Contagion dynamics in time-varying metapopulation networks

The metapopulation framework is adopted in a wide array of disciplines to describe systems of well separated yet connected subpopulations. The subgroups or patches are often represented as nodes in a network whose links represent the migration routes among them. The connections have been so far mostly considered as static, but in general evolve in time. Here we address this case by investigating simple contagion processes on time-varying metapopulation networks. We focus on the SIR process and determine analytically the mobility threshold for the onset of an epidemic spreading in the framework of activity-driven network models. We find profound differences from the case of static networks. The threshold is entirely described by the dynamical parameters defining the average number of instantaneously migrating individuals and does not depend on the properties of the static network representation. Remarkably, the diffusion and contagion processes are slower in time-varying graphs than in their aggregated static counterparts, the mobility threshold being even two orders of magnitude larger in the first case. The presented results confirm the importance of considering the time-varying nature of complex networks.

💡 Research Summary

The paper tackles a fundamental limitation of traditional metapopulation models, which assume a static network of migration routes among subpopulations. Recognizing that real‑world mobility patterns are highly time‑dependent, the authors adopt the activity‑driven network framework to model a metapopulation in which links appear and disappear at each discrete time step. In this setting each node (patch) is characterized by an activity rate a; with probability a it becomes active and creates m temporary connections to randomly chosen other patches. The connections exist only for the current time step, after which the network is reshuffled. This construction yields a time‑varying graph whose instantaneous average degree is ⟨k⟩ = m⟨a⟩, while the aggregated static representation would simply be the time‑average of these degrees.

Within each patch the authors consider a classic Susceptible–Infected–Recovered (SIR) process with infection rate β and recovery rate γ. Individuals may also migrate between patches: at each time step an individual moves with probability p along one of the currently active links. The central analytical goal is to determine the mobility threshold p_c that marks the onset of a global epidemic. By linearizing the coupled dynamics of infection prevalence and the number of infected migrants, the authors derive a closed‑form expression for p_c that depends solely on dynamical parameters:

 p_c ≈ (γ/β) ·