# Non-Parametric Model-Based Estimation of the Effective Reproduction Number for SARS-CoV-2 +
## 3 Results
### Application to SARS-CoV-2 Data from Germany  
After demonstrating the performance of the SIRD-AKS method for multiple-parameter time courses and model misspecifications in simulated data sets, the method was applied to COVID-19 data from Germany between January 2020 and August 2021. The data were taken from the COVID-19 Data Repository maintained by the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University [70] and is displayed in Figure 5A as green dots. Data pre-processing was performed with a moving average over seven days in order to reduce weekday effects and reporting delays in the raw data. Figure 5A shows the analyzed data and estimated time courses of the observed fluxes for the newly infected \(v_{I}\), newly recovered \(v_{R}\) and newly dead \(v_{D}\). The estimated time courses for the recovery rate \(\gamma_{t}\) and death rate \(\theta_{t}\) are shown in Figure 5B. The fluctuations in the death rate are in line with the time periods when more old people had been reported as being infected, i.e., from April to June and from November to February 2020 [71]. This also agrees with previous studies showing that the mortality of COVID-19 is heavily linked to the patient's age [72; 73]. The estimated time-dependent reproduction rate \(\mathcal{R}_{t}\) from the SIRD-AKS method is shown as a black line in Figure 5C. For comparison, two additional \(\mathcal{R}_{t}\) values are provided, which are both based on the previously discussed incidence-based method and assume a fixed serial interval of \(s=4\) days and an averaging window of \(\tau=7\) days. However, they use different data sources. The \(\mathcal{R}_{t}\) value shown as a blue line is calculated from the same data from [70] that were also analyzed by using the SIRD-AKS method. The other shown value in magenta is the officially issued and so-called _seven-day_\(\mathcal{R}_{t}^{4,7}\) value from the  
Figure 4: Comparison of SIRD-AKS estimation and incidence-based method with fixed serial interval time. Simulated data were generated from the SECIR model [27] with five-step function as time-dependent input for parameter \(R_{1}\). The corresponding true time course for \(\mathcal{R}_{t}\) is depicted as red line. The SIRD-AKS estimate as shown in black roughly reassembles the true SECIR reproduction number. The \(\mathcal{R}_{t}^{5,\tau=4}\) value calculated by the incidence-based method from Equation (24) with fixed \(s=4\) is shown in blue. It is biased towards \(\mathcal{R}_{t}=1\) (dashed line) and is not able to correctly estimate the true time course. For different choices of the fixed serial-interval time hyperparameter \(s\), the incidence-based method may yield better results for \(\mathcal{R}_{t}^{5,4}\), as shown by the gray lines for different values of \(s=2,...,10\). RKI [66]. The difference is that it does not use the raw data from the CSSE repository but instead applies an additional so-called _Nowcasting_ pre-processing method that copes with the redistribution of the case numbers for the elimination of reporting delays [74]. Both incidence-based methods show a similar time course for \(\mathcal{R}_{t}\) but with peaks that seem to be shifted by a couple of days, presumably because of the case-number redistribution. When compared to the estimate from the SIRD-AKS method, both incidence-based values again lie closer to \(\mathcal{R}_{t}=1\). The time course from the SIRD-AKS shows a larger variety of peak heights in both directions and more deviations from \(\mathcal{R}_{t}=1\) on longer  
Figure 5: Application of the SIRD-AKS to SARS-CoV-2 data of Germany from the CSSE repository. (**A**): Incidence data taken from [70] for March 2020 until August 2021 are shown in green. SIRD-AKS results for the fluxes of newly infected, newly recovered and newly deceased is shown as black line.