A Dynamic Transmission Model to Evaluate the Effectiveness of Infection Control Strategies

Abstract Background The advancement of knowledge about control of antibiotic resistance depends on the rigorous evaluation of alternative intervention strategies. The STAR*ICU trial examined the effects of active surveillance and expanded barrier precautions on acquisition of methicillin-resistant Staphylococcus aureus (MRSA) and vancomycin-resistant Enterococcus (VRE) in intensive care units. We report a reanalyses of the STAR*ICU trial using a Bayesian transmission modeling framework. Methods The data included admission and discharge times and surveillance test times and results. Markov chain Monte Carlo stochastic integration was used to estimate the transmission rate, importation, false negativity, and clearance separately for MRSA and VRE. The primary outcome was the intervention effect, which when less than (or greater than) zero, indicated a decreased (or increased) transmission rate attributable to the intervention. Results The transmission rate increased in both arms from pre- to postintervention (by 20% and 26% for MRSA and VRE). The estimated intervention effect was 0.00 (95% confidence interval [CI], −0.57 to 0.56) for MRSA and 0.05 (95% CI, −0.39 to 0.48) for VRE. Compared with MRSA, VRE had a higher transmission rate (preintervention, 0.0069 vs 0.0039; postintervention, 0.0087 vs 0.0046), higher importation probability (0.22 vs 0.17), and a lower clearance rate per colonized patient-day (0.016 vs 0.035). Conclusions Transmission rates in the 2 treatment arms were statistically indistinguishable from the pre- to postintervention phase, consistent with the original analysis of the STAR*ICU trial. Our statistical framework was able to disentangle transmission from importation and account for imperfect testing. Epidemiological differences between VRE and MRSA were revealed.

The dynamic tranmission model that we implemented in this work is an extension of the model described previously in (Thomas et al. (2013)). The model extensions have been implemented in part to allow for a more general transmission model to account for multiple modeling assumptions, as well as to build a framework for specifically addressing the performance of the intervention in the STAR*ICU trial. The model was implemented within a Bayesian framework, depending on data D, which consists of observed data and is also augmented with unobserved data. Examples of the augmented data are the times of colonization or loss of colonization. We assume that there are N I ICUs, and that the list of events ordered by time in ICU I is denoted by D I (D = ∪ I D I ), and model parameters are represented by θ. We can express the likelihood for the augmented data D, given the model parameters θ as g(e, t e ; θ)h(t e , t e − ; θ). (1) Here, we define t e and t − e as the times of the events e and the prior event, e − . g(e, t e ; θ) is a function that contributes to the likelihood according to the type of event (described in more detail below) e, while h(t e , t e − ; θ) represents probability that no event is observed between times t e − and t e .

Model parameters and assumptions
The term h(t e , t e − ; θ) which represents the probability that no events occur between the consecutive events e − and e, is given by the formula where S I (t) and C I (t) are the number of susceptible and colonized patients in ICU I at time t, λ I (t) is the transmission rate parameter in ICU I at time t, and η I is the rate of loss of colonization for patients in ICU I.

Transmission
We assume that acquisitions in ICU I at time t for a given uncolonized individual occur at a rate λ I (t)C I (t), which corresponds with the law of mass-action. In principal, we could allow λ I (t) to take on a number of functional forms, but in order to be consistent with the basic idea underlying the STAR*ICU trial, we assume that λ I (t) is piecewise constant taking one of two values,

Clearance
We assume that for a given colonized patient in ICU I, they lose colonization at a constant rate η I . Therefore, when a patient loses colonization in the model, the corresponding event contributes g(e; θ) = η I to the likelihood.

Surveillance tests
Negative test results can correspond with either a true negative or a false negative, which depends on the unobserved colonization status reflected in the augmented data. We define S(e) to be an indicator that the patient corresponding with event e is colonized at time t e . Given this notation, the formula for the probability g(e; θ) of the negative test is given by g(e; θ) = φ S(e) (1 − φ) (1−S(e)) , where φ is the probability of a false negative test.

Importation
The status of a patient at the time of admission contributes to the likelihood based on their probability of being colonized, or importing to the ICU at the time of admission. This importation probability depends on whether the admission event is the first admission or a readmission. Readmissions are modeled by a continuous time Markov chain in order to account for the possibility of switching colonization status during the period between consecutive admissions. In particular, between consecutive admissions, uncolonized become colonized at rate α I and colonized lose colonization at rate γ I . From these two parameters, we define the "steady-state", or first-admission importation probability as ν I = α I /(α I + γ I ). Therefore, for a first admission event, we assume a constant importation probability ν I into ICU I so that the contribution to the likelihood is For a patient whose admission event e is a readmission, their importation probability is computed using information from their previous discharge e d . In particular, we use standard methods for computing transition probabilities of a homogeneous CTMC over the during the period of time from t d to t e . It is straighforward to compute the probability of importation at a readmission, and the contribution to the likelihood is given by

In-situ probability
At the time of the beginning of the study, there will be patients present in ICU I that were admitted prior to the beginning of the study. For those individuals, their probability of colonization at the beginning of the study is treated as a nuisance parameter, σ I , called the in-situ probability. For a patient present at the onset of the study, their first observation which indicates presence in ICU I contributes g(e; θ) = σ

Non-random events
For all other events e assumed to be observed without error, g(e; θ) = 1.

Prior Distribution
Below we describe the prior distributions that help to complete the posterior distribution. We write the prior distribution as π I (·) can be further expressed as

Transmission parameters
We assume that (λ pre I , λ pre I ) are bivariate log-normal random variables. In particular, we assume that Here, µ pre and µ post are the mean log-transmission rates during the pre-intervention and post-intervention periods respectively across all ICUs, both assumed independent of the intervention. We define χ(I) to be an indicator variable that ICU I is in the intervention arm of the study, β is the effect of the intervention on mean log-transmission, and Σ is the covariance matrix for {(λ pre I , λ post I )} I .
The mean log-transmission rates (µ pre , µ pos ) and the corresponding covariance matrix (Σ) are assumed to have the normal-inverse-Wishart (NIW) distribution for their prior, which is the conjugate prior distribution corresponding with a multivariate normal likelihood.
The intervention effect parameter (β) is modeled having a normal prior distribution.

All other parameters
We use a gamma distribution as the prior distribution for outpatient colonization (α I ) and decolonization rates (γ I ), and for inpatient decolonization rates (η I ). The in-situ probability (σ I ) and false negative probability (φ) are given uninformative Beta priors.

MCMC Updates
Now we describe the sequence of steps used to update both the augmented data and the model parameters.

Augmented data
We use stochastic integration to account for the uncertainty in the augmented data set by updating the augmented data by sampling from the conditional distribution of each patient's history of colonization given the observed data, the current parameter values, and the current history of all other patients.
To sample the augmented data, we sequentially step through each patient admission and propose a new sequence of times that the patient might become colonized and decolonized. This proposal depends on the probability that the patient is an importation at the time of admission, as well as the current colonization and decolonization rates. The newly proposed augmented data is then accepted according to acceptance probability, following the Metropolis sampling scheme.
The complete history for a single patient admission can be specified by the initial state x (colonized or uncolonized), the number of times a change of state occurs y, and the sorted list of change times u = (u 1 , u 2 , . . . , u y ). Letting D − represent the current state of the augmented data for all but the episode being updated, we can write the conditional posterior for this episode as π(x, y, u|D − , θ) which can be calculated efficiently from the stored list of events as we describe them below. To implement an RJMCMC update, we first sample new values x and y from the distribution so that x and y are independent Bernoulli and Poisson variates. Atypically for RJMCMC this proposal does not depend on the current state. The values of γ and ρ can be chosen arbitrarily to tune the mixing performance of the sampler. We then sample y sorted U (t a , t d ) random variables v 1 , . . . , v y where a and d are the admission and discharge events for the episode. That is, we sample (v 1 , . . . , v y ) from the ordered uniform distribution Thus, (u 1 , . . . , u y , v 1 , . . . , v y ) forms the complete set of continuous variables, active and auxiliary, associated with the current state. We transform this to the variables (u 1 , . . . , u y , v 1 , . . . , v y ) required for the new state by setting u i = v i , i = 1 . . . y and v j = u j , j = 1 . . . y, in effect simply switching the role of the active and auxiliary variables. This transformation is clearly reversible and has Jacobian 1. Hence, the proposed state x , y , u is accepted with probability min 1, π(x , y , u |D − , θ) π(x, y, u|D − , θ)

Transmission parameter updates
To update λ pre I and λ post I for each ICU I, we use Metropolis sampling. To improve mixing, we sequentially generate 10 Metropolis sample updates, keeping the 10 th sample for the updated parameter values. For each Metropolis sample, we generate a pair of gamma(2, 1) random variables. Each pair of gamma random variates are then accepted or rejected according to the standard Metropolis likelihood ratio. This process is repeated for each ICU I.
We sample µ = (µ pre , µ post ) and Σ using Gibbs samples by sampling directly from the normal inverse Wishart distribution. In particular, we make the following random draws: The parameters necessary for sampling from the above distribution are given bȳ and we assume Ψ 0 to be the identity matrix.
We use Gibbs sampling to update β from a N(m 0 , s 2 0 ) distribution. The parameters m 0 and s 2 0 are given by and where N int is the number of ICUs in the intervention arm of the study, and Int is the list of intervention ICUs.

Inpatient clearance update
To update the inpatient decolonization rate in ICU I, η I , we use Gibbs sampling. We generate a Gamma(a I , b I ) random variable for each I, where a I gives the total number of decolonization events, and b I gives the total patient time at risk for decolonization according to the augmented data. Updates for each ICU are made sequentially.

Outpatient parameter updates
To update the outpatient colonization rate α I and decolonization rate γ I , we use Metropolis sampling. For each of the two parameters, we propose a new parameter value by adding a standard normal random variable with small variance to the current parameter value. The proposed sample is accepted or rejected according to the standard Metropolis sampling scheme. To improve mixing, this process was iterated 10 times, keeping the 10 th update for the new parameter value. Updates for each ICU are made sequentially.

False negative update
To update the value of the false negative parameter φ, we use Gibbs sampling, generating a beta(a, b) random variable where a is the number of false negative tests based on the augmented data, and b is the total number of tests in the dataset.

In-situ update
To update the in-situ probability of being colonized in ICU I σ I , we use Gibbs sampling, generating a Beta(a I , b I ) random variable for each I where a I gives the total number of individuals present in ICU I at the beginning of the study who are colonized according to the augmented data, and b I is the total number of individuals present in the ICU at the beginning of the study.