This is a great thesis to look at in this excellent story. Firstly, you might want to start by comparing the character of the mother with that of the father. She clearly loses all hope and as a result commits suicide, leaving her husband and son to face a desperately uncertain future by themselves. However, it is the father and the son who express hope and the desperate desire to survive, whatever the danger, and to believe in a better future somehow. To me, however, one of the most moving parts of this novel that deals with hope and its importance comes at the end as the father is dying and encouraging his son to live on:
I want to be with you.
You can't. You have to carry the fire.
I don't know how to.
Yes you do.
Is it real? The fire?
Yes it is.
Where is it? I don't know where it is.
Yes you do. It's inside you. It was always there. I can see it.
"The fire" is of course a symbol of the hope that the father has carried within him, no matter how bleak or desperate things have looked, and now has passed on to his son for him to keep on burning. It is this "fire," this hope, that has proved to be absolutely essential for survival, and this same "fire" that the boy's mother so clearly lost.