  We constrain cosmological parameters from a joint cosmic shear analysis of
peak-counts and the two-point shear correlation functions, as measured from the
Dark Energy Survey (DES-Y1). We find the structure growth parameter $S_8\equiv
\sigma_8\sqrt{\Omega_{\rm m}/0.3} = 0.766^{+0.033}_{-0.038}$, which at 4.8%
precision, provides one of the tightest constraints on $S_8$ from the DES-Y1
weak lensing data. In our simulation-based method we determine the expected
DES-Y1 peak-count signal for a range of cosmologies sampled in four $w$CDM
parameters ($\Omega_{\rm m}$, $\sigma_8$, $h$, $w_0$). We also determine the
joint covariance matrix with over 1000 realisations at our fiducial cosmology.
With mock DES-Y1 data we calibrate the impact of photometric redshift and shear
calibration uncertainty on the peak-count, marginalising over these
uncertainties in our cosmological analysis. Using dedicated training samples we
show that our measurements are unaffected by mass resolution limits in the
simulation, and that our constraints are robust against uncertainty in the
effect of baryon feedback. Accurate modelling for the impact of intrinsic
alignments on the tomographic peak-count remains a challenge, currently
limiting our exploitation of cross-correlated peak counts between high and low
redshift bins. We demonstrate that once calibrated, a fully tomographic joint
peak-count and correlation functions analysis has the potential to reach a 3%
precision on $S_8$ for DES-Y1. Our methodology can be adopted to model any
statistic that is sensitive to the non-Gaussian information encoded in the
shear field. In order to accelerate the development of these beyond-two-point
cosmic shear studies, our simulations are made available to the community upon
request.
