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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 R(0) 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.

An adaptation of the definition of the basic reproduction number R (0) to time-periodic seasonal models was suggested a few years ago. However, its biological interpretation remained unclear. The present paper shows that in demography, this R (0) is the asymptotic ratio of total births in two successive generations of the family tree. In epidemiology, it is the asymptotic ratio of total infections in two successive generations of the infection tree. This result is compared with other recent work.

The basic reproduction number R (0) has been used in population biology, especially in epidemiology, for several decades. But a suitable definition in the case of models with periodic coefficients was given only in recent years. The definition involves the spectral radius of an integral operator. As in the study of structured epidemic models in a constant environment, there is a need to emphasize the biological meaning of this spectral radius. In this paper we show that R (0) for periodic models is still an asymptotic per generation growth rate. We also emphasize the difference between this theoretical R (0) for periodic models and the "reproduction number" obtained by fitting an exponential to the beginning of an epidemic curve. This difference has been overlooked in recent studies of the H1N1 influenza pandemic.