The basic reproduction number, noted R0, is a concept that is often used in epidemiology. Originally developed by Lotka in a demographic context, it was popularized in epidemiology by Anderson & May. In simple models, it is the average number of secondary cases due to one primary case at the beginning of an epidemic. An epidemic occurs when R0 is bigger than 1.

For more complex models, the definition of R0 requires a little attention in order to keep R0=1 as the epidemic threshold. In constant environments, R0 is often the spectral radius of a "next-generation matrix".

However many infectious diseases, in particular air-borne and vector-borne diseases, are very much influenced by seasonality. We shall explain how the definition of R0 was adapted to time-periodic environments a few years ago and how classical results concerning R0 are modified in this case.

Finally we shall explain how the definition of R0 was recently adapted to random environments.

References:

[1] N. Bacaër, E. Ait Dads: On the biological interpretation of a definition for the parameter R0 in periodic population models. J Math Biol 65 (2012) 601-621.

[2] N. Bacaër, M. Khaladi: On the basic reproduction number in a random environment. J Math Biol, doi:10.1007/s00285-012-0611-0