no, the epidemiological model in question has two degrees of freedom, and one of those degrees is stipulated by the assumed shape of the curve (exponential, even though that's a poor fit overall, logistic being a better fit). R0 does not constrain the other degree of freedom; it only gives you the initial condition. it requires knowing that other degree of freedom, which you're making implicit assumptions about, that governs what you can say about the rapidness of spread and makes some statements about bounding. this is rudimentary calculus and linear modeling.