Two basic strategies have been proposed for using transgenic Aedes aegypti mosquitoes to decrease dengue virus transmission: population reduction and population replacement. Here we model releases of a strain of Ae. aegypti carrying both a gene causing conditional adult female mortality and a gene blocking virus transmission into a wild population to assess whether such releases could reduce the number of competent vectors. We find this "reduce and replace" strategy can decrease the frequency of competent vectors below 50% two years after releases end. Therefore, this combined approach appears preferable to releasing a strain carrying only a female-killing gene, which is likely to merely result in temporary population suppression. However, the fixation of anti-pathogen genes in the population is unlikely. Genetic drift at small population sizes and the spatially heterogeneous nature of the population recovery after releases end prevent complete replacement of the competent vector population. Furthermore, releasing more individuals can be counter-productive in the face of immigration by wild-type mosquitoes, as greater population reduction amplifies the impact wild-type migrants have on the long-term frequency of the anti-pathogen gene. We expect the results presented here to give pause to expectations for driving an anti-pathogen construct to fixation by relying on releasing individuals carrying this two-gene construct. Nevertheless, in some dengue-endemic environments, a spatially heterogeneous decrease in competent vectors may still facilitate decreasing disease incidence.