  Healthy and sick individuals (A and B particles) diffuse independently with
diffusion constants D_A and D_B. Sick individuals upon encounter infect healthy
ones (at rate k), but may also spontaneously recover (at rate 1/\tau). The
propagation of the epidemic therefore couples to the fluctuations in the total
population density. Global extinction occurs below a critical value \rho_{c} of
the spatially averaged total density. The epidemic evolves as the
diffusion--reaction--decay process
  A + B --> 2B, B --> A ,
  for which we write down the field theory. The stationary state properties of
this theory when D_A=D_B were obtained by Kree et al. The critical behavior for
D_A<D_B is governed by a new fixed point. We calculate the critical exponents
of the stationary state in an $\eps$ expansion, carried out by Wilson
renormalization, below the critical dimension d_{c}=4. We then go on to to
obtain the critical initial time behavior at the extinction threshold, both for
D_A=D_B and D_A<D_B. There is nonuniversal dependence on the initial particle
distribution. The case D_A>D_B remains unsolved.
