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Here we show that the requirement that all the local reproduction numbers\n \n \n <\/jats:inline-formula>\n be larger than unity is neither necessary nor sufficient for outbreaks to occur when local settlements are connected by networks of primary and secondary infection mechanisms. To determine onset conditions, we derive general analytical expressions for a reproduction matrix\n \n \n <\/jats:inline-formula>\n , explicitly accounting for spatial distributions of human settlements and pathogen transmission via hydrological and human mobility networks. At disease onset, a generalized reproduction number\n \n \n <\/jats:inline-formula>\n (the dominant eigenvalue of\n \n \n <\/jats:inline-formula>\n ) must be larger than unity. We also show that geographical outbreak patterns in complex environments are linked to the dominant eigenvector and to spectral properties of\n \n \n <\/jats:inline-formula>\n . Tests against data and computations for the 2010 Haiti and 2000 KwaZulu-Natal cholera outbreaks, as well as against computations for metapopulation networks, demonstrate that eigenvectors of\n \n \n <\/jats:inline-formula>\n provide a synthetic and effective tool for predicting the disease course in space and time. Networked connectivity models, describing the interplay between hydrology, epidemiology, and social behavior sustaining human mobility, thus prove to be key tools for emergency management of waterborne infections.\n <\/jats:p>","DOI":"10.1073\/pnas.1217567109","type":"journal-article","created":{"date-parts":[[2012,11,13]],"date-time":"2012-11-13T13:32:23Z","timestamp":1352813543000},"page":"19703-19708","update-policy":"https:\/\/doi.org\/10.1073\/pnas.cm10313","source":"Crossref","is-referenced-by-count":72,"title":["Generalized reproduction numbers and the prediction of patterns in waterborne disease"],"prefix":"10.1073","volume":"109","author":[{"given":"Marino","family":"Gatto","sequence":"first","affiliation":[{"name":"Dipartimento di Elettronica e Informazione, Politecnico di Milano, 20133 Milan, Italy;"},{"name":"Laboratory of Ecohydrology, School of Architecture, Civil and Environmental Engineering, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne, 1015 Lausanne, Switzerland;"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lorenzo","family":"Mari","sequence":"additional","affiliation":[{"name":"Dipartimento di Elettronica e Informazione, Politecnico di Milano, 20133 Milan, Italy;"},{"name":"Laboratory of Ecohydrology, School of Architecture, Civil and Environmental Engineering, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne, 1015 Lausanne, Switzerland;"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Enrico","family":"Bertuzzo","sequence":"additional","affiliation":[{"name":"Laboratory of Ecohydrology, School of Architecture, Civil and Environmental Engineering, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne, 1015 Lausanne, Switzerland;"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Renato","family":"Casagrandi","sequence":"additional","affiliation":[{"name":"Dipartimento di Elettronica e Informazione, Politecnico di Milano, 20133 Milan, Italy;"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lorenzo","family":"Righetto","sequence":"additional","affiliation":[{"name":"Laboratory of Ecohydrology, School of Architecture, Civil and Environmental Engineering, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne, 1015 Lausanne, Switzerland;"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ignacio","family":"Rodriguez-Iturbe","sequence":"additional","affiliation":[{"name":"Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ 08544; and"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andrea","family":"Rinaldo","sequence":"additional","affiliation":[{"name":"Laboratory of Ecohydrology, School of Architecture, Civil and Environmental Engineering, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne, 1015 Lausanne, Switzerland;"},{"name":"Dipartimento di Ingegneria Civile, Edile ed Ambientale, Universit\u00e0 di Padova, 35131 Padua, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"341","published-online":{"date-parts":[[2012,11,12]]},"reference":[{"key":"e_1_3_3_1_2","volume-title":"The Global Burden of Disease: 2004 Update","author":"World Health Organization","year":"2008","unstructured":"; World Health Organization The Global Burden of Disease: 2004 Update (WHO, Geneva, 2008)."},{"key":"e_1_3_3_2_2","doi-asserted-by":"publisher","DOI":"10.1029\/2011GL046823"},{"key":"e_1_3_3_3_2","doi-asserted-by":"publisher","DOI":"10.1098\/rsif.2011.0304"},{"key":"e_1_3_3_4_2","doi-asserted-by":"publisher","DOI":"10.7326\/0003-4819-154-9-201105030-00334"},{"key":"e_1_3_3_5_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1102149108"},{"key":"e_1_3_3_6_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1203333109"},{"key":"e_1_3_3_7_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1019712108"},{"key":"e_1_3_3_8_2","doi-asserted-by":"publisher","DOI":"10.1126\/science.286.5441.968"},{"key":"e_1_3_3_9_2","doi-asserted-by":"publisher","DOI":"10.1029\/2007WR006211"},{"key":"e_1_3_3_10_2","doi-asserted-by":"publisher","DOI":"10.1098\/rsif.2009.0204"},{"key":"e_1_3_3_11_2","doi-asserted-by":"publisher","DOI":"10.1126\/science.1172330"},{"key":"e_1_3_3_12_2","first-page":"121","article-title":"A mathematical model for the 1973 cholera epidemic in the European Mediterranean region","volume":"27","author":"Capasso V","year":"1979","unstructured":"V Capasso, SL Paveri-Fontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region. 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