Wastewater-based Epidemiology for COVID-19 Surveillance: A Survey.

The pandemic of COVID-19 has imposed tremendous pressure on public health systems and social economic ecosystems over the past years. To alleviate its social impact, it is important to proactively track the prevalence of COVID-19 within communities. The traditional way to estimate the disease prevalence is to estimate from reported clinical test data or surveys. However, the coverage of clinical tests is often limited and the tests can be labor-intensive, requires reliable and timely results, and consistent diagnostic and reporting criteria. Recent studies revealed that patients who are diagnosed with COVID-19 often undergo fecal shedding of SARS-CoV-2 virus into wastewater, which makes wastewater-based epidemiology (WBE) for COVID-19 surveillance a promising approach to complement traditional clinical testing. In this paper, we survey the existing literature regarding WBE for COVID-19 surveillance and summarize the current advances in the area. Specifically, we have covered the key aspects of wastewater sampling, sample testing, and presented a comprehensive and organized summary of wastewater data analytical methods. Finally, we provide the open challenges on current wastewater-based COVID-19 surveillance studies, aiming to encourage new ideas to advance the development of effective wastewater-based surveillance systems for general infectious diseases.

Distributions of 4-subtree patterns for uniform random unrooted phylogenetic trees.

Tree shape statistics based on peripheral structures have been utilized to study evolutionary mechanisms and inference methods. Partially motivated by a recent study by Pouryahya and Sankoff on modeling the accumulation of subgenomes in the evolution of polyploids, we present the distribution of subtree patterns with four or fewer leaves for the unrooted Proportional to Distinguishable Arrangements (PDA) model. We derive a recursive formula for computing the joint distributions, as well as a Strong Law of Large Numbers and a Central Limit Theorem for the joint distributions. This enables us to confirm several conjectures proposed by Pouryahya and Sankoff, as well as provide some theoretical insights into their observations. Based on their empirical datasets, we demonstrate that the statistical test based on the joint distribution could be more sensitive than those based on one individual subtree pattern to detect the existence of evolutionary forces such as whole genome duplication.

Distributions of cherries and pitchforks for the Ford model.

Distributional properties of tree shape statistics under random phylogenetic tree models play an important role in investigating the evolutionary forces underlying the observed phylogenies. In this paper, we study two subtree counting statistics, the number of cherries and that of pitchforks for the Ford model, the alpha model introduced by Daniel Ford. It is a one-parameter family of random phylogenetic tree models which includes the proportional to distinguishable arrangement (PDA) and the Yule models, two tree models commonly used in phylogenetics. Based on a non-uniform version of the extended Pólya urn models in which negative entries are permitted for their replacement matrices, we obtain the strong law of large numbers and the central limit theorem for the joint distribution of these two statistics for the Ford model. Furthermore, we derive a recursive formula for computing the exact joint distribution of these two statistics. This leads to exact formulas for their means and higher order asymptotic expansions of their second moments, which allows us to identify a critical parameter value for the correlation between these two statistics. That is, when the number of tree leaves is sufficiently large, they are negatively correlated for 0≤α≤1/2 and positively correlated for 1/2<α<1.

On asymptotic joint distributions of cherries and pitchforks for random phylogenetic trees.

Tree shape statistics provide valuable quantitative insights into evolutionary mechanisms underpinning phylogenetic trees, a commonly used graph representation of evolutionary relationships among taxonomic units ranging from viruses to species. We study two subtree counting statistics, the number of cherries and the number of pitchforks, for random phylogenetic trees generated by two widely used null tree models: the proportional to distinguishable arrangements (PDA) and the Yule-Harding-Kingman (YHK) models. By developing limit theorems for a version of extended Pólya urn models in which negative entries are permitted for their replacement matrices, we deduce the strong laws of large numbers and the central limit theorems for the joint distributions of these two counting statistics for the PDA and the YHK models. Our results indicate that the limiting behaviour of these two statistics, when appropriately scaled using the number of leaves in the underlying trees, is independent of the initial tree used in the tree generating process.