# Non-Parametric Model-Based Estimation of the Effective Reproduction Number for SARS-CoV-2 +
## 1 Introduction  
The last five decades have witnessed a significant rise in emerging infectious diseases, particularly zoonoses and vector-borne diseases [1; 2]. To gain a comprehensive understanding of their dynamics, mathematical models have become an indispensable tool [3; 4; 5; 6; 7]. The global pandemic caused by the novel SARS-CoV-2 in 2020/21 prompted numerous modeling efforts, ranging from early situation assessments [8; 9; 10] to evaluating the effectiveness of non-pharmaceutical interventions [11; 12; 13], analyzing undetected cases [14; 15] and employing agent-based model descriptions [16; 17; 18]. Infectious disease spread is often described by using the classical susceptible-infected-recovered (SIR) compartmental model where the model parameters are estimated from the observed data [19; 20; 21]. These parameters, specific to each disease, depend on the pathogen's characteristics and its contagiousness [6; 22]. To improve the description of infection dynamics, extensions to the classical SIR structure have been proposed, including differentiation between deceased (D) and recovered (R) individuals or the introduction of an intermediate exposed compartment (E) for infected but not yet contagious individuals. While more compartments can offer detailed descriptions, overly complex models may face challenges in parameter estimation due to limited data availability and parameter identifiability issues [23]. Nevertheless, comprehensive SIR-like model structures have been developed for COVID-19, covering essential features like asymptomatic infectious individuals, undetected infections and different hospitalization states [8; 24; 25; 26; 27]. The temporal variations in detected SARS-CoV-2 infection case numbers, with multiple epidemic waves observed in most countries [28; 29], necessitate time-dependent model parameters to accurately describe the dynamics [30; 31; 32]. Time dependencies in the parameters account for seasonal effects [33; 34] and changes in population behavior or the impact of non-pharmaceutical interventions. Among the various model parameters of interest, the effective reproduction number \(\mathcal{R}_{t}\) holds particular significance in epidemic models. It represents the average number of secondary infections caused by one infectious individual and varies over time, offering insight into the outbreak's current status and indicating turning points in infection growth. Estimating the effective reproduction number from measured data via epidemic models with time-dependent parameters has been a key focus for informing governmental decision makers during the SARS-CoV-2 pandemic. Different approaches have been proposed, such as specifying step functions to incorporate non-pharmaceutical interventions [9] or assuming spline functions to capture flexible time dependencies [35]. While these approaches offer valuable insights into pandemic scenarios, they depend on the selected function class and often require a high number of model parameters, leading to challenges in parameter estimation. Defining functions to adequately represent more diffuse changes, such as behavioral adjustments in the population, remains challenging. Addressing the task of estimating time-dependent parameters in high-dimensional ordinary differential equation (ODE) models, especially with sparse data, poses difficulties [36; 37; 38; 39]. In the field of epidemiology, established methods like Sequential Monte Carlo (SMC) or particle Markov chain Monte Carlo (pMCMC) have been successfully applied to estimate time-dependent transmission rates in SIR-type models [40; 41; 42; 43; 44; 45]. However, these methods often require the careful calibration of meta parameters and can be computationally expensive. Alternatively, function-class-free methods, like iterative Kalman-filter-based approaches, offer computationally simpler and cheaper alternatives [46].