Great question, and it's a matter of historical, archaeological, and epidemiological importance. Until recently, the prevailing view was that rats were introduced after the Roman Empire, but that view is changing.

Until recently, there were no Roman rats. As disease, ecological change, and their economic implications push their way to the top of the historian's agenda, ancient rodents have triggered controversy and new research, some of it in the pages of this journal. Rats are crucial to epidemics of bubonic plague, a disease that has been linked to the massive demographic changes that ushered medieval Europe into the modern age. Some historians implicate rodent-borne plague in the end of the ancient world. Today archaeology and zoology draw a picture of rats and their history that differs from even a decade ago. Tiny bones and DNA are yielding glimpses of the rat's migration from southeast Asia into the Roman empire and medieval Europe. The diffusion of the rat across Europe looks increasingly like an integral part of the Roman conquest. Its movements illuminate patterns of eco- nomic organization, communications, and urbanism, and carry significant implications for the history of disease and the ecology. The history of rats is tightly interwoven with the economic rise and fall of the ancient world, as well as the expansion of the medieval economy.

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Aside from isolated, and therefore controversial, archaeological evidence for rats in the Roman world [emerged] only during the last two decades. It comes in different gnawing marks on bones; owl or other predator remains preserved in situ. Why it was so scarce before simple: Archaeologists were not looking for it, bones easily escape the troweler's naked eye. Only laborious procedures can detect them.

Rats, Communications, and Plague: Toward an Ecological History

So now folks are looking for rat bones, and they're finding them.

There are some obvious mechanisms for the migration of rats from Asia to the Mediterranean. Roman Egypt traded to the East, and enormous grain fleets traveled from Egypt to Italy, nothing like a boatful of grain to make a nice home for a traveling rat.

How did Romans deal with pests? We don't have a sense of the rat burden at the time of the Republic or the Caesars, and the Romans used a latin word mus -- which doesn't tell us whether they're talking about a rat or a mouse. The Romans had household pets: dogs a great favorite, cats not so common as in Egypt, and even snakes. 20 centuries later, a cat, and some breeds of dog, are efficient remedies for a rodent problem. But the most frequently mentioned Roman household rodent killer is the weasel (mustela); these are often mentioned for killing mus, whether these might be rats or mice we don't know.

Sources:

Greek and Roman Household Pets

The Grain Trade under the Roman Empire

The Domestic Cat as a Factor in Urban Ecology

The Origin of Latin Mustela

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