Île Sans Fil (ÎSF) is a not-for-profit organization established in 2004 in Montreal, Canada, that operates a system of public Internet hotspots. Hotspots are located in cafes, community and recreation centers, salons, markets, and other small businesses and public places. They are maintained by ÎSF staff and volunteers with the Internet connection provided by the establishment. We analyzed the database of all connections to the system of 352 hotspots between August 2004 and March 2010. Raw data from the ÎSF database consisted of 2.18 million connection records. Each record included an anonymized user ID, latitude and longitude coordinates for each ÎSF hotspot location, connection and disconnection times, and the unique media access control address for the user’s wireless device. The data reported in this paper are available from the Community Resource for Archiving Wireless Data at Dartmouth (CRAWDAD) archive [27].

We built a contact network by interpreting each individual user as a node and concurrent ÎSF usage at the same hotspot as an edge. This preliminary network contained 114,810 nodes and 1.2 million edges. It contained both self-loops (users connecting multiple devices at once) and parallel edges (pairs of users with multiple overlapping hotspot visits) that we removed to produce a nonredundant network with 637,430 edges. We analyzed the largest connected component of this network, which consisted of 103,425 nodes and 630,893 edges.

Modularity (Q) quantifies the extent of community structure in a network relative to a comparable random network. Given a network and a particular partitioning of the nodes into communities, Q is defined as the number of edges contained within communities minus the number of edges expected to fall within communities if the edges were distributed randomly (preserving the degrees of all nodes), normalized for network size. Q ranges from zero for randomly connected networks to greater than 0.3 for networks with substantial community structure [28]. We used a heuristic method to divide the Montreal network into a set of communities that maximized Q using an algorithm [28] that initially assigned each node to its own community and then iteratively aggregated whichever pair of communities resulted in the largest increase in Q. We identified 1420 distinct communities associated with a Q value of 0.69.

Epidemic curves were simulated by EpiFire [29] using a chain-binomial [30] network epidemic simulator with a susceptible-infectious-recovered state progression. Each epidemic begins with all nodes susceptible except for a single, randomly selected, infectious node. Nodes remain infectious for a fixed period with arbitrary time units, after which they become recovered for the remainder of the simulation. Transmission from infectious nodes to susceptible neighbors is attempted once per time unit with transmission probability T_cb. We assumed an infectious period of 5 time units because it yielded sufficient temporal resolution to distinguish epidemic waves. T_cb relates to the percolation transmission probability as given by T=1–(1–T_cb)^d, where d is the infectious period. We simulated epidemics across 3 transmission scenarios defined by the value of the basic reproduction number, R₀, which is defined as the expected number of secondary cases produced by a typical infection in a completely susceptible population: a low R₀=1.9, similar to estimates for recent influenza pandemics [31-33]; a moderate R₀=2.4, which maximizes the probability of multiwave epidemics and is in the range estimated for the 1918 influenza pandemic in the United States [34,35]; as well as a high R₀=7.5, where spread is rapid both within and between communities. Finally, we performed a sensitivity analysis with respect to the length of the infectious period and considered values of 1, 5, 10, and 50 time steps.

To automatically identify epidemic curves exhibiting multiple waves, we defined a new 2-peak metric (TP), indicating the depth of the deepest valley in the epidemic time series (specifically, the geometric mean of the heights of on each side of the deepest valley). Additional details are provided in Multimedia Appendix 1. Example epidemic curves and their corresponding value of TP are shown in the supplementary information (see Figure S1 in Multimedia Appendix 2). The distribution of TP values tended to be bimodal with single-wave epidemics yielding values close to zero and multiwave epidemics yielding higher values (see Figure S2 in Multimedia Appendix 2).

We adapted methods from percolation theory [36] to estimate global and community-specific values of R₀, final epidemic sizes, and epidemiological connectivity among different communities to test the hypothesis that multiwave epidemics occur in the absence of between-community degree assortativity. The details and derivations of these percolation-based quantities can be found in Multimedia Appendix 1.

To examine the epidemiological impact of community structure we constructed “null” networks that shared many properties of the original Montreal network (eg, the degree distribution) but lacked community structure. Specifically, we iteratively randomized the network by shuffling connections; that is, we chose random pairs of connections and swapped the ends (eg, A-B, C-D became A-D, C-B). This slowly degraded community structure while preserving the number of contacts for each individual. We selected a fraction f of edges and broke them to form stubs (half-edges still attached to their nodes). Then, the list of stubs was randomized and each sequential pair of stubs was connected. We eliminated newly formed self-edges and redundant edges via edge swaps with randomly chosen edges, leading to randomization of slightly more than the intended fraction of edges (~1.01f). Modeling each community as a semirandom network with the observed within- and between-community degree distributions, we estimated the impact of random shuffling and variation in R₀ on the epidemiological proximity of communities. We built 2048 new networks by randomly shuffling from 0.1% of edges (631 edges) to 3% of edges (18,927 edges) in increments of 0.1%. All estimates are averages based on the networks built for each shuffling level.

Community structure is a prominent feature of the Montreal network, with the 3 major communities exhibiting substantially different epidemiological thresholds and dynamics. However, community structure alone is insufficient to produce multiwave epidemics because direct contacts between highly connected individuals in different communities can fuel rapid intercommunity transmission. It is the insularity of the most intraconnected community that makes multiple waves not just possible, but likely, when R₀ resembles that estimated for the 1918 influenza pandemic. The Montreal network shows anecdotally that both strongly connected and insular communities can emerge naturally in human social systems and exist side-by-side. However, insularity is fragile, disappearing with minimal perturbation to the network. In contrast, the presence of strong bridges between communities is more robust and arises by random (or other) processes that link highly connected individuals to one another. Conversely, we expect that social differences, geographic heterogeneity in contact patterns (eg, city vs suburbs), and polarizing events would generate insularity.

The 3 large Montreal hotspot communities exhibit substantially different epidemiological characteristics, including epidemic thresholds and sensitivity to the reproduction number. For mildly transmissible diseases just above the epidemic threshold, some communities may be fully protected by the sparseness of their within-community network. Well above the epidemic threshold, the attack rate may vary considerably among communities, depending again on the structure of their within-community network.

Ball and Neal [40] introduced a theoretical framework addressing the epidemiological interplay between local and global connectivity, and showed that transient long-range connections can fuel epidemics even when local network structure is too sparse to sustain epidemics. Although their model considers transient global contacts rather than fixed community structure, a similar approach may ultimately provide a theoretical perspective on the epidemiological phenomena we observed in the Montreal network.

The insularity of the most intraconnected Montreal community is likely to produce a multiwave epidemic when an outbreak originates in that community and the R₀ is moderate, close to that estimated for the 1918 influenza pandemic. For less contagious diseases, epidemics are unlikely to escape beyond this community; for highly contagious diseases, epidemics flow readily among communities. Our results thus support an alternative nontemporal explanation for multiwave epidemics—insular community structure—that arises naturally from social network structure fundamental to human interactions.

All historical influenza pandemics have produced multiple epidemic waves in many North American cities [1]. Specifically, Montreal experienced 2 distinct waves of influenza-related mortality during the 1957-1958 pandemic, which did not occur elsewhere in Quebec. McDonald [41] attributed this difference to population density. The 2009 pandemic also produced dual waves in Montreal [42]. Prior attempts to explain these patterns have largely overlooked population structure in favor of dynamic phenomena, such as pathogen evolution or host behavior [14,15,43,44]. This omission stems partly from the chronic lack of data and insight into the structure of urban contact networks. Although this remains a challenge, our study suggests that urban community structure may naturally produce multiwave pandemics, even without any temporal forcing.

The major limitation of our study lies in the inherent difficulty of capturing and characterizing human contact patterns at an individual level. The Montreal ÎSF network excludes many interactions important to disease transmission, such as those occurring in homes and schools. In addition, some of the colocation contacts in our dataset may not have been sufficient for disease transmission. For these 2 reasons, the ÎSF network does not fully reflect the human contact patterns responsible for disease spread; however, it reveals previously uncharacterized urban-scale community structure that may reflect fundamental geographic, economic, and cultural processes that shape physical contact networks.

Anticipating the emergence of secondary epidemic waves is vital to epidemic and pandemic preparedness and response. For example, in the midst of an outbreak, it may allow public health officials to target interventions toward subpopulations still at risk for significant transmission. Identifying informative indicators of urban community structure, including the insularity estimates proposed in this study, will allow us to construct temporal epidemiological risk maps and target interventions (eg, vaccination) toward highly connected individuals reaching outside their own communities that can potentially serve as bridge superspreaders. In the Montreal network, communities had distinct geographic signatures, suggesting that investigations of community structure may also facilitate the effective placement of surveillance or intervention sites such as vaccination clinics.