# Non-Parametric Model-Based Estimation of the Effective Reproduction Number for SARS-CoV-2 +
## 2 Materials and Methods  
### SIRD Model and Reparameterization  
The conventional method of characterizing the propagation of an infectious disease within a population involves employing the SIR model [20]. This model comprises three compartments: susceptible \(S\), infected \(I\) and removed \(R\). Frequently, the removed compartment is further divided into recovered \(R\) and deceased \(D\), leading to the SIRD model [52]. The time-continuous evolution of this model is described by the following system of ODEs:  
\[\dot{S} =-\beta\cdot\frac{S\cdot I}{N_{0}} \tag{1}\] \[\dot{I} =\beta\cdot\frac{S\cdot I}{N_{0}}-\gamma\cdot I-\theta\cdot I\] \[\dot{R} =\gamma\cdot I\] \[\dot{D} =\theta\cdot I\,.\]  
The model contains three time-constant transition rate parameters: \(\beta\), \(\gamma\) and \(\theta\), which represent the infection rate, recovery rate and mortality rate, respectively. Initially, at \(t=0\), the values of the four compartments are denoted as \(S(0)=S_{0}\), \(I(0)=I_{0}\), while \(R(0)=0\) and \(D(0)=0\), ensuring that the total number of individuals is given by \(N_{0}=S_{0}+I_{0}\). In the early stages of an epidemic, when \(S_{0}\approx N_{0}\), the disease-specific _basic reproduction number_ denoted as \(\mathcal{R}_{0}\), which represents the average number of secondary cases, can be directly calculated from the model parameters as \(\mathcal{R}_{0}=\beta/(\gamma+\theta)\). As the disease spreads and more individuals become infected, the assumption \(S(t)\approx N_{0}\) is no longer valid, and the time-dependent _effective reproduction number_ is defined in the literature as [53]  
\[\mathcal{R}_{t}^{*}=\frac{\beta}{\gamma+\theta}\cdot\frac{S(t)}{N_{0}} \tag{2}\]  
which depends on the change in the susceptible individuals \(S(t)\) over time. The effective reproduction number \(\mathcal{R}_{t}^{*}\) serves as a crucial metric for understanding the status of an epidemic, where \(\mathcal{R}_{t}^{*}>1\) indicates an increasing number of newly infected individuals and \(\mathcal{R}_{t}^{*}<1\) suggests a declining trend. In addition to intrinsic factors, external influences such as seasonal variations and behavioral changes in the population significantly impact viral transmission. The time dependencies in the infection rate \(\beta\) effectively capture the influence of increased hygiene standards and non-pharmaceutical interventions, which play a critical role in epidemic development. The manifestation of such effects can be observed through the emergence of epidemic scenarios featuring multiple waves of infections, as seen in seasonal influenza and COVID-19 outbreaks. To accommodate the dynamic nature of epidemics, an unambiguous model extension involves considering time-dependent transition rates: \(\beta_{t}\), \(\gamma_{t}\) and \(\theta_{t}\). By incorporating these temporal changes into the model, we can accurately define the effective reproduction number within the context of evolving epidemic patterns as  
\[\mathcal{R}_{t}=\frac{\beta_{t}}{\gamma_{t}+\theta_{t}}\cdot\frac{S(t)}{N_{0}}. \tag{3}\]  
In order to simplify the estimation of \(\mathcal{R}_{t}\), we substitute Equation (3) into Equation (1), yielding a reparameterized version of the SIRD model:\[ \begin{split}\dot{S}&=-\mathcal{R}_{t}\cdot(\gamma_{t}+ \theta_{t})\cdot I\\ I&=(\mathcal{R}_{t}-1)\cdot(\gamma_{t}+\theta_{t}) \cdot I\\ \dot{R}&=\gamma_{t}\cdot I\\ \dot{D}&=\theta_{t}\cdot I.\end{split}  \tag{4}\]  
It is important to highlight that as a result of this transformation, the susceptible compartment \(S\) no longer appears on the right-hand side of Equation (4). Consequently, the differential equation for \(S\) can be disregarded without impacting the solution of the remaining system. Parameter estimation for non-linear ODE models with time-constant parameters typically involves the utilization of efficient numerical optimization algorithms.