Social behavior is fundamental to the survival of many species. It allows the formation of social groups providing fitness advantages from greater access to resources and better protection from predators (Krause and Ruxton, 2002). Structure within these groups can be found in the way individuals communicate across space, cooperate in sexual or parental behavior, or clash in territorial or mating conflicts (Hinde, 1976). While animal societies are usually studied independently of each other, studying how they differ in these regards has potential to reveal new insights into the nature of social living (Sah et al., 2018; Dunbar and Shultz, 2010).

When social interaction requires shared physical space it can also be a conduit for the transmission of infectious disease (Altizer et al., 2003). In a typical infectious disease model, if the disease spreads through the environment then the transmission rate is assumed to scale proportionally to the local population density (de Jong et al., 1995; Hopkins et al., 2020). Alternatively, if transmission requires close proximity encounters that only occur between bonded individuals then we expect social connectivity to determine the outcome. These two paradigms are known in the literature as density-dependence and frequency-dependence (Silk et al., 2017).

The problem, however, is that real diseases are not so easy to categorize (Patterson and Ruckstuhl, 2013). For example, as social groups grow in size, new bonds must be created to maintain cohesiveness (Lehmann et al., 2007). To manage the time and cognitive effort required to create these bonds, individuals tend to interact mostly with a small number of close companions while maintaining cohesion with the wider group through less frequent contact (Silk, 2007; Sueur et al., 2011; Dakin and Ryder, 2020). For an infectious disease, this creates fewer transmission opportunities than we would expect to see in a group with highly fluid social dynamics. The extent to which group size amplifies the transmission rate therefore depends on how individuals choose to distribute their social effort between strong and weak ties (Karsai et al., 2014).

While transmission rate has been observed to scale non-linearly with group size for a number of disease systems (Cross et al., 2013; Smith et al., 2009; Silk et al., 2017), it remains unclear how much this dependency is related to the internal social structure of the group; few studies observe social dynamics at sufficient detail while simultaneously monitoring the disease status of each individual. In the absence of direct observations, our contribution to this discussion centers around modeling; incorporating empirical social data with computational simulations. We address two specific questions. Firstly, can we quantify the variability in how individuals choose to distribute their social effort within a group, and secondly, what will this tell us about the effect that population density has on disease transmission?

In the first part of this paper, we introduce a mathematical model founded on the concept of social fluidity which we define as variability in the amount of social effort the individual invests in each member of their social group. Using openly available data, we estimate the social fluidity of 57 human and animal social systems. In the second part, we derive an expression for the basic reproductive number of an infectious disease in the social fluidity model and demonstrate its accuracy in predicting simulated outcomes. Furthermore, social fluidity emerges as a coherent mathematical framework providing the smooth connection between density-dependent and frequency-dependent disease systems.

Social behavior model

Consider a closed system of $N$ individuals and a set of interactions between pairs of individuals that were recorded during some observation period. These observations can be represented as a network: each individual, $i$, is a node; an edge exists between two nodes $i$ and $j$ if at least one interaction was observed between them; the edge weight, $w_{i,j}$, denotes the number of times this interaction was observed. The total number of interactions of $i$ is denoted strength, $s_i={\sum }_j{w}_{i,j}$, and the number of nodes with whom $i$ is observed interacting is its degree, k_i (Barrat et al., 2004).

We define $x_{j|i}$ to be the probability that an interaction involving $i$ will also involve node $j$. Therefore, the probability that at least one of these interactions is with $j$ is $1-{(1-x_{j|i})}^{s_i}$. The main assumption of the model is that the values of $x_{j|i}$ over all $i,j$ pairs are distributed according to a probability distribution, $\rho (x)$. Thus, if a node interacts $s$ times, the marginal probability that an edge exists between that node and any other given node in the network is

(1) $\mathrm{\Psi }(s)=1-\int \rho (x){(1-x)}^s𝑑x.$

Technically, $\rho (x)$ is the distribution of marginal ${\displaystyle x_{i|j}}$ values of the joint probability distribution $\rho (x)$ where $\mathrm{X}$ is a matrix whose $\mathrm{i}\mathrm{,}\mathrm{j}$ entry is -1 if ${\displaystyle i=j}$ and $x_{j|i}$ otherwise. While the values of $x_{j|i}$ are subject to network interdependencies, specifically $\mathrm{A}\mathsf{}\mathrm{X}\mathrm{=}{\mathrm{X}}^{\mathrm{T}}\mathsf{}\mathrm{A}$ and ${\displaystyle \mathbf{1}=\mathbf{0}}$, where $\mathrm{A}$ is any diagonal matrix with positive entries, and ${\displaystyle \mathbf{1}}$ and ${\displaystyle \mathbf{0}}$ are column vectors of length $\mathrm{N}$ containing only 0 and 1, we do not take these constraints into account when estimating ${\displaystyle \rho }$. 

Our goal is to find a form of ρ that accurately reproduces network structure observed in real social systems. Motivated by our exploration of empirical interaction patterns from a variety of species, we propose that ρ has a power-law form:

(2) ${\displaystyle \rho (x)=\frac{\varphi {ϵ}^{\varphi }}{1-{ϵ}^{\varphi }}{x}^{-(1+\varphi )}\mathrm{f}\mathrm{o}\mathrm{r}ϵ<x<1,}$

where $\varphi$ controls the variability in the values of $x$, and $ϵ$ simply truncates the distribution to avoid divergence. The form of $\rho (x)$ was chosen for its analytical tractability but other heavy-tailed distributions produce a similar result (Figure 1—figure supplement 1 ). Combining (1) and (2) we find

(3) $\mathrm{\Psi }(s,\varphi ,ϵ)=1-\frac{\varphi {ϵ}^{\varphi }{(1-ϵ)}^{s+1}}{(1-{ϵ}^{\varphi })(s+1)}{}_{2}F_1(s+1,1+\varphi ,s+2,1-ϵ)$

where the notation ${}_{2}F_1$ refers to the Gauss hypergeometric function (Abramowitz and Stegun, 1975). It follows from ${\sum }_j{x}_{j|i}=1$ that

(4) $N=1+\frac{(1-\varphi )(1-{ϵ}^{\varphi })}{\varphi {ϵ}^{\varphi }(1-{ϵ}^{1-\varphi })},$

which can be solved numerically to find $ϵ$ for given values of $N$ and $\varphi$. The expectation of the degree is $\kappa (s,\varphi ,N)=(N-1)\mathrm{\Psi }(s,\varphi ,ϵ)$.

Figure 1 illustrates how the value of $\varphi$ can produce different types of social behavior. As $\varphi$ is the main determinant of social behavior in our model, we use the term social fluidity to refer to this quantity. Low social fluidity ($\varphi \ll 1$) produces what we might describe as ‘allegiant’ behavior: interactions with the same partner are frequently repeated at the expense of interactions with unfamiliar individuals. As $\varphi$ increases, the model produces more ‘gregarious’ behavior: interactions are repeated less frequently and the number of partners grows faster. While names like ‘social strategy’ and ‘loyalty’ have been applied to similar concepts (Valdano et al., 2015; Miritello et al., 2013), fluidity, as a property of matter, is a useful metaphor for communicating the main idea behind this model.

Figure 1 with 1 supplement see all

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Estimating social fluidity in empirical networks

To understand the results of the model in the context of real systems, we estimate $\varphi$ in 57 networks from 20 studies of human and animal social behavior (further details in the supplement) (Isella et al., 2011; Stehlé et al., 2011a; Mastrandrea et al., 2015; Vanhems et al., 2013; Modlmeier et al., 2019; Blonder and Dornhaus, 2011; Génois et al., 2015; Carter and Wilkinson, 2013; Grant, 1973; Levin et al., 2016; Sailer and Gaulin, 1984; Mourier et al., 2017; Massen and Sterck, 2013; Sade, 1972; Butovskaya et al., 1994; Takahata, 1991; Hass, 1991; Lott, 1979; Schein and Fohrman, 1955; Hobson and DeDeo, 2015; Gernat et al., 2018), focusing our attention to those interactions which are capable of disease transmission (i.e. those that, at the least, require close spatial proximity). The advantage of using this model over more detailed network descriptions is that we obtain a single parameter estimate, $\varphi$ that is easily compared across animal species and environments.

Each dataset provides the number of interactions that were observed between pairs of individuals. We assume that the system is closed, and that the total network size ($N$) is equal to the number of individuals observed in at least one interaction. To estimate social fluidity, we find the value of $\varphi$ that minimizes ${\sum }_i{[k_i-\kappa (s_i,\varphi ,N)]}^2$; the total squared squared error between the observed degrees and their expectation given by the model. Uncertainty is displayed using the 2.5th and 97.5th percentile of the distribution of $\varphi$ computed on a set of 1000 ‘bootstrap’ samples, created by sampling $N$ data points, $\{k_i,s_i\}$, with replacement, from the observed data. Being estimated from the relationship between strength and degree, and not their absolute values, social fluidity is a good candidate for comparing social behavior across different systems as it is independent of the distributions of s_i or k_i, and of the timescale of interactions.

Figure 2 shows the estimated values of $\varphi$ for all networks in our study. We organize the measurements of social fluidity by interaction type. Aggressive interactions have the highest fluidity (which implies that most interactions are rarely repeated between the same individuals), while grooming and other forms of social bonding have the lowest (which implies frequent repeated interactions between the same individuals). Social fluidity also appears to be related to species: ant systems cluster around $\varphi =1$, monkeys around $\varphi =0.5$, humans take a range of values that depend on the social environment. Sociality type does not appear to affect $\varphi$; sheep, bison, and cattle have different social fluidity compared to kangaroos and bats, although they are all categorized as fission-fusion species (Sah et al., 2018).

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Across the 57 networks, there is no evidence that social fluidity scales with the size of the network or the number of observations per individual. No correlation was found between the mean number of interactions per individual ($\overline{s}$) and social fluidity when testing for a monotonic relationship between the variables (Spearman $r^2=0.02$, $p=0.36$), and in general no correlation across sets of networks taken from the same study (Supplementary file 1: Table S2). Similarly, network size ($N$) does not correlate with $\varphi$ (Spearman $r^2=0.02$, $p=0.28$). To test for a non-monotonic relationship, we partition the set of networks into 10 equally sized groups according to each of the two measures being compared, and compute the adjusted mutual information (AMI) of the two groupings. We find AMI=0.15 for the relationship between $\varphi$ and $N$, and AMI=0.2 between $\varphi$ and $\overline{s}$. While non-negative values of AMI typically indicate a non-random relationship, an inherent amount of clustering is to be expected in data aggregated from a diverse range of sources.

Larger values of $\varphi$ correspond to higher mean degrees (Spearman $r^2=0.21$, $p<0.001$) and lower variability in the distribution of edge weights (measured as the index of dispersion of $w_{i,j}$; Spearman $r^2=0.46$, $p<0.001$). Weight variability and mean degree are uncorrelated in these data (Spearman $r^2=0.01$, $p=0.54$, AMI=0.01) implying that $\varphi$ combines these two entirely distinct features of social behavior. Finally, the modularity of the network (computed by the Louvain method on the unweighted network Blondel et al., 2008) is negatively correlated with $\varphi$ ($r^2=0.52$, $p<0.001$). This is expected as individuals tend to be loyal to those within the same module while maintaining weaker connections with the remaining network - in all but one network the mean weight of edges within modules is higher than the mean weight of edges between modules (supplementary document).

As with any applied modeling, the validity of these results depends on the extent to which each study system conforms to the assumptions of the model. The value of $N$, for example, might not represent the true group size if some individuals in the group did not have their interactions recorded, or if there are individuals who did not interact during the time-frame of observation. While we found that variation in the value of $N$ did not have a large impact on the estimated value of $\varphi$, as shown in Figure 2—figure supplement 1, we warn that the amount of consistency between model assumptions and the conditions of each study will vary, and close consideration should be given to the way data were collected when interpreting these results.

Disease transmission model

We consider the transmission of an infectious disease on the social behavior model introduced in the previous section. An infectious node $i$ interacting with a susceptible node $j$ will transmit the infection with probability β. The node will recover from infection with rate γ, assuming an exponential distribution of the length of the infectious period. The probability that the infection is transmitted from $i$ to any given $j$ is

(5) $T_{i\to j}(\beta ,\gamma ,s_i,\tau ,x_{j|i})=1-\exp (-s_i{x}_{j|i}\beta /\gamma \tau ),$

assuming that the interactions s_i of $i$ are distributed randomly across an observation period of duration τ.

By integrating Equation (5) over all possible values $x_{j|i}$ and infectious period durations and multiplying by the number of susceptible individuals ($N-1$) we obtain the expected number of infections caused by individual $i$, 

(6) $r(s_i)={\displaystyle \frac{1-\varphi }{\varphi ({ϵ}^{\varphi }-ϵ)}}[1-{ϵ}^{\varphi }+{ϵ}^{\varphi }{}_{2}F_1(-\varphi ,1,1-\varphi ;-\beta s_i/\gamma \tau )-{}_{2}F_1(-\varphi ,1,1-\varphi ;-ϵ\beta s_i/\gamma \tau )].$

The basic reproductive number (usually denoted R₀) is defined as the mean number of secondary infections caused by a typical infectious individual in an otherwise susceptible population (Diekmann et al., 1990). We will use the notation $R_0^{\varphi }$ to signify the social fluidity reproductive number, that is the analogue of R₀ derived from our social behavior model.

We assess the relation of the reproductive number with the population density by focusing on a special case where every node has the same strength, that is $s_i=s$ for all $i$, so that $R_0^{\varphi }=r(s)$. Furthermore, we choose $\beta =\gamma \tau R_0^{\mathrm{\infty }}/s$ where $R_0^{\mathrm{\infty }}$ is $R_0^{\varphi }$ as $\varphi \to \mathrm{\infty }$, that is a constant that represents what the basic reproductive number would be if every new interaction occurred between a pair of individuals who have not previously interacted with each other.

Figure 3 shows the effect of social fluidity on the density dependence of the disease. At small population sizes, $R_0^{\varphi }$ increases with $N$ and converges as $N$ goes to ∞ (Figure 3A). The rate of this convergence increases with $\varphi$, and the limit it converges to is higher, meaning that $\varphi$ determines the extent to which density affects the spread of disease. As $N\to \mathrm{\infty }$, we find that $R_0^{\varphi }\to R_0^{\mathrm{\infty }}$ for $\varphi >1$. When $\varphi <1$, $R_0^{\varphi }\to [(1-\varphi )/\varphi ][{}_{2}F_1(-\varphi ,1,1-\varphi ;-R_0^{\mathrm{\infty }})-1]$. At these values of $\varphi$ the disease is constrained by individuals choosing to repeat interactions despite having the choice of infinitely many potential interaction partners (Figure 3B).

Figure 3

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We proposed a measure of fluidity in social behavior which quantifies how much mixing exists within the social relationships of a population. While social networks can be measured with a variety of metrics including size, connectivity, contact heterogeneity and frequency, our methodology reduces all such factors to a single quantity allowing comparisons across a range of human and animal social systems. Social fluidity correlates with both the density of social ties (mean degree) and the variability in the weight of those ties, although these quantities do not correlate with each other. Social fluidity is thus able to combine these two aspects seamlessly in one quantity.

By measuring social fluidity across a range of human and animal systems we are able to rank social behaviors. We identify aggressive interactions as the most socially fluid; this indicates a possible learning effect whereby each aggressive encounter is followed by a period during which individuals avoid further aggression with each other (Parker, 1974). At the opposite end of the scale, we find interactions that strengthen bonds (and thus require repeated interactions) such as grooming in monkeys (Seyfarth and Cheney, 1984) and food-sharing in bats (Carter and Wilkinson, 2013). The fact that food-sharing ants are far more fluid than bats, despite performing the same kind of interaction, reflects their eusocial nature and the absence of any need to consistently reinforce bonds with their kin (Hölldobler and Wilson, 2009).

Our results contribute to a body of work examining the disproportionate distribution of social effort in both human and animal groups. This phenomena has been directly observed in human telecommunication (Mac Carron et al., 2016; Saramäki et al., 2014; Gonçalves et al., 2011; Tamarit et al., 2018). Quantifying this aspect of sociality in animal systems, however, has been held back by the limitations of the data, such as the bias introduced by variation in activity levels across the social group (Di Bitetti, 2000). Additionally, while heterogeneous interaction frequencies and temporal dynamics have become common in epidemiological models (Rocha and Blondel, 2013; Colman et al., 2018), our results highlight the importance of including variability in how the individual chooses to expend their social effort.

As with most studies that aim to describe and quantify social structure, there are a number of concerns that ought to be mentioned. The degree of an individual, for example, is known to scale with the length of the observation period (Perra et al., 2012). This is also true of the networks used here (Figure 2—figure supplement 1). Similarly, social fluidity can be affected by the length of the observation window. However, since our model focuses not on the absolute value of degree, but on how degree scales with the number of observations, the results we obtain are relatively robust against this variability (Figure 2—figure supplement 1). Additionally, observed interactions are typically assumed to persist over time (Perreault, 2010). In our model this is not the case; only the distribution of edge weights remains constant, an assumption consistent with growing evidence (Miritello et al., 2013; Centellegher et al., 2017).

We therefore consider the model to be applicable to the data analysed in this study, but advise caution when applying this approach to other data sources. If the duration of a study allows for substantial developments in the group structure, for example, then a model of edge formation and dissolution may be preferred.

Finally, we do not know the extent to which an interaction, as defined for each network, is capable of transmission which can depend on the pathogen’s transmission mode and the infectious dose required. Furthermore, the transmission probability is unlikely to be the same for all interactions within the group since, for example, the duration of contact is known to be important for disease spread (Stehlé et al., 2011b). We did not include explicitly the duration of each contact in our model as this information was only available in a fraction of the datasets (Barrat et al., 2014). There is therefore potential to improve the applicability of this model as more high resolution data becomes openly available.

Our estimate of reproductive number derived from social fluidity provides a better predictor for the epidemic risk of a host population, going beyond predictors based on density or degree only. To illustrate this point, the social network of individuals at a conference ($R_0^{\text{Est}}=1.60$; conference_0, supplementary document) is predicted to be at higher risk compared to the social network at a school ($R_0^{\text{Est}}=1.39$; highschool_0), despite having a smaller size and lower connectivity ($N=93$ vs. $N=312$, and $\overline{k}=5.63$ vs. $\overline{k}=6.78$, respectively). The discrepancy in the risk prediction comes from the lower frequency of repeated contacts between individuals in the conference, compared to the school. Interactions between infectious individuals and those they have previously infected are redundant in terms of transmission. This dynamic is nicely captured by the social fluidity, with $\varphi =0.66$ for the conference and $\varphi =0.40$ for the high school.

Unlike previous work that explores the disease consequences of population mixing (Volz and Meyers, 2007; Reluga and Shim, 2014), our analysis allows us to investigate this relation across a range of social systems. We see, for example, how the relationship between mixing and disease risk scales with group size. For social systems that have high values of social fluidity, $R_0^{\varphi }$ is highly sensitive to changes in $N$, whereas this sensitivity is not present at low values of $\varphi$. This corroborates past work on the scaling of transmission being associated to heterogeneity in contact (Begon et al., 2002; Ferrari et al., 2011). Going beyond previous work, our model captures in a coherent theoretical framework both density-dependence and frequency-dependence, and social fluidity is the measure to tune from one to the other in a continuous way. Since many empirical studies support a transmission function that is somewhere between these two modeling paradigms (Smith et al., 2009; Cross et al., 2013; Borremans et al., 2017; Hopkins et al., 2020), the modeling approaches applied in this paper can be carried forward to inform transmission relationships in future disease studies.

All scripts, data, and documentation used in this study are available through https://github.com/EwanColman/Social-Fluidity (copy archived at https://archive.softwareheritage.org/swh:1:rev:90b27e1b84ce4417633885cd260c89bbf1b07eac).

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Author details

1. Ewan Colman

   1. Department of Biology, Georgetown University, Washington, United States
   2. Roslin Institute, University of Edinburgh, Midlothian, United Kingdom

   Contribution

   Conceptualization, Data curation, Software, Formal analysis, Investigation, Visualization, Methodology, Writing - original draft, Writing - review and editing

   For correspondence

   ecolman@ed.ac.uk

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-2551-8589

2. Vittoria Colizza

   INSERM, Sorbonne Université, Institut Pierre Louis d’Épidémiologie et de Santé Publique (IPLESP UMRS 1136), F75012, Paris, France

   Contribution

   Methodology, Writing - review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-2113-2374

3. Ephraim M Hanks

   Department of Statistics, Eberly College of Science, Penn State University, State College, United States

   Contribution

   Funding acquisition, Methodology, Writing - review and editing

   Competing interests

   No competing interests declared

4. David P Hughes

   Department of Entomology, College of Agricultural Sciences, Penn State University, State College, United States

   Contribution

   Funding acquisition, Writing - review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-9954-8919

5. Shweta Bansal

   Department of Biology, Georgetown University, Washington, United States

   Contribution

   Conceptualization, Supervision, Funding acquisition, Investigation, Methodology, Writing - review and editing

   For correspondence

   shweta.bansal@georgetown.edu

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-1740-5421

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