Much recent empirical evidence shows that community structure is ubiquitous in the real-world networks. In this paper 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 Susceptive-Infected-Recovery-Susceptive (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 Qc approximately or equal to 0.835 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.