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J R Soc Interface. 2009 December 6; 6(41): 1121–1134.
Published online 2009 March 4. doi:  10.1098/rsif.2008.0524
PMCID: PMC2817154

Spread of infectious disease through clustered populations

Abstract

Networks of person-to-person contacts form the substrate along which infectious diseases spread. Most network-based studies of this spread focus on the impact of variations in degree (the number of contacts an individual has). However, other effects such as clustering, variations in infectiousness or susceptibility, or variations in closeness of contacts may play a significant role. We develop analytic techniques to predict how these effects alter the growth rate, probability and size of epidemics, and validate the predictions with a realistic social network. We find that (for a given degree distribution and average transmissibility) clustering is the dominant factor controlling the growth rate, heterogeneity in infectiousness is the dominant factor controlling the probability of an epidemic and heterogeneity in susceptibility is the dominant factor controlling the size of an epidemic. Edge weights (measuring closeness or duration of contacts) have impact only if correlations exist between different edges. Combined, these effects can play a minor role in reinforcing one another, with the impact of clustering the largest when the population is maximally heterogeneous or if the closer contacts are also strongly clustered. Our most significant contribution is a systematic way to address clustering in infectious disease models, and our results have a number of implications for the design of interventions.

Keywords: epidemic, clustering, reproductive ratio, epidemic probability, attack rate

1. Introduction

Recently, H5N1 avian influenza and SARS have raised the profile of emerging infectious diseases. Both can infect humans, but have a primary animal host. Typically, such zoonotic diseases emerge periodically into the human population and disappear (e.g. Ebola, hantavirus, rabies), but sometimes (e.g. HIV) the disease achieves sustained person-to-person spread. With the advent of modern transportation networks, diseases that formerly emerged in isolated villages and died out without further spread may now spread worldwide.

A number of interventions are available to control emerging diseases, each with distinct costs and benefits. To design optimal policies, we must address several related, but nevertheless distinct, questions. How fast would an epidemic spread? How likely is a single introduced infection to result in an epidemic? How many people would an epidemic infect? We quantify these using An external file that holds a picture, illustration, etc.
Object name is rsif20080524e60.jpg0, the basic reproductive ratio, which measures the average number of new cases each infection causes early in the outbreak; An external file that holds a picture, illustration, etc.
Object name is rsif20080524e61.jpg, the probability that a single infection sparks an epidemic; and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e62.jpg, the attack rate or fraction of the population infected in an epidemic. Understanding these different quantities and what affects them helps us to select policies with maximal impact for given cost.

Many different models are used to study disease spread. Perhaps the most important decision in developing a model is how the interactions of the population are represented. Owing to the complexity of the population, it is invariably necessary to make simplifying assumptions. The errors (and therefore the conclusions) resulting from many of these approximations are not well quantified. In this paper, we will focus on quantifying the impact of clustering (the tendency to interact in small groups) and individual-scale heterogeneity on the spread of an epidemic.

Based on how they handle clustering, models for population structure fit into a hierarchy of three classes (which in turn may be subdivided). At the simplest level, the population is assumed to mix without any clustering. Most existing models fall into this category. At the most complex level, agent-based models are used: the movements of each individual are tracked and people who are in the same location are able to infect one another. These models typically require significant resources to develop, and the clustering is explicitly included. An intermediate level of complexity attempts to introduce the clustering as a parameter (or several parameters). Usually these models consider clustering only in terms of the number of triangles in a network, but as we shall see, other structures may play a role.

Before introducing the details of our model, we review some previous work. All the models we consider are susceptible–infected–recovered (SIR) epidemic models (Anderson & May 1991), in which individuals begin susceptible, become infected by contacting infected individuals and finally recover with immunity.

For unclustered populations, ordinary differential equation (ODE) models were among the earliest models used (Kermack & McKendrick 1927) and remain the most common. They are deterministic, and so cannot directly calculate An external file that holds a picture, illustration, etc.
Object name is rsif20080524e63.jpg, but they give insights into the factors controlling An external file that holds a picture, illustration, etc.
Object name is rsif20080524e64.jpg0 and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e65.jpg. Because they assume mass-action mixing, it is difficult to incorporate individual heterogeneity in the number of contacts. More recently, some network-based models have been introduced for unclustered populations (Andersson 1998; Newman 2002; Meyers et al. 2005, 2006; Kenah & Robins 2007; Miller 2007; Meyers 2007). These models represent the population as nodes with edges between nodes representing contacts, along which disease spreads stochastically. Heterogeneity in the number of contacts is introduced by modifying the degree (number of edges) of each node. By neglecting clustering, these studies are able to make analytic predictions through branching process arguments. A recent sociological study (Mossong et al. 2008) has used surveys with participants recording the length and nature of their contacts. These data are valuable for providing the contact distribution needed for the above network models, and allow us to apply network results to real populations. However, these data do not directly tell us anything about the clustering of the population resulting from family/work/other groups. Other recent work by Kenah & Robins (2007) and Miller (2007) analytically addresses the impact of heterogeneity in infectiousness and susceptibility in unclustered networks.

Using agent-based simulations (Eubank et al. 2004; Barrett et al. 2005; Ferguson et al. 2005; Del Valle et al. 2006; Germann et al. 2006; Ajelli & Merler 2008) allows us to directly incorporate clustering. In these simulations, the population is a collection of individuals who move and contact one another. The modeller has complete control over the parameters governing interactions and how the disease spreads. This allows us to study many effects, but introduces many parameters. It is difficult to test the accuracy of the assumptions used to generate these models and to extract which parameters are essential to the disease dynamics. The expense of developing these simulations is frequently prohibitive.

In this paper, we introduce a systematic approach for calculating the impact of clustering and quantifying the error. Because our model investigates disease spread in clustered networks, we provide a more detailed review of previous work on clustering and disease. A few investigations have been made into the interaction of clustering with disease spread using network models. The attempts that have been made (Keeling 1999; Newman 2003a; Serrano & Boguñá 2006a,b; Britton et al. 2007; Eames 2008) typically use approximations whose errors are not quantified, resulting in apparently contradictory results. A few papers (Kuulasmaa 1982; Trapman 2007; Miller 2008) have considered clustering and heterogeneities, rigorously showing that increased heterogeneity tends to decrease An external file that holds a picture, illustration, etc.
Object name is rsif20080524e66.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e67.jpg, but without quantitative predictions. Recently, Eames (2008) has considered the spread of epidemics in a class of random networks for which the number of triangles could be controlled. It may be inferred from his fig. fig.33 that clustering decreases the growth rate and that sufficient clustering can increase the epidemic threshold. However, at small and moderate levels, clustering appears not to alter the final size of epidemics significantly. Similar observations have been made by Bansal (2008). At first glance, this contradicts the observations of Serrano & Boguñá (2006a,b) that clustering significantly reduces the size of epidemics, but that sufficiently strong clustering reduces the epidemic threshold (see also Newman 2003a), allowing epidemics at lower transmissibility. The discrepancy in epidemic size may be resolved by noting that the networks in Serrano & Boguñá (2006a,b) have low average degree. We will see that clustering affects the size only if the typical degree is small or clustering is very high. The apparent discrepancy in epidemic threshold with strong clustering may be resolved by noting that the form of strong clustering considered by Serrano & Boguñá (2006a,b) forces preferential contacts between high-degree nodes. The reduction in epidemic threshold is perhaps better understood in terms of degree–degree correlations than in terms of clustering.

Figure 3
Different options for paths of length 2 between nodes u and v: (a) nuv=4, xuv=1; (b) nuv=4, xuv=0.

In this paper, we develop techniques to incorporate general small-scale structure (beyond triangles) into the calculations of An external file that holds a picture, illustration, etc.
Object name is rsif20080524e68.jpg0, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e69.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e70.jpg. To calculate An external file that holds a picture, illustration, etc.
Object name is rsif20080524e71.jpg0, we develop a systematic series expansion that allows us to interpolate between unclustered and clustered results by including more terms. To calculate An external file that holds a picture, illustration, etc.
Object name is rsif20080524e72.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e73.jpg, we use a similar approach, but give only the estimates on the size of correction terms. Our methods give us a rigorous means to understand how the unclustered results relate to more realistic populations, and our results resolve the apparent discrepancies mentioned above. Our theory accurately predicts epidemic behaviour in a more realistic contact network derived from an agent-based simulation of Portland, Oregon, by EpiSimS (Del Valle et al. 2006). We expand this to investigate the interplay of clustering, heterogeneities in individual infectiousness or susceptibility, and variations in edge weights in their effects on An external file that holds a picture, illustration, etc.
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Object name is rsif20080524e75.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e76.jpg.

The paper is organized as follows: §2 describes our model and networks and summarizes earlier work on unclustered networks. These results will be the leading-order terms for our expansions for clustered networks in the remainder of the paper. Section 3 considers how epidemics spread in a clustered network assuming homogeneous transmission. We derive the corrections to An external file that holds a picture, illustration, etc.
Object name is rsif20080524e77.jpg0 and show that the corrections to An external file that holds a picture, illustration, etc.
Object name is rsif20080524e78.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e79.jpg are insignificant unless the typical degree is small or clustering very high. Section 4 considers epidemics in clustered networks with heterogeneous infectiousness or susceptibility, building on §3. Section 5 extends this further to consider epidemics spreading on clustered networks with weighted edges. Edges with large weights tend to occur in family or work groups, which magnifies the impact of clustering. Finally, §6 discusses the implications of our results, particularly for designing interventions. We conclude that, in general, heterogeneity significantly impacts An external file that holds a picture, illustration, etc.
Object name is rsif20080524e80.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e81.jpg, but not An external file that holds a picture, illustration, etc.
Object name is rsif20080524e82.jpg0, while clustering impacts An external file that holds a picture, illustration, etc.
Object name is rsif20080524e83.jpg0 significantly, but not An external file that holds a picture, illustration, etc.
Object name is rsif20080524e84.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e85.jpg. Heterogeneity or edge weights may enhance the impact of clustering.

2. Formulation

2.1. The disease model

We consider the spread of a disease using a discrete SIR model on a static network G. Nodes of G represent individuals and edges represent (potentially infectious) contacts. The contact structure of the network is fixed during the course of the outbreak. The degree k of a node u is the number of edges containing u. Figure 1 shows a sample outbreak. For a single infection, the index case is chosen uniformly from the population to begin an outbreak. Infection spreads along an edge from an infected node u to a susceptible node v with probability Tuv, the transmissibility. The time it takes for infection and recovery to occur may vary but does not affect our results. Once u recovers it cannot be reinfected. Typically, for a large random network with a population of N=|G| nodes, the final size of outbreaks is either large, with An external file that holds a picture, illustration, etc.
Object name is rsif20080524e86.jpg(N) cumulative infections, or small, with An external file that holds a picture, illustration, etc.
Object name is rsif20080524e87.jpg(log N) infections (Bollobás 2001). Large outbreaks are epidemics and small outbreaks are non-epidemics.

Figure 1
A sample network and several stages of an outbreak. Nodes begin susceptible (small circles), become infected (large open circles), possibly infecting others along edges, and then recover (large filled circles). The outbreak finishes when no infected nodes ...

2.1.1. Transmissibility

A number of factors influence the transmissibility from u to v such as the viral load and duration of infection of u, the vaccination history and general health of v, the duration and nature of the contact between u and v and characteristics of the disease.

For each node u, we denote its ability to infect others by An external file that holds a picture, illustration, etc.
Object name is rsif20080524e88.jpgu and its ability to be infected by An external file that holds a picture, illustration, etc.
Object name is rsif20080524e89.jpgu. Each edge has a weight wuv. The parameter α measures disease-specific quantities. In most of our calculations, we assume that these are scalars and follow Del Valle et al. (2007) and Miller (2007), setting

equation image
2.1

If all contacts are identical, wuv may be absorbed into α

equation image
2.2

Note that Tuv is a number assigned to an edge, while An external file that holds a picture, illustration, etc.
Object name is rsif20080524e92.jpg is a function that states what the transmissibility between two nodes would be if they shared an edge.

With mild abuse of notation, we denote the probability density functions (pdfs) of An external file that holds a picture, illustration, etc.
Object name is rsif20080524e93.jpg, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e94.jpg and w by P(An external file that holds a picture, illustration, etc.
Object name is rsif20080524e95.jpg), P(An external file that holds a picture, illustration, etc.
Object name is rsif20080524e96.jpg) and P(w), respectively. We assign An external file that holds a picture, illustration, etc.
Object name is rsif20080524e97.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e98.jpg independently, but allow w to be assigned either independently or based on observed contacts (i.e. by observing contacts in a population, we may create a static network with edge weights assigned based on the observed contact). If w is assigned independently, then it is possible to eliminate edge weights from the analysis by marginalizing over the distribution of weights. However, if weights are not independent (for example work or family contacts tend to have correlated weights), then the details of the distribution and the correlations are important.

Given the infectiousness An external file that holds a picture, illustration, etc.
Object name is rsif20080524e99.jpgu of node u, we follow Miller (2007, 2008) and define its out-transmissibility

equation image
2.3

This is the marginalized probability that u infects a randomly chosen neighbour given An external file that holds a picture, illustration, etc.
Object name is rsif20080524e101.jpgu. From the definition of Tout and the pdf P(An external file that holds a picture, illustration, etc.
Object name is rsif20080524e102.jpg), we can calculate the pdf Qout(Tout). We symmetrically define the in-transmissibility Tin and its pdf Qin(Tin).

We denote the average of a quantity by left angle bracket.right angle bracket. The average transmissibility left angle bracketTright angle bracket is

equation image
2.4

2.1.2. Epidemic percolation networks

Rather than studying outbreaks as dynamic processes on networks, we may consider them in the context of epidemic percolation networks (EPNs; Kenah & Robins 2007a,b; Miller 2008). The EPN framework allows us to study epidemics as static objects and is useful for quickly estimating An external file that holds a picture, illustration, etc.
Object name is rsif20080524e106.jpg0, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e104.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e105.jpg. In this section, we summarize properties of EPNs; more details are provided in Kenah & Robins (2007), Miller (2007, 2008) and in §A of the electronic supplementary material.

Once the properties of the nodes and edges are assigned, an EPN An external file that holds a picture, illustration, etc.
Object name is rsif20080524e107.jpg is created as follows: we place each node of G into An external file that holds a picture, illustration, etc.
Object name is rsif20080524e108.jpg. For each edge {u,v} in G, we place directed edges (u,v) and (v,u) into An external file that holds a picture, illustration, etc.
Object name is rsif20080524e109.jpg independently with probability Tuv and Tvu, respectively. The nodes infected in an outbreak correspond exactly to those nodes that may be reached from the index case following the edges of An external file that holds a picture, illustration, etc.
Object name is rsif20080524e110.jpg. More specifically, the distribution of out-components of a node u in different EPN realizations matches the distribution of outbreaks resulting from different epidemic realizations in the original model with u as the index case. It may be shown that the distributions of out- and in-component sizes give us information about the probability of nodes to start an epidemic or become infected in an epidemic. We will see that in a large population the structure of a single EPN can be used to accurately estimate An external file that holds a picture, illustration, etc.
Object name is rsif20080524e113.jpg0, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e111.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e112.jpg.

Once we create an EPN and choose the index case, we define the rank of node v as the length of the shortest directed path from the index case to v.1 If no such path exists, v is never infected.

Interchanging all arrow directions interchanges An external file that holds a picture, illustration, etc.
Object name is rsif20080524e114.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e115.jpg. This means that if we can calculate An external file that holds a picture, illustration, etc.
Object name is rsif20080524e116.jpg, then An external file that holds a picture, illustration, etc.
Object name is rsif20080524e117.jpg may be calculated by the same technique, but with the direction of infection reversed. Owing to this, we focus our attention on calculating An external file that holds a picture, illustration, etc.
Object name is rsif20080524e118.jpg and apply the same methodology to calculate An external file that holds a picture, illustration, etc.
Object name is rsif20080524e119.jpg. An important consequence is that if T is constant, then An external file that holds a picture, illustration, etc.
Object name is rsif20080524e120.jpg=An external file that holds a picture, illustration, etc.
Object name is rsif20080524e121.jpg (Newman 2002; Miller 2007).

2.1.3. The basic reproductive ratio

We expect that epidemics are possible if and only if the basic reproductive ratio An external file that holds a picture, illustration, etc.
Object name is rsif20080524e122.jpg0 is greater than 1. That is, if an average infection causes more than one new case, an epidemic may occur, but otherwise the outbreak dies out quickly. However, this use of An external file that holds a picture, illustration, etc.
Object name is rsif20080524e123.jpg0 is not consistent with the typical definition: the average number of new infections caused by a single infected individual introduced into a fully susceptible population, which gives An external file that holds a picture, illustration, etc.
Object name is rsif20080524e124.jpg. A more appropriate definition is the average number of new infections caused by infected individuals early in outbreaks. The distinction is subtle, but results from the fact that whether an outbreak can grow depends on whether the people of low rank infect more than one person each (Diekmann et al. 1990). Low-rank individuals may be different from the average individual. Most obviously, they have more contacts (Feld 1991; Newman 2002); but with clustering, they also have a disproportionately large fraction of neighbours infected or recovered.

In order to quantify An external file that holds a picture, illustration, etc.
Object name is rsif20080524e125.jpg0 more rigorously, we first define Nr to be the number of people of rank r for a given outbreak simulation. We then define the rank reproductive ratio

equation image
2.5

to be the expected number of new cases caused by a rank r node (averaged over all possible outbreak realizations). An external file that holds a picture, illustration, etc.
Object name is rsif20080524e127.jpg corresponds to the usual definition of An external file that holds a picture, illustration, etc.
Object name is rsif20080524e128.jpg0. In practice, we find that An external file that holds a picture, illustration, etc.
Object name is rsif20080524e129.jpg0,r reaches a plateau quickly as r increases before eventually decreasing as the finite size of the population becomes important. Consequently, an improved definition of An external file that holds a picture, illustration, etc.
Object name is rsif20080524e130.jpg0 is the limit of An external file that holds a picture, illustration, etc.
Object name is rsif20080524e131.jpg0,r as r grows, subject to the assumption that An external file that holds a picture, illustration, etc.
Object name is rsif20080524e132.jpg0,r is unaffected by the finite size of G. This gives (cf. Trapman 2007)

equation image
2.6

and generalizes the definition given by Diekmann et al. (1990) for ODE models. Under this definition, epidemics are possible if An external file that holds a picture, illustration, etc.
Object name is rsif20080524e134.jpg0>1, but not if An external file that holds a picture, illustration, etc.
Object name is rsif20080524e135.jpg0<1. We discuss this further in §B of the electronic supplementary material. In a large population, considering multiple index cases with a single EPN gives a good estimate of An external file that holds a picture, illustration, etc.
Object name is rsif20080524e136.jpg and hence An external file that holds a picture, illustration, etc.
Object name is rsif20080524e137.jpg0,r.

2.2. Configuration model networks

We consider two different types of networks. The first is a class of (unclustered) random networks for which we can derive analytic results based only on the degree distribution. These analytic results will form the leading-order term of our perturbation expansions. The second is a more complicated network resulting from an agent-based simulation, which we will use to demonstrate the accuracy of our perturbation expansions.

Our random networks are created by an algorithm that has been discovered independently a number of times (e.g. Molloy & Reed 1995). These have come to be called configuration model (CM; Newman 2003b) networks. These networks are maximally random given the degree distribution. As the number of nodes in a CM network grows, the frequency of short cycles becomes negligible. The resulting lack of clustering allows us to calculate analytic results for epidemics. We briefly discuss these results assuming T is constant. More details are in Andersson (1998), Newman (2002), Meyers et al. (2006), Kenah & Robins (2007), Marder (2007), Miller (2007) and Noël et al. (2009) and §C of the electronic supplementary material (which also addresses edge weights).

In the early stages of an outbreak in a CM network, the probability that a newly infected (non-index case) node has degree k is kP(k)/left angle bracketkright angle bracket. Clustering is unimportant and so the node will have k−1 susceptible neighbours, regardless of its rank. Thus, the expected number of infections caused by a newly infected node is

equation image
2.7

To calculate the probability An external file that holds a picture, illustration, etc.
Object name is rsif20080524e139.jpg that infection of a randomly chosen index case results in an epidemic, we instead calculate the probability f=1−An external file that holds a picture, illustration, etc.
Object name is rsif20080524e140.jpg that it does not. Then f is the probability that each neighbour of the index case is either not infected, or infected but does not start an epidemic. Defining h to be the probability that a secondary case does not start an epidemic,

equation image
2.8

We find a similar relationship for h, except that the probability for a secondary case to have degree k is kP(k)/left angle bracketkright angle bracket and only k−1 neighbours are susceptible

equation image
2.9

We solve this recurrence relationship for h numerically, and use the result to find f. An external file that holds a picture, illustration, etc.
Object name is rsif20080524e143.jpg follows immediately. Because T is constant, this also gives An external file that holds a picture, illustration, etc.
Object name is rsif20080524e144.jpg (Newman 2002; Miller 2007).

If T is not constant, the calculation becomes more difficult, and is discussed further in §C of the electronic supplementary material and Kenah & Robins (2007) and Miller (2007). In general, if T can vary for CM networks, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e145.jpg, while the values calculated assuming constant T give upper bounds for An external file that holds a picture, illustration, etc.
Object name is rsif20080524e146.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e147.jpg.

2.3. The EpiSimS network

We are interested in understanding the impact of clustering on disease spread. The term clustering is rather vague, and is usually measured by the number of triangles in a network (Watts & Strogatz 1998). However, any sufficiently short cycles impact the spread of an infectious disease. For our purposes, we think of a clustered network as a network with enough short cycles to impact disease dynamics.

It is relatively simple to measure the degree distribution of a population using survey methods. We can easily calculate An external file that holds a picture, illustration, etc.
Object name is rsif20080524e148.jpg0, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e149.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e150.jpg for a CM network with the same degree distribution, but the errors between these values and the values for the original clustered network are unknown. Our goal in this paper is to develop analytical techniques to quantify these errors.

To test our predictions, we turn to an agent-based network derived from a single EpiSimS (Eubank et al. 2004; Barrett et al. 2005; Del Valle et al. 2006) simulation of Portland, Oregon. The simulation includes roads, buildings and a statistically accurate (based on census data) population of approximately 1.6 million people who perform daily tasks based on population surveys. This gives a highly detailed knowledge of the interactions in the synthetic population. The degree distribution and contact structure emerge from the simulation. The resulting network has significant clustering and average degree of approximately 16. More details are in §D of the electronic supplementary material.

3. Clustered networks with homogeneous nodes

In this section, we assume that the population is homogeneous and all contacts are equally weighted. Consequently, transmissibility is constant: An external file that holds a picture, illustration, etc.
Object name is rsif20080524e151.jpg for all edges. It follows that An external file that holds a picture, illustration, etc.
Object name is rsif20080524e152.jpg=An external file that holds a picture, illustration, etc.
Object name is rsif20080524e153.jpg (Newman 2002; Miller 2007). We develop a predictive theory for An external file that holds a picture, illustration, etc.
Object name is rsif20080524e154.jpg0, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e155.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e156.jpg and test the theory with simulations on the EpiSimS network. We begin with An external file that holds a picture, illustration, etc.
Object name is rsif20080524e157.jpg0.

3.1. The basic reproductive ratio

The simulated rank reproductive ratio An external file that holds a picture, illustration, etc.
Object name is rsif20080524e158.jpg0,r is shown in figure 2 for 0≤r≤4. At all values of T, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e159.jpg is clearly distinct from An external file that holds a picture, illustration, etc.
Object name is rsif20080524e160.jpg0,r, r>0 (which are close together). For r>0, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e161.jpg0,r is asymptotic to the unclustered approximation Tleft angle bracketk2kright angle bracket/left angle bracketkright angle bracket as T→0. This is because at small T the disease only rarely follows all edges of short cycles and so clustering has no impact. As T increases, these curves lie significantly below the unclustered approximation, because clustering reduces the number of available susceptibles. An external file that holds a picture, illustration, etc.
Object name is rsif20080524e162.jpg0,4 peels away from An external file that holds a picture, illustration, etc.
Object name is rsif20080524e163.jpg0,1, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e164.jpg0,2 and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e165.jpg0,3 for larger T because the population is finite, and so the number of susceptibles available to infect after rank 4 is reduced. In larger populations, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e166.jpg0,4 would not deviate.

Figure 2
(a,b) Simulated values of the rank reproductive ratio An external file that holds a picture, illustration, etc.
Object name is rsif20080524e2.jpg for r=0, …, 4 using an EPN from the (fixed) EpiSimS network with a homogeneous population, compared with the unclustered prediction. (b) At small T, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e3.jpg0,1An external file that holds a picture, illustration, etc.
Object name is rsif20080524e4.jpg0,4 match the ...

We conclude that An external file that holds a picture, illustration, etc.
Object name is rsif20080524e167.jpg0,r converges quickly, and that An external file that holds a picture, illustration, etc.
Object name is rsif20080524e168.jpg0,1 is a good approximation to An external file that holds a picture, illustration, etc.
Object name is rsif20080524e169.jpg0, but An external file that holds a picture, illustration, etc.
Object name is rsif20080524e170.jpg0,0 is not. This implies that the network has an important structure contained in the paths of length 2, but not in the paths of length 3. This fortunate observation allows us to approximate An external file that holds a picture, illustration, etc.
Object name is rsif20080524e171.jpg0 by An external file that holds a picture, illustration, etc.
Object name is rsif20080524e172.jpg0,1, which we may analytically calculate with relative ease (An external file that holds a picture, illustration, etc.
Object name is rsif20080524e173.jpg0,r becomes combinatorially hard as r grows). To find An external file that holds a picture, illustration, etc.
Object name is rsif20080524e174.jpg, we first note that An external file that holds a picture, illustration, etc.
Object name is rsif20080524e175.jpg. Calculating An external file that holds a picture, illustration, etc.
Object name is rsif20080524e176.jpg is more difficult: consider all pairs of nodes u and v with at least one path of length 2 between them. Let nuv be the number of paths of length 2 between u and v and Χuv be an indicator function: Χuv=1 if {u,v} is an edge and Χuv=0 if it is not (figure 3). The probability that an infection of u results in infection of v in exactly two steps is An external file that holds a picture, illustration, etc.
Object name is rsif20080524e177.jpg. Summing this over all pairs yields (where N is the size of the population and each pair u and v appears twice)

equation image

which allows us to calculate An external file that holds a picture, illustration, etc.
Object name is rsif20080524e179.jpg0,1 exactly. This sum is straightforward to calculate, but we can increase our understanding with a small T expansion. We approximate An external file that holds a picture, illustration, etc.
Object name is rsif20080524e180.jpg for T[double less-than sign]1 by

equation image

where An external file that holds a picture, illustration, etc.
Object name is rsif20080524e182.jpg is the average number of triangles each node is in, and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e183.jpgis the average number of squares each node is in (cf. Hastings 2006). Higher order terms involve more complicated shapes. This gives

equation image
3.1

At the leading order, we recover the unclustered prediction for An external file that holds a picture, illustration, etc.
Object name is rsif20080524e185.jpg0, reflecting the fact that at small T the probability the outbreak follows all edges of a cycle is negligible. As T increases, the first corrections are due to triangles, then squares, then pairs of triangles sharing an edge and sequentially larger and larger structures made up of paths of length 2. A comparison of these approximations with the exact value is shown in figure 4.

Figure 4
(a,b) Comparison of first three asymptotic approximations for An external file that holds a picture, illustration, etc.
Object name is rsif20080524e11.jpg0,1 from equation (3.1) with the exact value for the EpiSimS network. (b) The comparison at small T is shown (solid curve, exact An external file that holds a picture, illustration, etc.
Object name is rsif20080524e12.jpg0,1; dotted curve, first approximation; dot-dashed curve, second ...

Although we have defined An external file that holds a picture, illustration, etc.
Object name is rsif20080524e186.jpg0 for an ensemble of realizations, figure 5 shows that An external file that holds a picture, illustration, etc.
Object name is rsif20080524e187.jpg0,1 accurately predicts the observed ratio Nr+1/Nr for individual simulations once the outbreaks are well established. Early in outbreaks, the behaviour is dominated by stochastic effects, and so the ratio of successive rank sizes is noisy. Once the outbreak has grown large enough, random events become unimportant and the ratio settles at An external file that holds a picture, illustration, etc.
Object name is rsif20080524e188.jpg0,1.2

Figure 5
The progression of 10 simulated epidemics for (a) T=0.1 and (b) T=0.2 in the EpiSimS network. (a) Nr+1/Nr against rank and (b) the cumulative fraction of the population infected are shown (dotted curve, unclustered An external file that holds a picture, illustration, etc.
Object name is rsif20080524e13.jpg0 prediction; dashed curve, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e14.jpg0,1).

3.2. Epidemic probability and size

In order to assess the effect of clustering on An external file that holds a picture, illustration, etc.
Object name is rsif20080524e189.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e190.jpg, we compare epidemics on the EpiSimS network with the analytic predictions derived assuming a CM network of the same degree distribution in figure 6. The epidemic threshold is not notably altered, and the values of An external file that holds a picture, illustration, etc.
Object name is rsif20080524e191.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e192.jpg are almost indistinguishable from the predictions made assuming no clustering, despite the large amount of clustering in the network.

Figure 6
Probability An external file that holds a picture, illustration, etc.
Object name is rsif20080524e15.jpg and attack rate An external file that holds a picture, illustration, etc.
Object name is rsif20080524e16.jpg of epidemics for the (clustered) EpiSimS network (pluses) versus T, compared with the prediction derived from the degree distribution assuming no clustering. Each data point is from a single EPN (the variation in An external file that holds a picture, illustration, etc.
Object name is rsif20080524e17.jpg resulting ...

Although initially surprising, these results may be understood intuitively as follows: if T is large enough that the disease follows all edges of a short cycle, then some other edge from a node of that cycle is likely to start an epidemic and the cycle does not prevent an epidemic. On the other hand, if T is smaller so that it does not follow all edges of a cycle, then the disease never sees the existence of the cycle, and the outbreak progresses as if there were no cycle.

To make this more rigorous, we first look at the epidemic threshold. We assume that An external file that holds a picture, illustration, etc.
Object name is rsif20080524e193.jpg0 is well approximated by An external file that holds a picture, illustration, etc.
Object name is rsif20080524e194.jpg0,1. Let T0=left angle bracketkright angle bracket/left angle bracketk2kright angle bracket be the threshold without clustering and T0+δT be the threshold found by including the correction due to triangles. From equation (3.1), it follows that

equation image
3.2

Because a given node of degree k is contained in at most (k2k)/2 triangles, we conclude 2left angle bracketn[big up triangle, open]right angle bracket/left angle bracketk2kright angle bracket≤1. So if left angle bracketkright angle bracket/left angle bracketk2kright angle bracket is small, then the leading-order term of equation (3.2) is small and triangles do not significantly alter the epidemic threshold regardless of the density of triangles. For the EpiSimS network, left angle bracketkright angle bracket/left angle bracketk2kright angle bracket takes the value 0.046, and so we do not anticipate clustering to play an important role in determining the threshold.

Above threshold, we assume that An external file that holds a picture, illustration, etc.
Object name is rsif20080524e196.jpg may be expanded much as (3.1)

equation image
3.3

where An external file that holds a picture, illustration, etc.
Object name is rsif20080524e198.jpg0 is the epidemic probability in a CM network of the same degree distribution. Although calculating An external file that holds a picture, illustration, etc.
Object name is rsif20080524e199.jpg0,1 only requires information about the nodes of distance at most two from the index case, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e200.jpg may depend on the effects occurring at larger distance, and so the expansion has many additional terms. In general, we expect that if the average degree is large, then the various coefficients of the correction terms are all small. The larger a structure is, the smaller we expect its corresponding coefficient to be. The coefficient for triangles An external file that holds a picture, illustration, etc.
Object name is rsif20080524e201.jpg1 may be found by

equation image

where An external file that holds a picture, illustration, etc.
Object name is rsif20080524e203.jpg is the probability that a given triangle prevents an epidemic if u is the index case (regardless of whether u is part of the triangle). Reversing the order of summation, we get

equation image

where N[big up triangle, open] is the number of triangles in G and left angle bracket.right angle bracket[big up triangle, open] is the average of the given quantity taken over all triangles. Thus

equation image

and we can find An external file that holds a picture, illustration, etc.
Object name is rsif20080524e206.jpg1 by considering the average effect of a single triangle in an unclustered network.

To calculate the impact of a triangle with nodes u, v and w on An external file that holds a picture, illustration, etc.
Object name is rsif20080524e207.jpg for a given network, we consider that triangle and a randomly chosen edge {x,y} elsewhere in the network. If we replace the edges {v,w} and {x,y} with {v,x} and {w,y}, then we have a new network without the triangle, but with the same degree distribution. We must estimate the expected change in An external file that holds a picture, illustration, etc.
Object name is rsif20080524e208.jpg caused by switching the edges.

We begin by assuming that u is the index case. The triangle can affect An external file that holds a picture, illustration, etc.
Object name is rsif20080524e209.jpg only if the infection tries to cross all three edges, that is if the infection process ‘loses’ an edge because of clustering. This may happen in three distinct ways. In the first, node u infects both v and w, and then v and/or w tries to infect the other. In the second, u infects v but not w, then v infects w and finally w tries to infect u. The third is symmetric to the second (with u infecting w).

To leading order we can ignore other short cycles, so the probability that an edge leading out of u (not to v or w) will not cause an epidemic is g=1−T+Th, where h (as before) is the probability that a randomly chosen secondary case does not cause an epidemic in an unclustered network and can be calculated using equation (2.9).

We perform a sample calculation with the first case: u infects both v and w. Assume that u has degree ku, v has degree kv and w has degree kw. The probability that u infects both v and w without some other edge leading from u, v or w starting an epidemic is An external file that holds a picture, illustration, etc.
Object name is rsif20080524e210.jpg. If the {v,w} edge were broken and v and w were joined to x and y, respectively (figure 7), then the new probability of u to infect both v and w without an epidemic becomes An external file that holds a picture, illustration, etc.
Object name is rsif20080524e211.jpg. The difference is An external file that holds a picture, illustration, etc.
Object name is rsif20080524e212.jpg, which is the product of three terms, all at most 1. If the sum ku+kv+kw is moderately large, then either An external file that holds a picture, illustration, etc.
Object name is rsif20080524e213.jpg or An external file that holds a picture, illustration, etc.
Object name is rsif20080524e214.jpg (if g is not close to 1 then the first term is small, otherwise the second term is small). Thus, the triangle has little impact on the epidemic probability in this case.3 Similar analysis applies to the other two cases where the w to u or v to u infections are lost. Provided the typical sum of degrees of nodes in a triangle is relatively large, the probability of an epidemic when the index case is in the triangle is not impacted significantly.

Figure 7
Replacing the edges {v,w} and {x,y} with {v,x} and {w,y} breaks the triangle and allows more infections, without affecting the degree distribution.

If the index case is not part of the triangle, then the above analysis is modified because we must also consider each node in the path from the index case to the triangle. We must first calculate the probability that infection reaches a node in the triangle while simultaneously no intermediate node sparks an epidemic, and then we calculate the probability as above that the triangle prevents an epidemic. If the index case is u1 and the path from u1 to the triangle goes through u2, …, un and then reaches u, then the probability that the triangle prevents an epidemic An external file that holds a picture, illustration, etc.
Object name is rsif20080524e216.jpg is given by An external file that holds a picture, illustration, etc.
Object name is rsif20080524e217.jpg. This falls off very quickly, and so nodes not in the triangle are unimportant, unless typical degrees are small.

By contrast, in a network with small average degree and a significant number of triangles, this becomes significant. This explains the observations of Serrano & Boguñá (2006a,b) who use networks with average degree less than 3 and find that clustering significantly alters An external file that holds a picture, illustration, etc.
Object name is rsif20080524e218.jpg.

It is tempting to generalize our conclusion and state that if the average degree is large, clustering has no impact on An external file that holds a picture, illustration, etc.
Object name is rsif20080524e219.jpg or An external file that holds a picture, illustration, etc.
Object name is rsif20080524e220.jpg. However, there are a number of counter-examples: consider a network made up of isolated cliques with Nc nodes, then in expansion (3.3), the coefficient for cliques of Nc nodes will not be small. Consequently, care must be taken when using such an expansion to ensure that neglected terms resulting from larger scale structures are in fact negligible. For social networks, we generally anticipate this highly segregated situation to be unimportant.

We conclude that for most reasonable networks, clustering is only important for An external file that holds a picture, illustration, etc.
Object name is rsif20080524e221.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e222.jpg if the typical degrees of nodes are low in which case An external file that holds a picture, illustration, etc.
Object name is rsif20080524e223.jpg0 is small. A consequence of these results is that if An external file that holds a picture, illustration, etc.
Object name is rsif20080524e224.jpg0 is moderately large, then An external file that holds a picture, illustration, etc.
Object name is rsif20080524e225.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e226.jpg are effectively unaltered by clustering. If An external file that holds a picture, illustration, etc.
Object name is rsif20080524e227.jpg0 is small, however, clustering may or may not play a role in determining An external file that holds a picture, illustration, etc.
Object name is rsif20080524e228.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e229.jpg, depending on whether An external file that holds a picture, illustration, etc.
Object name is rsif20080524e230.jpg0 is small because the degrees are small or T is small.

4. Clustered networks with heterogeneous nodes

When we drop the assumption of constant transmissibility, disease spread becomes more complicated. If An external file that holds a picture, illustration, etc.
Object name is rsif20080524e231.jpg is heterogeneous and u infects a neighbour, then the a posteriori expectation for Tout(u) becomes higher: it is likely to infect more neighbours. This accentuates the effect of short cycles, enhancing the impact of clustering on An external file that holds a picture, illustration, etc.
Object name is rsif20080524e232.jpg0, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e233.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e234.jpg. A similar argument applies with heterogeneity in An external file that holds a picture, illustration, etc.
Object name is rsif20080524e235.jpg: if v is not infected by one of its neighbours, then the a posteriori expectation for Tin(v) becomes lower: it is less likely to be infected by other neighbours, and so has multiple opportunities to prevent an epidemic. Again this accentuates the effect of short cycles.

In this section, we investigate how varying the infectiousness and susceptibility of nodes in the EpiSimS network enables clustering to alter the values of An external file that holds a picture, illustration, etc.
Object name is rsif20080524e400.jpg0, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e236.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e237.jpg. We will make use of the ordering assumption and its consequences from Miller (2008): if u1 is ‘more infectious’ than u2 in a given instance (or v1 ‘more susceptible’ than v2), then u1 is always more infectious than u2 (or v1 always more susceptible than v2). More specifically, the ordering assumption states that Tout(u1)>Tout(u2) if and only if An external file that holds a picture, illustration, etc.
Object name is rsif20080524e238.jpg for all An external file that holds a picture, illustration, etc.
Object name is rsif20080524e239.jpg, with inequality for some An external file that holds a picture, illustration, etc.
Object name is rsif20080524e410.jpg, and the corresponding statement for Tin. The results of Miller (2008) show that if the ordering assumption holds, heterogeneity tends to reduce An external file that holds a picture, illustration, etc.
Object name is rsif20080524e240.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e241.jpg, and the upper bounds on An external file that holds a picture, illustration, etc.
Object name is rsif20080524e242.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e243.jpg correspond to homogeneous populations (constant T).

For simulations in this section, we consider five different illustrative cases, which will be denoted throughout by the symbols given in table 1. In the first four cases, we use equation (2.2), so that An external file that holds a picture, illustration, etc.
Object name is rsif20080524e244.jpg with the distribution of An external file that holds a picture, illustration, etc.
Object name is rsif20080524e245.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e246.jpg varying for each. We vary α to change the average transmissibility. In the fifth case, the out-transmissibility is maximally heterogeneous: a fraction left angle bracketTright angle bracket of the population infect all neighbours, while the remaining 1−left angle bracketTright angle bracket infect no neighbours.

Table 1
For the calculations of §§4 and 5, we determine Tuv using equations (2.1) and (2.2) with the distributions of An external file that holds a picture, illustration, etc.
Object name is rsif20080524e49.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e50.jpg given in the first four rows, or by considering a maximally heterogeneous population for which left angle bracketTright angle bracket of the ...

The fifth case gives a lower bound on An external file that holds a picture, illustration, etc.
Object name is rsif20080524e247.jpg for a homogeneously susceptible population (Trapman 2007). It is hypothesized to remain a lower bound on An external file that holds a picture, illustration, etc.
Object name is rsif20080524e248.jpg if susceptibility is allowed to vary (Miller 2008). We could also consider maximal heterogeneity in susceptibility, but the results for An external file that holds a picture, illustration, etc.
Object name is rsif20080524e249.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e250.jpg merely correspond to interchanging their values for maximal heterogeneity in infectiousness, and so we do not need to consider it explicitly.

4.1. The basic reproductive ratio

We use simulations to calculate the rank reproductive ratio An external file that holds a picture, illustration, etc.
Object name is rsif20080524e251.jpg0,r for the cases of table 1 and plot the result for 0≤r≤4 in figure 8. Note that An external file that holds a picture, illustration, etc.
Object name is rsif20080524e252.jpg0,1 remains a good approximation to An external file that holds a picture, illustration, etc.
Object name is rsif20080524e253.jpg0. In the first four cases, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e254.jpg0 is again asymptotic to the unclustered approximation as left angle bracketTright angle bracket→0. There are small kinks for An external file that holds a picture, illustration, etc.
Object name is rsif20080524e411.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e412.jpg at left angle bracketTright angle bracket=0.5 and left angle bracketTright angle bracket=0.7, respectively, resulting from the nature of those distributions. The heterogeneities act to enhance the effect of clustering on An external file that holds a picture, illustration, etc.
Object name is rsif20080524e255.jpg0, but the effect is relatively small.

Figure 8
(a–e) An external file that holds a picture, illustration, etc.
Object name is rsif20080524e18.jpg calculated from EPNs for the heterogeneous examples of table 1 (black solid curve, unclustered An external file that holds a picture, illustration, etc.
Object name is rsif20080524e19.jpg0; black dashed curve, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e20.jpg0,0; grey solid curve, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e21.jpg0,1; dotted curve, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e22.jpg0,2; dot-dashed curve, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e23.jpg0,3; grey dashed curve, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e24.jpg0,4). (f) An external file that holds a picture, illustration, etc.
Object name is rsif20080524e25.jpg0,1 values for all ...

In the final, maximally heterogeneous case An external file that holds a picture, illustration, etc.
Object name is rsif20080524e415.jpg, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e256.jpg0,1 remains a good approximation to An external file that holds a picture, illustration, etc.
Object name is rsif20080524e257.jpg0. At small values of left angle bracketTright angle bracket, the heterogeneity causes clustering to have a larger impact than in a homogeneous population as seen in figure 8f, and so this is not asymptotic to the unclustered approximation. At larger values of left angle bracketTright angle bracket, the heterogeneous and homogeneous growth rates are similar.

As before, we can calculate An external file that holds a picture, illustration, etc.
Object name is rsif20080524e258.jpg0,1 analytically, which helps to explain our observations. If the ordering assumption holds, we may use a simplified notation T(Tout,Tin) to denote the transmissibility from a node with out-transmissibility Tout to a node with in-transmissibility Tin.4 We have An external file that holds a picture, illustration, etc.
Object name is rsif20080524e261.jpg and

equation image

and so we may express the growth rate as a perturbation about the unclustered case An external file that holds a picture, illustration, etc.
Object name is rsif20080524e263.jpg giving

equation image
4.1

For the second term, it may be shown that left angle bracketTright angle bracket3left angle bracketToutTinT(Tout,Tin)right angle bracketleft angle bracketTright angle bracket2. The minimum occurs when T is constant, suggesting that the maximum growth rate occurs in a homogeneous population. The maximum left angle bracketTright angle bracket2 occurs either for An external file that holds a picture, illustration, etc.
Object name is rsif20080524e415.jpg

equation image
4.2

i.e. when the out-transmissibility is maximally heterogeneous, or when the in-transmissibility is maximally heterogeneous

equation image
4.3

Consequently, we expect that for given left angle bracketTright angle bracket, the minimum growth rate occurs with maximally heterogeneous infectiousness or susceptibility. These two minima for An external file that holds a picture, illustration, etc.
Object name is rsif20080524e267.jpg0,1 have previously been hypothesized to give lower bounds on An external file that holds a picture, illustration, etc.
Object name is rsif20080524e268.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e269.jpg, respectively (Miller 2008).

We note that in the maximally heterogeneous case, the correction term in (4.1) is significant at the leading order in T. Consequently, if left angle bracketn[big up triangle, open]right angle bracket is comparable with left angle bracketk2kright angle bracket/2 (i.e. the clustering coefficient (Watts & Strogatz 1998) is comparable with 1), then the threshold value of left angle bracketTright angle bracket may be increased by clustering, and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e270.jpg0 is not asymptotic to the unclustered prediction as left angle bracketTright angle bracket→0.

4.2. Probability and size

Figure 9 shows that the unclustered predictions provide a good estimate of An external file that holds a picture, illustration, etc.
Object name is rsif20080524e271.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e272.jpg in the clustered EpiSimS network. We expect that in a network with sufficiently large average degree, the impact of clustering should once again be small.

Figure 9
Comparison of (a) An external file that holds a picture, illustration, etc.
Object name is rsif20080524e28.jpg and (b) An external file that holds a picture, illustration, etc.
Object name is rsif20080524e29.jpg observed from EPNs in the clustered EpiSimS network with heterogeneities (symbols) with that predicted by the unclustered theory (curves) using table 1. Each data point is based on a single EPN. For both An external file that holds a picture, illustration, etc.
Object name is rsif20080524e412.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e415.jpg, Tin(v)=left angle bracket ...

We use arguments similar to that before, taking a triangle with nodes u, v and w. The reasoning becomes more difficult because knowledge that u infects v increases the expectation that u infects w. Consequently, the lost edges in triangles are more frequently encountered by the outbreak. However, the knowledge that u infects v also increases the expectation that u infects its other neighbours. For a triangle to prevent an epidemic, we need both that no edge outside the triangle leads to an epidemic and that the lost edge would otherwise have caused an epidemic. If the typical degree of the network is not small, then the fact that the lost edge is encountered more frequently may be offset by the fact that when it is encountered, other edges are more likely to spark an epidemic.

For An external file that holds a picture, illustration, etc.
Object name is rsif20080524e415.jpg where nodes infect all or none of their neighbours, the effect of different triangles that share the index case cannot be separated easily. The probability that the index case directly infects a set of m nodes of interest is left angle bracketTright angle bracket, rather than Tm. Thus, expansions as in (3.3) do not work as well: terms that were previously higher order become significant. Close to the epidemic threshold, this can play an important role. However, well above the epidemic threshold, if the index case infects all of its neighbours, then an epidemic is almost guaranteed and so An external file that holds a picture, illustration, etc.
Object name is rsif20080524e273.jpg regardless of whether the network is clustered. Thus for An external file that holds a picture, illustration, etc.
Object name is rsif20080524e415.jpg, clustering affects An external file that holds a picture, illustration, etc.
Object name is rsif20080524e274.jpg only close to the epidemic threshold.

In the opposite case where nodes would be infected by any neighbour or else no neighbour, the values of An external file that holds a picture, illustration, etc.
Object name is rsif20080524e275.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e276.jpg are interchanged. Thus, for maximally heterogeneous susceptibility, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e277.jpg could be significantly altered close to the threshold. The reason for this is as follows: for the first step, the spread is indistinguishable from that of an outbreak with constant T. However, when infections of rank 1 attempt to infect their neighbours, they cannot infect any of the neighbours of the index case. By contrast, in the constant T case, any neighbour not infected by the index case would be susceptible at later steps. Consequently, the impact of triangles becomes much more important (by a factor of 1/left angle bracketTright angle bracket) and our earlier argument for neglecting them fails. The interaction of maximal heterogeneity with clustering in this case is larger, but it nevertheless becomes unimportant far from the threshold.

Our prediction that heterogeneity allows clustering to be more significant close to the threshold is borne out for An external file that holds a picture, illustration, etc.
Object name is rsif20080524e414.jpg where there is relatively strong heterogeneity in susceptibility just above the epidemic threshold. The epidemic threshold for An external file that holds a picture, illustration, etc.
Object name is rsif20080524e414.jpg is increased compared with the other cases. By contrast, there is much stronger heterogeneity in susceptibility for An external file that holds a picture, illustration, etc.
Object name is rsif20080524e411.jpg at left angle bracketTright angle bracket=0.5 and in infectiousness for An external file that holds a picture, illustration, etc.
Object name is rsif20080524e412.jpg at left angle bracketTright angle bracket=0.7. This results in a reduction in An external file that holds a picture, illustration, etc.
Object name is rsif20080524e278.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e279.jpg, respectively, but because it is far from threshold, there is little deviation from the unclustered predictions.

5. Clustered networks with weighted edges

When we allow edges to be weighted, new complications arise. The weights we use in our simulations are the durations of contacts from the EpiSimS simulation and are discussed in more detail in §D of the electronic supplementary material. If a contact in the original EpiSimS simulation is longer, then a higher weight is assigned. If the weights of different edges were independent, then we could simply take An external file that holds a picture, illustration, etc.
Object name is rsif20080524e280.jpg. However, edge weights are not independent: clustered connections tend to have larger weights. If brief contacts are negligible, then the disease spreads on a subnetwork of the original network. The new network has a comparable number of short cycles to the original, but lower typical degree. This should enhance the impact of clustering.

For our calculations in this section, we first isolate the impact of weighted edges by taking a homogeneous population (An external file that holds a picture, illustration, etc.
Object name is rsif20080524e281.jpg=An external file that holds a picture, illustration, etc.
Object name is rsif20080524e282.jpg=1) and using An external file that holds a picture, illustration, etc.
Object name is rsif20080524e283.jpg. We vary α in order to set left angle bracketTright angle bracket. We then investigate a heterogeneous population using equation (2.1) with the first four distributions of table 1.

Results for a homogeneous population are shown in figure 10. Because Tuv=Tvu for all pairs, it follows that An external file that holds a picture, illustration, etc.
Object name is rsif20080524e284.jpg=An external file that holds a picture, illustration, etc.
Object name is rsif20080524e285.jpg. If different edge weights were uncorrelated, then the value of An external file that holds a picture, illustration, etc.
Object name is rsif20080524e286.jpg0 would match with figure 2 and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e287.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e288.jpg would match with figure 6. We see, however, that An external file that holds a picture, illustration, etc.
Object name is rsif20080524e289.jpg0 is significantly reduced from the homogeneous unweighted population (but An external file that holds a picture, illustration, etc.
Object name is rsif20080524e290.jpg0,1 remains a good approximation). An external file that holds a picture, illustration, etc.
Object name is rsif20080524e293.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e294.jpg are mildly reduced close to the threshold. These observations are consistent with our expectation that clustering should be accentuated by incorporating edge weights. Although the predictions for An external file that holds a picture, illustration, etc.
Object name is rsif20080524e295.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e296.jpg are not far off, we expect that they would improve if we adjusted the degree distribution to match that of the effective network on which the disease spreads.

Figure 10
(a) An external file that holds a picture, illustration, etc.
Object name is rsif20080524e31.jpg0,r (black solid curve, unclustered An external file that holds a picture, illustration, etc.
Object name is rsif20080524e32.jpg0; black dashed curve, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e33.jpg0,0; grey solid curve, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e34.jpg0,1; dotted curve, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e35.jpg0,2; dot-dashed curve, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e36.jpg0,3; grey dashed curve, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e37.jpg0,4) and (b) An external file that holds a picture, illustration, etc.
Object name is rsif20080524e38.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e39.jpg for the weighted EpiSimS network with a homogeneous population.

When the population is moderately heterogeneous (figure 11), we still find that An external file that holds a picture, illustration, etc.
Object name is rsif20080524e297.jpg0,1 is a reasonable approximation to the true value of An external file that holds a picture, illustration, etc.
Object name is rsif20080524e298.jpg0; however, it slightly underestimates An external file that holds a picture, illustration, etc.
Object name is rsif20080524e299.jpg0 as left angle bracketTright angle bracket grows. Unfortunately, the analytic calculation of An external file that holds a picture, illustration, etc.
Object name is rsif20080524e300.jpg0,1 is much more difficult, and so it is more appropriate to use simulations to estimate its value. If there were no correlation between weights of different edges, then the calculation would reduce to that of §4.

Figure 11
(a–d) An external file that holds a picture, illustration, etc.
Object name is rsif20080524e40.jpg0,r with heterogeneous transmissibility and weighted edges on the EpiSimS network (black solid curve, unclustered An external file that holds a picture, illustration, etc.
Object name is rsif20080524e41.jpg0; black dashed curve, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e42.jpg0,0; grey solid curve, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e43.jpg0,1; dotted curve, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e44.jpg0,2; dot-dashed curve, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e45.jpg0,3; grey dashed curve, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e46.jpg0,4).

We consider An external file that holds a picture, illustration, etc.
Object name is rsif20080524e301.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e302.jpg in figure 12. The unclustered predictions are reasonable approximations of the actual values. The error is larger than before because we have combined two effects (edge weights and heterogeneity) that both accentuate the impact of clustering. In spite of this, the predicted values of An external file that holds a picture, illustration, etc.
Object name is rsif20080524e303.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e304.jpg are not far off, and the direction of the error is consistent: the unclustered prediction is always an overestimate.

Figure 12
Simulated (a) An external file that holds a picture, illustration, etc.
Object name is rsif20080524e47.jpg and (b) An external file that holds a picture, illustration, etc.
Object name is rsif20080524e48.jpg (symbols) for the weighted EpiSimS network compared with predictions in unclustered networks with the same edge weight distribution (curves).

6. Discussion

We have investigated the interplay of clustering, node heterogeneity and edge weights on the growth rate An external file that holds a picture, illustration, etc.
Object name is rsif20080524e305.jpg0, probability An external file that holds a picture, illustration, etc.
Object name is rsif20080524e306.jpg and size of epidemics An external file that holds a picture, illustration, etc.
Object name is rsif20080524e307.jpg in social networks. For unclustered networks with independently distributed edge weights, it is possible to predict all these quantities analytically. Under weak assumptions, we can accurately estimate An external file that holds a picture, illustration, etc.
Object name is rsif20080524e308.jpg0, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e309.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e310.jpg for clustered networks.

If the typical degrees are not small, then for a given average transmissibility and degree distribution, the following can be stated.

  •  The dominant effect controlling the growth rate of epidemics is clustering. Increased clustering reduces An external file that holds a picture, illustration, etc.
Object name is rsif20080524e311.jpg0.
  •  The dominant effect controlling the probability of epidemics is heterogeneity in infectiousness. Increased heterogeneity reduces An external file that holds a picture, illustration, etc.
Object name is rsif20080524e312.jpg.
  •  The dominant effect controlling the size of epidemics is heterogeneity in susceptibility. Increased heterogeneity reduces An external file that holds a picture, illustration, etc.
Object name is rsif20080524e313.jpg.

We are thus able to neglect clustering and still closely estimate An external file that holds a picture, illustration, etc.
Object name is rsif20080524e314.jpg based only on the degree distribution and the out-transmissibility pdf Qout. The estimate for An external file that holds a picture, illustration, etc.
Object name is rsif20080524e315.jpg depends only on the degree distribution and the in-transmissibility pdf Qin. The impact of clustering is significant in altering An external file that holds a picture, illustration, etc.
Object name is rsif20080524e316.jpg0, and its impact is mildly enhanced by heterogeneities. This enhancement occurs because the probability of following all edges of a cycle is increased if some of the edges are correlated owing to the heterogeneity. If heterogeneity is large, clustering may play a small role in moving the epidemic threshold, but otherwise its effect on the threshold is negligible. In networks with small typical degree, it has been observed that clustering can modify An external file that holds a picture, illustration, etc.
Object name is rsif20080524e317.jpg or An external file that holds a picture, illustration, etc.
Object name is rsif20080524e318.jpg (Serrano & Boguñá 2006a,b), which is consistent with our estimates.

If edge weights are included, but are independently distributed, then their impact is in modifying Qin(Tin) and Qout(Tout). The resulting modification may be calculated explicitly, and edge weights have no further effect. If edge weights are correlated, then they have a more important role in governing the behaviour of epidemics, particularly if higher weight edges tend to be the clustered edges (as frequently occurs in social networks). If this happens, then the impact of clustering is enhanced and the growth rate of epidemics is further reduced.

When we move from predicting An external file that holds a picture, illustration, etc.
Object name is rsif20080524e319.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e320.jpg to predicting An external file that holds a picture, illustration, etc.
Object name is rsif20080524e321.jpg0, we find that the growth rate is well approximated by An external file that holds a picture, illustration, etc.
Object name is rsif20080524e322.jpg. This may be calculated analytically in the homogeneous case (constant T). When heterogeneities are included, the calculation becomes harder, and when edge weights are included it becomes largely intractable. However, these are easily estimated through simulation.

These observations show that using An external file that holds a picture, illustration, etc.
Object name is rsif20080524e323.jpg0 to predict An external file that holds a picture, illustration, etc.
Object name is rsif20080524e324.jpg will generally be inadequate. In a homogeneous but clustered population, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e325.jpg0 is reduced but An external file that holds a picture, illustration, etc.
Object name is rsif20080524e326.jpg is unaffected, and so predictions of An external file that holds a picture, illustration, etc.
Object name is rsif20080524e327.jpg based on An external file that holds a picture, illustration, etc.
Object name is rsif20080524e328.jpg0 will be too small. In networks that are not clustered but have heterogeneities in susceptibility, An external file that holds a picture, illustration, etc.
Object name is rsif20080524e329.jpg0 is unaffected but An external file that holds a picture, illustration, etc.
Object name is rsif20080524e330.jpg is substantially reduced. Consequently, the value of An external file that holds a picture, illustration, etc.
Object name is rsif20080524e331.jpg predicted from An external file that holds a picture, illustration, etc.
Object name is rsif20080524e332.jpg0 will be too large.

Perhaps our most important conclusion about clustering is that it plays an important role in altering the growth of an epidemic, but it plays only a small role in determining whether an epidemic may occur or how big it would be. If the relevant questions are, ‘how likely is an epidemic and how large would it be?’, then the modeller may proceed ignoring clustering. If however, the question is ‘how fast will an epidemic grow initially?’, then clustering must be considered, but only enough to calculate An external file that holds a picture, illustration, etc.
Object name is rsif20080524e333.jpg0,1.

Our results have implications for designing intervention strategies. A number of strategies are available to control epidemic spread, including travel restrictions, quarantines and vaccination. Most of the mathematical theory predicting the effects of these strategies has been developed under the assumption of no clustering. Most immediately, if we measure An external file that holds a picture, illustration, etc.
Object name is rsif20080524e334.jpg0=2 at the early stages of an epidemic, traditional approaches will suggest that vaccinating just over half of the population will bring the epidemic below threshold. However, if the population is clustered, then the observed An external file that holds a picture, illustration, etc.
Object name is rsif20080524e335.jpg0 was already affected by the fact that some transmission chains were redundant. Following vaccination, some of these chains will no longer be redundant and the disease may still spread with An external file that holds a picture, illustration, etc.
Object name is rsif20080524e336.jpg0>1.

Achieving a better understanding of the effect of clustering further helps to guide our intuition when choosing between strategies. For example, let us assume that we have the choice between two strategies: in the first, we stagger work schedules in such a way that a typical person's contacts are reduced by one-third; in the second, we implement population-wide behaviour changes so that the same reduction in number of contacts is achieved, but the work contacts are unaltered. The first reduces clustering while the second increases the relative frequency of clustering. The value of An external file that holds a picture, illustration, etc.
Object name is rsif20080524e337.jpg0 is smaller in the second case than in the first because of the larger clustering, but An external file that holds a picture, illustration, etc.
Object name is rsif20080524e338.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e339.jpg are reduced by a comparable amount in both cases. Which strategy is best depends on our goals and relative costs.

Strategies that enhance heterogeneity in infectiousness or susceptibility can be important to help reduce An external file that holds a picture, illustration, etc.
Object name is rsif20080524e340.jpg or An external file that holds a picture, illustration, etc.
Object name is rsif20080524e341.jpg, even when there is little impact on An external file that holds a picture, illustration, etc.
Object name is rsif20080524e342.jpg0. Depending on which quantity we want to minimize, different choices will be optimal. Consider a choice between vaccinating all individuals with a vaccine that reduces Tuv by a factor of 1/2 for all pairs u and v or a contact tracing strategy that will remove half of all new infections before they have a chance to infect anyone. Both strategies reduce left angle bracketTright angle bracket by a half. However, the first reduces Tout uniformly, while the second increases heterogeneity in Tout. Thus, if we have the choice of the two strategies, then contact tracing is more likely to eliminate the disease before an epidemic can happen. If our choice is instead between a global vaccine reducing Tin by a factor of 1/2 for all individuals and a completely effective vaccine that is only available for half of the population, the latter choice will be more effective for reducing An external file that holds a picture, illustration, etc.
Object name is rsif20080524e343.jpg.

Acknowledgements

This work was supported by the Division of Mathematical Modeling at the UBC CDC under CIHR (grant nos. MOP-81273 and PPR-79231) and the BC Ministry of Health (Pandemic Preparedness Modeling Project), by DOE at LANL under contract DE-AC52-06NA25396 and the DOE Office of ASCR programme in Applied Mathematical Sciences and by the RAPIDD programme of the Science & Technology Directorate, Department of Homeland Security and the Fogarty International Center, National Institutes of Health. Luís M. A. Bettencourt contributed greatly to the early development of this work. I am grateful to Sara Y del Valle for providing the EpiSimS network data.

Footnotes

We follow Ludwig (1975) in using the term rank rather than generation which has been used elsewhere, but is potentially ambiguous. The rank is the smallest number of infectious contacts between the index case and a node. It is possible that a different path takes less time. The path infection actually follows is the path that is shorter in time, rather than number of links.

Early noise controls how quickly outbreaks become epidemics, and so once stochastic effects become small, the curves appear to be translations in time. We note that it is common to consider the temporal average of a number of outbreaks. However, prior to taking an average, the curves should be shifted in time so that they coincide once the stochastic effects are no longer important. Failure to do so underestimates the early growth, peak incidence and late decay, while it overestimates the epidemic duration. This can lead to an incorrect understanding of ‘typical’ outbreaks.

If An external file that holds a picture, illustration, etc.
Object name is rsif20080524e215.jpg is small, then the relative change may be large, but the absolute change is small.

We can use this notation because the ordering assumption allows us to uniquely identify An external file that holds a picture, illustration, etc.
Object name is rsif20080524e259.jpg from Tout and An external file that holds a picture, illustration, etc.
Object name is rsif20080524e260.jpg from Tin. If the ordering assumption fails, similar results hold, but the notation is more cumbersome.

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