The spread of excitations by Rydberg facilitation bears many similarities to
epidemics. Such systems can be modeled with Monte-Carlo simulations of
classical rate equations to great accuracy as a result of high dephasing. In
this paper, we analyze the dynamics of a Rydberg many-body system in the
facilitation regime in the limits of high and low temperatures. While in the
high-temperature limit a homogeneous mean-field behaviour is recovered,
characteristic effects of heterogeneity can be seen in a frozen gas. At large
temperatures the system displays an absorbing-state phase transition and, in
the presence of an additional loss channel, self-organized criticality. In a
frozen or low-temperature gas, excitations are constrained to a network
resembling an Erd\"os-Renyi graph. We show that the absorbing-state phase
transition is replaced with an extended Griffiths phase, which we accurately
describe by a susceptible-infected-susceptible model on the Erd\"os-Renyi
network taking into account Rydberg blockade. Furthermore, we expand upon an
existing macroscopic Langevin equation to more accurately describe the density
of Rydberg atoms in the frozen and finite temperature regimes.Comment: 14 pages, 11 figure