  Weak-lensing peak counts has been shown to be a powerful tool for cosmology.
It provides non-Gaussian information of large scale structures, complementary
to second order statistics. We propose a new flexible method to predict weak
lensing peak counts, which can be adapted to realistic scenarios, such as a
real source distribution, intrinsic galaxy alignment, mask effects, photo-$z$
errors from surveys, etc. The new model is also suitable for applying the
tomography technique and non-linear filters. A probabilistic approach to model
peak counts is presented. First, we sample halos from a mass function. Second,
we assign them NFW profiles. Third, we place those halos randomly on the field
of view. The creation of these "fast simulations" requires much less computing
time compared to $N$-body runs. Then, we perform ray-tracing through these fast
simulation boxes and select peaks from weak-lensing maps to predict peak number
counts. The computation is achieved by our \textsc{Camelus} algorithm, which we
make available at http://www.cosmostat.org/software/camelus/ . We compare our
results to $N$-body simulations to validate our model. We find that our
approach is in good agreement with full $N$-body runs. We show that the lensing
signal dominates shape noise and Poisson noise for peaks with SNR between 4 and
6. Also, counts from the same SNR range are sensitive to $\Omega_\mathrm{m}$
and $\sigma_8$. We show how our model can discriminate between various
combinations of those two parameters. In summary, we offer a powerful tool to
study weak lensing peaks. The potential of our forward model is its high
flexibility, making the use of peak counts under realistic survey conditions
feasible.
