... are not of Earthly origin. Bonus points if some or all of the cats that can deal with them aren't either.
Both the Hate Plague and the hamsters that spread them...
...are bio-weapons and the unknown party that created them uses the ship crews as guinea pigs.
Bio-weapon angle seems distinctly possible given the reveal that they also interfere with electronics
According to the reveal in the 8th episode, this plague (called the "Totalitarian Disease") and possibly the rodents were indeed artificially created on a laboratory submarine as a weapon. The virus creates a hive mind in his victims via some kind of electromagnetic waves disrupting electric devices nearby and making the victim try to spread it. The submarine sank, but was pushed out on an island by an active volcano. The military sent a party to retrieve the virus (under the guise of marine ecology studies attached to the Sarushima (the attacking ship from the first episode)) and to destroy the ship afterwards, but the scientists were infected by the virus and spread it to the crew of the Sarushima, who spread it to crews of other ships.