# Non-Parametric Model-Based Estimation of the Effective Reproduction Number for SARS-CoV-2 +
## 3 Results
### Influence of Potential Model Misspecifications
As expected, the peak in the observed infections appears later for higher latency periods corresponding to longer times that individuals remain in the exposed state \(E\). For all scenarios, the SIRD-AKS was applied to the SEIRD data sets, as shown in black in Figure 3A-D. It can be seen that the time courses of the parameters \(\gamma_{t}\) and \(\theta_{t}\) are throughout accurately obtained. Interestingly, the estimates for the effective reproduction number \(\mathcal{R}_{t}\) are also very close to the true values for the low-to-medium latency periods. In contrast, for higher latency periods, the estimates for \(\mathcal{R}_{t}\) are slightly shifted to later time points and are also biased towards the characteristic value of \(\mathcal{R}_{t}=1\). In total, this shows that the AKS method is able to cope with moderate deviations in the model structure, while more extreme model misspecifications, e.g., latency periods of three weeks, can noticeably influence the result of the estimated time-dependent parameters. Especially, for COVID-19, such a model misspecification does not negatively impact the SIRD-AKS estimation results since a latency period of 5-6 days was typically reported [69]. In order to formulate a model-misspecification problem that is closer to a real-world scenario, we additionally utilized a larger and more complicated model structure that covers a multitude of additional features. For this, we employed the so-called SECIR model that consists of sixteen model states and was developed to obtain a detailed description of the model structure. Figure 3: AKS shows robustness to model misspecifications. (**A**–**D**): The SEIRD model from Equation (25) was simulated with time-varying parameters \(\beta_{1}\) and \(\theta_{1}\) and for different latency periods increasing from \(\delta^{-1}=4d\) (**A**) to \(\delta^{-1}=21d\) (**D**). Calculated time courses for \(\mathcal{R}_{t}\) depend on the latency period parameter and thus changes in the four scenarios. Simulated data were generated from the SEIRD model fluxes with 5 % relative error. The data were analyzed by the SIRD-AKS algorithm, which assumes the reparameterized SIRD model as truth and, in particular, does not contain an exposed state \(E\). Despite the model misspecification between the data-generating process and the analyzing model, the SIRD-AKS estimates for the time-dependent parameters are accurately met for shorter latency periods in scenario (**A**,**B**). For larger deviations from the assumed SIRD model structure, the AKS results are slightly biased towards values of one (**C**,**D**), while the overall dynamics are retrieved sufficiently. of the SARS-CoV-2 epidemic in Germany in 2020, including, for example, several infected carrier states as well as diverse hospitalization stages [27]. This model was used for data generation by using parameters values within the allowed parameter ranges as in the original publication. The only deviation is that we replaced the originally estimated time course of the transmission parameter \(R_{1}\) by a simple five-step function. This parameter mainly influences the time course of the true value of the effective reproduction number \(\mathcal{R}_{t}\) within the SECIR model, which was calculated according to the respective equation in the original publication and is shown as a reference in red in Figure 4. Within the chosen parameter set for the simulation, the resulting true \(\mathcal{R}_{t}\) time course covers a broad range of values, in particular below and above the characteristic value of \(\mathcal{R}_{t}=1\). The generated data sets for the newly infected \(v_{t}\), newly recovered \(v_{R}\) and newly dead \(v_{D}\) were again provided as observations of the SIRD-AKS analysis.