in 3.8(a) with two or more positive swabs indicate persistent carriage detected by a series of true positive swabs. Patterns in 3.8(b) with one negative swab following one or more positive swabs may represent clearance or the isolation of a false negative swab. The probability of clearance increases as the number of negative swabs following one or more positive swabs increases. In each of the sub-tables, patterns with the first swab positive are more likely to indicate the patient was colonised on admission than those with the first swab negative.

52


(a) Patterns with positive (b) Patterns with positive swabs and no subsequent swabs followed by negative negative swabs swabs

Swab Pattern + ++ +++ ++++ +++++ +++++++ +++++++++++ -+ -++ -+++ -++++ -+++++ -++++++ –+ –++ —++ –+++ –++++ —+ —-+ —–+ ——+ ——-++

No. 55 27 8 6 2 1 2 19 6 3 3 3 1 6 4 1 1 2 5 2 2 3 1 163

Swab Pattern ++++++++++++++++-+–++-++++– ++– -+– –+– —-++– ++— +— –+— ——-+— -+++—+—— -+—— –+—— –+——+————

No. 12 4 1 1 2 3 1 1 2 1 1 1 1 1 2 1 1 1 1 1 1 1 1 42

(c) Patterns of positive swabs interrupted by intermittent negative swabs

Swab Pattern +-+++++++ -+-+ -+-+-+-++++-+ ++-++ -++-++ –+-+++++ -++-++–+++++ ++–+ ++–++++ ++—+ +—-+ –+——-++ -+-++++-+—– +–++———– –++—-+++-++++—-++-

No. 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 19

TABLE 3.9. Screening swab patterns and frequency for admissions to the London ICUs. Patterns describe the swab results for 224 admissions in which 7 transfers occurred and 2 admissions were to unknown ICUs.


53

Antibiotic administration The type of antibiotics administered to each patient was recorded daily. Antibiotics administered, in addition to properties are listed in Table 3.10. Antibiotic usage can be grouped according to: • usage of any, or no, antibiotics, • staphylococcal properties: anti-staph, anti-MRSA, • gram-negativity, • whether the antibiotic is an amenogycide, cephalosporin or quinolone, or • quantity of antibiotics taken. It has been highlighted that swab sensitivities for VRE may depend on antibiotic therapy, which can suppress carriage levels and make detection hard (Cooper et al., 2005). Antibiotic therapy suppresses the pathogenic load of MRSA (Aly et al., 1970) which may affect the sensitivity of swabs testing for MRSA. In Appendix C, plots of the numbers of patients in the 18-bed London ICU on antibiotics within each of the above categories are provided. The graphs show no visual evidence of a difference in antibiotic usage between phases, nor of a given antibiotic type having any effect on the sensitivity of the swabbing procedure. A more formal investigation is required and recommended as future research.

Gram +ive

Gram -ive

Ameno- Beta- Carba- Quino- Glyptogycides lactams regens lones cides

X X

Broadspectrums x X

x x

X X

-

X X

-

-

-

x (x) (x) (x) x XX XX (X)

X X X X X X X X

X X x x x x X X

x (X) x x x x X X

X -

X -

X

X -

-

x XX X XX

X X X X X

X X X X x

X x x x x

-

-

-

-

X

TABLE 3.10. Properties of antibiotics administered to the London ICU patients. Source: Barry Cookson.


Augmentin XX Other X Cephalosporin Cefuroxime XX Ciprofloxacin XX Clindamycin XX Erythromycin XX Flucloxacillin XX Fusidic Acid XX Gentamycin XX Imipenem & (X) Meropenem Piperacillin x Rifampicin XX Teicoplanin Trimethoprim X Vancomycin XX

AntiMRSA

54

AntiMSSA

3.4 Discussion

3.4

55

Discussion

This chapter presented details of admissions, the size of the problem posed by MRSA and infection control procedures at the ICUs of the Princess Alexandra hospital and London hospitals.

56


CHAPTER 4

Mechanistic description of the transmission process

4.1

Introduction

This chapter introduces the epidemic models which will be used in later chapters to describe the spread of communicable pathogens in hospital-ward patient populations. The underlying model is defined in Section 4.2 by considering factors relevant to transmission. The assumptions on which the model is constructed are outlined. The deterministic formulation, mentioned in Section 4.3, underlies the stochastic model which is presented in Section 4.4. The data required to infer values of model parameters for a given hospital ward is discussed in Section 4.5. A critical examination of factors that may influence the values of the true model parameters is given in Section 4.6.

4.2

Model definition and assumptions

Epidemic models can be used to describe the transmission of infectious diseases. For an epidemic model to be valuable an understanding of the underlying epidemic process is required and factors relevant to transmission and infection control should be contained within the model (Massanari, 1997). This thesis is concerned with modelling the transmission of an endemic communicable pathogen within a population of hospital-ward patients, some of whom may be asymptomatically colonised. A compartmental model (Jacquez, 1996) is used to model the infectious process by a succession of states. At a given time t, a ward patient is characterised as being susceptible (but not colonised) (S), or non-isolated (and colonised) (C), or isolated from other patients due to being detected as colonised (Q), or (4) removed or discharged from the ward (R). Once colonised, it is assumed that patients remain so until discharge. Patients discharged from the ward play no further role in the epidemic.

58

Chapter 4. Mechanistic description of the transmission process

The number of patients in each of these compartments (susceptible, non-isolated and isolated) at time t is denoted S(t), C(t) and Q(t), respectively. The population size N (t) is the number of patients in the ward at any given time t and is equal to S(t) + C(t) + Q(t). Upon admission, an individual may be susceptible or asymptomatically colonised with the pathogen. The term ‘importation probability’ is used to refer to the probability, ϕ, that a patient is colonised on admission to the ward. Patients admitted in a susceptible (non-colonised) state can acquire the pathogen via indirect contact with colonised patients or by background transmission, which is defined to include all nosocomial transmission arising from outside the ward. Transmission from colonised HCWs and from equipment and HCWs transiently contaminated within the hospital but outside the ward are examples of possible background contamination sources. Transmission mechanisms are illustrated in Figure 4.1. Upon acquisition of the pathogen, patients are asymptomatically colonised with detection only possible via swabs testing for the presence of the pathogen. Colonised patients will not be detected if, after colonisation, they are not swabbed or subsequent swabs are falsenegative swabs. Patients are isolated once notification is received that a positive swab was cultured from them, regardless of location. Susceptible patients can acquire the pathogen via indirect contact with isolated patients, however it is expected that the rate will be lower than for non-isolated patients. The admission, colonisation, isolation and discharge of patients are referred to as events. Each event has the effect of either decreasing or increasing the number of individuals in one or more compartments. For example, the admission of a susceptible patient will increase the number of patients in the susceptible compartment, whilst the colonisation of a susceptible patient will decrease the number of patients in the susceptible compartment and increase the number of patients in the colonised compartment. The model is an adaption of the SEIR model. A discussion of the SEIR model and numerous well-known variants is provided by Hethcote (1994). A graphical repreColonised Health Care Worker

Environment

Colonised Patient

Health Care Worker

Susceptible Patient

F IGURE 4.1. Transmission dynamics of a nosocomial antibiotic resistant bacterium. sentation of the compartmental model is provided in Figure 4.2 and an instance of a clinical time-frame in Figure 4.3. An implicit assumption of the model is that the level of mixing between a susceptible and colonised patient depends on whether or not the patient is isolated (i.e. if it is known that the colonised patient is colonised). It is expected that the rate of mixing will be less between susceptible and isolated patients than between susceptible and non-isolated (colonised) patients. Patients within each compartment are assumed to be homogenous and homogenously mixing. The possibility of falsenegative swabs are factored into the model, however false-positive swabs are not, i.e.

4.2 Model definition and assumptions

59

imperfect sensitivity is allowed but imperfect specificity is not. The period that colonised patients remain non-isolated is a factor of both imperfect sensitivity (denoted ρ) and the reporting delay. If a patient i colonised at time ci , tij−1 < ci < tij is swabbed at times ti = {ti1 , ti2 , . . .}, and the swab cultured at time tij is true-positive, then it will not be known until some time tij +dij that patient i is colonised. The period dij represents the reporting delay. With a swabbing process that is 100% sensitive the colonisation time would be known within an interval of censoring [tij−1 , tij ). With imperfect sensitivity (and perfect specificity) it can be deduced only

Admission

Admission Colonisation

Colonised C(t)

Susceptible S(t) Discharge

Isolation

Discharge

Isolated Q(t) Discharge

Removed

F IGURE 4.2. Stochastic compartmental model of the infection dynamics of nosocomial antibiotic resistant bacteria. A patient within the ward may have one of several states, namely susceptible, colonised or isolated. Once a patient is discharged from the ward, they play no further part in the epidemic and are considered to be removed. Admissions, colonisations, isolations and discharges are events at which the number and composition of patients within any state may change. S

Patient enters ward Colonisation event

Q

C

Patient colonised

R

Patient discharged Patient isolated

Routine swab Swab results received

F IGURE 4.3. Example clinical time-line for a hospital ward patient who is colonised with antibiotic resistant bacteria during his/her stay in the ward. Routine swabs testing for the presence of the bacteria are administered. The patient is isolated once staff have been notified that the patient is colonised. The time between the administration of the swab and notification of the swab status represents a reporting delay.

60


that the patient was colonised at some time ci < tij .

4.3 Deterministic epidemic model

4.3

61

Deterministic epidemic model

A deterministic model (Section 2.2.1) of the epidemic process would assume that the number of events in a short time interval is proportional to the number of individuals in each compartment and that the discrete numbers of susceptible, colonised and isolated patients can be approximated by real values. For small population sizes, such as those seen in hospital wards, population sizes are not well approximated by real values (Cooper et al., 1999; Grundmann et al., 2002; Bonten et al., 2001) and a stochastic approach should be taken.

4.4

Stochastic epidemic model

In a stochastic epidemic model, probability distributions of the numbers of susceptible and colonised individuals occurring at any instant replace the real-values of deterministic treatments. Assuming homogeneity and no variation in susceptibility nor infectivity over time, the probability that a susceptible patient i is colonised in some small time interval dci > 0, given no colonisation up to time ci , can be described by the hazard function h(ci ). If β0 , β1 and β2 are the rates of transmission from background and non-isolated and isolated (colonised) patient sources, respectively, the hazard function is 1 P (ci < Ci ≤ ci + dci |Ci > ci ) dci →0 dci = β0 + β1 C(ci ) + β2 Q(ci ).

h(ci ) = lim

(4.1a) (4.1b)

The hazard function (4.1a) is sometimes referred to as the force-of-colonisation (or more generally as the force-of-infection). Formulation of the force-of-colonisation in terms of the number of patients is referred to as the “pseudo mass-action” assumption (de Jong et al., 1995). If the number of transmissions between a colonised patient and each of the susceptible patients is expected to vary with the number of patients in the ward, β1 and β2 should be divided by the number of patients in the unit (de Jong et al., 1995). Non-isolated colonised patients are considered homogenous with respect to infectivity and each non-isolated colonised patient is expected to mix uniformly with each susceptible patient. Likewise, isolated colonised patients are considered homogenous and to mix uniformly with each susceptible patient. The rate of mixing between a susceptible patient and a non-isolated colonised patient is expected to differ to that between a susceptible patient and an isolated colonised patient. An implicit assumption is that patients remain susceptible for exponentially distributed periods and that the colonisation process is Markovian. The conditional distribution of a patient being colonised at a time ci given the colonisation times of patients colonised at a time prior to ci depends only on the colonisation time of the last patient to have been colonised. The Selke construction (Selke, 1983) provides an alternative interpretation of the hazard function. Each susceptible patient is considered to have a threshold to colonisation which is exponentially distributed with unit mean. Once the colonisation pressure exerted by other patients exceeds the threshold level, the patient is colonised. The total colonisation pressure on a susceptible patient at time ci is given by the integral of the hazard from exposure to time ci .

62


If only colonisation events are subject to stochastic variation, equation 4.1a describes a form of the general stochastic epidemic model (Bailey, 1975; Andersson and Britton, 2000). A variation of the model would allow for stochastic admission, isolation and discharge events. Using survival analysis theory (Collett, 1994), the hazard function (4.1a) can be examined in terms of the density, distribution and survivor functions. To formulate these functions, it is postulated that if patients remained in the ward indefinitely, each patient i would have a colonisation time ci . The random variable for colonisation time of patient i is denoted Ci . Density function, f (ai , ci ) The probability density that a patient is colonised at time ci , ci ≥ ai . Distribution function, F (ai , ci ) The probability that a patient is colonised before time ci , ai ≤ Ci ≤ ci , Z ci F (ai , ci ) = P (Ci ≤ ci ) = f (ai , u) du. ai

Survivor function, X(ai , ci ) The probability that a patient is colonised after time ci , ai ≤ ci < Ci , X(ai , ci ) = 1 − F (ai , ci ) = P (Ci > ci ≥ ai ). The hazard function can be expressed in terms of the density, distribution and survivor functions as follows: {P (ci < Ci ≤ ci + dci )|(Ci > ci )} h(ci ) = lim dci →0 dci F (ai , ci + dci ) − F (ai , ci ) 1 = lim dci →0 dci X(ai , ci ) f (ai , ci ) = (4.2a) X(ai , ci ) d = − {log X(ai , ci )}. (4.2b) dci From Equation 4.2a and Equation 4.2b it follows that n Z ci o f (ai , ci ) = h(ci ) exp − h(u) du

(4.3)

ai

and X(ai , ci ) = exp

n

−

Z

ci

ai

o h(u) du .

(4.4)

If C(t) and Q(t) are constant for t ∈ [tj−1 , tj ), then the colonisation hazard h(ci ) defined in Equation 4.1a is constant for that time interval and is denoted λj , − λj = β0 + β1 C(t− (4.5) j−1 ) + β2 Q(tj−1 ) ,

where t− j is time infinitesimally preceding tj . It follows that if the hazard is piecewiseconstant over the intervals t0 = ai < t1 < . . . < tj < . . ., then the survivor function (4.4) and density function (4.3) are, respectively, X(t0 , t) = exp{−

j−1 X k=1

λk (tk − tk−1 ) − λj (t − tj−1 )}

4.5 Statistical inference

63

and f (t0 , t) = λj exp{−

j−1 X k=1

4.5

λk (tk − tk−1 ) − λj (t − tj−1 )}.

Statistical inference

Statistical inference can be used to estimate the parameters of a deterministic or stochastic epidemic model given data generated from a process governed by the model. The parameters of the epidemic models introduced in the preceding sections include the transmission rate parameters β = {β0 , β1 , β2 }, the swab sensitivity ρ and the importation probability ϕ. Statistical inference techniques include maximum likelihood estimation, estimating equations using Martingales and Bayesian inference using Markov chain Monte Carlo techniques (which were discussed in Section 2.3). Inference for infectious diseases is complicated by dependencies within the epidemic process and partial observation of the epidemic process (Becker and Britton, 1999; Andersson and Britton, 2000; O’Neill et al., 2000). For example, the risk of colonisation depends on the number of patients who are already colonised and it is usually unknown who colonises whom and at what time colonisation occurs. Regular swabs detecting for the presence of colonisation may allow knowledge of the colonisation time within an interval of censoring provided that the swabs are not subject to imperfect sensitivity. In the following chapters, methods are developed and described to infer the model parameters from observed data. The type of data required to infer the parameters θ = {β, ρ, ϕ} is described below.

4.5.1 Data and notation Knowledge of the admission, isolation and discharge times is required for each patient admitted to the ward during the observation period. Patients in the ward at the beginning of the observation period are assumed to have entered the ward on that day. If a patient is detected as colonised, the positive swab time is also required. The colonisation time ci and final colonisation state si of each patient i will be unknown. Three possible values for the final colonisation state of a patient i are considered, namely   ss if the patient remains susceptible, si = sd if the patient is colonised in the ward,   sp if the patient is colonised prior to admission.

The colonisation time is censored by the admission time for patients colonised on admission and by the discharge time for patients remaining susceptible. The unknown colonisation time and final colonisation state are referred to as latent data. For a patient i, i = 1, 2, . . ., let ai , vi , qi , ri denote respectively the admission, positive swab, isolation and discharge times. The positive swab times of patients without positive isolates and the isolation times for patients not isolated are effectively censored by the discharge times of such patients. The discharge times for patients in

64


the ward at the end of the observation period are set to that censored time. Swabs to test for the presence of the pathogen are cultured from each patient i at times ti = ti1 , ti2 , . . .. The times of swabs from all patients {t1 , t2 , . . .} are denoted ts . The observed data {a, v, q, r, ts } is denoted D. The subset Dd = {a, q, r, ts } is assumed to emanate from deterministic processes.

4.6 Discussion The parameters that will be estimated in later chapters include the background transmission rate, β0 , the rate of transmission from a non-isolated (colonised) patient, β1 , the rate of transmission from an isolated (colonised) patient, β2 , the probability of a patient being colonised on admission, ϕ and the probability of a false-negative swab when testing for the presence of the pathogen, ρ. The parameters β0 , β1 and β2 combine a multitude of epidemiological, environmental and social factors. Transient carriage on HCW hands is the main mechanism for patient-to-patient transmission of MRSA (Thompson et al., 1982; Bauer et al., 1990). The parameters β0 , β1 and β2 will therefore depend on the number of effective contacts between patients and HCWs. HCWs who dress MRSA infected wounds may carry the organism on their hands for up to three hours. However, this can be eradicated by handwashing after patient contact (Cooper et al., 2003). The number of effective contacts will depend on pathogenic factors and infection control procedures. Non-epidemic strains of MRSA are less likely to cross-contaminate other patients, particularly among community acquired MRSA patients (Hoefnagels-Schuermans et al., 1997). Infection control procedures such as surveillance, isolation or barrier precautions, eradication of carriage, hand-hygiene and restricted or modified use of antibiotics, aim to reduce the reservoirs and transmission of nosocomial pathogens. Compliance to infection control practices will depend on staff levels, workload, bed occupancy and education (Haley and Bregman, 1982; Fridkin et al., 1996; Pittet et al., 1999; Vicca, 1999; Grundmann et al., 2002; Blatnik and Leˇsniˇcar, 2006). Contact patterns and compliance to infection control procedures vary between different types of HCWs, with physicians having lower adherence to hand disinfection (McBryde et al., 2004; Nijssen et al., 2003). The epidemic models introduced in this chapter assume uniformly mixing patients. In a hospital ward, indirect mixing of patients occurs through HCWs. The existence of patients requiring more hands on activity or a higher proportion of physician visits will violate the assumption of uniform mixing. Furthermore, staff-patient contact patterns may lead to increased transmission between adjacent patients. The contact between susceptible and isolated patients is expected to be less than between susceptible and non-isolated patients. This expectation is incorporated into the model by the use of the two transmission parameters β1 and β2 . The level of patient susceptibility will also impact the values of these parameters. The risk of MRSA, for example, may increase with illness severity (Ibelings and Bruining, 1998), the use of intravascular devices (Pujol et al., 1994), and other risk factors inherent to the individual patient. The models introduced in this chapter are based on the assumption that patients are equally susceptible to colonisation, which does not hold true for MRSA. The parameters are likely to be higher in ward populations with larger proportions of patients with MRSA risk factors. Background transmission β0 includes all nosocomial transmission not arising from colonised patients within the ward. In addition to the factors mentioned above,

4.6 Discussion

65

the rate of background transmission may depend on environmental contamination (Bhalla et al., 2004; Griffiths et al., 2002; Bures et al., 2000), HCW colonisation (Thompson et al., 1982; Jernigan et al., 1996), MRSA reservoirs combined with the extent of HCW duties outside the ward and antibiotic usage. Antibiotic pressures can cause pre-existing resistant flora to flourish and therefore represents a source of endogenous acquisition. Furthermore, antibiotic therapy may destroy the protective flora of the patient potentially increasing the susceptibility of the patient to colonisation. The rate of transmission from colonised patients depends on the number of effective contacts between susceptible and colonised patients, patient susceptibility and the “infectiousness”, or transmission ability, of the colonised patient. Factors affecting the number of effective contacts and patient susceptibility were mentioned earlier. The infectiousness of an MRSA patient may depend on the pathogenic load and rate of dispersal. Persistent carriers have higher loads of S. aureus, resulting in increased dispersal and a higher risk of endogenous infection (Nouwen et al., 2005) and it seems plausible that such patients will have higher cross-contamination rates. Antibiotic usage has been shown to suppress the pathogenic load of MRSA nasal carriers (Aly et al., 1970). Carriage site may also be important and there is some evidence that perineal carriers are more likely to be dispersers (Coello et al., 1994). Although no clinical evidence was provided, Baird and Hawley (2000) state that patients with conditions such as productive cough, expectoration of sputum, exuding wounds and loss of bodily fluids all lead to contamination of the environment. The importation probability parameter ϕ is the probability of a patient being admitted to the ward in a colonised state. Published rates for the proportion of patients colonised with MRSA on admission include 6.8% for an Australian ICU (Marshall et al., 2003), 6.9% (range 3.7 to 20%) among 14 French ICUs (Lucet et al., 2003) and 10% for an English ICU (Thompson, 2004). More recent studies (Troché et al., 2005; Eveillard et al., 2005) in France detected MRSA in 4.2 to 10.1% of patient admissions. The model assumes a constant importation probability for each patient, however the rate is likely to vary for each patient according to the prevalence of the pathogen at the patient’s prior location and patient risk factors. ICU patients with longer lengths of stay in the hospital prior to admission are more likely to be colonised with MRSA on admission to the ICU (Marshall et al., 2003; Lucet et al., 2003). For direct admission ICU patients, a history of ICU stay, age and open skin lesions are risk factors for MRSA colonisation on admission (Lucet et al., 2003). The sensitivity parameter ρ will depend on the number and location of patient sites swabbed, the pathogenic load within the patient and laboratory procedures. MRSA carriage sites include the nares, throat, perineum and axillae (Coello et al., 1994; Solberg, 2000) and so, for example, patients colonised only in the perineum will not be detected from nasal swabs. The nares is the principle carriage site and cultures taken from the nares have been found to detect between 77 to 93% of MRSA patients (Sewell et al., 1993; Coello et al., 1994; Sanford et al., 1994). As previously discussed, the pathogenic load may depend on the type of carrier (intermittent or persistent) and antibiotic usage. Antibiotic usage may affect the carriage level of VRE making detection more difficult (Harbarth et al., 2002; Cooper et al., 2005) and it is possible that antibiotic usage has a similar affect on the detectability of MRSA. Laboratory methods using a PCR method which involved pre-incubation in broth for four hours for the detection of MRSA from the nose, groin and other sites had a 66.7%[CI95 51.9,83.3] sensitivity (Hope et al., 2004). Modifications to the PCR method such as increased broth time and use of selective broth may improve sensitivity. In a separate

66


study, the PCR method was 81% sensitivity for detecting S. aureus nasal colonisation and 87% sensitivity for detecting S. aureus tracheal colonisation (Keene et al., 2005). Detection can be improved by taking more than one specimen from multiple sites (Coello et al., 1994; Sewell et al., 1993). The assumption of perfect specificity for the screening process seems valid for MRSA; detection of MRSA from the nose, groin and other sites using PCR methods have approximately 99% specificity (Hope et al., 2004; Keene et al., 2005). An assumption of the model is that once colonised, patients remain so until discharged. In practice, MRSA carriers may be intermittent or occasional carriers and decolonisation may occur with treatment. Options for treatment of MRSA include mupirocin for nasal colonisation, chlorhexidine for skin colonisation and rifampicin and sodium fusidate for throat colonisation (Cox et al., 1995). True negative swabs taken after a true positive swab will lead to underestimation of the sensitivity parameter. Furthermore, the model will not provide an accurate description of the number of patients in each compartment; the number of susceptible patients will be lower and the number of colonised patients according to the model will be higher than in reality. The impact of this assumption will depend on the rate of decolonisation and patient lengths of stays. With shorter lengths of stays, it is more probable that patients will be discharged prior to decolonisation. This chapter presented a deterministic and stochastic epidemic model to describe the transmission of asymptomatic nosocomial pathogens in a hospital ward. Data required to infer parameters of the model was also mentioned. The following chapters introduce statistical inference techniques to estimate parameters of the model using observed data describing the occurrence of methicillin-resistant Staphylococcus aureus. Knowledge of the parameters will allow quantification of the relative importance of cross-contamination, spontaneous acquisition (as measured by background transmission) and of the effect of isolation of colonised patients on the rate of cross-contamination.

CHAPTER 5

Generalised linear model (GLM) and inference

This chapter forms the basis of a paper published in Infection Control and Hospital Epidemiology (Forrester and Pettitt, 2005)

Statement of joint authorship M.L. Forrester (Candidate): determined required methodology, developed Bayesian model, formatted data, performed computational work, investigated effects of the mass-action assumption and bed occupancy, interpreted the results and wrote and proofread the manuscript. A.N. Pettitt: conceived conceptual model, assisted with formulation of the methodology, assisted with interpretation of results and proofread the manuscript.

68

Chapter 5. Generalised linear model (GLM) and inference

5.1 Introduction Generalised linear models can to be used to approximate the stochastic epidemic model by treating time as discrete rather than continuous. Generalised linear models have been used to model infectious diseases within the community (Becker, 1989; Brookhart et al., 2002). In this chapter a generalised linear model is adapted for analysis of data describing the observed occurrence of a nosocomial pathogen within a hospital ward. In particular, the methods are applied to data describing methicillinresistant Staphylococcus aureus (MRSA) occurrence within the Princess Alexandra Hospital (PAH) intensive care unit (ICU).

5.2 Model and methodology The hazard function of the stochastic epidemic model for nosocomial pathogens (Section 4.4) describes the probability of transmission. For a time interval [tj−1 , tj ) the hazard function, given in Equation 4.5 is − λj = β0 + β1 C(t− j−1 ) + β2 Q(tj−1 ) .

If the time unit is chosen so that not many colonisations occur within the interval [tj−1 , tj ) the conditional probability of a given susceptible escaping colonisation in the interval, given that the individual is susceptible at a time just prior to tj−1 , is approximately exp{−λj (tj − tj−1 )}. It follows that conditional probability of colonisation in the interval [tj−1 , tj ), denoted πj , is πj = 1 − exp{−λj (tj − tj−1 )} = 1 − exp{− β0 +

β1 C(t− j−1 ) +

(5.1a) β2 Q(t− j−1 )

(tj − tj−1 )}.

(5.1b)

An inherent assumption is that individuals are homogenous with respect to risk factors. The expression for πj (5.1) can also be derived using theory of interval censored events for survival data (Collett, 1994) as follows: πj = Pr{(tj−1 ≤ ci < tj )|(ci ≥ tj−1 )} X(t0 , tj−1 ) − X(t0 , tj ) = X(t0 , tj−1 ) = 1 − exp{−λj (tj − tj−1 )}.

(5.2)

Since the probability of the event {tj−1 ≤ ci < tj } is conditioned on the event ci ≥ tj−1 , λj will not depend on the history of the process preceding tj−1 . Note that the probability pj that patient i is colonised in [tj−1 , tj ) can be represented as a sequence of j Bernoulli trials consisting of j −1 failures followed by success, with

5.2 Model and methodology

69

πj probability of success for the j th event: pj = Pr(tj−1 ≤ ci < tj )

= X(t0 , tj−1 ) − X(t0 , tj )

= Pr{Ci ≥ t1 |Ci > t0 } Pr{Ci ≥ t2 |Ci > t1 } . . . Pr{Ci ≥ tj−1 |Ci > tj−2 }

× Pr{tj−1 ≤ Ci < tj |Ci ≥ tj−1 }

= πj (1 − π1 )(1 − π2 ) . . . (1 − πj−1 ) for j = 1, 2, . . . .

At j = 1, p1 = π1 . The probability that a patient is not colonised at time tj is repreQ sented by a sequence of j failures and is equal to j−1 k=0 (1 − πk+1 ).

Patients within a hospital ward may be routinely swabbed to test for the presence of asymptomatic pathogens. If all patients are swabbed at the same times, with swab times ts = {ts1 , ts2 , . . .}, and swabs have one hundred percent sensitivity and specificity then the number of patients colonised in each interval [tsj−1 , tsj ) is known and denoted NjC . If it is assumed that, within a given swabbing interval, patients escape colonisation, or become colonised independently, then the distribution of the number of colonisations in the routine swab interval [tsj−1 , tsj ) is binomially distributed with index S(ts− j−1 ) and parameter πj , NjC ∼ Bin(S(ts− j−1 ), πj ).

(5.3)

In this model, the probability of a patient being colonised on admission is not considered, rather it is assumed that the colonisation status of patients is known on admission to the ward. Furthermore the sensitivity of the swabbing process is assumed to be perfect. The generalised linear model is an approximation to the stochastic epidemic model described in Chapter 4. To illustrate the generalised linear model in the form of a stochastic compartmental model, let Sj , Cj and Qj be the number of susceptible, ˙S ˙C ˙Q non-isolated and isolated patients in the ward at time ts− j , Aj , Aj and Aj represent the number of susceptible, colonised and isolated patients admitted to the ward in the j th interval [tsj−1 , tsj ), and R˙ jS , R˙ jC and R˙ jQ be the number of susceptible, colonised and isolated patients discharged from the ward in the j th interval. The total number of admissions and discharges per swab interval, denoted A˙ j and R˙ j , respectively, are known. However, the proportion of these patients which are susceptible or colonised is unknown. The number of patients isolated in the interval [tsj−1 , tsj ) is NjQ . A graphical representation of the generalised linear model in the form of a compartmental model is given in Figure 5.1. The illustration is an adaption of the stochastic compartmental model in Figure 4.2.

70


Susceptible Sj−1

Admission A˙ S

Susceptible Discharge Sj ˙S Colonisation Rj C Nj

Admission A˙ C

Colonised Discharge Cj R˙ jC Isolation NjQ

j

Colonised Cj−1

j

Isolated Qj−1

Admission A˙ Q j

Isolated Qj

Discharge R˙ jQ

F IGURE 5.1. Relationships between data of the generalised linear model. The number of susceptible, colonised and isolated patients at interval j are determined by the number of respective patients in each compartment in addition to the number of admission, isolation and discharge events.

5.3 Statistical inference

5.3

71

Statistical inference

In the following subsections a description of the data required to allow inference of the generalised linear model parameters is given. Statistical inference techniques of ML-estimation and MCMC within a Bayesian framework are defined. Methods to ascertain the fit of the model to data are given, followed by an introduction to model comparison.

5.3.1 Data and notation Inference for the generalised linear model requires knowledge of the number of susceptible, colonised and isolated patients, in addition to the number of colonisations, in each time interval j. Swab intervals are defined according to the routine swab times ts = {ts1 , ts2 , . . .}. The data required for the generalised linear model can be derived as follows (and in accordance with the relationships summarised in Figure 5.1),

Sj = Cj =

j X k=0 j X k=0

Qj = Rj =

j X

A˙ Sk − NkC − R˙ kS ,

(5.4a)

Q C ˙C A˙ C k + Nk − Nk − Rk ,

(5.4b)

Q ˙Q A˙ Q k + Nk − Rk ,

(5.4c)

k=0 R˙ jS +

R˙ jC + R˙ jQ .

(5.4d)

Additionally, it should be noted that Sj = Sj−1 + A˙ j − NjC − R˙ jS , such that NjC = Sj−1 − Sj + A˙ j − R˙ jS . The method to determine values for the numbers of susceptible, colonised and isolated patients being admitted and discharged in each time interval in addition to the Q ˙S C ˙Q number of patients being colonised and isolated, i.e. for A˙ Sj , A˙ C j , Aj , Nj , Nj , Rj , R˙ jC , R˙ jQ , is described in Appendix F. The calculations require only the data outlined in Section 4.5.1, namely the admission, positive swab, isolation and discharge times for each patient.

5.3.2 Maximum likelihood estimation Let yj be the number of patients escaping colonisation in the j th interval. This is equal to the number of susceptible patients in the (j − 1)th interval not colonised in the j th interval, yj = Sj−1 − NjC . Following Equations 5.1 and 5.3, yj is binomially distributed, yj ∼ Bin(Sj−1 , 1 − πj ),

72


with log(1 − πj ) = −(β0 + β1 Cj−1 + β2 Qj−1 )(tsj − tsj−1 ). If x1 = −(tsj − tsj−1 ),

x2 = −Cj−1 (tsj − tsj−1 ) and x3 = −Qj−1 (tsj − tsj−1 ), then log(1 − πj ) = β0 x1 + β1 x2 + β2 x3 . Here x1 , x2 and x3 can be determined from the data and β0 , β1 and β2 are the unknown parameters. ML-estimation to infer β0 , β1 and β2 can be implemented using standard statistical packages to fit the GLM to the data (McCullagh and Nelder, 1989). A log link function together with a user-written script (see Appendix G) to calculate link, inverse and derivative functions may be used.

5.3.3 Bayesian inference using MCMC techniques Bayesian inference (see Section 2.3.3) can be used to determine the posterior distribution P (θ|D) of the model parameters θ = {β0 , β1 , β2 }, given data D ={S, C, Q, NC }. The likelihood of the data P (D|θ), prior distribution of the parameters P (θ), posterior distribution P (θ|D) and a Markov chain Monte Carlo (MCMC) algorithm to sample from the posterior distribution are described below. Likelihood Following Equation 5.3 and assuming that the number of colonisations per interval, NjC , are conditionally independent, the likelihood of colonisation events across all intervals is Y NC ∼ Bin(Sj−1 , πj ), (5.5) j≥2

where NC is understood to exclude the case for the first swab interval. Equivalently, n Y S o C C j−1 P (D|θ) ∝ πj Nj (1 − πj )Sj−1 −Nj . (5.6) C Nj j≥2

From Equations 4.5 and 5.1, πj = 1 − exp (β0 + β1 Cj−1 + β2 Qj−1 )(tsj − tsj−1 ) . Prior distributions The transmission rate parameters β0 , β1 and β2 are given uniform priors with lower bounds aβ0 , aβ1 , aβ2 and upper bounds bβ0 , bβ1 , bβ2 , respectively. The prior distributions are given by P (θ) ∝ 1aβ0