Wastewater surveillance for the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is an emerging approach to help identify the risk of a coronavirus disease (COVID-19) outbreak. This tool can contribute to public health surveillance at both community (wastewater treatment system) and institutional (e.g., colleges, prisons, and nursing homes) scales. This paper explores the successes, challenges, and lessons learned from initial wastewater surveillance efforts at colleges and university systems to inform future research, development and implementation. We present the experiences of 25 college and university systems in the United States that monitored campus wastewater for SARS-CoV-2 during the fall 2020 academic period. We describe the broad range of approaches, findings, resources, and impacts from these initial efforts. These institutions range in size, social and political geographies, and include both public and private institutions. Our analysis suggests that wastewater monitoring at colleges requires consideration of local information needs, sewage infrastructure, resources for sampling and analysis, college and community dynamics, approaches to interpretation and communication of results, and follow-up actions. Most colleges reported that a learning process of experimentation, evaluation, and adaptation was key to progress. This process requires ongoing collaboration among diverse stakeholders including decision-makers, researchers, faculty, facilities staff, students, and community members.
Background Wastewater surveillance for SARS-CoV-2 is an emerging approach to help identify the risk of a COVID-19 outbreak. This tool can contribute to public health surveillance at both community (wastewater treatment system) and institutional (e.g., colleges, prisons, nursing homes) scales. Objectives This research aims to understand the successes, challenges, and lessons learned from initial wastewater surveillance efforts at colleges and university systems to inform future research, development and implementation. Methods This paper presents the experiences of 25 college and university systems in the United States that monitored campus wastewater for SARS-CoV-2 during the fall 2020 academic period. We describe the broad range of approaches, findings, resource needs, and lessons learned from these initial efforts. These institutions range in size, social and political geographies, and include both public and private institutions. Discussion Our analysis suggests that wastewater monitoring at colleges requires consideration of information needs, local sewage infrastructure, resources for sampling and analysis, college and community dynamics, approaches to interpretation and communication of results, and follow-up actions. Most colleges reported that a learning process of experimentation, evaluation, and adaptation was key to progress. This process requires ongoing collaboration among diverse stakeholders including decision-makers, researchers, faculty, facilities staff, students, and community members.
Using a 30 day time series of aphid Aphis helianthi and coccinellid counts on 107 mapped racemes of Yucca glauca, we demonstrate progressive, predation-induced self-organization of aphid colonies on individual racemes into extremely low and extremely high population sizes. This was driven by a two-attractor structure of density dependence that developed only in the presence of coccinellid predators. Foraging movements of the coccinellids among plants produced a power law relationship (average power 0.142) between aphid and coccinellid numbers. This resulted in increased predation pressure on mid-size colonies and decreased predation pressure on small and large populations. A field-parameterized mathematical model predicts a two-attractor structure in broad agreement with our observations. The overall system was integrated by the influence of the largest aphid populations, which determined the total number of coccinellids present, and thus the predation pressure throughout the system. Our study provides clear evidence of predator-driven self-organization of prey populations in a patchy environment.
In this paper, we consider the second-order Jacobi differential expressionhere, the Jacobi parameters are α > −1 and β = −1. This is a nonclassical setting since the classical setting for this expression is generally considered when α, β > −1. In the classical setting, it is well-known that the Jacobi polynomials {P (α,β) n } ∞ n=0 are (orthogonal) eigenfunctions of a self-adjoint operator T α,β , generated by the Jacobi differential expression, in the Hilbert space L 2 ((−1, 1); (1 − x) α (1 + x) β ). When α > −1 and β = −1, the Jacobi polynomial of degree 0 does not belong to the Hilbert space L 2 ((−1, 1); (1 − x) α (1 + x) −1 ). However, in this paper, we show that the full sequence of Jacobi polynomials {P (α,−1) n } ∞ n=0 forms a complete orthogonal set in a Hilbert-Sobolev space Wα, generated by the inner productWe also construct a self-adjoint operator Tα, generated by α,−1 [·] in Wα, that has the Jacobi polynomials {P (α,−1) n } ∞ n=0 as eigenfunctions. Mathematics Subject Classification (2010). Primary 33C45, 34B30, 47B25; Secondary 34B20, 47B65. 284 A. Bruder and L. L. Littlejohn Results. Math.
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