Parametric modeling of univariate cumulative incidence functions and logistic models have been studied extensively. However, to the best of our knowledge, there is no study using logistic models to characterize cumulative incidence functions. In this paper, we propose a novel parametric model which is an extension of a widely-used four-parameter logistic function for dose-response curves. The modified model can accommodate various shapes of cumulative incidence functions and be easily implemented using standard statistical software. The simulation studies demonstrate that the proposed model is as efficient as or more efficient than its nonparametric counterpart when it is correctly specified, and outperforms the existing Gompertz model when the underlying cumulative incidence function is sigmoidal. The practical utility of the modified three-parameter logistic model is illustrated using the data from the Cache County Study of dementia.