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.