{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,10]],"date-time":"2026-10-10T00:36:36Z","timestamp":1791592596393,"version":"4.3.4"},"update-to":[{"DOI":"10.1371\/journal.pcbi.1009347","type":"new_version","label":"New version","source":"publisher","updated":{"date-parts":[[2021,9,17]],"date-time":"2021-09-17T00:00:00Z","timestamp":1631836800000}}],"reference-count":50,"publisher":"Public Library of Science (PLoS)","issue":"9","license":[{"start":{"date-parts":[[2021,9,7]],"date-time":"2021-09-07T00:00:00Z","timestamp":1630972800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000265","name":"Medical Research Council","doi-asserted-by":"publisher","award":["MR\/R015600\/1"],"award-info":[{"award-number":["MR\/R015600\/1"]}],"id":[{"id":"10.13039\/501100000265","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["www.ploscompbiol.org"],"crossmark-restriction":false},"short-container-title":["PLoS Comput Biol"],"abstract":"<jats:p>We construct a recursive Bayesian smoother, termed EpiFilter, for estimating the effective reproduction number, R, from the incidence of an infectious disease in real time and retrospectively. Our approach borrows from Kalman filtering theory, is quick and easy to compute, generalisable, deterministic and unlike many current methods, requires no change-point or window size assumptions. We model R as a flexible, hidden Markov state process and exactly solve forward-backward algorithms, to derive R estimates that incorporate all available incidence information. This unifies and extends two popular methods, EpiEstim, which considers past incidence, and the Wallinga-Teunis method, which looks forward in time. We find that this combination of maximising information and minimising assumptions significantly reduces the bias and variance of R estimates. Moreover, these properties make EpiFilter more statistically robust in periods of low incidence, where several existing methods can become destabilised. As a result, EpiFilter offers improved inference of time-varying transmission patterns that are advantageous for assessing the risk of upcoming waves of infection or the influence of interventions, in real time and at various spatial scales.<\/jats:p>","DOI":"10.1371\/journal.pcbi.1009347","type":"journal-article","created":{"date-parts":[[2021,9,7]],"date-time":"2021-09-07T13:55:53Z","timestamp":1631022953000},"page":"e1009347","update-policy":"https:\/\/doi.org\/10.1371\/journal.pcbi.corrections_policy","source":"Crossref","is-referenced-by-count":91,"title":["Improved estimation of time-varying reproduction numbers at low case incidence and between epidemic waves"],"prefix":"10.1371","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7806-3605","authenticated-orcid":true,"given":"Kris 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preliminary statistics of certain localities","volume":"34","author":"W Frost","year":"1919","journal-title":"Public Health Rep"},{"key":"pcbi.1009347.ref043","unstructured":"WHO. WHO Coronavirus Disease (COVID-19) Dashboard; 2020. Available from: https:\/\/covid19.who.int\/."},{"key":"pcbi.1009347.ref044","doi-asserted-by":"crossref","first-page":"1474","DOI":"10.1016\/S0140-6736(20)31097-7","article-title":"New Zealand eliminates COVID-19","volume":"395","author":"S Cousins","year":"2020","journal-title":"Lancet"},{"key":"pcbi.1009347.ref045","unstructured":"Ritchie H, Ortiz-Ospina E, Beltekian D, Mathieu E, Hasell J, Macdonald B, et al. Coronavirus Pandemic (COVID-19); 2020. Available from: https:\/\/ourworldindata.org\/coronavirus."},{"issue":"11","key":"pcbi.1009347.ref046","doi-asserted-by":"crossref","first-page":"e612","DOI":"10.1016\/S2468-2667(20)30225-5","article-title":"COVID-19 in New Zealand and the impact of the national response: a descriptive epidemiological study","volume":"5","author":"S Jefferies","year":"2020","journal-title":"Lancet Public Health"},{"key":"pcbi.1009347.ref047","volume-title":"Impact of non-pharmaceutical interventions (NPIs) to reduce COVID- 19 mortality and healthcare demand","author":"N Ferguson","year":"2020"},{"issue":"17","key":"pcbi.1009347.ref048","doi-asserted-by":"crossref","DOI":"10.2807\/1560-7917.ES.2020.25.17.2000257","article-title":"Estimating the generation interval for coronavirus disease (COVID-19) based on symptom onset data, March 2020","volume":"25","author":"T Ganyani","year":"2020","journal-title":"Euro Surveill"},{"key":"pcbi.1009347.ref049","volume-title":"Reproduction number (R) and growth rate (r) of the COVID-19 epidemic in the UK: methods of estimation, data sources, causes of heterogeneity, and use as a guide in policy formulation","author":"R Anderson","year":"2020"},{"issue":"2","key":"pcbi.1009347.ref050","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/S1473-3099(20)30785-4","article-title":"The temporal association of introducing and lifting non-pharmaceutical interventions with the time-varying reproduction number (R) of SARS-CoV-2: a modelling study across 131 countries","volume":"21","author":"Y Li","year":"2020","journal-title":"Lancet Infect Dis"}],"updated-by":[{"DOI":"10.1371\/journal.pcbi.1009347","type":"new_version","label":"New version","source":"publisher","updated":{"date-parts":[[2021,9,17]],"date-time":"2021-09-17T00:00:00Z","timestamp":1631836800000}}],"container-title":["PLOS Computational Biology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pcbi.1009347","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,8]],"date-time":"2023-11-08T08:38:14Z","timestamp":1699432694000},"score":1,"resource":{"primary":{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pcbi.1009347"}},"subtitle":[],"editor":[{"given":"Sergei L.","family":"Kosakovsky Pond","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"editor"}]}],"short-title":[],"issued":{"date-parts":[[2021,9,7]]},"references-count":50,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2021,9,7]]}},"URL":"https:\/\/doi.org\/10.1371\/journal.pcbi.1009347","relation":{"has-preprint":[{"id-type":"doi","id":"10.1101\/2020.09.14.20194589","asserted-by":"object"}]},"ISSN":["1553-7358"],"issn-type":[{"value":"1553-7358","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,9,7]]}}}