It is also worth noting that there are many results derived through these "approximate" approaches that are the same as exact results or are numerically indistinguishable from exact results and simulation. We give some examples below and also note that the dynamical PGF approach [129] is numerically indistinguishable from the exact model (14) above for certain parameter values [130] . What is currently lacking is a rigorous mathematical proof of exactness for ODE models other than those outlined in Section 4.2.4 above. While for many practical purposes the absence of such a proof will not matter, we preserve here the conceptual distinction between results that are provably exact, and those that are numerically exact in all cases tested so far.