Endemic infectious states below the epidemic threshold and beyond herd immunity

Understanding the intricacies behind epidemic spreading models is of great relevance to reach the point in which trustable predictions on the propagation patterns can be attained. In the recent COVID-19 pandemic, we have assisted at a sequence of epidemic peaks as prevention measures were enforced and relaxed. However, even immediately after lifting the most stringent home confinement of the first wave the number of new cases remained low but non-zero. Some previous works have gone so far as considering this regime as critic. Inspired by this phenomenon, we study here the paradigmatic Susceptible-Infected-Recovered (SIR) model in a meta-population framework with inflow of infected individuals from a reservoir. Focusing on a regime where this external seeding is so small that cannot be detected from the analysis of epidemic curves, we find that outbreaks of finite duration percolate in time resulting in an overall endemic state for a broad parameter area. Using a two-state description of the local dynamics we are able to extract analytical predictions for the phase space. The main findings hold for a variety of epidemic and mobility models, network topologies connecting the subpopulations and demographic distributions. While we concentrate here on grasping the basic mechanisms behind this phenomenon, the consequences about the presence of these endemic states can be immediately translated to applications and to epidemic forecasting.