Baricitinib for Reducing HIV in the Brain

Inclusion Criteria: 1. HIV infected on continuous ART with plasma HIV RNA \<200 copies/ml for at least 12 months (on at least two previous clinic visits and confirmed at screening). If a viral load is documented from a CLIA-certified laboratory 14 days before screening, then this result can be used in place of the screening lab result. 2. Current CD4+ \> 350 cells/microliter for at least twelve months (on at least two previous clinic visits and confirmed at screening). If a CD4 count is documented from a CLIA-certified laboratory 30 days before screening, then this result can be used in place of the screening lab result. 3. Women of reproductive age will have a negative pregnancy test at study entry and agree to contraception while on the study drug. Women who are at least 50 years of age and who have been amenorrheic for at least 12 months will not be required to agree to contraception to participate. Exclusion Criteria: 1. \< 18 years of age or \> 65 years of age 2. Pregnancy or breastfeeding 3. Significant hematological abnormalities at screening (ANC \< 1000, Hgb\<10, platelet\< 100,000) 4. History of progressive multifocal leukoencephalopathy 5. Untreated latent tuberculosis infection (which will be screened for before entry). If there is a prior positive test, the test does not need to be repeated at screening. 6. Immunosuppressive medications (including corticosteroids) and anticoagulants (aspirin acceptable) within 1 month. A partial list is provided in the SOP for staff, but otherwise, if there is a question it will be adjudicated by the Investigator(s). 7. History of deep venous thrombosis 8. Cardiovascular disease: 1. Coronary artery disease or history of myocardial infarction, no exclusion if greater than 3 months 2. Congestive heart failure with left ventricular ejection fraction ≤40% per American Heart Association guidelines-- no exclusion if greater than 3 months 3. Ever a history of stroke 9. Hematologic malignancies including lymphoma and leukemia which have no evidence of cure or are at least in remission for \> 5 years 10. Major surgery within 8 weeks before screening or will require major surgery during the study 11. Current or recent (\<4 weeks before randomization) clinically serious viral (including COVID-19), a bacterial, fungal, or parasitic infection or any other active or recent infection. History of untreated syphilis infection. If an RPR was negative in the 3 months before screening, then an RPR is not needed at screening 12. Symptomatic herpes simplex at the time of randomization 13. Symptomatic herpes zoster infection within 12 weeks before randomization. 14. History of disseminated/complicated herpes zoster (for example, ophthalmic zoster or CNS involvement). 15. Positive test for hepatitis B virus (HBV) defined as: 1. positive for hepatitis B surface antigen (HBsAg), or 2. positive for hepatitis B core antibody (HBcAb) and positive for hepatitis B virus deoxyribonucleic acid (HBV DNA) 16. Hepatitis C virus (HCV) chronic infection (hepatitis C antibody-positive and HCV ribonucleic acid \[RNA\]-positive), ever. 17. Cirrhosis of the liver from any cause 18. Any of the following specific abnormalities on screening laboratory tests: 1. ALT or AST \>2 x upper limits of normal (ULN) 2. alkaline phosphatase (ALP) ≥2 x ULN 3. total bilirubin ≥1.5 x ULN (except patients on atazanavir, who must have total bilirubin \<2 x ULN) 4. International Normalized Ratio (INR) \> 1.5 5. Absolute Neutrophil Count (ANC) \<1000 cells/mm3, confirmed on repeat testing. 19. Chronic kidney disease with eGFR \<40 mL/min/1.73 m2 (note that the dose of baricitinib will be reduced to 1 mg daily in participants with GFR between 40 and 60). Specifically, the CKD-EPI without race-based equation is used 20. Current dependence on illicit drugs except for marijuana 21. Bleeding disorders such as Von Willebrand's Disease, hemophilia, or other coagulopathies as determined by history. 22. Any evidence of a mass lesion by history that could lead to increased intracranial pressure and evidence of trauma to the lumbar vertebra (see LP exclusion criteria above) by history. - no lumbar trauma or surgery in the last 60 days, but this will be adjudicated by Investigator(s) if need be. 23. Population: The study team will not include any of the following groups: * Adults unable to consent * Individuals who are not yet adults (infants, children, teenagers) * Pregnant women * Prisoners * Cognitively impaired or Individuals with Impaired Decision-Making Capacity * Individuals who are not able to clearly understand English * Community Participation (if applicable)