# How to assess the relative importance of different colonization routes of pathogens within hospital settings

^{*}Department of Mathematics, University of Utrecht, Budapestlaan 6, 3584 CD Utrecht, The Netherlands; and

^{†}Department of Internal Medicine, Division of Infectious Diseases and AIDS, University Medical Center Utrecht, Heidelberglaan 100, 3584 CX Utrecht, The Netherlands

^{‡}To whom reprint requests should be addressed. E-mail: ln.uza.dgid@netnob.m.j.m.

## Abstract

The emergence of antibiotic resistance among nosocomial pathogens has reemphasized the need for effective infection control strategies. The spread of resistant pathogens within hospital settings proceeds along various routes of transmission and is characterized by large fluctuations in prevalence, which are typical for small populations. Identification of the most important route of colonization (exogenous by cross-transmission or endogenous caused by the selective pressure of antibiotics) is important for the design of optimal infection control strategies. Such identification can be based on a combination of epidemiological surveillance and costly and laborious as well as time-consuming methods of genotyping. Furthermore, analysis of the effects of interventions is hampered by the natural fluctuations in prevalence. To overcome these problems, we introduce a mathematical algorithm based on a Markov chain description. The input is longitudinal prevalence data only. The output is estimates of the key parameters characterizing the two colonization routes. The algorithm is tested on two longitudinal surveillance data sets of intensive care patients. The quality of the estimates is determined by comparing them to accurate estimates based on additional information obtained by genotyping. The results warrant optimism that this algorithm may help to quantify transmission dynamics and can be used to evaluate the effects of infection control interventions more carefully.

Infections with
antibiotic-resistant bacteria are emerging in hospitalized patients,
especially those treated in intensive care units (ICUs) (1). Nosocomial
pathogens with resistance to almost all commercially available
antibiotics [e.g., vancomycin-resistant enterococci (VRE) and
vancomycin-intermediate *Staphylococcus aureus*] have
confronted physicians with the possibility of a postantibiotic era (2,
3). Therefore, prevention of transmission of such microorganisms has
become even more important (4).

Patient colonization with antibiotic-resistant microorganisms within hospital settings is determined by admission and discharge rates of colonized and noncolonized patients and on the likelihood that noncolonized patients acquire colonization. Several distinct routes of colonization can be distinguished. Colonization may result from exogenous acquisition—i.e., from the environment or from other patients treated in the ward, usually via temporarily contaminated hands of health care workers (5). Alternatively, colonization may be undetectable until, because of the selective growth advantage provided by antimicrobial treatment, bacterial growth of resistant pathogens is such that detection limits of culture methods are exceeded. In these cases, colonization is regarded to be from endogenous sources. Because of the small patient populations involved (typically 10–20 patients in one ICU) large fluctuations in prevalence frequently occur just by chance. In longitudinal studies, fluctuations in point prevalences of colonization with antibiotic-resistant microorganisms in ICUs from 0% to 80% have been observed, in periods without changes in infection control or antibiotic prescription policies (6, 7).

The prevalence of antibiotic-resistant microorganisms in hospital settings is the resultant of the “traffic” along the various routes. Prevention of colonization and infection should be tailored on the relative importance of the transmission routes. Unfortunately, assessing the importance seems to require the use of microbiological surveillance in combination with genotyping of microorganisms. Microbiological surveillance is necessary to determine prevalence, and genotyping is essential to demonstrate that acquired colonization is a result of either cross-transmission or endogenous colonization. As genotyping techniques are costly and laborious, wide-scaled use is precluded. As a result, the relative importance of the different routes of colonization in ICUs is generally not known. Such knowledge is also necessary to evaluate the effects of interventions reliably. Because of the natural occurrence of large fluctuations in prevalence, it is difficult to ascribe a change in prevalence to a specific intervention. Separate quantification of exogenous and endogenous incidences will provide more reliable information.

In the present study, therefore, we develop a method for estimating the relative importance on the basis of readily available longitudinal surveillance data. We do so within the framework of a Markov chain model, which involves assumptions concerning the admission and discharge of patients as well as pathogen transmission.

Recently, several mathematical models have been used to investigate
transmission dynamics of antibiotic-resistant pathogens within
hospitals and hospital wards (6, 8–10). For example, Austin *et
al.* (6) used the Ross model of vector-borne diseases to describe
the dynamics of VRE in ICUs and validated the model on epidemiological
data obtained in an ICU where colonization with VRE was endemic.
Special emphasis was put on the quantitative effects of classical
infection control measures, such as hand disinfection and staff
cohorting (6). Independently, Lipsitch *et al.* (8)
constructed a model of the spread of antibiotic resistance in
hospitals, and qualitatively predicted the effects of different
antibiotic strategies. Both models were basically deterministic, using
differential equations to describe bacterial transmission. Such
deterministic models may not be reliable when populations are small,
such as those in hospital settings, as opposed to general populations.
In some models the conclusions were, therefore, further tested against
Monte Carlo simulations (6, 10).

We will show that our algorithm for data analysis, which is based on a
stochastic (Markov chain) model, provides a useful tool to analyze the
dynamics of transmission and to determine the relative importance of
different routes of colonization within hospital settings by using only
longitudinal data on the number of patients being colonized. Our
assumptions concerning discharge and colonization formulated below are
identical to those of Austin *et al.* (6). The difference is
that we explicitly represent the patients in the actual number of beds
in a ward, rather than concentrating on the mean and the variance of
the number of colonized patients.

To test the method we have applied it to two data sets, both containing detailed information obtained by genotyping (6, 7). For this purpose we have initially ignored the information based on genotyping and have used only the time series of colonized patients. The accuracy of the model predictions, however, was assessed by comparison with the conclusions that were derived from the full data set, including genotyping results. In other words, the results of genotyping were used as a gold standard, to check how accurately we can assess the situation when such information is not available.

## Formulation of the Model

We consider the spread of one species of bacteria in an ICU
composed of *N* beds, which, reflecting the resources needed
to maintain them, are assumed to be occupied to full capacity at all
times. The patients are either colonized or noncolonized. A patient is
considered colonized when colonization has been demonstrated by
standard microbiological culture techniques. Rates of discharge are
denoted as 1/*d* (in days) for noncolonized patients and
1/*d*′ for colonized patients. A change in the number of
colonized patients in ICU, say *i*, can occur in the following
ways:

(*i*) A noncolonized patient is replaced by a colonized
patient. If admitted patients are colonized with probability
*q*, this replacement occurs with per bed rate
*q*/*d*.

(*ii*) A colonized patient is replaced by a noncolonized
patient. This replacement occurs with per bed rate (1 −
*q*)/*d*′.

(*iii*) A noncolonized patient acquires colonization
spontaneously—i.e., endogenous colonization occurs. The
chance for this event per unit time per patient is denoted by α.
Explosive growth under antibiotic treatment of preexisting small
numbers of resistant microorganisms is the most likely explanation, but
the development of resistance *de novo* is an alternative or
additional possibility. We assume that every patient may acquire
colonization spontaneously and disappearance of evident colonization
will not occur (true for typical lengths of stay).

(*iv*) A noncolonized patient acquires colonization from
another patient [cross-colonization, usually via contaminated hands of
health care workers (HCWs)]. The rate of transmission is assumed to be
equal to θ*i*/*N*, with θ a constant depending
on the details of patient–HCW interactions, but independent of the
number of colonized patients, *i,* whenever the duration of
contamination for HCWs is small.

We can, and do, ignore the replacement of a patient by another patient with the same colonization status. With these transition rates specified we can now formulate a bookkeeping equation (the so-called master equation) for this continuous-time Markov process:

with *p _{i}*(

*t*) denoting the probability of having

*i*colonized patients at time

*t*and the parameters

*a*and

*c*defined in terms of

*q*,

*d*,

*d*′, and α introduced above, by

The compound parameter *a* is the spontaneous
colonization rate, including replacement of noncolonized by colonized
patients; *c* is the “decolonization” rate (i.e.,
replacement of colonized by noncolonized patients), and θ is the
transmission rate. The changes in the course of time of the number of
colonized patients in the model are completely determined by these
three parameters.

## The Stationary Distribution

The distribution of colonized patients will, in the long run,
converge to the stationary distribution
*p*^{s}, which is determined by the
condition *dP _{i}*/

*dt*= 0 and given explicitly by the formula

Note that the stationary distribution depends only on the ratio
*a*/*c* between the spontaneous colonization rate
(*a*) and the “decolonization” rate (*c*), and
the ratio θ/*a* between the transmission rate (θ) and
the spontaneous colonization rate (*a*). The key point to note
about the meaning of the stationary distribution is that
*p* gives the probability to find
precisely *i* colonized patients in the ICU at any given
moment of time. The stationary distribution for *n* = 10
and a variety of parameter values *a*/*c* and
*a*/θ are depicted in Fig.
Fig.1.1. An almost normal distribution occurs
when *a*/*c* = 1 and θ/*a* =
1, but for other parameter values the shape of the distribution is very
different. Note that the distributions are not very sharply peaked,
meaning that in the course of time nonnegligible fluctuations in the
actual number of colonized patients are to be expected. Therefore, an
observed increase in the number of colonized patients does not
necessarily indicate a deterioration of hygienic standards, as it may
be a chance fluctuation. Neither should low colonization rates be a
reason for complacency. Note that large θ/*a* values give
large “tails” to the distribution corresponding to outbreak
situations, but increases in *a* also shift the distribution
rightwards. Incidentally, we remark that for *q* tending to
zero the stationary distribution should converge to the quasistationary
distribution as studied by Nasell (11), but we did not explore this
connection.

Concentrating for the moment on the mean number of colonized patients,
we see (Fig. (Fig.2)2) that it goes to 0 when
*a*/*c* decreases (i.e., when decolonization occurs
much more frequently than spontaneous colonization), but rapidly
increases to *N* when *a*/*c* increases.
Even for large θ/*a*, when transmission is relatively
important, the number of colonized patients increases only linearly
with slope *N*, for small *a*/*c*. This
observation confirms the importance of infection control strategies
that separate colonized patients from noncolonized patients as quickly
as possible (leading to a small length of stay *d*′ and thus
to a small *a*/*c*).

## Estimating Parameters from Observational Data

Two sets of data on the epidemiology of, respectively, VRE and
*Pseudomonas aeruginosa* were obtained from different ICUs (7,
12). In both studies extensive surveillance of colonization and
genotyping were combined to determine prevalences of colonization and
the contributions of endogenous and exogenous colonization
to the incidence. Pulsed-field gel electrophoresis (PFGE) techniques
were used for genotyping (13). Cross-transmission was defined as a case
of acquired colonization with a pathogen with a genotype identical to
that of an isolate of another patient treated in the same ward and in
the same time period. Endogenous colonization was defined
as acquired colonization with a genotype that was not found to be
associated with colonization in any other patient. With these
definitions, cross-transmission was the responsible route of
colonization in 85% of the patients acquiring colonization with VRE
and in 13% of the patients acquiring colonization with *P.
aeruginosa* (7, 12). Table Table1
summarizes1
summarizes the main epidemiological determinants for these studies. The
parameters *c, a,* and θ of the model can be computed
directly from the information provided in Table Table1,1, and are represented
under “Observed” in Table Table22.

Independently, we determined the parameters *c, a,* and θ by
maximizing the *a priori* probability of occurrence of the
observed time series of the number of colonized patients. This
probability follows from the model as follows: if we write Eq.
1 in the form *dp*/*dt =
Ap*, then the chance of finding

*j*colonized patients at time

*t*+ Δ

*t*, given the fact that

*i*patients are colonized at time

*t*, is given by (

*e*

^{ΔtA})

_{ij}; so, given a list {

*c*

_{1}, …,

*c*} of the number of colonized patients at times {

_{n}*t*

_{1}, …,

*t*} it is easy to calculate the above-mentioned probability as

_{n}
where ** A**, and thereby

*P*

_{obs}, depends on

*a, c,*and θ. Using standard numerical techniques, we then determined the parameters

*a*,

*c,*and θ that maximize

*P*

_{obs}. The outcome is listed in Table Table2 under2 under “Maximum likelihood estimate (MLE) fit to model.” As shown, the two methods to estimate

*a*,

*c,*and θ (genotyping and MLE) yield comparable results. The key point is that the MLE yields the relevant information with a minimal input of data. The only data that should be available for this analysis are the surveillance data.

*There is no need for bacterial genotyping*. On the other hand, the data must encompass a sufficiently long period (how long is sufficient depends on the time scales in the ICU, but typically months) to reduce the uncertainty in the determination of the parameters.

Fig. Fig.33 shows contours of equal
*P*_{obs} for VRE. Given the observed
longitudinal data on the numbers of patients colonized with VRE, the
combination of parameters *a, c,* and θ that maximizes
*P*_{obs} is depicted by the central
contour (and listed as MLE in Table Table2).2). Indicated with crosses are the
parameter values obtained by using the complete data set (i.e., those
listed under “Observed” in Table Table2).2). As noted before, the
“directly observed parameters” (crosses) and the maximum
likelihood parameter estimates derived from the Markov chain model are
consistent. The contours of equal probability in Fig. Fig.33
*Right* are stretched in one direction along the line of
constant colonization pressure, defined by α +
*X*_{1}θ (with
*X*_{1} the mean prevalence of colonized
patients). The shape of the contours reflects the inherent
difficulty of distinguishing between two different routes of
colonization (spontaneous and cross-transmission), because both
increase the number of colonized patients. We can, however, quantify
the relative importance of both colonization routes: Plotted in Fig. Fig.33
*Right* is the line of equal importance of
endogenous colonization and cross-transmission
(approximately
θ/α*X*_{1} = 1). For
VRE the probability of the observations is greater above this line
[*P*(θ/α*X*_{1}
> 1) = 0.75], indicating that cross-transmission was at least as
important as, but probably more important than endogenous
colonization, in agreement with the conclusions based on bacterial
genotyping
(θ*X*_{1}/α ≈
5 ± 1.5).

*P*

_{obs}of

*a–c*and

*a*–θ slices of parameter space at θ = 0.91 and

*c*= 0.113, respectively, for VRE. Contours are linear in

*P*

_{obs}. The dotted line indicates equality between spontaneous colonization

**...**

Similarly, the values of *P*_{obs} are
depicted for *P. aeruginosa* in Fig.
Fig.4.4. Again, there is good agreement between
the parameter estimates and the observations. The results show a much
lower value for θ, which is in agreement with the direct
observations. The probability of the observations above the equality
line is low
[*P*(θ/α*X*_{1}
> 1) = 0.08], which is in accordance with the fact that
cross-transmission was relatively unimportant for *P.
aeruginosa* in this ICU
(θ*X*_{1}/α ≈
0.1 ± 0.02).

## Discussion

We have presented an algorithm to determine the relative importance of different routes of bacterial colonization within the context of a simple Markov chain model for small closed populations, such as ICUs or other hospital wards. It uses only readily available longitudinal data on prevalences of colonization in the ICU. We have demonstrated that it is possible to determine the relative importance of endogenous colonization and cross-transmission without resorting to costly and time-consuming methods of genotyping and despite the inherent intricacies of disentangling mechanisms that have the same effect.

The emergence of antibiotic resistance and the possibility of a
“postantibiotic” era have reemphasized the need for effective
measures to limit the spread of these pathogens. Classic infection
control practices have focused on improving antibiotic policies and
compliance with infection control measures, such as hand disinfection.
Although these measures remain cornerstones of infection control, they
have been insufficient to prevent the emergence of antibiotic
resistance. A better understanding of the dynamics of colonization and
infection may contribute to more successful infection control
strategies. For example, the mean endemic prevalences of VRE and
*P. aeruginosa* in the two different ICUs were almost
identical. Yet, both fingerprinting of multiple isolates and our
estimates from the longitudinal data clearly demonstrated that
cross-transmission was the most important transmission route for VRE,
whereas endogenous colonization was much more important for
*P. aeruginosa*. Such insights are crucial when designing
infection control measures.

Most of the models of nosocomial spread of pathogens are deterministic
(6, 8–10)—i.e., the dynamic behavior of the system is completely
determined by the initial conditions. However, because the populations
within closed hospital settings are usually small, chance effects
cannot be neglected (and that is why some authors supplement the
analysis of a deterministic model by simulations of a stochastic
variant—e.g., refs. 6 and 10). A Markov chain description has at least
two advantages over more conventional deterministic models, even when
incorporating Gaussian correction terms extends the latter. Markov
chain descriptions capture much of the chance fluctuations that are due
to the fact that we describe a finite, but often small, population of
discrete entities—i.e., the individuals. In addition, there is a
direct correspondence between observable quantities and model
variables, which allows extracting maximal information about the model
parameters from the available data. In typical pre- and
postintervention studies, outcomes (such as numbers of patients
colonized with a resistant microorganism) are usually compared by
standard statistical tests, such as χ^{2} tests or
logistic regression analyses. However, in doing so it is assumed that
patients are affected independently, which is not true for transmitted
pathogens. The proportion of other patients being colonized, also
called colonization pressure, amplifies the risk for noncolonized
patients to acquire colonization (14–17). Therefore, statistical tests
may yield significant differences between two periods when these are in
fact caused by chance. Estimation of the parameters of the Markov chain
model before and after an intervention would avoid such problems. The
estimates also show in what way the intervention was effective: Was
transmission reduced or did the occurrence of spontaneous cases
decline?

A central concept in infectious disease dynamics is the basic
reproduction number, *R*_{0}, which
corresponds to the average number of secondary infected cases (18, 19).
When applied to the dynamics of nosocomial pathogens,
*R*_{0} represents the number of
secondary colonized patients because of cross-transmission generated by
a primary case in a pathogen-free ward. Infection prevention aims to
decrease the effective *R*_{0} below unity
[it is helpful to speak about the effective reproduction number when
infection control measures have been applied (6)]. In our model the
effects of infection control measures are captured in θ, and the
effective *R* is equal to *d*′θ, so to
0.7 ± 0.2 for VRE and 0.14 ± 0.04 for *P.
aeruginosa*. Previously we found an effective *R* for VRE
of 0.7, using the same data set but a different method (6). This
finding implies that neither of the pathogens can persist in the ICU by
transmission alone, and that the continuous admission of already
colonized patients is needed for endemicity.
*R*_{0} for VRE was estimated to be ≈3.8
In the absence of infection prevention, however (6).

Our model encompasses a number of simplifications, which may limit its
practical applicability. We assumed that the patient population was
homogeneous: all patients were considered to be identical with regard
to the probability of their length of stay and susceptibility for
colonization. Moreover, we have assumed that the various processes
occur with certain constant probabilities in each time interval and
that the transmission of bacteria from patient to patient occurred
instantaneously. In reality some time will probably be needed for
bacterial multiplication after an event of colonization before
pathogens can be transmitted further. Moreover, some patients may be
more likely to act as sources for bacterial spread, because they
receive more patient contacts or such contacts are more likely to
result in cross-transmission. In addition, our model does not allow for
differences in the severity of colonization, as our patients are either
considered noncolonized and thus not contagious or colonized and fully
contagious. However, a more detailed model capturing all these
uncertainties would have more parameters, and so would not necessarily
be more reliable. It is also obvious that rapid changes in *q*
(i.e., admission prevalences), for example during epidemics, will
influence, and probably decrease, the reliability of the predictions.
We assumed *q* to be constant, which we think is reasonable
for the combination of time periods and the kind of nosocomial events
we consider. Finally, some important general aspects of data analysis
were neglected. For example, we just gave point estimates of the
parameters without trying to obtain more information about the
probability distribution of the parameters, given the data (20, 21).

In addition, all surveillance techniques will have a lower detection limit, which we have ignored even though it might have an impact on the conclusions. However, we think that the different routes of colonization, and thus relative impacts, would be equally affected. The use of more sensitive tests may identify more patients with presumed endogenous colonization, thereby diminishing the relative impact of exogenous colonization. On the other hand, such a test may identify more patients already colonized on admission, thereby diminishing the relative impact of endogenous colonization. And finally, a more sensitive test may identify more cases of cross-transmission, as patients may initially be colonized with very low numbers of microorganisms. Furthermore, more specific methods of microbial genotyping may diminish incidences of cross-transmission. For example, phenotypes of antimicrobial resistance profiles may not have sufficient discriminatory power to distinguish clonal difference (13). More work is needed to quantify the definition of colonization and its relation to transmissibility. Here, we have assumed that all colonized patients could be a source of transmission, which, we think, is a fair assumption.

In conclusion, we described a method that can identify the relative importance of various transmission routes of microorganisms on the basis of a sufficiently long time series of the number of colonized patients in hospital settings with, typically, a small total number of patients. Such information may both guide infection control strategies and help to carefully evaluate the outcome of intervention studies.

## Acknowledgments

We thank M. Lipsitch and M. C. M. de Jong for stimulating discussions and helpful comments.

## Abbreviations

- ICU
- intensive care unit
- VRE
- vancomycin-resistant enterococci

## Footnotes

This paper was submitted directly (Track II) to the PNAS office.

## References

*Stochastic Epidemic Models and Their Statistical Analysis*.

**National Academy of Sciences**

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