Summary: So far, in some standard rumor spreading models, the transition probability from ignorants to spreaders is always treated as a constant. However, from a practical perspective, the case that individual whether or not be infected by the neighbor spreader greatly depends on the trustiness of ties between them. In order to solve this problem, we introduce a stochastic epidemic model of the rumor diffusion, in which the infectious probability is defined as a function of the strength of ties. Moreover, we investigate numerically the behavior of the model on a real scale-free social site with the exponent \(\gamma = 2.2\). We verify that the strength of ties plays a critical role in the rumor diffusion process. Specially, selecting weak ties preferentially cannot make rumor spread faster and wider, but the efficiency of diffusion will be greatly affected after removing them. Another significant finding is that the maximum number of spreaders max \((S)\) is very sensitive to the immune probability \(\mu\) and the decay probability \(v\). We show that a smaller \(\mu\) or \(v\) leads to a larger spreading of the rumor, and their relationships can be described as the function ln(max \((S)) = Av + B\), in which the intercept \(B\) and the slope \(A\) can be fitted perfectly as power-law functions of \(\mu \). Our findings may offer some useful insights, helping guide the application in practice and reduce the damage brought by the rumor.