Necessary and sufficient conditions for R₀ to be a sum of contributions of fertility loops.

Recently, de-Camino-Beck and Lewis (Bull Math Biol 69:1341-1354, 2007) have presented a method that under certain restricted conditions allows computing the basic reproduction ratio R₀ in a simple manner from life cycle graphs, without, however, giving an explicit indication of these conditions. In this paper, we give various sets of sufficient and generically necessary conditions. To this end, we develop a fully algebraic counterpart of their graph-reduction method which we actually found more useful in concrete applications. Both methods, if they work, give a simple algebraic formula that can be interpreted as the sum of contributions of all fertility loops. This formula can be used in e.g. pest control and conservation biology, where it can complement sensitivity and elasticity analyses. The simplest of the necessary and sufficient conditions is that, for irreducible projection matrices, all paths from birth to reproduction have to pass through a common state. This state may be visible in the state representation for the chosen sampling time, but the passing may also occur in between sampling times, like a seed stage in the case of sampling just before flowering. Note that there may be more than one birth state, like when plants in their first year can already have different sizes at the sampling time. Also the common state may occur only later in life. However, in all cases R₀ allows a simple interpretation as the expected number of new individuals that in the next generation enter the common state deriving from a single individual in this state. We end with pointing to some alternative algebraically simple quantities with properties similar to those of R₀ that may sometimes be used to good effect in cases where no simple formula for R₀ exists.