  This paper introduces a run-and-tumble model with self-reinforcing
directionality and rests. We derive a single governing hyperbolic partial
differential equation for the probability density of random walk position, from
which we obtain the second moment in the long time limit. We find the criteria
for the transition between superdiffusion and diffusion caused by the addition
of a rest state. The emergence of superdiffusion depends on both the parameter
representing the strength of self-reinforcement and the ratio between mean
running and resting times. The mean running time must be at least $2/3$ of the
mean resting time for superdiffusion to be possible. Monte Carlo simulations
validate this theoretical result. This work demonstrates the possibility of
extending the telegrapher's (or Cattaneo) equation by adding self-reinforcing
directionality so that superdiffusion occurs even when rests are introduced.
