  We consider the Diffusive Epidemic Process (DEP), a two-species
reaction-diffusion process originally proposed to model disease spread within a
population. This model exhibits a phase transition from an active epidemic to
an absorbing state without sick individuals. Field-theoretic analyses suggest
that this transition belongs to the universality class of Directed Percolation
with a Conserved quantity (DP-C). However, some exact predictions derived from
the symmetries of DP-C seem to be in contradiction with lattice simulations.
Here we revisit the field theory of both DP-C and DEP. We discuss in detail the
symmetries present in the various formulations of both models, some of which
had not been identified previously. We then investigate the DP-C model using
the derivative expansion of the non-perturbative renormalization group
formalism. We recover previous results for DP-C near its upper critical
dimension $d_c=4$, but show how the corresponding fixed point seems to no
longer exist below $d \lesssim 3$. Consequences for the DEP universality class
are considered.
