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Natural Human Mobility Patterns and Spatial Spread of Infectious Diseases Natural Human Mobility Patterns and Spatial Spread of Infectious Diseases The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Belik, Vitaly, Theo Geisel, and Dirk Brockmann. “Natural Human Mobility Patterns and Spatial Spread of Infectious Diseases.” Physical Review X 1, no. 1 (August 2011). As Published http://dx.doi.org/10.1103/PhysRevX.1.011001 Publisher American Physical Society Version Final published version Citable link http://hdl.handle.net/1721.1/89013 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW X 1, 011001 (2011) Natural Human Mobility Patterns and Spatial Spread of Infectious Diseases Vitaly Belik,1,* Theo Geisel,1,2 and Dirk Brockmann3,4 1Max Planck Institute for Dynamics and Self-Organization, Go¨ttingen, Germany 2Faculty of Physics, University of Go¨ttingen, Go¨ttingen, Germany 3Northwestern Institute on Complex Systems, Northwestern University, Evanston, Illinois, USA 4Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois, USA (Received 13 October 2010; revised manuscript received 23 May 2011; published 8 August 2011) We investigate a model for spatial epidemics explicitly taking into account bidirectional movements between base and destination locations on individual mobility networks. We provide a systematic analysis of generic dynamical features of the model on regular and complex metapopulation network topologies and show that significant dynamical differences exist to ordinary reaction-diffusion and effective force of infection models. On a lattice we calculate an expression for the velocity of the propagating epidemic front and find that, in contrast to the diffusive systems, our model predicts a saturation of the velocity with an increasing traveling rate. Furthermore, we show that a fully stochastic system exhibits a novel threshold for the attack ratio of an outbreak that is absent in diffusion and force of infection models. These insights not only capture natural features of human mobility relevant for the geographical epidemic spread, they may serve as a starting point for modeling important dynamical processes in human and animal epidemiology, population ecology, biology, and evolution. DOI: 10.1103/PhysRevX.1.011001 Subject Areas: Complex Systems, Interdisciplinary Physics, Statistical Physics The geographic spread of emergent infectious diseases, diffusion processes on networks of metapopulations. The epitomized by the 2009 H1N1 outbreak and subsequent key assumptions of diffusive transport [Fig. 1(a)] are that pandemic [1], the worldwide spread of SARS in 2003 (a) individuals behave identically, (b) movements are sto- [2,3], and recurrent outbreaks of influenza epidemics chastic, (c) spatial increments are local and as consequence [4–6], is determined by a combination of disease relevant individuals eventually visit every location in the system. human interactions and mobility across multiple spatial Although it is intuitively clear that these assumptions are scales [7]. While infectious contacts yield local outbreaks idealizations and in conflict with everyday experience, the and proliferation of a disease in single populations, multi- difficulty is how to refine them when data on mobility is scale human mobility is responsible for spatial propagation lacking, insufficient, or incomplete. Fortunately, a series of [8]. Therefore, a prominent lineage of mathematical mod- recent studies [16–19] substantially advanced our knowl- els has evolved that is based on reaction-diffusion dynam- edge on multiscale human mobility. An important discov- ics [9,10] in which the combination of local exponential ery that emerged from these studies are individual mobility growth and diffusive dispersal captures qualitative aspects networks, i.e., individuals typically only visit a limited of observed dynamics. Typically these systems exhibit number of places frequently, predominantly performing constant velocity epidemic wave fronts. A related class commutes between home and work locations and possibly of phenomenological models is based on the concept of a few other locations. Consequently, individuals or groups an effective force of infection across distance and thus does of individuals exhibit spatially constrained movement pat- not require explicit modeling of dispersal [11,12]. terns despite their potentially high mobility rate. It has Although more sophisticated models [2,4,13,14] have remained elusive how this novel and important empirical been developed to describe the dynamics of recent emer- insight on individual mobility networks can be reconciled gent epidemics such as the H1N1 pandemic or SARS, with epidemiological models, to what extent it may impact taking into account long distance travel and multiscale spatial disease dynamics, and how it may alter spreading mobility networks [15], the majority of these models are scenarios and predictions promoted by ordinary reaction- still based on the interplay of local reaction kinetics and diffusion models, in which mobile hosts can reach every location in the system. In this paper we demonstrate how individual mobility *belik@mit.edu Present address: Massachusetts Institute of Technology, networks can be incorporated into a class of models for Department of Civil and Environmental Engineering, spatial disease dynamics, and address the questions of Cambridge, MA, USA. whether and how significantly the existence of spatially constrained individual mobility networks impacts on key Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- features of disease dynamics. To this end we investigate a bution of this work must maintain attribution to the author(s) and model that explicitly accounts for the bidirectional mobil- the published article’s title, journal citation, and DOI. ity of individuals between their unique base location 2160-3308=11=1(1)=011001(5) 011001-1 Published by the American Physical Society VITALY BELIK, THEO GEISEL, AND DIRK BROCKMANN PHYS. REV. X 1, 011001 (2011) populations is diffusive dispersal among those populations defined by hopping rates wnm > 0 from population m to n, which yields a metapopulation reaction-diffusion system, i.e., for infected and susceptible individuals, X s @tIn ¼ SnIn=Nn À In þ ðwnmIm À wmnInÞ; X m s (2) @tSn ¼SnIn=Nn þ ðwnmSm À wmnSnÞ; m s where Nn is the number of individuals in population n FIG. 1. Metapopulational mobility models. Patches and arrows in diffusive equilibrium. Travel rates !mn are usually represent individual populations and travel. (a) Diffusive dis- s estimated by gravitylike laws [23]. Nn is determined by persal: indistinguishable individuals travel randomly between Ns=Ns ¼ w =w different locations governed by the set of transition rates !nm. detailed balance, i.e., n m nm mn and the conser- (b) Dispersal capturing individual mobility patterns: individuals vationP of the number of individuals in the metapopulation k k N ¼ Ns with label k possess travel rates !mk and !km at which they m m. These types of models have been employed travel from their base location k to connected locations m and in numerous recent studies [2,14,24,25]. The tight connec- back. tion to spatially continuous reaction-diffusion systems is revealed for the special case of a linear grid of populations placed at regular intervals l at positions xm ¼ ml and (e.g. their home) and a small set of other locations w ¼ [Fig. 1(b)]. The entire population is therefore represented mobility between neighboring populations, i.e., nm ! þ ! by a set of overlapping individual mobility networks asso- nÀ1;m nþ1;m which yields 2 2 ciated with each base location, see, e.g., [20–22]. We focus @tj ¼ js À j þ D@xj; @ts ¼js þ D@xs; (3) on the analysis of epidemics on regular lattices and com- s s 2 in which jðx; tÞ¼In=Nn, sðx; tÞ¼Sn=Nn, and D ¼ l !. plex metapopulation networks and systematically compare Equation (3) is related to a classic form of a Fisher equation the dynamics to reaction-diffusion systems as well as the which has been deployed in mathematical epidemiology force of infection models. In conflict with reaction- [9,10]. For sufficiently localized initial conditions this diffusion models, in which front velocities increase with systems exhibits traveling waves with speed travel rates unboundedly, we show that our model predicts qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi a saturation of wave front velocities, a direct consequence c ¼ 2 Dð1 À =Þ !: (4) of the rank of locations in individual mobility patterns. This suggests that estimates for propagation speeds may Note that this velocity increases monotonically with the have been considerably overestimated in the past. mobility rate !. Furthermore, we analyze a fully stochastic model to In order to account for individual mobility patterns we show that both, in regular lattices as well as complex propose the following generalization: individuals possess metapopulation networks, the global outbreak of a disease two indices n and k, characterizing their current and their is determined by a novel type of threshold for the attack base location, respectively. The dispersal dynamics is gov- ratio of the disease which depends on the characteristic erned by a Markov process time spent at distant locations. Finally, we show that in the !k k mn k limit of low and high
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