{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,15]],"date-time":"2026-09-15T07:02:30Z","timestamp":1789455750032,"version":"build-2803163510"},"reference-count":55,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2026,7,22]],"date-time":"2026-07-22T00:00:00Z","timestamp":1784678400000},"content-version":"vor","delay-in-days":202,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000266","name":"UK Engineering and Physical Sciences Research Council","doi-asserted-by":"crossref","award":["EP\/H012133\/1"],"award-info":[{"award-number":["EP\/H012133\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100000266","name":"UK Engineering and Physical Sciences Research Council","doi-asserted-by":"crossref","award":["EP\/K039083\/1"],"award-info":[{"award-number":["EP\/K039083\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100000266","name":"UK Engineering and Physical Sciences Research Council","doi-asserted-by":"crossref","award":["EP\/V030515\/1"],"award-info":[{"award-number":["EP\/V030515\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Royal Academy of Engineering Chair in Emerging Technologies","award":["BB\/Y512916\/1"],"award-info":[{"award-number":["BB\/Y512916\/1"]}]},{"name":"Royal Academy of Engineering Chair in Emerging Technologies","award":["BB\/Y008332\/1"],"award-info":[{"award-number":["BB\/Y008332\/1"]}]},{"name":"Royal Academy of Engineering Research Fellowship","award":["RF\/202021\/20\/352"],"award-info":[{"award-number":["RF\/202021\/20\/352"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,1,14]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>We propose a suite of simple equations to estimate the probability and duration of two important processes in microbial ecology: immigration and extinction. Our work is based on the gambler\u2019s ruin equation, which determines the probability that a number of immigrants (i) can attain an abundance N given the ratio of the probabilities of death q and division (or birth) p. We estimate the probability of an organism attaining a value of N in the context of bioaugmentation, transplantation, infection, mutation, and extinction. For example, an inoculum of 108 bacteria with a q\/p of 1.00000001 has a 10\u221243 chance of attaining an abundance of 1010. The ratio of deaths to births controls the immigration parameter used in neutral models (m), and infectious dose in pathogens. We use Vibrio cholerae infections to demonstrate that the gambler\u2019s ruin equation can be used to estimate the infectious dose in naturally occurring infections. We calculated the long-term average value of m and q\/p in a wastewater treatment plant. All values of q\/p were \u22651. We expect the long-term average value of q\/p to be ~1 in all stable microbial communities. In the absence of migration, bacterial populations with q\/p\u00a0\u2265\u20091 will go extinct with probability 1. We use the ratio q\/p and simple recurrence relationships to estimate the time for a given change in abundance to occur. When q\/p\u00a0=\u20091, extinction in even a small microbial population will take thousands of years. Our simple mechanistic models could play a powerful role in theory and practice.<\/jats:p>","DOI":"10.1093\/ismejo\/wrag195","type":"journal-article","created":{"date-parts":[[2026,7,22]],"date-time":"2026-07-22T12:07:55Z","timestamp":1784722075000},"source":"Crossref","is-referenced-by-count":0,"title":["Probability, parameters, and duration of immigration and extinction in microbial communities"],"prefix":"10.1093","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9009-1748","authenticated-orcid":false,"given":"Thomas P","family":"Curtis","sequence":"first","affiliation":[{"name":"School of Engineering, Newcastle University , Newcastle NE1 7RU ,","place":["United Kingdom"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5978-0812","authenticated-orcid":false,"given":"Ben","family":"Allen","sequence":"additional","affiliation":[{"name":"School of Engineering, Newcastle University , Newcastle NE1 7RU ,","place":["United Kingdom"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0627-3937","authenticated-orcid":false,"given":"Mathew","family":"Brown","sequence":"additional","affiliation":[{"name":"School of Engineering, Newcastle University , Newcastle NE1 7RU ,","place":["United Kingdom"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Amy","family":"Bell","sequence":"additional","affiliation":[{"name":"School of Engineering, Newcastle University , Newcastle NE1 7RU ,","place":["United Kingdom"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Donna","family":"Swan","sequence":"additional","affiliation":[{"name":"School of Engineering, Newcastle University , Newcastle NE1 7RU ,","place":["United Kingdom"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3272-4778","authenticated-orcid":false,"given":"Russell","family":"Davenport","sequence":"additional","affiliation":[{"name":"School of Engineering, Newcastle University , Newcastle NE1 7RU ,","place":["United Kingdom"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9450-7384","authenticated-orcid":false,"given":"William","family":"Sloan","sequence":"additional","affiliation":[{"name":"Department of Civil Engineering, Glasgow University , Glasgow G12 8QQ ,","place":["United Kingdom"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2026,7,22]]},"reference":[{"key":"2026091502020297400_ref1","doi-asserted-by":"publisher","first-page":"1942","DOI":"10.1111\/mec.13075","article-title":"The influence of numbers on invasion success","volume":"24","author":"Blackburn","year":"2015","journal-title":"Mol Ecol"},{"key":"2026091502020297400_ref2","doi-asserted-by":"publisher","first-page":"1651","DOI":"10.1890\/02-0571","article-title":"Predicting invasions: propagule pressure and the gravity of allee effects","volume":"85","author":"Leung","year":"2004","journal-title":"Ecology."},{"key":"2026091502020297400_ref3","volume-title":"The Unified Neutral Theory of Biodiversity and Biogeography","author":"Hubbell","year":"2001"},{"key":"2026091502020297400_ref4","doi-asserted-by":"publisher","first-page":"732","DOI":"10.1111\/j.1462-2920.2005.00956.x","article-title":"Quantifying the roles of immigration and chance in shaping prokaryote community structure","volume":"8","author":"Sloan","year":"2006","journal-title":"Environ Microbiol"},{"key":"2026091502020297400_ref5","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1016\/j.copbio.2019.02.021","article-title":"Microbial immigration in wastewater treatment systems: analytical considerations and process implications","volume":"57","author":"Frigon","year":"2019","journal-title":"Curr Opin Biotechnol"},{"key":"2026091502020297400_ref6","doi-asserted-by":"publisher","first-page":"e2021589118","DOI":"10.1073\/pnas.2021589118","article-title":"Mass-immigration determines the assembly of activated sludge microbial communities","volume":"118","author":"Dottorini","year":"2021","journal-title":"Proc Natl Acad Sci U S A"},{"key":"2026091502020297400_ref7","doi-asserted-by":"publisher","first-page":"144","DOI":"10.1186\/s40168-019-0760-0","article-title":"Quantifying the contribution of microbial immigration in engineered water systems","volume":"7","author":"Mei","year":"2019","journal-title":"Microbiome."},{"key":"2026091502020297400_ref8","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1038\/ismej.2015.117","article-title":"The activated sludge ecosystem contains a core community of abundant organisms","volume":"10","author":"Saunders","year":"2016","journal-title":"ISME J"},{"key":"2026091502020297400_ref9","doi-asserted-by":"publisher","first-page":"e2038856","DOI":"10.1080\/19490976.2022.2038856","article-title":"The potential utility of fecal (or intestinal) microbiota transplantation in controlling infectious diseases","volume":"14","author":"Ghani","year":"2022","journal-title":"Gut Microbes"},{"key":"2026091502020297400_ref10","volume-title":"Sanitation and Disease: Health Aspects of Excreta and Wastewater Management","author":"Feachem","year":"1983"},{"key":"2026091502020297400_ref11","doi-asserted-by":"publisher","first-page":"288","DOI":"10.1038\/170288a0","article-title":"Bacterial invasion","volume":"170","author":"Druet","year":"1952","journal-title":"Nature."},{"key":"2026091502020297400_ref12","doi-asserted-by":"publisher","first-page":"38","DOI":"10.1099\/00221287-16-1-38","article-title":"Some hypotheses on the aetiology of fatal infections in partially resistant hosts and their application to mice challenged with Salmonella paratyphi-b or Salmonella typhimurium by intraperitoneal injection","volume":"16","author":"Meynell","year":"1957","journal-title":"Microbiology."},{"key":"2026091502020297400_ref13","volume-title":"Theory and Practice in Experimental Bacteriology","author":"Meynell","year":"1970"},{"key":"2026091502020297400_ref14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1214\/aoms\/1177730285","article-title":"On the generalized birth-and-death process","volume":"19","author":"Kendall","year":"1948","journal-title":"Ann Math Stat"},{"key":"2026091502020297400_ref15","doi-asserted-by":"publisher","first-page":"338","DOI":"10.1111\/j.2517-6161.1965.tb01501.x","article-title":"The basic birth-death model for microbial infections","volume":"27","author":"Williams","year":"1965","journal-title":"J R Stat Soc Ser B Methodol"},{"key":"2026091502020297400_ref16","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1111\/j.2517-6161.1961.tb00414.x","article-title":"Some stochastic processes with absorbing barriers","volume":"23","author":"Saaty","year":"1961","journal-title":"J R Stat Soc Ser B Methodol"},{"key":"2026091502020297400_ref17","doi-asserted-by":"publisher","first-page":"562","DOI":"10.2307\/2528542","article-title":"A stochastic model for distributions of biological response times","volume":"21","author":"Shortley","year":"1965","journal-title":"Biometrics."},{"key":"2026091502020297400_ref18","doi-asserted-by":"publisher","first-page":"102","DOI":"10.1128\/br.29.1.102-141.1965","article-title":"Independent-action and birth-death models in experimental microbiology","volume":"29","author":"Shortley","year":"1965","journal-title":"Bacteriol Rev"},{"key":"2026091502020297400_ref19","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1038\/214473a0","article-title":"Time-dependence and count-dependence in microbial infection","volume":"214","author":"Williams","year":"1967","journal-title":"Nature."},{"key":"2026091502020297400_ref20","doi-asserted-by":"publisher","first-page":"76","DOI":"10.1002\/j.1551-8833.1991.tb07252.x","article-title":"Modeling the risk from giardia and viruses in drinking water","volume":"83","author":"Regli","year":"1991","journal-title":"J AWWA"},{"key":"2026091502020297400_ref21","volume-title":"Introduction to Probability Models","author":"Ross","year":"2003"},{"key":"2026091502020297400_ref22","doi-asserted-by":"publisher","first-page":"264","DOI":"10.1016\/j.watres.2019.01.003","article-title":"Coupled virus - bacteria interactions and ecosystem function in an engineered microbial system","volume":"152","author":"Brown","year":"2019","journal-title":"Water Res"},{"key":"2026091502020297400_ref23","doi-asserted-by":"publisher","first-page":"695","DOI":"10.1128\/aem.59.3.695-700.1993","article-title":"Profiling of complex microbial-populations by denaturing gradient gel-electrophoresis analysis of polymerase chain reaction-amplified genes-coding for 16S ribosomal-RNA","volume":"59","author":"Muyzer","year":"1993","journal-title":"Appl Environ Microbiol"},{"key":"2026091502020297400_ref24","doi-asserted-by":"publisher","first-page":"1621","DOI":"10.1038\/ismej.2012.8","article-title":"Ultra-high-throughput microbial community analysis on the Illumina Hiseq and Miseq platforms","volume":"6","author":"Caporaso","year":"2012","journal-title":"ISME J."},{"key":"2026091502020297400_ref25","volume-title":"Models in Ecology","author":"Maynard-Smith","year":"1974"},{"key":"2026091502020297400_ref26","doi-asserted-by":"publisher","first-page":"7160","DOI":"10.1073\/pnas.88.16.7160","article-title":"A constant rate of spontaneous mutation in DNA-based microbes","volume":"88","author":"Drake","year":"1991","journal-title":"Proc Natl Acad Sci U S A"},{"key":"2026091502020297400_ref27","doi-asserted-by":"publisher","first-page":"516","DOI":"10.1109\/jproc.2015.2428213","article-title":"Linking statistical and ecological theory: Hubbell's unified neutral theory of biodiversity as a hierarchical dirichlet process","volume":"105","author":"Harris","year":"2017","journal-title":"Proc IEEE"},{"key":"2026091502020297400_ref28","doi-asserted-by":"publisher","first-page":"2473","DOI":"10.1111\/1462-2920.15453","article-title":"Drift dynamics in microbial communities and the effective community size","volume":"23","author":"Sloan","year":"2021","journal-title":"Environ Microbiol"},{"key":"2026091502020297400_ref29","doi-asserted-by":"publisher","first-page":"908","DOI":"10.1128\/iai.39.2.908-916.1983","article-title":"Experimental Campylobacter jejuni infection of adult mice","volume":"39","author":"Blaser","year":"1983","journal-title":"Infect Immun"},{"key":"2026091502020297400_ref30","volume-title":"Quantitative Microbial Risk Assessment"},{"key":"2026091502020297400_ref31","first-page":"361","article-title":"Study of induced typhoid fever in man. I. Evaluation of vaccine effectiveness","volume":"79","author":"Hornick","year":"1966","journal-title":"Trans Assoc Am Phys"},{"key":"2026091502020297400_ref32","first-page":"267","volume-title":"Quantitative Microbial Risk Assessment","author":"Haas"},{"key":"2026091502020297400_ref33","first-page":"737","article-title":"Studies on cholera carriers","volume":"37","author":"Dizon","year":"1967","journal-title":"Bull World Health Organ"},{"key":"2026091502020297400_ref34","doi-asserted-by":"publisher","first-page":"e1002533","DOI":"10.1371\/journal.pbio.1002533","article-title":"Revised estimates for the number of human and bacteria cells in the body","volume":"14","author":"Sender","year":"2016","journal-title":"PLoS Biol"},{"key":"2026091502020297400_ref35","doi-asserted-by":"publisher","first-page":"877","DOI":"10.1038\/nature07084","article-title":"Inapparent infections and cholera dynamics","volume":"454","author":"King","year":"2008","journal-title":"Nature."},{"key":"2026091502020297400_ref36","volume-title":"Modelling Fluctuating Populations","author":"Nisbet","year":"1982"},{"key":"2026091502020297400_ref37","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511624094","volume-title":"Modelling Biological Populations in Space and Time","author":"Renshaw","year":"1991"},{"key":"2026091502020297400_ref38","doi-asserted-by":"crossref","DOI":"10.2307\/j.ctvjghw98","volume-title":"Evolutionary Dynamics: Exploring the Equations of Life","author":"Nowak","year":"2006"},{"key":"2026091502020297400_ref39","doi-asserted-by":"publisher","first-page":"230","DOI":"10.1128\/aem.01527-09","article-title":"Probabilistic model of microbial cell growth, division, and mortality","volume":"76","author":"Horowitz","year":"2010","journal-title":"Appl Environ Microbiol"},{"key":"2026091502020297400_ref40","doi-asserted-by":"publisher","first-page":"e32976","DOI":"10.7554\/eLife.32976","article-title":"Antibiotic-induced population fluctuations and stochastic clearance of bacteria","volume":"7","author":"Coates","year":"2018","journal-title":"Elife."},{"key":"2026091502020297400_ref41","doi-asserted-by":"publisher","first-page":"20212486","DOI":"10.1098\/rspb.2021.2486","article-title":"Stochastic establishment of \u03b2-lactam-resistant Escherichia coli mutants reveals conditions for collective resistance","volume":"289","author":"Saebelfeld","year":"2022","journal-title":"Proc Biol Sci"},{"key":"2026091502020297400_ref42","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1099\/mic.0.001365","article-title":"Quantifying stochastic establishment of mutants in microbial adaptation","volume":"169","author":"Alexander","year":"2023","journal-title":"Microbiology."},{"key":"2026091502020297400_ref43","doi-asserted-by":"publisher","first-page":"2748","DOI":"10.1038\/s41396-018-0202-1","article-title":"Stochastic processes govern invasion success in microbial communities when the invader is phylogenetically close to resident bacteria","volume":"12","author":"Kinnunen","year":"2018","journal-title":"ISME J"},{"key":"2026091502020297400_ref44","doi-asserted-by":"publisher","first-page":"11594","DOI":"10.1073\/pnas.1505204112","article-title":"Dynamics of an experimental microbial invasion","volume":"112","author":"Acosta","year":"2015","journal-title":"Proc Natl Acad Sci U S A"},{"key":"2026091502020297400_ref45","doi-asserted-by":"publisher","first-page":"1743","DOI":"10.1002\/ecy.1852","article-title":"Biotic resistance shapes the influence of propagule pressure on invasion success in bacterial communities","volume":"98","author":"Jones","year":"2017","journal-title":"Ecology."},{"key":"2026091502020297400_ref46","first-page":"18","article-title":"Stability in model ecosystems","volume":"6","author":"May","year":"1971","journal-title":"Proc Ecol Soc Aust"},{"key":"2026091502020297400_ref47","doi-asserted-by":"publisher","first-page":"1056","DOI":"10.1073\/pnas.62.4.1056","article-title":"On population growth in a randomly varying environment","volume":"62","author":"Lewontin","year":"1969","journal-title":"Proc Natl Acad Sci U S A"},{"key":"2026091502020297400_ref48","volume-title":"Natural Inheritance","author":"Galton","year":"1889"},{"key":"2026091502020297400_ref49","doi-asserted-by":"publisher","first-page":"561","DOI":"10.1038\/s44319-025-00681-5","article-title":"From domination to partnership","volume":"27","author":"de Lorenzo","year":"2026","journal-title":"EMBO Rep"},{"key":"2026091502020297400_ref50","doi-asserted-by":"publisher","first-page":"5901","DOI":"10.1128\/aem.01383-14","article-title":"Agreement between amoA gene-specific quantitative PCR and fluorescence in situ hybridization in the measurement of ammonia-oxidizing bacteria in activated sludge","volume":"80","author":"Baptista","year":"2014","journal-title":"Appl Environ Microbiol"},{"key":"2026091502020297400_ref51","doi-asserted-by":"publisher","first-page":"537","DOI":"10.1038\/s41586-023-06161-x","article-title":"Unveiling the transition from niche to dispersal assembly in ecology","volume":"618","author":"Loke","year":"2023","journal-title":"Nature."},{"key":"2026091502020297400_ref52","doi-asserted-by":"publisher","first-page":"488","DOI":"10.1038\/nrmicro1455","article-title":"Microbial ecologists: It's time to 'go large'","volume":"4","author":"Curtis","year":"2006","journal-title":"Nat Rev Microbiol"},{"key":"2026091502020297400_ref53","doi-asserted-by":"publisher","first-page":"1024","DOI":"10.1038\/s41586-025-08947-7","article-title":"Bioremediation of complex organic pollutants by engineered Vibrio natriegens","volume":"642","author":"Su","year":"2025","journal-title":"Nature."},{"key":"2026091502020297400_ref54","doi-asserted-by":"publisher","first-page":"e3001116","DOI":"10.1371\/journal.pbio.3001116","article-title":"Ecological rules for the assembly of microbiome communities","volume":"19","author":"Coyte","year":"2021","journal-title":"PLoS Biol"},{"key":"2026091502020297400_ref55","doi-asserted-by":"publisher","first-page":"3598","DOI":"10.1021\/es980200z","article-title":"Pilot-scale evaluation of bioaugmentation for in-situ remediation of a carbon tetrachloride contaminated aquifer","volume":"32","author":"Dybas","year":"1998","journal-title":"Environ Sci Technol"}],"container-title":["The ISME Journal"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/ismej\/advance-article-pdf\/doi\/10.1093\/ismejo\/wrag195\/69802331\/wrag195.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/ismej\/article-pdf\/20\/1\/wrag195\/69802331\/wrag195.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/ismej\/article-pdf\/20\/1\/wrag195\/69802331\/wrag195.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,15]],"date-time":"2026-09-15T06:02:17Z","timestamp":1789452137000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/ismej\/article\/doi\/10.1093\/ismejo\/wrag195\/8740005"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1]]},"references-count":55,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,1,14]]}},"URL":"https:\/\/doi.org\/10.1093\/ismejo\/wrag195","relation":{},"ISSN":["1751-7362","1751-7370"],"issn-type":[{"value":"1751-7362","type":"print"},{"value":"1751-7370","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2026,1]]},"published":{"date-parts":[[2026,1]]},"article-number":"wrag195"}}