Cross referencing between columns and rows?

I'm not sure how best to explain this so have attached a spreadsheet that I hope will make things a bit clearer. Basically what I have in column A is a list of names, and along the top in row 1, are dates on a month by month basis from column B to F.

Underneath each month I have input a number.

Now things get tricky, you'll see that Ive entered dates in column's H&I. These dates represent the date the figure 1st appeared for that particular name, and the last date in the range (regardless of whether a number appears or not)

What I need to be able to do, is come up with a formula that "finds" the beginning date for each "name" and then initiates a sum from that date to the end of the range, eg =sum(b2:e2) for the 1st name.

Any ideas?