Assessing the Impact of Intervention Delays on Stochastic Epidemics

A stochastic model of disease transmission among a population partitioned into groups is defined. The model is of SEIR (Susceptible-Exposed-Infective-Removed) type and features intervention in response to the progress of the disease, and moreover includes a random delay before the intervention occurs. A threshold parameter for the model, which can be used to assess the efficacy of the intervention, is defined. The threshold parameter can be calculated either in closed form or via recursion, for a number of different choices of exposed, infectious and delay period distributions, both for the epidemic model itself and also a large-group approximation. In particular both constant and Erlang-distributed delay periods are considered. Sufficient conditions under which a constant delay gives the least effective intervention are presented. For a given mean delay, it is shown that the two-point delay distribution provides the optimal intervention.