The Computational Model
A two-level compartmental model with deterministic framework was employed. Nursing home residents are considered as hosts of MRSA, and HCWs are considered as transmission vectors. Residents are categorized into two mutually exclusive states: (a) uncolonized or (b) colonized; whereas HCWs are categorized into two mutually exclusive states: (c) contaminated or (d) uncontaminated. The model features the residents’ admission from community/hospital and discharge/death of the residents who are susceptible and colonized. Different routes of transmission were taken into account via the close contact among HCWs and residents: (i) HCWs-residents contacts, (ii) residents-residents contacts, and (iii) residents-HCWs contacts. Two types of transmission were defined in the model: Colonization and Contamination. MRSA can be transmitted to uncolonized residents by contact made with colonized residents or contaminated HCWs. Uncontaminated HCWs can be contaminated by contact made with colonized residents. Schematic of the transmission model is illustrated in Fig. 1.
The model is described with the following set of nonlinear differential equations:
$$\frac{dU\left(t\right)}{dt}= -\frac{{\beta }_{r-r}U\left(t\right) C\left(t\right)}{{N}_{r}}-\frac{{\beta }_{h-r}U\left(t\right){H}_{c}\left(t\right)}{{N}_{h}}+\omega C\left(t\right)+\left(1-\lambda \right)\varLambda -{\gamma }_{u}U\left(t\right)$$
$$\frac{dC\left(t\right)}{dt}= \frac{{\beta }_{r-r}U\left(t\right) C\left(t\right)}{{N}_{r}}+\frac{{\beta }_{h-r}U\left(t\right) {H}_{c}\left(t\right)}{{N}_{h}}-\omega C\left(t\right)+\lambda \varLambda -{\gamma }_{c}C\left(t\right)$$
$$\frac{dH\left(t\right)}{dt}= -\frac{{\beta }_{r-h}H\left(t\right) C\left(t\right)}{{N}_{r}}-\frac{{\beta }_{h-h}H\left(t\right) {H}_{c}\left(t\right)}{{N}_{h}}+\mu {H}_{c}\left(t\right)$$
$$\frac{d{H}_{c}\left(t\right)}{dt}=\frac{{\beta }_{r-h}H\left(t\right) C\left(t\right)}{{N}_{r}}-\mu {H}_{c}\left(t\right)$$
where U(t), C(t), H(t) and Hc(t) denoted the population size of uncolonized-residents, colonized-residents, uncontaminated-HCWs and contaminated-HCWs respectively in a nursing home at time t. 1/µ denoted the mean duration of contamination while 1/ω denoted the mean duration of colonization. Nr denoted the population of residents and Nh denoted the population of healthcare-workers, where Nr = U(t) + C(t) and Nh = H(t) + Hc(t).
The average length of stay of colonized residents and uncolonized residents are denoted by γu and γc respectively. Residents are admitted at the rate of Λ, and the probability of admitted residents being colonized is λ. It is assumed that the number of residents and HCWs remain constant, and the occupancy rate is 100%. Therefore, residential admission rate equals to residents discharge rate; i.e. Λ = γuU + γcC.
Three transmission rates were defined in the model. βr−r and βh−r denoted the residents-to-residents transmission rate and HCWs-to-residents transmission rate respectively, whereas βr−h denoted the residents-to-HCWs transmission rate. The values of these three transmission rates were defined as the estimated number of contacts multiplied by the transmission probability via contact, as summarized in Table 1a and 1b. Three types of contact and transmission probability were defined. ar−r, ar−h and ah−r denote the average number of residents-to-residents contact, residents-to-HCWs contact and HCWs-to-residents contact respectively. pr−r, pr−h and ph−r denote the transmission probability via residents-to-residents contact, residents-to-HCWs contact and HCWs-to-residents contact respectively.
Table 1
a. The three transmission rates defined in the computational model.
Description
|
Symbol
|
Value
|
Residents-to-residents transmission rate
|
βr−r
|
ar−rpr−r
|
HCWs-to-residents transmission rate
|
βh−r
|
ah−rph−r
|
Residents-to-HCWs transmission rate
|
βr−h
|
ar−hpr−h
|
Table 1
b. The six parameters used in the three transmission rates.
Description
|
Symbol
|
Average number of residents-to-residents contact
|
ar−r
|
Average number of HCWs-to-residents contact
|
ah−r
|
Average number of residents-to-HCWs contact
|
ar−h
|
Transmission probability via residents-to-residents contact
|
pr−r
|
Transmission probability via HCWs-to-residents contact
|
ph−r
|
Transmission probability via residents-to-HCWs contact
|
pr−h
|
Parameterization
Parameters Identification
15 parameters were incorporated in the model. Values of 12 of the 14 parameters were identified in the parameterization process. Values of 2 parameters cannot be identified and were estimated. Table 2 summarizes the 14 parameters.
Table 2
List of 12 of the 14 parameter values used in the computational model.
Description
|
Symbol
|
Value
|
Reference
|
Number of residents
|
Nr
|
75.10
|
[16]
|
Number of HCWs
|
Nhcw
|
8.61
|
[16]
|
Probability of admission of colonized residents
|
Λ
|
15.8%
|
[17]
|
Average duration of colonization (days)
|
1/ω
|
268.8
|
[18]
|
Average length of stay for uncolonized residents (days)
|
1/γu
|
233.89*
|
[17, 19]
|
Average length of stay for colonized residents (days)
|
1/γc
|
233.89*
|
[17, 19]
|
Average contamination duration (hours)
|
1/µ
|
4.1*
|
[16]
|
Average number of residents-to-residents contact
|
ar−r
|
0.2
|
[16]
|
Average number of HCWs-to-residents contact
|
ah−r
|
1.2
|
[16]
|
Average number of residents-to-HCWs contact
|
ar−h
|
12.7
|
[16]
|
Average number of HCWs-to-HCWs contact
|
ah−h
|
0.6
|
[16]
|
Transmission probability via residents-to-residents contact
|
pr−r
|
Not identified
|
|
Transmission probability via HCWs-to-residents contact
|
ph−r
|
Not identified
|
|
Transmission probability via residents-to-HCWs contact
|
pr−h
|
20%
|
[20]
|
* Average length of stay for uncolonized residents and average length of stay for colonized residents were calculated based on data presented in Kwok et al. [19]. Average contamination duration was calculated based on data presented in Kwok et al. [16]. Detail of calculations of these three parameters is included in Supplementary Material S1.
Parameter Estimation
No best estimates of two key parameters were identified from prior study: (a) transmission probability via residents-to-residents contact and (b) transmission probability via HCWs-to-residents contact. We estimated these two parameters by fitting the mathematical transmission model to the prevalence data. Four MRSA point-prevalence rates and their 95% confidence interval (CI) of nursing homes in Hong Kong were identified from prior studies (Table 3) [17, 21–23]. We denoted the date of these four point-prevalence rates as Day 1, Day 108, Day 1600 and Day 2239.
Table 3
Point-prevalence of MRSA in Hong Kong nursing homes.
Study period
|
Mid-date of study period
|
Day count
|
Prevalence
|
95% CI
|
Reference
|
3 Mar – 26 Sep 2011
|
14 Jun 2011
|
#1
|
17.8%
|
16.2% − 19.4%
|
[21]
|
1 Jul – 31 Dec 2011
|
30 Sep 2011
|
#108
|
21.6%
|
19.8% − 23.4%
|
[17]
|
1 Sep – 31 Dec 2015
|
31 Oct 2015
|
#1600
|
30.1%
|
25.1% − 35.6%
|
[22]
|
1 Jul – 31 Aug 2017
|
31 Jul 2015
|
#2239
|
37.9%
|
33.7% − 42.15
|
[23]
|
Simulations using the computational model described in the previous section were conducted by varying the two unknown transmission probabilities and the prevalence of Day 1. The combinations of the three variables used in the simulations are constructed based on 3-dimensional hypercube, varying the two transmission probabilities from 0–100% (with 0.1% increments) and the prevalence of Day 1 from 16.2–19.4% (with 0.1% increment). 33,066,033 simulations (1,001 × 1,001 × 33) were conducted. Each simulation was conducted to simulate prevalence for 2,500 days where the prevalence rates reached equilibrium. 487 of the 33,066,033 simulation generated results fitted within the 95% confidence interval of the four point-prevalence rate on their respective day. These 487 sets of parameters were used to further conduct simulations of interventions. Figure 2 illustrates simulation results of these 487 simulations.
Simulations of Interventions
Three types of intervention scenarios were evaluated using the computational model: (1) hand-hygiene compliance by HCWs, (2) screening-and-isolation upon all admission, and (3) implementing both interventions at the same time. Assuming interventions implemented on Day 2501, we simulated the long term impact of different hypothetical intervention scenarios on MRSA transmission dynamics. The 487 sets of parameters that fitted within 95% CI of the four point-prevalence rates were used as the input parameters for the computational model. Effects of interventions were evaluated by comparing the point prevalence of Day 2500 and Day 7500 for each of the simulations.
Intervention Scenario 1: Hand-Hygiene Compliance by HCWs
Hand-hygiene compliance by HCWs was simulated by reducing the average contamination duration 1/µ. Five levels of hand-hygiene compliance by HCWs were evaluated by reducing the average contamination duration by 10%, 20%, 30%, 40% or 50%. Computational model defined in the previous section was used to simulate effects of hand-hygiene by HCWs compliance to prevalence in our study. 2,435 simulations (5 levels of compliance x 487 set of parameters) were conducted for this scenario.
Intervention Scenario 2: Screening-and-Isolation upon Admission
The model was further extended in order to study effects of screening-and-isolation upon admission to prevalence rate. We assume a 100% screening success rate, a 5-day isolation period, and a 100% decolonization rate after 5 days in isolation. An additional compartment was added in the model in order to represent the screening-and-isolation upon admission. All colonized admissions were put into an isolation compartment I, and they were forwarded to compartment U after 5 days. Schematic of the extended model is illustrated in Fig. 3. 487 simulations were conducted for this scenario.
Intervention Scenario 3: Implementing Both Interventions at the Same Time
Simulation of intervention scenario 2 was run with five levels of hand-hygiene compliance by HCWs (reducing the average contamination duration by 10%, 20%, 30%, 40% or 50%). 2,435 simulations were conducted for this scenario.