  In most cities, transit consists solely of fixed-route transportation, whence
the inherent limited Quality of Service for travellers in suburban areas and
during off-peak periods. On the other hand, completely replacing fixed-route
(FR) with demand-responsive (DR) transit would imply a huge operational cost.
It is still unclear how to integrate DR transportation into current transit
systems to take full advantage of it. We propose a Continuous Approximation
model of a transit system that gets the best from fixed-route and DR
transportation. Our model allows deciding whether to deploy a FR or a DR
feeder, in each sub-region of an urban conurbation and each time of day, and to
redesign the line frequencies and the stop spacing of the main trunk service.
Since such a transit design can adapt to the spatial and temporal variation of
the demand, we call it Adaptive Transit. Numerical results show that, with
respect to conventional transit, Adaptive Transit significantly improves
user-related cost, by drastically reducing access time to the main trunk
service. Such benefits are particularly remarkable in the suburbs. Moreover,
the generalized cost, including agency and user cost, is also reduced. These
findings are also confirmed in scenarios with automated vehicles. Our model can
assist in planning future-generation transit systems, able to improve urban
mobility by appropriately combining fixed and DR transportation.
