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American Journal of Epidemiology Advance Access originally published online on July 21, 2007
American Journal of Epidemiology 2007 166(7):841-851; doi:10.1093/aje/kwm149
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American Journal of Epidemiology © The Author 2007. Published by the Johns Hopkins Bloomberg School of Public Health. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org.

PRACTICE OF EPIDEMIOLOGY

An Algorithm to Estimate the Importance of Bacterial Acquisition Routes in Hospital Settings

MCJ Bootsma1, MJM Bonten2,3,4, S Nijssen2, AC Fluit3 and O Diekmann1

1 Department of Mathematics, Utrecht University, Utrecht, Kingdom of the Netherlands
2 Department of Internal Medicine and Dermatology, University Medical Center Utrecht, Utrecht, Kingdom of the Netherlands
3 Eijkman Winkler Institute for Microbiology, Infectious Diseases, and Inflammation, University Medical Center Utrecht, Utrecht, Kingdom of the Netherlands
4 Julius Center for Health Research and Primary Care, University Medical Center Utrecht, Utrecht, Kingdom of the Netherlands

Correspondence to Dr. Martin Bootsma, Mathematical Institute, Utrecht University, Budapestlaan 6, 3508 TA Utrecht, Kingdom of the Netherlands (e-mail: bootsma{at}math.uu.nl).

Received for publication December 4, 2006. Accepted for publication April 4, 2007.


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 References
 
An algorithm is presented to calculate likelihoods of acquisition routes using only individual patient data concerning period of stay and microbiologic surveillance (without genotyping). The algorithm also produces estimates for the prevalence and the number of acquisitions by each route. The algorithm is applied to colonization data of third-generation cephalosporin-resistant Enterobacteriaceae (CRE) from September 2001 to May 2002 in two intensive care units (ICUs) (n = 277 and n = 180, respectively) of Utrecht, Kingdom of the Netherlands. Genotyping and epidemiologic linkage are used as the reference standard. Surveillance cultures were obtained on admission and twice weekly thereafter. All CREs were genotyped. According to the reference standard, the daily prevalence of CRE in ICU-1 and ICU-2 was 26.1% (standard deviation: 15.4) and 15.1% (standard deviation: 13.4), respectively, with five of 23 (21.7%) and six of 21 (28.6%) cases of acquired colonization being of exogenous origin, respectively. On the basis of the algorithm, the endogenous route was responsible for more acquisitions than the exogenous route (p = 0.003 and p < 0.001 for ICU-1 and ICU-2, respectively). The estimated number of acquisitions is 30 and 27, and the estimated prevalence is 27.6% and 17.6% for ICU-1 and ICU-2, respectively. By use of longitudinal colonization data only, the algorithm determines the relative importance of acquisition routes taking patient dependency into account.

algorithms; disease transmission; Enterobacteriaceae; maximum likelihood; parameters; small population

Abbreviations: AFLP, amplified fragment-length polymorphism; CRE, cephalosporin-resistant Enterobacteriaceae; ICU(s), intensive care unit(s); MLE, maximum likelihood estimate; SD, standard deviation


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 References
 
Within health-care settings, antibiotic resistance increasingly hampers successful treatment of infections, especially in intensive care units (ICUs) (1). For some pathogens (e.g., vancomycin-resistant Staphylococcus aureus, pan-resistant Pseudomonas aeruginosa, and Acinetobacter species), the post-antibiotic era is approaching. With a limited armamentarium of antibiotics remaining available for treatment, infection prevention becomes more and more important. The epidemiology of antibiotic resistance in hospital settings, however, is complex, and quantitative understanding of the dynamics is essential for designing efficient infection control strategies. As only a fraction of colonized patients will develop infections (2), the true volume of antibiotic resistance is best represented by asymptomatic carriage (i.e., colonization). Changes in the prevalence of colonization with antibiotic-resistant microorganisms within hospital settings may occur through different processes: admission and discharge of colonized and noncolonized patients; mutations, changing susceptible bacteria into resistant ones, followed by selection due to antibiotic pressure; and cross-transmission, usually via temporarily contaminated hands of health-care workers (3). A key characteristic of cross-transmission is dependence among patients. The risk of acquisition (also called "colonization pressure") is influenced by the colonization status of other patients (4). This has been demonstrated for methicillin-resistant S. aureus (5), vancomycin-resistant enterococci (6), and Enterobacteriaceae (7).

Because of the typically small patient populations in ICUs (usually <20) and the rapid patient turnover, large fluctuations in proportions of colonized patients occur naturally (3). In addition, the dependence created by cross-transmission leads to overdispersion and autocorrelation in the number of colonized patients per day (8).

As the quantification of infection routes is relevant for the design of infection control strategies, as well as for the interpretation of the observed effects of interventions (8, 9), our aim here is to determine from the available data the relative importance of the various routes leading to detectable colonization. The Markov model proposed by Pelupessy et al. (10) uses longitudinal data concerning the number of patients colonized with a certain pathogen as input for maximum likelihood estimation of acquisition parameters. The extension introduced here uses data on individual patients, with the advantages that we

  • can explicitly distinguish rates of admission of colonized patients from endogenous selection rates.
  • can use the actual changes in bed occupancy (11, 12); that is, there is no need to assume that all beds are occupied and that the length of stay is exponentially distributed.
  • can take the moments of obtaining cultures and the results of these cultures as the bookkeeping cornerstone of the model, while a stochastic model estimates the status of patients between culture-sampling moments.
  • can allow for incorporation of other patient characteristics, for example, antibiotic use.

So the model formulation is data driven from the very beginning and incorporates all the information that is available. It yields maximum likelihood estimates (MLEs), as well as confidence regions, for acquisition parameters, thus enabling the identification of the dominant acquisition route. Moreover, the probability that a specific patient is colonized at a given time can be determined. This allows for calculation of relevant quantities, for example, the prevalence and the expected number of acquisitions in the unit.

As the modifications from previously published methods (10, 11) are based on incorporating known information (duration of stay per patient, the exact moments of culturing, and the results of the culturing) explicitly into the model without making additional assumptions, this method stays closer to reality and, hence, is expected to produce more reliable results. Here, we apply the method to data from two ICUs concerning third-generation cephalosporin-resistant Enterobacteriaceae (CRE) and test the performance by comparing conclusions concerning the relative importance of endogenous and exogenous acquisitions with those obtained from genotyping data that, in this particular case and for the limited period of 8 months, are available. We have chosen CRE as the marker pathogen because, according to the literature, both endogenous and exogenous acquisitions can contribute to its epidemiology (13, 14). This is in contrast to the epidemiology of methicillin-resistant S. aureus and vancomycin-resistant enterococci, where exogenous acquisition is known to be a much more dominant acquisition route.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS
 DISCUSSION
 APPENDIX
 References
 
The underlying acquisition model
An important building block for our algorithm is a mechanistic acquisition model, which governs the changes in colonization status. It incorporates the different infection routes and, preferably, requires the specification of only a few parameters. The mechanistic model that we use in the present study is characterized by the following assumptions.

  • Patients can be in two states. Either a patient is colonized, that is, he/she carries the pathogen of interest at a level that is, in principle, detectable, or a patient is uncolonized, that is, he/she does not carry the infective agent at a detectable level.
  • Once a patient becomes colonized, he/she remains colonized during the rest of the stay.
  • Uncolonized patients can acquire colonization exogenously by transmission or can go through an endogenous process in which the (already present) pathogen grows to detectable levels.
  • When we know the colonization status of all patients at a certain time, we know the probability per unit of time for uncolonized patients to acquire colonization. More precisely, we assume that the probability per susceptible patient to turn into a colonized patient per infinitesimal small unit of time {Delta}t is ({alpha} + ß I/n){Delta}t, where {alpha} represents the endogenous term and ßI/n the cross-transmission term, with I the number of colonized patients in the ICU and n the total number of patients in the ICU. Both {alpha} and ß should be nonnegative and are to be estimated from the data. (Note incidentally that the parameter for cross-transmission ß can also be expressed in terms of a reproduction number RN that gives the average number of secondary infected cases if all individuals in the ICU are noncolonized (15). For small values of ß, RN is approximately equal to ßD, with D the average length of stay in the unit.)
  • As only the days of culturing/admission/discharge are known, and not the exact moments, we use a day as the smallest time unit in our model and pretend that admission and discharge always occur at one and the same hour (e.g., 12:00 a.m.).
  • We assume that uncolonized patients can acquire colonization only from those patients who were already colonized at the hour of admission and discharge and not in a two-step procedure during one and the same day (i.e., patients who become colonized cannot infect other patients during the same day). This leads to a per diem probability per susceptible patient to acquire colonization of Formula .

The algorithm
As input data, we need the following for each patient:

  • day of admission
  • day of discharge
  • days at which a sample is taken (which is cultured)
  • results of cultures (assumed, for the time being, to be 100 percent reliable)
  • the colonization status at admission.

The output consists of MLEs (and confidence intervals) of the acquisition parameters {alpha} and ß, MLEs for the probabilities that patients are colonized on each day, and related quantities as the total number of acquisitions and the fraction of the acquisitions that can be ascribed to each acquisition route.

For patients whose colonization status is unknown, we work with the probability of being colonized. From day to day these probabilities evolve according to the mechanistic acquisition model. So once a culture result becomes available, we know how likely it was, given the parameters in the acquisition model. This "knowledge" then serves as the basis for the MLE. The rest is bookkeeping: We need to incorporate that patients are discharged and that new patients are admitted (note that it is difficult to assign a probability of being colonized to a newly admitted patient, but that it is straightforward if patients are cultured on admission).

A detailed technical description of the representation of the ICU state and the various operations that update this state in our C-code on the basis of the mechanistic model and the data is given in the Appendix. There we also explain how to use the algorithm to calculate relevant epidemiologic quantities, for example, the prevalence per day and the expected number of acquisitions that can be ascribed to each acquisition route.

Justification of the use of the algorithm
The standard likelihood ratio method (16) to calculate confidence areas using a {chi}2 test is only asymptotically correct (when the length of the study period approaches infinity) and requires that the true parameters, which are assumed to exist, are not on the boundary of the domain, hence, that both of the colonization routes are of importance. To test how well the asymptotic theory performs for finite study periods and when the true acquisition parameters are on the boundary of the domain, we simulated an ICU with 10 beds, which are always occupied. We varied the relative importance of the acquisition routes but kept the mean prevalence in the ICU constant at 20 percent, while 5 percent of the patients were colonized on admission. The length of stay in the ICU was exponentially distributed with a mean of 8 days. Observation of colonization was assumed to be perfect. Results are based on 100,000 simulations. For each simulation, we applied our algorithm to calculate 95 percent confidence areas for the acquisition parameters, and we calculated the fraction of the simulations for which the calculated confidence areas contained the parameters used in the simulation.

For the clinical study, a goodness-of-fit {chi}2 test (with two free parameters) based on the MLE was performed to test whether the model fitted the data accurately.

Setting of the clinical study
Colonization with CRE was studied in a medical (ICU-1) and a neurosurgical (ICU-2) ICU of the University Medical Center Utrecht, Utrecht, Kingdom of the Netherlands. This study was approved by the institutional review board. No informed consent was required. ICU-1 has 10 beds, four of which are in separate rooms, and ICU-2 has eight beds, one in a separate room. Nursing and medical staffs are not shared between these ICUs. Standard infection control measures were used in both units, and these did not change during the period of study.

Microbiologic surveillance and genotyping
During an 8-month period from September 2001 to May 2002, rectal colonization with CRE was determined for all patients admitted to the two ICUs. Rectal swabs were obtained on admission and twice weekly thereafter. Swabs were plated on Chromogenic UTI agar (Oxoid Limited, Basingstoke, United Kingdom) supplemented with 8 µg/ml of cefpodoxime (Aventis Pharma, Paris, France) and 6 µg/ml of vancomycin to suppress growth of Gram-positive bacteria. All morphologically different colonies were further processed. Species identification was performed by use of VITEK II (bioMérieux Benelux B.V., 's Hertogenbosch, Kingdom of the Netherlands). Additional susceptibility testing was performed by microdilution according to guidelines from the Clinical and Laboratory Standards Institute and, subsequently, all isolates not resistant to either cefpodoxime or ceftazidime were excluded from analysis. Two morphologically different isolates per species per patient (if available) were genotyped by use of an amplified fragment-length polymorphism (AFLP) (17). If more than two isolates of one species were available, the first and last isolates were selected. Genetic relatedness was determined by both visual and computerized interpretations of AFLP patterns of isolates of epidemiologically linked patients. A similarity of more than 80 percent, based on similarities in AFLP patterns among multiple isolates obtained from individual patients, was used as the cutoff point.

Colonization with CRE was classified as "present on admission" when CRE was demonstrated in cultures obtained less than 48 hours after admission and as "acquired" when demonstrated in cultures obtained 48 hours or more after admission with a previous negative culture. Two patients in the same ICU were considered to be epidemiologically linked when either these patients had an overlapping period of stay or, to allow for survival of pathogens in unidentified reservoirs (18), the time between discharge from the ICU of one of the patients and admission to the ICU of the other patient was at most 7 days. We evaluated the effect of a change in the length of this time window. Possible unidentified reservoirs are health-care workers, environmental contamination, and other patients, which are not sampled at the site of colonization. Cross-transmission was defined as acquired colonization with a CRE that is genetically similar to one previously found in an epidemiologically linked patient. Acquired colonization without epidemiologic linkage or genetic relatedness was considered to be endogenous.


   RESULTS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
DISCUSSION
 APPENDIX
 References
 
Colonization characteristics
In all, 457 patients were studied (277 admitted to ICU-1 and 180 to ICU-2), and 1,243 rectal swabs were obtained (753 in ICU-1 and 490 in ICU-2) (table 1). Adherence to the surveillance protocol was 97 percent. Forty-eight patients in ICU-1 and 35 patients in ICU-2 were colonized during their stay. In ICU-1, 23 patients were colonized on admission, and 23 patients acquired colonization. In ICU-2, 10 patients were colonized on admission, and 21 patients acquired colonization. Routes of acquisition could not be determined for six patients (two in ICU-1 and four in ICU-2), because the first cultures were taken 48 hours or more after admission or because patients had been admitted to the ICU before the start of the study. The mean daily prevalence of colonization with CRE was 26.1 (standard deviation (SD): 15.4) percent in ICU-1 and 15.1 (SD: 13.4) percent in ICU-2. Acquisition rates were, respectively, 17/1,000 and 18/1,000 patient-days at risk in ICU-1 and ICU-2. The mean time to acquire colonization for patients who acquired colonization was 6 (SD: 8) days in ICU-1 and 8 (SD: 11) days in ICU-2 (table 1). In total, 174 isolates (107 patients from ICU-1 and 67 patients from ICU-2) were genotyped. On the basis of AFLP results and epidemiologic linkage, five patients in ICU-1 and six patients in ICU-2 acquired colonization via cross-transmission. Therefore, five of 23 (21.7 percent) cases and six of 21 (28.6 percent) cases of acquired colonization resulted from cross-colonization in ICU-1 and ICU-2, respectively, representing cross-transmission rates of 3.6 and 5.3 per 1,000 patient-days at risk in ICU-1 and ICU-2, respectively. The ratios between endogenous and exogenous acquisitions were 3.6:1 for ICU-1 and 2.5:1 for ICU-2. The time interval in the definition of epidemiologic linkage hardly influenced the number of acquisitions that were ascribed to cross-transmission. Indeed, in both ICUs, only one case of cross-transmission would have been misclassified if the length of the time window were zero (actual durations between recipients and presumed donor patients were 3 and 4 days). More cases of acquisition would have been considered as cross-transmission only if the time window allowed exceeded 21 days.


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  		TABLE 1. Colonization characteristics of patients admitted to the two intensive care units, Utrecht, Kingdom of the Netherlands, 2001–2002*

 
Model predictions
The MLEs for the parameters {alpha}, describing endogenous processes, and ß, describing cross-transmission, with their 95 percent confidence areas and lines of equal importance of both acquisition routes are depicted in figure 1. In ICU-1, MLEs for {alpha} and ß were 0.022 (95 percent confidence interval: 0.013, 0.032) and 0 (95 percent confidence interval: 0.0, 0.035), respectively (figure 1A). In ICU-2, MLEs for {alpha} and ß were 0.024 (95 percent confidence interval: 0.015, 0.035) and 0 (95 percent confidence interval: 0.0, 0.054), respectively (figure 1B). A reanalysis of the current data with the "old" model (10) yielded {alpha} = 0.027 (95 percent confidence interval: 0.016, 0.036) and ß = 0 (95 percent confidence interval: 0, 0.050) for ICU-1 and {alpha} = 0.019 (95 percent confidence interval: 0.012, 0.027) and ß = 0 (95 percent confidence interval: 0, 0.056) for ICU-2, where the parameter {alpha} for the endogenous route also includes admission of colonized patients.


Figure 1
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  		FIGURE 1. Contour plots of the likelihood of the acquisition parameters {alpha} (endogenous acquisition) and ß (exogenous acquisition) for cephalosporin-resistant Enterobacteriaceae in a medical (A) and a surgical (B) intensive care unit, respectively (ICU-1 and ICU-2), Utrecht, Kingdom of the Netherlands, 2001–2002. The black area represents the 95% confidence area. The line represents the parameters for which the endogenous route and the exogenous route are equally important. The white dot represents the maximum likelihood estimators. The development of the confidence areas in both intensive care units in time is demonstrated in Web videos 1 and 2. (This process is illustrated in two supplementary videos posted on the Journal's website (http://aje.oxfordjournals.org/).)

 
The estimated numbers of acquisitions were 30 (95 percent confidence interval: 28, 32) and 27 (95 percent confidence interval: 26, 29) for ICU-1 and ICU-2, respectively, which exceed the observed numbers of acquisitions by 30 percent. The proportion of acquisitions due to cross-transmission was estimated to be 0 percent for both ICUs (95 percent confidence interval: 0, 30 (ICD-1) and 0, 25 (ICU-2)). The calculated proportions based on epidemiologic linkage and genotyping are 22 and 29 percent for ICU-1 and ICU-2, respectively (table 2), and so are included in the confidence interval only in the case of ICU-1. By use of the profile likelihood method and comparison of the maximum likelihood with the maximum likelihood constrained to the parameter space for which the relative importance of cross-transmission is more than 50 percent, the algorithm established, with a {chi}2 test, that less than 50 percent of the acquisitions were due to cross-transmission with p = 0.003 and p < 0.001 for ICU-1 and ICU-2, respectively.


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  		TABLE 2. Epidemiologic variables of cephalosporin-resistant Enterobacteriaceae according to genotyping in combination with epidemiologic linkage (observed) and model predictions, Utrecht, Kingdom of the Netherlands, 2001–2002

 
The estimated endemic prevalences based on the MLEs for {alpha} and ß were 27.6 percent and 17.6 percent for ICU-1 and ICU-2, respectively. Both values slightly exceed the observed endemic prevalence (26.1 (SD: 15.4) percent for ICU-1 and 15.1 (SD: 13.4) percent for ICU-2). Calculated RN values (expected number of secondary cases through cross-transmission generated by a primary case in a pathogen-free ward) were 0 (95 percent confidence interval: 0.0, 0.25) and 0 (95 percent confidence interval: 0, 0.44) for ICU-1 and ICU-2, respectively. A goodness-of-fit test gave no reason to question our mechanistic transmission model (p = 0.29 and p = 0.28 for ICU-1 and ICU-2, respectively).

Simulations show that the confidence intervals calculated by our algorithm are conservative when, as with our MLEs, one of the colonization routes is of no importance (refer to figure 2). When only the endogenous or the exogenous acquisition route is present, the calculated 95 percent confidence areas in fact represent 97 and 99 percent confidence areas, respectively. For other parameter combinations, 95 percent confidence areas will cover the true parameter indeed in 95 percent of the cases when the study period is sufficiently long. In the worst case, for study periods of 6 months, the calculated 95 percent confidence areas still cover the true parameters in 93.5 percent of the simulations.


Figure 2
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  		FIGURE 2. Fraction of simulations for which the true parameters are contained in the calculated 95% confidence area obtained by our algorithm. The lines represent different values of the relative importance (in %) of cross-transmission. Results are based on 100,000 simulations of an intensive care unit (ICU) with 10 beds, which are always occupied. The mean prevalence in the ICU was kept constant at 20%, while 5% of the patients are colonized on admission. The length of stay in the ICU was exponentially distributed with a mean of 8 days. Observation of colonization was assumed to be perfect. A perfect method to determine 95% confidence areas would yield the constant 0.95 irrespectively of the duration of the study period.

 

   DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
APPENDIX
 References
 
In this study, we applied the Markov chain method to real data (i.e., colonization with CRE in ICU patients) and compared model predictions with traditionally used definitions of exogenous and endogenous colonization. From this limited data set, both the traditional method and the new method drew the same conclusion that the endogenous route is predominant. The Markov model, therefore, seems a promising tool for disentangling the contributions of various acquisition routes on the basis of longitudinal data, without requiring labor-intensive and costly genotyping procedures.

The Markov algorithm ascribed less cases to cross-transmission than found by way of genotyping and epidemiologic linkage data, but the confidence intervals for both methods overlap. Note that the reference standard is not cast iron. The definition of epidemiologic linkage contains an arbitrary time window of 7 days, but more importantly, if two patients are colonized with the same genotype and are epidemiologically linked, this does not necessarily imply that one patient acquired colonization from the other. For instance, a widely spread clone in the extramural population could give the false impression that many acquisitions are exogenous. On the other hand, genotyping and epidemiologic linkage may also underestimate the amount of transmission. This may happen, for example, when the genotyping method is too discriminatory. So it may in fact be that the algorithm provides more reliable estimates than the reference standard does.

Although the details of the algorithm as presented in the Appendix may seem complicated, the input to the model consists of a database with only the moments of culturing and the results of these cultures combined with the admission and discharge data from the patients. The output of the algorithm provides results with a clear medical interpretation. Hence, when made user friendly, the software can be a valuable tool that can be used routinely in settings where colonization data are collected.

The framework allows adaptation to alternative Markovian transmission models and, importantly, individual patient characteristics, such as antibiotic use, the room in which the patient is treated, scores for the severity of illness, or multiple sites of colonization, can be incorporated (19).

The Markov methodology may also improve the reliability of the interpretation of results of interventions. Many infection control interventions (such as improving hand hygiene, use of gloves and gowns, and antibiotic cycling) have been analyzed in quasiexperimental designs, such as before-and-after studies (6, 9, 20–23). Results were evaluated by standard statistical tests, such as the {chi}2 test, Student's t test, and regression analysis, that neglect dependence among patients. If cross-transmission is relevant, differences between the baseline and intervention periods, considered to be significant according to these statistical tests, do not necessarily show causality between intervention and outcome; for a quantitative example of how wrong conclusions can be when dependence is simply ignored, refer to the report by Nijssen et al. (24). The Markov model provides estimates of confidence intervals for endogenous and exogenous transmissions, in itself correcting for autocorrelation when cross-transmission is relevant and for chance processes, such as a temporarily lower admission rate of colonized patients.

Our model has some limitations. First, the incorporation of more complex acquisition routes (environmental contamination (25) and persistently colonized health-care workers (26, 27)) requires that the "state" is adjusted such that the Markov property is retained.

Second, colonization is assumed to remain until discharge, which holds true for many but not all antibiotic-resistant nosocomial pathogens. Yet, the possibility of intermittent colonization and eradication can easily be included.

Third, the running time of the algorithm increases with an increasing number of patients with unknown colonization status (e.g., when incorporating the possibility of false positive and false negative culture results (12)). The actual unit size, on the other hand, can be very large, as long as the number of patients for whom the actual colonization status is not known does not become much larger than 10. If it does, the method can still be used, but techniques to approximate the likelihood (e.g., expectation-maximization algorithm (28)) have to be used.

Fourth, only a limited number of acquisition routes can be incorporated in the model, as otherwise there will be too many parameters that have to be estimated from the (limited amount of) data.

After the initial work of Pelupessy et al. (10), estimation of the relative importance of the different acquisition processes of antibiotic resistance was pursued in several studies. Cooper and Lipsitch (8), building on the work of Pelupessy et al. (10), proposed a "hidden Markov model" to analyze infection data, whereby the previous Markov model (10) governs the unobserved dynamics of colonization and colonized patients have a constant probability per day of developing an infection. As infection prevalence only represents the tip of the iceberg, long surveillance periods (during which the parameters should remain constant) are needed to derive reliable estimates of the parameters in the underlying transmission process. The counterbalance is that longitudinal data on infection are easier to obtain than data on colonization. A recent study of Mikolajczyk et al. (29) uses the mean time between different outbreaks to determine the importance of the endogenous route. The advantage is that the calculations are easy, but the application is limited to settings with a peak-trough prevalence pattern, and long study periods are needed. The studies of Forrester and Pettitt (11) and Forrester et al. (12), also based on the model of Pelupessy et al. (10), used Monte Carlo Markov chain methods (30–32). They estimated transmission rates for methicillin-resistant S. aureus in an ICU where cultures were performed twice weekly. No cultures were performed on admission, all patients were swabbed on the same days of the week (which was required in their analysis), and they assumed that all acquisitions of colonization between two successive culture moments were independent of each other. The Monte Carlo Markov chain method is a useful and flexible tool, but it requires choosing a prior distribution of the parameters (although this need not be a disadvantage when good prior information is available), and choosing the burn-in period and bad mixing properties of the Markov chain may thwart its application.

In this paper, we focused on the methodology and the data analysis, rather than on the clinical effects of candidate infection control measures in the considered units or on risk factors for colonization. Thus, one should not apply our findings concerning the unimportance of cross-transmission too readily to other settings with different patient populations, infrastructure, ecology, antibiotic use, infection control adherence, patient:staff ratio, and colonization pressure. Our aim here has been to introduce a reliable method for obtaining clear conclusions from data that are not too difficult to collect. We hope that the method will be fruitfully applied to investigate obscure details of acquisition for many other nosocomial pathogens in a variety of hospitals/ICUs.


   APPENDIX
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
References
 
Definitions
We divide the period of stay of a patient retrospectively into (at most) three periods based on the results of the culturing (refer to Appendix figure 1).

Patient is known to be uncolonized: t0 ≤ t ≤ t1.
Patient may or may not be colonized: t1 < t < t2.
Patient is known to be colonized: t2 ≤ t ≤ te.


Figure 3
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  		APPENDIX FIGURE 1. Example of a timeline of a patient. "+" and "–" signs correspond to positive and negative cultures, respectively.

 
Per day, we have three categories of patients, uncolonized patients, patients whose colonization status is uncertain, and colonized patients. We label these three categories Formula , Formula (for "questionable"), and Formula , respectively. The numbers of patients in the categories are represented by, respectively, u, q, and c. For later convenience, for every time t, we order the patients in category Formula in increasing order of the time they are already in category Formula ; that is, a patient entering category Formula will be the first one in the ordering.

By definition of the categories, we are certain about the colonization status of the patients in Formula and Formula . For each day, the number of patients in these two categories can be determined from the data directly. Hence, u and c are treated as (time-dependent) parameters and, for computational reasons, patients in Formula and Formula are not included in our definition of the state space. As each of the patients in Formula can be colonized or not, the total number of possible states for the ICU is Formula = {0, 1}q. Note that the cardinality of the state space, 2q, changes over time as q may change from day to day.

Each ICU state is denoted by a vector v = (v1, v2, ..., vq), with vk isin {0, 1}, where vk denotes whether the kth patient in Formula is colonized or not. The state (v1, v2, ..., vq) can also be represented by the binary number v1v2 ... vq and, therefore, we have a natural labeling j of each of the 2q states (0 ≤ j ≤ 2q – 1).

For notational convenience, we would like to be able to switch back and forth between a state represented as a finite sequence of 0s and 1s, that is, as an element of Formula , and its corresponding number. Therefore, we introduce the numbering function defined as follows:

Formula (1)
The inverse of the numbering function, N–1, relates a state number m (0 ≤ m ≤ 2q – 1) to the colonization status of the individuals in Formula . Specifically, the component N–1(m)k shows whether in ICU state m individual k (1 ≤ k ≤ q) is colonized or not. We also introduce an ordering on Formula :

Formula (2)
and the l1 norm:

Formula

As the state is actually uncertain, we want to use a stochastic description and assign to each state a probability that it is the actual (unknown) state. So we introduce the probability vector p(t) = {p0(t), p1(t), ..., p2q–1(t)} of length 2q in which pj(t) denotes the likelihood that the ICU is in state j.

We consider a period of observation from time 0 until time T. The observations can be divided into two parts: those before or at time t and those after time t. We define the "forward" vector vf, a column vector, on the basis of the observations until time t. The forward vector has as its components the unnormalized (i.e., relative) probabilities that the system is in a specific state at time t if we take into account only the observations until time t. However, the best estimate for p(t) is no longer vf(t) when results of cultures performed after time t become available. To take such additional information into account, we define the "backward" vector vb(t), a row vector, for which the ith component is the unnormalized probability that, given that the ICU is at time t in the state numbered i, the ICU will develop in a way that is compatible with all observations after time t. With these definitions, the ith component of the probability vector p(t) that takes all observations into account is

Formula (3)
All observations before or at the start of the study period are incorporated in vf(0). In the case that all patients are cultured at t = 0, we know the ICU state at time t = 0 with certainty, and the component of vf(0) corresponding to this state will be one, while all other components of vf(0) are zero. When t = T, all states are compatible with the observations after time T as there are no such observations, and hence vb(T) = 1, with 1 the row vector with all elements equal to one.

We now construct an algorithm to calculate the forward and the backward vector for all 0 ≤ t ≤ T. We need four operators to describe, respectively:

  1. evolution according to the mechanistic model.
  2. incorporation of the culture results. That is, states that are not compatible with the results of the culturing are given zero a posteriori likelihood.
  3. removal of the patients that leave state Formula , in particular, adaptation of the state space. When a patient leaves state Formula , the number of possible states is reduced by a factor 2.
  4. incorporation of the new patients in Formula , in particular, adjustment of the state space. With each additional patient in Formula , the state space increases in size with a factor 2.
The updating of the forward and backward vector can be expressed in terms of these four operators.

Evolution in the absence of new observations
To calculate the time evolution of the forward vector vf, we need the mechanistic model. The mechanistic model gives probabilities Amn, (0 ≤ m, n ≤ 2q – 1), which describe how likely state m is at time t + 1 just before culturing, discharge, and admission, given that the system was in state n at time t just after the culturing, discharge, and admission. At this point, we do not yet express in the notation that Amn depends on t, simply since q does; note that it depends on q what the numbers m and n tell us about the ICU state. The evolution can then be defined in terms of matrix multiplication:

Formula (4)
The matrix A has a special structure. Let {pi}(k) be the probability that an uncolonized patient acquires colonization during a day, given that there are k colonized patients in the ward. Each transition probability in column m is either zero, when the transition to state m is not allowed by the mechanistic model, or it can be written as a product of powers of {pi}(c + j) and (1 – {pi}(c + j)), with c the number of colonized patients in Formula and j the number of patients in Formula that are colonized when the system is in state n. Explicitly,

Formula (5)
For instance, in the case that there are two c and u patients in Formula , Formula , and Formula , respectively, with {rho}(k) = 1 – {pi}(k), the matrix A becomes:

Formula

Note that, when u != 0, the matrix A does not preserve the norm of the vector on which it acts. (This is due to the fact that we leave out of consideration all transitions that could in principle have happened to the Formula category.)

Incorporation of the results of the culturing
We now will use the culture results, the discharge data, and the admission data. Suppose that the kth patient in Formula is cultured. By the definition of the category Formula , this culture will be positive. Therefore, only the states m, 0 ≤ m ≤ 2q – 1, with N–1(m)k = 1 are allowed by the data, and the other states have zero a posteriori probability. Mathematically, culturing of patient k in Formula amounts to projecting the vector Aw on a linear subspace isomorphic to Formula . The diagonal matrix

Formula (6)
is given by

Formula

Example: In the case that q = 3 and we culture the second patient in Formula and before the culturing the state vector is w = (w0, w1, ..., w7), then after the culturing, the vector will be (0, 0, w2, w3, 0, 0, w6, w7).

If several category Formula patients are cultured at the same time, the operator C(t) consists of a product of Cks. When a category Formula patient "leaves" Formula , either because he/she was cultured or because he/she leaves the unit without being cultured, the number of possible states is reduced by a factor 2.

Removal of patients who leave Formula
For 1 ≤ k ≤ q, we can define the operator Rk that removes the kth patient in Formula via

Formula (7)
where the components of w' are defined by w'N(v1, ..., vq–1) = {sum}iisin{0,1}wN(v1, ..., vk–1, i, vk, ..., vq–1). This operator Rk adds the probabilities of the two states for which the colonization status of the remaining q – 1 patients is identical.

If several category Formula patients "leave" Formula at the same time, the operator R(t) consists of a product of Rks. To avoid confusion about which of the patients in Formula "leaves" Formula , we should use some convention, for instance, order the operators Rk such that we do the removal in decreasing order of the patient number in Formula .

Incorporation of patients who enter Formula
Suppose now that l patients enter category Formula at a certain time t. By the definition of the category Formula , patients enter category Formula directly after their last negative culture, so we know that these patients enter category Formula uncolonized. As we ordered the patients in category Formula according to the day they entered this category, these l patients correspond to the first l digits in the binary expansion. Because of this ordering, the function Il that deals with the admission of l new patients to Formula is defined as follows:

Formula (8)
where the elements in the vector w' are given by (0 ≤ k ≤ 2q+l – 1):

Formula (9)
Note that R(t) and Il involve a change of the dimension of the state space. Indeed, we "glue" together state spaces of different sizes according to the need as exposed by observed events.

The forward process
With the previous definition of the operators, the "forward vector" can be written thus:

Formula (10)
where we used the convention that the terms in the product are ordered according to Formula The forward vector can also be derived iteratively from the recursion:

Formula (11)

The likelihood of the observed events during 1 day is the norm of the final state vector (assuming that the initial state vector had norm 1). More precisely, the likelihood is given by |CAvf|/|vf|. The likelihood of the observed events over several days is the product of the relevant 1-day likelihoods, and the overall likelihood is given by |vf(T)|. This likelihood function leads to maximum likelihood estimates of parameters and to confidence regions (16).

Note that, for calculation of the likelihood of the observations, it suffices to consider only the forward vector. When during a certain period the dynamics of acquisition in an ICU are followed, the new observations that become available each day can be processed by the algorithm to improve the estimates of the transmission parameters as more data become available.

The backward process
The "backward vector" can be written as follows:

Formula (12)
or derived iteratively from the backward recursion:

Formula (13)
To explain equation 13, we consider the special case that both R(t + 1) and I(t + 1) are the identity operator or, in other words, the case that neither discharge nor admission occurs at t + 1. (The general case differs from the special case in bookkeeping aspects; all truly probabilistic considerations are incorporated in A(t) and C(t + 1).) Let S(t) denote the state at time t, and let Formula denote the observations for {tau}1 ≤ t ≤ {tau}2. We can write the following notation:

Formula
which is exactly the ith component of the identity 13. Here, we have used in particular that, when we look at times ≥t + 2 and condition on the state being j at t + 1, knowledge about the state at t is irrelevant.

Combining both processes
Note first of all that the inner product vb(t)vf(t) is independent of t and, hence, is equal to 1vf(T) = |vf(T)|, the overall likelihood of the observations. Once we know vf(t) and vb(t) and, hence, given all the information that we possess, the true probabilities for each state at all times, we can calculate the expected prevalence by averaging over all the states. Calculation of the number of acquisitions per day and the subdivision of this number according to the routes requires one additional step.

Fix t. Let P(j, i) be the probability that the ICU is in state j at day t and in state i at day t + 1. With Km the projection operator on the mth component, we can write the next equation:

Formula (14)
To explain equation 14, we again focus on the case in which there is neither discharge nor admission:

Formula (15)
We can write as follows:

Formula (16)
and

Formula (17)
Combining the identities 15, 16, and 17, we obtain equation 14. (Strictly speaking, the derivation above applies only when P(j, i) > 0.)

The expected number of acquisitions during day t is {sum}i, jP(j, i)gij, with gij the expected number of acquisitions during the transition from ICU state j to i. Clearly, gij depends on whether patients who were discharged at day t + 1 without being cultured at discharge acquired colonization during day t or not. Therefore, it is convenient to count the number of acquisitions before performing the bookkeeping operations. So, we adapt the evolution matrix and define the matrix m(t) by equation 18:

Formula (18)
with aij(t) the components of the evolution matrix A(t) and fij(t) the expected number of acquisitions by a route or combination of routes in the case of a transition from state j to state i. Note carefully that several choices of fij are relevant; also note that these numbers depend on t simply because the state space and, hence, the precise meaning of i and j change with time. In case we are interested in the number of acquisitions by the endogenous route, we define as follows:

Formula (19)

The expected number of acquisitions {theta}(t) during day t by the route(s) under consideration is given by the following inner product:

Formula (20)
When we sum over all days, we obtain the expected total number of acquisitions per route during the study period. We use these expressions to calculate the relative importance of each route. Confidence intervals for the relative importance of each route can be obtained by the profile likelihood method using a {chi}2 test. If the relative importance of cross-transmission for the maximum likelihood parameters is less than 50 percent, we can establish a confidence level that less than 50 percent of the acquisitions were due to cross-transmission by comparing the maximum likelihood with the maximum likelihood constrained to the parameter space for which the relative importance of cross-transmission is more than 50 percent.


   ACKNOWLEDGMENTS
 
Financial support was provided by Nederlandse Organisatie voor Wetenschappelijk Onderzoek grants ZonMW 2100.0051 and CLS 635.100.002.

The authors thank Richard Gill for very helpful advice and Hans Metz for a useful conversation.

Conflict of interest: none declared.


   References
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 References
 

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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.
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