{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,11,13]],"date-time":"2026-11-13T04:05:47Z","timestamp":1794542747329,"version":"4.9.1"},"reference-count":53,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2019,6,11]],"date-time":"2019-06-11T00:00:00Z","timestamp":1560211200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,6,11]],"date-time":"2019-06-11T00:00:00Z","timestamp":1560211200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Math. Biol."],"published-print":{"date-parts":[[2019,8,15]]},"DOI":"10.1007\/s00285-019-01390-z","type":"journal-article","created":{"date-parts":[[2019,6,11]],"date-time":"2019-06-11T07:03:09Z","timestamp":1560236589000},"page":"1077-1104","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":61,"title":["Optimal control of vaccination in a vector-borne reaction\u2013diffusion model applied to Zika virus"],"prefix":"10.1007","volume":"79","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2735-3044","authenticated-orcid":false,"given":"Tiago Yuzo","family":"Miyaoka","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Suzanne","family":"Lenhart","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jo\u00e3o F. C. A.","family":"Meyer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2019,6,11]]},"reference":[{"key":"1390_CR1","first-page":"1","volume":"76","author":"H Abboubakar","year":"2017","unstructured":"Abboubakar H, Kamgang JC, Nkamba LN, Tieudjo D (2017) Bifurcation thresholds and optimal control in transmission dynamics of arboviral diseases. J Math Biol 76:1\u201349","journal-title":"J Math Biol"},{"key":"1390_CR2","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1016\/j.ecocom.2016.12.007","volume":"29","author":"F Agusto","year":"2017","unstructured":"Agusto F, Bewick S, Fagan W (2017) Mathematical model for Zika virus dynamics with sexual transmission route. Ecol Complex 29:61\u201381","journal-title":"Ecol Complex"},{"key":"1390_CR3","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pmed.1002157","author":"CL Althaus","year":"2016","unstructured":"Althaus CL, Low N (2016) How relevant is sexual transmission of Zika virus? PLoS Med. https:\/\/doi.org\/10.1371\/journal.pmed.1002157","journal-title":"PLoS Med"},{"key":"1390_CR4","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1016\/j.matcom.2015.06.014","volume":"133","author":"R Anguelov","year":"2017","unstructured":"Anguelov R, Dufourd C, Dumont Y (2017) Simulations and parameter estimation of a trap-insect model using a finite element approach. Math Comput Simul 133:47\u201375","journal-title":"Math Comput Simul"},{"issue":"2","key":"1390_CR5","doi-asserted-by":"publisher","first-page":"219","DOI":"10.3934\/mbe.2008.5.219","volume":"5","author":"E Asano","year":"2008","unstructured":"Asano E, Gross LJ, Lenhart S, Real LA (2008) Optimal control of vaccine distribution in a rabies metapopulation model. Math Biosci Eng 5(2):219\u2013238","journal-title":"Math Biosci Eng"},{"issue":"13","key":"1390_CR6","doi-asserted-by":"publisher","first-page":"20,751","DOI":"10.2807\/1560-7917.ES2014.19.13.20751","volume":"19","author":"M Besnard","year":"2014","unstructured":"Besnard M, Lastere S, Teissier A, Cao-Lormeau V, Musso D (2014) Evidence of perinatal transmission of Zika virus, French Polynesia, December 2013 and February 2014. Eurosurveillance 19(13):20,751","journal-title":"Eurosurveillance"},{"key":"1390_CR7","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718577","volume-title":"Practical methods for optimal control and estimation using nonlinear programming","author":"JT Betts","year":"2010","unstructured":"Betts JT (2010) Practical methods for optimal control and estimation using nonlinear programming, vol 19. SIAM, Philadelphia"},{"key":"1390_CR8","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pone.0185540","author":"E Bonyah","year":"2017","unstructured":"Bonyah E, Khan MA, Okosun K, Islam S (2017) A theoretical model for Zika virus transmission. PLoS ONE. https:\/\/doi.org\/10.1371\/journal.pone.0185540","journal-title":"PLoS ONE"},{"issue":"7075","key":"1390_CR9","doi-asserted-by":"publisher","first-page":"462","DOI":"10.1038\/nature04292","volume":"439","author":"D Brockmann","year":"2006","unstructured":"Brockmann D, Hufnagel L, Geisel T (2006) The scaling laws of human travel. Nature 439(7075):462","journal-title":"Nature"},{"issue":"1","key":"1390_CR10","doi-asserted-by":"publisher","first-page":"298","DOI":"10.1016\/j.jmaa.2016.06.037","volume":"444","author":"AL De Araujo","year":"2016","unstructured":"De Araujo AL, Boldrini JL, Calsavara BM (2016) An analysis of a mathematical model describing the geographic spread of dengue disease. J Math Anal Appl 444(1):298\u2013325","journal-title":"J Math Anal Appl"},{"issue":"3","key":"1390_CR11","first-page":"1049","volume":"22","author":"KR De Silva","year":"2017","unstructured":"De Silva KR, Phan TV, Lenhart S (2017) Advection control in parabolic PDE systems for competitive populations. Discrete Contin Dyn Syst Ser B 22(3):1049\u20131072","journal-title":"Discrete Contin Dyn Syst Ser B"},{"issue":"3","key":"1390_CR12","doi-asserted-by":"publisher","first-page":"503","DOI":"10.1137\/0716039","volume":"16","author":"J Douglas Jr","year":"1979","unstructured":"Douglas J Jr, Dupont T, Ewing RE (1979) Incomplete iteration for time-stepping a Galerkin method for a quasilinear parabolic problem. SIAM J Numer Anal 16(3):503\u2013522","journal-title":"SIAM J Numer Anal"},{"issue":"22","key":"1390_CR13","doi-asserted-by":"publisher","first-page":"2195","DOI":"10.1056\/NEJMc1604449","volume":"374","author":"E D\u2019Ortenzio","year":"2016","unstructured":"D\u2019Ortenzio E, Matheron S, de Lamballerie X, Hubert B, Piorkowski G, Maquart M, Descamps D, Damond F, Yazdanpanah Y, Leparc-Goffart I (2016) Evidence of sexual transmission of Zika virus. N Engl J Med 374(22):2195\u20132198","journal-title":"N Engl J Med"},{"key":"1390_CR14","volume-title":"Partial differential equations","author":"LC Evans","year":"1998","unstructured":"Evans LC (1998) Partial differential equations. American Mathematical Society, Providence"},{"issue":"2","key":"1390_CR15","first-page":"341","volume":"23","author":"R Fister","year":"1997","unstructured":"Fister R (1997) Optimal control of harvesting in a predator-prey parabolic system. Houst J Math 23(2):341\u2013355","journal-title":"Houst J Math"},{"issue":"1","key":"1390_CR16","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1186\/s12976-017-0051-z","volume":"14","author":"W Fitzgibbon","year":"2017","unstructured":"Fitzgibbon W, Morgan J, Webb G (2017) An outbreak vector-host epidemic model with spatial structure: the 2015\u20132016 zika outbreak in rio de janeiro. Theoretical Biology and Medical Modelling 14(1):7","journal-title":"Theoretical Biology and Medical Modelling"},{"key":"1390_CR17","doi-asserted-by":"publisher","DOI":"10.1038\/srep28070","author":"D Gao","year":"2016","unstructured":"Gao D, Lou Y, He D, Porco TC, Kuang Y, Chowell G, Ruan S (2016) Prevention and control of zika as a mosquito-borne and sexually transmitted disease: a mathematical modeling analysis. Sci Rep. https:\/\/doi.org\/10.1038\/srep28070","journal-title":"Sci Rep"},{"issue":"7196","key":"1390_CR18","doi-asserted-by":"publisher","first-page":"779","DOI":"10.1038\/nature06958","volume":"453","author":"MC Gonzalez","year":"2008","unstructured":"Gonzalez MC, Hidalgo CA, Barabasi AL (2008) Understanding individual human mobility patterns. Nature 453(7196):779","journal-title":"Nature"},{"issue":"3","key":"1390_CR19","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1007\/BF02251947","volume":"20","author":"W Hackbusch","year":"1978","unstructured":"Hackbusch W (1978) A numerical method for solving parabolic equations with opposite orientations. Computing 20(3):229\u2013240","journal-title":"Computing"},{"key":"1390_CR20","unstructured":"Health Office (2017) Rio Grande do Norte epidemiological bulletin, 2016. http:\/\/www.saude.rn.gov.br. Accessed 16 August 2017"},{"issue":"2","key":"1390_CR21","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1002\/gamm.201010010","volume":"33","author":"M Hinze","year":"2010","unstructured":"Hinze M, Schulz V (2010) Preface of the guest-editors. GAMM-Mitteilungen 33(2):131\u2013132","journal-title":"GAMM-Mitteilungen"},{"key":"1390_CR22","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-30785-5","volume-title":"Advances in mathematical modeling. Optimization and optimal control","author":"JB Hiriart-Urruty","year":"2016","unstructured":"Hiriart-Urruty JB, Korytowski A, Maurer H, Szymkat M (2016) Advances in mathematical modeling. Optimization and optimal control, vol 109. Springer, Berlin"},{"issue":"6","key":"1390_CR23","doi-asserted-by":"publisher","first-page":"1203","DOI":"10.1590\/S0102-311X2009000600003","volume":"25","author":"NA Hon\u00f3rio","year":"2009","unstructured":"Hon\u00f3rio NA, Castro MG, Barros FSMd, Magalh\u00e3es MdAFM, Sabroza PC (2009) The spatial distribution of Aedes aegypti and Aedes albopictus in a transition zone, Rio de Janeiro, Brazil. Cadernos de Sa\u00fade P\u00fablica 25(6):1203\u20131214","journal-title":"Cadernos de Sa\u00fade P\u00fablica"},{"key":"1390_CR24","volume-title":"The finite element method: linear static and dynamic finite element analysis","author":"TJ Hughes","year":"2012","unstructured":"Hughes TJ (2012) The finite element method: linear static and dynamic finite element analysis. Courier Corporation, North Chelmsford"},{"key":"1390_CR25","doi-asserted-by":"publisher","DOI":"10.1515\/9781400841035","volume-title":"Modeling infectious diseases in humans and animals","author":"MJ Keeling","year":"2008","unstructured":"Keeling MJ, Rohani P (2008) Modeling infectious diseases in humans and animals. Princeton University Press, Princeton"},{"key":"1390_CR26","doi-asserted-by":"crossref","unstructured":"Kon CML, Labadin J, Tiga J (2016) Generic reaction-diffusion model for transmission of mosquito-borne diseases: results of simulation with actual cases. In: ECMS, pp 93\u201399","DOI":"10.7148\/2016-0093"},{"key":"1390_CR27","doi-asserted-by":"crossref","DOI":"10.1201\/9781420011418","volume-title":"Optimal control applied to biological models","author":"S Lenhart","year":"2007","unstructured":"Lenhart S, Workman JT (2007) Optimal control applied to biological models. Crc Press, New York"},{"key":"1390_CR28","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898717839","volume-title":"Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems","author":"RJ LeVeque","year":"2007","unstructured":"LeVeque RJ (2007) Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems, vol 98. SIAM, Philadelphia"},{"key":"1390_CR29","volume-title":"Optimal control theory for infinite dimensional systems","author":"X Li","year":"2012","unstructured":"Li X, Yong J (2012) Optimal control theory for infinite dimensional systems. Springer, Berlin"},{"key":"1390_CR30","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-65024-6","volume-title":"Optimal control of systems governed by partial differential equations","author":"JL Lions","year":"1971","unstructured":"Lions JL (1971) Optimal control of systems governed by partial differential equations. Springer, Berlin"},{"issue":"4","key":"1390_CR31","doi-asserted-by":"publisher","first-page":"543","DOI":"10.1007\/s00285-010-0346-8","volume":"62","author":"Y Lou","year":"2011","unstructured":"Lou Y, Zhao XQ (2011) A reaction-diffusion malaria model with incubation period in the vector population. J Math Biol 62(4):543\u2013568","journal-title":"J Math Biol"},{"key":"1390_CR32","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pntd.0004530","author":"RW Malone","year":"2016","unstructured":"Malone RW, Homan J, Callahan MV, Glasspool-Malone J, Damodaran L, Schneider ADB, Zimler R, Talton J, Cobb RR, Ruzic I, Smith-Gagen J, Janies D, Wilson J (2016) Zika virus: medical countermeasure development challenges. PLoS Negl Trop Dis. https:\/\/doi.org\/10.1371\/journal.pntd.0004530","journal-title":"PLoS Negl Trop Dis"},{"issue":"1","key":"1390_CR33","doi-asserted-by":"publisher","first-page":"4","DOI":"10.1590\/0037-8682-0220-2015","volume":"49","author":"CB Marcondes","year":"2016","unstructured":"Marcondes CB, Ximenes MdFFd (2016) Zika virus in Brazil and the danger of infestation by Aedes (Stegomyia) mosquitoes. Revista da Sociedade Brasileira de Medicina Tropical 49(1):4\u201310","journal-title":"Revista da Sociedade Brasileira de Medicina Tropical"},{"issue":"1","key":"1390_CR34","doi-asserted-by":"publisher","first-page":"178","DOI":"10.1016\/j.jtbi.2008.04.011","volume":"254","author":"S Marino","year":"2008","unstructured":"Marino S, Hogue IB, Ray CJ, Kirschner DE (2008) A methodology for performing global uncertainty and sensitivity analysis in systems biology. J Theor Biol 254(1):178\u2013196","journal-title":"J Theor Biol"},{"issue":"2","key":"1390_CR35","first-page":"253","volume":"18","author":"TY Miyaoka","year":"2017","unstructured":"Miyaoka TY, Meyer JFCA, Souza JMR (2017) A general boundary condition with linear flux for advection\u2013diffusion models. Trends Appl Comput Math 18(2):253\u2013272","journal-title":"Trends Appl Comput Math"},{"key":"1390_CR36","doi-asserted-by":"publisher","first-page":"951","DOI":"10.1056\/NEJMoa1600651","volume":"374","author":"J Mlakar","year":"2016","unstructured":"Mlakar J, Korva M, Tul N, Popovi\u0107 M, Polj\u0161ak-Prijatelj M, Mraz J, Kolenc M, Resman Rus K, Vesnaver Vipotnik T, Fabjan Vodu\u0161ek V, Vizjak A, Jo\u017ee P, Petrovec M (2016) Av\u0161i\u010d \u017dupanc T (2016) Zika virus associated with microcephaly. New England Journal of Medicine 374:951\u2013958","journal-title":"New England Journal of Medicine"},{"issue":"10","key":"1390_CR37","doi-asserted-by":"publisher","first-page":"2244","DOI":"10.1016\/j.matcom.2011.05.006","volume":"81","author":"C Mocenni","year":"2011","unstructured":"Mocenni C, Madeo D, Sparacino E (2011) Linear least squares parameter estimation of nonlinear reaction diffusion equations. Math Comput Simul 81(10):2244\u20132257","journal-title":"Math Comput Simul"},{"issue":"11","key":"1390_CR38","doi-asserted-by":"publisher","first-page":"1101","DOI":"10.1056\/NEJMc1607262","volume":"375","author":"IJ Motta","year":"2016","unstructured":"Motta IJ, Spencer BR, Cordeiro da Silva SG, Arruda MB, Dobbin JA, Gonzaga YB, Arcuri IP, Tavares RC, Atta EH, Fernandes RF, Costa DA, Ribeiro LJ, Limonte F, Higa LM, Voloch CM, Brindeiro RM, Tanuri A, Ferreira OC (2016) Evidence for transmission of Zika virus by platelet transfusion. N Engl J Med 375(11):1101\u20131103","journal-title":"N Engl J Med"},{"key":"1390_CR39","doi-asserted-by":"publisher","DOI":"10.1007\/b98868","volume-title":"Mathematical biology. Vol. 2: spatial models and biomedical applications","author":"JD Murray","year":"2002","unstructured":"Murray JD (2002) Mathematical biology. Vol. 2: spatial models and biomedical applications. Springer, New York"},{"issue":"2","key":"1390_CR40","doi-asserted-by":"publisher","first-page":"603","DOI":"10.1016\/j.jmaa.2010.12.035","volume":"378","author":"RM Neilan","year":"2011","unstructured":"Neilan RM, Lenhart S (2011) Optimal vaccine distribution in a spatiotemporal epidemic model with an application to rabies and raccoons. J Math Anal Appl 378(2):603\u2013619","journal-title":"J Math Anal Appl"},{"issue":"4","key":"1390_CR41","first-page":"1013","volume":"37","author":"I Neitzel","year":"2008","unstructured":"Neitzel I, Tr\u00f6ltzsch F (2008) On convergence of regularization methods for nonlinear parabolic optimal control problems with control and state constraints. Control Cybernet 37(4):1013\u20131043","journal-title":"Control Cybernet"},{"key":"1390_CR42","unstructured":"NIAID (2017) National Institute of Allergy and Infectious Diseases. https:\/\/www.niaid.nih.gov\/news-events\/nih-begins-testing-investigational-zika-vaccine-humans. Accessed 26 Oct 2017"},{"key":"1390_CR43","doi-asserted-by":"publisher","DOI":"10.2807\/1560-7917.ES2014.19.9.20720","author":"E Oehler","year":"2014","unstructured":"Oehler E, Watrin L, Larre P, Leparc-Goffart I, Lastere S, Valour F, Baudouin L, Mallet H, Musso D, Ghawche F (2014) Zika virus infection complicated by Guillain-Barre syndrome\u2013case report, French Polynesia, December 2013. Eurosurveillance. https:\/\/doi.org\/10.2807\/1560-7917.ES2014.19.9.20720","journal-title":"Eurosurveillance"},{"issue":"5","key":"1390_CR44","doi-asserted-by":"publisher","first-page":"1297","DOI":"10.1007\/s11538-008-9300-y","volume":"70","author":"M Otero","year":"2008","unstructured":"Otero M, Schweigmann N, Solari HG (2008) A stochastic spatial dynamical model for Aedes aegypti. Bull Math Biol 70(5):1297","journal-title":"Bull Math Biol"},{"key":"1390_CR45","unstructured":"PAHO (2017) Pan American Health Organization. https:\/\/www.paho.org\/zika. Accessed 26 Oct 2017"},{"key":"1390_CR46","unstructured":"Powell MJ (2009) The BOBYQA algorithm for bound constrained optimization without derivatives. Cambridge NA Report NA2009\/06, University of Cambridge, Cambridge"},{"issue":"02n03","key":"1390_CR47","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1142\/S0218339016500108","volume":"24","author":"SM Raimundo","year":"2016","unstructured":"Raimundo SM, Yang HM, Massad E (2016) Modeling vaccine preventable vector-borne infections: yellow fever as a case study. J Biol Syst 24(02n03):193\u2013216","journal-title":"J Biol Syst"},{"key":"1390_CR48","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.mbs.2013.10.006","volume":"247","author":"HS Rodrigues","year":"2014","unstructured":"Rodrigues HS, Monteiro MTT, Torres DF (2014) Vaccination models and optimal control strategies to dengue. Math Biosci 247:1\u201312","journal-title":"Math Biosci"},{"issue":"2","key":"1390_CR49","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1007\/s10144-013-0415-0","volume":"56","author":"S Soubeyrand","year":"2014","unstructured":"Soubeyrand S, Roques L (2014) Parameter estimation for reaction\u2013diffusion models of biological invasions. Popul Ecol 56(2):427\u2013434","journal-title":"Popul Ecol"},{"issue":"3","key":"1390_CR50","doi-asserted-by":"publisher","first-page":"598","DOI":"10.1007\/s11538-018-0393-7","volume":"80","author":"W Valega-Mackenzie","year":"2018","unstructured":"Valega-Mackenzie W, R\u00edos-Soto KR (2018) Can vaccination save a Zika virus epidemic? Bull Math Biol 80(3):598\u2013625","journal-title":"Bull Math Biol"},{"key":"1390_CR51","doi-asserted-by":"publisher","DOI":"10.1038\/s41598-017-07264-y","author":"L Wang","year":"2017","unstructured":"Wang L, Zhao H, Oliva SM, Zhu H (2017) Modeling the transmission and control of Zika in brazil. Sci Rep. https:\/\/doi.org\/10.1038\/s41598-017-07264-y","journal-title":"Sci Rep"},{"key":"1390_CR52","unstructured":"WHO (2017) World Health Organization. https:\/\/www.who.int\/topics\/zika. Accessed 26 Oct 2017"},{"key":"1390_CR53","first-page":"1","volume":"76","author":"B Zheng","year":"2017","unstructured":"Zheng B, Tang M, Yu J, Qiu J (2017) Wolbachia spreading dynamics in mosquitoes with imperfect maternal transmission. J Math Biol 76:1\u201329","journal-title":"J Math Biol"}],"container-title":["Journal of Mathematical Biology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00285-019-01390-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00285-019-01390-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00285-019-01390-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,7,29]],"date-time":"2022-07-29T16:23:59Z","timestamp":1659111839000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00285-019-01390-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,11]]},"references-count":53,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2019,8,15]]}},"alternative-id":["1390"],"URL":"https:\/\/doi.org\/10.1007\/s00285-019-01390-z","relation":{},"ISSN":["0303-6812","1432-1416"],"issn-type":[{"value":"0303-6812","type":"print"},{"value":"1432-1416","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,6,11]]},"assertion":[{"value":"12 December 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 May 2019","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 June 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}