Generalized reproduction numbers and the prediction of patterns in waterborne disease. Proc Natl Acad Sci USA

Dipartimento di Elettronica e Informazione, Politecnico di Milano, 20133 Milan, Italy.
Proceedings of the National Academy of Sciences (Impact Factor: 9.67). 11/2012; 109(48). DOI: 10.1073/pnas.1217567109
Source: PubMed


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 $${R}_{\mathbf{0}}$$ 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 $${\mathit{G}}_{\mathbf{0}}$$, explicitly accounting for spatial distributions of human settlements and pathogen transmission via hydrological and human mobility networks. At disease onset, a generalized reproduction number $${\hbox{ \Lambda }}_{\mathbf{0}}$$ (the dominant eigenvalue of $${\mathit{G}}_{\mathbf{0}}$$) 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 $${\mathit{G}}_{\mathbf{0}}$$. 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 $${\mathit{G}}_{\mathbf{0}}$$ 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.

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    • "Furthermore, seasonality can influence the spatial patterns of pathogen invasion. The importance of generalizing the concept of basic reproduction number not only to spatially explicit (Gatto et al. 2012, 2013) but also to time-varying systems (this study) is also demonstrated, in fact, by the ability of our approach to describe the geography of disease outbreaks, which is shown to be well characterized by the dominant eigenvector of the monodromy matrix. This result can be easily applied to other realistic landscapes described by networks of any given complexity (of which the present examples act as a proof of concept), ranging from a few to thousands of nodes. "
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    • "structures for the hydrological network range from simple one-dimensional lattices to more realistic mathematical characterizations, such as Peano basins (as in Gatto et al. 2012), optimal channel networks (Rinaldo et al. 1992; Rodriguez-Iturbe et al. 1992; see below for details), or real river systems (e.g., Bertuzzo et al. 2008; Mari et al. 2012a; Rinaldo et al. 2012). As for the human-mobility network, we assume that the nodes of this second layer correspond to those of the hydrological layer, whereas edges are defined by connections among communities. "
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