We present and discuss the design details of an extensible, modular, open-source software framework called EXOSIMS (Exoplanet Open-Source Imaging Mission Simulator), which creates end-to-end simulations of space-based exoplanet imaging missions. We motivate the development and baseline implementation of the component parts of this software with models of the wide-field infrared survey telescope-astrophysics focused telescope assets (WFIRST-AFTA) coronagraph and present initial results of mission simulations for various iterations of the WFIRST-AFTA coronagraph design. We present and discuss two sets of simulations. The first compares the science yield of completely different instruments in the form of early competing coronagraph designs for WFIRST-AFTA. The second set of simulations evaluates the effects of different operating assumptions, specifically the assumed postprocessing capabilities and telescope vibration levels. We discuss how these results can guide further instrument development and the expected evolution of science yields.