# Non-Parametric Model-Based Estimation of the Effective Reproduction Number for SARS-CoV-2 +
## 3 Results
### Influence of Potential Model Misspecifications  
Up to this point, we have only shown applications of the SIRD-AKS method to simulated data from the same SIRD model structure. In more realistic applications, however, the true model structure is not known, or as in all real-world scenarios, the chosen model structure is only an appropriate simplification of the real underlying biological process. Therefore, we next investigated the performance of the SIRD-AKS method in light of model misspecifications. To this aim, data were generated from non-SIRD model structures but  
Figure 2: Performance of SIRD-AKS is robust towards high noise levels in the data. (**A–D**): The SIRD model was simulated with a time-varying \(\beta_{t}\) and time-constant \(\gamma_{t}\) and \(\theta_{t}\), as shown in red. The parameters were chosen as in parameter setting II in Figure 1. The data were simulated by using increasing noise levels from scenario (**A–D**), as shown as green dots for the observed fluxes \(v_{D},v_{l}\) and \(v_{R}\). The SIRD-AKS estimates are shown as black lines with gray error bands for their uncertainties and are compared to the corresponding true time courses. Again, instead of parameter \(\beta_{t}\) from the data-generating SIRD model, \(\mathcal{R}_{t}\) was calculated from the true parameter time courses and compared to the SIRD-AKS result. While the precision of the SIRD-AKS estimates decreases with higher noise levels, the average dynamics of the parameter time courses are satisfactorily met. were analyzed by the AKS method by using the _inappropriate_ SIRD model structure. We tried to reconstruct the time courses of the effective reproduction number in two examples of such intentionally misspecified model structures for the SIRD-AKS method. First, the so-called SEIRD model was considered for data generation. It constitutes an extension of the SIRD model augmented by a model state of exposed (E) corresponding to individuals that are infected but not yet infectious. This extension is commonly considered when the disease progression underlies an incubation period. The ODE system of the SEIRD model reads  
\[\dot{S} =-\beta\cdot\frac{S\cdot I}{N_{0}} \tag{25}\] \[\dot{E} =\beta\cdot\frac{S\cdot I}{N_{0}}-\delta\cdot E\] \[\dot{I} =\delta\cdot E-\gamma\cdot I-\theta\cdot I\] \[\dot{R} =\gamma\cdot I\] \[\dot{D} =\theta\cdot I\,,\]  
where the reciprocal of the additional parameter \(\delta^{-1}\) is called the latency period. Note that the limiting case of the SEIRD model for \(\delta^{-1}\to 0\) is the SIRD model, while for higher values of \(\delta^{-1}\), the difference between the two models increases and distinct dynamics are expected. To illustrate this transition, four different analyses with latency periods varying from 4 to 21 days were performed. The SEIRD model Equation (25) was simulated for data generation with time-dependent parameters \(\beta_{t}\) and \(\theta_{t}\) as well as the time-constant parameter \(\gamma_{t}\), following the previous parameter setting I. Again, the true values for the reproduction number \(\mathcal{R}_{t}\) were calculated from the model parameters for the SEIRD model [68]. To mimic a realistic application of the presented method in an epidemic scenario, where the true underlying process is not fully known, data from the SEIRD model were analyzed by the SIRD-AKS, and only the daily numbers of the newly infected \(v_{I}\), newly recovered \(v_{R}\) and newly dead \(v_{D}\) were provided as observations to the algorithm. In consequence, the SIRD-AKS is able to estimate the SIRD model states and parameters but cannot infer the transition rate \(\delta\) or the exposed state \(E\). The resulting time courses of the SIRD-AKS estimates are shown in the upper part of Figure 3 with increasing values for the latency period \(\delta^{-1}\) from panel A to D. Comparing the four simulation scenarios, the different latency periods are reflected both in different dynamics of the observables and in the time courses of the reproduction number \(\mathcal{R}_{t}\).