Exact solution of the dynamic epidemic model on the Bethe lattice

The dynamic epidemic model considers the spreading of a cluster in a medium containing a fraction x of mobile particles which are pushed by the propagation front. This model is analytically studied on the Bethe lattice for any branching rate z. We give the exact solution xc = (z2 − 1)/z2 for the percolation threshold. This is in contrast with the xc = (z − 1)/z result for static particles. Moreover, we calculate the critical exponents γ = 1 and ν = 1 characterizing respectively the divergence of the cluster mass and the correlation length at xc. These exponents are found to be the same as for the case of static particles, i.e. for random percolation on the Bethe lattice.