Natural Human Mobility Patterns and Spatial Spread of Infectious Diseases

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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 *[email protected] 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 In þ ðwnmIm wmnInÞ; X m s (2) @tSn ¼SnIn=Nn þ ðwnmSm 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- n1;m nþ1;m which yields 2 2 ciated with each base location, see, e.g., [20–22]. We focus @tj ¼ js 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 =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 travel rates our model agrees with Xn Ð Xm; (5) k reaction-diffusion models and the direct force of the in- !nm fection class of models, respectively. where X stands for each infectious state S, I, and R. M m Consider a system of populations labeled and Equation (5) implies that individuals with base location k assume that in each an epidemic outbreak can be described k possess a unique mobility rate wnm that determines how by a compartmental SIR model, i.e., they travel between locations n and m. This yields a gen- eralization of (2) Im þ Sm!2Im;Im!Rm; (1) X X k k m k k k k k @tIn ¼ Sn In In þ ð!nmIm !mnInÞ; where Sm, Im, and Rm label and quantify susceptible, Nn m m infected, and recovered individuals of population m. X X (6) @ Sk ¼ Sk Im þ ð!k Sk !k Sk Þ; Infections and recovery events occur at rates and , t n n n nm m mn n Nn m m respectively, with >. The number of individuals in k k population m is given by Nm ¼ Sm þ Im þ Rm, which, where In and Sn are the number of infected and susceptible however, is conserved only in the statistical sense at equi- individuals of type k that are currently located in popula- librium. This system of reactions yields the mean-field tion n. Nn is the total number of individuals at n, i.e., Nn ¼ P k k k k descriptions @tI ¼ IS=N I and @tS ¼IS=N.A kðIn þ Sn þ RnÞ.If!nm are k independent, we recover natural and plausible extension to a system of M coupled the reaction-diffusion case described above. In the

011001-2 NATURAL HUMAN MOBILITY PATTERNS AND SPATIAL ... PHYS. REV. X 1, 011001 (2011) following we consider the case of overlapping star-shaped the forward rate !þ. For a systematic comparison to the networks corresponding to commuting between base and ordinary reaction-diffusion system, we have to establish a k destination locations only. This implies that !nm ¼ 0 if relation between ! and ! in Eq. (4). For the sake of either k n or k m. We further assume the constant simplicity we consider the symmetric case, i.e., !þ ¼ k return rate wmk ¼ w for all k and m. Realistically we ! ¼ !, yielding have !n =! 1, implying individuals belonging to n pffiffiffiffiffiffiffiffi kn 2 6D! remain at their base most of the time. c ¼ : (9) If mobility rates are large compared to the rates associ- ð þ 3!Þ ated with the infection and recovery, i.e., !k , ! , mk Numerical simulations of a fully stochastic system nicely , the detailed balance condition is fulfilled for infected agree with the analytical predictions (Fig. 2). The essential and susceptible individuals separately and the last term in pffiffiffiffi feature of cð!Þ is its saturation as ! !1whereas c ! Eq. (6) vanishes in which case the above model is equiva- for reaction-diffusion systems. This effect is a direct con- lent to the remote-force-of-infection model—see also sequence of the spatial constraints imposed by individual [11,21]. mobility patterns, i.e., increasing the mobility rate ! yields In general, it is difficult, if not impossible, to extract the a higher frequency of travel events but is restricted to the set dynamic differences between the systems defined by of locations connected to the base location and thus is Eqs. (2) and (6) for an arbitrary metapopulation. Yet, it is universal and also holds for more complex metapopulation crucial to understand the key dynamic differences that topologies [26]. For high rates, the deviation between ordi- emerge when individual mobility patterns replace ordinary nary reaction-diffusion and bidirectional mobility increases diffusion. We therefore investigate the impact of individual without bounds. This is an important insight as the expres- mobility patterns in the instructive one-dimensional lattice sion for wave front speeds for reaction-diffusion have been case. We assume that only infected individuals can travel used in the past to estimate wave front speeds from mobility and that recovery is absent (SI model), i.e., ¼ 0 (relaxing rates and vice versa. Our results indicate that if constrained these two restrictions does not change the main results but mobility patterns and bidirectional movement patterns are clarifies the analysis). We denote the number of infected at work these estimates must be treated with particular care. individuals with base location n that are located on site n n Note that in the limit ! !1Eq. (7) reduces to the heuristic (i.e. their base) and sites n 1 by In and In , respectively. equation in Ref. [27]. At rates !þ and ! individuals leave and return to their A key question in epidemiological contexts concerns base. In order to obtain a spatially continuous description 2 conditions under which an epidemic outbreak propagates I ! Iðx lÞI lrI þ l I we approximate n1 2 or wanes. Outbreak dynamics are usually triggered by u ¼ I =N v ¼ and introduce relative concentrations n n , n crossing thresholds inherent in the system’s dynamics ðIþ þ IÞ=N n n leading to [28]. For example the basic reproduction number R0 of þ a disease, i.e., the expected number of secondary cases @tu ¼ ð1uvÞðuþvþDvÞþ! v2! u; þ (7) @tv ¼ 2! u! v; 5 where D ¼ l2=2. The function uðx; tÞ is the fraction of 4 SI individuals based on and located at x whereas vðx; tÞ is the fraction of individuals based on but not located at x. SI Note that we discarded here a third equation for w ¼ 3 SIR ðIþ IÞ=N independent from (7) with a solution SI (immobile S) vanishing at long times. This does not change the main 2 result (8). The nontrivial steady state is given by Front velocity, c ? þ ? þ þ u ¼ ! =ð2! þ ! Þ, v ¼ 2! =ð2! þ ! Þ with 1 u? þ v? ¼ 1 as expected. One of the key questions is if the system (7) exhibits 0 stable propagating waves as solutions and how their veloc- 0246810 ω ity depends on system parameters. The ansatz uðx; tÞ¼ Travel rate, u~ðx ctÞ v , and likewise for , leads to a stable solution with FIG. 2. Front velocity cð!Þ of the model defined by Eq. (6)as a velocity given by a function of mobility rate ! in comparison to ordinary reaction- pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi diffusion dynamics [Eq. (2)]. For stochastic simulations we used 2!þ Dð2 þ !=!þÞ 102 N ¼ 104 c ¼ : (8) a lattice with sites and individuals/site. Mean ð þ ! þ 2!þÞ velocities are indicated by red (bidirectional mobility) and blue (reaction-diffusion) symbols. Analytical results, i.e., þ Fixing ! , this implies that c ! 0 as ! ! 0. On the other Eqs. (9) and (4) are shown by dashed and dash-dotted lines, hand, if ! is small, the system is entirely determined by respectively.

011001-3 VITALY BELIK, THEO GEISEL, AND DIRK BROCKMANN PHYS. REV. X 1, 011001 (2011) caused by a single infected individual in a susceptible for a wide range of local population sizes (50–2000) the population represents such a threshold [8]. In the single attack ratio collapses onto a universal curve population SIR dynamics introduced above R0 ¼ =.If ð! =NÞ which suggests that the basic mechanism of R0 > 1, an outbreak occurs, otherwise the epidemic dies the invasion threshold and its functional dependence on out. In the extended metapopulation system with diffusive return rate ! can be understood theoretically. host movements a second threshold, known as the global To estimate the observed invasion threshold on a one- invasion threshold [24], is induced by the flux of individu- dimensional lattice for R0 * 1 we calculated the number als between populations, which must be sufficiently high of infected individuals that originate from an affected for an epidemic to spread throughout the metapopulation. location and can seed an outbreak in an unaffected one An interesting and important feature of the model we [28]. These seeders are approximately given by the total propose is the existence of another invasion threshold number of individuals entering the new location per unit which is determined by the return rate ! or equivalently time, i.e., N! multiplied by the typical time they remain by the time an individual spends outside the base location. active in the destination location. This time is given by the This finding is illustrated in Fig. 3, which depicts the attack inverse rate r of exiting the seeders class. Since two ratio as a function of ! for a bidirectional SIR epidemic processes (returning to the base location and recovery) on a lattice. The attack ratio is defined as the fraction of can trigger this exit, r ¼ þ ! . Therefore, the overall population affected during an epidemic. We N!=ð þ ! Þ. In the tree approximation the threshold fixed the entire interlocation flux at a value above the condition for an epidemic on a network with an average global invasion threshold ensuring a global outbreak in degree k is given by ðk 1ÞðR0 1Þ > 1 [24]. Thus, for the reaction-diffusion system. a one-dimensional lattice the threshold condition reads   This result is universal for the bidirectional system and ! !þ & 2NðR 1Þ 1 ; is independent of the particular topology. Insets (a) and 0 (10) (b) of Fig. 3 display simulation results for uncorrelated þ scale-free and Erdo˝s-Re´nyi networks. For low return rates where we kept the total interlocation flux ! ¼ 2! ! = ð2!þ þ !Þ constant (this relation and positivity of travel the attack ratio is close to unity. However, with growing þ values of the return rate, the attack ratio decreases steeply rates impose the restrictions: ! >!and !

ρ N=500 0 −1 0 1 2 ! 0.6 10 10 10 10 typically assumed by the overall particle flux . In the context of disease dynamics this implies that more efficient 1 (a) containment strategies could be potentially devised that do 0.4 not target the overall mobility but rather, modify mobility

Attack ratio, N=200 0.5 patterns in an asymmetric way. N=100 0.2 Note that the transition takes place only in a system with 0 −1 0 1 2 N=50 10 10 10 10 a finite number of agents per site; in the system with an infinite number of individuals this effect would disappear, 0 −1 0 1 2 10 10 10 10 as also confirmed in Fig. 3. Indeed from Eq. (10) for − N! Travel rate, ω /N ! the scaling ð! Þ follows—with an increas- ing number of individuals per site N, the threshold value of FIG. 3. Results of stochastic simulations. Attack ratio ! ð!=NÞ ! the return rate increases. as a function of the return travel rate for SIR Recently, an unprecedented amount of detailed informa- epidemics on a lattice with 102 sites with N agents/site. tion on human activity became available, requiring the Epidemic parameters: ¼ 1, ¼ 0:1. Insets: (a) ð! =NÞ for a scale-free network ( ¼ 1:5)of103 nodes populated revision of established models and the development of uniformly with hNi¼250=site, with kmin ¼ 5 and kmax ¼ 50; new more sophisticated ones. We investigate an epidemio- (b) ð!=NÞ for an Erdo˝s-Re´nyi network with 500 nodes and logical model explicitly incorporating such important prop- k ¼ 10. Results were averaged over 50 realizations. erties of human mobility as individual mobility networks Interlocation flux rate was h!i¼1. and frequent bidirectional movements between home and

011001-4 NATURAL HUMAN MOBILITY PATTERNS AND SPATIAL ... PHYS. REV. X 1, 011001 (2011) destination locations. It manifests surprising dynamical [13] L. A. Rvachev and I. M. Longini, A Mathematical Model features such as the existence of bounded front velocity for the Global Spread of Influenza, Math. Biosci. 75,3 and a novel propagation threshold, which are universal for (1985). any metapopulation topology. As more detailed data con- [14] V. Collizza, R. Pastor-Satorras, and A. Vespignani, tinue to become available our approach is a promising Reaction-Diffusion Processes and Metapopulation Models in Heterogeneous Networks, Nature Phys. 3, 276 foundation for further research. (2007). D. B. and V.B. acknowledge support from the [15] A. Vespignani, Predicting the Behavior of Techno-Social Volkswagen Foundation, D. B. acknowledges EU-FP7 Systems, Science 325, 425 (2009). [16] D. Brockmann, L. Hufnagel, and T. Geisel, The Scaling grant Epiwork. V.B. and D. B. would like to thank W. 439 Noyes and H. Schla¨mmer for invaluable comments. Laws of Human Travel, Nature (London) , 462 (2006). [17] M. C. Gonza´lez, C. A. Hidalgo, and A.-L. Baraba´si, Understanding Individual Human Mobility Patterns, Nature (London) 453, 779 (2008). [1] C. Fraser et al., Pandemic Potential of a Strain of Influenza [18] C. Song, Z. Qu, N. Blumm, and A.-L. Baraba´si, Limits of A (H1N1): Early Findings, Science 324, 1557 (2009). Predictability in Human Mobility, Science 327, 1018 [2] L. Hufnagel, D. Brockmann, and T. Geisel, Forecast and (2010). Control of Epidemics in a Globalized World, Proc. Natl. [19] C. Song, T. Koren, and P. Wang et al., Modelling the Acad. Sci. U.S.A. 101, 15124 (2004). Scaling Properties of Human Mobility, Nature Phys. 6, [3] V. Colizza, A. Barrat, and M. Barthelemy et al., 818 (2010). Predictability and Epidemic Pathways in Global [20] L. Sattenspiel and K. Dietz, A Structured Epidemic Model Outbreaks of Infectious Diseases: The SARS Case Study, Incorporating Geographic Mobility Among Regions, BMC Med. 5, 34 (2007). Math. Biosci. 128, 71 (1995). [4] D. Balcan, H. Hu, and B. Goncalves et al., Seasonal [21] M. J. Keeling and P. Rohani, Estimating Spatial Coupling Transmission Potential and Activity Peaks of the New in Epidemiological Systems: A Mechanistic Approach, Influenza A(H1N1): A Monte Carlo Likelihood Analysis Ecol. Lett. 5, 20 (2002). Based on Human Mobility, BMC Med. 7, 45 (2009). [22] J. Arino and P. van den Driessche, The Basic Reproduction [5] D. Balcan, B. Goncalves, and H. Hu et al., Modeling the Number in a Multi-City Compartmental Epidemic Model, Spatial Spread of Infectious Diseases: The GLobal Lect. Notes in Control and Information Science Vol. 294 Epidemic and Mobility Computational Model,J. Comp. (Springer, New York, 2003) p. 135. Sci. 1, 132 (2010). [23] V.V. Belik and D. Brockmann, Accelerating Random [6] D. Balcan, V. Colizza, and B. Goncalves et al., Multiscale Walks by Disorder, New J. Phys. 9, 54 (2007). Mobility Networks and the Spatial Spreading of Infectious [24] V. Colizza and A. Vespignani, Invasion Threshold in Diseases, Proc. Natl. Acad. Sci. U.S.A. 106, 21484 Heterogeneous Metapopulation Networks, Phys. Rev. (2009). Lett. 99, 148701 (2007). [7] S. Riley, Large-Scale Spatial-Transmission Models of [25] N. M. Ferguson, D. A. Cummings, and C. Fraser et al., Infectious Disease, Science 316, 1298 (2007). Strategies for Containing an Emerging Influenza [8] R. M. Anderson and R. M. May, Infectious Diseases of Pandemic in Southeast Asia, Nature (London) 437, 209 Humans (Oxford University Press, New York, 1991). (2005). [9] A. Kolmogorov, I. Petrovsky, and N. Piscounov, Study of [26] We assume here that individuals return to their base the Diffusion Equation with Growth of the Quantity of location k before traveling somewhere else. The initial Matter and its Application to a Biology Problem, Bull. de model defined by Eq. (5) allows us to explore more l’univ. d’e´tat a` Moscou, Se´r. internat., sect. A 1, 1 (1937), general mobility patterns by relaxing this assumption. k reprinted in Dynamics of Curved Fronts, edited by P. Pelce´ However, as long as the individual rates !nm are domi- and A. Libchaber (Academic Press, San Diego, 1988). nated by the dynamics that incorporate a base location, [10] R. A. Fisher, The Wave of Advance of Advantageous propagation speeds do not deviate significantly from the Genes, Ann. Eugenics 7, 355 (1937). model we investigate here. [11] S. Rushton and A. J. Mautner, The Deterministic Model of [27] U. Naether, E. B. Postnikov, and I. M. Sokolov, Infection a Simple Epidemic for More than One Community, Fronts in Contact Disease Spread, Eur. Phys. J. B 65, 353 Biometrika 42, 126 (1955). (2008). [12] T. J. Hagernaas, C. A. Donnelly, and N. M. Ferguson, [28] F. Ball, D. Mollison, and G. Scalia-Tomba, Epidemics with Spatial Heterogeneity and the Persistence of the Two Levels of Mixing, Ann. Appl. Prob. 7,46 Infectious Diseases, J. Theor. Biol. 229, 349 (2004). (1997).

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