Standard epidemiological models apply the vaccination rate to the entire susceptible population. Everyone who hasn't been infected or vaccinated is assumed to be available for vaccination. This is mathematically convenient and empirically wrong.
Glenn Ledder shows what happens when you ignore that some fraction of the susceptible population refuses vaccination. The error is not small, and more importantly, it cannot be corrected by reducing the vaccination rate constant. On endemic timescales, the equilibrium predictions are substantially wrong. On epidemic timescales, the errors persist, particularly for less infectious diseases with slower vaccination programs.
The instinct to fix this by tuning the rate is natural. If 20% refuse, decrease the rate by 20%. But this doesn't work because a rate applies to a population uniformly. Decreasing the rate delays vaccination for everyone — willing and unwilling alike. The willing receive their vaccines too slowly; the unwilling still receive them eventually. The actual dynamics require partitioning the susceptible compartment: willing susceptibles and unwilling susceptibles, with vaccination applied only to the first.
The distinction is between parametric error and structural error. A parametric error is a wrong number in the right equation. A structural error is the right number in the wrong equation. No amount of parameter adjustment corrects a structural error because the adjustment operates within the structure that's causing the problem.
This is a general principle disguised as an epidemiological detail. Any model that treats a heterogeneous population as homogeneous generates errors that cannot be absorbed by adjusting the rate constants. The error lives in the partition — in which entities the equation applies to — not in the coefficients. The fix is not a better number. It's a better equation.