Critical-curve conjecture for the generalized deterministic-cure contact process

Determine whether, for every fixed resetting probability p∈[0,1], the generalized contact process with deterministic cures CPDC(λ,p) has a phase transition at a critical infection rate λ_c(p), and whether its critical curve is the weighted harmonic interpolation λ_c(p)=λ_c(0)λ_c(1)/[(1−p)λ_c(1)+pλ_c(0)].

Background

The paper studies CPDC(λ,p) on Z^d, in which infected vertices recover exactly one time unit after infection unless an incoming infection attempt is accepted as a reset with probability p. The endpoint p=0 is the non-resetting Contact Process with Deterministic Cure, while p=1 is the fully resetting Contact Process with Restarting Deterministic Cure. Because deterministic recovery deadlines can shield later infection attempts, the process is generally non-attractive for p<1, making standard monotonicity-based phase-transition arguments unavailable.

The paper proves uniform extinction for λ<1/(2d) and, in dimensions d≥2, uniform survival above an oriented-percolation-based bound. It establishes the existence of a phase transition in λ for p=1 using attractiveness, but does not establish the corresponding phase transition for every p∈[0,1] or the proposed interpolation formula. The conjectured numerical values in dimension two are λ_c(0)≈0.40755 and λ_c(1)≈0.33247.