The role of chain-branching cross-over temperatures in shock-induced ignition of reactive materials is studied by numerical simulation, using a three-step chain-branching reaction model. In order to provide insight into shock initiation, the simpler problem of a spatially homogeneous explosion is first considered. It is shown that for ratios of the cross-over temperature to the initial temperature, $T_B$, sufficiently less than unity, the homogeneous explosion can be quantitatively described by a widely used two-step model, while for $T_B$ sufficiently above unity the homogeneous explosion can be effectively described by the standard one-step model. From the matchings between these homogeneous-explosion solutions, the parameters of the reduced models are identified in terms of those of the three-step model. When $T_B$ is close to unity, all the reactions of the three-step model have a leading role, and hence in this case the model cannot be reduced further. In the case of shock initiation, for $T_B$ (which is now the ratio of the cross-over temperature to the initial shock temperature) sufficiently below unity, the three-step solutions are qualitatively described by those of the matched two-step model, but there are quantitative differences due to the assumption in the reduced model that a purely chain-branching explosion occurs instantaneously. For $T_B$ sufficiently above unity, the matched one-step model is found to effectively describe the way in which the heat release and fluid dynamics couple. For $T_B$ close to unity, the competition between chain branching and chain termination is important from the outset. In these cases the speed at which the forward moving explosion wave that emerges from the piston is sensitive to $T_B$, and changes from supersonic to subsonic for a value of $T_B$ just below unity.