Annoy at/by/with

What I know is that "annoy with person" and "annoy at something". But when I see sentences using all three "annoy at/by/with", I got confused.

There are some examples:

1. I am being annoyed at/by/with her.

2. I was annoyed with/at/by him for being so late.

3. He was annoyed with/at children.

4. The traveler and farmer are at times annoyed by the mosquito.

5. Obviously he was annoyed by her caustic tone.

Can you please explain what is difference between their uses?