Infectious diseases have been recognized as major public health concerns for decades. Close contact discovery is playing an indispensable role in preventing epidemic transmission. In this light, we study the continuous exposure search problem: Given a collection of moving objects and a collection of moving queries, we continuously discover all objects that have been directly and indirectly exposed to at least one query over a period of time. Our problem targets a variety of applications, including but not limited to disease control, epidemic pre-warning, information spreading, and co-movement mining. To answer this problem, we develop an exact group processing algorithm with optimization strategies. Further, we propose an approximate algorithm that substantially improves the efficiency without false dismissal. Extensive experiments offer insight into effectiveness and efficiency of our proposed algorithms.