The Streptobacillosis form of rat-bite fever is known by the alternative names Haverhill Fever and epidemic arthritic erythema. It is a severe disease caused by Streptobacillus moniliformis, transmitted either by rat bite or ingestion of contaminated products. After an incubation period of 2–10 days, Haverhill fever begins with high prostrating fevers, rigors (shivering), headache and polyarthralgia (joint pain). Soon an exanthem (widespread rash) appears, either maculopapular (flat red with bumps) or petechial (red or purple spots) and arthritis of large joints can be seen. The organism can be cultivated in blood or articular fluid. The disease can be fatal if untreated in 20% of cases due to malignant endocarditis, meningoencephalitis or septic shock.
Thats basically the disease as a whole; however instead of waiting the two days for it to kick in, the Genjutsu emulates instant effects after the bites are made. Red/purple bumps on their skin is essentially the rash, but you can add more detail into since it creates a false diseases so you can accurately implement it.
And yes you can create any amount, once they bite them the disease kicks in and the start experience it, along with the rat bites. You can add the smoke to make it look like a summoning instead of just randomly creating rats out of thin air