The higher-order interactions emerging in the network topology affect the effectiveness of digital contact tracing (DCT). In this paper, we propose a mathematical model in which we use the hypergraph to describe the gathering events. In our model, the role of DCT is modeled as individuals carrying the app. When the individuals in the hyperedge all carry the app, epidemics cannot spread through this hyperedge. We develop a generalized percolation theory to investigate the epidemic outbreak size and threshold. We find that DCT can effectively suppress the epidemic spreading, i.e., decreasing the outbreak size and enlarging the threshold. DCT limits the spread of the epidemic to larger cardinality of hyperedges. On real-world networks, the inhibitory effect of DCT on the spread of epidemics is evident when the spread of epidemics is small.