Thinking back about the initial Q posts before Q became known as Q, the Q signature could be a reference to the rare disease Q fever.
Q fever is highly contagious from spores and has been weaponized for biological warfare.
https://en.wikipedia.org/wiki/Q_fever
Q fever is associated with infertility and breeding problems in cattle, which presumably would also extend to humans if women were the ones to get infected more often. Q fever would make possible a large global population reduction without collapsing the economy too badly if sprayed from planes worldwide.

Once the Rapture happens, the door of Heaven will be closed to everyone. I think this means that children will stop being born because the flow of new souls from God will stop. There is ample Biblical precedent for God using infertility as a punishment and not letting children to be born after the Rapture would protect them from the great evil that would otherwise target them.