The Human Immunodeficiency Virus (HIV) infects helper-T cells, and takes advantage of the naturally occurring quiescent phenotype of T cells to persist even under effective treatment conditions. If an infected cell does not produce virus and enters this quiescent state, it forms a natural reservoir that is not targeted by either the existing antiretroviral drugs or the immune system. These quiescent cells intermittently switch to an activated phenotype and begin to produce virus, and are the primary source of viral rebound following treatment cessation. Recent experimental results have shown that, despite this reservoir having a years-long half-life under treatment, most of the cells in the reservoir were infected in a few weeks prior to the start of treatment. This can only be explained by assuming that this reservoir has a short half-life off treatment and a very long half-life on treatment. In this paper, we introduce a novel model of reservoir formation and turnover explaining this difference as a result of antigen-dependent activation. We introduce a second control input through infusion of HIV antigen, mimicking the non-infection pseudovirus (PV) produced by protease inhibitor therapy. This model is coupled to an existing model of immune response to HIV. We fit the parameters of this model to the existing clinical observations of latency. We show that the use of antigen infusion therapy can result in order-of-magnitude decrease in the size of the quiescent reservoir, and that this may provide a way to rapidly stabilize a post-treatment control state in treated HIV infected individuals.