Wilson renormalization of a reaction-diffusion process

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/τ). The propagation of the epidemic therefore couples to the fluctuations in the total population density. Global extinction occurs below a critical value ρ 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 ε 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.