You can use UTM query string parameters and assign each of them to a different campaign, which will mark the affiliate. On GA you will be able to analyze the traffic according to this specific campaign and see how many converted.
For example, you can do:
domain.com/?utm_campaign=affiliate1/2/3&utm_source={affiliate_domain}&utm_medium=affiliate-link&ref=code1/2/3
(1/2/3 - a number for each affiliate)
In the default settings of GA, once a user arrives with a new campaign code, he is from that moment assigned to the new campaign.
You will have to ask your affiliate to add these parameters to the target URL, or to do a URL rewrite.