  The basic reproduction number R0 is a concept which originated in population
dynamics, mathematical epidemiology, and ecology and is closely related to the
mean number of children in branching processes.We offer below three new
contributions to the literature: 1) We order a universal algorithmic definition
of a (F, V) gradient decomposition (and hence of the resulting R0), which
requires a minimal input from the user, namely the specification of an
admissible set of disease/infection variables. We also present examples where
other choices may be more reasonable, with more terms in F, or more terms in V .
  2) We glean out from the works of Bacaer a fixed point equation (8) for the
extinction probabilities of a stochastic model associated to a deterministic
ODE model, which may be expressed in terms of the (F, V ) decomposition. The
fact that both R0 and the extinction probabilities are functions of (F, V )
underlines the centrality of this pair, which may be viewed as more fundamental
than the famous next generation matrix FV^{-1}.
  3) We suggest introducing a new concept of sufficient/minimal
disease/infection set (sufficient for determining R0). More precisely, our
universal recipe of choosing "new infections" once the "infections" are
specified suggests focusing on the choice of the latter, which is also not
unique. The maximal choice of choosing all compartments which become 0 at the
given boundary point seems to always work, but is the least useful for analytic
computations, therefore we propose to investigate the minimal one. As a bonus,
this idea seems useful for understanding the Jacobian factorization approach
for computing R0 . Last but not least, we offer Mathematica scripts and
implement them for a large variety of examples, which illustrate that our
recipe others always reasonable results, but that sometimes other reasonable
(F, V ) decompositions are available as well.
