# Non-Parametric Model-Based Estimation of the Effective Reproduction Number for SARS-CoV-2 +
## 2 Materials and Methods
### SIRD-AKS Method for Estimating the Effective Reproduction Number \(\mathcal{R}_{t}\)  
Based on the general derivation in Equation (19), the time-discrete state-space formulation of the SIRD model from Equation (4) can be written as  
\[x_{k}=\begin{pmatrix}S_{k}\\ I_{k}\\ R_{k}\\ D_{k}\\ \mathcal{R}_{k}\\ \gamma_{k}\\ \theta_{k}\end{pmatrix}=b(x_{k-1})+\epsilon_{k-1}=\begin{pmatrix}S_{k-1}\\ I_{k-1}\\ R_{k-1}\\ D_{k-1}\\ \mathcal{R}_{k-1}\\ \gamma_{k-1}\\ \theta_{k-1}\end{pmatrix}+\begin{pmatrix}-\mathcal{R}_{k-1}\cdot(\gamma_{k-1} +\theta_{k-1})\cdot I_{k-1}\\ (\mathcal{R}_{k-1}-1)\cdot(\gamma_{k-1}+\theta_{k-1})\cdot I_{k-1}\\ \gamma_{k-1}\cdot I_{k-1}\\ \theta_{k-1}\cdot I_{k-1}\\ 0\\ 0\\ 0\end{pmatrix}+\epsilon_{k-1}\,, \tag{20}\]  
where the first four rows are the four states of the time-continuous ODE and \(\epsilon_{k-1}\) corresponds to the process noise, as introduced in Equation (5). To incorporate the desired time dependence of the three SIRD model parameters, namely \(\mathcal{R}_{k}\), \(\gamma_{k}\) and \(\theta_{k}\), the AKS method treats these parameters as additional states within the state-space formulation, leading to the SIRD-AKS model. As a result, the SIRD-AKS not only provides time-course estimates for the model states \(S_{k}\), \(I_{k}\), \(R_{k}\) and \(D_{k}\), but also for the parameters \(\mathcal{R}_{k}\), \(\gamma_{k}\) and \(\theta_{k}\).  
In real-world outbreak scenarios, data on the number of currently infectious individuals in state \(I\) per day are often unavailable. However, the incidence, representing the number of newly infected cases per time period, is commonly reported and easily accessible. In the SIRD model, these incidence numbers correspond to the flux of individuals into the infectious state, given by \(v_{I,k}=\mathcal{R}k\cdot(\gamma k+\theta_{k})\cdot I_{k}\), transitioning from time step \(k\) to \(k+1\). Similarly, published data for the number of newly recovered \(v_{R,k}=\gamma_{k}\cdot I_{k}\) and newly deceased \(v_{D,k}=\theta_{k}\cdot I_{k}\) individuals form the fluxes into their respective states. Hence, the fluxes \(v_{I,k}\), \(v_{R,k}\) and \(v_{D,k}\) serve as the observables \(y_{k}\) for the SIRD-AKS method.  
However, the observation matrix \(H\) introduced in Equation (5) only allows linear relationships between observables and states. To address this limitation and map the data to the appropriate variables of the state space for the SIRD-AKS method, the state-space formulation is expanded once more by introducing three additional states corresponding to the three fluxes, resulting in \[x_{k}=\left(\begin{array}{c}S_{k}\\ I_{k}\\ R_{k}\\ D_{k}\\ \mathcal{R}_{k}\\ \gamma_{k}\\ \theta_{k}\\ v_{l,k}\\ v_{R,k}\\ v_{D,k}\end{array}\right)=b(x_{k-1})+\epsilon_{k-1}=\left(\begin{array}{c}S_{k-1 }\\ I_{k-1}\\ R_{k-1}\\ D_{k-1}\\ \mathcal{R}_{k-1}\\ \gamma_{k-1}\\ \theta_{k-1}\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right)+\left(\begin{array}{c}-v_{l,k-1}\\ v_{l,k-1}-v_{R,k-1}-v_{D,k-1}\\ v_{R,k-1}\\ v_{D,k-1}\\ 0\\ 0\\ \mathcal{R}_{k-1}\cdot(\gamma_{k-1}+\theta_{k-1})\cdot I_{k-1}\\ \gamma_{k-1}\cdot I_{k-1}\\ \theta_{k-1}\cdot I_{k-1}\end{array}\right)+\epsilon_{k-1} \tag{21}\]  
and an observation matrix  
\[H=\left(\begin{array}{cccc}1&0&0\\ 0_{M_{7,3}}&0&1&0\\ 0&0&1\end{array}\right). \tag{22}\]  
that links the incidence data to the respective fluxes \(v_{l,k}\), \(v_{R,k}\) and \(v_{D,k}\). The SIRD-AKS method utilizes incidence data to estimate the state-space vector \(x_{k}\) through the state-space formulation defined in Equation (21). The entries of \(x_{k}\) are interpreted as discretized time series of the model states, time-dependent parameters and fluxes, i.e., incidence numbers. To ensure that all the variables remain strictly positive, the entire analysis is conducted on a logarithmic scale for \(x_{k}\). Employing the logarithmic scale offers advantages in terms of numerical stability.