Much recent empirical evidence shows that textit{community structure} is ubiquitous in the real-world networks. In this Letter, we propose a growth model to create scale-free networks with the tunable strength (noted by $Q$) of community structure and investigate the influence of community strength upon the collective synchronization induced by SIRS epidemiological process. Global and local synchronizability of the system is studied by means of an order parameter and the relevant finite-size scaling analysis is provided. The numerical results show that, a phase transition occurs at $Q_csimeq0.835$ from global synchronization to desynchronization and the local synchronization is weakened in a range of intermediately large $Q$. Moreover, we study the impact of mean degree $<k>$ upon synchronization on scale-free networks.
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