Several studies have suggested that despite suppressive combination antiretroviral therapy (ART) in people who are HIV-positive, in some individuals there may be on-going viral replication. Clarifying the extent of on-going viral replication on ART is important for the development of HIV-1 curative strategies and for reducing HIV-1 associated immune activation. The investigators hypothesize that treatment intensification with dolutegravir will inhibit residual virus replication in HIV-1 infected patients on ART. The primary objective of this study is to determine the effects of dolutegravir intensification on residual virus replication in circulating cluster of differentiation 4 (CD4+) T cells.