The sailors aboard these 2 ships cleaned the decks and likely ingested contaminated water and food. US Military officials say they made no attempt to move the ships out of the plume immediately or limit exposure because their nuclear detection equipment was inoperative.
However, this excuse seems very improbable considering how the USS Ronald Reagan is a nuclear powered aircraft carrier. All nuclear reactors have nuclear physics personnel who have callibrated equipment to measure any exposure and contamination of the ship and crew.
We know the USS Ronald Reagan and Essex were contaminated because several ports refused to allow these ships to enter their port. The personnel were forced to hang out for several months our at sea. On the bright side, of the 4000-member-crew, only 70 are sick so far. Those with strong immune systems resisted the exposure.