# Non-Parametric Model-Based Estimation of the Effective Reproduction Number for SARS-CoV-2 +
## 3 Results
### Validation of Parameter Time-Course Estimates in an ODE Model  
The presented SIRD-AKS method is based on the transition of the model structure from the time-continuous ODE system to the time-discrete and recursive formulation in the state space. Further, estimates for the time-dependent parameters are in fact augmented states in the state space, interpreted as parameter time courses. We asked whether these parameter time courses produce coherent dynamics when being incorporated into the corresponding ODE system as input functions. To perform this consistency check, we plugged in the estimated time courses for the parameters \(\mathcal{R}_{t}\), \(\gamma_{t}\) and \(\theta_{t}\) from Figure 5B,C into the ODEs of the reparameterized SIRD model in Equation (21). To choose appropriate initial values for the numerical solution of this ODE system, the values of the SIRD-AKS estimates at time point \(k=62\), i.e., 24 March 2020, were chosen. This time point corresponds to the first time point where the estimated uncertainties of the time-dependent parameters become adequately small. It also coincides with the starting point of the large initial decrease in the time course of \(\mathcal{R}_{t}\), c.f. Figure 5C. The results of the numerical simulation of the time-continuous ODE system with time-dependent parameters are shown in Figure 6 as a black line. It is remarkable that all three trajectories are very well in agreement with the flux data for \(v_{I}\), \(v_{R}\) and \(v_{D}\) that went into the previous AKS analysis, shown as green dots in Figure 6.  
Figure 6: ODE solutions for the SIRD model with estimated time-dependent parameters. The fluxes for \(v_{I}\), \(v_{R}\) and \(v_{D}\) are calculated from the numerically solved ODEs of the reparameterized SIRD model from Equation (4) and are compared to Germany data from Figure 5A, as depicted by green dots. The black lines indicate the ODE solutions when the three estimated time courses for \(\gamma_{t},\theta_{t}\) and  
\(\mathcal{R}_{t}\) from Figure 5B,C are utilized as time-dependent parameters in the SIRD ODE system. Identical ODE simulations were performed and are shown as blue line, where the \(\mathcal{R}_{t}\) time course is taken from the incidence-based method with fixed \(s=4\) and \(\tau=7\) (c.f. Figure 5C blue line). While the SIRS-AKS estimates yield almost perfect ODE trajectories, the incidence-based method underestimates the dynamics of the pandemic in an SIRD model. This confirms that the parameter time courses estimated from the incidence data by the AKS method are indeed able to reassemble the correct dynamics in the ODE solutions. The presented AKS method thus qualifies as a potential input-estimation approach for ODE models in general. For comparison, the same ODE simulation was performed by utilizing the alternative \(\mathcal{R}_{t}\) time course calculated from the incidence-based method with a fixed serial-interval time of \(s=4\) days and an averaging interval of \(\tau=7\) days, as presented above in Figure 5C as a blue line. Since this methods lacks estimates for parameters \(\gamma_{t}\) and \(\theta_{t}\) as required to solve the ODE system, time-course estimates for \(\gamma_{t}\) and \(\theta_{t}\) from the SIRD-AKS method were chosen. The resulting incidence-based ODE trajectories are displayed as blue lines in Figure 6.  
In contrast to the ODE trajectories with SIRD-AKS estimates, the ODE solutions with incidence-based \(\mathcal{R}_{t}\) do not agree well with the data. Although the overall dynamic activity of the data is approximately covered, it does not capture certain details of the data series well, e.g., the partial drop in the infected flux during the second wave in February 2021. Furthermore, the predicted amplitudes of newly infected individuals are not adequately met and thus the overall pandemic situation would be underestimated.