Lagging in R when a condition is matched

I have a dataframe with just dates of medical checkups and presence of infection (yes/no), and I want to add a third column representing the date of the last infection. The new last_infection column should have NA if the patient has had no previous infection. If they have had previous infection, it should show the date of the most recent visit where they tested "yes" for infection.

I want the output to look like this:

date      infection   last_infection
01-01-18  no          NA
06-01-18  no          NA
07-01-18  yes         NA
09-01-18  no          07-01-18
01-01-19  no          07-01-18
02-01-19  yes         07-01-18
03-01-19  yes         02-01-19
04-01-19  no          03-01-19
05-01-19  no          03-01-19

How can I do this in R? Can a function like lag() check for a condition, or should I do something else entirely?