Push-belt (or Van Doorne-type) CVT systems are used for power transmission in automotive applications, including notably in engine-transmission subsystems. In order to characterize the physics of a Van Doorne CVT, two modeling options are commonly used. High fidelity models track each push-belt block as well as the dynamics of the bands that connect the blocks. The main disadvantage of this technique lies in its large number of degrees of freedom and resulting long CPU time. A second approach relies on a lesser-fidelity model with few degrees of freedom that can subsequently be used in long simulations, e.g. vehicle drive-cycles. In this work, we review different modeling techniques at this modeling level, and propose a fast-running model that overcomes some of the limitations of lesser-fidelity models yet is still suitable for long simulations. Typical fast-running models enforce kinematic constraints between the pulleys, i.e. the CVT bands and blocks are assumed to be rigid. In order to overcome some of the limitations associated with a rigid CVT, a fast-running flexible variant is proposed. The model has been implemented within a general-purpose tool in which a complete vehicle system can be modeled. Several examples are analyzed to validate the proposed model. First, a validation example and also comparison of the rigid and elastic models are presented. Next, numerical and experimental results are compared for vehicle transients.