  Recently Dantas, Oliveira and Stilck [J. Stat. Mech. (2007) P08009] studied
how the one-dimensional diffusive contact process crosses over from the
critical behavior of directed percolation to an effective mean field behaviour
when the diffusion rate is sent to infinity. They showed that this crossover
can be described in terms of a crossover exponent $\phi$, finding the
boundaries 3 <= $\phi$ <= 4 in one spatial dimension. In the present work we
refine and extend this result up to four spatial dimensions by a
field-theoretic calculation and extensive numerical simulations.
