Background: The ($alpha$,n) and ($alpha$,$gamma$) reactions on $^{17,18}$O have significant impact on the neutron balance in the astrophysical $s$-process. In this scenario stellar reaction rates are required for relatively low temperatures below $T_9 lesssim 1$. Purpose: The uncertainties of the $^{17,18}$O($alpha$,n)$^{20,21}$Ne reactions are investigated. Statistical model calculations are performed to study the applicability of this model for relatively light nuclei in extension to a recent review for the $20 le A le 50$ mass range. Method: The available experimental data for the $^{17,18}$O($alpha$,n)$^{20,21}$Ne reactions are compared to statistical model calculations. Additionally, the reverse $^{20}$Ne(n,$alpha$)$^{17}$O reaction is investigated, and similar studies for the $^{17}$F mirror nucleus are provided. Results: It is found that on average the available experimental data for $^{17}$O and $^{18}$O are well described within the statistical model, resulting in reliable reaction rates above $T_9 gtrsim 1.5$ from these calculations. However, significant experimental uncertainties are identified for the $^{17}$O($alpha$,n$_0$)$^{20}$Ne(g.s.) channel. Conclusions: The statistical model is able to predict astrophysical reaction rates for temperatures above 1 GK with uncertainties of less than a factor of two for the nuclei under study. An experimental discrepancy for the $^{17}$O($alpha$,n)$^{20}$Ne reaction needs to be resolved.
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