On a new perspective of the basic reproduction number in heterogeneous environments

Although its usefulness and possibility of the well-known definition of the basic reproduction number R (0) for structured populations by Diekmann, Heesterbeek and Metz (J Math Biol 28:365-382, 1990) (the DHM definition) have been widely recognized mainly in the context of epidemic models, originally it deals with population dynamics in a constant environment, so it cannot be applied to formulate the threshold principle for population growth in time-heterogeneous environments. Since the mid-1990s, several authors proposed some ideas to extend the definition of R (0) to the case of a periodic environment. In particular, the definition of R (0) in a periodic environment by BacaA << r and Guernaoui (J Math Biol 53:421-436, 2006) (the BG definition) is most important, because their definition of periodic R (0) can be interpreted as the asymptotic per generation growth rate, which is an essential feature of the DHM definition. In this paper, we introduce a new definition of R (0) based on the generation evolution operator (GEO), which has intuitively clear biological meaning and can be applied to structured populations in any heterogeneous environment. Using the generation evolution operator, we show that the DHM definition and the BG definition completely allow the generational interpretation and, in those two cases, the spectral radius of GEO equals the spectral radius of the next generation operator, so it gives the basic reproduction number. Hence the new definition is an extension of the DHM definition and the BG definition. Finally we prove a weak sign relation that if the average Malthusian parameter exists, it is nonnegative when R (0) > 1 and it is nonpositive when R (0) < 1.