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Research Article

Generalized reproduction numbers and the prediction of patterns in waterborne disease

Marino Gatto, Lorenzo Mari, Enrico Bertuzzo, Renato Casagrandi, Lorenzo Righetto, Ignacio Rodriguez-Iturbe, and Andrea Rinaldo
PNAS November 27, 2012 109 (48) 19703-19708; https://doi.org/10.1073/pnas.1217567109
Marino Gatto
aDipartimento di Elettronica e Informazione, Politecnico di Milano, 20133 Milan, Italy;
bLaboratory of Ecohydrology, School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland;
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Lorenzo Mari
aDipartimento di Elettronica e Informazione, Politecnico di Milano, 20133 Milan, Italy;
bLaboratory of Ecohydrology, School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland;
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Enrico Bertuzzo
bLaboratory of Ecohydrology, School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland;
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Renato Casagrandi
aDipartimento di Elettronica e Informazione, Politecnico di Milano, 20133 Milan, Italy;
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Lorenzo Righetto
bLaboratory of Ecohydrology, School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland;
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Ignacio Rodriguez-Iturbe
cDepartment of Civil and Environmental Engineering, Princeton University, Princeton, NJ 08544; and
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Andrea Rinaldo
bLaboratory of Ecohydrology, School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland;
dDipartimento di Ingegneria Civile, Edile ed Ambientale, Università di Padova, 35131 Padua, Italy
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  • For correspondence: andrea.rinaldo@epfl.ch
  1. Contributed by Andrea Rinaldo, October 10, 2012 (sent for review July 14, 2012)

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Abstract

Understanding, predicting, and controlling outbreaks of waterborne diseases are crucial goals of public health policies, but pose challenging problems because infection patterns are influenced by spatial structure and temporal asynchrony. Although explicit spatial modeling is made possible by widespread data mapping of hydrology, transportation infrastructure, population distribution, and sanitation, the precise condition under which a waterborne disease epidemic can start in a spatially explicit setting is still lacking. Here we show that the requirement that all the local reproduction numbers Graphic 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 Graphic, explicitly accounting for spatial distributions of human settlements and pathogen transmission via hydrological and human mobility networks. At disease onset, a generalized reproduction number Graphic (the dominant eigenvalue of Graphic) 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 Graphic. 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 Graphic 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.

  • ecohydrology
  • aquatic ecosystems
  • invasion
  • bifurcations

Footnotes

  • ↵1To whom correspondence should be addressed. E-mail: andrea.rinaldo{at}epfl.ch.
  • Author contributions: M.G., L.M., E.B., R.C., I.R.-I., and A.R. designed research; M.G., L.M., E.B., R.C., L.R., and A.R. performed research; M.G., L.M., E.B., R.C., L.R., and A.R. analyzed data; and M.G., L.M., I.R.-I., and A.R. wrote the paper.

  • The authors declare no conflict of interest.

  • This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1217567109/-/DCSupplemental.

Freely available online through the PNAS open access option.

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Reproduction numbers in waterborne disease
Marino Gatto, Lorenzo Mari, Enrico Bertuzzo, Renato Casagrandi, Lorenzo Righetto, Ignacio Rodriguez-Iturbe, Andrea Rinaldo
Proceedings of the National Academy of Sciences Nov 2012, 109 (48) 19703-19708; DOI: 10.1073/pnas.1217567109

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Reproduction numbers in waterborne disease
Marino Gatto, Lorenzo Mari, Enrico Bertuzzo, Renato Casagrandi, Lorenzo Righetto, Ignacio Rodriguez-Iturbe, Andrea Rinaldo
Proceedings of the National Academy of Sciences Nov 2012, 109 (48) 19703-19708; DOI: 10.1073/pnas.1217567109
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Proceedings of the National Academy of Sciences: 109 (48)
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