Inhomogeneous contact processes on trees

We consider an inhomogeneous contact process on a tree double-struck T sign k of degree k, where the infection rate at any site is λ, the death rate at any site in S ⊂ double-struck T sign_k is δ (with 0 < δ ≤ 1) and that at any site in double-struck T sign_k - S is 1. Denote by λ_c(double-struck T sign_k) the critical value for the homogeneous model (i.e., δ = 1) on double-struck T sign_k and by 0(δ, λ) the survival probability of the inhomogeneous model on double-struck T sign_k. We prove that when k > 4, if S = double-struck T sign_σ, a subtree embedded in double-struck T sign_k, with 1 ≤ σ ≤ √k, then there exists δ^σ_c strictly between λ_c(double-struck T sign_k)/λ_c(double-struck T sign_σ) and 1 such that 0(δ λ_c(double-struck T sign_k)) = 0 when δ > δ^σ_c and 0(δ, λ_c(double-struck T sign_k)) > 0 when δ < δ^σ_c; if S = {0}, the origin of double-struck T sign_k, then θ(δ, λ_c(double-struck T sign_k)) = 0 for any δ ∈ (0, 1).