Biostatistics Advance Access originally published online on August 22, 2006
Biostatistics 2007 8(2):383-401; doi:10.1093/biostatistics/kxl017
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Bayesian inference of hospital-acquired infectious diseases and control measures given imperfect surveillance data
Queensland University of Technology, Brisbane QLD, Australia a.pettitt{at}qut.edu.au
Heriot-Watt University, Edinburgh, United Kingdom
* To whom correspondence should be addressed.
 |
SUMMARY |
|---|
 TOP SUMMARY 1. INTRODUCTION2. MODEL, DATA, AND...3. MCMC ALGORITHM AND...4. APPLICATION: MRSA...5. DISCUSSIONREFERENCES |
|---|
This paper describes a stochastic epidemic model developed to infer transmission rates of asymptomatic communicable pathogens within a hospital ward. Inference is complicated by partial observation of the epidemic process and dependencies within the data. The epidemic process of nosocomial communicable pathogens can be partially observed by routine swabs testing for the presence of the pathogen. False-negative swab results must be accounted for and make it difficult to ascertain the number of patients who were colonized. Reversible jump Markov chain Monte Carlo methods are used within a Bayesian framework to make inferences about the colonization rates and unknown colonization times. The methods are applied to routinely collected data concerning methicillin-resistant Staphylococcus Aureus in an intensive care unit to estimate the effectiveness of isolation on reducing transmission of the bacterium.
Keywords: Bayesian inference; False negatives; Imperfect detectability; Infectious diseases; Markov chain Monte Carlo methods; MRSA; Reversible jump methods; Screening; Sensitivity; Staphylococcus; Stochastic epidemic models
|
1. INTRODUCTION |
|---|
|
TOPSUMMARY1. INTRODUCTION2. MODEL, DATA, AND...3. MCMC ALGORITHM AND...4. APPLICATION: MRSA...5. DISCUSSIONREFERENCES |
|---|
The emergence of multidrug-resistant nosocomial pathogens such as methicillin-resistant Staphylococcus Aureus (MRSA) and vancomycin-resistant enterococci (VRE) has emphasized the importance of transmission prevention within hospitals. Typically, simulation studies, e.g. Sébille and others (1997)
, Austin and others (1999), Lipsitch and others (2000), are used to investigate the effectiveness of widely accepted infection control procedures. Such studies are limited by the assumptions made about model parameters and the associated lack of knowledge and uncertainty. In particular, their findings are largely influenced by assumptions regarding the transmission rate parameters (Cooper and others, 1999).
Estimation of transmission parameters is complicated because the epidemic process can only be partially observed (Becker and Britton, 1999). 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. Dependencies within the epidemic process further complicate the task of transmission parameter estimation. Dependencies arise because the risk of acquisition depends on the number of others who are colonized.
Jernigan and others (1996) gave transmission rate estimates for isolated and non-isolated patients in a neonatal intensive care unit (ICU). The analysis required the correct identification of sources and times of transmissions, which is rarely feasible. Furthermore, it did not consider that different realizations of the same process could have occurred; that is, it did not consider the element of randomness. In small populations such as those in an ICU, significant fluctuations in the incidence and prevalence of colonization and infection will occur by chance and therefore a stochastic analysis should be undertaken (Bonten and others, 2001; Renshaw, 1999).
Pelupessy and others (2002) proposed a Markov model to allow for a stochastic analysis of routine hospital surveillance data. Using maximum likelihood techniques, the transmission rates of VRE colonization were estimated. The model assumed a sequence of surveillance swabs capable of detecting carriage with certainty. The model formed a basis of the hidden Markov model proposed by Cooper and Lipsitch (2004) where colonizations were not observed but inferred by the number of patients being infected, rather than colonized. The underlying Markov model described the number of patients harboring the organism. The observed number of infections was assumed to follow a Poisson distribution conditional on the unknown number of patients harboring the organism.
In an earlier paper (Forrester and Pettitt, 2005), an interval-censored approach was used to estimate the transmission rate of MRSA within an ICU. The number of patients detected with MRSA in a given swab interval was described as a binomial distribution given the number of patients susceptible to MRSA in the preceding swab interval. The probability of colonization was described as a function of the number of non-isolated (colonized) and isolated (colonized) patients in the preceding swab interval. A weakness of the approach is the inherent assumption that colonizations within each swabbing interval are independent.
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, 2004) for analyzing data on partially observed infectious diseases within the community. Unlike the methods proposed by Pelupessy and others (2002), Cooper and Lipsitch (2004), and Forrester and Pettitt (2005), MCMC methods can be applied to infer colonization times from interval-censored data leading to greater accuracy and precision in inference by allowing for dependent transmissions within intervals. The methods appear well suited to routinely collected hospital data but must be adapted to allow for patient admission and discharge. This is in contrast to community populations which have relatively small turnover and are typically assumed to be closed. Cooper (2000) applied this approach to hospital infection data on VRE. The importance of considering imperfect sensitivity was emphasized 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).
The objective of this study is to develop methodology to estimate the transmission rate parameters of a transmissible nosocomial pathogen. MCMC methods are used and adapted for routine surveillance data and extended to allow for imperfect sensitivity of the surveillance process. We do this by utilizing reversible jump Markov chain Monte Carlo (RJMCMC) (Green, 1995, 2003). The methodology is applied to data of the ICU of the Princess Alexandra Hospital (PAH), Brisbane, Australia, to estimate the transmission rate of MRSA from colonized to non-colonized (susceptible) patients and to quantify the effect of isolating colonized patients when a background source of MRSA burden exists. Our analysis suggests that within this ICU, isolation of patients colonized with MRSA is an effective infection control procedure.
In Section 2, the model and framework for statistical inference are introduced. The MCMC methodology is described in Section 3. In Section 4, the methods are applied to data of the PAH ICU. The paper concludes with a discussion on limitations of the current model and possible extensions for future research.
|
2. MODEL, DATA, AND BAYESIAN FRAMEWORK |
|---|
|
TOPSUMMARY1. INTRODUCTION2. MODEL, DATA, AND...3. MCMC ALGORITHM AND...4. APPLICATION: MRSA...5. DISCUSSIONREFERENCES |
|---|
We consider a transient hospital-ward population of patients, some of whom may be asymptomatically colonized with a communicable pathogen. Epidemics are initiated by the admission of colonized patients to the ward or by background contamination. Background contamination is defined to include all nosocomial transmission arising from outside the ward. Transmissions from colonized health care workers (HCWs), and from equipment and HCWs transiently contaminated within the hospital but outside the ward, are examples of possible background contamination sources. Patients admitted in a susceptible (non-colonized) state can acquire the pathogen by indirect contact with colonized patients via HCWs or by background transmission. Colonized patients may be detected via routine swabbing procedures and placed in isolation. 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 term "importation probability" is used to refer to the probability 
that a patient is colonized on admission to the ward. We assume that the colonization status of patients on admission to the ward is independent of the colonization status of other patients. The routine swabbing procedure may be subject to imperfect sensitivity, so that some false-negative swabs are possible. The sensitivity (or detectability) of the routine swabbing procedure is denoted 
. We assume 100% specificity.
At a given time t, a ward patient is characterized as being (1) susceptible (but not colonized), (2) non-isolated (and colonized), (3) isolated from other patients due to being detected as colonized, or (4) removed or discharged from the ward. Once colonized, patients are assumed to remain so until discharged. Patients discharged from the ward play no further role in the epidemic. The number of patients in each of these compartments (susceptible, colonized, isolated, and removed) at time t is denoted S(t), C(t), Q(t), and R(t), respectively. We use t– to describe a time just prior to time t, so that, for example, S(t–) is the number of susceptible patients in the ward just prior to time t.
Upon discharge, a patient i will be in one of the following states,
 |
Assuming homogeneity and no variation in susceptibility or infectivity over time, the probability that a susceptible patient i is colonized in some small time interval dci > 0, given no colonization up to time ci, can be described by the hazard function h(ci), where
 | (2.1) |
Here ß0 is referred to as the background rate, ß1 as the non-isolated (colonized) rate, and ß2 as the isolated (colonized) rate. The background transmission rate captures colonizations acquired in the ward that were not transmitted from a colonized patient in the ward at the time of acquisition. Formulation of the hazard function in terms of the number of patients is referred to as the "pseudo-mass action" assumption (de Jong and others, 1995). If the number of transmissions between a colonized patient and each of the susceptible patients is expected to vary with the number of patients in the ward, ß1 and ß2 should be divided by the number of patients in the unit (de Jong and others, 1995).
Patient admission, colonization, isolation, and discharge times constitute the set of event times (see Figure 1). We consider that colonizations are stochastic events and that the remaining events are governed by deterministic dynamics. A variation of the model would allow for stochastic admission, isolation, and discharge events. The colonization process hazard h(ci) defined in (2.1) is assumed to be piecewise constant over each event interval. The variable e is the vector of event times e0 
e1 e2
eNe at which the susceptible, colonized, and isolated patient population numbers can change. The described model is a form of the general stochastic epidemic model (Bailey, 1975; Andersson and Britton, 2000). A graphical representation is provided in Figure 1.
|
The admission, isolation, and discharge times are known 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 been admitted to the ward on that day. If a patient is detected as colonized, the positive swab time is also known. The colonization time ci is unknown. For a patient i, we let ai, vi, qi, and ri denote, respectively, the admission, positive swab, isolation, and discharge times. Swabs to test for the presence of the pathogen are routinely cultured from patients at times ts, where ts = t
,t
,...,t
,... . We assume that isolates are cultured from all patients in the ward at times ts with 100% compliance. The observed data {a,v,q,r,ts} are denoted D. The subset Dd = {a,q,r,ts} is given and arises from events which are not modeled stochastically. The colonization time is censored by the admission time for patients colonized on admission and by the discharge time for patients remaining susceptible.
Our objective is to make inferences about the parameters; the transmission rates ß (see (2.1)), the swab sensitivity , and the importation probability , based on the observed data. To do so we explore the joint posterior density p(ß,,|D). The likelihood p(v|ß,,,Dd) is intractable since it involves integrating over all possible values of the unknown colonization times. Consequently, we consider the likelihood of the observed data augmented with both the unobserved colonization times and final patient states, p(c,s,v|ß,,,Dd).
From Bayes theorem, the posterior density p(ß,,,c,s|D) is proportional to the product of the parameter likelihood for the observed and augmented data jointly and the parameter prior density, i.e.
 |
Here 
(ß), (), and () are the marginal prior distributions for the parameters ß, , and , respectively, and are assumed independent.
An expression for the likelihood can be obtained using the Poisson process (Davison, 2003
). With perfect sensitivity and no importation events, the likelihood is
 | (2.2) |
where 1x is equal to one if x is true and t
is the time of last negative routine isolate cultured from patient i. The first term on the right-hand side (the term involving the indicator function) restricts the unobserved colonization time according to the admission, last negative swab, positive swab (if a positive isolate was obtained), and discharge times observed for each patient. The index of the exponential function sums the log-hazard of each colonization event and the log-survival function for the duration that each patient remains susceptible, i.e. until colonization or discharge, whichever occurs first.
Imperfect sensitivity changes the likelihood in (2.2) by a factor of NTP(1 – )NFN(c,D), where NFN(c,D) and NTP are the numbers of false-negative and true-positive swabs, respectively. In the application that follows we do not have, and therefore do not assume, information on swabs cultured from patients after the first positive isolate. Without this information, an informative prior distribution is required for the swab sensitivity parameter . We must also account for the possibility of patients being colonized on admission. The likelihood of the set of importation events is Nsp(s)(1 – )NA – Nsp(s) where Nsp(s) is the number of patients colonized on admission and NA is the number of admissions to the ward during the observation period.
Adapting (2.2) for imperfect detectability and the set of importation events, the likelihood of the joint data is
 | (2.3) |
To compare (2.3) with (2.2), the first term involving the indicator function now allows the unobserved colonization time ci to take place any time immediately after admission and before the sooner of the positive swab or discharge time. The next set of terms involving and give the probabilities, as explained earlier, for imperfect sensitivity and colonization on admission. The exponential terms remain unchanged from (2.2).
The transmission rates are constrained to be greater than zero. If, for example, we wanted to allow for a null background transmission rate, that is ß0 = 0, we would have to multiply (2.3) by 
i:si = sd1C(c
) + Q(c) > 0. With a null background transmission rate, there must be at least one colonized or isolated patient in the ward for another patient to be colonized in the ward.
We assume proper uniform priors for the transmission rate parameters and beta priors for the sensitivity and importation probability parameters.
|
3. MCMC ALGORITHM AND CONVERGENCE ASSESSMENT |
|---|
|
TOPSUMMARY1. INTRODUCTION2. MODEL, DATA, AND...3. MCMC ALGORITHM AND...4. APPLICATION: MRSA...5. DISCUSSIONREFERENCES |
|---|
MCMC is used to approximate the posterior p(ß,,,c,s|D) by iteratively drawing samples of (ß,,,c,s) values. Metropolis (Metropolis and others, 1953) update steps are used for the transmission rate parameters and Gibbs (Geman and Geman, 1984; Gelfand and Smith, 1990) update steps for the swab sensitivity and importation probability parameters. The final patient state and colonization time for each admission are updated jointly according to Metropolis–Hastings (Metropolis and others, 1953; Hastings, 1970). The update step for the final patient state and colonization time may require RJMCMC, depending on the current and proposed final state.
To generate the Markov chain ß
,ß
,...,ß
,... , for each of the transmission rates ßk where k
{0,1,2}, given some current value ß, we first propose a new value ß
. The value is set to the absolute value of a random number generated from a Gaussian density centered on ß. The proposed transmission rate, ß, is accepted with probability
 |
as the proposal density is symmetric. Here p(ß|ß – k,c,D) is the full conditional of ß, which is proportional to
 |
for k = 0, 1, and 2, respectively.
To generate a Markov chain for the swab sensitivity and importation probability, values are sampled from their full conditional distributions,
 |
The colonization time and final state are updated concurrently for each patient. To do so, a patient i is chosen at random and one of several proposals is made dependent on whether a positive isolate was obtained from that patient during his/her stay in the ward. This step is repeated for a given fraction of admissions, sampled at random, to the ward during the observation period. Following Green (1995, 2003), we label each possible proposal and consider that certain proposals (or moves) occur in pairs. A pair consists of a move from one state to another and a move in reverse. Figure 2 illustrates possible moves depending on the current patient state and defines the proposal probabilities and the corresponding reverse moves.
|
Proposing to change a patient's final state from colonized in the ward (sd) to either never colonized (ss) or colonized prior to admission (sp) and vice versa requires either the proposal of a colonization time or the removal of the currently assumed colonization time for that patient. Evaluation of proposals within these move pairs requires comparisons between realizations of the process with different numbers of colonization events. As the colonization times are model parameters and the number of colonization times is unknown, we require a framework for jumping between parameter subspaces of differing dimensionality. We employ RJMCMC to move between the different dimensional spaces.
When changing patient i‘s final state from si to some state s
, we can show using the ideas of Green (1995, 2003) that detailed balance will be preserved if the new state is accepted with probability
 | (3.1) |
A description of p(c,s|ß,,,c – i,s – i,D), the target distribution; jm(s|si), the state proposal probability; and qs(c|D), the parameter proposal density is given below. The Jacobian 
is equal to one in all instances and not discussed further.
,s|ß,,,c – i,s – i,D).The target distribution is the full conditional of the proposed state s
and associated censored colonization time c for the selected patient i. It is proportional to the joint posterior,
 | (3.2) |
Note that if s = sp, then the summations in the braces are not required.
|s).The state proposal probability is the probability of proposing to update the selected patient's state from si to s
. The probabilities are defined in terms of the parameters K1, K4, K5, K
, K
, and K
introduced in Figure 2. For example, the state proposal probability for move type (i), j(i)(sp|sd), is K1.
(c|D).The parameter proposal density is the probability density of proposing a given colonization time. Proposing a final state of susceptible (ss), or colonized prior to admission (sp), is equivalent to proposing no colonization event during a patient's stay in the ward. If a final state of colonized in the ward (sd) is proposed, a colonization time c
is proposed from an exponential distribution truncated at the patient's admission time.
Cowles and Carlin (1996) review a number of approaches to assess MCMC convergence. We define a set of realizations according to the number of patients inferred (rather than observed) to have been colonized in the ward. Convergence of the transmission rate, sensitivity, and importation probability parameters in addition to the number of realizations can be ascertained using the convergence diagnostics proposed by Geweke (1992) and Gelman and Rubin (1992). These approaches will not detect lack of convergence within each realization. This problem is overcome by the method of Brooks and Giudici (1998). Like the method proposed by Gelman and Rubin (1992), Brooks and Giudici (1998) propose running parallel chains and splitting the total variation not only between chains but also between models or realizations. A weakness of the approach is that the last two sources of variation are unstable if a rare model (i.e. number of patients inferred) is visited by one of the chains. An alternative approach has been proposed by Castelloe and Zimmerman (2002) in which visits to rare models have a lesser impact on the convergence diagnostics. The convergence diagnostic utilizes multiple chains and detects between-chain variation, between-model variation that differs from one chain to another, and significant differences in the frequencies of model visits from one chain to another.
|
4. APPLICATION: MRSA TRANSMISSION WITHIN THE PAH ICU |
|---|
|
TOPSUMMARY1. INTRODUCTION2. MODEL, DATA, AND...3. MCMC ALGORITHM AND...4. APPLICATION: MRSA...5. DISCUSSIONREFERENCES |
|---|
We consider a data set (previously analyzed by Forrester and Pettitt, 2005
) of routinely collected information concerning admissions and MRSA occurrence of the PAH ICU between 1 January 1995 and 28 March 1997. Table 1 and Figure 3 summarize key characteristics of the data. During this period, the ICU had 12 beds: 2 isolation rooms, 2 x 2-bed bays, and 2 x 3-bed bays.
|
|
For each admission i to the ICU, the date and time of admission and discharge were recorded. The data concerning MRSA colonization outcome were collected from routine nasal or groin swabs. The swabs were taken on mondays and thursdays. We assume they were taken at 11 AM with 100% compliance. For patients notified as MRSA positive, the positive swab time (vi) and the notification time (qi) (assumed to occur at 11 AM on the recorded date) were recorded. For some patients, the date of the swab returning a positive isolate was either not recorded or listed as occurring after notification of the swab outcome. In case of discrepancy, the swab date resulting in a positive isolate is set to be 2 weekdays prior to the notification date.
Upon notification of a patient being colonized with MRSA, the patient was placed in isolation. When there were three or more detected MRSA patients, they were grouped in a two- or three-bed bay and referred to as cohorted. We consider that detected patients were isolated as soon as notification was received, regardless of the delay between notification and isolation and whether the patient was placed in isolation or cohorted. In contrast to the general ward, the isolation room had a sink for each patient bed and the hand-washing 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.
We fitted the described model to the PAH ICU data with uniform Unif(0,1] priors for the transmission rates, an informative Beta(127.2,31.8) (mean 0.8, variance 0.001) prior for the sensitivity parameter, and an uninformative Beta(1,1) prior for the importation probability parameter. A discussion concerning the choice of these priors is left to Section 5.
To make inference, 380 000 MCMC iterates were used following a burn-in of 20 000 iterates. The 380 000 iterations were then thinned by a factor of 40, leaving 9500 iterations for inference. In each iteration, the colonization times were updated for 10% of admissions during the observation period, with each admission chosen at random. Probabilities (K1,K4,K5,K,K,K) of the allowable proposal states given the current patient state were weighted equally. The transmission rates are measured in terms of new colonizations per day.
The observed and augmented data for July 1995 taken from one iteration of the Markov chain are shown in Figure 4. Table 2 and Figure 5 present the estimated transmission rate, sensitivity, and importation probability parameters, in addition to estimates of the number of acquired and imported MRSA cases. The 95% credible interval for the difference in transmission between non-isolated and isolated patients, ß1 – ß2, includes the value zero. The posterior mean sensitivity of the swabbing process is lower than the prior mean. It is estimated that 46.0% [95% credible interval 36.0,61.7] of MRSA patients were detected and 45.5% [95% credible interval 41.3,48.7] of MRSA cases were imported. A strong negative posterior correlation was found between the non-isolated (ß1) and background (ß0) transmission rates ( – 0.57). Background transmission (ß0) was weakly correlated with both isolated transmission (ß2) ( – 0.32) and the importation probability () ( – 0.23).
|
|
|
Convergence was verified using the Geweke (1992)
and Gelman and Rubin (1992) convergence diagnostics provided within CODA (Version 0.40bs1, MRC Biostatistics Unit, Cambridge, United Kingdom). The Castelloe and Zimmerman (2002) convergence diagnostics were used to verify within-model convergence.
The posterior predictive assessment method (Rubin, 1984
), (Gelman and others, 1996), (Gelman and others, 2000) compares observed Dobs and replicated Drep data sets on the basis of test statistics T(D) and test discrepancies T(D,
). The algorithm is to construct a replicated data set for each draw from the posterior distribution, evaluate the test statistic or discrepancy for the drawn parameter values and each of the data sets, real and replicated, and produce a scatter plot with the discrepancy of the observed data on the x-axis and the discrepancy of the replicated data on the y-axis. If the model fits, replicated data generated under the model should look similar to observed data. The predictive p-value is the proportion of times the predictive discrepancy D(Drep;) is greater than or equal to the observed discrepancy D(Dobs;), or the proportion of points in the scatter plot which lie above the line of unit slope. A p-value of 0.5 is optimal. We use posterior prediction to compare the number of detected patients during the observation period in the observed and simulated data sets (see Figure 6).
|
The cross-validation technique (Gelfand and others, 1992
) uses existing data, rather than hypothetical realizations, to validate the model. It compares the observed responses yj to those expected, Yj|y – j, from the data y – j with the jth response missing. Various checking functions are possible. We use the Freeman–Tukey residual and the tail-area probability. The Freeman–Tukey residuals, 
, are appropriate when counts are small (Brooks and others, 2000). In practice, the residuals can be approximated,
 | (4.1) |
Unless the data set is small and yj is an extreme outlier, this approximation should be adequate (Carlin and Louis, 2000). Plotting the residuals dj against the swab times tj should reveal patterns of over- /underfitting. The residuals should be approximately normal with large absolute values causing concern. The tail-area probabilities of the response, d
= P(Yj < yj|y – j), should be uniformly distributed.
The observed number of detected patients yj per swab interval can be derived directly from the data and is straightforward to calculate. The expected number of detected patients per swab interval for a given Markov chain iteration g is considered to be binomially distributed,
 |
where n
is the number of colonized patients swabbed in the jth swab interval. The posterior expectation of this value is approximated by
 |
These values are substituted into (4.1) to obtain the Freeman–Tukey residuals. The lower and upper tail-area probabilities are
 |
and
 |
respectively. The two-sided tail-area probability is obtained by selecting twice the minimum of the lower and upper values. The Freeman–Tukey residuals and tail-area probabilities are provided in Figure 7. The distribution of the expected detection outcome is concentrated around the observed detection value somewhat more than expected. Overall, there do not appear to be any unusually poorly fitted cases or time trends suggesting lack of fit. If hypothesized true parameter values were used with continuous data, we would expect the tail-area probabilities to have a (0,0.5) uniform distribution. With discrete data, the two-sided tail-area probabilities are distributed over an interval (0,a), a(0.5,1]. For a unimodal distribution with a large modal probability close to one, a itself is close to one corresponding to observing the modal value. The distribution of the two-sided tail-area probabilities for the data is highly skewed toward one suggesting a reasonable fit.
|
Alternative methods to assess goodness of fit include conditional Bayesian p-values (Bayarri and Berger, 1998
), posteriors from the simulation (Dey and others, 1995), (Sinha and Dey, 1997), Bayesian latent residuals (Aslanidou and others, 1998; Ibrahim and others, 2001), and prequential (Arjas and Gasbarra, 1997; Ibrahim and others, 2001) approaches but we have not investigated these further.
An informative Beta(127.2,31.8) (mean 0.8, variance 0.001) prior distribution was used for the sensitivity parameter. We assess the impact on the posterior distributions when Beta(146.3,62.7) (mean 0.7, variance 0.001), Beta(80.1,8.9) (mean 0.9, variance 0.001), and Unif(0.6,1) prior distributions are used. These prior distributions have been chosen to reflect information about the sensitivity of the swabbing process (see Section 5).
The posterior mean value of the estimated decrease in transmission per day by isolating a colonized patient is provided in Table 3 for various informative sensitivity () priors. A higher prior mean for the sensitivity will lead to a lower estimated non-isolated transmission (ß1) rate. The isolated transmission (ß2) rate remains more or less the same regardless of the prior mean. The estimated decrease in mean transmission resulting from isolation, (ß1 – ß2), is smaller for higher prior sensitivity means. These results are understandable because the number of patients colonized in the ward, C(t), must increase as the assumed sensitivity, , decreases. For ß1C(t) to remain constant, ß1 must decrease as C(t) increases. Additional colonizations due to the sensitivity decreasing are explained by ß0 increasing. The number of patients isolated is known and so ß2Q(t) remains constant. We stress that conclusions concerning the effectiveness of isolation are dependent on what is assumed about the imperfect sensitivity.
|
|
5. DISCUSSION |
|---|
|
TOPSUMMARY1. INTRODUCTION2. MODEL, DATA, AND...3. MCMC ALGORITHM AND...4. APPLICATION: MRSA...5. DISCUSSIONREFERENCES |
|---|
We have considered a model for the analysis of communicable hospital pathogens using routinely collected surveillance data. The methodology is applicable to pathogens which conform to the susceptible-colonized-removed paradigm. The algorithm permits inference about nosocomial transmission and importation probability even when the surveillance data are subject to false-negative results.
The methods are applied to MRSA data from the ICU in the PAH. MRSA transmission is primarily via transiently contaminated HCWs. The role of HCWs as vectors of transmission is implicit and not modeled explicitly.
An MCMC approach within a Bayesian framework is used to determine posterior distributions of transmission rates from background, non-isolated colonized patient, and isolated (colonized) patient sources, in addition to detection sensitivity and the probability of being colonized on admission. For all parameters excluding the detection sensitivity, non-informative priors were chosen.
The prior distribution for the detection sensitivity was chosen to reflect current information concerning the probability that a patient colonized with MRSA is detected at a routine swabbing time. This value is likely to depend on the number and location of patient swabs, compliance to swabbing and laboratory procedures. Routine swabs taken at the PAH ICU are taken from the nares or groin and will not detect throat or perineum colonization. One study (Coello and others, 1994) found that 9.9% of carriers in a university hospital were colonized in the perineum alone and 5% in the throat alone. Compliance to swabbing within the PAH ICU during the observation period is unknown; however, an unpublished report suggests it was approximately 93% in 1993. Sensitivity of laboratory methods for detecting MRSA was 66.7% [95%CI 51.9,83.3] in one study (Hope and others, 2004) and 81% for detecting S. aureus nasal colonization and 87% for detecting S. aureus tracheal colonization in another study (Keene and others, 2005).
The likelihood of the data has a strong effect on the sensitivity parameter giving high posterior values to small values of the sensitivity parameter. With a posterior mean sensitivity of 71.7% (95% credible interval 64.1,79.0), the mean rate of transmission of MRSA from isolated patients was lower (0.0045 transmissions per day) than the mean rate from non-isolated patients (0.0131 transmissions per day). The posterior mean absolute risk of non-isolation (0.0086 transmissions per day, 95% credible interval – 0.0035,0.0209) is higher than that estimated by Forrester and Pettitt (2005) using the same data. Forrester and Pettitt (2005) estimated that the absolute risk of non-isolation was 0.0036 transmissions per day (95% credible interval – 0.0011,0.0086), whereas the model assumed independence of colonizations within the same swabbing interval and 100% sensitivity. A higher estimate for the absolute risk of non-isolation (0.131 transmissions per day) was obtained during an epidemic outbreak within a neonatal ICU (Jernigan and others, 1996). Our model is applied to endemic data in which background transmission can take place when no colonized patient is present. The background transmission rate was found to be 0.0103 transmissions per day.
The estimated mean importation probability (4.7%, 95% credible interval 3.0,6.7) is supported by the literature. Published rates for the proportion of patients colonized with MRSA on admission include 6.8% for an Australian ICU (Marshall and others, 2003), 6.9% (range 3.7 to 20%) among 14 French ICUs (Lucet and others, 2003), and 10% for an English ICU (Thompson, 2004). More recent studies (Troché and others, 2005; Eveillard and others, 2005) in France detected colonization on admission rates varying from 4.2 to 10.1%.
The MCMC methodology employed here facilitated the imputation of the unobserved colonization times of patients admitted to a hospital ward. Non-aggregated patient-level data were used thereby exploiting the full information content of the data. The framework can be readily extended to allow for heterogeneity in susceptibility. By introducing a latent parameter to identify the source of infection for each infected individual, MCMC sampling on the transmission rate parameters can be performed using Gibbs steps (Geman and Geman, 1984; Gelfand and Smith, 1990) rather than Metropolis–Hastings steps (Metropolis and others, 1953; Hastings, 1970).
Transmission from a colonized or isolated patient was assumed to remain constant until discharge. This assumption can be investigated statistically by modeling the hazard as being dependent on the colonization time of the patient.
We did not have data on compliance during the period of observation (1995 to 1997); 100% compliance was assumed. With less than perfect compliance, the number of swabs taken would be lower than calculated and it is reasonable to expect that the true sensitivity value may be higher than estimated by the model.
A recent review (Cooper and others, 2003) highlighted a pressing need for research to determine the effect of isolation wards in hospitals. We have presented an approach which allows the effect on isolation in hospital wards to be quantified. When applied to data from the PAH ICU, we found that the transmission rate from isolated patients is lower than from non-isolated patients. In prospective studies, it is clear that detectability will have a critical role in determining the transmission rates for patients in isolation.
Findings from the presented approach will be advantageous to simulation studies requiring knowledge of the transmission rate parameters, for which there is a lack of information. Additionally, analysis of simulated data using the methods described here could assist in the design of future studies, e.g. to determine the ideal study duration and swab frequency. The methodology presented here can be used to quantify the effect of infection control interventions by providing pre- and post-intervention estimates of key transmission parameters. Evidence-based decisions can therefore be made on the impact of infection control procedures.
|
ACKNOWLEDGMENTS |
|---|
We thank Dr M. Whitby and Dr N.A. Ismail for providing and collating the original data set, L. Bradley for providing additional data, information concerning policies, and procedures of the ICU within the PAH, and in conjunction with Dr D.A. Cook, Dr G. Streftaris, Dr P.D. O'Neill and the editors for valuable comments. We thank Dr B.A. Reeves, R. Evans and T. Moroney for C++ programming assistance. Computational resources and services used in this work were provided by the High Performance Computing and Research Support Unit, Queensland University of Technology, Brisbane, Australia. This work was supported financially by the Australian Research Council and PAH. Conflict of Interest: None declared.
|
REFERENCES |
|---|
|
TOPSUMMARY1. INTRODUCTION2. MODEL, DATA, AND...3. MCMC ALGORITHM AND...4. APPLICATION: MRSA...5. DISCUSSIONREFERENCES |
|---|
-
Andersson H and Britton T. (2000) Stochastic Epidemic Models and Their Statistical Analysis(Springer, New York) Volume 151: Lecture notes in statistics.
Arjas E and Gasbarra D. (1997) On prequential model assessment in life history analysis. Biometrika 84:505–22.
Aslanidou H, Dey D, Sinha D. (1998) Bayesian analysis of multivariate survival data using Monte Carlo methods. Canadian Journal of Statistics 26:33–48.
Austin DJ, Bonten MJM, Weinstein RA, Slaughter S, Anderson RM. (1999) Vancomycin-resistant enterococci in intensive-care hospital settings: transmission dynamics, persistence, and the impact of infection control programs. Proceedings of the National Academy of Sciences USA 96:6908–13.
Bailey NTJ. (1975) The Mathematical Theory of Infectious Diseases and Its Applications 2nd edition (Griffin, London).
Bayarri M and Berger J. (1998) Quantifying surprise in the data and model verification. In Bernardo J, Berger J, Dawid A, Smith A (Eds.). Bayesian Statistics 6(Oxford University Press, Oxford, UK) pp. 53–82.
Becker NG and Britton T. (1999) Statistical studies of infectious disease incidence. Journal of the Royal Statistical Society Series B Statistical Methodology 61:287–307.[CrossRef][Web of Science]
Bonten MJM, Austin DJ, Lipsitch M. (2001) Understanding the spread of antibiotic resistant pathogens in hospitals: mathematical models as tools for control. Clinical Infectious Diseases 33:1739–46.[CrossRef][Web of Science][Medline]
Brooks S, Catchpole E, Morgan B. (2000) Bayesian animal survival estimation. Statistical Science 15:357–76.[CrossRef][Web of Science]
Brooks SP and Giudici P. (1998) Convergence assessment for reversible jump MCMC simulations. In Bernardo J, Berger J, Dawid A, Smith A (Eds.). Bayesian Statistics 6(Oxford University Press, Oxford, UK) pp. 733–42.
Carlin B and Louis TA. (2000) Bayes and Empirical Bayes methods for data analysis. Chapman & Hall texts in statistical science series 2nd edition (Chapman & Hall, Boca Raton).
Castelloe JM and Zimmerman DL. (2002) Convergence assessment for reversible jump MCMC samplers. Technical Report 313(SAS Institute, Cary, NC).
Coello R, Jiménez J, García M, Arroyo P, Minguez D, Fernández C, Cruzet F, Gaspar C. (1994) Prospective study of infection, colonization and carriage of methicillin-resistant Staphylococcus aureus in an outbreak affecting 990 patients. European Journal of Clinical Microbiology and Infectious Diseases 13:74–81.[CrossRef][Web of Science][Medline]
Cooper B. (2000) The transmission dynamics of methicillin-resistant Staphylococcus aureus and vancomycin-resistant enterococci in hospital wards, [dissertation](University of Warwick, Coventry, UK).
Cooper B and Lipsitch M. (2004) The analysis of hospital infection data using hidden Markov models. Biostatistics 5:223–37.[Abstract]
Cooper BS, Medley GF, Scott GM. (1999) Preliminary analysis of the transmission dynamics of nosocomial infections: stochastic and management effects. Journal of Hospital Infection 43:131–47.[CrossRef][Web of Science][Medline]
Cooper BS, Stone SP, Kibbler CC, Cookson BD, Roberts JA, Medley GF, Duckworth GJ, Lai R, Ebrahim S. (2003) Systematic review of isolation policies in the hospital management of methicillin resistant Staphylococcus aureus: a review of the literature with epidemiological and economic modelling. Health Technology Assessment 7:1–194.[Medline]
Cowles MK and Carlin BP. (1996) Markov chain Monte Carlo convergence diagnostics: a comparative review. Journal of the American Statistical Association 91:883–904.[CrossRef][Web of Science]
Davison A. (2003) Statistical models, Chapter 6. Stochastic models. Cambridge series in statistical and probabilistic mathematics(Cambridge University Press, Cambridge, UK) pp. 274–91.
de Jong MCM, Diekmann O, Heesterbeek H. (1995) How does transmission of infection depend on population size? In Mollison D (Ed.). Epidemic Models: Their Structure & Relation to Data, No. 5 in Publications of the Newton Institute(Cambridge University Press, Cambridge, UK) pp. 84–94.
Dey D, Gelfand A, Swartz T, Vlachos P. (1995) A simulation-intensive based model checking for hierarchical methods. Technical Report 95-29Department of Statistics, University of Connecticut.
Eveillard M, Lancien E, Barnaud G, Hidri N, Gaba S, Benlolo J, Joly-Giullou ML. (2005) Impact of screening for MRSA carriers at hospital admission on risk-adjusted indicators according to the imported MRSA colonization pressure. Journal of Hospital Infection 59:254–8.[CrossRef][Web of Science][Medline]
Forrester M and Pettitt AN. (2005) Use of stochastic epidemic modeling to quantify transmission rates of colonization with methicillin-resistant Staphylococcus aureus in an intensive care unit. Infection Control and Hospital Epidemiology 26:598–606.[CrossRef][Web of Science][Medline]
Gelfand A, Dey D, Chang H. (1992) Model determination using predictive distributions with implementation via sampling-based methods. In Bernardo J, Berger J, Dawid A, Smith A (Eds.). Bayesian Statistics 4(Oxford University Press, Oxford, UK) pp. 147–67.
Gelfand A and Smith A. (1990) Sampling-based approaches to calculating marginal densities. Journal of American Statistical Association 85:398–409.[CrossRef]
Gelman A, Carlin JB, Stern HS, Rubin DB. (2000) Bayesian Data Analysis(Chapman & Hall, Boca Raton).
Gelman A, Meng XL, Stern H. (1996) Posterior predictive assessment of model fitness via realized discrepancies. Statistica Sinica 6:733–807.[Web of Science]
Gelman A and Rubin D. (1992) Inference from iterative simulation using multiple sequences. Statistical Science 7:457–511.[CrossRef]
Geman S and Geman D. (1984) Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence 6:721–41.[CrossRef][Web of Science]
Geweke J. (1992) Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. In Bernardo J, Berger J, Dawid A, Smith A (Eds.). Bayesian Statistics 4(Oxford University Press, Oxford, UK) pp. 169–93.
Gibson GJ and Renshaw E. (1998) Estimating parameters in stochastic compartmental models using Markov chain methods. IMA Journal of Mathemetics Applied in Medicine and Biology 15:19–40.
Gibson GJ and Renshaw E. (2001) Likelihood estimation for stochastic compartmental models using Markov chain methods. Statistical Computation 11:347–58.[CrossRef]
Green PJ. (1995) Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82:711–32.
Green PJ. (2003) Trans-dimensional Markov chain Monte Carlo. In Green PJ, Hjort NL, Richardson S (Eds.). Highly Structured Stochastic Systems, Volume 27 of Oxford Statistical Science(Oxford University Press, Oxford, UK) pp. 179–98.
Hastings W. (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57:97–109.
Hope WW, Morton AP, Looke DFM, Schooneveldt JM, Nimmo GR. (2004) A PCR method for the identification of methicillin resistant Staphylococcus aureus (MRSA) from screening swabs. Pathology 36:265–8.[CrossRef][Web of Science][Medline]
Ibrahim JG, Chen MH, Sinha D. (2001) Bayesian Survival Analysis. Springer series in statistics(Springer, New York).
Jernigan JA, Titus MG, Gröschel DHM, Getchell-White SI, Farr BM. (1996) Effectiveness of contact isolation during a hospital outbreak of methicillin-resistant Staphylococcus aureus. American Journal of Epidemiology 143:496–504.
Keene A, ad M-H Lee PV, Finnerty K, Nicolls D, Cespedes C, Quagliarello B, Chiasson M, Chong D, Lowy FD. (2005) Staphylococcus aureus colonization and the risk of infection in critically ill patients. Infection Control and Hospital Epidemiology 26:622–8.[CrossRef][Web of Science][Medline]
Lipsitch M, Bergstrom CT, Levin BR. (2000) The epidemiology of antibiotic resistance in hospitals: paradoxes and prescriptions. Proceedings of the National Academy of Sciences USA 97:1938–43.
Lucet JC, Chevret S, Durand-Zaleski I, Chastang C, Régnier B. (2003) Prevalence and risk factors for carriage of methicillin-resistant Staphylococcus aureus at admission to the Intensive Care Unit. Archives of Internal Medicine 163:181–8.
Marshall C, Harrington G, Wolfe R, Fairley CK, Wesselingh S, Spelmen D. (2003) Acquisition of methicillin-resistant Staphylococcus aureus in a large intensive care unit. Infection Control and Hospital Epidemiology 24:322–6.[CrossRef][Web of Science][Medline]
Metropolis N, Rosenbluth A, Rosenbluth M, Teller A, Teller E. (1953) Equations of state calculations by fast computing machines. Journal of Chemical Physics 21:1087–92.[CrossRef]
O'Neill PD and Roberts GO. (1999) Bayesian inference for partially observed stochastic epidemics. Journal of the Royal Statistical Society Series A 162:121–9.
Pelupessy I, Bonten MJM, Diekmann O. (2002) How to assess the relative importance of different colonization routes of pathogens within hospital settings. Proceedings of the National Academy of Sciences USA 99:5601–5.
Renshaw E. (1999) Stochastic effects in population models. In McGlade J (Ed.). Advanced Ecological Theory: Principles & Applications, Chapter 2(Blackwell Science, Malden, Mass) pp. 23–63.
Rubin D. (1984) Bayesianly justifiable and relevant frequency calculations for the applied statistician. Annals of Statistics 12:1151–72.[CrossRef][Web of Science]
Sébille V, Chevret S, Valleron AJ. (1997) Modeling the spread of resistant nosocomial pathogens in an Intensive Care Unit. Infection Control and Hospital Epidemiology 18:84–92.[Web of Science][Medline]
Sinha D and Dey D. (1997) Semiparametric Bayesian analysis of survival data. Journal of the American Statistical Association 92:1195–212.[CrossRef][Web of Science]
Smith T and Vounatsou P. (2003) Estimation of infection and recovery rates for highly polymorphic parasites when detectability is imperfect, using hidden Markov models. Statistical Medicine 22:1709–24.[CrossRef]
Streftaris G and Gibson GJ. (2004) Bayesian inference for stochastic epidemics in closed populations. Statistical Model 4:63–75.
Thompson D. (2004) Methicillin-resistant Staphylococcus aureus in a general intensive care unit. Journal Royal Society of Medicine 97:521–6.[CrossRef]
Troché G, Joly LM, Guibert M, Zazzo JF. (2005) Detection and treatment of antibiotic-resistant bacterial carriage in a surgical intensive care unit: a 6-year prospective study. Infection Control and Hospital Epidemiology 26:161–5.[CrossRef][Web of Science][Medline]
Trotter C and Gay N. (2003) Analysis of longitudinal bacterial carriage studies accounting for sensitivity of swabbing: an application to Neisseria meningitidis. Epidemiology and Infection 130:201–5.[CrossRef][Medline]
Received July 25, 2005; revised March 24, 2006; revised July 7, 2006; accepted for publication August 9, 2006.
CiteULike 
Connotea 
Del.icio.us What's this?
This article has been cited by other articles:
|
|
 |
|
  B. S. Cooper, G. F. Medley, S. J. Bradley, and G. M. Scott An Augmented Data Method for the Analysis of Nosocomial Infection Data Am. J. Epidemiol., September 1, 2008; 168(5): 548 - 557. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
M. Bootsma, M. Bonten, S Nijssen, A. Fluit, and O Diekmann An Algorithm to Estimate the Importance of Bacterial Acquisition Routes in Hospital Settings Am. J. Epidemiol., October 1, 2007; 166(7): 841 - 851. [Abstract] [Full Text] [PDF]
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||