The non-equilibrium phase transition in models for epidemic spreading with long-range infections in combination with incubation times is investigated by field-theoretical and numerical methods. Here the spreading process is modelled by spatio-temporal Levy flights, i.e., it is assumed that both spreading distance and incubation time decay algebraically. Depending on the infection rate one observes a phase transition from a fluctuating active phase into an absorbing phase, where the infection becomes extinct. This transition between spreading and extinction is characterized by continuously varying critical exponents, extending from a mean-field regime to a phase described by the universality class of directed percolation. We compute the critical exponents in the vicinity of the upper critical dimension by a field-theoretic renormalization group calculation and verify the results in one spatial dimension by extensive numerical simulations.

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