Fractional dynamics has experienced a firm upswing during the past few years, having been forged into a mature framework in the theory of stochastic processes. A large number of research papers developing fractional dynamics further, or applying it to various systems have appeared since our first review article on the fractional Fokker–Planck equation (Metzler R and Klafter J 2000a, Phys. Rep. 339 1–77). It therefore appears timely to put these new works in a cohesive perspective. In this review we cover both the theoretical modelling of sub- and superdiffusive processes, placing emphasis on superdiffusion, and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function. We also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems.

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The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics

Networks and the epidemiology of directly transmitted infectious diseases are fundamentally linked. The foundations of epidemiology and early epidemiological models were based on population wide random-mixing, but in practice each individual has a finite set of contacts to whom they can pass infection; the ensemble of all such contacts forms a ‘mixing network’. Knowledge of the structure of the network allows models to compute the epidemic dynamics at the population scale from the individual-level behaviour of infections. Therefore, characteristics of mixing networks—and how these deviate from the random-mixing norm—have become important applied concerns that may enhance the understanding and prediction of epidemic patterns and intervention measures. Here, we review the basis of epidemiological theory (based on random-mixing models) and network theory (based on work from the social sciences and graph theory). We then describe a variety of methods that allow the mixing network, or an approximation to the network, to be ascertained. It is often the case that time and resources limit our ability to accurately find all connections within a network, and hence a generic understanding of the relationship between network structure and disease dynamics is needed. Therefore, we review some of the variety of idealized network types and approximation techniques that have been utilized to elucidate this link. Finally, we look to the future to suggest how the two fields of network theory and epidemiological modelling can deliver an improved understanding of disease dynamics and better public health through effective disease control.

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Networks and epidemic models.

The concept of self-organized criticality was introduced to explain the behaviour of the sandpile model. In this model, particles are randomly dropped onto a square grid of boxes. When a box accumulates four particles they are redistributed to the four adjacent boxes or lost off the edge of the grid. Redistributions can lead to further instabilities with the possibility of more particles being lost from the grid, contributing to the size of each ‘avalanche’. These model ‘avalanches’ satisfied a power-law frequency‐area distribution with a slope near unity. Other cellular-automata models, including the slider-block and forest-fire models, are also said to exhibit self-organized critical behaviour. It has been argued that earthquakes, landslides, forest fires, and species extinctions are examples of self-organized criticality in nature. In addition, wars and stock market crashes have been associated with this behaviour. The forest-fire model is particularly interesting in terms of its relation to the critical-point behaviour of the sitepercolation model. In the basic forest-fire model, trees are randomly planted on a grid of points. Periodically in time, sparks are randomly dropped on the grid. If a spark drops on a tree, that tree and adjacent trees burn in a model fire. The fires are the ‘avalanches’ and they are found to satisfy power-law frequency‐area distributions with slopes near unity. This forest-fire model is closely related to the site-percolation model, that exhibits critical behaviour. In the forest-fire model there is an inverse cascade of trees from small clusters to large clusters, trees are lost primarily from model fires that destroy the largest clusters. This quasi steady-state cascade gives a power-law frequency‐area distribution for both clusters of trees and smaller fires. The site-percolation model is equivalent to the forest-fire model without fires. In this case there is a transient cascade of trees from small to large clusters and a power-law distribution is found only at a critical density of trees.

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Self-organized criticality

We report that human walk patterns contain statistically similar features observed in Levy walks. These features include heavy-tail flight and pause-time distributions and the super-diffusive nature of mobility. Human walks are not random walks, but it is surprising that the patterns of human walks and Levy walks contain some statistical similarity. Our study is based on 226 daily GPS traces collected from 101 volunteers in five different outdoor sites. The heavy-tail flight distribution of human mobility induces the super-diffusivity of travel, but up to 30 min to 1 h due to the boundary effect of people's daily movement, which is caused by the tendency of people to move within a predefined (also confined) area of daily activities. These tendencies are not captured in common mobility models such as random way point (RWP). To evaluate the impact of these tendencies on the performance of mobile networks, we construct a simple truncated Levy walk mobility (TLW) model that emulates the statistical features observed in our analysis and under which we measure the performance of routing protocols in delay-tolerant networks (DTNs) and mobile ad hoc networks (MANETs). The results indicate the following. Higher diffusivity induces shorter intercontact times in DTN and shorter path durations with higher success probability in MANET. The diffusivity of TLW is in between those of RWP and Brownian motion (BM). Therefore, the routing performance under RWP as commonly used in mobile network studies and tends to be overestimated for DTNs and underestimated for MANETs compared to the performance under TLW.

/pdf/on-the-levy-walk-nature-of-human-mobility-tdkmsro9ui.pdf

On the levy-walk nature of human mobility

Spatio-temporal travelling waves are striking manifestations of predator-prey and host-parasite dynamics. However, few systems are well enough documented both to detect repeated waves and to explain their interaction with spatio-temporal variations in population structure and demography. Here, we demonstrate recurrent epidemic travelling waves in an exhaustive spatio-temporal data set for measles in England and Wales. We use wavelet phase analysis, which allows for dynamical non-stationarity--a complication in interpreting spatio-temporal patterns in these and many other ecological time series. In the pre-vaccination era, conspicuous hierarchical waves of infection moved regionally from large cities to small towns; the introduction of measles vaccination restricted but did not eliminate this hierarchical contagion. A mechanistic stochastic model suggests a dynamical explanation for the waves-spread via infective 'sparks' from large 'core' cities to smaller 'satellite' towns. Thus, the spatial hierarchy of host population structure is a prerequisite for these infection waves.

/pdf/travelling-waves-and-spatial-hierarchies-in-measles-172zzvx46l.pdf

Travelling waves and spatial hierarchies in measles epidemics

Using conventional radio-tracking techniques employed by field ecologists, evidence for scale-free (fractal) behaviour in the foraging trajectories of a species of African jackal is presented. It is believed that the particular form of foraging strategy observed here is a response by the jackal to its need to locate resources in an unpredictable environment. The methods used in this study are completely general and can be applied to other radio-tracked species, thus beginning a systematic investigation of foraging strategies in mammals.

Scale‐free dynamics in the movement patterns of jackals

TEMPORAL changes in the incidence of measles virus infection within large urban communities in the developed world have been the focus of much discussion in the context of the identification and analysis of nonlinear and chaotic patterns in biological time series1–11. In contrast, the measles records for small isolated island populations are highly irregular, because of frequent fade-outs of infection12–14, and traditional analysis15 does not yield useful insight. Here we use measurements of the distribution of epidemic sizes and duration to show that regularities in the dynamics of such systems do become apparent. Specifically, these biological systems are characterized by well-defined power laws in a manner reminiscent of other nonlinear, spatially extended dynamical systems in the physical sciences16–19. We further show that the observed power-law exponents are well described by a simple lattice-based model which reflects the social interaction between individual hosts.

Power laws governing epidemics in isolated populations

https://www.cs.princeton.edu/courses/archive/fall09/cos323/papers/RhodesAnderson97.pdf

Epidemic Thresholds and Vaccination in a Lattice Model of Disease Spread

We show how ideas and models which were originally introduced to gain an understanding of critical phenomena can be used to interpret the dynamics of epidemics of communicable disease in real populations. Specifically, we present an analysis of the dynamics of disease outbreaks for three common communicable infections from a small isolated island population. The strongly fluctuating nature of the temporal incidence of disease is captured by the model, and comparisons between exponents calculated from the data and from simulations are made. A forest-fire model with sparks is used to classify the observed scaling dynamics of the epidemics and provides a unified picture of the epidemiology which conventional epidemiological analysis is unable to reproduce. This study suggests that power-law scaling can emerge in natural systems when they are driven on widely separated time-scales, in accordance with recent analytic renormalization group calculations.

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C. J. Rhodes

Papers

Scale‐free dynamics in the movement patterns of jackals

Power laws governing epidemics in isolated populations

Epidemic Thresholds and Vaccination in a Lattice Model of Disease Spread

On the critical behaviour of simple epidemics

Persistence and Dynamics in Lattice Models of Epidemic Spread