{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T12:29:19Z","timestamp":1787228959943,"version":"build-2736575974"},"reference-count":14,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM Rev."],"published-print":{"date-parts":[[2025,5,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>We present and discuss the Streeter\u2013Phelps equations, which were the first river quality model. If the parameters are constants, then the model in its linear formulation can be solved explicitly. This reveals, however, that depending on parameters and initial data, the model might predict negative oxygen concentrations, which marks a breakdown of the model. To address this shortcoming, we introduce a nonlinear modification which, in the case of constant parameters, we can study in the phase plane. In real-world applications, parameters are never constant and are usually known not exactly, but instead with some uncertainty. We show how we can use the solutions for the constant parameter case to obtain estimates for the unknown solutions from estimates of the model parameters, using differential inequalities.<\/jats:p>","DOI":"10.1137\/23m1616406","type":"journal-article","created":{"date-parts":[[2025,5,8]],"date-time":"2025-05-08T03:36:49Z","timestamp":1746675409000},"page":"375-398","source":"Crossref","is-referenced-by-count":0,"title":["Uncertainty Analysis of a Simple River Quality Model Using Differential Inequalities"],"prefix":"10.1137","volume":"67","author":[{"given":"Grace","family":"D\u2019Agostino","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, University of Guelph, Guelph, ON, N1G2W1, Canada."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5558-0872","authenticated-orcid":true,"given":"Hermann J.","family":"Eberl","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, University of Guelph, Guelph, ON, N1G2W1, Canada."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2025,5,8]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-007-5509-3"},{"key":"ref2","volume-title":"Elementary Differential Equations and Boundary Value Problems","author":"Boyce W.","year":"2005"},{"key":"ref3","unstructured":"Canadian Council of Ministers of the Environment, Canadian Water Quality Guidelines for the Protection of Aquatic Life: Dissolved Oxygen (Freshwater), https:\/\/ccme.ca\/en\/res\/dissolved-oxygen-freshwater-en-canadian-water-quality-guidelines-for-the-protection-of-aquatic-life.pdf (accessed on December 21, 2024). (Cited on p. 376)"},{"key":"ref4","unstructured":"Chesswatch, Water Quality Indicator: Dissolved Oxygen, https:\/\/www.qmul.ac.uk\/chesswatch\/media\/chesswatch\/Dissolved-Oxygen-leaflet.pdf. (Cited on p. 376)"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1016\/0025-5564(80)90113-3"},{"key":"ref6","volume-title":"Water Environment Modelling","author":"Liu C.","year":"2022"},{"key":"ref7","volume-title":"Introduction to Environmental Engineering and Science","author":"Masters G.","year":"1997"},{"key":"ref8","volume-title":"Maximum Principles in Differential Equations","author":"Protter M.","year":"1967"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.2166\/wst.2001.0241"},{"key":"ref10","volume-title":"Modeling and Control of River Quality","author":"Rinaldi S.","year":"1979"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511801204"},{"key":"ref12","unstructured":"W. Streeter and E. Phelps, A Study of the Pollution and Natural Purification of the Ohio River, Technical report, United States Public Health Service, 1925. (Cited on p. 376)"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.2105\/AJPH.75.9.1059"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-57240-1"}],"container-title":["SIAM Review"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/23M1616406","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T12:09:28Z","timestamp":1787227768000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/23M1616406"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,8]]},"references-count":14,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,5,31]]}},"alternative-id":["10.1137\/23M1616406"],"URL":"https:\/\/doi.org\/10.1137\/23m1616406","relation":{},"ISSN":["0036-1445","1095-7200"],"issn-type":[{"value":"0036-1445","type":"print"},{"value":"1095-7200","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,5,8]]}}}