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.PACS 05.40+j, 05.70.Ln, 82.20.Mj
We consider two stochastic processes, the Gribov process and the general epidemic process, that describe the spreading of an infectious disease. In contrast to the usually assumed case of shortrange infections that lead, at the critical point, to directed and isotropic percolation respectively, we consider long-range infections with a probability distribution decaying in d dimensions with the distance as 1/R d+σ . By means of Wilson's momentum shell renormalization-group recursion relations, the critical exponents characterizing the growing fractal clusters are calculated to first order in an ε-expansion. It is shown that the long-range critical behavior changes continuously to its short-range counterpart for a decay exponent of the infection σ = σc > 2.
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