Temporal percolation of a susceptible adaptive network

In the last decades, many authors have used the susceptible-infected-recovered model to study the impact of the disease spreading on the evolution of the infected individuals. However, few authors focused on the temporal unfolding of the susceptible individuals. In this paper, we study the dynamic of the susceptible-infected-recovered model in an adaptive network that mimics the transitory deactivation of permanent social contacts, such as friendship and work-ship ties. Using an edge-based compartmental model and percolation theory, we obtain the evolution equations for the fraction susceptible individuals in the susceptible biggest component. In particular, we focus on how the individual’s behavior impacts on the dilution of the susceptible network. We show that, as a consequence, the spreading of the disease slows down, protecting the biggest susceptible cluster by increasing the critical time at which the giant susceptible component is destroyed. Our theoretical results are fully supported by extensive simulations.

💡 Research Summary

This paper addresses a largely overlooked aspect of epidemic modeling: the temporal evolution of the susceptible (healthy) sub‑network in an adaptive contact structure. While most SIR (susceptible‑infected‑recovered) studies focus on the dynamics of infected and recovered individuals, the authors ask how the network of healthy people—its connectivity, size, and resilience—changes as the disease spreads. To capture realistic social behavior, they introduce a “susceptible adaptive network” in which permanent social ties (friendships, workplace relationships) can be temporarily deactivated. Concretely, each edge that connects a susceptible node to an infected node is deactivated with probability (p_d); a deactivated edge remains inactive for a fixed duration (\tau) before it is restored. This mechanism mimics real‑world actions such as temporary social distancing, quarantine, or reduced communication.

The analytical framework combines two powerful tools. First, an edge‑based compartmental model (EBCM) is employed. Unlike traditional node‑based mean‑field equations, EBCM tracks the probability (\theta(t)) that a randomly chosen edge has not transmitted infection up to time (t). The evolution of (\theta(t)) is governed by a set of differential equations that incorporate the infection rate (\beta), recovery rate (\gamma), deactivation probability (p_d), and restoration time (\tau). Second, percolation theory is used to link (\theta(t)) to the size (S(t)) of the giant susceptible component (GSC), i.e., the largest connected cluster of healthy nodes. The critical time (t_c) at which the GSC disintegrates satisfies (\theta(t_c)=\theta_c), where (\theta_c) depends on the underlying degree distribution (average degree (\langle k\rangle) and higher moments). The authors show analytically that larger (p_d) and longer (\tau) slow the decay of (\theta(t)), thereby postponing (t_c). In other words, more aggressive or prolonged edge deactivation protects the susceptible network, giving it a larger “survival window” before fragmentation.

To validate the theory, extensive Monte‑Carlo simulations are performed on two canonical network families: Erdős–Rényi random graphs and scale‑free networks. Parameter sweeps cover a range of infection rates (\beta), recovery rates (\gamma), deactivation probabilities (p_d), and restoration times (\tau). The simulation outcomes match the analytical predictions across all tested regimes. Notably, scenarios with high (p_d) and long (\tau) exhibit a substantial delay in GSC collapse, even when the epidemic peak occurs relatively early. This creates a clear temporal gap between the maximum number of infected individuals and the moment the healthy network loses its giant component. The authors argue that this gap is crucial for public‑health interventions: policies that temporarily suppress contacts can flatten the infection curve while preserving the structural integrity of the social fabric, allowing a smoother post‑epidemic recovery.

The paper’s contributions are threefold. (1) It introduces a dynamic, edge‑centric perspective on the susceptible sub‑network, extending epidemic modeling beyond the traditional focus on infection trajectories. (2) By integrating EBCM with percolation theory, it provides a closed‑form expression for the critical time of giant component destruction, linking individual behavioral parameters directly to network robustness. (3) The work offers actionable insights for epidemic control: targeted, temporary reduction of contacts not only reduces transmission but also prolongs the lifespan of the healthy social core, which can be vital for maintaining essential services and societal cohesion during an outbreak. The thorough analytical derivations, complemented by large‑scale simulations, make the study a robust addition to the literature on adaptive networks and disease dynamics, and it opens avenues for future research on multi‑layered, temporally evolving contact structures.