This chapter provides an outlook of the issues, born out of the material presented in the book, that will likely be at the center of future developments of the field.They are organized following the scheme set by the preceding chapters.

This chapter illustrates the implications of river network structure for the spread of waterborne diseases. Human mobility is also added as a driver and a network of interaction. The water-related (WR) diseases considered are epidemic cholera, endemic schistosomiasis, and proliferative kidney disease in fish. After reviewing the basic (space-implicit) epidemiological models for micro- and micro-parasites, general space-explicit models for both kinds of parasites are studied. Both the hydrologic and the human mobility network are included, and the general conditions for disease establishment are derived, including also the case of seasonal forcings. Conditions for transient (though possibly large) epidemics are also found. The microparasitic model is applied to several cholera epidemics, including the one that has been devastating Haiti. Spatially explicit macroparasitic models of schistosomiasis are then analyzed and applied to the cases of Senegal and Burkina Faso. Proliferative kidney disease (PKD) in salmonid fish, a pathology linked to global warming, is modeled. The space-explicit approach is used for the study of PKD spread in the basin of the river Wigger (Switzerland).

Three separate appendices, rather technical, compose this chapter. The first deals with the stability of dynamical systems and bifurcation analysis, the second with a background and rationale for optimal channel networks. The third deals with computational tools: Matlab code is explained and run, and results are highlighted. The source codes may be downloaded at www.epfl.ch/labs/echo/.

The chapter explores the problem of species spread in river systems. It introduces the basic quantitative tools for dealing with the topic: first advection-diffusion-reaction traditional equations, then more modern approaches via interacting particle systems and reaction random walks on graphs. The Mississippi-Missouri river system is once again taken as the epitomizing river network: the dramatic invasion of the zebra mussel (Dreissena polymorpha), seeded by the accidental discharge of ballast water from European cargo ships in the Great Lakes region, is studied via a multilayer network model, including not only hydrology but also the displacement due to anthropic activity. Advanced laboratory experiments are then described and modeled in which the response of the alga Euglena gracilis when exposed to controlled light fields is analyzed. This allows insight into the problem of how spatial resource availability shapes the invasion patterns of riverine populations. Finally, a thorough analysis of mixing and dispersion in river systems is the basic tool to tackle a problem of enormous interest, namely, estimating species distribution and abundance using environmental DNA.

The introductory chapter outlines the leit-motiv of the book – dendritic substrates for ecological interactions, chief and foremost river networks in our case, bear important consequences for a number of processes, from patterns of biodiversity to controls of spreading of waterborne disease. In this chapter we discuss important methodological aspects of spatially explicit ecology that are used throughout the book.

The chapter is not a complete review of the subject but rather contains a specific choice of topics relevant to the general concepts of biodiversity in river networks. It first explores the fish diversity of the Mississippi-Missouri river system via a hierarchical metacommunity model that includes river hydrology and habitat suitability. Illustrated is the role of the frequency distribution of shear stresses in determining the spatially explicit probability distribution functions of benthic invertebrate habitat suitability. The viewpoint is then broadened by addressing general metapopulation persistence in river networks; this is achieved by analyzing stage-structured populations that exploit different dispersal pathways, both alongstream and overland. Examples are given with reference to amphibians. Gradients of biodiversity in fluvial landscapes, as determined by changes in elevation, is the next topic. The characteristic features of hump-shaped patterns of species richness along elevational gradients are derived, using both idealized (but realistic) fluvial landscapes and real landscapes of the Swiss Alps. The approach allows also the outline of possible consequences of climate change.

The epidemiology of mosquito-borne diseases is changing. This is a fact. Diseases that were mostly ‘confined’ to tropical regions are now becoming real threats for temperate countries – for instance, the chikungunya outbreak in Northern Italy in 2007, the chikungunya epidemic in the Americas and the Caribbean beginning in late 2013, the major Zika epidemic in 2015, which also led to local transmission in the US, and the recent surge of yellow fever cases in previously unaffected areas of Brazil. In this chapter, we propose a mathematical framework to emulate the dynamics of the spreading of a mosquito-borne disease in an urban environment. The main innovation of the proposed modelling framework is to improve on the current compartmental epidemiological models by considering not only vector-to-human transmission at fixed favourite cells, but also the probability of transmission along mobility pathways. This modus operandi allow us to improve the understanding of the interplay between human mobility and mosquito-borne disease in urban environments.

Introduction

What do biological evolution, the physics of glassy materials, a superfluid near its phase transition, the brain, a slowly driven sandpile, and a fluvial network have in common? Little, it might seem. Nevertheless, they give rise to similar theoretical notions and statistical features (Bak, 1996). Central to their similitude is the power-law nature of the probability distribution that describes key geometric and topologic quantities, or the descriptors of time activity that yield signals (usually termed “ 1/f noise,” from the algebraic decay with exponent –f of the power spectrum) having components of all durations. The algebraic decay (the so-called fat tail) of the distribution ensures that extreme events will be much more likely than in any “regular” case, where the exponential decay of the probability of large events makes catastrophic events (the extremes) prohibitively unlikely. It is this feature that is responsible for the enhanced probability of individual, anomalously large events.

Power laws are also the essential ingredients of scaling arguments; that is, they describe a scale-free arrangement of the parts and the whole such that no characteristic scale is present in the growth of the structure and such a scale can arise only through the size of the system. The foregoing description postulates that in infinite domains, no embedded scale typical of the process is found. In perhaps simpler terms, this means that “events” (or values for the quantity under study) range from infinitely small to infinitely large without preference, and thus they scale with the system size.

A fundamental question among scientists in diverse disciplines is related to the dynamic reasons behind scale-free growth.