Hi Lisa
First of all welcome to the forum and I really feel for your situation (you and your brother).
First of all, the fear of him never getting better is understandable but I know he will get better, I have been there in the past and got myself through it with the help of friends, family and professional help. If I could put myself in the mindset of your brother (which is exactly where I was) he feels that home is his safety blanket and his safe haven. This is probably the only place he feels in control and away from the outside world - as you will see for yourself anxiety is a terrible condition and only those who have experienced it or seen someone close to them suffering will actually understand. We are all different but I will list some of the actions which I took to overcome my anxiety and leave it in my past:-
Is your brother sleeping? I struggled to sleep when suffering from anxiety and this was one of the first things I addressed. I spoke with my doctor who gave me a short term stock of pills to take the edge off my anxiety which slowly improved my sleeping pattern. A lack of sleep is one of the main components of any mental health condition - unless addressed at an early stage it can make things seem many many times worse.
Speaking about your concerns is also one of the first steps - does you brother open up to you? Maybe he would open up to his doctor or a Cognitive Behavioural Therapist (CBT)?
Taking the focus away from his anxiety - one thing which helped me immensely was having something else to focus on, whether this is a friend in trouble, something I could do to help a friend or family member, exercise and I have to say my little dog was a godsend to spend time with. Every minute that I was not focused upon my anxiety was a step in the direction and very quickly the minutes grew in number and then suddenly I was not focused upon my anxiety 24/7 and was then able to be rational about what I was worried about - nothing.
One of the first questions my therapist asked me was "So what are you worried about?" I told him I was worried about going to the shops where there are many people and he asked why. I stopped and thought, mmmm what am I worried about? I said I was concerned about talking to people and he said all you have to do is say "Hi" or even smile and nod your head when passing. This was a major step for me, what was I really worried about? Breaking the cycle of worry is vital.
I will add more comments about my experiences as we go along (I hope they help you) but I don't want to overload you with comments in one go.