Epidemics and percolation in small-world networks

We study some simple models of disease transmission on small-world networks, in which either the probability of infection by a disease or the probability of its transmission is varied, or both. The resulting models display epidemic behavior when the infection or transmission probability rises above the threshold for site or bond percolation on the network, and we give exact solutions for the position of this threshold in a variety of cases. We confirm our analytic results by numerical simulation.Comment: 6 pages, including 3 postscript figure

Phase transitions in contagion processes mediated by recurrent mobility patterns

Human mobility and activity patterns mediate contagion on many levels, including the spatial spread of infectious diseases, diffusion of rumors, and emergence of consensus. These patterns however are often dominated by specific locations and recurrent flows and poorly modeled by the random diffusive dynamics generally used to study them. Here we develop a theoretical framework to analyze contagion within a network of locations where individuals recall their geographic origins. We find a phase transition between a regime in which the contagion affects a large fraction of the system and one in which only a small fraction is affected. This transition cannot be uncovered by continuous deterministic models due to the stochastic features of the contagion process and defines an invasion threshold that depends on mobility parameters, providing guidance for controlling contagion spread by constraining mobility processes. We recover the threshold behavior by analyzing diffusion processes mediated by real human commuting data.Comment: 20 pages of Main Text including 4 figures, 7 pages of Supplementary Information; Nature Physics (2011

Random graphs with arbitrary degree distributions and their applications

Recent work on the structure of social networks and the internet has focussed attention on graphs with distributions of vertex degree that are significantly different from the Poisson degree distributions that have been widely studied in the past. In this paper we develop in detail the theory of random graphs with arbitrary degree distributions. In addition to simple undirected, unipartite graphs, we examine the properties of directed and bipartite graphs. Among other results, we derive exact expressions for the position of the phase transition at which a giant component first forms, the mean component size, the size of the giant component if there is one, the mean number of vertices a certain distance away from a randomly chosen vertex, and the average vertex-vertex distance within a graph. We apply our theory to some real-world graphs, including the world-wide web and collaboration graphs of scientists and Fortune 1000 company directors. We demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world, while in others there is a measurable discrepancy between theory and reality, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph.Comment: 19 pages, 11 figures, some new material added in this version along with minor updates and correction

The spread of epidemic disease on networks

The study of social networks, and in particular the spread of disease on networks, has attracted considerable recent attention in the physics community. In this paper, we show that a large class of standard epidemiological models, the so-called susceptible/infective/removed (SIR) models can be solved exactly on a wide variety of networks. In addition to the standard but unrealistic case of fixed infectiveness time and fixed and uncorrelated probability of transmission between all pairs of individuals, we solve cases in which times and probabilities are non-uniform and correlated. We also consider one simple case of an epidemic in a structured population, that of a sexually transmitted disease in a population divided into men and women. We confirm the correctness of our exact solutions with numerical simulations of SIR epidemics on networks.Comment: 12 pages, 3 figure

The spatial resolution of epidemic peaks

The emergence of novel respiratory pathogens can challenge the capacity of key health care resources, such as intensive care units, that are constrained to serve only specific geographical populations. An ability to predict the magnitude and timing of peak incidence at the scale of a single large population would help to accurately assess the value of interventions designed to reduce that peak. However, current disease-dynamic theory does not provide a clear understanding of the relationship between: epidemic trajectories at the scale of interest (e.g. city); population mobility; and higher resolution spatial effects (e.g. transmission within small neighbourhoods). Here, we used a spatially-explicit stochastic meta-population model of arbitrary spatial resolution to determine the effect of resolution on model-derived epidemic trajectories. We simulated an influenza-like pathogen spreading across theoretical and actual population densities and varied our assumptions about mobility using Latin-Hypercube sampling. Even though, by design, cumulative attack rates were the same for all resolutions and mobilities, peak incidences were different. Clear thresholds existed for all tested populations, such that models with resolutions lower than the threshold substantially overestimated population-wide peak incidence. The effect of resolution was most important in populations which were of lower density and lower mobility. With the expectation of accurate spatial incidence datasets in the near future, our objective was to provide a framework for how to use these data correctly in a spatial meta-population model. Our results suggest that there is a fundamental spatial resolution for any pathogen-population pair. If underlying interactions between pathogens and spatially heterogeneous populations are represented at this resolution or higher, accurate predictions of peak incidence for city-scale epidemics are feasible

Real-time numerical forecast of global epidemic spreading: Case study of 2009 A/H1N1pdm

Background Mathematical and computational models for infectious diseases are increasingly used to support public-health decisions; however, their reliability is currently under debate. Real-time forecasts of epidemic spread using data-driven models have been hindered by the technical challenges posed by parameter estimation and validation. Data gathered for the 2009 H1N1 influenza crisis represent an unprecedented opportunity to validate real-time model predictions and define the main success criteria for different approaches. Methods We used the Global Epidemic and Mobility Model to generate stochastic simulations of epidemic spread worldwide, yielding (among other measures) the incidence and seeding events at a daily resolution for 3,362 subpopulations in 220 countries. Using a Monte Carlo Maximum Likelihood analysis, the model provided an estimate of the seasonal transmission potential during the early phase of the H1N1 pandemic and generated ensemble forecasts for the activity peaks in the northern hemisphere in the fall/winter wave. These results were validated against the real-life surveillance data collected in 48 countries, and their robustness assessed by focusing on 1) the peak timing of the pandemic; 2) the level of spatial resolution allowed by the model; and 3) the clinical attack rate and the effectiveness of the vaccine. In addition, we studied the effect of data incompleteness on the prediction reliability. Results Real-time predictions of the peak timing are found to be in good agreement with the empirical data, showing strong robustness to data that may not be accessible in real time (such as pre-exposure immunity and adherence to vaccination campaigns), but that affect the predictions for the attack rates. The timing and spatial unfolding of the pandemic are critically sensitive to the level of mobility data integrated into the model. Conclusions Our results show that large-scale models can be used to provide valuable real-time forecasts of influenza spreading, but they require high-performance computing. The quality of the forecast depends on the level of data integration, thus stressing the need for high-quality data in population-based models, and of progressive updates of validated available empirical knowledge to inform these models

Dengue epidemics and human mobility

In this work we explore the effects of human mobility on the dispersion of a vector borne disease. We combine an already presented stochastic model for dengue with a simple representation of the daily motion of humans on a schematic city of 20x20 blocks with 100 inhabitants in each block. The pattern of motion of the individuals is described in terms of complex networks in which links connect different blocks and the link length distribution is in accordance with recent findings on human mobility. It is shown that human mobility can turn out to be the main driving force of the disease dispersal.Comment: 24 pages, 13 figure

The effect of travel restrictions on the spread of a moderately contagious disease

BACKGROUND: Much research in epidemiology has been focused on evaluating conventional methods of control strategies in the event of an epidemic or pandemic. Travel restrictions are often suggested as an efficient way to reduce the spread of a contagious disease that threatens public health, but few papers have studied in depth the effects of travel restrictions. In this study, we investigated what effect different levels of travel restrictions might have on the speed and geographical spread of an outbreak of a disease similar to severe acute respiratory syndrome (SARS). METHODS: We used a stochastic simulation model incorporating survey data of travel patterns between municipalities in Sweden collected over 3 years. We tested scenarios of travel restrictions in which travel over distances >50 km and 20 km would be banned, taking into account different levels of compliance. RESULTS: We found that a ban on journeys >50 km would drastically reduce the speed and geographical spread of outbreaks, even when compliance is < 100%. The result was found to be robust for different rates of intermunicipality transmission intensities. CONCLUSION: This study supports travel restrictions as an effective way to mitigate the effect of a future disease outbreak

Evaluating the Combined Effectiveness of Influenza Control Strategies and Human Preventive Behavior

Control strategies enforced by health agencies are a major type of practice to contain influenza outbreaks. Another type of practice is the voluntary preventive behavior of individuals, such as receiving vaccination, taking antiviral drugs, and wearing face masks. These two types of practices take effects concurrently in influenza containment, but little attention has been paid to their combined effectiveness. This article estimates this combined effectiveness using established simulation models in the urbanized area of Buffalo, NY, USA. Three control strategies are investigated, including: Targeted Antiviral Prophylaxis (TAP), workplace/school closure, community travel restriction, as well as the combination of the three. All control strategies are simulated with and without regard to individual preventive behavior, and the resulting effectiveness are compared. The simulation outcomes suggest that weaker control strategies could suffice to contain influenza epidemics, because individuals voluntarily adopt preventive behavior, rendering these weaker strategies more effective than would otherwise have been expected. The preventive behavior of individuals could save medical resources for control strategies and avoid unnecessary socio-economic interruptions. This research adds a human behavioral dimension into the simulation of control strategies and offers new insights into disease containment. Health policy makers are recommended to review current control strategies and comprehend preventive behavior patterns of local populations before making decisions on influenza containment