HELP!!! Blowflies everywhere!?

  I recently noticed a lot of flies in my apartment. I keep killing them, but it seems they just appear out of nowhere. I inspected my whole apartment for anything that they might be feeding o, n but there is nothing. I know they are not coming in from the outside because the doors and windows are all closed. They are mainly the green Blowflies though some of them are the common house flies. I've checked all the drains and nothing. The trash gets taken out everyday and the apartment is very clean. I do have 2 cats but i keep up the litter cleaning. I check anywhere where they might be having babies, but i can't find anything. It is very frustrating because there are so many. Yesterday I killed like 20 and didn't see any more, but when I woke up there were another 15 or so on the windows. I am sure i would notice the scent of a dead animal if it was in the wall or something. So please, if anyone has some information of where they could be coming from or have you ever experienced anything like this, it would be helpful. Thank you.
That is actually really scary, it sounds like something from a horror movie!: (
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Poor you. I think you may want a pest inspector in there. That or a priest!