Characterizing an outbreak of vancomycin-resistant enterococci using hidden Markov models

*Author for c

Received 17 JAccepted 9 F

Characterizing an outbreak ofvancomycin-resistant enterococci using

hidden Markov models

E. S. McBryde1,*, A. N. Pettitt1, B. S. Cooper2 and D. L. S. McElwain1

1School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434,Brisbane, Queensland 4001, Australia

2Modelling and Bioinformatics Department, Centre for Infections Health Protection Agency,61 Colindale Avenue, London NW9 5EQ, UK

Background. Antibiotic-resistant nosocomial pathogens can arise in epidemic clusters orsporadically. Genotyping is commonly used to distinguish epidemic from sporadicvancomycin-resistant enterococci (VRE). We compare this to a statistical method todetermine the transmission characteristics of VRE.

Methods and findings. A structured continuous-time hidden Markov model (HMM) wasdeveloped. The hidden states were the number of VRE-colonized patients (both detected andundetected). The input for this study was weekly point-prevalence data; 157 weeks of VREprevalence. We estimated two parameters: one to quantify the cross-transmission of VREand the other to quantify the level of VRE colonization from sporadic sources. We comparedthe results to those obtained by concomitant genotyping and phenotyping.

We estimated that 89% of transmissions were due to ward cross-transmission while 11%were sporadic. Genotyping found that 90% had identical glycopeptide resistance genes and84% were identical or nearly identical on pulsed-field gel electrophoresis (PFGE).

There was some evidence, based on model selection criteria, that the cross-transmissionparameter changed throughout the study period. The model that allowed for a change intransmission just prior to the outbreak and again at the peak of the outbreak was superior toother models. This model estimated that cross-transmission increased at week 120 anddeclined after week 135, coinciding with environmental decontamination.

Significance. We found that HMMs can be applied to serial prevalence data to estimate thecharacteristics of acquisition of nosocomial pathogens and distinguish between epidemic andsporadic acquisition. This model was able to estimate transmission parameters despiteimperfect detection of the organism. The results of this model were validated against PFGEand glycopeptide resistance genotype data and produced very similar results. Additionally,HMMs can provide information about unobserved events such as undetected colonization.

Keywords: HMM; nosocomial pathogens; genotyping; statistical modelling; VRE

1. INTRODUCTION

There has been an alarming worldwide increase inthe rate of infection from vancomycin-resistantenterococci (VRE) in the last 15 years (Murray2006). Enterococci are part of the normal gastro-intestinal flora and VRE colonization often isasymptomatic and undetected. However, in patientswith compromised immune systems and breachedintegument, enterococci can become pathogenic,causing, for example, urinary tract infection, bacter-aemia and endocarditis. Large teaching hospitals andintensive care units (ICUs) have the highest rate ofinfection with VRE (Weinstein 2005). Infection withenterococci harbouring a vancomycin resistance geneis associated with higher mortality (Lodise et al.

orrespondence ([email protected]).

anuary 2007ebruary 2007 745

2002) and many strains of VRE are resistant to allknown antibiotics.

Acquisitions of VRE colonization can be groupedbroadly into those that come from cross-transmissionwithin the ward which we call transmitted, and VREthat comes from other sources which we call sporadic.Ward transmission of multi-resistant organisms isbelieved to be predominantly from patient to patientvia the transiently contaminated hands of health careworkers (Boyce 2001). The sources of sporadic VREinclude patients’ gastrointestinal tract, prior coloniza-tion with VRE and transmission from outside the ward.The presence of VRE on admission is often initially notdetected owing to infrequent swabbing, poor sensitivityof swabs or undetectable quantities of organism. VREmay exist in subdetectable numbers in human gut sothat exposure of patients to antibiotics which facilitateVRE growth (Donskey et al. 2002) may lead to an

J. R. Soc. Interface (2007) 4, 745–754

doi:10.1098/rsif.2007.0224

Published online 13 March 2007

This journal is q 2007 The Royal Society

746 Transmission model of VRE E. S. McBryde et al.

apparently new case of VRE. VRE is also known tospread from other hospital wards via patient and staffmovements (Trick et al. 1999).

To select the most appropriate infection controlinterventions, one needs to be able to estimate howmuch of the new acquisition is transmitted and howmuch is sporadic. Restricting antibiotic exposure isthought to control sporadic VRE, by reducing selectionpressure in patients’ endogenous flora, while handhygiene, cohorting, patient isolation and limitingadmission of colonized patients are thought to impacton transmitted VRE.

Outbreak investigation often involves time intensivemethods to characterize the mode of VRE acquisition.Genotyping techniques such as pulsed-field gel electro-phoresis (PFGE), distinguish clonal outbreaks, whichare presumed to be due to transmitted VRE, frommultiple new strain introductions, which are presumedto be due to sporadic VRE. There are occasions whenthis technique breaks down, when horizontal transfer ofthe resistance gene, vanA or vanB, can lead to severaldifferent genotypes being detected when in fact a singletransposon is being transmitted (Suppola et al. 1999;Bradley et al. 2002; Weinstein 2005).

Attempts have been made to distinguish betweenthese two processes of colonization based on statisticalanalysis of surveillance data. Pelupessy et al. (2002)used a Markov model, without hidden states, toestimate transmission parameters; finding estimateswere similar to those using full event data andgenotyping (PFGE). Cooper & Lipsitch (2004) usedstructured and unstructured hidden Markov models(HMMs) to describe infection incidence time-series dataand to estimate transmission parameters. Collinearitybetween parameter estimates, failure of convergenceand computational difficulties were identified aspotential problems using HMMs for sparse data suchas is typically found in time-series infection control data.Forrester & Pettitt (2005) compared background rateswith cross-transmission rates of methicillin-resistanceStaphylococcus aureus, finding background rates werelarger than cross-transmission rates.

Estimating transmission coefficients using hospitalinfection control data has a number of challenges. Thereare unobserved processes occurring; the time of newacquisition of colonization is not observed.Additionally,when relying on routine swabs to determine the numberof colonized patients, the sensitivity of swabs is lessthan 100%.

This study uses an epidemic model structure tocharacterize transmission of VRE during an outbreak atan 800 bed Australian teaching hospital. The currentpaper extends the work by Pelupessy et al. (2002) byestimating epidemiological parameters in the presence ofsuboptimal swab sensitivities. It also allows for the factthat new colonizations are not immediately detectable.We use an HMM structure to estimate transmission inthe face of incomplete datasets and unobserved events.This framework distinguishes between rates of trans-mitted and sporadic VRE acquisitions. This study alsoconsiders that the transmission rates may change overtime. Section 2.1 describes the data used to estimateVREepidemicdeterminants. Section 2.4describes themodel of

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VREtransmission,while§2.5describes theHMMand themethodology behind it. Section 3 gives the results of theparameter estimates, comparison of model estimates andgenotyping data and model selection.

2. METHODS

2.1. Description of outbreak and infectioncontrol interventions

VRE was first isolated at the Princess AlexandraHospital, Brisbane, Australia in October 1996 and aVRE screening programme commenced in January 1997,the beginning of the data collection period for this study.Data used in this study are VRE colonization data fromthe ICU, renal and infectious diseases units. VREcolonized patients and were identified by clinical isolates,weekly routine screening and contact tracing swabs.Infection control interventions introduced from the startof the study period were restriction of vancomycin andthird-generation cephalosporin use and isolation ofcolonized patients. Fromweek 125 of this study, infectioncontrol teams were aware of an increased prevalence ofVRE and further measures were taken. Dedicatedequipment was used in patient rooms and patients werecohorted. VRE patients requiring haemodialysis used adialysis facility within the infection control unit. Medicaland nursing staff wore disposable aprons and latex glovesfor patient contacts. An environmental audit wasperformed in August 1999, approximately week 135 ofthe study period and an aggressive cleaning programmewas instituted (Bartley et al. 2001).

2.2. Serial surveillance data used for statisticalanalysis

Input data for the statistical model in this study were:

—Weekly prevalence data for VRE colonization.—Mean length of stay of colonized patients: 15 days.

This was calculated as the time from first identifi-cation of colonization to discharge.

—The total number of beds in the wards, NZ68.

The data were collected from 1 January 1997 to 31December 1999. The weekly prevalence data are shownin figure 1.

2.3. Data used for cluster analysis

Microbiological and clinical data were collected, includ-ing admission dates and discharge dates of VREcolonized patients, as well as the date of first positiveisolate. Additionally, we had information on thecolonization status on admission of three of the patientstransferred from other hospitals. Genotype data, bothPFGE and glycopeptide resistance genotyping, werecompared with the results of the statistical analysis aspart of the study validation. Presumptive VRE colonieswere identified using standard techniques. Speciation(distinguishingEnterococcus faecium andEnterococcusfaecalis) was initially achieved by carbohydrate fermen-tation reactions of arabinose, mannose and raffinosethen confirmed by a multiplex PCR assay based on

50 100 1500

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10

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14

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week of study

week 120 week 135

prev

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ce o

f VR

E c

olon

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pat

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s

Figure 1. Prevalence data for VRE over 157 weeks. Arrows show times in which changes in transmission rates may havetaken place.

CN–C

mC

bC (N–C) +u(N–C)

Figure 2. The transmission of bacterial pathogens in thehospital ward.

Transmission model of VRE E. S. McBryde et al. 747

specific detection of genes encoding D-alanine: D-alanineligases (Bartley et al. 2001). VRE phenotype wasidentified based on vancomycin and teichoplanin meaninhibitory concentrations using theE -test method. Thispresumptively distinguishes vanA VRE, resistant toboth vancomycin and teichoplanin, from vanB VRE,resistant to vancomycin but sensitive to teichoplanin.This phenotype result was confirmed by glycopeptideresistance genotyping, achieved through a modifiedmultiplex PCR assay, described in detail by Bartleyet al. (2001).

In the study on this outbreak by Bartley et al.(2001), isolates were also characterized using PFGE.Electrophoretic band patterns were analysed accordingto the criteria established by Tenover et al. (1995).Computer comparison using GEL COMPAR v. 4.1(Applied Maths Kortrijk, Belgium) was based on thealgorithm of the unweighted pair group method forarithmetic averages and using the Dice coefficient with1.5% band tolerance (Bartley et al. 2001). Thisinformation was used to estimate the proportion ofisolates that were from the same strain.

2.4. Model of transmission

We based our ward transmission model on theSusceptible-infected model with migration, describedby Bailey (1975). Modified versions of this model havebeen used previously to analyse nosocomial trans-mission data (Pelupessy et al. 2002; Cooper & Lipsitch2004; Forrester & Pettitt 2005).

A schematic of themodel is shown in figure 2. The rateof cross-transmission of VRE colonization (per colonizedper susceptible patient per day) is denoted by b. It isassumed that the ward is of fixed size, N, hence thenumber of uncolonized patients is NKC. Colonizedpatients are assumed to remain colonized for their entirehospital stay, therefore, transition from colonized touncolonized occurs via discharge of a colonized patientand replacement with an uncolonized patient, whichoccurs at a rate m. Duration of stay of colonized patientswas available from the dataset. Acquisition of VRE, thatis transmitted, is described by the mass-action term,bC(NKC). VRE acquisition, that is sporadic, can arise

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through ward admission of a colonized patient or anyother process that is not related to the number ofcolonized patients, and occurs at a rate, n(NKC). Eachof the processes that lead to sporadic acquisition (forexample, prior colonization or colonization from out-of-ward sources, endogenous gastrointestinal coloniza-tion) can reasonably be assumed to be independent of thenumber of colonized patients in the ward.

The probability of a change in the number of colonizedpatients, C, in a short time period, h, is given by

Pr½CðtChÞZ iC1jCðtÞZ i�Z biðNKiÞhCnðNKiÞhCoðhÞ;

Pr½CðtChÞZ iK1jCðtÞZ i�ZmihCoðhÞ;Pr½CðtChÞZ ijCðtÞZ i�

Z 1KbiðNKiÞhKnðNKiÞhKmihCoðhÞ;Pr½CðtChÞZ jðjsiK1; i; iC1ÞjCðtÞZ i�Z oðhÞ:

ð2:1ÞThe number of colonized patients in the ward, C(t),forms a Markov process on state space 0, ., N, whereN is the number of patients on the ward. Reflectingboundaries occur at states iZ0 and iZN, providednO0, otherwise 0 is an absorbing state, and providedmO0, otherwise N is an absorbing state.

2.5. Hidden Markov model

We aim to estimate parameters associated withsporadic colonization, n, and the colonization causedby ward transmission, b, using the structured HMMillustrated in figure 3.

Table 1. Parameters used in the model. Fitted values arediscussed in §3.

parameter symbol value source

number ofpatients

N 68 directly fromdataset

removal rateof colonizedpatient

m 1/15 dayK1 directly fromdataset

transmissionrate

b 1.0!10K3 fitted usingHMM

sporadicacquisitionrate

n 2.0!10K4 fitted usingHMM

detectionprobability

D 0.58–0.97 literaturereview

C1

Y1

C2

Y2

C4

Y4

C3

Y3

Figure 3. Hidden Markov model. Here, C represents thenumber of colonized patients in the ward (detected orundetected), Y represents the number of patients detectedat each time point. The horizontal arrows represent thetransition from one state to the next, and the vertical arrowsrepresent the relationship between the hidden state and thecorresponding observation.

748 Transmission model of VRE E. S. McBryde et al.

Our HMM consists of: observations, Y, the numberof patients detected at each time point; underlyinghidden states, C, the number of colonized patients inthe ward; a transition model linking each hidden statewith its adjacent states, represented by horizontal linesin figure 3; an observation model linking the data withthe hidden state, represented by the vertical lines infigure 3. There is one hidden state for each observation,denoted C1, C2,., Cn.

The full conditional probability of any node dependsonly on neighbouring nodes to which it is connecteddirectly. The observation component of the HMM,denoted by Y, consists of 157 data inputs of weeklyVREprevalence taken over 3 years and the vector of timepoints, tZt1,., tn, corresponding to each observationtime. The vectorC consists of the nZ157 hidden states.The transitionprobabilitymatrix, giving the relationshipbetween the hidden states, is described in appendix A.The observation model, giving the relationship betweenthe observed and hidden states, is described in §2.6.

The parameters used in themodel are given in table 1.

2.5.1. Model assumptions. The model makes thefollowing assumptions

(i) The ward is of fixed size, N.(ii) The model parameters are time invariant (this

assumption is relaxed later in the study).(iii) Each colonized patient remains colonized for

the remainder of their stay.(iv) Each observation of patients not known to be

colonized is conditionally independent giventhe corresponding hidden state.

(v) The hidden states follow a first order timehomogenous Markov process, that is

PrðCðtkÞjCðt1Þ;.;CðtkK1ÞÞZPrðCðtkÞjCðtkK1ÞÞZPrðCðtkK tkK1ÞjCð0ÞÞ:

(vi) Homogenous mixing of patients takes place.(vii) Uncolonized patients are identical with respect

to susceptibility to colonization.(viii) Colonized patients are identical with respect to

transmission of VRE.(ix) Time from colonization to discharge is expo-

nentially distributed. Review of patienthistories confirms that this is approximatelythe case.

These assumptions are discussed in §4.

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2.6. Observation model

The probability of being known to be colonized (andtherefore being included in the prevalence data) giventhat a patient is colonized, d, was unknown. Literaturesources regarding the sensitivity of rectal swabs indetecting VRE were used to develop an expression forthe uncertainty in this parameter. Estimates of thesensitivity of a rectal swab for VRE range from 0.58(D’Agata et al. 2002) to 0.97 (Reisner et al. 2000) withvalues in between (Lemmen et al. 2001; Trick et al.2004). We allowed for the uncertainty regarding thedetection by assigning a uniform [0.58, 0.97] priordistribution to d. The probability of detection at agiven prevalence check, d, used in this study waspatient related rather than simply swab related. If apatient was known from previous swabs to be colonized,the patient was automatically detected, thus dwould beexpected to be at least as high as the sensitivity of asingle swab. The observation model assumed that eachweek’s observed prevalence is independent of theprevious week’s observed prevalence, given the under-lying true prevalence. This is an approximation as thetrue detection is the known colonized patients fromthe previous week and the new colonizations from thecurrent week.

The probability relationship between the states andthe data is described by the binomial distributionYkwBin(Ck, d ), where Yk is the k th observed coloniza-tion prevalence and Ck is the actual number ofcolonized patients, the hidden state, at time tk. Thisassumes that the probability, d, remains constant overthe study period (for each iteration) and the probabilityof detection of each colonized patient is independent ofthe number of other colonized patients.

Alternative observation models with greater dis-persion could have been used. For example, the Poissonor negative binomial distribution could have beenchosen, had we been dealing with incidence ratherthan prevalence data. We chose the binomial distri-bution because it has a sound probabilistic basis(assuming fixed detection) and, unlike the Poisson,ensures that the hidden state (number colonized) isalways larger than the observation (number detected),a necessary result when using prevalence data.

Transmission model of VRE E. S. McBryde et al. 749

2.7. Bayesian framework

The parameters for transmitted VRE, b and sporadicVRE, n were estimated using a Bayesian framework.Let qpZ{b, n, d} be the vector of model parameters.Baum et al.’s (1970) recursion formula, summarized inappendix B, was used to determine the likelihood of thedata, l(Y jqp). UniformU [0, 0.1] priors were assigned tob and n, because little was known about theseparameters other than that negative values or valueshigher than 0.1 were completely implausible. Theposterior probability distributions

PrðqpjY ÞfpðqÞlðY jqpÞ; ð2:2Þwere estimated using a Monte-Carlo Markov chainalgorithm, described in appendix C.

The Bayesian framework can provide estimates (andfull posterior probability density) of any function ofmodel parameters including functions which dependupon knowledge of hidden states. Let qh be the vector ofn inferred hidden states C1,., Cn and let qZ{qp, qh}.The expected number of within-ward transmissionsfor the week, following week k is bCk(NKCk), whilethe total number of transmissions is bCk(NKCk)Cn(NKCk). The expected proportion of VRE acqui-sitions due to ward transmission over the time of thestudy, f(q), is approximated by

f ðqÞZ

PnkZ1

bCkðNKCkÞ

PNkZ1

bCkðNKCkÞCnðNKCkÞ: ð2:3Þ

We evaluate the expectation, E [ f (q)jY ], by drawingsamples qk, kZ1,., m from p(qjY ) and using theapproximation of Gilks et al. (1996, ch. 1)

E½ f ðqÞjY �z 1

m

XmkZ1

f ðqkÞ: ð2:4Þ

The algorithm for this Monte-Carlo integration isgiven in appendix C.

2.8. Comparison of cluster analysis results usinggenotyping with statistical analysis

A genotyping study was performed on the VRE isolatesby Bartley et al. (2001). Of the 49 isolates available foranalysis, 44 were found to be E. faecium vanA usingglycopeptide resistance genotyping. The estimatednumber of isolates having identical or closely relatedpatterns on PFGE using the criteria of Tenover et al.(1995) was 41 of 49.

2.8.1. Cluster analysis based on genotypic relatedness.We compared the proportion of ‘identical isolates’(presumed to be part of a cluster) with the estimatedproportion of transmitted VRE derived from the HMMand prevalence data. The posterior probability distri-bution of the proportion of VRE cases that are identicalby genotype can readily be derived using a Bayesianframework and conjugate prior distribution (Gelmanet al. 2004). Denote the parameter of interest, theproportion of VRE acquisitions that are identical, by p.Assume the form Beta(1, 1) for the prior distribution

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for the proportion; this is the same as the uniform[0, 1]prior. The probability of the data is given by thebinomial Bin(a; (aCb), p), where a is the number ofidentical isolates and b is the number of non-identicalisolates, as detected by the laboratory methods. Theposterior probability density of p is Beta(1Ca, 1Cb).

3. RESULTS

3.1. Transmission parameter estimation

Theestimatedvalue for the transmissioncoefficient,bwas10!10K4 (CI95 7.9!10K4, 13!10K4) and the sporadicacquisition rate n was 2.0!10K4 (CI95 0.85!10K4,3.8!10K4). The coefficient of correlation between b

and n was estimated to be K0.24. These results wereobtained using a Markov chain Monte-Carlo algo-rithm with a burn-in period of 50 000 as described inappendix C.

The basic reproduction ratio, R0, is ‘the averagenumber of persons directly infected by an infectiouscase during its entire infectious period, after entering atotally susceptible population’ (Giesecke 1994). In thismodel it can be shown to be R0ZbN/m. This formulaforR0 is an approximation as there is a finite populationin this setting. The basic reproduction ratio isestimated to be 1.07 (CI95 0.78–1.34).

The mean value for the estimated detection rate, d,was 0.75 with a 95% credible interval of 0.59–0.93.

3.2. Comparison of statistical model andgenotyping data

The proportion of VRE acquisitions due to trans-mission, was estimated to be 89% (CI95Z78–95%),using Bayesian inference applied to the HMMstructure. This compares with 84% (41/49) of isolatesobserved to be identical or nearly identical using PFGEgenotyping and 90% (44/49) using glycopeptide resist-ance genotyping. The posterior distribution of theestimated proportion of colonizations due to wardtransmission compared with those found to be identicalby glycopeptide resistance genotype and PFGEmethods are displayed in figure 4.

3.3. Sensitivity analysis

The length of stay following colonization could begreater than the estimated 15 days because acquisitioncould have preceded detection. Conversely, the lengthof stay could have been less than 15 days becauseundetected colonized patients are likely to have shorterstays than detected colonized patients. We thereforeperformed a sensitivity analysis on the discharge rateparameter, m. We took upper and lower values for m

which we believed at the extreme ends of plausibility.We then repeated the estimation of the proportion ofVRE acquisitions due to within-ward transmission.Results are given in table 2.

Table 2 shows there is a small change in the estimatefor large changes in the discharge parameter, m.

0.2 0.4 0.6 0.8 1.00

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proportion of colonization due to cross-transmission

post

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PFGE dataglycopeptideresistance data

Figure 4. Posterior distribution of proportion of VREacquisitions that are due to ward transmission. The histogramgives the posterior distribution from the Bayesian analysis ofthe HMM, the solid curve gives the posterior distributionbased on the observed proportion of identical strains usingPFGE genotype data and the broken line gives the posteriordistribution based on observed proportion of identical strainsusing glycopeptide resistance phenotype and genotype data(Bartley et al. 2001).

Table 2. Analysis of sensitivity of the model outcome tochanges in the discharge rate, m.

m estimated proportion (%)

1/10 91.41/15 891/20 86.5

750 Transmission model of VRE E. S. McBryde et al.

3.4. Model selection

The values of the deviance information criterion (DIC)were used to assess the optimum model to fit the data(Gelman et al. 2004). Results are given in table 3.

Several models were explored. Setting either b or n tozero led to much higher values for the DIC, givingsubstantial statistical support to amixedmodel, inwhichVRE colonization arose both from cross-transmission intheward and sporadically. Themodel inwhich b changedafter week 120 was a superior fit to the model with time-invariant parameters. Allowing for a further change in b

after week 135 provided the best fit of those modelsinvestigated. The effective number of parameters in alatent variable model depends on the collinearity of theparameters and the influence of the latent variables.

4. DISCUSSION

The aim of this study was to characterize transmissionof VRE using statistical methods and simple serialsurveillance data. We included a term for sporadiccolonization because we believe that new acquisitions ofVRE could occur through means other than within-ward patient to patient cross-transmission. Sources ofsporadic colonization have been labelled in the past asendogenous, spontaneous (Pelupessy et al. 2002) or

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background (Forrester & Pettitt 2005). Our statisticalmethods were designed to quantify the rates of sporadicand cross-transmitted VRE. Previous attempts haveencountered difficulties especially with identifiability ofvariables (Cooper & Lipsitch 2004).

Full patient histories, PFGE and glycopeptideresistance genotype data were used for validation butwere not included in the statistical analysis in thisstudy. Estimates of the proportion of VRE resultingfrom cross-transmission based on statistical methods(HMMs) in this study were very similar to those basedon vancomycin resistance genotype data.

The proportion of clustered isolates based on PFGEanalysis was lower than both the vancomycin resistancegenotype data and the statistical analysis. This couldbe due to horizontal transfer of resistance gene to newstrains of enterococci, which has been reported pre-viously (Suppola et al. 1999; Weinstein 2005). Ifhorizontal transfer of resistance genes occurs duringan outbreak, cross-transmitted strains have identicalglycopeptide resistance genotypes but different PFGEpatterns, hence PFGE underestimates clustering.

Using a structured HMM, one can estimate thehidden states behind the data, the number of patientscolonized on the ward (both detected and undetected).We estimated the basic reproduction ratio to be close tounity, the threshold value that could lead to endemicVRE. We were able to make estimates of transmissionin the face of imperfect datasets in which transmissiontimes and patient histories were unknown and swabsensitivity was considerably less than 100%. Thisapproach is similar to that of Cooper & Lipsitch(2004), who observed monthly infection incidence andassumed a Poisson relationship with the numbercolonized, the hidden state. The current study avoidsthe ambiguity of the relationship between the obser-vations and the hidden state using prevalence (observednumber detected at time points) which relates directlyto the hidden state, the number colonized, through abinomial relationship.

For simplicity, this study assumed homogenousmixing of staff and patients. Future studies couldextend this model to include ward coupling, however,dividing the data to incorporate ward structure wouldlead to reduced precision in parameter estimates andincreased model complexity. We incorporated thisuncertainty into our parameter estimates and modelconclusions were robust to changes in its value.

The time to discharge was estimated by taking themean time from first identification of colonized patientsto discharge (15 days). The discharge rate was taken asthe reciprocal of the mean time to colonization. Thisassumes that the time to discharge was exponentiallydistributed which was indeed approximately the casefor those known to be colonized. The true time todischarge of colonized patients could have been longerthan estimated in this study if patients were colonizedfor significant time-intervals prior to detection or theycould have been shorter if a substantial number of theundetected colonized patients had shorter durations ofstay. We performed a sensitivity analysis on thedischarge rate parameter, m, and found large changesin m (G33%) resulted in small changes in the estimates

Table 3. Comparison of different models using the deviance information criterion. Pd : effective number of parameters.

model estimate of b (95% CI)!10K4 estimate of n (95% CI)!10K4 DIC Pd

one value for n and three values for b withchange points at the end of week 120 and 135

b13.4(0.28–8.8) 2.2(0.96–4.0) 251 4.0b215.3(13.5–17.1)b310.9(7.1–13.0)

one value for n and two values for b withchange point at the end of week 120

b13.4(0.28–8.7) 2.2(0.96–4.0) 253 2.3b211.9(10.2–13.5)

one value for n and one value for b 10(7.9–13) 2.0(0.85–3.8) 261 2.6one value for n and two values for b with

change point at the end of week 135b111(7.6–14.6) 2.0(0.88–3.7) 261 2.6b29.6(7.9–11.4)

bZ0 and one value for n 0 9.7(7.7–11.7) 393 1.2nZ0 and one value for b 8.7(6.9–10.1) 0 531 1.5

Transmission model of VRE E. S. McBryde et al. 751

of proportion of patients colonized within the ward andis therefore unlikely to have influenced the conclusionsof this study.

The model presented in this study postulated thatVRE acquisition arose from both cross-transmissionand sporadic sources. Model comparison techniquesfound this model to be a far superior fit to the datacompared with models which relied on either cross-transmission or sporadic sources of VRE acquisitionalone, strongly supporting that both modes of acqui-sition were taking place.

We investigated changes in transmission over timeusing a structured epidemic model. Model comparisonshowed that there was evidence supporting theconclusion that there was an increase in cross-trans-mission just prior to the outbreak. There was limitedevidence that the cross-transmission rate reduced afterthe epidemic peak at week 135, coinciding with theenvironmental cleaning intervention. Future studiesusing larger surveillance datasets could extend themethodology presented to consider more models inwhich parameters are time-dependent. One approachto this would be to use the reversible jumpMonte-CarloMarkov chain method (Green 1995) or the birth–deathMarkov process model (Stephens 2000).

Inaccuracies in PFGE cluster analysis can arise fromthe horizontal transfer of resistance genes. Glycopep-tide resistance genotype analyses are not subject toinaccuracies due to gene transfer but cannot distinguishdifferent strains that might all be of the same resistancegenotype. Statistical methods are not subject to theseproblems and have the additional advantage that theyare not resource intensive. It is interesting to speculatewhether they also have the potential to be used in realtime, within a control-chart outbreak alert system.

The model presented here can be used to model thetransmission of other bacterial pathogens in small scalesettings of healthcare institutions, such as methicillin-resistant Staphylococcus aureus, extended spectrumbeta-lactamase producing and other multi-resistantGram-negative pathogens.

This work was partially supported by a grant under theAustralian Research Council Linkage Scheme (LP0347112)and NHMRC scholarship number 290541. The authorswould like to thank Dr Mike Whitby for providing dataand Dr Paul Bartley for helpful comments. The authorswould like to thank the anonymous reviewers for theirconstructive comments.

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APPENDIX A. CONSTRUCTING A TRANSITIONPROBABILITY MATRIX

Following the theory of Cox & Miller (1965), wedeveloped a transition probability matrix, GðtkKtkK1Þ.The ij th element of GðtkKtkK1Þ gives the probabilityof having j colonized patients on the ward at time tk,given that there were i colonized patients on the wardat time tkK1.

To construct the transition probability matrix for anarbitrary time-interval, first we developed a discretetime transition probability matrix, A, for a small time-interval, h. Let A be the matrix in which the ij thelement is given by Pr(C(tCh)ZjjC(t)Zi ). A is givenusing the system of equation (2.1). Here, i and j are thenumber of patients colonized in the ward and can takeon values 0, ., N.

Let p(t) be the (NC1) vector of probabilities of thenumber colonized at time t. The generator matrix,G isa square, (NC1)!(NC1), matrix that has theproperty that

dpðtÞdt

ZGpðtÞ: ðA 1Þ

The ij th element of the generator matrix, G, isthe instantaneous rate of change of probability ofbeing in state j, given a beginning in state i. Then Gis given by

GZ limh/0

1

hðAKI Þ; ðA 2Þ

where I is the identity matrix.Following from expression (A 2), we have

pðtkC1ÞZpðtkÞeðtkC1KtkÞG; ðA 3Þ

in general. Specifically, after a time-interval tk–tkK1, theprobability of being in state j having begun in state i isthe ij th element of the transition probability matrix,given by

GðtkKtkK1Þij ZPrðCk Z jjCkK1Z iÞZðeðtkKtkK1ÞGÞij : ðA4Þ

Cox & Miller (1965, ch. 4.5) and MacDonald &Zucchini (1997) give an expanded explanation. The

matrix exponential eðtkKtkK1ÞG was calculated using theMATLAB ‘expm’ function.

752 Transmission model of VRE E. S. McBryde et al.

APPENDIX B. LIKELIHOOD COMPUTATION

The probability of the full dataset and a particularsequence of hidden states, C1, C2,., Cn is given by

PrðY1;.;Yn;C1;.;Cnjb; nÞ

ZPrðC1ÞPrðY1jC1ÞYnkZ2

GCkK1CkPrðYk jCkÞ; ðB1Þ

with GCkK1Ckas defined in appendix A.

The likelihood calculation of this single permutationof hidden states requires 2n computations even after thematrix exponential has been evaluated. The fulllikelihood of the data over all the states is

PrðY1;.;Ynjb;nÞ

ZXNC1

C1Z1

;.;XNC1

CnZ1

PrðY1;.;Yn;C1;.;Cnjb;nÞ; ðB2Þ

which requires 2n(NC1)n computations for onelikelihood evaluation (Le Strat & Carrat 1999). Thisintractable calculation (with nZ157 and NZ68) can besimplified using Baum’s recursion technique (Baumet al. 1970) as shown below.

The forward recursion involves simplifying thelikelihood computations by considering a partialobservation sequence and a single state sequence. Letfk(i ) be the probability of the partial observationsequence (Y1, Y2,., Yk) produced by all possiblestate sequences that end in state i. The probability isgiven by

fkðiÞZLðY1;.;Yk ;Ck Z ijn; bÞ; k%n: ðB 3ÞLet d be the (size NC1) vector of probabilities of the

first state, (diZPr(C1Zi )). In the forward recursionmethod of likelihood computation, the value of d needsto be determined in the absence of data. The stationarydistribution of the transition matrix can be used for this(MacDonald & Zucchini 1997). The probability of thefirst state and first observation, Y1, is given by

f1ðiÞZ diPrðY1jC1 Z iÞ: ðB 4ÞThe forward recursion formula is then applied. We

multiply every state probability, fkK1(i ), by thetransition probability Gij and by the probability of thek th data point, given the hidden state j. This results ina vector of probabilities which is then summed todetermine fk( j ). Thus, the probability of subsequentstates is given by

fkðjÞZXNiZ0

fkK1ðiÞGij

" #PrðYk jCk Z jÞ: ðB 5Þ

At each step in the forward recursion, the procedurecan be terminated and the probability of the partialobservation sequence is determined by

PrðY1;.;Yk jn; bÞZXNiZ0

fkðiÞ: ðB 6Þ

The likelihood of the data can then be determined by

PrðY1;.;Ynjb; nÞZXNiZ0

fnðiÞ: ðB 7Þ

See Petrushin (2000) for a detailed discussion of theforward and backward recursion formulae.

J. R. Soc. Interface (2007)

APPENDIX C. MONTE-CARLO MARKOV CHAINALGORITHM

The algorithm for this Monte-Carlo integration used toestimate the proportion of VRE acquisitions due toward cross-transmission, f(q), is given below.

The MCMC algorithm has the following steps:

(i) Assume the prior probability for b and n, to be(U [0, 0.1]). These priors were used as littleprior information was known except thatnegative values and values greater than 0.1are completely implausible.

(ii) Initialize b, n and d. Different initial valueswere chosen from (bZ10K5 to bZ10K2, fromnZ10K5 to nZ10K2 ) and from dZ0.58 todZ0.97.

(iii) Assign the prior probability of the hiddenstates. A discrete uniform distribution on(0, ., N ) was used.

(iv) Initialize each hidden state using its corre-sponding observation and the (binomial) obser-vation model YkwBin(Ck, d ).

(v) Determine the probability of the data andsequence of hidden states using equation (B 1).

(vi) Propose a new b 0 using a simple random walk,the step size wN(0, 10K4).

(vii) Accept b 0 using a Metropolis–Hastings stepwith the acceptance probability

a Zmin 1;pðb0ÞPrðY ;C jb0Þqðb0/bÞpðbÞPrðY ;C jbÞqðb/b0Þ

� �;

ðC 1Þ

where q(b/b0) is the proposal probability forb 0 from b which is the normal density for b 0

with mean b and variance 10K4.(viii) Repeat for n 0 and d 0.(ix) Update each hidden state using a Gibbs

update, drawing from the distributions givenby the conditional probability of the states,determined by neighbouring states and obser-vations, as described below.

(x) Determine f(q) for the particular sequence ofhidden states and parameters b and n usingexpression (A 3).

(xi) Iterate by returning to step (iv).(xii) Burn-inusing50 000 iterations.Use the following

50 000 updates to estimate the posterior prob-ability distribution (using the ergodic average) ofthe hidden states (C1,., Cn) and f(q).

(xiii) Repeat steps 2–12 to construct 10 such Markovchains each with different initial values.Convergence tests showed that 50 000 updateswere sufficient to get precise estimates of theparameters (R̂Z1:02 for estimates of logit(proportion)) (Gelman et al. 2004, ch. 11.6).

(xiv) Use 10!50 000 updates to determine theposterior probability densities of the modelparameters.

The Gibbs update involves determining theconditional probability of the hidden states (giveneverything else). The assumption that the hidden states

Transmission model of VRE E. S. McBryde et al. 753

are a first order Markov process means that theconditional probability of the hidden states is basedonly on neighbouring states and the correspondingdatum. The full conditional probability of the hiddenstate, Cn(kZ2,., nK1), is given by

PrðCk Z ijCnk ;yÞfPrðCkC1 Z jjCk Z iÞ!PrðCk Z ijCkK1 Z hÞPrðYk jCk Z iÞ;

ðC2Þ

where C\k is the set of all states other than Ck; and i isthe proposed value of the k th hidden state; and h and jare the current values of the hidden states kK1 andkC1, respectively.

The first and last states depend only on a singleneighbour and the dataassociatedwith that state.That is

PrðC1ZijCn1;Y ÞfPrðC2ZjjC1ZiÞPrðY1jC1ZiÞ;ðC3Þ

andPrðCnZijCnn;Y ÞfPrðCnZijCnK1ZhÞPrðYnjCnZiÞ:

ðC4ÞThe conditional probability of the states can be

determined and this becomes the sampling distributionfor thehidden state.Eachof then states canbeupdated ina forward, backward or random manner. To estimatevalues of n and b, we do not need to infer hidden states.The simplifiedMCMC algorithm has the following steps:

(i) Assign the prior probability for b and n using(U [0, 0.1]).

(ii) Initialize b, n and d.(iii) Determine the likelihood of the data usingBaum’s

recursion formula.(iv) Propose a new b0 using a simple randomwalk, the

step sizewN(0, 0.0001).(v) Accept b0 using a Metropolis–Hastings step with

the acceptance probability

aZmin 1;pðb0ÞlðY jb0Þqðb0/bÞpðbÞlðY jbÞqðb/b0Þ

� �: ðC5Þ

(vi) Repeat for n and d.(vii) Iterate as above.

REFERENCES

Bailey, N. 1975 The biomathematics of malaria. London, UK:Charles Griffin.

Bartley, P. B., Schooneveldt, J. M., Looke, D. F., Morton, A.,Johnson, D. W. & Nimmo, G. R. 2001 The relationship of aclonal outbreak of Enterococcus faecium VanA to methi-cillin-resistant Staphylococcus aureus incidence in anAustralian hospital. J. Hosp. Infect. 48, 43–54. (doi:10.1053/jhin.2000.0915)

Baum, L., Petrie, T., Soules, G. & Weiss, N. 1970 Amaximisation technique occurring in the statistical analysisof probabilistic functions of Markov chains. Ann. Math.Stat. 41, 164–171.

Boyce, J. M. 2001 MRSA patients: proven methods to treatcolonization and infection. J. Hosp. Infect. 48(Suppl. A),S9–S14. (doi:10.1016/S0195-6701(01)90005-2)

Bradley, S. J., Kaufmann, M. E., Happy, C., Ghori, S., Wilson,A. L. & Scott, G. M. 2002 The epidemiology of glycopep-tide-resistant enterococci on a haematology unit: analysisby pulsed-field gel electrophoresis. Epidemiol. Infect. 129,57–64. (doi:10.1017/S0950268802007033)

J. R. Soc. Interface (2007)

Cooper, B. & Lipsitch, M. 2004 The analysis of hospitalinfection data using Hidden Markov models. Biostatistics 5,223–237. (doi:10.1093/biostatistics/5.2.223)

Cox, D. R. & Miller, H. D. 1965 The theory of stochasticprocesses. London, UK: Methuen.

D’Agata, E. M., Gautam, S., Green,W. K. &Tang, Y.W. 2002High rate of false-negative results of the rectal swab culturemethod in detection of gastrointestinal colonization withvancomycin-resistant enterococci. Clin. Infect. Dis. 34,167–172. (doi:10.1086/338234)

Donskey, C. J., Hoyen, C. K., Das, S. M., Helfand, M. S. &Hecker, M. T. 2002 Recurrence of vancomycin-resistantenterococcus stool colonization during antibiotic therapy.Infect. Control Hosp. Epidemiol. 23, 436–440. (doi:10.1086/502081)

Forrester, M. & Pettitt, A. N. 2005 Use of stochastic epidemicmodeling to quantify transmission rates of colonization withmethicillin-resistant Staphylococcus aureus in an intensivecare unit. Infect. Control Hosp. Epidemiol. 26, 598–606.(doi:10.1086/502588)

Gelman, A., Carlin, J., Stern, H. & Rubin, D. B. 2004Bayesiandata analysis 2nd edn., p. Fla. Boca Raton, FL: Chapmanand Hall/CRC.

Giesecke, J. 1994 Modern infectious disease epidemiology.London, UK: Edward Arnold.

Gilks, W., Richardson, S. & Spiegelhalter, D. 1996 MarkovChain Monte Carlo in practice. London, UK: Chapmanand Hall.

Green, P. 1995 Reversible jump Markov Chain MonteCarlo computation and Bayesian Model determination.Biometrika 82, 711–732. (doi:10.1093/biomet/82.4.711)

Le Strat, Y. & Carrat, F. 1999 Monitoring epidemiologicsurveillance data using hidden Markov models. Stat. Med.18, 3463–3478. (doi:10.1002/(SICI)1097-0258(19991230)18:24!3463::AID-SIM409O3.0.CO;2-I)

Lemmen, S. W., Hafner, H., Zolldann, D., Amedick, G. &Lutticken, R. 2001 Comparison of two sampling methodsfor the detection of Gram-positive and Gram-negativebacteria in the environment: moistened swabs versus rodacplates. Int. J. Hyg. Environ. Health 203, 245–248. (doi:10.1078/S1438-4639(04)70035-8)

Lodise, T. P., McKinnon, P. S., Tam, V. H. & Rybak, M. J.2002 Clinical outcomes for patients with bacteremia causedby vancomycin-resistant enterococcus in a level 1 traumacenter. Clin. Infect. Dis. 34, 922–929. (doi:10.1086/339211)

MacDonald, I. & Zucchini, W. 1997 Hidden Markov models fordiscrete valued time series. London,UK: Chapman andHall.

Murray, B. 2006 Overview of enterococci. In UpToDate (ed.B. D. Rose). Massachusetts, MA: UpToDate. (CD-rom.)

Pelupessy, I., Bonten, M. J. & Diekmann, O. 2002 How toassess the relative importance of different colonization routesof pathogens within hospital settings. Proc. Natl Acad. Sci.USA 99, 5601–5605. (doi:10.1073/pnas.082412899)

Petrushin, V. 2000 Hidden Markov models: fundamentals andapplications. Part2 Discrete and continuous hiddenMarkovmodels. In Online Symp. for Electronics Engineers. http://www.techonline.com/osee/.

Reisner, B. S., Shaw, S., Huber, M. E., Woodmansee, C. E.,Costa, S., Falk, P. S. & Mayhall, C. G. 2000 Comparison ofthree methods to recover vancomycin-resistant enterococci(VRE) from perianal and environmental samples collectedduring a hospital outbreak of VRE. Infect. Control Hosp.Epidemiol. 21, 775–779. (doi:10.1086/501734)

Stephens, M. 2000 Bayesian analysis of mixture models withan unknown number of components-an alternative toreversible jump methods. Ann. Stat. 28, 40–74. (doi:10.1214/aos/1016120364)

Suppola, J. P., Kolho, E., Salmenlinna, S., Tarkka, E., Vuopio-Varkila, J. & Vaara, M. 1999 vanA and vanB incorporate

754 Transmission model of VRE E. S. McBryde et al.

into an endemic ampicillin-resistant vancomycin-sensitiveEnterococcus faecium strain: effect on interpretation ofclonality. J. Clin. Microbiol. 37, 3934–3939.

Tenover, F., Arbeit, R., Goering, R., Mickelsen, P., Murray,B., Persing, D. & Swaminathan, B. 1995 Interpretingchromosomal DNA restriction patterns produced by pulsedfield gel electrophoresis: criteria for bacterial strain typing.J. Clin. Microbiol. 33, 2233–2239.

Trick, W. E., Kuehnert, M. J., Quirk, S. B., Arduino, M. J.,Aguero, S. M., Carson, L. A., Hill, B. C., Banerjee, S. N. &Jarvis, W. R. 1999 Regional dissemination of vancomycin-resistant enterococci resulting from interfacility transfer of

J. R. Soc. Interface (2007)

colonized patients. J. Infect. Dis. 180, 391–396. (doi:10.1086/314898)

Trick, W. E., Paule, S. M., Cunningham, S., Cordell, R. L.,Lankford, M., Stosor, V., Solomon, S. L. & Peterson, L. R.2004 Detection of vancomycin-resistant enterococci beforeand after antimicrobial therapy: use of conventional cultureand polymerase chain reaction. Clin. Infect. Dis. 38,780–786. (doi:10.1086/381552)

Weinstein, J. 2005 Hospital-acquired (nosocomial) infectionswith vancomycin-resistant enterococci. In UpToDate (ed.B. D. Rose), p. 2006. Massachusetts, MA: UpToDate.(CD-rom.)