In this article, a fuzzy clustering model for multivariate time series based on the quantile cross-spectral density and principal component analysis is extended by including: 1) a weighting system which assigns a weight to each principal component in accordance with its importance concerning the underlying clustering structure and 2) a penalization term allowing to take into account the spatial information. The iterative solutions of the new model, which employs the exponential distance in order to gain robustness against outlying series, are derived. A simulation study shows that the weighting system substantially enhances the effectiveness of the former approach. The behavior of the extended model in terms of the spatial penalization term is also analyzed. An application involving multivariate time series of mobility indicators concerning COVID-19 pandemic highlights the usefulness of the proposed technique.

The quantification of the SARS-CoV-2 load in wastewater has emerged as a useful method to monitor COVID-19 outbreaks in the community. This approach was implemented in the metropolitan area of A Coruña (NW Spain), where wastewater from the treatment plant of Bens was analyzed to track the epidemic’s dynamic in a population of 369,098 inhabitants. We developed statistical regression models that allowed us to estimate the number of infected people from the viral load detected in the wastewater with a reliability close to 90%. This is the first wastewater-based epidemiological model that could potentially be adapted to track the evolution of the COVID-19 epidemic anywhere in the world, monitoring both symptomatic and asymptomatic individuals. It can help to understand with a high degree of reliability the true magnitude of the epidemic in a place at any given time and can be used as an effective early warning tool for predicting outbreaks.