# Non-Parametric Model-Based Estimation of the Effective Reproduction Number for SARS-CoV-2 +
## 3 Results  
### AKS Performance for Multiple Time-Varying SIRD Model Parameters  
As a first simulation setting, the classical SIRD model from Equation (1) with the initial values \(S_{0}=10^{8}\), \(I_{0}=1\) was used for data generation. Furthermore, three different parameter settings for the three model parameters \(\beta_{t}\), \(\theta_{t}\) and \(\gamma_{t}\) were simulated. In the first parameter setting shown in Figure 1A, time-varying parameters were chosen as \(\beta_{t}=0.05+0.04\cdot\sin\left(t/50\right)\) and \(\theta_{t}=0.003+0.0015\cdot\sin\left(t/33\right)\), while \(\gamma=0.02\) was chosen to be constant over time. The resulting dynamics for \(\mathcal{R}_{t}\) reflect the combination of the sinusoidal input \(\beta_{t}\) and the decrease in the susceptible \(S\), c.f. Equation (3). To generate data sets, the fluxes \(v_{I}=\beta_{t}\frac{SI}{N_{0}}\), \(v_{D}=\theta_{t}I\) and \(v_{R}=\gamma_{t}I\) between the model's states were evaluated at 375 equidistant time points, mimicking approximately one year of daily observations. Gaussian noise with a relative error of 5% was added to the three time series of observables, shown as green dots in Figure 1.  
Figure 1: Estimation of multiple time-dependent parameters. (**A**): The classical SIRD model was simulated with a time-constant parameter \(\gamma\) and time-dependent parameters \(\beta_{t}\) and \(\theta_{t}\). From these parameters, the effective reproduction number \(\mathcal{R}_{t}\) was calculated via Equation (3). True parameter time courses and states are depicted as red solid lines and red dashed lines, respectively. Therefore, data were simulated at 375 equidistant time points for the observed fluxes \(v_{D}\), \(v_{l}\) and \(v_{R}\), as indicated by green dots. The SIRD-AKS is based on the reparameterized SIRD model and thus does not yield estimates for \(\beta_{t}\) or S but instead directly for \(\mathcal{R}_{t}\). The estimates of the SIRD-AKS are shown as black solid lines with gray bands for their uncertainties. Results are shown on logarithmic scale. The results of the the SIRD-AKS algorithm are able to describe the data and are in good agreement with the true parameter time courses, both for the time-constant and time-dependent parameters. Note that the SIRD-AKS estimates for the states overlap with the true time courses of the model states. (**B,C**): In different parameter settings, simulation was performed with time-constant parameters \(\gamma\) and \(\theta\), whereas \(\beta_{t}\) was either chosen as a stochastic process (**B**) or as a step function (**C**), respectively. The corresponding values for \(\mathcal{R}_{t}\) were computed by using Equation (3). In both cases, the SIRD-AKS results retrieve the true parameter time courses adequately. Next, the noisy data were provided to the SIRD-AKS in order to estimate the model states and time-dependent parameters. For this, the EM algorithm part of the SIRD-AKS was initialized with \(I_{0}=100,\ R_{0}=0.1,\ D_{0}=0.1,\ v_{I,0}=1,\ v_{R,0}=0.1,\ v_{D,0}=0.1\), \(\gamma_{0}=\log(0.1),\theta_{0}=\log(0.01)\) and \(\mathcal{R}_{t,0}=\log(6)\), and the convergence threshold was chosen to be \(\alpha=0.001\). The estimation results are shown as black lines in Figure 1A, where the gray error bands of the estimates correspond to the estimated covariance matrices \(P_{k|k}\). Note that the time-dependent parameters are shown on a logarithmic scale throughout this work. The SIRD-AKS is able to simultaneously capture the dynamics of the simulated values for parameters, states and observations. In particular, the time-dependent parameters \(\gamma_{t}\), \(\mathcal{R}_{t}\) and \(\theta_{t}\) were obtained without providing any information on their function class or dynamical behavior. It can be observed that the gray error bands for the estimated parameters in Figure 1 are larger in the beginning of the time series, where also the parameter estimates slightly deviate from the true values due to missing dynamics in the data.