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The scaling of human contacts and epidemic processes in metapopulation networks
We study the dynamics of reaction-diffusion processes on heterogeneous metapopulation networks where interaction rates scale with subpopulation sizes. We first present new empirical evidence, based on the analysis of the interactions of 13 million users on Twitter, that supports the scaling of human interactions with population size with an exponent γ ranging between 1.11 and 1.21, as observed in recent studies based on mobile phone data. We then integrate such observations into a reaction- diffusion metapopulation framework. We provide an explicit analytical expression for the global invasion threshold which sets a critical value of the diffusion rate below which a contagion process is not able to spread to a macroscopic fraction of the system. In particular, we consider the Susceptible-Infectious-Recovered epidemic model. Interestingly, the scaling of human contacts is found to facilitate the spreading dynamics. This behavior is enhanced by increasing heterogeneities in the mobility flows coupling the subpopulations. Our results show that the scaling properties of human interactions can significantly affect dynamical processes mediated by human contacts such as the spread of diseases, ideas and behaviors
Approximation of epidemic models by diffusion processes and their statistical inference
Multidimensional continuous-time Markov jump processes on
form a usual set-up for modeling -like epidemics. However,
when facing incomplete epidemic data, inference based on is not easy
to be achieved. Here, we start building a new framework for the estimation of
key parameters of epidemic models based on statistics of diffusion processes
approximating . First, \previous results on the approximation of
density-dependent -like models by diffusion processes with small diffusion
coefficient , where is the population size, are
generalized to non-autonomous systems. Second, our previous inference results
on discretely observed diffusion processes with small diffusion coefficient are
extended to time-dependent diffusions. Consistent and asymptotically Gaussian
estimates are obtained for a fixed number of observations, which
corresponds to the epidemic context, and for . A
correction term, which yields better estimates non asymptotically, is also
included. Finally, performances and robustness of our estimators with respect
to various parameters such as (the basic reproduction number), ,
are investigated on simulations. Two models, and , corresponding to
single and recurrent outbreaks, respectively, are used to simulate data. The
findings indicate that our estimators have good asymptotic properties and
behave noticeably well for realistic numbers of observations and population
sizes. This study lays the foundations of a generic inference method currently
under extension to incompletely observed epidemic data. Indeed, contrary to the
majority of current inference techniques for partially observed processes,
which necessitates computer intensive simulations, our method being mostly an
analytical approach requires only the classical optimization steps.Comment: 30 pages, 10 figure
Spreading processes in Multilayer Networks
Several systems can be modeled as sets of interconnected networks or networks
with multiple types of connections, here generally called multilayer networks.
Spreading processes such as information propagation among users of an online
social networks, or the diffusion of pathogens among individuals through their
contact network, are fundamental phenomena occurring in these networks.
However, while information diffusion in single networks has received
considerable attention from various disciplines for over a decade, spreading
processes in multilayer networks is still a young research area presenting many
challenging research issues. In this paper we review the main models, results
and applications of multilayer spreading processes and discuss some promising
research directions.Comment: 21 pages, 3 figures, 4 table
A Diffusion Approximation for an Epidemic Model
Influenza is one of the most common and severe diseases worldwide. Devastating epidemics actuated by a new subtype of the influenza A virus occur again and again with the most important example given by the Spanish Flu in 1918/19 with more than 27 million deaths. For the development of pandemic plans it is essential to understand the character of the dissemination of the disease. We employ an extended SIR model for a probabilistic analysis of the spatio-temporal spread of influenza in Germany. The inhomogeneous mixing of the population is taken into account by the introduction of a network of subregions, connected according to Germany's commuter and domestic air traffic. The infection dynamics is described by a multivariate diffusion process, the discussion of which is a major part of this report. We furthermore present likelihood-based estimates of the model parameters
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