  The hypothesis that pulsar wind nebulae (PWNe) can significantly contribute
to the excess of the positron ($e^+$) cosmic-ray flux has been consolidated
after the observation of a $\gamma$-ray emission at TeV energies of a few
degree size around Geminga and Monogem PWNe, and at GeV energies for Geminga at
a much larger extension. The $\gamma$-ray halos around these PWNe are
interpreted as due to electrons ($e^-$) and $e^+$ accelerated and escaped by
their PWNe, and inverse Compton scattering low-energy photons of the
interstellar radiation fields. The extension of these halos suggests that the
diffusion around these PWNe is suppressed by two orders of magnitude with
respect to the average in the Galaxy. We implement a two-zone diffusion model
for the propagation of $e^+$ accelerated by the Galactic population of PWNe. We
consider pulsars from the ATNF catalog and build up simulations of the PWN
Galactic population. In both scenarios, we find that within a two-zone
diffusion model, the total contribution from PWNe and secondary $e^+$ is at the
level of AMS-02 data, for an efficiency of conversion of the pulsar spin down
energy in $e^\pm$ of $\eta\sim0.1$. For the simulated PWNe, a $1\sigma$
uncertainty band is determined, which is of at least one order of magnitude
from 10 GeV up to few TeV. The hint for a decreasing $e^+$ flux at TeV energies
is found, even if it is strongly connected to the chosen value of the radius of
the low diffusion bubble around each source.
