Signal-Analytics Modeling of Fluorescence Time-to-Detection for E. coli in Treated Wastewater: A Joint Censoring and...

Coliform bacteria are widely used as indicators of fecal contamination, and pathogenic strains of Escherichia coli may pose substantial risks to human health. Virulent E. coli can cause clinical manifestations ranging from self-limiting gastroenteritis to severe systemic disease, including in otherwise healthy individuals. Accordingly, numerous studies have pursued strategies for the early detection and quantification of these pathogens in environmental and clinical settings. Motivated by a miniaturized fluorescence-based incubation, the Optical Bacterial Growth Sensor-Fluorescence Detector, which provides minute-resolved time-to-detection (TTD) readouts for E. coli in treated wastewater, we propose a new probabilistic model tailored to describe the distribution of TTD values. In practical terms, the model is intended to indicate when a fluorescence-signal increase should be treated as a detection event and how this information can support routine operation. Using 47 independent treated-wastewater assays, we extracted minute-resolved fluorescence-signal TTD values and fitted the proposed model by maximum likelihood. The proposed distribution provides a more faithful description of fluorescence-signal TTD behavior under treated-wastewater operating conditions, supports fit-for-purpose specification of the lower operational boundary of the assay counting window, and provides operational guidance for routine monitoring under the present sensing conditions. The fitted model indicates that most TTD values fall within an operational window of approximately 7-10 h, with a very low probability of detections beyond 12 h. We show that the model outperforms classical distributions commonly used to describe TTD data for E. coli. Key mathematical properties are derived to support the empirical results, and Monte Carlo evidence based on a parametric bootstrap corroborates the main findings. Overall, this framework expands the statistical toolkit for modeling sensor-based TTD phenomena associated with E. coli and supports routine monitoring.