Connectivity disruption sparks explosive epidemic spreading

We investigate the spread of an infection or other malfunction of cascading nature when a system component can recover only if it remains reachable from a functioning central component. We consider the susceptible-infected-susceptible model, typical of mathematical epidemiology, on a network. Infection spreads from infected to healthy nodes, with the addition that infected nodes can only recover when they remain connected to a predefined central node, through a path that contains only healthy nodes. In this system, clusters of infected nodes will absorb their noninfected interior because no path exists between the central node and encapsulated nodes. This gives rise to the simultaneous infection of multiple nodes. Interestingly, the system converges to only one of two stationary states: either the whole population is healthy or it becomes completely infected. This simultaneous cluster infection can give rise to discontinuous jumps of different sizes in the number of failed nodes. Larger jumps emerge at lower infection rates. The network topology has an important effect on the nature of the transition: we observed hysteresis for networks with dominating local interactions. Our model shows how local spread can abruptly turn uncontrollable when it disrupts connectivity at a larger spatial scale.