Vent about a Cat

URGHHH there is a stray cat that is hanging around outside. The kids next door are feeding and watering it and have kinda become attached. I know that talking to the mother of the kids, she doesn't particularly care for the cat hanging around and I know the landlord doesn't because when I mentioned it to him, he said that he has told them before not to feed stray cats. DS is allergic to cats and I am servrely allergic to them to the point where I had a near ana rection from just walking into the room at the pet store where they are. I felt my thoart closing up and I started gagging. Took some benadryl but still got a horrible headache from it. Anyway my allergies and DS allergies have all of a sudden gone HIGHWIRE. I am miserable and so is he. I've also had a headache everyday for over a week (one of my reactions). He's also not sleeping and is cranky. I know cleaning out my dad's old dusty house isn't helping which is why I'm not spending much time over there. My friend mentioned yesturday that it could be this cat that is hanging around that is causing the all of a sudden increase in our reactions. I didn't really think about it until yesterday but she has a point. As we are probably tracking in dander on out shoes through out the house. UGHHHHHHHHHHHHHHHHH So now I have to figure out a way to get rid of this cat without breaking these kids hearts. URGHHHHHHHHHHHHHHHHHHHHHH On top of that I'm miserable and so is DS. Anyone know how to get rid of a cat without actually touching the cat? LOL I can't touch it or I will have a major reaction.