Time evolution of infection state correlation of nodes in an SIR model on a random contact network

Traditional network epidemic models often assume that the infection states of a node's neighbours are independent, an assumption that is commonly used to approximate triple-node interactions by pairwise ones. However, this assumption is generally invalid for infectious nodes, which limits the ability of pairwise approximations to study SIS dynamics and contact tracing. In this paper, we propose a comprehensive edge-based SIR compartmental model on configuration model networks that explicitly incorporates the infection age of infected nodes. By tracking the time evolution of edges connecting infected nodes to their neighbours, we derive a system of partial differential equations for the age-dependent conditional probabilities of neighbour states. The model admits explicit solutions for these correlation probabilities. Numerical simulations reveal a strong local depletion effect: the probability that an infected node has a susceptible neighbour decays monotonically with infection age. In contrast, the probability that an infected node has an infected neighbour exhibits a non-monotonic peak, reflecting the synchronization of local secondary infections. This framework rigorously links microscopic stochastic transmission events with macroscopic epidemic dynamics, showing that individuals with older infections are effectively shielded by localized herd immunity. These findings provide a theoretical foundation for understanding age-dependent transmission heterogeneity and for optimizing the timing of interventions such as contact tracing.