Emergent spatiotemporal heterogeneity in networked epidemics: Turing instability driven by topology and mobility.

While pattern formation in reaction-diffusion systems has been widely explored for epidemics in continuous media, its manifestation in networked populations remains poorly understood. We propose a theoretical framework integrating metapopulation networks with susceptible-infected-susceptible epidemic dynamics, revealing how topology and mobility jointly drive emergent spatiotemporal heterogeneity through Turing instability. Linear stability analysis identifies critical thresholds where eigenvector localization in scale-free networks amplifies heterogeneity by destabilizing low-degree nodes. Numerical simulations demonstrate that an infection rate (β) governs epidemic magnitude and pattern geometry, while a network degree distribution shapes a hierarchical phenomenon. Analytical solutions quantify how hub nodes suppress local instability, yet enhance global transmission. This work establishes Turing mechanisms as fundamental to networked epidemic patterns, bridging network science with reaction-diffusion theory. Our findings offer predictive tools for identifying high-risk zones in real-world mobility systems and informing targeted intervention strategies.