Higher-order adaptive behaviors outperform pairwise strategies in mitigating contagion dynamics

TL;DR

The paper investigates how adaptive behaviors based on risk perception affect contagion on hypergraphs, comparing six strategies that use absolute/relative and pairwise/higher-order information. Through stochastic simulations and an individual-based mean-field framework, it demonstrates that absolute higher-order information (ng, nw) most effectively suppresses contagion while incurring lower social cost, by creating heterogeneous risk awareness that concentrates protection on high-hyperdegree nodes and large groups. These strategies disrupt the conventional hubs-and-groups pathways that sustain spread, and can even remove bistability in higher-order contagion, unlike pairwise-relay strategies which are less efficient and sometimes costly. The results highlight the importance of exploiting higher-order interaction structure for mitigation, with implications for designing targeted interventions in real-world social systems.

Abstract

When exposed to a contagion phenomenon, individuals may respond to the perceived risk of infection by adopting behavioral changes, aiming to reduce their exposure or their risk of infecting others. The social cost of such adaptive behaviors and their impact on the contagion dynamics have been investigated in pairwise networks, with binary interactions driving both contagion and risk perception. However, contagion and adaptive mechanisms can also be driven by group (higher-order) interactions. Here, we consider several adaptive behaviors triggered by awareness of risk perceived through higher-order and pairwise interactions, and we compare their impact on pairwise and higher-order contagion processes. By numerical simulations and a mean-field analytic approach, we show that adaptive behaviors driven by higher-order information are more effective in limiting the spread of a contagion, than similar mechanisms based on pairwise information. Meanwhile, they also entail a lower social cost, measured as the reduction of the intensity of interactions in the population. Indeed, adaptive mechanisms based on higher-order information lead to a heterogeneous risk perception within the population, producing a higher alert on nodes with large hyperdegree (i.e., participating in many groups), on their neighborhoods, and on large groups. This in turn prevents the spreading process to exploit the properties of these nodes and groups, which tend to drive and sustain the dynamics in the absence of adaptive behaviors.

Figures (7)

• Figure 1: Schematic representation of the contagion and adaptive mechanisms.a: an example of a hypergraph $\mathcal{H}$ and its weighted pairwise projection $\mathcal{G}$. b: schematic representation of the SIS contagion mechanisms for the node $i$, using higher-order contagion (left panel) or pairwise contagion (right panel). c: schematic representation of the six adaptive strategies considered for node $i$: the first and second rows show respectively the strategies based on relative (fraction) and absolute (number) information; the columns identify different nature of information: pairwise, higher-order and hybrid. In this case (considering $\theta=0.3$): $f_i^{fn}(t)=4/7$, $f_i^{nn}(t)=4/\langle k \rangle$, $f_i^{fw}(t)=5/11$, $f_i^{nw}(t)=5/\langle s \rangle$, $f_i^{fg}(t)=3/5$, $f_i^{ng}(t)=3/\langle D \rangle$.
• Figure 2: Phase diagram for pairwise and higher-order processes.a, b: epidemic prevalence in the asymptotic steady state, $I_{\infty}$, as a function of the effective infection rate, $r=\lambda_0^2/\mu$, for the non-adaptive case (NAD) and for the six adaptation strategies, respectively for the pairwise and higher-order contagion process. We show the results of numerical simulations (markers) and of the integration of the mean-field equations (lines). For the higher-order contagion case, we consider simulations and numerical integration starting with a low (triangles and dashed lines) and a high (dots and solid lines) fraction of infected nodes. We consider the hospital dataset, the numerical results are averaged over $300$ simulations, $\nu=4$, $\theta=0.3$. The insets represent a zoom on the mean-field results in the area marked by a dashed rectangle and the vertical line indicates the pairwise epidemic threshold $r_C^{NAD,p}$. Note that the error-bars on numerical simulations are smaller than the corresponding markers (considering one standard deviation).
• Figure 3: Impact of adaptive behaviors on the endemic state. We consider the pairwise contagion process on the hospital dataset, with $r=0.05$, $\theta=0.3$. For the NAD case and for each adaptive strategy, we show (through bar-plots): a: the endemic epidemic prevalence $I_{\infty}$; b: the average value of the transmission parameter $\langle \lambda_i^{\infty} \rangle/\lambda_0$ in the steady state; c: the standard deviation of the transmission parameter $\sigma(\lambda_i^{\infty})$ in the endemic state. In all panels, the left dark-colored bars are obtained by averaging over 300 numerical stochastic simulations, while the right light-colored bars are obtained through numerical integration of the mean-field equations (see legend).
• Figure 4: Heterogeneous risk perception of nodes and groups. We consider the pairwise contagion process on the hospital dataset, $r=0.05$, $\theta=0.3$. We focus on the asymptotic steady state and we show the distribution $\rho(\lambda_i^{\infty}/\lambda_0)$ of the transmission parameter for nodes (panel a) and the distribution $\varrho(\overline{\lambda}_e^{\infty}/\lambda_0)$ of the average transmission parameter within a group (panel d), both obtained by averaging over 300 numerical simulations. We divide nodes into hyperdegree classes $D$ and for each class we estimate: b: the average reduction in the transmission parameter $\langle \lambda_i^{\infty} \rangle_D/\lambda_0$ in the steady state; c: the average fraction of time nodes spent infected $\langle \tau^{\infty}_i \rangle_D$ in the steady state. For each hyperedge size $m$ we estimate: e: the mean reduction of the transmission parameter within a group $\langle \overline{\lambda}_e^{\infty} \rangle_m/\lambda_0$ in the steady state; f: the average fraction of infected nodes within a group $\langle \eta^{\infty}_e \rangle_m$ in the steady state. All quantities are estimated for each node or hyperedge by averaging over 300 numerical simulations (markers) or by integrating numerically the mean-field equations (dashed curves). In all panels, we consider the NAD case and the six adaptive strategies.
• Figure 5: Impact of adaptive behaviors on the epidemic transient. We consider the pairwise contagion process on the hospital dataset, with $r=0.05$, $\theta=0.3$. a: temporal evolution of the fraction of infected nodes $I(t)$; b: temporal evolution of the mean transmission parameter $\langle \lambda_i(t) \rangle/\lambda_0$; c: temporal evolution of the standard deviation of the transmission parameter $\sigma(\lambda_i(t))$. Solid lines in a-c are obtained by averaging over 300 numerical simulations; dashed lines are obtained through numerical integration of the mean-field equations. d: distribution of the transmission parameter $\rho(\lambda_i(t)/\lambda_0)$ at different times $t$, obtained by averaging over 300 numerical simulations. In all panels we consider the NAD case and the six adaptive strategies.