Critical reaction time during a disease outbreak

We consider the problem of disease surveillance, early detection, and the requirement for rapid response. If it is important to implement a certain intervention before a threshold number of individuals become infected, then we must determine the expected time (after detection) at which this will occur. We define this as the critical reaction time. Suppose a disease outbreak within a closed population N may be detected when a certain number I0 individuals are infected. Now assume that a specific intervention should be applied before this number increases to a level I1 > I0. The first time at which I1 is reached is called the critical reaction time. We derive simple asymptotic formulae for the expectation, variance and full distribution of the reaction time T(I1, I0, N) in the framework of a stochastic SI (susceptible/infected) model. Asymptotically these results are in agreement with the mean-field model and Gaussian approximations. Further corrections to these approximations are obtained that give a better fit to the tails of the distribution of reaction times, which are important for the modelling of extreme events.

► The discrete stochastic SI epidemic is analysed in detail.
► We consider the problem of disease surveillance and initial outbreak detection.
► We consider a need for rapid reaction before a threshold outbreak size is reached.
► The critical reaction time is calculated using the mean-field approximation.
► We find rules of thumb for the chance that reaction times fall below critical values.