Before revision:  In a bid to resolve this conundrum we are inspired by de Finetti's Theorem, and we exhibit equivalence classes of meta-population models that are orbits of the symmetric group. We illustrate with a mixture of stochastic SIR models in which growth can be inferred from a classic bond pricing formula of Vasicek. This approach exploits the symmetry of model observables, and then uses convexity adjustments to directly determine the degree of variation that is needed to locate the orbit of nature's model . Convexity adjustments are also useful, and material, for cross-sectional comparison.
After revision:  In a bid to resolve this conundrum we are inspired by de Finetti's Theorem, to show how the modeling discrepancy can be explained. Moreover the difference between predictions of classic constant parameter epidemiological models and those with variation and stochastic evolution can be reduced to simple "convexity" formulas. A novel feature of this approach is that we do not attempt to locate a true model but only a model that is equivalent after permutations . Convexity adjustments are also useful, and material, for cross-sectional comparison.