  We investigate how site dilution, as would be introduced by immunization,
affects the properties of the active-to-absorbing non-equilibrium phase
transition in the paradigmatic Susceptible-Infectious-Recovered (SIR) model on
regular cubic lattices. According to the Harris criterion, the critical
behavior of the SIR model, which is governed by the universal scaling exponents
of the dynamic isotropic percolation (DyIP) universality class, should remain
unaltered after introducing impurities. However, when the SIR reactions are
simulated for immobile agents on two- and three-dimensional lattices subject to
quenched disorder, we observe a wide crossover region characterized by varying
effective exponents. Only after a sufficient increase of the lattice sizes does
it becomes clear that the SIR system must transition from that crossover regime
before the effective critical exponents asymptotically assume the expected DyIP
values. We attribute the appearance of this exceedingly long crossover to a
time lag in a complete recovery of small disconnected clusters of susceptible
sites which are apt to be generated when the system is prepared with
Poisson-distributed quenched disorder. Finally, we demonstrate that this
transient region becomes drastically diminished when we significantly increase
the value of the recovery rate or enable diffusive agent mobility through
short-range hopping.
