# Non-Parametric Model-Based Estimation of the Effective Reproduction Number for SARS-CoV-2 +
## 4 Discussion  
The general task of estimating time-dependent parameters in ODE models, also referred to as input-function estimation, is known to be conceptually difficult. Either a particular function class, i.e., a parameterization of a general input function, needs to be provided, or the initial value problem of solving the ODE needs to be transferred into a boundary value problem for which no good optimizers are available [36].  
To remedy this, we present in this work a non-parametric model-based estimation for all time-dependent parameters in epidemiological ODE models by combining an Expectation-Maximization algorithm with the Augmented Kalman Smoother (AKS) algorithm.  
The approach is able to estimate the time dependencies in multiple model parameters and independently from the particular input function class. While other approaches exist that also take advantage of Kalman filter methods for the estimation of a single time dependency in compartmental models [76], this approach is more general and enables the estimation of all time-dependent model parameters simultaneously.  
We showed the performance of our AKS method by using an SIRD core model structure (SIRD-AKS) in diverse simulation settings. The SIRD-AKS performs well even with high noise levels and under different degrees of model misspecification. Thus, the method also yields meaningful estimates in situations where the details of the data-generating process remain unknown, meaning that the assumed AKS model structure does not perfectly cover the true process structure. In particular, the presented approach only utilizesincidence data of infected, recovered and deceased individuals and, as a major advantage compared to existing methods [27; 66], does not require any further prior knowledge on the analyzed disease.  
When applying the SIRD-AKS method to COVID-19 data from Germany, it can be observed that all the parameter time-course estimates are plausible, and in particular, the effective reproduction number \(\mathcal{R}_{t}\) estimate displays the impact of all relevant non-pharmaceutical interventions and applied control measures during the SARS-CoV-2 pandemic in Germany. Further, the simultaneously estimated death rate \(\theta_{t}\) matches well with the time pattern of how many elderly people were infected by the disease.  
We further compared our SIRD-AKS method to an alternative approach for the estimation of \(\mathcal{R}_{t}\), which considerably depends on the choice for the value of the assumed Dirac \(\delta\)-distributed serial interval.  
In contrast, the SIRD-AKS method is independent of the distribution of the serial interval, and therefore, it does not require any information on the assumed serial interval time as a hyperparameter. Instead, our approach internally copes with the serial interval through the ODE model and is not affected by any inadequate assumptions on the serial-interval distribution. Even if the serial-interval distribution is also subject to change over time, as, for example, reported in [77], this is compensated by the general time dependency e.g., of the transmission parameter \(\beta_{t}\) in the SIRD model.  
Compared to the incidence-based method, the SIRD-AKS estimate for the effective reproduction number \(\mathcal{R}_{t}\) shows the pandemic situation more clearly as it is less biased towards \(\mathcal{R}_{t}=1\). Furthermore, we checked the appropriateness of the estimated parameter time courses obtained by using the SIRD-AKS method by plugging them back into the original ODE model. While the incidence-based estimates yielded case numbers that were orders of magnitude below the original data, we were able to correctly reassemble the case numbers by the ODE model from the SIRD-AKS estimates.