what do you call the kind of life that the person is still alive but doesn't react? I don't mean comma or brain death?
I am not a teacher but this might be your answer.
PARALYTIC
Coma, not comma.
I agree that it would be helpful to know what you mean!
My suggestion is "persistent vegetative state". As far as I'm aware, that is (in medical terms) different from a coma.
When someone is brain dead, then it's generally time to pull the plug or stop trying to resuscitate etc. You wouldn't make the same decision (necessarily) for someone in a coma or a PVS.
With regard to an earlier answer, "paralytic" would certainly not work. I can only assume the poster meant "paralysed" but that is completely different. "Paralytic" in BrE means that you are so drunk that you can't function.
It also depends on how the OP means "doesn't react".
Doesn't react voluntarily? - in a coma, unconcious. WIll still react to pain, by withdrawal of limb, etc.
Doesn't react via deep or superficial reflexes? Deeper coma. Patients in a semi-vegetative state can chew and swallow by the stimulus of food put in their mouths.
Many patients in vegetative states will react to various primitive reflexes - eg. constriction of pupils to light.
Even in brain dead patients, the body can react to infection by inflammation.
So there's a wide range of how a body can react, depending on which systems are still functional.