In nearly all people with human immunodeficiency virus (HIV) infection, immunity cannot either control or eradicate the infection. There are good medicinal treatments, collectively called "ART" (antiretroviral therapy) which control HIV infection by suppressing the virus in the bloodstream. ART is needed for life, and if a person stops taking ART the HIV infection returns in the bloodstream. So, there is good treatment, but no cure. The researchers want to test whether a period of treatment with vedolizumab can be used to control HIV infection in the bloodstream in persons with HIV on ART, after stopping ART.