Suppose person 1 is sick on day 1. We can show that after recovering they won't fall sick again.
For person 1 to be sick again, they need to be not immune when any of their friends are sick. But all of their friends fall sick the very next day while person 1 is immune.
Using the same argument, we can show that the friends of person 1 won't fall sick again after recovering.
As every person falls sick atmost once so the process must be finitely long.
Considering a graph where friendships are undirected edges, then:
On day 1 only person 1 falls sick.
On day 2 all the people at a shortest distance of 1 from person 1 fall sick.
On day n all the people at a shortest distance of n-1 from person 1 fall sick.