Spatial correlations in susceptible-infected-susceptible processes on random regular graphs

In network-based susceptible-infected-susceptible (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.