Model formulation

The model proposed is for a single ward setting and comprises of: (i) a ward-level patient arrival process; (ii) an individual-based model for patient transitions in the ward; and (iii) a time-series model for the level of environmental contamination.

At any time t, patients in the ward are categorized based on their MRSA status where they can be in the susceptible group [S(t)], the undetected MRSA colonized group [C _xd (t)], the detected with MRSA colonization and undergoing appropriate treatment group [C _d (t)], the undetected MRSA infected group [I _xd (t)], or the detected with MRSA infection and undergoing appropriate treatment group [I _d (t)]. A schematic illustration of the model is provided in Figure 1 with E(t) representing the ward environmental contamination levels.

Fig. 1. Compartmental diagram for the MRSA transmission model incorporating environmental contamination. The solid black lines represent patient transitions between the different states as well as admissions and discharges [only for the S(t) and C _xd (t) compartments]. The red dashed lines denote the contribution from the various compartments to the colonization process while the black dashed lines show the compartments contributing to the evolution of the E(t) compartment.

The model is an example of Discrete Event Simulation (DES), a technique that is widely used in healthcare research [Reference Fone20–Reference Pitt22]. While perhaps more commonly used in scheduling problems, DES has also been applied to investigate pathogen transmission [Reference Mielczarek and Uzia Uziałko-Mydlikowska21]. DES provides a flexible modelling approach to represent individual patient transitions during their hospitalization episode, allowing for the inclusion of stochastic variability (important for small population studies such as in a hospital ward) and effects of individual patient information.

Patient admissions into the ward are modelled as a right-censored (at ward capacity M) Poisson process [ $\!A\left(t \right)\sim Po\left( \lambda \right)]$ with a binomial variable to separate arrivals to either susceptibles [AS(t)] or colonized (but not detected, i.e.C _xd ) [AC(t)]. It is assumed that patients cannot be infected on admission (as infected patients are typically isolated or cohorted to reduce transmission risk to other patients). Excess arrivals, beyond the ward capacity M, are assumed to be allocated to a separate ward thus creating the right-censoring in the arrival process.

The likelihood for the admissions at time t can therefore be written as:

$$\eqalign{ P{\rm (}A\left( t \right) & = i,\; AS\left( t \right) = j,\; AC\left( t \right) = i - j\; {\rm \vert} Y\left( {t - 1} \right)) = \; \cr&{\hskip-32pt}\left\{ {\matrix{ {\displaystyle{{\lambda ^i} \over {i!}}{\rm exp}\left\{ { - \lambda} \right\}\left( {\matrix{ i \cr j \cr}} \right)\vartheta ^j \left( {1 - \vartheta} \right)^{i - j}} & {0 \le i \lt Y\left( {t - 1} \right)} \cr {\mathop \sum \limits_{l = Y\left( {t - 1} \right)}^\infty \displaystyle{{\lambda ^l} \over {l!}}{\rm exp}\left\{ { - \lambda} \right\}\left( {\matrix{ l \cr j \cr}} \right)\vartheta ^j \left( {1 - \vartheta} \right)^{l - j}} & {i = Y\left( {t - 1} \right),} \cr}} \right.}$$

where Y(t) is the number of empty beds in the ward at time t and $\vartheta $ is the proportion of admissions that arrive susceptible.

The admissions at time t will then be assigned to the empty beds in the ward but will not undergo the individual patient transitions until the next time point.

The individual-based model, which is for patient transitions in the ward, processes each patient present in the ward at each time point based on the patient's current MRSA status. The following assumptions were used to formulate the individual-based model patient transitions:

1. (1) Each patient can only undergo one transition (discharge, colonization, infection, recovery, detection) per time period.

2. (2) Susceptible patients have to be colonized before developing an infection.

3. (3) Patient colonization will always be undetected when first colonized.

4. (4) Colonized patients will not return to the susceptible state.

5. (5) Undetected colonized patients cannot transition directly to the detected infected state as it counts as two transitions (detection and infection).

6. (6) Detected colonized and infected patients cannot return to the undetected state.

7. (7) Detected colonized patients are placed under the decolonisation treatment and cannot develop an infection.

8. (8) Infected patients only recover to the colonized state, and not to the susceptible state.

9. (9) Detected infected patients are placed under an appropriate treatment which increases their probability of recovery over their infection duration.

10. (10) Undetected infected patients cannot recover as they have not received appropriate treatment yet.

At each time point t, each susceptible patient S can either leave the ward as susceptible with probability p _L , become colonized (but not detected) with probability p _C , or remain susceptible with probability p _s such that p _L  + p _C  + p _s  = 1.

The probability of being colonized is modelled as p _C  = f _E (1 − p _L ), where f _E is an increasing function of E(t), C _xd (t − 1), C _d (t − 1), I _xd (t − 1) and I _d (t − 1). Specifically, the following form for f _E was used

$$f_E \left( t \right) = 1 - \exp \left\{ { - \nu \left( t \right){\rm \Delta} t} \right\},\; $$

where ν(t) = β ₀ + β ₁ C _xd (t − 1) + β ₂ C _d (t − 1) + β ₃ I _xd (t − 1) + β ₄ I _d (t − 1) + β ₅ E(t) is the instantaneous hazard of being colonized, or also known as the force of infection for this model, and $0 \le f_E \left( t \right) \lt 1\; \forall \; t$ . Last, p _S  = (1 − f _E )(1 − p _L ).

Each undetected colonized patient C _xd is detected with probability ρ (assumed to be the screening test sensitivity). Otherwise, the undetected colonized patient can either leave the ward with probability q _L , develop an infection with probability q _I , or remain colonized in the ward with probability q _C such that q _L  + q _I  + q _C  = 1. No additional structure is imposed on these probabilities values as it is assumed that each colonized patients will have the same probability values.

Each detected colonized patient C _d can either leave the ward with probability q _L or remain colonized and detected with probability 1 − q _L . Due to a lack of information to differentiate the probability of leaving for undetected and detected colonized patients, these were assumed to be same. One of the interventions considered (DECOL) increases the probability of leaving for just the detected colonized patients.

Each undetected infected patient I _xd can either be detected with probability ρ or remain undetected with probability 1 − ρ.

Each detected infected patient I _d will have a probability r _C of recovering (transitioning to C _d ) where

$$r_C \left( {t{\rm \vert} \psi, \; ti_k} \right) = \; 1 - \exp \left\{ { - \psi \left( {t - ti_k} \right)} \right\},$$

is an increasing function of the difference of the current time (t) and the time the individual k became infected (ti _k ). In other words, it is assumed that the longer a patient is infected, the more likely the patient will recover at the next time point. An infected patient remains infected with probability 1 − r _C .

By definition, only the (approximate) date that a patient is detected to be colonized or infected is available from hospital surveillance databases. The transition times from susceptible to undetected with MRSA colonization (tc _k ), and subsequently undetected infection (ti _k ) are typically imputed from a range of plausible values between the patient's admission date (a _k ) and first positive screening test date (d _k ) where the full conditional for (tc _k , ti _k ) can be written as

$$\eqalign{& \left( {1 - \rho} \right)^{N_F \left( {ti_k} \right)} \cr & \quad {\rm exp}\left\{ {\mathop \sum \limits_b \log \nu \left( {t_b} \right) - \; \mathop \sum \limits_d S\left( {t_d} \right)\nu \left( {t_d} \right)\left( {t_{d + 1} - t_d} \right)} \right\} \cr & \quad\times q_I {\rm exp}\left\{ { - \; q_I \left( {ti_k - tc_k} \right)} \right\},} $$

where tc _k  < ti _k , N _F (ti _k ) is the number of false-negative screening test results for patient k given ti _k , the b subscript indexes time points where a susceptible patient becomes colonized between tc _k to patient k’s discharge and the d subscript indexes the time points where v(t) changes between a _k and tc _k . The expression can be evaluated for all potential (tc _k , ti _k ) values to obtain a discrete distribution to be used in a Metropolis–Hastings step within a Markov chain Monte Carlo algorithm to impute these unobserved quantities and estimate the remaining model parameters [Reference Forrester, Pettitt and Gibson4, Reference McBryde, Pettitt and McElwain14, Reference Streftaris and Gibson23].

An autoregressive-moving average time series model with exogenous covariates (ARMAX) [Reference Hyndman and Athanasopoulos24] is used to describe the environmental contamination levels E(t). The exogenous covariates assumed to be contributing to the levels of environmental contamination at time t are the C _xd and I _xd patients in the ward at time t − 1. It is assumed that detected (colonized and infected) MRSA patients undergo the decolonization treatment which halts shedding from the patient to the environment. The orders of the ARMAX model are determined using the auto.arima() function in the R package forecast [Reference Hyndman and Khandakar25].

Parameter values

The model parameter values used for the normal burden setting simulations are summarized in Table 1. Additional details of the parametrization are provided in the Supplementary material. The normal burden setting is reflective of MRSA burden in a typical hospital ward in a developed country. These parameters values are also used in the high burden setting simulations with the following modifications:

1. (1) there is an additional factor of two multiplying ν(t);

2. (2) the probability of a colonized patient developing an infection q _I is doubled and q _C is reduced accordingly to ensure q _L  + q _I  + q _c  = 1;

3. (3) there is decreased sensitivity in the screening test, ρ = 0·6;

that is, we assumed that in this setting, the hypothetical pathogen is more likely to colonize susceptible patients, colonized patients more readily develop an infection and it is harder to detect the presence of the pathogen. The high burden setting attempts to mimic either the MRSA dynamics in a developing country [Reference Allegranzi26] or a novel strain of pathogen that is more virulent and less readily detected by routine surveillance.

Table 1. Parameter values for the stochastic model describing multidrug-resistant organisms' transmission in a hospital ward

AR, Autoregressive; MA, moving average.

* Unpublished observations are estimates obtained from fitting a non-homogeneous Poisson process to the data. More details provided in the supplementary material.

There was no available source to estimate the parameter ω which represents the difference between colonized and infected patients on the force of infection. The ω value in the Results section was 1 as a reflection of the lack of information on the parameter. Alternative values of 0·1 and 1·9 were also investigated in the parameter sensitivity analysis (provided in the Supplementary material). We found that the AR, C _xd and C _d outcomes (defined below) were particularly sensitive to a low value of ω (giving a stronger influence to colonized patients) in both normal and high burden settings. Distributions of AR outcome for the different values of ω are provided in Figure 2. Similar plots for the other outcomes and parameters are provided in the Supplementary material.

Fig. 2. Attack rate outcome for normal burden (left plot) and high burden (right plot) settings. The x-axis denotes the baseline, low ω value and high ω value (moving from left to right).

Interventions

Five common intervention strategies were considered in the model investigation below:

1. (1) Not colonized on admission (COA) ( $\vartheta = 1$ ), where all patients who are colonized on admission are assumed to be detected on admission and isolated elsewhere, i.e. universal screening [Reference Harbarth27].

2. (2) Improved environmental cleaning (ENV), which halved the intercept term in the environmental time-series model (α ₁) [Reference Dancer28].

3. (3) Improved contact precaution practices (CP), which decreases ν by a factor of ξ where ξ was set to 0·75 [Reference Kypraios29].

4. (4) Perfect screening test sensitivity (SENS), where test sensitivity ρ was set to 1 [Reference McBryde, Pettitt and McElwain14].

5. (5) Improved decolonization treatment for colonized patients (DECOL) where the probability for a C _d patient leaving the ward is now q _L  + Δ (with the probability of staying adjusted accordingly) [Reference McBryde, Pettitt and McElwain14].

We considered six outcome measures for the investigations. They are the attack rate (AR) defined as the average of the force of infection ν(t) [Reference McBryde, Pettitt and McElwain14] as well as the cumulative numbers of

• • patients who were colonized on admission (AC),

• • patients who were colonized but not detected (C _xd ),

• • detected, colonized patients (C _d ),

• • patients who were infected but not detected (I _xd ),

• • detected, infected patients (I _d ).

Note that there is a slight abuse of notation where C _xd , C _d , I _xd and I _d refer to the cumulative number of patients in each group for the outcome measures, but the time-varying prevalence of the groups in the model.

Due to the stochastic model formulation, each intervention setting was simulated 1000 times and we compared the distributional differences of the outcomes rather than just point estimates of the outcomes.

Pairs of distributions (denoted generally by X and Y here) were assessed using the generalized Mann–Whitney statistic which estimates the parameter

$$\theta = P\left( {Y \gt X} \right) + \; \textstyle{1 \over 2}\; P\left( {Y = X} \right)\,{\rm using}\,\hat \theta = \; \displaystyle{U \over {mn}},$$

where $U = \; \mathop \sum \nolimits_{i = 1}^m \mathop \sum \nolimits_{j = 1}^n {\open 1}\left( {Y_j \gt X_i} \right) + \; \displaystyle{1 \over 2}{\rm \;} {\open 1}\left( {Y_j = X_i} \right)$ with {Y _j ; j = 1,…, n} and {X _i ; i = 1,…, m} being samples from the Y and X distributions, respectively. Confidence intervals (CIs) for $\hat \theta $ were computed based on method 5 of Newcombe [Reference Newcombe30].

Following the definition above, values of θ larger than 0·5 indicate that the Y is stochastically larger than X and, conversely, values of θ less than 0·5 indicate X is stochastically larger than Y. For the results below, θ values between 0 and 0·2 (and similarly between 0·8 and 1) are considered strong evidence that the two distributions are substantially different. Intermediate θ values between 0·2 and 0·4 (or 0·6 and 0·8) are assumed to provide weak evidence of a difference between the distributions. Values of θ close to 0·5 (between 0·4 and 0·6) indicate that there is no evidence that the two distributions being compared are dissimilar.