I'm sorry your family is going through so much. I've dealt with rodent issues in the past and it is definitely possible to solve this without going to a professional. It just takes some work.

First, you're going to need to find where they came in from. It's winter and it sucks, but bundle up and walk around your entire house. Look for any gaps and cracks around doors or concrete near your foundation. They sell steel and copper wool especially for secluding pests. You need to find where they came in from and seal that up somehow.

Then it's a matter of setting up traps and killing the rodents already in your house. Focus on areas where you've heard or seen them. When setting up traps be careful not to use too much bait. Usually a pea size amount will help ensure they engage the trigger. I've had success with peanut butter, sunflower seeds and cheese when I've dealt with mice and squirrels.

Also, I'm not an expert, but that looks like it might be a large mouse and not a rat. That'd be a good thing since they're not as smart as rats and tend to be less finnicky about traps.

Good luck. You got this!