  We study the continuous absorbing-state phase transition in the
one-dimensional diffusive epidemic process via mean-field theory and Monte
Carlo simulation. In this model, particles of two species (A and B) hop on a
lattice and undergo reactions B -> A and A + B -> 2B; the total particle number
is conserved. We formulate the model as a continuous-time Markov process
described by a master equation. A phase transition between the (absorbing)
B-free state and an active state is observed as the parameters (reaction and
diffusion rates, and total particle density) are varied. Mean-field theory
reveals a surprising, nonmonotonic dependence of the critical recovery rate on
the diffusion rate of B particles. A computational realization of the process
that is faithful to the transition rates defining the model is devised,
allowing for direct comparison with theory. Using the quasi-stationary
simulation method we determine the order parameter and the survival time in
systems of up to 4000 sites. Due to strong finite-size effects, the results
converge only for large system sizes. We find no evidence for a discontinuous
transition. Our results are consistent with the existence of three distinct
universality classes, depending on whether A particles diffusive more rapidly,
less rapidly, or at the same rate as B particles.
