In the sequential change-point detection problem for multi-stream data, it is assumed that there are M processes in a system and at some unknown time, an occurring event impacts one unknown local process in the sense of changing the distribution of observations from that affected local process. In this paper, we consider such problem under the sampling control constraint, in which one is able to take observations from only one of the local processes at each time step. Our objective is to design an adaptive sampling policy and a stopping time policy that is able to raise a correct alarm as quickly as possible subject to the false alarm and sampling control constraint. We develop an efficient sequential change-point detection algorithm under the sampling control that turns out to be second-order asymptotically optimal under the full data scenario. That is, with the sampling rate that is only 1/M of the full data scenario, our proposed algorithm has the same performance up to second-order as the optimal procedure under the full data scenario.