{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,9]],"date-time":"2026-09-09T16:32:07Z","timestamp":1788971527865,"version":"build-2803163510"},"reference-count":51,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2011,3,15]],"date-time":"2011-03-15T00:00:00Z","timestamp":1300147200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Math. Biol."],"published-print":{"date-parts":[[2012,2]]},"DOI":"10.1007\/s00285-011-0417-5","type":"journal-article","created":{"date-parts":[[2011,3,14]],"date-time":"2011-03-14T13:50:33Z","timestamp":1300110633000},"page":"403-422","source":"Crossref","is-referenced-by-count":44,"title":["The model of Kermack and McKendrick for the plague epidemic in Bombay and the type reproduction number with seasonality"],"prefix":"10.1007","volume":"64","author":[{"given":"Nicolas","family":"Baca\u00ebr","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2011,3,15]]},"reference":[{"key":"417_CR1","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1017\/S0022172400003016","volume":"6","author":"Advisory Committee appointed by the Secretary of State for India","year":"1906","unstructured":"Advisory Committee appointed by the Secretary of State for India: (1906a) Reports on plague investigations in India, I, Experiments upon the transmission of plague by fleas. J Hyg 6: 425\u2013482","journal-title":"J Hyg"},{"key":"417_CR2","doi-asserted-by":"crossref","first-page":"524","DOI":"10.1017\/S0022172400003120","volume":"6","author":"Advisory Committee appointed by the Secretary of State for India","year":"1906","unstructured":"Advisory Committee appointed by the Secretary of State for India: (1906b) Reports on plague investigations in India, IX, On the quantitative estimation of the septicaemia in human plague. J Hyg 6: 524\u2013529","journal-title":"J Hyg"},{"key":"417_CR3","doi-asserted-by":"crossref","first-page":"472","DOI":"10.1017\/S0022172400033441","volume":"7","author":"Advisory Committee appointed by the Secretary of State for India","year":"1907","unstructured":"Advisory Committee appointed by the Secretary of State for India: (1907a) Reports on plague investigations in India, XX, A note on man as a host of the Indian rat flea. J Hyg 7: 472\u2013476","journal-title":"J Hyg"},{"key":"417_CR4","doi-asserted-by":"crossref","first-page":"724","DOI":"10.1017\/S0022172400033684","volume":"7","author":"Advisory Committee appointed by the Secretary of State for India","year":"1907","unstructured":"Advisory Committee appointed by the Secretary of State for India: (1907b) Reports on plague investigations in India, XXII, Epidemiological observations in Bombay City. J Hyg 7: 724\u2013798","journal-title":"J Hyg"},{"key":"417_CR5","doi-asserted-by":"crossref","first-page":"266","DOI":"10.1017\/S0022172400003314","volume":"8","author":"Advisory Committee appointed by the Secretary of State for India","year":"1908","unstructured":"Advisory Committee appointed by the Secretary of State for India: (1908) Reports on plague investigations in India, XXXI, On the seasonal prevalence of plague in India. J Hyg 8: 266\u2013301","journal-title":"J Hyg"},{"key":"417_CR6","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1017\/S0022172400016648","volume":"11","author":"Advisory Committee appointed by the Secretary of State for India","year":"1911","unstructured":"Advisory Committee appointed by the Secretary of State for India: (1911) Reports on plague investigations in India, XLVII, Observations on the breeding of Mus Rattus in captivity. J Hyg 11: 193\u2013206","journal-title":"J Hyg"},{"key":"417_CR7","first-page":"221","volume":"12","author":"Advisory Committee appointed by the Secretary of State for India","year":"1912","unstructured":"Advisory Committee appointed by the Secretary of State for India: (1912) Reports on plague investigations in India, XLIX, Statistics of the occurrence of plague in man and rats in Bombay. J Hyg 12: 221\u2013226","journal-title":"J Hyg"},{"key":"417_CR8","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198545996.001.0001","volume-title":"Infectious diseases in humans: dynamics and control","author":"RM Anderson","year":"1991","unstructured":"Anderson RM, May RM (1991) Infectious diseases in humans: dynamics and control. Oxford University Press, Oxford"},{"key":"417_CR9","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1007\/s00285-006-0015-0","volume":"53","author":"N Baca\u00ebr","year":"2006","unstructured":"Baca\u00ebr N, Guernaoui S (2006) The epidemic threshold of vector-borne diseases with seasonality. J Math Biol 53: 421\u2013436","journal-title":"J Math Biol"},{"key":"417_CR10","doi-asserted-by":"crossref","first-page":"1067","DOI":"10.1007\/s11538-006-9166-9","volume":"69","author":"N Baca\u00ebr","year":"2007","unstructured":"Baca\u00ebr N (2007) Approximation of the basic reproduction number R 0 for vector-borne diseases with a periodic vector population. Bull Math Biol 69: 1067\u20131091","journal-title":"Bull Math Biol"},{"key":"417_CR11","doi-asserted-by":"crossref","first-page":"1781","DOI":"10.1007\/s11538-009-9426-6","volume":"71","author":"N Baca\u00ebr","year":"2009","unstructured":"Baca\u00ebr N (2009) Periodic matrix population models: growth rate, basic reproduction number, and entropy. Bull Math Biol 71: 1781\u20131792","journal-title":"Bull Math Biol"},{"key":"417_CR12","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-85729-115-8","volume-title":"A short history of mathematical population dynamics","author":"N Baca\u00ebr","year":"2011","unstructured":"Baca\u00ebr N (2011) A short history of mathematical population dynamics. Springer, London"},{"key":"417_CR13","doi-asserted-by":"crossref","unstructured":"Baca\u00ebr N, Ait Dads E (2011) Genealogy with seasonality, the basic reproduction number, and the influenza pandemic. J Math Biol. doi: 10.1007\/s00285-010-0354-8","DOI":"10.1007\/s00285-010-0354-8"},{"key":"417_CR14","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-03052-3","volume-title":"Growth and diffusion phenomena","author":"RB Banks","year":"1994","unstructured":"Banks RB (1994) Growth and diffusion phenomena. Springer, Berlin"},{"key":"417_CR15","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611971262","volume-title":"Nonnegative matrices in the mathematical sciences","author":"A Berman","year":"1994","unstructured":"Berman A, Plemmons RJ (1994) Nonnegative matrices in the mathematical sciences. SIAM, Philadelphia"},{"key":"417_CR16","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-4360-1","volume-title":"Differential equations and their applications","author":"M Braun","year":"1993","unstructured":"Braun M (1993) Differential equations and their applications, 4th edn. Springer, New York","edition":"4"},{"key":"417_CR17","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4471-0049-2","volume-title":"Essential mathematical biology","author":"NF Britton","year":"2003","unstructured":"Britton NF (2003) Essential mathematical biology. Springer, London"},{"key":"417_CR18","volume-title":"Cultural transmission and evolution: a quantitative approach","author":"LL Cavalli-Sforza","year":"1981","unstructured":"Cavalli-Sforza LL, Feldman MW (1981) Cultural transmission and evolution: a quantitative approach. Princeton University Press, Princeton"},{"key":"417_CR19","doi-asserted-by":"crossref","first-page":"703","DOI":"10.1086\/532493","volume":"107","author":"SK Cohn Jr","year":"2002","unstructured":"Cohn SK Jr (2002) The black death: end of a paradigm. Am Hist Rev 107: 703\u2013738","journal-title":"Am Hist Rev"},{"key":"417_CR20","volume-title":"The black death transformed","author":"SK Cohn Jr","year":"2010","unstructured":"Cohn SK Jr (2010) The black death transformed. Bloomsbury Academic, London"},{"key":"417_CR21","volume-title":"Mathematical epidemiology of infectious diseases","author":"O Diekmann","year":"2000","unstructured":"Diekmann O, Heesterbeek JAP (2000) Mathematical epidemiology of infectious diseases. Wiley, Chichester"},{"key":"417_CR22","doi-asserted-by":"crossref","first-page":"873","DOI":"10.1098\/rsif.2009.0386","volume":"7","author":"O Diekmann","year":"2010","unstructured":"Diekmann O, Heesterbeek JAP, Roberts MG (2010) The construction of next-generation matrices for compartmental epidemic models. J R Soc Interface 7: 873\u2013885","journal-title":"J R Soc Interface"},{"key":"417_CR23","doi-asserted-by":"crossref","first-page":"97","DOI":"10.3103\/S1068362309020034","volume":"44","author":"K Dietz","year":"2009","unstructured":"Dietz K (2009) Epidemics: the fitting of the first dynamic models to data. J Contemp Math Anal 44: 97\u2013104","journal-title":"J Contemp Math Anal"},{"key":"417_CR24","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898719147","volume-title":"Mathematical models in biology","author":"L Edelstein-Keshet","year":"2005","unstructured":"Edelstein-Keshet L (2005) Mathematical models in biology. SIAM, Philadelphia"},{"key":"417_CR25","unstructured":"Gatacre WF (1897) Report on the bubonic plague in Bombay 1896\u20131897. Plague Committee, Bombay. http:\/\/digital.nls.uk"},{"key":"417_CR26","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-2731-9","volume-title":"Population biology: concepts and models","author":"A Hastings","year":"1997","unstructured":"Hastings A (1997) Population biology: concepts and models. Springer, New York"},{"key":"417_CR27","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/j.mbs.2004.10.013","volume":"206","author":"JAP Heesterbeek","year":"2007","unstructured":"Heesterbeek JAP, Roberts MG (2007) The type-reproduction number T in models for infectious disease control. Math Biosci 206: 3\u201310","journal-title":"Math Biosci"},{"key":"417_CR28","volume-title":"La Math\u00e9matisation du R\u00e9el","author":"G Israel","year":"1996","unstructured":"Israel G (1996) La Math\u00e9matisation du R\u00e9el. Seuil, Paris"},{"key":"417_CR29","doi-asserted-by":"crossref","first-page":"903","DOI":"10.1038\/35038073","volume":"407","author":"MJ Keeling","year":"2000","unstructured":"Keeling MJ, Gilligan CA (2000) Metapopulation dynamics of bubonic plague. Nature 407: 903\u2013906","journal-title":"Nature"},{"key":"417_CR30","doi-asserted-by":"crossref","first-page":"2219","DOI":"10.1098\/rspb.2000.1272","volume":"267","author":"MJ Keeling","year":"2000","unstructured":"Keeling MJ, Gilligan CA (2000) Bubonic plague: a metapopulation model of a zoonosis. Proc R Soc Lond B 267: 2219\u20132230","journal-title":"Proc R Soc Lond B"},{"key":"417_CR31","doi-asserted-by":"crossref","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":"417_CR32","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1098\/rspa.1927.0118","volume":"115","author":"WO Kermack","year":"1927","unstructured":"Kermack WO, McKendrick AG (1927) Contributions to the mathematical theory of epidemics. Proc R Soc A 115: 700\u2013721","journal-title":"Proc R Soc A"},{"key":"417_CR33","first-page":"33","volume":"53","author":"WO Kermack","year":"1991","unstructured":"Kermack WO, McKendrick AG (1991) Contributions to the mathematical theory of epidemics (reprint). Bull Math Biol 53:33\u201355","journal-title":"Bull Math Biol"},{"key":"417_CR34","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1093\/biomet\/33.3.183","volume":"33","author":"PH Leslie","year":"1945","unstructured":"Leslie PH (1945) On the use of matrices in certain population mathematics. Biometrika 33: 183\u2013212","journal-title":"Biometrika"},{"key":"417_CR35","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511819872","volume-title":"The theoretical biologist\u2019s toolbox","author":"M Mangel","year":"2006","unstructured":"Mangel M (2006) The theoretical biologist\u2019s toolbox. Cambridge University Press, Cambridge"},{"key":"417_CR36","doi-asserted-by":"crossref","first-page":"582","DOI":"10.1016\/j.ijmm.2009.05.003","volume":"299","author":"S Monecke","year":"2009","unstructured":"Monecke S, Monecke H, Monecke J (2009) Modelling the black death, a historical case study and implications for the epidemiology of bubonic plague. Int J Med Microbiol 299: 582\u2013593","journal-title":"Int J Med Microbiol"},{"key":"417_CR37","doi-asserted-by":"crossref","DOI":"10.1007\/b98868","volume-title":"Mathematical biology I: an introduction","author":"JD Murray","year":"2002","unstructured":"Murray JD (2002) Mathematical biology I: an introduction, 3rd edn. Springer, Berlin","edition":"3"},{"key":"417_CR38","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198504184.001.0001","volume-title":"Virus dynamics: mathematical principles of immunology and virology","author":"MA Nowak","year":"2000","unstructured":"Nowak MA, May RM (2000) Virus dynamics: mathematical principles of immunology and virology. Oxford University Press, Oxford"},{"key":"417_CR39","volume-title":"An introduction to mathematical models in the social and life sciences","author":"M Olinick","year":"1978","unstructured":"Olinick M (1978) An introduction to mathematical models in the social and life sciences. Addison-Wesley, Reading"},{"key":"417_CR40","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1017\/S002217240001812X","volume":"32","author":"L Otten","year":"1932","unstructured":"Otten L (1932) The problem of the seasonal prevalence of plague. J Hyg 32: 396\u2013405","journal-title":"J Hyg"},{"key":"417_CR41","unstructured":"Pollitzer R (1954) Plague. WHO Monograph Series No. 22, Geneva. http:\/\/whqlibdoc.who.int\/monograph\/WHO_MONO_22_(chp1).pdf"},{"key":"417_CR42","doi-asserted-by":"crossref","first-page":"1359","DOI":"10.1098\/rspb.2003.2339","volume":"270","author":"MG Roberts","year":"2003","unstructured":"Roberts MG, Heesterbeek JAP (2003) A new method for estimating the effort required to control an infectious disease. Proc R Soc Lond B 270: 1359\u20131364","journal-title":"Proc R Soc Lond B"},{"key":"417_CR43","doi-asserted-by":"crossref","first-page":"949","DOI":"10.1098\/rsif.2007.1031","volume":"4","author":"MG Roberts","year":"2007","unstructured":"Roberts MG (2007) The pluses and minuses of R 0. J R Soc Interface 4: 949\u2013961","journal-title":"J R Soc Interface"},{"key":"417_CR44","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198548522.001.0001","volume-title":"Biological invasions: theory and practice","author":"N Shigesada","year":"1997","unstructured":"Shigesada N, Kawasaki K (1997) Biological invasions: theory and practice. Oxford University Press, Oxford"},{"key":"417_CR45","first-page":"881","volume":"15","author":"R St John Brooks","year":"1917","unstructured":"St John Brooks R (1917) Reports on plague investigations in India, LXXXIV, The influence of saturation deficiency and of temperature on the course of epidemic plague. J Hyg 15: 881\u2013899","journal-title":"J Hyg"},{"key":"417_CR46","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1080\/07036338408428855","volume":"7","author":"HR Thieme","year":"1984","unstructured":"Thieme HR (1984) Renewal theorems for linear periodic Volterra integral equations. J Integr Equ 7: 253\u2013277","journal-title":"J Integr Equ"},{"key":"417_CR47","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1137\/080732870","volume":"70","author":"HR Thieme","year":"2009","unstructured":"Thieme HR (2009) Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity. SIAM J Appl Math 70: 188\u2013211","journal-title":"SIAM J Appl Math"},{"key":"417_CR48","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/S0025-5564(02)00108-6","volume":"180","author":"P Driessche van den","year":"2002","unstructured":"van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180: 29\u201348","journal-title":"Math Biosci"},{"key":"417_CR49","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-80820-3","volume-title":"Deterministic threshold models in the theory of epidemics","author":"P Waltman","year":"1974","unstructured":"Waltman P (1974) Deterministic threshold models in the theory of epidemics. Springer, Berlin"},{"key":"417_CR50","doi-asserted-by":"crossref","first-page":"699","DOI":"10.1007\/s10884-008-9111-8","volume":"20","author":"W Wang","year":"2008","unstructured":"Wang W, Zhao XQ (2008) Threshold dynamics for compartmental epidemic models in periodic environments. J Dyn Differ Equ 20: 699\u2013717","journal-title":"J Dyn Differ Equ"},{"issue":"12","key":"417_CR51","doi-asserted-by":"crossref","first-page":"e8401","DOI":"10.1371\/journal.pone.0008401","volume":"4","author":"MR Welford","year":"2009","unstructured":"Welford MR, Bossak BH (2009) Validation of inverse seasonal peak mortality in medieval plagues, including the Black Death, in comparison to modern Yersinia pestis-variant diseases. PLoS ONE 4(12): e8401. doi: 10.1371\/journal.pone.0008401","journal-title":"PLoS ONE"}],"container-title":["Journal of Mathematical Biology"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00285-011-0417-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00285-011-0417-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00285-011-0417-5","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,4,5]],"date-time":"2024-04-05T04:46:12Z","timestamp":1712292372000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00285-011-0417-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,3,15]]},"references-count":51,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2012,2]]}},"alternative-id":["417"],"URL":"https:\/\/doi.org\/10.1007\/s00285-011-0417-5","relation":{},"ISSN":["0303-6812","1432-1416"],"issn-type":[{"value":"0303-6812","type":"print"},{"value":"1432-1416","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,3,15]]}}}