We consider a class of endomorphisms which contains a set of piecewise partially hyperbolic dynamics semi-conjugated to non-uniformly expanding maps. The aimed transformation preserves a foliation which is almost everywhere uniformly contracted with possible discontinuity sets, which are parallel to the contracting direction. We apply the spectral gap property and the $zeta$-Holder regularity of the disintegration of its physical measure to prove a quantitative statistical stability statement. More precisely, under deterministic perturbations of the system of size $delta$, we show that the physical measure varies continuously with respect to a strong $L^infty$-like norm. Moreover, we prove that for certain interesting classes of perturbations its modulus of continuity is $O(delta^zeta log delta)$.
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