HIV virus replication?

An interesting question. Well, the virus does only live for 1.5 days, but its DNA is still transcribed in the white blood cells in blood. Unfortunately, these will still be alive when the blood is administered during the transfusion (otherwise, whats the point of giving someone blood with dead cells?). When the blood is transfused, the white blood cells begin creating more viruses (whether or not the virus has survived the time). The virus does not need to be necessarily alive for this phenomena, its DNA has already been injected into the host cell.