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.
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