  We present a study of the connection between the escape fraction of
Lyman-alpha (Ly$\alpha$) and Lyman-continuum (LyC) photons within a sample of
N=152 star-forming galaxies selected from the VANDELS survey at
$3.85<z_{spec}<4.95$. By combining measurements of H$\alpha$ equivalent width
$(W_\lambda(\rm{H\alpha}))$ derived from broad-band photometry with Ly$\alpha$
equivalent width $(W_\lambda(Ly\alpha))$ measurements from VANDELS spectra, we
individually estimate $f_{\rm{esc}}^{Ly\alpha}$ for our full sample. In
agreement with previous studies, we find a positive correlation between
$W_\lambda(Ly\alpha)$ and $f_{\rm{esc}}^{Ly\alpha}$, increasing from
$f_{\rm{esc}}^{Ly\alpha}\simeq0.04$ at $W_\lambda(Ly\alpha)=10$\r{A} to
$f_{\rm{esc}}^{Ly\alpha}\simeq0.1$ at $W_\lambda(Ly\alpha)=25$\r{A}. For the
first time at $z\sim4-5$, we investigate the relationship between
$f_{\rm{esc}}^{Ly\alpha}$ and $f_{\rm{esc}}^{\rm{LyC}}$ using
$f_{\rm{esc}}^{\rm{LyC}}$ estimates derived using the equivalent widths of
low-ionization, FUV absorption lines in composite VANDELS spectra. Our results
indicate that $f_{\rm{esc}}^{\rm{LyC}}$ rises monotonically with
$f_{\rm{esc}}^{Ly\alpha}$, following the relation
$f_{\rm{esc}}^{\rm{LyC}}\simeq 0.15^{+0.06}_{-0.04}f_{\rm{esc}}^{Ly\alpha}$.
Based on composite spectra of sub-samples with roughly constant
$W_\lambda(Ly\alpha)$, but very different $f_{\rm{esc}}^{Ly\alpha}$, we show
that the $f_{\rm{esc}}^{\rm{LyC}}-f_{\rm{esc}}^{Ly\alpha}$ correlation is not
driven by a secondary correlation between $f_{\rm{esc}}^{Ly\alpha}$and
$W_\lambda(Ly\alpha)$. The $f_{\rm{esc}}^{\rm{LyC}}-f_{\rm{esc}}^{Ly\alpha}$
correlation is in good qualitative agreement with theoretical predictions and
provides further evidence that estimates of $f_{\rm{esc}}^{\rm{LyC}}$ within
the Epoch of Reionization should be based on proxies sensitive to neutral gas
density/geometry and dust attenuation.
