Hi,
First rule of war, know thy enemy. Make certain that you are dealing with fleas. Get some on sticky tape/Sellotape and grab a magnifying glass. Google images of Ctenocephalides, the flea group. Wikipedia has good images. If they match, then consider the following.
Most fleas humans encounter are Ctenocephalides canis, the Dog Flea, which is quite happy living on a wide variety of hosts. Humans rarely sustain the population, but are frequently affected by populations created on other animals. In most circumstances a cat or dog is involved. If you have pets, they are now part of the problem and need a flea treatment product applied for the next 6 months. The host animal can left up to 12m prior to the problem occurring. Adult fleas which do the biting are obligate parasites, meaning they don't survive long off their host. Their eggs and larvae tho, which they lay several thousand of each, lay dormant for up to 12 months, then hatch when there is vibration. So you move in and fleas magically appear!
Work out where your host lived, or still lives. A single rat could not create that burden of fleas. Your host lived in or under the house, like pet dogs or a wild cat. It built a huge flea egg population in the carpet/floor board cracks/dry soil which are now hatching progressively and will do for months. Residual insecticidal sprays or those containing insect growth regulators will deal with this burden. Vacuuming the living space is also helpful. It will take 1-2 months to clear them.
And if you didn't bring animals to the house, there's no way those fleas are not your problem to solve.
