Our expert says:
Undoubedly, you are describing a terrible situation you are having to put up with, enormously distressing and frustrating. I'm sure you realize, especially when feeling less awfully angry, that while it is indeed important that you should find a way to free yourself from experiencing this area of life as living as a victim, that taking steps towards harming your brother would not only be wrong, but would increase your amount of suffering, as well. However angry thios situation makes you, do NOT act in any way so as to harm your brother --- that will be seriously harmful to yourself and your gran as well.
A major part of your problem is that too many other people are not meeting their responsibilities. Your gran sounds like she's doing the best she can, but your mom seems to be avoiding the real issues, with as much Denial of the reality of the situation as she can manage.
But the people who sound as if they're most thoroughly failing in their duty, are the doctors and nurses at the clinic or hospital where your brother is under treatment. They are supposed to be supervising his treatment, and making absolutely sure that he receives his injections and takes his medication, and to be repeatedly assessing his state of mind, as they surely know that he can be very disordered in his thoughts and irresponsible in his actions. They should not let someone like this slip between the cracks without noticing that he is not getting them medicines he should ; and they should have access to community nurses who can and ought to do home-visits to see why he isn't coming in, and to be sure what's happening.
You really need to get into contact with the doctor in charge of his case, and maybe the head nurse at the clinic or ward where he is being treated, and make it entirely clear that he is not getting his bedication, is being dangerously threatening and indicating possible arson, rape and assault towards you and other members of your family, and needs to be sectioned and re-admitted to the hospital, and that you will make sure that that doctor and nurse and their colleagues will be held directly responsible for any further damage he causes due to their lack of proper levels of care.
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