PHYSICAL REVIEW E 88, 012809 (2013)

Epidemic fronts in complex networks with metapopulation structure

Jason Hindes,1,* Sarabjeet Singh,2 Christopher R. Myers,1,3 and David J. Schneider4,5

1Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York2Theoretical and Applied Mechanics, Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York

3Institute of Biotechnology, Cornell University, Ithaca, New York4Plant-Microbe Interactions Research Unit, Robert W. Holley Center for Agriculture and Health, Agricultural Research Service,

United States Department of Agriculture, Ithaca, New York5Department of Plant Pathology and Plant-Microbe Biology, Cornell University, Ithaca, New York

(Received 17 April 2013; published 15 July 2013)

Infection dynamics have been studied extensively on complex networks, yielding insight into the effects of het-erogeneity in contact patterns on disease spread. Somewhat separately, metapopulations have provided a paradigmfor modeling systems with spatially extended and “patchy” organization. In this paper we expand on the use ofmultitype networks for combining these paradigms, such that simple contagion models can include complexity inthe agent interactions and multiscale structure. Using a generalization of the Miller-Volz mean-field approximationfor susceptible-infected-recovered (SIR) dynamics on multitype networks, we study the special case of epidemicfronts propagating on a one-dimensional lattice of interconnected networks—representing a simple chain ofcoupled population centers—as a necessary first step in understanding how macroscale disease spread dependson microscale topology. Applying the formalism of front propagation into unstable states, we derive the effectivetransport coefficients of the linear spreading: asymptotic speed, characteristic wavelength, and diffusion coeffi-cient for the leading edge of the pulled fronts, and analyze their dependence on the underlying graph structure. Wealso derive the epidemic threshold for the system and study the front profile for various network configurations.

DOI: 10.1103/PhysRevE.88.012809 PACS number(s): 89.75.Fb, 64.60.aq, 87.19.X!, 87.23.Cc

I. INTRODUCTION

Network theory has proven a powerful framework forstudying the effects of randomness and heterogeneity on thedynamics of interacting agents with nontrivial connectivitypatterns [1]. One of the most important applications ofthis work is to the spread of infectious diseases amonghuman populations, where the interaction structure is highlycomplex, showing salient features such as power-law degreedistributions, small average path lengths, and modularity [2,3].Various models have been proposed, primarily with randomgraph configuration, that incorporate these complex featureswhile remaining theoretically tractable. Within the context ofdisease dynamics, graph nodes are generally taken to representindividuals, and edges to represent interactions between them,through which infection can spread. Both deterministic andstochastic infection dynamics have been studied on networksas well as bond percolation for the associated branchingprocess [4–8]. How various thermodynamic quantities ofinterest, such as the total proportion infected, the epidemic(percolation) threshold, and the distribution of small outbreaksizes, depend upon network topology is of great interest.

Often these approaches disregard the multiscale organi-zation of many real systems, in which agents can be mostnaturally thought of as partitioned into densely connectedcommunities with sparser coupling among neighboring com-munities. In some cases, it may be useful to conceptualizethe topology as a network of networks, where agent-to-agent interactions and community-to-community interactionsare both useful representations depending on the scale ofresolution [9]. The latter has been successfully developed

*jmh486@cornell.edu

in ecology, with a network of interconnected populationsreferred to as a “metapopulation” [10,11]. This frameworkis very useful in studying large-scale propagation of dis-eases where most infection transmission occurs in localizedregions, but can be transported on larger scales by themobility of individuals, traveling among population centers[12]. However, most metapopulation models assume thatpopulations are fully mixed, with no inherent complexityin the connectivity between agents. Much less understoodis how the multiscale structure of agent interactions affectsthe larger-scale propagation of infectious processes throughinterconnected networks [13,14].

In this paper, we expand on a possible avenue for addressingthis question using a multitype generalization of randomgraphs with simple, metalevel topology [9,15], and a dynami-cal mean-field theory for the SIR infection model in multitypeconfiguration model networks. Putting these together, weanalyze the average infection dynamics and propagating frontprofile on a simple metapopulation composed of coupledpopulation centers on a one-dimensional lattice and calculatethe phenomenological transport properties of the system asfunctions of the underlying network’s degree distributions.Our results are compared to stochastic simulations of theinfection kinetics on various networks and found to be ingood agreement in the thermodynamic limit. Broadly, wepresent this work as an illustration of how well-developedideas from different areas of statistical physics and ecologycan be naturally combined.

II. MULTITYPE CONFIGURATION MODEL NETWORKS

In order to incorporate relevant node attribute informationinto our network models, (generically applicable for suchthings as age, sex, ethnicity, and place of residence), we

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HINDES, SINGH, MYERS, AND SCHNEIDER PHYSICAL REVIEW E 88, 012809 (2013)

use a generalization of configuration model random graphs,wherein nodes are assigned a type from an arbitrary set of Mpossible types and a degree to each type from an arbitrary jointdistribution for degree types, Pi(k1,k2, . . . ,kM ) = Pi("k), withdegree kj denoting the number of connections to nodes of typej [1,9,15]. Additionally, nodes of type i occupy a fractionof the total network wi , where

!i wi = 1. Following the

configuration model prescription, we consider graphs chosenuniformly at random from the ensemble of possible graphswith the prescribed degree distributions and self-consistentedge constraint: wi

!"k kjPi("k) = wj

!"k# k#

iPj ( "k#),$(i,j )[1,15,16].

From this formalism, a variety of quantities can bedescribed compactly using generating functions [1,17]. Thegenerating function for the probability of a randomly selectednode of type i to have degree "k, is given by

Gi("x) ="

"k

Pi("k)M#

l=1

xkl

l : (1)

written as a power series in "x, an auxiliary variable definedover the unit interval, with expansion coefficients equal to therespective probabilities. Moments of the degree distributionscan be represented simply as derivatives of the correspondinggenerating function. For example, the average degree of a typei node to a type j node is

"

"k

kjPi("k) = !xjGi("x)|"1 % &kj 'i . (2)

Since node interactions occur along edges, an importantquantity in network models is the excess degree: the numberof neighbors a node has which can be reached by selecting arandomly chosen edge, and not including the neighbor on theend of the selected edge. For a multitype configuration modelnetwork, the probability that a randomly chosen edge from atype i node leads to a type j node with degree "k is proportionalto kiPj ("k), and thus the probability for the correspondingexcess degree is generated by !xi

Gj ("x)/!xiGj ("x)|"1 [15], with

average degree to type l nodes,

&kl'i!j = !xl!xi

Gj ("x)|"1!xi

Gj ("x)|"1= &klki'j

&ki'j! "il . (3)

By construction, this framework lacks two-point correlations,in which the excess degree distributions depend on the degreesof both nodes sharing an edge [3].

III. MILLER-VOLZ MEAN-FIELD SIRIN MULTITYPE NETWORKS

In this report we consider simple dynamics for dis-ease spread: the susceptible-infected-recovered (SIR) model,wherein each individual is assigned a disease state, Y ({S,I,R}, and may undergo reactions to other states dependingon its state and the state of its neighbors. In this model, if anode of type i is susceptible and has a single infected neighborof type j , then it will change its state to infected with aconstant probability per unit time #ji . Likewise, an infectednode of type i will recover with a constant probability per unittime $i . Since the underlying dynamics is a continuous time

Markov process, a complete analysis would describe the fullprobability distribution for all system trajectories. Howeverfor our purposes, it will be sufficient to focus on the behaviorof extensive outbreaks (i.e., those which scale with the systemsize), the average dynamics of which can be derived in thelimit when the number of nodes tends to infinity, by applyinga generalization of the mean-field technique developed byMiller and Volz [18]. Below, we follow the basic structureof the derivations presented in Refs. [20,21].

In the thermodynamic limit, configuration model randomgraphs are locally treelike [16], which by construction allowsthem to satisfy many of the generic criteria for the applicabilityof mean-field theory assumptions [22]. In our case, we assumethat nodes are differentiated by their degree and disease statealone and that susceptible nodes feel a uniform force ofinfection along every edge, related to the average number ofedges connecting susceptible and infected nodes at any giventime in the network: a Curie-Weiss type approximation [23].Furthermore, from the perspective of susceptible nodes, allinfection attempts along different edges can be treated asuncorrelated—a consequence of the local treelike property[5,16,20]—and thus we assume that the states of neighbors ofsusceptible nodes are effectively independent.

Let the probability that a node of type j has not transmittedthe infection to a node of type i along a randomly chosen i-jedge, be %ij . This quantity is interpretable as the complementof the average cumulative hazard function along such edges.Given %ij , it follows that the fraction of susceptible nodes oftype i at time t is

Si(t) ="

"k

Pi("k)M#

j=1

%kj

ij (t)

= Gi[%i1(t),%i2(t), . . . ,%iM (t)]

% Gi["%i(t)]. (4)

The fractions of infected and recovered nodes of type ifollow from probability conservation, Si + Ii + Ri = 1, anda constant recovery rate for infected nodes $i

dIi

dt= !d "%i

dt· "!Gi("x)

$$$$"%i

! $iIi ,dRi

dt= $iIi , (5)

with the total fraction of susceptible nodes

S ="

i

wiGi("%i) % "w · "G(% ). (6)

The central probability and order parameter, %ij , can besubdivided into three compartments depending on the diseasestate of the terminal node j ,

%ij = %Sij + % I

ij + %Rij , (7)

and its dynamics determined by tracking the fluxes amongthese compartments. Since % can only change when an infectednode transmits the disease, the rate at which %ij changes isequal to the rate at which a corresponding neighbor infects,and therefore d%ij = !#ji%

Iij dt . Similarly, since %R can only

change if an infected node recovers, the rate at which %Rij

changes is equal to the rate at which a corresponding neighborrecovers, and thus d%R

ij = $j%Iij dt . Lastly, the probability that

a type j neighbor of a type i node has not transmitted and

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is susceptible, %Sij , is simply the probability that the corre-

sponding neighbor is susceptible. Because this neighbor couldnot have been infected along any of its other edges and hasexcess degree distribution generated by !xi

Gj ("x)/!xiGj ("x)|"1,

it follows that %Sij = !xi

Gj ("x)| "%j/!xi

Gj ("x)|"1. Combining thelatter with the two flux relations and the initial conditions (7),%ij (0) = 1 and %R

ij (0) = 0, we find

d%ij

dt= #ji

%!xi

Gj ("x)|"%j

!xiGj ("x)|"1

! %ij

&+ $j (1 ! %ij ). (8)

These M2, first-order, and coupled ODEs, % = F (%), definethe full system’s approximate mean dynamics, and form thebasis of our subsequent analysis. For a more detailed derivationof the analogous results for the special case of a single typenetwork, see [18–21].

The steady state is given by the fixed point of (8),

%ij = (1 ! Tji) + Tji

!xiGj ("x)| "%j

!xiGj ("x)|"1

, (9)

which upon substitution into (4), gives the cumulative infec-tion, P = 1 ! S, at equilibrium (i.e., the final epidemic size),with Tji = #ji/(#ji + $j ) the corresponding bond percolationprobability, or transmissibility [15]. This can have a nontrivialsolution corresponding to the existence of extensive outbreaks,if the disease-free state, %ij = 1 $(i,j ), is unstable. Thethreshold or phase transition, which signifies the region inparameter space that separates the epidemic and nonepidemicphases, can be obtained through a stability analysis of thedisease-free state, where the eigenvalue of the Jacobian for (8)with the largest real part, is real and vanishes when

det(N ! I ) = 0, with N(i,j )(k,l) = Tji"jk&kl'i!j (10)

an M2xM2 matrix [24]. Similar results for the equilibriumproperties are derivable from a multitype bond percolationapproach [15].

IV. FRAMEWORK FOR MULTISCALE NETWORKS

Of interest to us are systems where type structure adds anadditional scale of relevant topology, and not just demographiccomplexity [9,15]. For instance, we can apply the multitypenetwork formalism to a simple model for a metapopulationby affiliating population centers with node types and couplingamong populations with edges connecting their constituentnodes. In this way, a complex topology can be encodedon a microscale with a macroscale adjacency matrix, A,describing which populations are directly connected throughnode interactions [9]. We envisage example systems whereA describes the connectivity among urban centers, such ascities, towns, or villages, facilitated by roads or airlines. Byconceptualizing the topology in this manner, we can study thephenomenology of infection propagation among populationcenters and describe how the propagation properties depend onthe underlying connectivity patterns. A schematic is shown inFig. 1(a) for a simple system with the pertinent structure. Morebroadly, we advance this approach as an avenue for combiningthe frameworks of network theory, metapopulations, and frontpropagation, which will be particularly useful if the interactiontopology is coherent after some level of coarse graining.

FIG. 1. (Color online) (a) A schematic of SIR dynamics on ametapopulation, where infection spreads along edges connectingnodes of various types at the finest scale (shown with integer,population labels), and the macroscale topology identifies whichpopulations are connected through agent interactions. (b) An exampleof this framework, in which the macroscale topology takes the formof a one-dimensional (1D) lattice. In Sec. V, we focus on a simplecase with configuration model construction, where each site has anidentical degree distribution, specifying the probability of having agiven number of internal (0), right (+), and left (!) external edges(shown above with labels for site i), and calculate the velocity, v),growth rate, s), and spatial decay rate, q), for infection fronts.

V. 1D LATTICE METAPOPULATION DYNAMICS

To illustrate this approach, we consider a special case ofthe above where the macroscale topology is an infinite one-dimensional (1D) lattice, M * +, in which agents interactwith other agents of the same type and agents of neighbor-ing types, Anj = ("j,n + "j,n+1 + "j,n!1) [12]. If infection isstarted at a single site (e.g., site 0) in a fully susceptiblesystem, a strict directionality applies: in order for site i to beinfected, sites i ! 1,i ! 2, . . . must be infected first. In such acase, we expect a well-defined infection front to propagatethrough the lattice. In keeping with the above, we focus

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on an effective force of infection model among populationswith static configuration networks having prescribed degreedistributions, a generalization of the paradigmatic, spatialSIR model in one dimension, where the assumption of wellmixed populations is relaxed to include complexity in agentinteractions [11,25]. A schematic is shown in Fig. 1(b).

Since each node has three edge types, the mean equa-tions of motion describe a three-component field, "%n(t) %(%nn,%nn+1,%nn!1) % [%0

n (t),%+n (t),%!

n (t)], where (0), (+), and(!) denote internal, right-external, and left-external edges,at the corresponding site. For simplicity, homogeneity isassumed, with #, w, $ , and G all uniform, reducing the fieldequations to

d%0n

d&=

'1 ! %0

n

(+ T

%G0("%n)

G0("1)! 1

&

(11)d%±

n

d&= (1 ! %±

n ) + T

%G,("%n±1)

G,("1)! 1

&,

where the time, & , is measured in units of 1/(# + $ ), andthe subscript in G denotes a partial derivative with respectto the corresponding variable. For edge number consistency,G+("1) = G!("1), but in general we allow for other asymmetriesin the degree distributions.

A. Dispersion relation and transport coefficients

To understand the spatiotemporal dynamics (11), wefirst quantify how perturbations away from the unstablestate propagate by linearizing the dynamics around thedisease-free equilibrium, "%n(t) = "1 ! "'n(t), and decouplingthe perturbations into basis modes using the inverse discreteFourier transform, "'n(t) = 1

M

!M!1(=0 "'((t)ei(2)(n/M), (IDFT).

The dispersion relation can be found by substituting the IDFTinto (11) and using the basis properties of orthogonality andcompleteness. In the limit M * + the site perturbationsapproach the integral, "'n(t) = 1

2)

) 2)

0 "'(k)ei[kn!*(k)t]dk.With this prescription, we find the dispersion relation takes

the form of a cubic, characteristic equation

det%

Ke(q) ! 1+s(q)T

I

&= 0, with

Ke(q) =

*

+++,

&k20 '

&k0' ! 1 &k0k+'&k0'

&k0k!'&k0'

&k!k0'&k!' e!q &k!k+'

&k!' e!q' &k2

!'&k!' !1

(e!q

&k+k0'&k+' eq

' &k2+'

&k+' !1(eq &k+k!'

&k+' eq

-

.../, (12)

which for convenience, is written in terms of s and q, where* = is and k = iq [Fig. 1(b)]. Interestingly, this methodreveals a generalization of the average excess degree matrix,Ke(0), whose elements are found by selecting a randomlychosen edge in a particular direction, and counting the averagenumber of reachable neighbors of a particular type, for theinterconnected network system, Ke(q), which incorporates therelative states of adjacent sites on the lattice for each mode q.We expect this operator to emerge in similar problems oninterconnected networks.

Combining the above with the behavior of infection nearthe phase transition, where there is no exponential growth intime and each site has the same field value, "% (n) = "1: s * 0and q * 0 (12), we find a simple condition for the criticaltransmissibility Tc

Tc = 1+k

m(0), (13)

where +km(q) is the maximum eigenvalue of Ke(q), with +k

m(0)corresponding to Ke(0). Because the addition of externaledges increases the spreading capacity of the disease, thecritical transmissibility in the coupled system is less than theuncoupled case, implying that transport-mediated infectionsfrom neighboring sites can sustain epidemics even whenindividual populations on their own cannot [13,14].

Also from the dispersion relation, we can find the asymp-totic transport coefficients for rightward moving disturbancesby making a standard saddle-point approximation of the per-turbations’ integral representation in Fourier space: expandingthe integrand around its dominant contribution, k), in thecomoving frame, , = n ! v)t ,

ei[kn!*(k)t] - eik,eit(kv)!*(k))! d*dk

|k) (k!k)))e!it(k!k) )2

2d2*

dk2 |k)

and taking the infinite time limit while enforcing approximateconstancy with no exponential growth and , finite, where v)

is the asymptotic speed at which perturbations to the unstablestate propagate [26]. This procedure uncovers an exponentialmoving pulse for the leading edge of the infection profile witha diffusive correction [26]

1 ! % - e!q),e!, 2/4D)t

.D)t

, (14)

where q), v), and D) satisfy the saddle-point relations

v) = ds

dq

$$$$q)

= s(q))q) = T

d+km

dq

$$$$q)

= !1 + T +km(q))

q) (15)

and D) = 12

d2s

dq2

$$$$q)

= T

2d2+k

m

dq2

$$$$q)

, (16)

giving a transcendental equation for q), which correspondsto the selection of the minimum velocity, v), from the front’sspectrum of exponential modes, est!qn, with velocities, v(q) =s(q)/q [Fig. 1(b)] [25,26]. If multiple solutions exist for theminimum, the fastest solution is selected [26].

When the average excess degree matrix is irreducible (thedomain of interest to us), the dominant growth exponentfor each q is real and corresponds to a positive and uniqueeigenvector (12) [24], and thus we expect the same selectedvelocity for all field components, "%(n) [27]. Furthermore, thefields propagate in this regime with approximate proportions"1 ! '(t) "Q(q),s)), where "Q(q),s)) is the corresponding modeof Ke(q)) (12). The characteristic wavelength, 1/q), is relatedto the asymptotic size of the front’s leading edge, and divergesnear the phase transition. The diffusion coefficient, D), givesthe effective widening of the mean-field pulse in the comovingframe and is proportional to the largest finite-size correctionto v) in the limit where the number of nodes at each site tendsto infinity.

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In order to uncover the principal dependencies of thetransport coefficients, we study (12)–(16) near the phasetransition, where the power series expansion for the dispersionrelation is a convenient representation; the latter is foundby substituting [s(q) + 1]/T = a + bq + c

2q2 + · · · into (12),and equating powers in q. When s) and q) are small in thevicinity of Tc, the expansion can be truncated at low order,giving a Fisher-Kolmogorov-like dispersion relation with theapproximate scaling

s) -0T +k

m(0) ! 11

v) -0T +k

m(0) ! 11 1

2 D) 12

q) -0T +k

m(0) ! 11 1

2 D)! 12

D)

T +km(0)

- " (17)

" =&k!k0'&k0k+'

&k!'&k0' + &k!k+'&k!'

'+k

m(0) ! &k20 '

&k0' +1(

0+k

m(0) ! +k2(0)

10+k

m(0) ! +k3(0)

1 ,

where +k2(0) and +k

3(0) are the subdominant eigenvalues ofKe(0). In this regime, we find an effective reaction-diffusionbehavior with the generic dependence of the shape and speedof the propagating front’s leading edge on the reproductivenumber T +k

m(0) (a product of the spreading capacity alongedges and the magnitude of topological fluctuations) and onthe normalized diffusion coefficient ": measuring the relativestrength of connection between lattice sites (17). We see thatthe effective reaction rate is equal to the distance from the phasetransition, T +k

m(0) ! 1, and that all coefficients grow from zerowith this distance, except for D), which varies discontinuouslythrough Tc. Furthermore, the normalized diffusion coefficientincreases from zero with &k!k0'&k0k+' and &k!k+' – thecorrelation moments of the degree distribution which encodethe propensity for transport from the i , 1 site to the i ± 1site (both of which cannot be zero, otherwise epidemics arelocally confined), and with the viability of subdominant modesto support growth. In general, we find that as " increases v) andD) increase, q) decreases, and s) remains constant, implyingfaster transport and greater similarity among sites, as moreedge-type pairs allow for traversing the lattice, but with littlechange in the growth exponent.

The above demonstrates the typical trend for these mod-els, that the front dynamics is strongly influenced by thejoint degree distribution’s second moments (i.e., the relevantexcess degree properties are generally amplified by correlationamong degree types and degree heterogeneity). For example,in analogy with the single network case, fast transport can beachieved with the presence of a few nodes with large internaland external degrees, or “transport hubs”, even if the averagedegrees in the network are small [2,3].

B. Simple mixing example

Additional understanding of the basic form of the transportcoefficients is gained by looking at a special case of themicroscale degree distribution, where the generating functiontakes the form G[px0 + 1!p

2 (x+ + x!)], with total degreedescribed by G, and a given edge connecting nodes of thesame site with probability p, and nodes of left and rightneighboring sites with equal probability (1 ! p)/2, where1 ! p is an effective mixing parameter among populations.With this prescription, the critical transmissibility is reduced

to the inverse of the total-edge excess degree, Tc = G#(1)G##(1) ,

and the normalized diffusion coefficient, to the fraction ofexternal edges in a each direction, " = (1 ! p)/2. Moreover,the dispersion relation takes the instructive form

s(q) = !1 + T G##(1)G#(1)

[p + (1 ! p) cosh(q)], (18)

where s + 1 is given by the basic reproductive numbermultiplied by the average relative incidence, e!-xq , on theend of a randomly selected edge, illustrating the intuitivegeneralization of the single network case, where different edgetypes are more and less likely to connect to infected nodesdepending on their place in the lattice, and thus to contributeto local growth.

Likewise, from (15) and (16), we find the speed anddiffusion coefficient,

v) = T G##(1)G#(1)

(1 ! p) sinh(q)) (19)

and

D) = T G##(1)2G#(1)

(1 ! p) cosh(q)), (20)

where q) satisfies (15), and v) is given by the basic repro-ductive number multiplied by the average product of relativeposition and incidence, !-xe!-xq)

, on the end of a randomlyselected edge. Figure 2 shows the transport coefficients (18)–(20), as functions of T/Tc and p, with partial scaling collapse(17) for the corresponding class of network configurations.The expected reaction-diffusion scaling can be observed nearthe critical point, and far away from the critical region,when T / Tc, q), s)G#(1)

T G##(1) ,v)G#(1)T G##(1) , and D)G#(1)

T G##(1) tend to limiting

FIG. 2. (Color online) The transport coefficients for a one-dimensional lattice of configuration model networks with arbitrarytotal-degree distribution and inter-population mixing parameter1 ! p, shown as functions of the latter (Sec. V B): q) (a), s) (b), v) (c),and D) (d). The colored regions mark the range of each coefficient,which are bounded by the critical-region scaling (T ! Tc), and thelimiting behavior (T / Tc), delineated by dashed and solid curvesrespectively; the former are straight lines, signifying agreement withthe predicted scaling (17). Each panel’s arrow indicates the directionof increase in the distance from the phase transition, T/Tc ! 1.

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HINDES, SINGH, MYERS, AND SCHNEIDER PHYSICAL REVIEW E 88, 012809 (2013)

curves which depend only on 1 ! p, suggesting the intuitiveasymptotic proportionality to the reproductive number.

C. Pulled front classification

In order to connect the transport properties of the linearequations to the full nonlinear system, we refer to theclassification of fronts propagating into unstable states, whichin our system is the fully susceptible metapopulation lyingahead of the infection front. In general, there are two types ofdeterministic fronts: pulled and pushed, with the former havingfronts with asymptotic speed equal to the linear spreadingspeed and the latter having fronts with asymptotic speedgreater than the linear spreading speed [26]. Pushed frontsoccur because nonlinearities in the equations of motion tendto increase the growth of perturbations on the unstable state,resulting in nontrivial front shape dependence of the speed.However, in our system all nonlinearities are proportional toprobability generating functions, -G#(% )/G#(1) (11), whichare monotonically increasing over the unit interval. Therefore,all nonlinearities tend to increase % , and consequently dampenthe growth of infection, a sufficient condition for pulled fronts[28], and thus we anticipate fronts in this model to be pulled;this agrees with the intuition that epidemic propagation isgoverned by its behavior in a fully susceptible population. Inpractice, the classification has importance for control strategiesin systems with similar structure, implying that to mitigatethe spread of infection among populations, efforts should befocused on the leading edge of the front, and not on largeroutbreaks occurring farther behind.

D. Relaxation properties

In addition to quantifying the transport, the front speed canbe used to extract information about the dynamics away fromthe unstable state. As shown above, the "% field settles ontoa solution with translational similarity, "%n±x(t) = "%n(t , x

v) ),after an initial transient period. Behind the leading edge of thefront, the behavior resembles a relaxation to the stable equilib-rium (9), "%n(t) 0 "% + ".(t ! n

v) ) 0 "% + ".e!z)(n!v)t), where thespatial rate, |z)|, is the dominant eigenvalue of the nonlineareigenvalue equation

det%

G#e("% ,z)) ! 1+v)z)

TI

&= 0, with

G#e("%,z) =

*

+++,

G00

G0("1)G0+G0("1)

G0!G0("1)

G!0

G!("1)e!z G!+

G!("1)e!z G!!

G!("1)e!z

G+0

G+("1)ez G++

G+("1)ez G+!

G+("1)ez

-

.../

$$$$$$$$$"%

. (21)

The latter is the analog of Ke(q) at the stable state, which doesnot depend on the first two moments of the degree distributiondirectly, but on the generating function’s properties near theequilibrium (9). In general, the two characteristic spatial ratesfor this system are not equal, |z)| 1= q), and when theirdifference is large, it often signifies a significant separationin the time scales of growth, 1/s), and relaxation 1/v)|z)|.The latter provides an estimate for the amount of time a site is

infectious, with 1/|z)| yielding a related estimate for the widthof the propagating front (i.e., the typical spatial extent of anoutbreak at a given time). In particular, when the front speedis very fast and the degree distribution’s second moments arelarge with the first moments O(1), we find that |z)| 2 q),which suggests broad front profiles. In this case, propagationand relaxation can be thought of as approximately distinctprocesses.

VI. COMPARISON WITH STOCHASTIC SIMULATIONS

The above predictions for the mean-field dynamics on theone-dimensional metapopulation were compared to stochas-tic simulations of SIR dynamics on random instances ofmultiscale, metapopulation networks, using Gillespie’s directmethod [11,29,30]. The graphs were constructed using themultitype configuration model by first generating a degreesequence from the desired degree distribution and thenconnecting pairs of edge “stubs”, selected uniformly at random[2,15,16]. An outbreak was started by choosing one node fromthe first lattice site to be infected with all others susceptible.Only outbreaks which lead to epidemics withO(N ) cumulativeinfection were considered for comparison with mean-fieldpredictions. In order to ignore fluctuations in the initialtransients, time was zeroed after the first 100 reactions.

We are interested in the average shape of the front thatconnects the fully susceptible unstable state lying ahead ofthe infectious wave and the fully recovered (equilibrium) statelying behind it. The average shape was computed by takinginstantaneous “snapshots” of the profile for each stochasticrealization, conditioned on the middle lattice site havingcumulative infection equal to half the equilibrium value (9),and averaging the cumulative infection of the other sites overdifferent realizations. In general, the snapshots did not occur atthe same instant; however, aligning fronts based on a commonlevel of infection eliminates some of the effects of diffusivewandering [Fig. 3(b)]. The measured fronts were compared tothe mean-field profiles by integrating the lattice equations (11).A comparison is shown in Fig. 3 for two graphs with scale-freeand Poisson degree distributions, with generating functions

GS.F.("x) = Li/[e!1/Kx0(1 ! ( + (x+)(1 ! ( + (x!)]

and

GP ("x) = exp{C[x0(1 ! ( + (x+)(1 ! ( + (x!) ! 1]},

where Li/ is the polylogarithm function with exponent / [2].The parameters for the degree distributions were chosen suchthat each network had the same average degree and cloningparameter, ( (i.e., given a specified internal degree distribution,each of a node’s internal edges is copied to form an externaledge with probability (), but with different inherent levels ofheterogeneity.

We see in Fig. 3 that the epidemic front is broader in thescale-free network than in the Poisson. This difference comesfrom the much larger front speed of the former, which hadaverage excess degrees an order of magnitude larger thanthe latter, (12) and (15), and the relatively similar relaxationtimes (21) for the two classes of networks (implying that thetime scale over which a site is infectious in each network

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FIG. 3. (Color online) (a) A comparison between the averagecumulative infection profile for stochastic simulations of SIR dy-namics on a one-dimensional lattice of scale-free (blue/lower) andPoisson (green/upper) networks and mean-field predictions. Varioussite sizes are shown with different symbols and color shades, varyingfrom light to dark for 103 to 4 3 105 respectively. Front shapes forincreasingly large sizes are found to converge to the mean-field front.The parameters for the graphs were chosen to be: K = 100, / = 2and ( = 0.3 for the scale-free, and C = 2.901 57 and ( = 0.3 for thePoisson (Sec. VI). A lattice of 50 sites was used, which was largeenough to ensure uniformity with the above parameters and reactionrates # = $ = 1. The arrow indicates the propagation direction.(b) An ensemble of stochastic fronts conditioned on the middle latticesite having cumulative infection equal to half the equilibrium value(9). Because the infection fronts for different stochastic realizationsare offset from one another in time, we align all fronts based on a com-mon level of infection, and then average over realizations (Sec. VI).

is roughly the same). In the more homogeneous Poissonnetworks, the front is more narrow and propagates through thelattice on the same time scales as the local infection dynamics;whereas in the scale-free case, the leading edge of the frontpropagates quickly through the lattice, followed by a slowerrelaxation to the stable equilibrium state behind the front.This comparison shows that assumptions of homogeneity candrastically underestimate the speed and extent of fronts insystems with heterogenous interactions.

Additionally, the front speed v was numerically estimatedfrom the average time &&prog' required for the leading edge ofthe front to move forward by one lattice site (where the leadingedge was defined as that site where the incidence first reacheda set O(1) level) and averaging over such levels; i.e., 1/v =&&prog', once the initial spatial variation had decayed. Figure 4shows the convergence of the measured speed from simulationsto the mean-field prediction for each graph as a function ofthe steady state, cumulative infected population size at everylattice site, N = PN (Sec. III), with total size N . The linesrepresent fits to the expected scaling of the largest finite-sizecorrection for pulled fronts, v) ! vN - D)q))2

ln2(N), obtained from

a general 1/N cutoff in the mean-field equations [26,28,31];the coefficients are found to beO(1) of the expected scaling. Ingeneral, higher-order corrections in N must be calculated from

FIG. 4. (Color online) (a) Convergence of the average velocities,vN , to the mean-field predictions, v), for scale-free and Poissonnetworks (Fig. 3) as functions of the cumulative number of infectednodes at each site, shown with fits to the expected pulled front scaling,v) ! vN - D)q))2

ln2(N)(Sec. VI).

an analysis of the full, stochastic system [28]. The very slowconvergence in N comes from the transport dependence onthe linearized equations where infinitesimal infection levelsapply and sensitivity to stochastic effects is highest. Thiscan be seen in the fairly large finite-size corrections to thevelocity, particularly for the scale-free network, leading toa more narrow conditionally averaged front relative to themean-field, with fewer sites initiated at a given time (Fig. 3).

Finally, the average epidemic profile and transmissibil-ity threshold, (9) and (13), were compared to simulations.Figure 5 plots those comparisons for a system with left-right

FIG. 5. (Color online) The average epidemic fraction at each sitefor a network with asymmetric generating function (Sec. VI) andvarying transmissibilities. The point where the epidemic vanishesagrees with the prediction, Tc = 0.115 (13). Sites were occupied by20 000 nodes on a lattice of 100 sites.

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asymmetric generating function

GAsym("x) = 13

'2x2

0x2+x4

! + x40x6

+x2!(,

on a lattice of 100 sites, with 20 000 nodes on each site.Both the epidemic size for various transmissibilities and thethreshold were found to be in good agreement with mean-fieldpredictions, though the finite-size effects became increasinglyimportant as the critical region was approached, leading tosignificantly smaller outbreaks near the edges of the lattice.

VII. CONCLUSION

In this paper we developed a framework for studyinginfection dynamics in multiscale metapopulations, combiningmultitype networks and a mean-field theory, to explore howmacroscale disease propagation depends on microscale inter-action structure. As a necessary first step in this direction, weapplied the approach to a simple metapopulation model for achain of coupled populations, and derived the transport proper-ties for infection, including their scaling with the disease trans-missibility and the statistical properties of the underlying net-work. We also found a threshold for the viability of epidemics,and calculated the relaxation properties of the propagatingfront. These were compared for different network models, withheterogeneous networks having considerably higher speedsand broader fronts than their homogeneous counterparts,

illustrating the importance of including complexity in thefine-scale topology in order to accurately capture transportphenomenology.

Various extensions of the work presented, both in terms ofanalyses carried out and systems studied, could be considered.We have addressed here only the average dynamics of the one-dimensional, homogeneous system, without any descriptionof finite-size fluctuations, or consideration of the dynamics inhigher-dimensional generalizations. Greater complexity couldbe introduced through the spatiotemporal dependence of net-work parameters, and/or more general network configurations[32]. An interesting extension of the model discussed herewould include dynamic contacts between nodes and explicitmobility, instead of the assumed time-scale separation betweentopology and the overlying process [12,33,34]. However, thebasic formalism presented here can enable one to study suchfactors and build more realistic models for infectious processesin multiscale problems.

ACKNOWLEDGMENTS

This work was supported by the Science and TechnologyDirectorate of the US Department of Homeland Security viathe interagency Agreement No. HSHQDC-10-X-00138. Wethank Drew Dolgert for his assistance with our implementationof stochastic simulations.

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