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An epidemic process mediated by a decaying diffusing signal

We study a stochastic epidemic model consisting of elements (organisms in a community or cells in tissue) with fixed positions, in which damage or disease is transmitted by diffusing agents ("signals") emitted by infected individuals. The signals decay as well as diffuse; since they are assumed to be produced in large numbers, the signal concentration is treated deterministically. The model, which includes four cellular states (susceptible, transformed, depleted, and removed), admits various interpretations: spread of an infection or infectious disease, or of damage in a tissue in which injured cells may themselves provoke further damage, and as a description of the so-called radiation-induced bystander effect, in which the signals are molecules capable of inducing cell damage and/or death in unirradiated cells. The model exhibits a continuous phase transition between spreading and nonspreading phases. We formulate two mean-field theory (MFT) descriptions of the model, one of which ignores correlations between the cellular state and the signal concentration, and another that treats such correlations in an approximate manner. Monte Carlo simulations of the spread of infection on the square lattice yield values for the critical exponents and the fractal dimension consistent with the dynamic percolation universality class.