In a previous paper, see [14], we generalized the parameterization method of Cabré, Fontich and De la Llave to center manifolds of discrete dynamical systems. In this paper, we extend this result to several different settings. The natural setting in which center manifolds occur is at bifurcations in dynamical systems with parameters. Our first results will show that we can find parameter-dependent center manifolds near bifurcation points. Furthermore, we will generalize the parameterization method to center manifolds of fixed points of ODEs. Finally, we will apply our method to a reaction diffusion equation. In our application, we will show that the freedom to obtain the conjugate dynamics in normal form makes it possible to obtain detailed qualitative information about the center dynamics.

SUMMARY Background Surveillance of SARS-CoV-2 in wastewater offers an unbiased and near real-time tool to track circulation of SARS-CoV-2 at a local scale, next to other epidemic indicators such as hospital admissions and test data. However, individual measurements of SARS-CoV-2 in sewage are noisy, inherently variable, and can be left-censored. Aim We aimed to infer latent virus loads in a comprehensive sewage surveillance program that includes all sewage treatment plants (STPs) in the Netherlands and covers 99.6% of the Dutch population. Methods A multilevel Bayesian penalized spline model was developed and applied to estimate time- and STP-specific virus loads based on water flow adjusted SARS-CoV-2 qRT-PCR data from 1-4 sewage samples per week for each of the >300 STPs. Results The model provided an adequate fit to the data and captured the epidemic upsurges and downturns in the Netherlands, despite substantial day-to-day measurement variation. Estimated STP virus loads varied by more than two orders of magnitude, from approximately 1012 (virus particles per 100,000 persons per day) in the epidemic trough in August 2020 to almost 1015 in many STPs in January 2022. Epidemics at the local levels were slightly shifted between STPs and municipalities, which resulted in less pronounced peaks and troughs at the national level. Conclusion Although substantial day-to-day variation is observed in virus load measurements, wastewater-based surveillance of SARS-CoV-2 can track long-term epidemic progression at a local scale in near real-time, especially at high sampling frequency.

Background Surveillance of SARS-CoV-2 in wastewater offers a near real-time tool to track circulation of SARS-CoV-2 at a local scale. However, individual measurements of SARS-CoV-2 in sewage are noisy, inherently variable and can be left-censored. Aim We aimed to infer latent virus loads in a comprehensive sewage surveillance programme that includes all sewage treatment plants (STPs) in the Netherlands and covers 99.6% of the Dutch population. Methods We applied a multilevel Bayesian penalised spline model to estimate time- and STP-specific virus loads based on water flow-adjusted SARS-CoV-2 qRT-PCR data for one to four sewage samples per week for each of the more than 300 STPs. Results The model captured the epidemic upsurges and downturns in the Netherlands, despite substantial day-to-day variation in the measurements. Estimated STP virus loads varied by more than two orders of magnitude, from ca 1012 virus particles per 100,000 persons per day in the epidemic trough in August 2020 to almost 1015 per 100,000 in many STPs in January 2022. The timing of epidemics at the local level was slightly shifted between STPs and municipalities, which resulted in less pronounced peaks and troughs at the national level. Conclusion Although substantial day-to-day variation is observed in virus load measurements, wastewater-based surveillance of SARS-CoV-2 that is performed at high sampling frequency can track long-term progression of an epidemic at a local scale in near real time.

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