  Semi-numeric methods have made it possible to efficiently model the epoch of
reionisation (EoR). While most implementations involve a reduction to a simple
three-parameter model, we introduce a new mass-dependent ionising efficiency
parameter that folds in physical parameters that are constrained by the latest
numerical simulations. This new parameterization enables the effective modeling
of a broad range of host halo masses containing ionising sources, extending
from the smallest Population III host halos with $M \sim 10^6 M_\odot$, which
are often ignored, to the rarest cosmic peaks with $M \sim 10^{12} M_\odot$
during EoR. We compare the resulting ionising histories with a typical
three-parameter model and also compare with the latest constraints from the
Planck mission. Our model results in a optical depth due to Thomson scattering,
$\tau_{\mathrm{e}}$ = 0.057, that is consistent with Planck. The largest
difference in our model is shown in the resulting bubble size distributions
which peak at lower characteristic sizes and are broadened. We also consider
the uncertainties of the various physical parameters and comparing the
resulting ionising histories broadly disfavors a small contribution from
galaxies. As the smallest haloes cease a meaningful contribution to the
ionising photon budget after $z = 10$, implying they play a role in determining
the start of EoR and little else.
