Interesting

It would perhaps be helpfull if we (we not the viewers) have an idea how it spreads, by contact or by beeing in the area, how long you can carry it without problems.
I think that the easyest way is to let it speard by thouching the blood of an infected person, but you can carry it for a while before showing the symptones (explains why they don't think that it speads by blood). Further on I think that it shouldn't be a virus but a new kind of germ, that is completely invisible and not to track. It would create an unsolvable mystery but at least we know what we are talking about.
This whole theory fits in properly since the neighbor was infected and the girls of the family would be infected bacause of the blood on the knife and the other spilled blood that they touched.