## 2 Methodology
### Construction of the COVIX Index  
For a given observation period, the construction of COVIX can be divided into three steps: the estimation of the hospitalisation rates \(k_{N}(t)\) and \(k_{ICU}(t)\) at a reference date \(t_{ref}\), the adjustment of the historically detected new infections according to their relevance for hospitalisations in comparison to a reference date \(t_{ref}\), and normalisation to scale the COVIX to a standardised value of 100% at \(t_{ref}\). To improve the clarity of presentation, we only describe the objective of the fitting procedures in Step 1 and Step 2. A mathematical  
Figure 1: Hospital beds model and its parameters. description of the fitting procedures as well as its implementation in Python can be found in the supplementary material to this work. **Step 1:** Using the hospital beds model from Section 2.1 and data on new infections and hospitalisations, we obtain a least-squares estimate for the hospitalisation rates. By performing this estimation on a moving-window basis, we ensure that varying determinants, such as changes in testing frequency, age structure of the infected individuals, vaccination rate, as well as the circulating viral variant, are reflected in the estimate of the hospitalisation rates (see Figure 2a). **Step 2:** We keep the hospitalisation rates constant and at the level estimated in Step 1 on the reference date \(t_{ref}\) throughout the whole observation period, i.e., we set \(k_{N}(t)=k_{N}(t_{ref})\) and \(k_{ICU}(t)=k_{ICU}(t_{ref})\) for all time points \(t\) in the observation period. Given this parameter choice and the observed hospital bed occupancy, we estimate the new infections that are implied by our model. Thus, the resulting estimates for the new infections are obtained relative to the hospitalisation rates that were observed on the reference date \(t_{ref}\) and the estimates are accordingly called \"relative\" new infections \(I_{rel}(t,t_{ref})\) (see Figure 2b). In particular, we can view the relative new infections \(I_{rel}(t,t_{ref})\) as an adjustment of the new infections \(I_{D}(t)\), which accounts for changes in the hospitalisation rates at time \(t\) in comparison to \(t_{ref}\). **Step 3:** The COVIX at time point \(t\) with respect to \(t_{ref}\) is finally defined by normalising the relative new infections as the following. (Note that the hospital beds model is not a perfect representation of reality. Therefore, the calibration of the relative infections may yield \(I_{rel}(t_{ref},t_{ref})\
eq I_{D}(t_{ref})\). That is why we normalise by \(I_{rel}(t_{ref},t_{ref})\) instead of \(I_{D}(t_{ref})\) to ensure a consistent interpretation of COVIX.) \[\text{COVIX}(t,t_{ref}):=\frac{I_{rel}(t,t_{ref})}{I_{rel}(t_{ref},t_{ref})}.\]  
As opposed to Step 1, where we take the new infections and hospitalisation data as given and estimate the hospitalisation rates implied by the hospital beds model, in Step 2, we take the hospitalisation rates and hospitalisation data as given and estimate (relative) new infections implied by the hospital beds model. This creates a new time series of daily relative infections \(I_{rel}(t,t_{ref})\) over the whole observation period. During the calibration of the relative infections, we keep the hospitalisation rates constant at \(k_{N}(t_{ref})\) and \(k_{ICU}(t_{ref})\) throughout the whole observation period. Thus, the relative new infections \(I_{rel}(t,t_{ref})\)  
Figure 2: **A conceptual comparison of the two quantities fitted in the construction of COVIX. ** In both (**a**,**b**), the yellow-coloured quantity is fitted, whereas the white-colored quantities are given. are adjusted to the same relevance for the hospital system at time \(t\) as the new infections \(I_{D}(t_{ref})\) at the reference date \(t_{ref}\).