When a novel infectious disease emerges, enhanced contact tracing and isolation are implemented to prevent a major epidemic, and indeed, they have been successful for the control of severe acute respiratory syndrome (SARS) and Middle East respiratory syndrome (MERS), which have been greatly reduced without causing a global pandemic. Considering that asymptomatic and pre-symptomatic infections are substantial for the novel coronavirus disease (COVID-19), the feasibility of preventing the major epidemic has been questioned. Using a two-type branching process model, the present study assesses the feasibility of containing COVID-19 by computing the probability of a major epidemic. We show that if there is a substantial number of asymptomatic transmissions, cutting chains of transmission by means of contact tracing and case isolation would be very challenging without additional interventions, and in particular, untraced cases contribute to lowering the feasibility of containment. Even if isolation of symptomatic cases is conducted swiftly after symptom onset, only secondary transmissions after the symptom onset can be prevented.