Spatial correlations in SIS processes on random regular graphs

In network-based SIS models of infectious disease transmission, infection can only occur between directly connected individuals. This constraint naturally gives rise to spatial correlations between the states of neighboring nodes, as the infection status of connected individuals becomes interdependent. Although mean-field approximations and the standard pairwise model are commonly used to simplify disease forecasting on networks, they inadequately capture spatial correlations; mean-field frameworks assume that populations are well-mixed, while the pairwise model neglects correlations beyond nearest-neighbor connections, which leads to inaccurate predictions of infection numbers over time. As such, the development of approximations that account for higher order spatially correlated infections is of great interest, as they offer a compromise between accurate disease forecasting and analytic tractability. Here, we use existing corrections to mean-field theory on the regular lattice to construct a more general framework for equivalent corrections on random regular graph topologies. We derive and simulate a hierarchical system of ordinary differential equations for the time evolution of the spatial correlation function at various geodesic distances on random networks. Solving these equations allows us to predict the time-dependent global infection density, which agrees well with numerical simulations. Our results substantially improve on existing corrections to mean-field theory for infectious individuals in SIS processes and provide an in-depth characterization of how structural randomness in networks affects the dynamical trajectories of infectious diseases on networks.

💡 Research Summary

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The paper tackles a long‑standing limitation in the analytical treatment of SIS (Susceptible‑Infected‑Susceptible) epidemic dynamics on networks: conventional mean‑field (MF) approximations and the standard pairwise model (SPM) fail to capture spatial correlations that extend beyond immediate neighbors. While MF assumes a perfectly mixed population, SPM only tracks densities of adjacent S‑I and I‑I pairs, neglecting any dependence on longer shortest‑path distances. This shortfall becomes especially pronounced on low‑degree regular structures, where infection clustering is strong.

To overcome this, the authors build on recent lattice‑based corrections to MF theory and generalize them to random regular graphs (RRGs), which are networks where every node has the same degree (k_0) but connections are otherwise random. They introduce distance‑dependent correlation functions \