Mosquito-borne diseases such as malaria and dengue fever continue to be a major health problem through much of the world. Several new potential approaches to disease control utilize gene drive to spread anti-pathogen genes into the mosquito population. Prior to a release, these projects will require trials in outdoor cages from which transgenic mosquitoes may escape, albeit in small numbers. Most genes introduced in small numbers are very likely to be lost from the environment; however, gene drive mechanisms enhance the invasiveness of introduced genes. Consequently, introduced transgenes may be more likely to persist than ordinary genes following an accidental release. Here, we develop stochastic models to analyze the loss probabilities for several gene drive mechanisms, including homing endonuclease genes, transposable elements, Medea elements, the intracellular bacterium Wolbachia, engineered underdominance genes, and meiotic drive. We find that Medea and Wolbachia present the best compromise between invasiveness and containment for the six gene drive systems currently being considered for the control of mosquito-borne disease.