Epidemic detection using CUSUM

We consider the problem of detecting a proportional change in the intensity of a counting process with a continuous compensator. We prove that if the counting process does not have explosions, the CUSUM test is optimal in a Lorden sense. In particular, the CUSUM test minimizes the worst-case conditional expected number of events that occur after the change and before raising an alarm given the worst possible history up to the time of the change. We apply this result to the detection of the onset of epidemics and we discuss modifications of the optimal detection rule when the processes of interest are monitored in discrete times.