Unraveling the Disease Consequences and Mechanisms of Modular

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ECOLOGY AB This is because, in principle, we expect the maximum modularity Qmax of all networks to be equal to one. However, for smaller networks, Qmax tends to be a much smaller value (Fig. 2A). We therefore estimated relative modularity (obtained by normalizing Q by Qmax ) to facilitate comparison of modular subdivision strength across different animal groups. Animal social networks in our database ranged from homogeneous (Qrel = 0) to highly subdivided structure (Fig. 2A). Animal interactions are dynamic in nature, and therefore social networks of the same set of individuals can also fluctuate between being relatively homogeneous to highly modular over time C D (Fig. 2B). To gain intuition about how modular organization in animal social networks influences epidemiological outcomes, we first generated homogeneous null networks for each empirical animal social network in our database. In homogeneous null networks, higher-order organization in social networks (such as modular structure, clustering coefficient, cliquishness, and degree homophily) are randomized, weakening relative modularity (SI Appendix, Fig. S1), but individual contact heterogeneity (degree distribution) of the empirical network is preserved. We next performed susceptible–infected–recovered (SIR) disease simulations of a moderately transmissible infectious disease (basic reproduction number, R0 = 1.2) through these empirical networks and their homogeneous null network counter- Fig. 1. Schematic representation of a homogeneous social network with parts. Based on previous work suggesting a protective effect no modular subdivision (A) and highly modular social networks with high of modular structure on disease risk (3, 6), we expected a cohesion within subgroups (B), high network fragmentation (C), and high lower disease burden (measured as the proportion of popu- subgroup size variation (D). Each network has 60 nodes. Orange edges lation infected) in empirical networks than in homogeneous indicate interactions within a subgroup, and purple edges are between- null networks. Reduced disease burden, however, was apparent subgroup interactions. Q is Newman modularity (11) and Qrel is relative only in social networks with Qrel > 0.6 for moderately spread- modularity. ing infectious disease (Fig. 2C and SI Appendix, Fig. S2). Addi- tionally, none of the empirical networks demonstrated a major reduction in disease burden for a highly transmissible infec- between subgroups. Higher values of Newman modularity indi- tious disease (R0 = 4.8, SI Appendix, Fig. S3). This implies that cate stronger subdivisions of social networks. However, using the protective effect of modular subdivisions is realized only Newman modularity to compare modular subdivision across net- at high values of relative modularity, but the presence of a works is problematic because the measure is inherently cor- threshold depends on pathogen transmissibility (SI Appendix, related to the size and number of links in the network (12). Fig. S3).

ABC

Fig. 2. Animal social networks with modular subdivisions. (A) Values of Newman modularity, Q, and maximum possible modularity given the network configuration, Qmax , across all animal groups in the database. Point color represents the relative modularity estimated as the ratio of Q over Qmax . The dashed line represents Q = Qmax and the maximum value of Qrel = 1. (B) Relative modularity values of dynamic interactions in ants (Camponotus fellah), raccoons (Procyon lotor), and field voles (Microtus agrestis), where time points represent consecutive days, weeks, and trapping sessions, respectively. (C) Comparisons between real (filled points) and their homogeneous null networks (tips of arrows) with respect to the percentage of infected individuals (outbreak size) due to an outbreak with basic reproduction number, R0 = 1.2. Point color corresponds to the taxonomic class of the animal group. Social networks with nonsignificant modular subdivision (as indicated by t test analysis) have been excluded in C. The generated homogeneous null networks preserve only the local heterogeneity of contacts among individuals; the arrows therefore indicate the change in direction and magnitude of outbreak size under the scenario in which all higher-order structural complexities (including modular subdivisions) are removed from animal social networks. The shaded area represents the

region where empirical networks tend to experience reduced outbreak size (at Qrel > 0.6; SI Appendix, Fig. S2). The networks that experienced lower disease burden included social networks of raccoons (P. lotor), elephant seals (Mirounga angustirostris), ants (Camponotus pennsylvanicus), bottlenose dolphins (Tursiops truncatus), and Australian sleepy lizards (Tiliqua rugosa). In A and C, we used a randomly selected network snapshot for groups with temporal interaction data.

4166 | www.pnas.org/cgi/doi/10.1073/pnas.1613616114 Sah et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 a tal. et Sah o estimated. not VIF 0.66 average size Subgroup variation cohesion Subgroup cohesion* Subgroup fragmentation* Network size Network Intercept variable Explanatory orga- modular of (Q determinants nization on analysis Multivariable 1. network Table of level the change can interactions social season- with driven species or ally fission–fusion individuals, in nomadic with members species However, species, group efforts. com- between sampling consistent interactions unequal for dynamic with database our species in across group parison animal each for tions state reproductive female 16). and 15, (13, hierarchy, resource dominance in as ratio), (such changes and sex factors by demographic dolphins, about aggression, brought intragroup bottlenose modularity. be availability, can network chimpanzees, and in lions) on (e.g., elks, African species effect social hyenas, many negative in spotted common weak subgroup are sizes in a subgroup variation Fluid have found to we fragmen- network cohesion, sizes to subgroup contrast and In [e.g., (15)]. tation competition monkeys intragroup spider brown high den- in or resource distribution, on constraints resource by sity, fragmen- caused network be can High conversely, (14)]. tation, elephants African matrilineal in or [e.g., together, in foraging descent [e.g., spent requirements time nutrient of (13)], similar hyenas result spotted to direct due a cohe- bonds be effects social subgroup can strong positive Cohesiveness factors, strong organization. three had modular on fragmentation the network Of and 1). siveness (Table variation subgroup size and net- fragmentation, social network animal cohesion, in subgroup works: organization modular driving features work correla- networks temporal intragroup these below). avoid analyze one separately we to (and selected metrics analysis network randomly in the we tion for network, network social tem- a snapshot multiple subgroups had of that among snapshots groups individual poral contacts For of in variation). degree number variation (subgroup in and variation (degree); in and contacts variation in average and variation average sizes; subgroups); network; subgroup within the occur across cohesion that subgroup contacts total propor- the of as measured tion cohesion, (subgroup the subgroup in own fea- association subgroups with social preferential of network fragmentation); the number (network these the) in network of social present of (log individuals size); some of (network there- to network number We driv- attention the networks. patterns including our tures, social interaction turned animal of first in features fore This subdivision varying networks. modular to mod- across ing the due inconsistent above be fairly reduction could was size outbreak threshold Addi- of ularity threshold. extent certain a the above tionally, only and strength modularity, the relative on of depending context-dependent, Networks. are structure Social ular Animal in 2C Organization Fig. Modular of Determinants ugopdge aito 0.01 Sociality 0.07 — effects Random variation degree Subgroup variation degree Individual average degree Individual variation* size Subgroup aooi order Taxonomic seik n odtx niaesgicne xlntr aibe with variables Explanatory significance. indicate text bold and Asterisks interac- of snapshot single a only considered have we far, Thus, net- three identified model beta-regression mixed-effects The > eedopdfo h oe,adteeoeterefc ie were sizes effect their therefore and model, the from dropped were 5 ugssta h pdmooia osqecso mod- of consequences epidemiological the that suggests rel cosgop fdfeetspecies different of groups across ) fetsz 5 ofiec intervals confidence 95% size Effect −0.03 −0.07 −0.42 −0.13 1.14 — aineetmt (σ estimate Variance <0.001 0.066 .4t 0.08 to −0.14 0.03 to −0.17 .8t 0.10 to −0.08 .9to −0.59 .6to −0.26 .3t 1.25 to 1.03 0.82 to 0.51 .4t 0.18 to 0.04 — — −0.25 −0.01 2 ) aacsteoealdsaesra,adtu oua structure, modular thus and spread, disease tradeoff overall This the 3C). net- (Fig. balances transmission these disease in global within contacts reduce contacts works intersubgroup of low density high conversely, a homoge- subgroups; to with due transmission Compared local higher 3C). 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ECOLOGY infection in modular social networks. The structural trapping effect (albeit weak) has been observed in a ﬁeld study of pneumonia transmission in highly subdivided networks of bighorn lambs (SI Appendix, Fig. S13) (21) and others (SI Appendix, Table S2). We also found empirical evidence of structural delay in the spread of mycoplasma in the low-modular network, but highly cohesive subgroups, of house ﬁnches described in ref. 22 (SI Appendix, Fig. S14). A high probability of extinction associated A B with the structural trapping effect has been reported in theoretical studies of disease spread in spatially structured metapopulation models (7, 8). We note that the magnitude of structural trapping and delay depends on the contagiousness of the infection—moderately transmissible pathogens spread slowly through populations and are therefore able to perceive modular structures present in networks. For rapidly spreading pathogens, highly fragmented and cohesive networks do not contain, but delay, the spread of disease to other subgroups, creating only a structural delay effect.

CDIs Knowledge of Modular Structure Necessary (and Sufﬁcient) to Predict Disease Outcomes? Our investigations of disease consequences on synthetic networks suggest that only modular subdivision beyond a certain threshold can reduce disease burden of moderately transmissible pathogens. Other topological features Fig. 3. Overall disease implications of modular subdivisions in synthetic of the network—and, in particular, heterogeneity in host con- modular networks. (A) Average outbreak size, measured as the percentage of infected individuals, over increasing subdivided social networks and tact (23)—are, however, bound to confound the effect of mod- pathogen transmissibility. Outbreak size values have been normalized to the ularity on infectious disease spread. Our work thus far assumes maximum observed outbreak. The solid line indicates epidemic threshold— high variation in contacts among individuals (24), although fac- namely, the threshold value of pathogen contagiousness below which there tors including sociality (25) and nonpersistent space use (26) are is no risk of a large outbreak (> 10% outbreak size; Materials and Methods). known to reduce variation in individual contact rate. Does indi- (B) Epidemic robustness of networks with increasing value of relative mod- vidual contact heterogeneity then affect the way modular organi- ularity, measured as the percentage reduction in outbreak size compared zation inﬂuences disease spread? We found that for social net-

with outbreak size experienced by homogeneous (Qrel = 0) networks. The works with low contact heterogeneity, high modularity brings solid line indicates the modularity threshold, where networks experience at about a greater reduction in disease burden compared with net- least a 10% reduction in outbreak size. (C) Infection transmission events, works with high contact heterogeneity (SI Appendix, Fig. S8). expressed as the percentage of total outbreak size, within subgroups (local) These results imply that the local contact patterns should also and between subgroups (global); pathogen transmissibility = 0.18. (D) Dis- be considered when estimating the reduction of disease burden ease implications of modular subdivisions as a function of subgroup cohe- in extremely subdivided social networks. sion and network fragmentation (measured as the log-number of subgroups Although modular subdivision is a common feature observed in present in the network). many animal groups, it is often correlated with other higher-order network structures, such as clustering coefﬁcient and degree homophily (27). In empirical networks, it is therefore impor- under most conditions, does not lower disease burden. The con- tant to ask whether modular organization has any epidemio- ditions under which modularity does reduce disease burden of logical consequence above and beyond these network proper- social networks is largely determined by the two factors underly- ties, after accounting for contact heterogeneity. To answer this ing network modularity—network fragmentation and subgroup cohesion (Table 1). Synthetic networks that possess both high subgroup cohesion and high network fragmentation experience lower outbreak size compared with homogeneous and low mod- 70 A B ular social networks (Fig. 3C and SI Appendix, Fig. S6). In addi- Cohesion Percent outbreaks 60 60 tion, high variation in subgroup sizes along with high subgroup 1.00 5 0.75 cohesion and fragmentation further reduce outbreak size and 0.50 10 0.25 15 50 0.00 increase outbreak duration in social networks (SI Appendix, Fig. 40 S7). Disease consequences in real animal social networks, however, tend to be driven by subgroup cohesion because they typi- 40 cally exhibit a much lower range of fragmentation and subgroup 20 size variation compared with subgroup cohesion (SI Appendix, 30 Figs. S4 and S5).

Time to disease invasion 20 Uninfected subgroups (%) Uninfected 0 Mechanisms That Drive the Impact of High Modular Structure on Disease Spread: Structural Delay and Trapping. We next examined 12345678910 0.00 0.25 0.50 0.75 1.00 the mechanisms by which highly fragmented social networks Ordered subgroups Subgroup cohesion with cohesive subgroups experience reduced disease burden. The Fig. 4. Mechanisms behind the effect of modular organization on disease presence of only a few interactions between highly cohesive sub- spread of a moderately transmissible pathogen (= 0.1). (A) Structural delay groups increased the amount of time it took for an infection effect in a moderately fragmented network (fragmentation = 2.30): We con- to spread from one subgroup to another (the structural delay sidered the time to disease invasion for a subgroup as the number of time effect; Fig. 4A), causing longer disease outbreaks. When net- steps it takes for at least 2% of individuals to acquire infection. Subgroups works were fragmented into highly cohesive subgroups, the struc- at the x axis are sorted according to the increasing disease invasion time. tural delay imposed by cohesion caused infection to be local- (B) Structural trapping effect in a highly fragmented network (fragmenta- ized to a small proportion of subgroups before dying out (Fig. tion = 4.83): High fragmentation and subgroup cohesion localize infection to 4B). Popularly known as structural trapping (20), this effect also a small proportion of subgroups in the social network. Structural trapping reduces the overall likelihood of a major outbreak by a novel also increases the likelihood of stochastic extinction of disease outbreaks.

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ECOLOGY This study suggests the presence of a high modularity thresh- size, etc., toward network modularity, we ran a mixed-effects beta regres- old, above which social networks experience reduced outbreak sion model using the glmmADMB package (Version 0.8.3.3) in R (Version size. However, we caution against the use of a single mod- 3.2.3). We treated the species nested within the taxonomic order of the ani- ularity threshold that is applicable to all systems. Multiple mal group as a random effect in the model to account for any correlations behavioral, ecological, and epidemiological factors—including in network metrics. However, the variance accounted for by the nesting of 2 pathogen infectiousness (Fig. 3B), infectious period, local con- species was negligible (σ ≈ 0), and we therefore considered only taxo- tact heterogeneity, seasonality of contacts, and transmission, nomic order as the random effect in the analysis. We also accounted for coupled with recruitment of susceptible through births—can differences between the social organization of different species by defining three broad levels of sociality—relative solitary, social, and fission-fusion— influence the Qrel threshold and the extent to which network fragmentation and subgroup cohesiveness structurally trap and and included sociality as a random effect in the model. delay infection spread. Therefore, to assess the disease implica- Generation of Synthetic Modular and Null Networks. We used the network tions of modular structure observed in specific social groups, we generator described in ref. 17 to generate networks of a specified modu- recommend the use of null networks, as presented in this study. larity and degree sequence, while keeping other network properties close Comparison of disease consequences on empirical network data to homogeneous null networks. For Figs. 3 and 4, we generated synthetic with those on a hierarchy of null networks clarifies whether mod- modular networks with 10,000 nodes, with an exponential degree distribu- ular structure is important. If natural history and limited obser- tion with a mean degree of 10. Two types of null networks were generated vations suggest a species’ network will be highly modular, then for each animal social network. Modular null networks were created by ran- models that incorporate modularity should be used; otherwise, domizing within- and between-subgroup connections. The modular null net- homogeneous null models (based on basic knowledge about networks therefore had identical modular configuration and degree sequence as work size and local heterogeneity) may be sufficient when data- empirical networks, but were random with respect to other higher-order net- limited estimates of epidemic consequences are necessary. work properties. We generated homogeneous null networks by performing simultaneous edge swaps over within- and between-subgroup connections, Materials and Methods which preserves the local contact heterogeneity, but randomizes all higher- order features of the network, including network modularity. Extended methods are provided in SI Appendix, SI Text. Disease Simulations. We performed Monte Carlo simulations of a discrete- Measuring Modularity. We used modularity (Q) proposed by Newman (11) time SIR model of disease spread. Because we were interested only in major to measure the strength of modular organization in networks. Modularity outbreaks, we considered only those simulations in our calculations where Lw 2 PK k Lk can be defined as Q = k=1 L − L ), where Lk is the total number at least 10% of the population acquired infection. Transmission occurred according to a pathogen transmissibility, defined as the probability of trans- of edges in a subgroup k, of which Lw are the edges within the subgroup, k mission from an infected to susceptible host during the period when the and L is the number of total edges in the network. Community structure, or the number and composition of subgroups, for each animal social network infected host is infectious. The recovery probability was assumed to be con- was estimated by using the Louvain method (30). The highest possible mod- stant across hosts. ularity in a network (Q ) is achieved when all individuals in a subgroup k max ACKNOWLEDGMENTS. We thank Phil O’Neill for feedback on a previous only interact with each other and no edges are present between subgroups version of the manuscript. We thank all the researchers who have made their (i.e., subgroups are disjointed). In other words, Qmax of a network is when animal social network and disease data openly accessible, without which w PK Lk Lk Lk = Lk, and can be written as Qmax = k=1 L 1 − L . We measured this study would not have been possible. This work was supported by the National Science Foundation Ecology and Evolution of Infectious Diseases the relative modularity of networks as Q = Q . rel Qmax Grant 1216054. S.T.L. was supported by a postdoctoral Endeavour Research Fellowship from the Australian Department of Education and Training. Any Mechanisms of Modular Organization. To examine the relative contribution mention of trade, firm, or product names is for descriptive purposes only of factors such as network size, network fragmentation, average subgroup and does not imply endorsement by the U.S. government.

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