  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.
