On the probability of extinction in a periodic environment

For a certain class of multi-type branching processes in a continuous-time periodic environment, we show that the extinction probability is equal to (resp. less than) 1 if the basic reproduction number is less than (resp. bigger than) 1. The proof uses results concerning the asymptotic behavior of cooperative systems of differential equations. In epidemiology the extinction probability may be used as a time-periodic measure of the epidemic risk. As an example we consider a linearized SEIR epidemic model and data from the recent measles epidemic in France. Discrete-time models with potential applications in conservation biology are also discussed.