Anisotropic contact processes on homogeneous trees

Sufficient conditions are given for the existence of a weak survival phase in a homogeneous but not necessarily isotropic contact process on a homogeneous tree. These require that the contact process be homogeneous, that is, for any two vertices x,y of the tree there is an automorphism mapping x to y leaving the infection rates invariant; and that the contact process be weakly symmetric, that is, for each vertex there should be at least two incident edges with the same infection rate.