Epidemic spreading with long-range infections and incubation times

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. In this class of models the infection is assumed to spread isotropically over long distances r whose probability distribution decays algebraically as P(r) similar to r(-d-sigma), where d is the spatial dimension. Moreover, a freshly infected individual can infect other individuals only after a certain incubation time, modelled here as a waiting time Delta t, which is distributed probabilistically as P(Delta t) similar to (Delta t)(-1-kappa). Tuning the balance between spreading and spontaneous recovery one observes a continuous phase transition from a fluctuating active phase into an absorbing phase, where the infection becomes extinct. Depending on the parameters sigma and kappa 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. Specifying the phase diagram in terms of s and. we compute the critical exponents in the vicinity of the upper critical dimension d(c) = sigma(3 - kappa(-1)) by a field-theoretic renormalization group calculation and verify the results in one spatial dimension by extensive numerical simulations.