The time frame in which hydrogen reionization occurred is highly uncertain, but can be constrained by observations of Lyman-alpha (Ly$alpha$) emission from distant sources. Neutral hydrogen in the intergalactic medium (IGM) attenuates Ly$alpha$~photons emitted by galaxies. As reionization progressed the IGM opacity decreased, increasing Ly$alpha$~visibility. The galaxy Ly$alpha$~luminosity function (LF) is thus a useful tool to constrain the timeline of reionization. In this work, we model the Ly$alpha$~LF as a function of redshift, $z=5-10$, and average IGM neutral hydrogen fraction, $overline{x}_textsc{hi}$. We combine the Ly$alpha$~luminosity probability distribution obtained from inhomogeneous reionization simulations with a model for the UV LF to model the Ly$alpha$~LF. As the neutral fraction increases, the average number density of Ly$alpha$~emitting galaxies decreases, and are less luminous, though for $overline{x}_textsc{hi} lesssim 0.4$ there is only a small decrease of the Ly$alpha$~LF. We use our model to infer the IGM neutral fraction at $z=6.6, 7.0, 7.3$ from observed Ly$alpha$~LFs. We conclude that there is a significant increase in the neutral fraction with increasing redshift: $overline{x}_textsc{hi}(z=6.6)=0.08^{+ 0.08}_{- 0.05}, , overline{x}_textsc{hi}(z=7.0)=0.28 pm 0.05$ and $overline{x}_textsc{hi}(z=7.3)=0.83^{+ 0.06}_{- 0.07}$. We predict trends in the Ly$alpha$~luminosity density and Schechter parameters as a function of redshift and the neutral fraction. We find that the Ly$alpha$~luminosity density decreases as the universe becomes more neutral. Furthermore, as the neutral fraction increases, the faint-end slope of the Ly$alpha$~LF steepens, and the characteristic Ly$alpha$~luminosity shifts to lower values, concluding that the evolving shape of the Ly$alpha$~LF -- not just its integral -- is an important tool to study reionization.
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