  The study of the Sivers effect, describing correlations between the
transverse polarization of the nucleon and its constituent (unpolarized)
parton's transverse momentum, has been the topic of a great deal of
experimental, phenomenological and theoretical effort in recent years.
Semi-inclusive deep inelastic scattering measurements of the corresponding
single spin asymmetries (SSA) at the upcoming CLAS12 experiment at JLab and the
proposed Electron-Ion Collider will help to pinpoint the flavor structure and
the momentum dependence of the Sivers parton distribution function describing
this effect. Here we describe a modified version of the $\tt{PYTHIA}$ Monte
Carlo event generator that includes the Sivers effect. Then we use it to
estimate the size of these SSAs, in the kinematics of these experiments, for
both one and two hadron final states of pions and kaons. For this purpose we
utilize the existing Sivers parton distribution function (PDF) parametrization
extracted from HERMES and COMPASS experiments. Using this modified version of
$\tt{PYTHIA}$, we also show that the the leading order approximation commonly
used in such extractions may provide significantly underestimated values of
Sivers PDFs, as in our Monte Carlo simulations the omitted parton showers and
non-DIS processes play an important role in these SSAs, for example in the
COMPASS kinematics.
