We introduce a new class of heavy-tailed distributions for which any weighted average of independent and identically distributed random variables is larger than one such random variable in (usual) stochastic order. We show that many commonly used extremely heavy-tailed (i.e., infinite-mean) distributions, such as the Pareto, Fréchet, and Burr distributions, belong to this class. The established stochastic dominance relation can be further generalized to allow negatively dependent or non-identically distributed random variables. In particular, the weighted average of non-identically distributed random variables dominates their distribution mixtures in stochastic order.

Temporal variability and methodological differences in data normalization, among other factors, complicate effective trend analysis of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) wastewater surveillance data and its alignment with coronavirus disease 2019 (COVID-19) clinical outcomes. As there is no consensus approach for these analyses yet, this study explored the use of piecewise linear trend analysis (joinpoint regression) to identify significant trends and trend turning points in SARS-CoV-2 RNA wastewater concentrations (normalized and non-normalized) and corresponding COVID-19 case rates in the greater Las Vegas metropolitan area (Nevada, USA) from mid-2020 to April 2023. The analysis period was stratified into three distinct phases based on temporal changes in testing protocols, vaccination availability, SARS-CoV-2 variant prevalence, and public health interventions. While other statistical methodologies may require fewer parameter specifications, joinpoint regression provided an interpretable framework for characterization and comparison of trends and trend turning points, revealing sewershed-specific variations in trend magnitude and timing that also aligned with known variant-driven waves. Week-level trend agreement corroborated previous findings demonstrating a close relationship between SARS-CoV-2 wastewater surveillance data and COVID-19 outcomes. These findings guide future applications of advanced statistical methodologies and support the continued integration of wastewater-based epidemiology as a complementary approach to traditional COVID-19 surveillance systems.

Human toxocariasis is a worldwide parasitic disease caused by zoonotic roundworms of the genus Toxocara, which can cause blindness and epilepsy. The aim of this study was to estimate the risk of food-borne transmission of Toxocara spp. to humans in the UK by developing mathematical models created in a Bayesian framework. Parameter estimation was based on published experimental studies and field data from southern England, with qPCR Cq values used as a measure of eggs in spinach portions and ELISA optical density data as an indirect measure of larvae in meat portions. The average human risk of Toxocara spp. infection, per portion consumed, was estimated as 0.016% (95% CI: 0.000–0.100%) for unwashed leafy vegetables and 0.172% (95% CI: 0.000–0.400%) for undercooked meat. The average proportion of meat portions estimated positive for Toxocara spp. larvae was 0.841% (95% CI: 0.300–1.400%), compared to 0.036% (95% CI: 0.000–0.200%) of spinach portions containing larvated Toxocara spp. eggs. Overall, the models estimated a low risk of infection with Toxocara spp. by consuming these foods. However, given the potentially severe human health consequences of toxocariasis, intervention strategies to reduce environmental contamination with Toxocara spp. eggs and correct food preparation are advised.

We prove an ergodic theorem for Markov chains indexed by the Ulam–Harris–Neveu tree over large subsets with arbitrary shape under two assumptions: (i) with high probability, two vertices in the large subset are far from each other, and (ii) with high probability, those two vertices have their common ancestor close to the root. The assumption on the common ancestor can be replaced by some regularity assumption on the Markov transition kernel. We verify that these assumptions are satisfied for some usual trees. Finally, with Markov chain Monte Carlo considerations in mind, we prove that when the underlying Markov chain is stationary and reversible, the Markov chain, that is the line graph, yields minimal variance for the empirical average estimator among trees with a given number of nodes. In doing so, we prove that the Hosoya–Wiener polynomial is minimized over $[{-}1,1]$ by the line graph among trees of a given size.

We analyse a Markovian SIR epidemic model where individuals either recover naturally or are diagnosed, leading to isolation and potential contact tracing. Our focus is on digital contact tracing via a tracing app, considering both its standalone use and its combination with manual tracing. We prove that as the population size n grows large, the epidemic process converges to a limiting process, which, unlike with typical epidemic models, is not a branching process due to dependencies created by contact tracing. However, by grouping to-be-traced individuals into macro-individuals, we derive a multi-type branching process interpretation, allowing computation of the reproduction number R. This is then converted to an individual reproduction number $R^\mathrm{(ind)}$, which, in contrast to R, decays monotonically with the fraction of app-users, while both share the same threshold at 1. Finally, we compare digital (only) contact tracing and manual (only) contact tracing, proving that the critical fraction of app-users, $\pi_{\mathrm{c}}$, required for $R=1$ is higher than the critical fraction manually contact-traced, $p_{\mathrm{c}}$, for manual tracing.