More generally, how would slave masters deal with flu season and other outbreaks?
essentially you seem to inadvertently be asking for the historiography of the last 30 or so years in regards to slavery and the sucessors to the Dunning School of thought. So your question may be more complex than you realize since it's going to involve questions about how slaves manipulated the system by faking sickness of being lazy (slavery is perhaps the best example of the principal-agent problem) and how slave owners were able to coerce slaves to do their bidding.
also dealing with the deep south many plantation masters in south carolina weren't at the plantation during flue season as they went into Charleston or some other healthier place for that part of the year and it was left to either black or white overseer (though this might not be the right term) to deal with the nuts and bolts of this relationship then.
not really an answer as much an attempt to add a bit of perspective.