  Current models predict the hose instability to crucially limit the
applicability of plasma-wakefield accelerators. By developing an analytical
model which incorporates the evolution of the hose instability over long
propagation distances, this work demonstrates that the inherent drive-beam
energy loss, along with an initial beam energy spread detune the betatron
oscillations of beam electrons, and thereby mitigate the instability. It is
also shown that tapered plasma profiles can strongly reduce initial hosing
seeds. Hence, we demonstrate that the propagation of a drive beam can be
stabilized over long propagation distances, paving the way for the acceleration
of high-quality electron beams in plasma-wakefield accelerators. We find
excellent agreement between our models and particle-in-cell simulations.
