<p>Estimating the number of nodes that can be influenced by an initial set of active seeds is one of the fundamental problems in social networks. In this paper, we consider individual interactions in groups of three or more vertices and study the threshold model on directed hypergraphs. We propose a two-stage branching process to approximate the propagation dynamics. This yields a nonlinear one-dimensional recursion that approximately describes the evolution of the dynamics of the threshold model on most directed hypergraphs. We demonstrate that the active fraction is arbitrarily close to the output of the above recursion on directed hypergraphs, except for a small fraction of directed hypergraphs that vanishes as the graph size grows large. Through numerical experiments conducted on synthetic and empirical networks, we verify that the simulation results show good adherence to the theoretical predictions.</p>