This paper investigates the containment control problem of multi-agent systems with semi-Markovian switching topologies. Different from the related works, this paper considers a more general case that the control inputs of leaders are nonzero, the sojourn time distributions of the semi-Markovian switching topologies are mode transition-dependent, and the states of agents are not measurable. Reduced-order observer-based control protocols are proposed, respectively, for two cases that the transition rates are completely known and partially unknown, which ensures that the containment of the multi-agent systems can be achieved in the mean square sense. Finally, simulation examples are given to illustrate the validity of the theoretical results.