Does the critical infection rate for epidemic survival in the avoidance-isolation model on Z^d exhibit a sharp phase transition that depends continuously on the strength of behavioral response?

Consider an epidemic model on the integer lattice Z^d where individuals not only spread infection but also actively avoid contact with visibly infected neighbors and isolate when symptomatic. The central open problem is to determine whether the critical threshold separating extinction from endemic persistence changes continuously as the avoidance and isolation parameters are tuned, or whether there exist discontinuous jumps suggesting a first-order phase transition. Specifically, one wants to characterize the full phase diagram in the joint parameter space of infection rate, avoidance strength, and isolation rate, and determine the regularity of the critical surface separating these regimes.