2. The SIR model of infection with homophily

We model infection in a population in which individuals can be in one of three states: susceptible, infected and recovered. When susceptibles interact with infected individuals, they become infected with a rate equal to the effective transmissibility of the disease, τ. Infected individuals recover with a constant probability ρ per infection per unit time. This is the well-known SIR model of epidemics (Kermack & McKendrick, Reference Kermack and McKendrick1927; Tolles & Luong, Reference Tolles and Luong2020). The baseline model assumes random interactions governed by mass action, and the dynamics are described by well-known differential equations. This model yields the classic dynamics in which the susceptible and recovered populations appear as nearly mirrored sigmoids, while the rate of infected individuals rises and falls. The threshold for the epidemic is given by the basic reproduction number, R ₀, which is a measure of the expected number of secondary cases caused by a single, typical primary case at the outset of an epidemic in a population entirely composed of uninfected individuals. An epidemic occurs when R ₀ > 1. For the basic SIR model in a closed population, R ₀ = τ/ρ.

Our analysis will focus on scenarios where individuals assort based on identity. In this case, assume that individuals all belong to one of two identity groups, indicated with the subscript 1 or 2. Let w_i be the probability that interactions are with one's ingroup, i ϵ {1,2}. It is therefore a measure of homophily; populations are homophilous when w_i > 0.5. It is important to recognise that groups can differ in their homophily (Morris, Reference Morris1991). For example, if groups differ in socioeconomic class and group 1 tends to employ members of a group 2 as service workers, homophily will be higher for group 1; a member of group 2 is more likely to encounter members of group 1 than the reverse. We can update the equations governing infection dynamics for members of group 1, with analogous equations governing members of group 2.

$$\matrix{ {\displaystyle{{{\rm d}S_1} \over {{\rm d}t}}} \hfill & { = {-}\tau S_1\lpar {w_1I_1 + \lpar {1-w_1} \rpar I_2} \rpar \qquad } \hfill & {} \hfill & \qquad \hfill \cr {\displaystyle{{{\rm d}I_1} \over {{\rm d}t}}} \hfill & { = \tau S_1\lpar {w_1I_1 + \lpar {1-w_1} \rpar I_2} \rpar -\rho I_1\qquad } \hfill & {} \hfill & \qquad \hfill \cr {\displaystyle{{{\rm d}R_1} \over {{\rm d}t}}} \hfill & { = \rho I_1\qquad } \hfill & {} \hfill & \qquad \hfill \cr } $$

We assume the disease breaks out in one of the two groups, so the initial number infected in group 1 is small but non-zero, while the initial number infected in group 2 is exactly zero. Without loss of generality, we have assumed that group 1 is always infected first. When homophily is low, the model exhibits standard SIR dynamics approximating a single unified population. When an infection breaks out in group 1, homophily can delay the outbreak of the epidemic in group 2. Homophily for each group works somewhat synergistically, but the effect is dominated by w ₂. This is because the infection spreads rapidly in a homophilous group 1, and if group 2 is not homophilous, its members will rapidly become infected. However, if group 2 is homophilous, its members can avoid the infection for longer, particularly when group 1 is also homophilous. If only group 2 is homophilous, the initial outbreak will be delayed, but the peak infection rate in group 2 can actually be higher than in group 1, as the infection is driven by interactions with both populations (Figure 1).

Figure 1. Dynamics of the infected population of each group under low and high homophily (w_i = 0.6, 0.99). Other parameters used were τ = 0.3, ρ = 0.07, I ₁(0) = 0.01, I ₂(0) = 0. R ₀ ≈ 4.28 in the absence of homophily.

We also considered the case in which the transmissibility of the infection can be reduced to very near the recovery rate, so that R ₀ is very close to 1. In this case, homophily can protect groups where infection did not originally break out by keeping members relatively separated from the infection group (Figure S2).

3. Behavioural contagion with outgroup aversion

We model behaviour adoption as a susceptible–infectious–susceptible (SIS) process, in which individuals can oscillate between adoption and non-adoption of the behaviour indefinitely. We view this as more realistic than an SIR process for preventative-but-transient behaviours like social distancing or wearing face masks. To avoid confusion with infection status, we denote individuals who adopted the preventative behaviour as careful (C), and those who have not as uncareful (U). Unlike a disease, which is reasonably modelled as equally transmissible between any susceptible–infected pairing, where behaviour is concerned, susceptible individuals are more likely to adopt when interacting with ingroup adopters, but less likely to adopt when interacting with outgroup adopters. We model the behavioural dynamics for members of group 1 are as follows, with analogous equations governing members of group 2:

$$\matrix{ {\displaystyle{{{\rm d}U_1} \over {{\rm d}t}}} \hfill & { = {-}\lpar {\alpha_1 + \beta C_1} \rpar U_1 + \lpar {\gamma C_2 + \delta } \rpar C_1\qquad } \hfill & {} \hfill & \qquad \hfill \cr {\displaystyle{{{\rm d}C_1} \over {{\rm d}t}}} \hfill & { = \lpar {\alpha_1 + \beta C_1} \rpar U_1-\lpar {\gamma C_2 + \delta } \rpar C_1\qquad } \hfill & {} \hfill & \qquad \hfill \cr } $$

Members of group i may spontaneously adopt the behaviour independent of direct social influence, and do so at rate α_i. This adoption may be due to individual assessment of the behaviour's utility, to influences separate from peer mixing, such as from media sources, or to socioeconomic factors that make behaviour adoption more or less easy for certain groups. For these reasons, we assume that groups can differ in their rates of spontaneous adoption. In reality, it is possible for groups to differ on all four model parameters, all of which can influence differences in adoption rates. For simplicity, we restrict our analysis to differences in spontaneous adoption.

Uncareful individuals are positively influenced to become careful by observing careful individuals of their own group, with strength β. However, this is countered by the force of outgroup aversion, γ, whereby individuals may cease being careful when they observe this behaviour among members of the outgroup. The behaviour is eventually discarded at rate δ, representing the financial and/or psychological costs of continuing to adopt preventive behaviours like social distancing and wearing face masks.

This model assumes no explicit homophily in terms of behavioural influence. On the one hand, it seems obvious that we observe and communicate with those in our own group more than other groups. On the other hand, opportunities for observing outgroup behaviours are abundant in a digitally connected world, which alter the conditions for cultural evolution (Acerbi, Reference Acerbi2019). For simplicity, we do not add explicit homophily terms to this system. Instead, we simply adjust the relative strengths of ingroup influence and outgroup aversion, β/γ. When this ratio is higher, it reflects stronger homophily for behavioural influence.

Numerical simulations that illustrate the influence of outgroup aversion are depicted in Figure 2. In all cases, the careful behaviour is first adopted by group 1. In the absence of outgroup aversion, both groups adopt the behaviour at saturation levels, with group 2 being slightly delayed. When outgroup aversion is added, the delay increases, but more importantly, overall adoption declines for both groups. This decline continues as long as the strength of outgroup aversion is less than the strength of positive ingroup influence. A phase transition occurs here (Figure 2c, d). Although group 2 may initially adopt the behaviour, adoption is subsequently suppressed, resulting in a polarising behaviour that is abundant in group 1 but nearly absent in group 2.

Figure 2. Dynamics of the behavioural adoption. (a–c) Behaviour adoption dynamics in each group for different levels of outgroup aversion, γ. Parameters used were α ₁ = α ₂ = 0.001, β = 0.3, δ = 0, C ₁(0) = 0.01, C ₂(0) = 0. (d) Equilibrium adoption rates for each group as a function of outgroup aversion, γ. A bifurcation occurs when outgroup aversion overpowers the forces of positive influence. (e) Behaviour adoption dynamics for γ = 0.2 where group 1 has a higher spontaneous adoption rate, α ₁ = 0.1. Here, the two groups converge to different equilibrium adoption rates. (f) Equilibrium adoption rates for each group as a function of outgroup aversion, γ, when α ₁ = 0.1.

We also consider the case in which one group has a higher intrinsic adoption rate, which could be driven by differences in personality types, norms or media exposure between the two groups. When α ₁ > α ₂, the equilibrium adoption rate for group 1 could be considerably higher than for group 2, even when ingroup positive influence was greater than outgroup aversion (Figure 2e, f). Note that these differences arise entirely because of outgroup aversion. When γ = 0, both groups adopt at maximum levels.

Outgroup aversion has a strong influence on adoption dynamics. It can delay adoption, reduce equilibrium adoption rates and even suppress adoption entirely in the later-adopting group. As we will see, when the behaviour being adopted influences disease transmission, quite interesting dynamics can emerge.

4. Coupled contagion with homophily and outgroup aversion

Before we explore the coupled dynamics of this system, we must add one more consideration to the model. We focus on the adoption of preventative behaviours that decrease the effective transmission rate of the infection, such as social distancing or wearing face masks. We model this by asserting that the transmission rate is τ _C for careful individuals and τ _U for uncareful individuals, such that τ _U ≥ τ _C. When considering the interaction between careful and uncareful individuals, we use the geometric mean, so the transmissibility between SU and IU (that is, between susceptible and infected individuals who are both uncareful) is $\sqrt {\tau _{\rm U}\tau _{\rm C}}$. We use the geometric mean so that if either population reduces its transmissibility to zero, transmission among its members becomes impossible.

The full model has six compartments, with two-letter abbreviations denoting the disease and behavioural state (Figure 3). The coupled dynamics for members of group 1 are as follows, with analogous equations governing members of group 2, such that the full system is defined by 12 coupled differential equations. A list of all parameters is presented in Table 1.

$$\eqalign{\displaystyle{{\rm d}\lpar SU_1\rpar \over {\rm d}t} & = \lsqb \delta + \gamma \lpar {SC_2 + IC_2 + RC_2} \rpar \rsqb \lpar SC_1\rpar -\lsqb \alpha_1 + \beta \lpar SC_1 + IC_1 + RC_1\rpar \rsqb \lpar SU_1\rpar \cr & \quad - \tau _{\rm U}\lpar SU_1\rpar \lsqb {w_1\lpar IU_1\rpar + \lpar 1-w_1\rpar IU_2} \rsqb -\sqrt {\tau _{\rm U}\tau _{\rm C}} \lpar SU_1\rpar \lsqb {w_1\lpar IC_1 + \lpar 1-w_1\rpar IC_2} \rsqb \cr \displaystyle{{{\rm d}\lpar SC_1\rpar } \over {{\rm d}t}} & = {-}\lsqb {\delta + \gamma \lpar {SC_2 + IC_2 + RC_2} \rpar } \rsqb \lpar SC_1\rpar + \lsqb {\alpha_1 + \beta \lpar SC_1 + IC_1 + RC_1\rpar } \rsqb \lpar SU_1\rpar \cr & \quad - \sqrt {\tau _{\rm U}\tau _{\rm C}} \lpar SC_1\rpar \lsqb {w_1\lpar IU_1\rpar + \lpar 1-w_1\rpar IU_2} \rsqb -\tau _{\rm C}\lpar SC_1\rpar \lsqb {w_1\lpar IC_1 + \lpar 1-w_1\rpar IC_2} \rsqb \cr \displaystyle{{{\rm d}\lpar IU_1\rpar } \over {{\rm d}t}} & = \lsqb {\delta + \gamma \lpar {SC_2 + IC_2 + RC_2} \rpar } \rsqb \lpar IC_1\rpar -\lsqb {\alpha_1 + \beta \lpar SC_1 + IC_1 + RC_1\rpar } \rsqb \lpar IU_1\rpar \cr & \quad + \tau _{\rm U}\lpar SU_1\rpar \lsqb {w_1\lpar IU_1\rpar + \lpar 1-w_1\rpar IU_2} \rsqb + \sqrt {\tau _{\rm U}\tau _{\rm C}} \lpar SU_1\rpar \lsqb {w_1\lpar IC_1 + \lpar 1-w_1\rpar IC_2} \rsqb -\rho \lpar IU_1\rpar \cr \displaystyle{{{\rm d}\lpar IC_1\rpar } \over {{\rm d}t}} & = {-}\lsqb {\delta + \gamma \lpar {SC_2 + IC_2 + RC_2} \rpar } \rsqb \lpar IC_1\rpar + \lsqb {\alpha_1 + \beta \lpar SC_1 + IC_1 + RC_1\rpar } \rsqb \lpar IU_1\rpar \cr & + \sqrt {\tau _{\rm U}\tau _{\rm C}} \lpar SC_1\rpar \lsqb {w_1\lpar IU_1\rpar + \lpar 1-w_1\rpar IU_2} \rsqb + \tau _{\rm C}\lpar SC_1\rpar \lsqb {w_1\lpar IC_1 + \lpar 1-w_1\rpar IC_2} \rsqb -\rho \lpar IC_1\rpar \cr \displaystyle{{{\rm d}\lpar RU_1\rpar } \over {{\rm d}t}} & = \lsqb {\delta + \gamma \lpar {SC_2 + IC_2 + RC_2} \rpar } \rsqb \lpar RC_1\rpar -\lsqb {\alpha_1 + \beta \lpar SC_1 + IC_1 + RC_1\rpar } \rsqb \lpar RU_1\rpar + \rho \lpar IU_1\rpar \cr \displaystyle{{{\rm d}\lpar RC_1\rpar } \over {{\rm d}t}} & = -\lsqb {\delta + \gamma \lpar {SC_2 + IC_2 + RC_2} \rpar } \rsqb \lpar RC_1\rpar + \lsqb {\alpha_1 + \beta \lpar SC_1 + IC_1 + RC_1\rpar } \rsqb \lpar RU_1\rpar + \rho \lpar IC_1\rpar} $$

Figure 3. Illustration of the dynamics for the coupled contagion model. (a) Transition probabilities between compartments for members of group 1. For simplicity these probabilities do not include the influence of homophily. (b) Homophilous interactions. Members of group i have physical contact with members of their own group with probability w _i and members of the outgroup with probability 1 − w _i.

Table 1. Model parameters

Behavioural adoption is independent of infection status in this model. This may not be a realistic assumption for some systems, such as Ebola, where the both the infection status of the adopter and the perceived incidence in the population are likely to influence behaviour. The assumption seems more realistic for infections like influenza and COVID-19, where infection status is not always transparent and decisions are likely to be made on the basis of more abstract socially-transmitted information. There are intermediate cases, however, such as where media reports of disease prevalence or the perceived availability may influence the adoption of preventative behaviours (Lau et al., Reference Lau, Griffiths, Choi and Lin2010; Zhang et al., Reference Zhang, Kong and Chang2015; Seale et al., Reference Seale, Heywood, Leask, Sheel, Thomas, Durrheim and Kaur2020). We do not consider such cases here.

To make the behavioural adoption most meaningful, we focus on the case where instantaneous and universal adoption of the careful behaviour would decrease the disease transmissibility so that R ₀ < 1. That is, if everyone immediately adopted the behaviour, the epidemic would fizzle out. However, behaviour adoption doesn't typically work this way. We have already noted that under assumptions of between-group variation and outgroup aversion, a behaviour is likely to be adopted neither instantaneously nor universally. The question we tackle now is how those socially driven facets of behavioural adoption influence disease dynamics.

Figure 4 illustrates the wide range of possible disease dynamics under varying assumptions of homophily and outgroup aversion. A wider range of homophily values are explored in the Supplemental Materials (Figures S4 and S5). In the absence of either homophily or outgroup aversion, our results mirror previous work on coupled contagion in which the adoption of inhibitory behaviours reduces peak infection rates, flattening the curve of infection. Owing to differences in spontaneous adoption rates, however, group 2 may see a higher peak infection rate even when the infection breaks out in group 1, because the inhibitory behaviour spreads more slowly in that group (Figure 4a).

Figure 4. Coupled contagion dynamics when the behaviour leads to highly effective reduction in transmissibility, under varying conditions of homophily and outgroup aversion. Notice difference in y-axis scale for infection rate between top and bottom sets of graphs. Parameters used: τ _U = 0.3, τ _C = 0.069, ρ = 0.07, α ₂ = 0.1, α ₂ = 0.001, β = 0.3, δ = 0, SU ₁(0) = 0.98, SC ₁(0) = 0.01, IU ₁(0) = 0.01, IC ₁(0) = RU ₁(0) = RC ₁(0) = 0, SU ₂(0) = 1.0, SC ₂(0) = IU ₂(0) = IC ₂(0) = RU ₂(0) = RC ₂(0) = 0.

Homophilous interactions further lower infection rates. If group 1 alone is homophilous, the infection rate declines in that group, while peak infections actually increase in group 2 (Figure 4c). This is because group 1 adopts the careful behaviour early, decreasing their transmission rate and simultaneously avoiding contact with the less careful members of group 2, who become infected through their frequent contact with group 1. If group 2 alone is homophilous, on the other hand, the infection is staved off even more so than if both groups are homophilous (Figure 4b, d). This is because members of group 2 avoid contact with group 1 until the careful behaviour has been widely adopted, while members of group 1 diffuse their interactions with some members of group 2, and these are less likely to lead to new infections.

Outgroup aversion considerably changes these dynamics. First and foremost, outgroup aversion leads to less widespread adoption of careful behaviours, dramatically increasing the size of the epidemic. Moreover, because under many circumstances there will be between-group differences in equilibrium behaviour-adoption rates, this can lead to dramatic group differences in infection dynamics. In the absence of outgroup aversion, we saw that homophily in group 2 could lead to an almost total suppression of the epidemic. Not so with outgroup aversion, in which the peak infection rates increase relative to the low-homophily case (Figure 4e, f). This occurs because homophily causes a delay in the infection onset in group 2. Behavioural adoption slows the epidemic initially in both groups. However, when the infection finally reaches group 1, behavioural adoption has decreased past its maximum owing to the outgroup aversion, causing peak infections in both groups to soar.

The dynamics are particularly interesting for the case where the group in which the epidemic first breaks out (group 1 in our analyses) is also strongly homophilous. Owing to homophily along with rapid behaviour adoption, the epidemic is initially suppressed in this group. However, owing to slower and incomplete behaviour adoption, the infection spreads rapidly in group 2. As the infection peaks in group 2 while group 1 decreases its behaviour adoption rate, we observe a delayed ‘second wave’ of infection in group 1, well after the infection has peaked in group 2 (Figure 4g). This effect is exacerbated when both groups are homophilous, as the epidemic runs rampant in the less careful group 2 (Figure 4h). As shown in the Supplementary Material, the timing of the second wave is also delayed to a greater extent when the adopted behaviour is more efficacious at reducing transmission (Figure S6).

We explored the differences in the timing of the infection peaks between the two groups, as illustrated in Figure 5. As noted, homophily in group 1 has a larger effect than homophily in group 2 because the infection first breaks out in group 1. Without outgroup aversion, the infection peak in group 1 is usually closely timed to the infection peak in group 2, usually coming slightly later owingto group 2's lagged adoption of the preventative behaviour (Figure 5a). If group 1 has very strong homophily, however, the infection can peak earlier there, as its spread to group 2 is impeded. When outgroup aversion is strong, however, group 2's adoption of the preventative behaviour is severely impeded, which causes its infection rate to peak much earlier than in group 1, and this effect is only bolstered by strong homophily in group 1 (Figure 5b). The effect of outgroup aversion on the differential timing between groups of infection rate peaks is non-monotonic (Figure 5c), peaking at intermediate values of γ.

Figure 5. Difference in the timing of the peak infection rates between groups. These plots show the extend to which the peak in group 1 lags behind the peak in group 2. The first two plots show the peak delay for group 1 as a function of group 1 homophily, (a) with and (b) without outgroup aversion, γ. The third plot (c) more systematically varies outgroup aversion, for several values of group 1 homophily and moderate group 2 homophily, w ₂ = 0.7. Other parameters used: τ _U = 0.3, τ _C = 0.069, ρ = 0.07, α ₂ = 0.1, α ₂ = 0.001, β = 0.3, δ = 0.