Mathematical modelling
In order to quantify the effect of the change in RPE upon cases in red
wards, we developed a mathematical model, considering the numbers of
cases observed among HCWs as arising from a combination of ward-specific
infection risks, which relate directly to working on a red or green
ward, and non-ward-specific risks, which include infections arising from
the community. We first wrote expressions for the infection risk facing
workers in red and green wards on week i. For HCWs on green wards we
write
\begin{equation}
\lambda_{i}^{G}=\left(kC_{i-1}+g\right)W_{G,i}\nonumber \\
\end{equation}
while for HCWs on red wards we write
\begin{equation}
\lambda_{i}^{R}=\begin{matrix}\left(kC_{i-1}+r_{1}\right)W_{R,i}\ \ \ \ \ i<9\\
\left(kC_{i-1}+r_{2}\right)W_{R,i}\ \ \ \ \ i\geq 9\\
\end{matrix}\nonumber \\
\end{equation}
Here the term k is a constant, while the value Ci-1describes the number of observed cases in the local community in the
previous week. Our use of data from the previous week reflects a
generation time for SARS-CoV-2 of approximately seven days [20]; we
assumed that HCWs diagnosed with COVID-19 infection during this study
would have been infected by individuals who were diagnosed in the
previous week. The model parameters g, r1, and
r2 describe ward-specific infection risks; FFP3 masks
were used on red wards from week 9 onwards.
Model parameters were optimised using a likelihood framework,
identifying the maximum value of the term; here the number of cases on
each type of ward each week, Gi and Ri,
were represented as emissions from a Poisson distribution with parameter
equal to the total risk of infection.
\begin{equation}
L=\sum_{i}\left[\log{\frac{{\lambda_{i}^{G}}^{G_{i}}}{G_{i}!}\ }+\log\frac{{\lambda_{i}^{R}}^{R_{i}}}{R_{i}!}\right]\nonumber \\
\end{equation}
Confidence intervals for each parameter were then obtained using the
likelihood function. Constrained likelihood optimisations were performed
in which the likelihood was optimised subject to a fixed value of the
parameter in question. Confidence intervals were defined as the region
of parameter space in which the likelihood L was within 2 units of the
maximum. Similarly, constrained optimisation was used to identify a
confidence interval for parameter ratios such as
r2/r1, which describes the relative risk
to HCWs on red wards with, as opposed to prior to, the introduction of
FFP3.