Bayesian Modeling of Enteric Virus Density in Wastewater Using Left-Censored Data

Stochastic models are used to express pathogen density in environmental samples for performing microbial risk assessment with quantitative uncertainty. However, enteric virus density in water often falls below the quantification limit (non-detect) of the analytical methods employed, and it is always difficult to apply stochastic models to a dataset with a substantially high number of non-detects, i.e., left-censored data. We applied a Bayesian model that is able to model both the detected data (detects) and non-detects to simulated left-censored datasets of enteric virus density in wastewater. One hundred paired datasets were generated for each of the 39 combinations of a sample size and the number of detects, in which three sample sizes (12, 24, and 48) and the number of detects from 1 to 12, 24 and 48 were employed. The simulated observation data were assigned to one of two groups, i.e., detects and non-detects, by setting values on the limit of quantification to obtain the assumed number of detects for creating censored datasets. Then, the Bayesian model was applied to the censored datasets, and the estimated mean and standard deviation were compared to the true values by root mean square deviation. The difference between the true distribution and posterior predictive distribution was evaluated by Kullback-Leibler (KL) divergence, and it was found that the estimation accuracy was strongly affected by the number of detects. It is difficult to describe universal criteria to decide which level of accuracy is enough, but eight or more detects are required to accurately estimate the posterior predictive distributions when the sample size is 12, 24, or 48. The posterior predictive distribution of virus removal efficiency with a wastewater treatment unit process was obtained as the log ratio posterior distributions between the posterior predictive distributions of enteric viruses in untreated wastewater and treated wastewater. The KL divergence between the true distribution and posterior predictive distribution of virus removal efficiency also depends on the number of detects, and eight or more detects in a dataset of treated wastewater are required for its accurate estimation.