[hanche]

If the false positive rate is p and the false negative rate is q, and the infection rate is r, then you will have p·(1-r) false positives (as proportion of the tested population) and (1-q)·r true positives. Your hypothesis p>r is not enough to settle which of those two numbers is bigger.

(Edited to fix a silly mistake: The phone rang while I was posting, so I ended up being hasty.)

Edit the 2nd: Even in the simplified case q=0, you can't easily tell.