The evoSIR model is a modification of the usual SIR process on a graph $G$ in which $S-I$ connections are broken at rate $ρ$ and the $S$ connects to a randomly chosen vertex. The evoSI model is the same as evoSIR but recovery is impossible. In \cite{DOMath} the critical value for evoSIR was computed and simulations showed that when $G$ is an Erd\H os-Rényi graph with mean degree 5, the system has a discontinuous phase transition, i.e., as the infection rate $λ$ decreases to $λ_c$, the fraction of individuals infected during the epidemic does not converge to 0. In this paper we study evoSI dynamics on graphs generated by the configuration model. We show that there is a quantity $Δ$ determined by the first three moments of the degree distribution, so that the phase transition is discontinuous if $Δ>0$ and continuous if $Δ<0$.