  We formulate and solve a quickest detection problem with false negatives. A
standard Brownian motion acquires a drift at an independent exponential random
time which is not directly observable. Based on the observation in continuous
time of the sample path of the process, an optimiser must detect the drift as
quickly as possible after it has appeared. The optimiser can inspect the system
multiple times upon payment of a fixed cost per inspection. If a test is
performed on the system before the drift has appeared then, naturally, the test
will return a negative outcome. However, if a test is performed after the drift
has appeared, then the test may fail to detect it and return a false negative
with probability $\epsilon\in(0,1)$. The optimisation ends when the drift is
eventually detected. The problem is formulated mathematically as an optimal
multiple stopping problem and it is shown to be equivalent to a recursive
optimal stopping problem. Exploiting such connection and free boundary methods
we find explicit formulae for the expected cost and the optimal strategy. We
also show that when $\epsilon = 0$ our expected cost coincides with the one in
Shiryaev's classical optimal detection problem.
