# Non-Parametric Model-Based Estimation of the Effective Reproduction Number for SARS-CoV-2 +
## 3 Results
### Influence of Potential Model Misspecifications
The estimated time course of the effective reproduction number \(\mathcal{R}_{t}\) from the SIRD-AKS with SECIR data is shown as a black line in Figure 4. Except for the level of the first plateau until \(t=80\), the \(\mathcal{R}_{t}\) value from the SIRD-AKS method shows a similar shape as the true \(\mathcal{R}_{t}\) time course and yields almost correct levels on the other plateaus. However, the estimated \(\mathcal{R}_{t}\) time course is slightly biased to \(\mathcal{R}_{t}=1\) after \(t=80\) days. For a comparison of the different methods, Figure 4 also shows the results for the reproduction number \(\mathcal{R}_{t}^{s,\tau}\) calculated by using the incidence-based method from Equation (24) for different values of the fixed serial interval time \(s=2,...,10\) days in unit steps and with averaging windows of \(\tau=4\) days. It can be clearly seen that the choice of the fixed serial interval time heavily influences the resulting levels of the reproduction number. The Robert Koch Institute (RKI) as the German center for disease control provided officially issued reproduction number values during the SARS-CoV-2 pandemic for Germany based on this method with a chosen fixed serial interval value of \(s=4\) days. The accordingly calculated \(\mathcal{R}_{t}^{4,4}\) value for the SECIR data set is highlighted in blue in Figure 1.0 Comparing the blue and red line, it can be concluded that at least for the simulations of the SECIR model and the chosen parameterization, a fixed serial interval of \(s=4\) days is not an adequate choice. While the RKI-issued value roughly covers the general form of the time course, it is noticeably biased towards the characteristic value of \(\mathcal{R}_{t}=1\). As a consequence, the rate of increase or decay in newly infected individuals within an ongoing disease outbreak would be definitely underestimated. An appropriate choice of the serial-interval distribution or at least an optimal choice of the fixed serial interval time \(s\) would solve this issue, as can be seen from the gray lines in Figure 1.1 However, it should be noted that the selection of the optimal value for \(s\) is difficult in real-world applications and requires additional data or assumptions. That is, the simplicity of the calculation and interpretation by using the incidence-based method comes at the price of required assumptions for the serial interval times. Some deviations from the true \(\mathcal{R}_{t}\) value also appear in the results of the SIRD-AKS method. However, we surmise that these originate from the misspecified model structure. We argue that a clear advantage of the SIRD-AKS method is that it does not rely on additional hyperparameters or distributional assumptions, such as the incidence-based method does on the serial interval time. This renders the SIRD-AKS method to be a de facto non-parametric approach for the estimation of the reproduction number and model parameters in general. As shown in this example, the SIRD-AKS method also yields reasonable results when the internal state-space model does not meet the model structure of the data-generating process. It therefore qualifies for the application to real-world data sets, where the true underlying process is masked and thus the optimal choice of an appropriate fixed serial interval time for the incidence-based method becomes even more difficult.