  We investigate the first-passage properties of bursty random walks on a
finite one-dimensional interval of length L, in which unit-length steps to the
left occur with probability close to one, while steps of length b to the right
-- "bursts" -- occur with small probability. This stochastic process provides a
crude description of the early stages of virus spread in an organism after
exposure. The interesting regime arises when b is of the order of but less than
1, where the conditional exit time to reach L, corresponding to an infected
state, has a non-monotonic dependence on initial position. Both the exit
probability and the infection time exhibit complex dependences on the initial
condition due to the interplay between the burst length and interval length.
