HIV curative strategies currently under development aim to eradicate latent provirus, or prevent viral replication, progression to AIDS, and transmission. The impact of implementing curative programs on HIV epidemics has not been considered. We developed a mathematical model of heterosexual HIV transmission to evaluate the independent and synergistic impact of ART, HIV prevention interventions and cure on HIV prevalence and incidence. The basic reproduction number was calculated to study the potential for the epidemic to be eliminated. We explored scenarios with and without the assumption that patients enrolled into HIV cure programs need to be on antiretroviral treatment (ART). In our simulations, curative regimes had limited impact on HIV incidence if only ART patients were eligible for cure. Cure implementation had a significant impact on HIV incidence if ART-untreated patients were enrolled directly into cure programs. Concurrent HIV prevention programs moderately decreased the percent of ART treated or cured patients needed to achieve elimination. We project that widespread implementation of HIV cure would decrease HIV prevalence under all scenarios but would only lower rate of new infections if ART-untreated patients were targeted. Current efforts to identify untreated HIV patients will gain even further relevance upon availability of an HIV cure.