The first two children die in infancy because of improper medical treatment. The other children are used by Rufus like "a bit in her mouth" to show Alice he has complete control over her. The children were pawns in the game Rufus played with Alice, and his most powerful threat was to sell them to someone else. Alice resigned herself to give in to Rufus as long as he held this power. Alice hopes that her children will be free one day, and she names them Joseph and Hagar to express this hope.
When Rufus pretends that he's sold Alice's children, Alice loses all hope and has no reason to live any longer. She committs suicide, taking away the power that Rufus held over her. She never knew that Rufus had only sent the children away.