{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,30]],"date-time":"2026-01-30T04:32:29Z","timestamp":1769747549028,"version":"3.49.0"},"reference-count":49,"publisher":"EDP Sciences","license":[{"start":{"date-parts":[[2022,8,1]],"date-time":"2022-08-01T00:00:00Z","timestamp":1659312000000},"content-version":"vor","delay-in-days":212,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/04106\/2020"],"award-info":[{"award-number":["UIDB\/04106\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/04106\/2020"],"award-info":[{"award-number":["UIDB\/04106\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/04106\/2020"],"award-info":[{"award-number":["UIDB\/04106\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["PD\/BD\/114184\/2016"],"award-info":[{"award-number":["PD\/BD\/114184\/2016"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["CEECIND\/00564\/2018"],"award-info":[{"award-number":["CEECIND\/00564\/2018"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Model. Nat. Phenom."],"accepted":{"date-parts":[[2022,6,25]]},"published-print":{"date-parts":[[2022]]},"abstract":"<jats:p>We improve a recent mathematical model for cholera by adding a time delay that represents the time between the instant at which an individual becomes infected and the instant at which he begins to have symptoms of cholera disease. We prove that the delayed cholera model is biologically meaningful and analyze the local asymptotic stability of the equilibrium points for positive time delays. An optimal control problem is proposed and analyzed, where the goal is to obtain optimal treatment strategies, through quarantine, that minimize the number of infective individuals and the bacterial concentration, as well as treatment costs. Necessary optimality conditions are applied to the delayed optimal control problem, with a <jats:italic>L<\/jats:italic><jats:sup>1<\/jats:sup> type cost functional. We show that the delayed cholera model fits better the cholera outbreak that occurred in the Department of Artibonite - Haiti, from 1 November 2010 to 1 May 2011, than the non-delayed model. Considering the data of the cholera outbreak in Haiti, we solve numerically the delayed optimal control problem and propose solutions for the outbreak control and eradication.<\/jats:p>","DOI":"10.1051\/mmnp\/2022027","type":"journal-article","created":{"date-parts":[[2022,6,28]],"date-time":"2022-06-28T19:01:24Z","timestamp":1656442884000},"page":"25","source":"Crossref","is-referenced-by-count":12,"title":["A SIQRB delayed model for cholera and optimal control treatment"],"prefix":"10.1051","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4592-3426","authenticated-orcid":false,"given":"Ana P.","family":"Lemos-Pai\u00e3o","sequence":"first","affiliation":[]},{"given":"Helmut","family":"Maurer","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7238-546X","authenticated-orcid":false,"given":"Cristiana J.","family":"Silva","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8641-2505","authenticated-orcid":false,"given":"Delfim F. M.","family":"Torres","sequence":"additional","affiliation":[]}],"member":"250","published-online":{"date-parts":[[2022,8,1]]},"reference":[{"key":"R1","unstructured":"Asian Scientist, Math in a time of cholera (2017). https:\/\/www.asianscientist.com\/2017\/08\/health\/mathematical-model-yemen-cholera-outbreak"},{"key":"R2","doi-asserted-by":"crossref","first-page":"219","DOI":"10.12732\/ijpam.v111i2.8","volume":"111","author":"Beryl","year":"2016","journal-title":"Int. J. Pure Appl. Math"},{"key":"R3","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1006\/jmaa.1998.6109","volume":"228","author":"Boese","year":"1998","journal-title":"J. Math. Anal. Appl"},{"key":"R4","doi-asserted-by":"crossref","first-page":"109639","DOI":"10.1016\/j.chaos.2020.109639","volume":"133","author":"Calatayud","year":"2020","journal-title":"Chaos Solit. Fractals"},{"key":"R5","first-page":"121","volume":"27","author":"Capasso","year":"1979","journal-title":"Rev. Epidemiol. Sant\u00e9 Publique"},{"key":"R6","doi-asserted-by":"crossref","first-page":"1107","DOI":"10.1007\/s00285-014-0849-9","volume":"71","author":"Capone","year":"2015","journal-title":"J. Math. Biol"},{"key":"R7","unstructured":"Centers for Disease Control and Prevention, Cholera \u2014 vibrio cholerae infection (2018). Available from https:\/\/www.cdc.gov\/cholera\/general\/index.html"},{"key":"R8","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1186\/1471-2334-1-1","volume":"1","author":"Codeco","year":"2001","journal-title":"BMC Infect. Dis"},{"key":"R9","first-page":"77","volume":"29","author":"Cooke","year":"1986","journal-title":"Funkcial. Ekvac"},{"key":"R10","first-page":"16","volume":"2014","author":"Cui","year":"2014","journal-title":"J. Appl. Math"},{"key":"R11","doi-asserted-by":"crossref","first-page":"53","DOI":"10.11648\/j.acm.20150402.14","volume":"4","author":"Edward","year":"2015","journal-title":"Appl. Comput. Math"},{"key":"R12","unstructured":"Fourer R., Gay D.M. and Kernighan B.W., AMPL: A Modeling Language for Mathematical Programming. Scientific Press series, Thomson, Brooks, Cole (2003)."},{"key":"R13","doi-asserted-by":"crossref","unstructured":"Gay D.M., The AMPL modeling language: an aid to formulating and solving optimization problems, in: Numerical analysis and optimization, Vol. 134 of Springer Proc. Math. Stat., Springer, Cham (2015), pp. 95\u2013116.","DOI":"10.1007\/978-3-319-17689-5_5"},{"key":"R14","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1002\/oca.843","volume":"30","author":"Gollmann","year":"2009","journal-title":"Opt. Control Appl. Methods"},{"key":"R15","doi-asserted-by":"crossref","first-page":"413","DOI":"10.3934\/jimo.2014.10.413","volume":"10","author":"G\u00f6llmann","year":"2014","journal-title":"J. Ind. Manag. Optim"},{"key":"R16","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1007\/BF00933818","volume":"18","author":"Guinn","year":"1976","journal-title":"J. Optim. Theory Appl"},{"key":"R17","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1371\/journal.pmed.0030063","volume":"3","author":"Hartley","year":"2006","journal-title":"PLOS Med"},{"key":"R18","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.jtbi.2018.04.041","volume":"451","author":"He","year":"2018","journal-title":"J. Theoret. Biol"},{"key":"R19","doi-asserted-by":"crossref","first-page":"1583","DOI":"10.1016\/j.mcm.2010.11.060","volume":"53","author":"Hove-Musekwa","year":"2011","journal-title":"Math. Comput. Model"},{"key":"R20","unstructured":"Index Mundi, Demographics: Birth rate Haiti (2015). https:\/\/www.indexmundi.com\/g\/g.aspx?c=ha&v=25"},{"key":"R21","unstructured":"Index Mundi, Demographics: Death rate Haiti (2015). https:\/\/www.indexmundi.com\/g\/g.aspx?c=ha&v=26"},{"key":"R22","doi-asserted-by":"crossref","first-page":"845","DOI":"10.1007\/s11538-008-9384-4","volume":"71","author":"Joh","year":"2009","journal-title":"Bull. Math. Biol"},{"key":"R23","doi-asserted-by":"crossref","first-page":"2004","DOI":"10.1128\/AEM.01739-07","volume":"74","author":"Kirschner","year":"2008","journal-title":"Appl. Environ. Microbiol"},{"key":"R24","unstructured":"Kuang Y., Delay differential equations with applications in population dynamics. Vol. 191 of Mathematics in Science and Engineering. Academic Press, Inc., Boston, MA (1993)."},{"key":"R25","doi-asserted-by":"crossref","first-page":"95","DOI":"10.3934\/mbe.2004.1.95","volume":"1","author":"Ledzewicz","year":"2004","journal-title":"Math. Biosci. Eng"},{"key":"R26","unstructured":"Ledzewicz U. and Sch\u00f6attler H., On optimal singular controls for a general SIR-model with vaccination and treatment. Discrete Contin. Dyn. Syst. (2011) 981\u2013990."},{"key":"R27","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1016\/j.cam.2016.11.002","volume":"318","author":"Lemos-Paiao","year":"2017","journal-title":"J. Comput. Appl. Math"},{"key":"R28","doi-asserted-by":"crossref","first-page":"448","DOI":"10.3934\/Math.2018.4.448","volume":"3","author":"Lemos-Paiao","year":"2018","journal-title":"AIMS Math"},{"key":"R29","doi-asserted-by":"crossref","first-page":"7486","DOI":"10.1016\/j.jfranklin.2018.11.056","volume":"356","author":"Liu","year":"2019","journal-title":"J. Franklin Inst"},{"key":"R30","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1007\/BF02192163","volume":"86","author":"Maurer","year":"1995","journal-title":"J. Optim. Theory Appl"},{"key":"R31","doi-asserted-by":"crossref","first-page":"2004","DOI":"10.1007\/s11538-010-9521-8","volume":"72","author":"Miller Neilan","year":"2010","journal-title":"Bull. Math. Biol"},{"key":"R32","unstructured":"Mukandavire Z., Mutasa F.K., Hove-Musekwa S.D., Dube S. and Tchuenche J.M., Mathematical analysis of a cholera model with carriers and assessing the effects of treatment. Nova Science Publishers, Inc. (2008), pp. 1\u201337."},{"key":"R33","doi-asserted-by":"crossref","first-page":"8767","DOI":"10.1073\/pnas.1019712108","volume":"108","author":"Mukandavire","year":"2011","journal-title":"Proc. Natl. Acad. Sci"},{"key":"R34","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1016\/j.biosystems.2011.04.001","volume":"105","author":"Mwasa","year":"2011","journal-title":"Biosystems"},{"key":"R35","doi-asserted-by":"crossref","first-page":"399","DOI":"10.2217\/fmb-2017-0244","volume":"13","author":"Nishiura","year":"2018","journal-title":"Future Microbiol"},{"key":"R36","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1186\/s12976-017-0061-x","volume":"14","author":"Nishiura","year":"2017","journal-title":"Theor. Biol. Med. Model"},{"key":"R37","doi-asserted-by":"crossref","first-page":"131","DOI":"10.3354\/cr00730","volume":"36","author":"Pascual","year":"2008","journal-title":"Clim. Res"},{"key":"R38","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1111\/j.1574-6976.2002.tb00605.x","volume":"26","author":"Reidl","year":"2002","journal-title":"FEMS Microbiol. Rev"},{"key":"R39","doi-asserted-by":"crossref","first-page":"2916","DOI":"10.1007\/s11538-011-9652-6","volume":"73","author":"Sanches","year":"2011","journal-title":"Bull. Math. Biol"},{"key":"R40","first-page":"2657","volume":"19","author":"Schattler","year":"2014","journal-title":"Discrete Contin. Dyn. Syst. Ser. B"},{"key":"R41","doi-asserted-by":"crossref","first-page":"2423","DOI":"10.1007\/s11538-012-9759-4","volume":"74","author":"Shuai","year":"2012","journal-title":"Bull. Math. Biol"},{"key":"R42","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/j.cnsns.2016.10.007","volume":"45","author":"Sun","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul"},{"key":"R43","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1007\/s10107-004-0559-y","volume":"106","author":"Wachter","year":"2006","journal-title":"Math. Program"},{"key":"R44","first-page":"255","volume":"19","author":"Wang","year":"2011","journal-title":"Can. Appl. Math. Q"},{"key":"R45","doi-asserted-by":"crossref","first-page":"568","DOI":"10.1080\/17513758.2012.658089","volume":"6","author":"Wang","year":"2012","journal-title":"J. Biol. Dyn"},{"key":"R46","unstructured":"World Health Organization, Yemen: Weekly epidemiological bulletin W26 2018 (2018). Available at http:\/\/www.emro.who.int\/images\/stories\/yemen\/week_26.pdf?ua=1"},{"key":"R47","unstructured":"World Health Organization, Global Task Force on Cholera Control, Cholera country profile: Haiti (2011). http::\/\/www.who.int\/cholera\/countries\/HaitiCountryProfileMay2011.pdf"},{"key":"R48","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/0898-1221(96)00129-0","volume":"32","author":"Yang","year":"1996","journal-title":"Comput. Math. Appl"},{"key":"R49","doi-asserted-by":"crossref","first-page":"931","DOI":"10.1007\/s11071-020-06151-y","volume":"105","author":"Zhou","year":"2021","journal-title":"Nonlinear Dyn"}],"container-title":["Mathematical Modelling of Natural Phenomena"],"original-title":[],"link":[{"URL":"https:\/\/www.mmnp-journal.org\/10.1051\/mmnp\/2022027\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,1]],"date-time":"2022-08-01T07:56:15Z","timestamp":1659340575000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mmnp-journal.org\/10.1051\/mmnp\/2022027"}},"subtitle":[],"editor":[{"given":"Vitaly","family":"Volpert","sequence":"first","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2022]]},"references-count":49,"alternative-id":["mmnp210214"],"URL":"https:\/\/doi.org\/10.1051\/mmnp\/2022027","relation":{},"ISSN":["0973-5348","1760-6101"],"issn-type":[{"value":"0973-5348","type":"print"},{"value":"1760-6101","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022]]}}}