Praise be to Allaah.
If the matter is as you have described, and the house is old and the mother is exposed to sickness because of staying there, then the son should strive to convince her to move out. If she insists, he cannot force her to move, so long as she is of sound mind and understands what is being said to her, because she has the right to stay and live where she wants.
The son must strive to clean the house as much as he can, and he should supply her with what she needs of food, drink and other services, whether he does that himself or he hires someone to serve her. He must strive to honour her and uphold ties with her and check on her regularly, especially as she is so old, as Allaah says (interpretation of the meaning):
“And your Lord has decreed that you worship none but Him. And that you be dutiful to your parents. If one of them or both of them attain old age in your life, say not to them a word of disrespect, nor shout at them but address them in terms of honour.
24. And lower unto them the wing of submission and humility through mercy, and say: My Lord! Bestow on them Your Mercy as they did bring me up when I was young.”
[al-Isra’ 17:23,24]And Allaah knows best.