Bubonic plague spread by fleas.
Plague is endemic to Central Asia, primarily in Marmot populations.
Plague spread along Silk Road, in part after Silk Road was diverted into Marmot territory as a result of Mongol conquests disrupting normal trade routes.
Eventually Fleas spread plague to rats which carried disease easily into human cities and ships.
More cats might have helped.
Even if it didn't, at least they had cats.