One way is to explain that your loved one needs specialized care that you cannot provide at home and that it is critical that he/she be somewhere that they can get that care. If a doctor or nurse can reinforce that message, even better. Good luck.
We used to be able to just tell him that he needed extra care. He has been at long term care for 1 yr now. Now he gets upset and says he is fine and that there is nothing wrong with him. He keeps saying he wants to divorce my Mom because she won't take him home with him which obviously really upsets my Mom.