This study is exploring how inflammation affects depression in people with HIV (Human Immunodeficiency Virus). HIV is a virus that attacks the immune system. The study will last for 10 weeks and involves 60 participants. They will be randomly given either the anti-inflammatory drug baricitinib or a placebo (a pill that looks like medicine but isn't real) to see if it helps with symptoms like anhedonia (difficulty feeling pleasure) and psychomotor slowing (slow movements or thoughts). Participants will have tests like blood work, brain scans, and may have an optional spinal tap. They will visit the study site for different assessments over about 5 months.

• Participation involves several medical tests and assessments.
• The study period is approximately 5 months long.
• Participants won't know if they are taking the actual drug or a placebo.

To join, participants must have controlled HIV, be between 18 and 65 years old, and have depression. They cannot take certain medications or have certain health conditions. This study helps understand depression in people with HIV and could lead to better treatments.