This paper offers a new method for calculating the reaction rate based on the pore collapse "hot-spot" ignition mechanism in multi-component polymer bonded explosives (PBXs), and proposes a mesoscopic reaction rate model that allows us to predict the shock initiation and detonation behaviors of multi-component PBXs with any explosive component proportion and explosive particle size. The pressure-time histories in PBXC03 (87% HMX, 7% TATB, and 6% Viton by weight) and PBXC10 (25% HMX, 70% TATB, and 5% Kel-F800 by weight) are calculated by using the new mesoscopic reaction rate model, and the numerical results are found to be in good agreement with the experimental data. It is found that the shock initiation and detonation behaviors of PBXC03 with the dominant component of HMX is mainly controlled by the hot-spot ignition. The subsequent combustion reacts fast once the hot-spot is ignited and shows an accelerated reaction characteristic. While, with the dominant component of insensitive TATB, the critical initiation pressure of PBXC10 is high enough to make almost saturated reactive hot spots just behind the precursory shock-wave front, and the shock initiation behavior is basically determined by the combustion reaction, which is characterized by a stable reaction due to the slow combustion-wave velocity.