Article

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

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

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 $${R}_{\mathbf{0}}$$ 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. "
##### Article: Floquet theory for seasonal environmental forcing of spatially explicit waterborne epidemics
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ABSTRACT: The transmission of waterborne pathogens is a complex process that is heavily linked to the spatial characteristics of the underlying environmental matrix as well as to the temporal variability of the relevant hydroclimatological drivers. In this work, we propose a time-varying, spatially explicit network model for the dynamics of waterborne diseases. Applying Floquet theory, which allows to extend results of local stability analysis to periodic dynamical systems, we find conditions for pathogen invasion and establishment in systems characterized by fluctuating environmental forcing, thus extending to time-varying contexts the generalized reproduction numbers recently obtained for spatially explicit epidemiology of waterborne disease. We show that temporal variability may have multifaceted effects on the invasion threshold, as it can either favor pathogen invasion or make it less likely. Moreover, environmental fluctuations characterized by distinctive geographical signatures can produce diversified, highly nontrivial effects on pathogen invasion. Our study is complemented by numerical simulations, which show that pathogen establishment is neither necessary nor sufficient for large epidemic outbreaks to occur in time-varying environments. Finally, we show that our framework can be used to reliably characterize the early geography of epidemic outbreaks triggered by fluctuating environmental conditions.
Theoretical Ecology 11/2014; 7(4):351-365. DOI:10.1007/s12080-014-0223-y · 1.55 Impact Factor
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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. "
##### Article: Spatially Explicit Conditions for Waterborne Pathogen Invasion
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ABSTRACT: Abstract Waterborne pathogens cause many possibly lethal human diseases. We derive the condition for pathogen invasion and subsequent disease outbreak in a territory with specific, space-inhomogeneous characteristics (hydrological, ecological, demographic, and epidemiological). The criterion relies on a spatially explicit model accounting for the density of susceptible and infected individuals and the pathogen concentration in a network of communities linked by human mobility and the water system. Pathogen invasion requires that a dimensionless parameter, the dominant eigenvalue of a generalized reproductive matrix J0, be larger than unity. Conditions for invasion are studied while crucial parameters (population density distribution, contact and water contamination rates, pathogen growth rates) and the characteristics of the networks (connectivity, directional transport, water retention times, mobility patterns) are varied. We analyze both simple, prototypical test cases and realistic landscapes, in which optimal channel networks mimic the water systems and gravitational models describe human mobility. Also, we show that the dominant eigenvector of J0 effectively portrays the geography of epidemic outbreaks, that is, the areas of the studied territory that will be initially affected by an epidemic. This is important for planning an efficient spatial allocation of interventions (e.g., improving sanitation and providing emergency aid and medicines).
The American Naturalist 09/2013; 182(3):328-46. DOI:10.1086/671258 · 3.83 Impact Factor
• ##### Article: Examining rainfall and cholera dynamics in Haiti using statistical and dynamic modeling approaches
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ABSTRACT: Haiti has been in the midst of a cholera epidemic since October 2010. Rainfall is thought to be associated with cholera here, but this relationship has only begun to be quantitatively examined. In this paper, we quantitatively examine the link between rainfall and cholera in Haiti for several different settings (including urban, rural, and displaced person camps) and spatial scales, using a combination of statistical and dynamic models. Statistical analysis of the lagged relationship between rainfall and cholera incidence was conducted using case crossover analysis and distributed lag nonlinear models. Dynamic models consisted of compartmental differential equation models including direct (fast) and indirect (delayed) disease transmission, where indirect transmission was forced by empirical rainfall data. Data sources include cholera case and hospitalization time series from the Haitian Ministry of Public Health, the United Nations Water, Sanitation and Health Cluster, International Organization for Migration, and Hôpital Albert Schweitzer. Rainfall data was obtained from rain gauges from the U.S. Geological Survey and Haiti Regeneration Initiative, and remote sensing rainfall data from the National Aeronautics and Space Administration Tropical Rainfall Measuring Mission. A strong relationship between rainfall and cholera was found for all spatial scales and locations examined. Increased rainfall was significantly correlated with increased cholera incidence 4–7 days later. Forcing the dynamic models with rainfall data resulted in good fits to the cholera case data, and rainfall-based predictions from the dynamic models closely matched observed cholera cases. These models provide a tool for planning and managing the epidemic as it continues.
Epidemics 01/2013; 5(4):197–207. · 1.87 Impact Factor