My apartment is infested with rats. What is the likelihood that they are carriers of the plague?

Plague is rare. However , the the presence of vector and intermediate host is fertile ground for outbreak of communicable illness like plague. Likelihood is very low but no one can say impossible .
Maybe but not likely. Large numbers of rats in an apartment complex in the U.S. are not likely to carry the plague. However, the situation is "plague friendly", so things could take a quick turn for the worse if an infected animal joined the rats. The incubation period is only a few days, so dead rats may appear in unusual numbers. Cats, dogs, and their fleas can transmit the plague from the dead rats to humans.