Social Confinement and Mesoscopic Localization of Epidemics on Networks

Recommendations around epidemics tend to focus on individual behaviors, with much less efforts attempting to guide event cancellations and other collective behaviors since most models lack the higher-order structure necessary to describe large gatherings. Through a higher-order description of contagions on networks, we model the impact of a blanket cancellation of events larger than a critical size and find that epidemics can suddenly collapse when interventions operate over groups of individuals rather than at the level of individuals. We relate this phenomenon to the onset of mesoscopic localization, where contagions concentrate around dominant groups.

Figure 1

Framework for contagions on higher-order networks. Nodes are assigned to $m$ groups and groups are of various sizes $n$, distributed according to $g_m$ and $p_n$. We consider a SIS dynamics where infected nodes transmit the disease in a group of size $n$ at rate $\beta n^{-\nu }$ with $\nu \in [0,1]$, and recover at rate $\mu$. We characterize the phenomenon of mesoscopic localization, namely the concentration of infected nodes in large groups.

Figure 2

Epidemic localization in networks with heterogeneous higher-order structures. We use $\nu =0$ for all panels and power-law distributions for the memberships $g_m\propto m^{-{\gamma }_m}$ and group sizes $p_n\propto n^{-{\gamma }_n}$. (a),(b) Solid lines represent the stationary group prevalence and dashed lines represent the stationary global prevalence. (a) For strongly coupled groups $({\gamma }_m={\gamma }_n=2.2)$, we find a collective phase transition. (b) For weakly coupled groups $({\gamma }_m=4,{\gamma }_n=3.5)$, we find a phenomenon of mesoscopic localization. While the global prevalence in the population can remain extremely low, larger groups can self-sustain the epidemic. The vertical dotted line is an estimation of the delocalization threshold, where the epidemic invades the whole network (see Ref. [11]). (c) Mesoscopic localization is the rule rather than the exception. The solid line, ${\gamma }_m+{\gamma }_n=5$, separates the delocalized regime (blue area of strong group coupling) from the mesoscopic localization regime (green area of weak group coupling), obtained from Eqs. (8a) and (8b). The circle and diamond markers correspond to the networks in panels (a),(b).

Figure 3

Local impact of structural interventions in delocalized and localized epidemics. (a),(b) Prevalence within groups of size $n$ against the intervention strength [Eq. (9)]. We use the networks from Figs. 2 and 2 with transmission rates adjusted to have similar global prevalence for both regimes without intervention. (a) In the delocalized regime, using $\beta \approx 0.0041$, we find a similar benefit of the intervention among all groups. (b) In the mesoscopic localization regime, using $\beta =0.07$, we find a different story. Large groups that have not been removed by the intervention are barely affected, until the intervention is strong enough to cause a global collapse of the epidemic. Square markers indicate when groups of a particular size are removed.

Figure 4

Global impact of structural interventions in different localization regimes of SIS and SIR dynamics. (a) We compare the global prevalence of epidemics against the intervention strength [Eq. (9)] using the same networks as in Fig. 3. We use $\beta \in \{0.004,0.005,0.006,0.007\}$ and $\beta \in \{0.07,0.09,0.11,0.13\}$ for the delocalized and localized regimes. (b) A similar behavior is observed for the final size (fraction of recovered nodes at $t\to \infty$) in simulations of the SIR dynamics [18]. We generate 1000 networks of size $N={10}^6$ and run ${10}^4$ SIR simulations on each network, starting with a single random infected node, with $\beta =0.0036$ in the delocalized regime and $\beta =0.083$ in the localized regime. We only kept macroscopic outbreaks, with a final size over ${10}^{-3}$. Solid and dashed lines are sample means and the shaded regions represent twice the standard deviation. (c) Three examples of $c_{n,i}$ distributions for (1) strongly localized, (2) weakly localized, and (3) delocalized epidemics, using $\nu$ equal 0, 0.6, and 1 respectively. We used networks with ${\gamma }_m={\gamma }_n=3.5$ and $\beta =0.3/⟨k⟩$, where $⟨k⟩$ is the average weighted degree before interventions. (d) We look at the impact of interventions for intermediate localization regimes by varying $\nu$ with the same networks as in (c). (e) Schematic representation of how the impact of interventions varies as it passes through different localization regimes.