Yes, inmates in the prison infirmary generally retain the ability to make calls and receive mail. Being housed in the medical unit does not automatically strip those privileges the way a disciplinary placement in the SHU would.
That said, the practical reality depends on how sick or incapacitated the inmate is and how the specific facility manages its medical unit. Someone who is ambulatory and stable is in a very different situation than someone recovering from surgery or dealing with a serious acute condition. If they are well enough to get to a phone or request writing materials, access is typically available.
Some infirmaries have phones available within the unit itself. Others require inmates to be well enough to access the general population phone area, which may not be realistic depending on their condition. Mail continues to be delivered through the facility's normal mail process regardless of where someone is housed, so letters and postcards should still reach them as long as they are addressed correctly with the standard facility address.
If your loved one has been moved to the infirmary and communication has slowed or stopped, it may simply be a matter of their physical condition rather than any restriction on their privileges. The best move is to contact the facility directly, ask to speak with someone in the medical unit or the inmate's case manager, and get an update on their status. Facilities are limited in what medical information they can share, but they can generally confirm whether someone is able to receive calls and correspondence.
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