  In this paper, we investigate the fermionic current densities induced by a
magnetic flux running along the idealized cosmic string in a four-dimensional
spacetime, admitting that the coordinate along the string's axis is
compactified. In order to develop this investigation we construct the complete
set of fermionic mode functions obeying a general quasiperiodicity condition
along the compactified dimension. The vacuum expectation value of the azimuthal
current density is decomposed into two parts. The first one corresponds to the
uncompactified cosmic string geometry and the second one is the correction
induced by the compactification. For the first part we provide a closed
expression which includes various special cases previously discussed in the
literature. The second part is an odd periodic function of the magnetic flux
along the string axis with the period equal to the flux quantum and it is an
even function of the magnetic flux enclosed by the string axis. The
compactification of the cosmic string axis in combination with the
quasiperiodicity condition leads to the nonzero axial current density. The
latter is an even periodic function of the magnetic flux along the string axis
and an odd periodic function of the magnetic flux enclosed by the string axis.
The axial current density vanishes for untwisted and twisted fields in the
absence of the magnetic flux enclosed by the string axis. The asymptotic
behavior of the vacuum fermionic current is investigated near the string and at
large distances from it. In particular, the topological part of the azimuthal
current and the axial current are finite on the string's axis.
