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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 (reflecting the fact that the phenomena of interest are well approximated via branching processes, at their inception). Despite the very extensive literature around R0 for deterministic epidemic models, we believe there are still aspects which are not fully understood. Foremost is the fact that R0 is not a function of the original ODE model, unless we also include in it a certain (F,V) gradient decomposition, which is not unique. This is related to the specification of the “infected compartments”, which is also not unique. A second interesting question is whether the extinction probabilities of the natural continuous time Markovian chain approximation of an ODE model around boundary points (disease-free equilibrium and invasion points) are also related to the (F,V) gradient decomposition. We offer below several new contributions to the literature: (1) A universal algorithmic definition of a (F,V) gradient decomposition (and hence of the resulting R0). (2) A fixed point equation 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. Last but not least, we offer Mathematica scripts and implement them for a large variety of examples, which illustrate that our recipe offers always reasonable results, but that sometimes other reasonable (F,V) decompositions are available as well.