Generalized reproduction numbers and the prediction of patterns in waterborne disease

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 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 G 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 Λ 0 (the dominant eigenvalue of G 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 G 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 G 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.ecohydrology | aquatic ecosystems | invasion | bifurcations W aterborne diseases, mainly due to protozoa or bacteria and often resulting in profuse diarrhea (cholera is a prominent example), are one of the leading causes of death and especially strike infants and children in low-income countries (1). Therefore, it is fundamental to develop realistic dynamical models that can provide insights into the course of past and ongoing epidemics, assist in emergency management, and allocate health-care resources via an assessment of alternative intervention strategies (2-7). These models must properly account for the relevant disease and transport timescales (8-10). Whereas some diarrheal infections, like Rotavirus, have basically a fecal-oral transmission and their spread can thus be modeled as susceptible-infected-recovered (SIR) systems (11), traditional waterborne disease models (12, 13) include, in addition to susceptibles (S) and infectives (I) (14), the population dynamics of bacteria (B) in water reservoirs. More recent modeling has considered hyperinfectivity of newly excreted vibrios (15), prey-predator interactions with phages (16), and seasonal and climate forcings (17)(18)(19)(20)(21)(22). All these models, however, have not considered explicitly the ...

Spatially Explicit Conditions for Waterborne Pathogen Invasion

“…S6). In other terms, approximating a river network with a Peano construct (as done in Gatto et al 2012), rather than with a more realistic OCN, can lead to an overestimation of the actual likelihood of waterborne pathogen invasion (and epidemic outbreak).…”

“…Here we extend, generalize, and make more explicit the approach outlined by Gatto et al (2012). Specifically, in this work we explicitly account for spatial heterogeneities of both hydrological conditions and pathogen ecology, which were not considered in previous approaches.…”

“…Gatto et al (2012) analyzed epidemic outbreaks in Peano basins with spatially homogeneous rates of pathogen growth and hydrological transport. Peano basins are deterministic, space-filling fractals with a tree-like structure whose topological measures are similar to those of real rivers but fail to satisfy the statistics of aggregation and upstream/downstream distances (Rodriguez-Iturbe and Rinaldo 1997; Rinaldo et al 1999).…”

“…The transport process is assumed to be conservative, that is, . Possible topological n P p 1 ij jp1 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 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.…”

“…While metapopulation dynamics was incorporated into traditional SI or SIR models many years ago (Bolker and Grenfell 1995;Swinton et al 1998;Gog et al 2002;Arino et al 2005), the addition of human mobility to spatial models of waterborne diseases is more recent. This has been accomplished via either diffusion-based or gravity-like models Chao et al 2011;Tuite et al 2011;Gatto et al 2012;Mari et al 2012aMari et al , 2012bRinaldo et al 2012). The wide availability, via geographic information systems, of spatial data on hydrology, road networks, population distribution, and sanitation makes these models applicable to specific situations in each country and for each disease to be studied.…”