  We study large deviation events in the timing of disease recurrence. In
particular, we are interested in modeling cancer treatment failure due to
mutation-induced drug resistance. We first present a two-type branching process
model of this phenomenon, where an initial population of cells that are
sensitive to therapy can produce mutants that are resistant to the therapy. In
this model, we investigate two random times, the recurrence time and the
crossover time. Recurrence time is defined as the first time that the
population size of mutant cells exceeds a given proportion of the initial
population size of drug-sensitive cells. Crossover time is defined as the first
time that the resistant cell population dominates the total population. We
establish convergence in probability results for both recurrence and crossover
time. We then develop expressions for the large deviations rate of early
recurrence and early crossover events. We characterize how the large deviation
rates and rate functions depend on the initial size of the mutant cell
population. We finally look at the large deviations rate of early recurrence
conditioned on the number of mutant clones present at recurrence in the special
case of a deterministically decaying sensitive population. We find that if
recurrence occurs before the predicted law of large numbers limit then there
will likely be an increase in the number of clones present at recurrence time.
