The opportunistic network is based on the idea of "Store-Carry-Forward" between nodes which haven't complete communication links. It uses the encounter opportunity presented by node mobility to establish a route to realize communication. The traditional Epidemic algorithm uses the flooding mechanism which greatly increases the energy consumption of nodes, and it is often unable to supplement the energy of nodes at any time in the application environment of the opportunistic network. Therefore, this paper proposes an asynchronous sleep mechanism, which allows nodes to judge whether to enter the sleep state according to their own energy situation, real-time state and other information. It ensures that the nodes do not miss the possible communication opportunities by dynamically controlling the sleep time and wake-up time. The simulation results show that the proposed EASE algorithm can guarantee the network performance and greatly reduce the energy consumption of nodes compared with the classical opportunistic network routing algorithms.