We investigate the critical behavior of a stochastic lattice model describing the hopping of two species that undergo reactions B→A and A+B→2B. We simulate the model defined on a 3D lattice and determine the threshold of the absorbing phase transition to the state at which species B is extinct. Using steady state and short-time dynamics simulations, we calculate the order parameter, order parameter fluctuations, correlation length and their critical exponents. We focus in the case of species A diffusing much faster than species B. We did not find signatures of a first-order transition that has been conjectured in the literature. We report a continuous transition with the perpendicular correlation exponent ν⊥≈0.61(6), in agreement with ν⊥=2∕D. Also, for this diffusion regime, we estimate β∕ν⊥=1.48, very close to the ϵ expansion prediction β∕ν⊥=D−ε∕82 (ε=4−D) for the regime of equally diffusing particles. •Stationary and dynamic critical behavior of a diffusive epidemic process is investigated.•The absorbing state phase transition in a 3D lattice is explored in the case of fast diffusion of healthy individuals.•The dynamic transition to the absorbing state with no sick individuals is shown to be continuous.•The critical exponents are compared with those of related universality classes.