Necessary and sufficient conditions for $$R_{0}$$ to be a sum of contributions of fertility loops

Abstract

As a present to Odo Diekmann for his 65th birthday: the answer to a question that he posed in a conversation sometime ago.Abstract Recently, de-Camino-Beck and Lewis (2007, Bull Math Biol 69:1341-1354 have presented a method that under certain restricted conditions allows computing the basic reproduction ratio R 0 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 graphreduction 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 R 0 allows a simple interpretation as the expected number of new individuals that in the next generation enter the common state deriving from a single in- Rueffler and Metz dividual in this state. We end with pointing to some alternative algebraically simple quantities with properties similar to those of R 0 that may sometimes be used to good effect in cases where no simple formula for R 0 exists.

Generational Spreading Speed and the Dynamics of Population Range Expansion

“…First, in many biologically relevant cases, it is possible to calculate an algebraic expression for c * G . This is because reproduction often occurs from or to a single life-history stage, and the next-generation matrix takes a simplified form (see "Order of the Characteristic Equation"; de-Camino-Beck and Rueffler and Metz 2013). In the case of Gaussian dispersal, this leads to an explicit analytic form for the generational spreading speed (see "Simplification and Approximation").…”

“…In the case of Gaussian dispersal, this leads to an explicit analytic form for the generational spreading speed (see "Simplification and Approximation"). Second, even if we cannot generate an explicit expression for generational spreading speed, we can examine the influence on generational spreading speed of reproductive pathways, elucidated via graph reduction (de-Camino-Beck and Lewis 2007) or algebra (Rueffler and Metz 2013). This can yield insight into life-cycle events of particular importance for a species' ability to spread.…”

“…That is, F will have nonzero entries only in a single row or column. As a result, although calculating l involves an Nth-order polynomial, calculating R 0 often involves a first-order polynomial (de-Camino-Beck and Lewis 2007;Rueffler and Metz 2013). This can allow for intuitive insight through a closed-form, analytical expression for R 0 , expressed as the sum of contributions from different reproduction pathways in the life cycle (de-Camino-Beck and .…”

The many guises of R0 (a didactic note)

“…As indicated in the main text, computation of R 0 is often facilitated by identifying the possible states at birth, i.e., by studying the range of F. A more detailed analysis of the life cycle graph may help to further reduce the computational burden, see Rueffler and Metz (2013), de Camino-Beck et al (2009) and the references given there.…”

“…We say that the first have state 1 at birth and the second state 2. Following Rueffler and Metz (2013) we might call "seedling" a hidden state that any seed must go through before becoming a plant that produces seeds. This feature was exploited when we derived (6).…”