As classically defined by Macdonald in the early 1950s, for the case of diseases with one vector and one host, the Basic Reproduction Number, R0, is defined as the number of secondary infections caused by a single infective of the same type (vector or host) during its infectiousness period in an entirely susceptible population. In the case of a disease which has one vector and one host, it is easy to show that R0 coincides with the threshold for the establishment of an endemic state: if R0 > 1 (< 1), the disease can invade (cannot invade) the host population. In this paper we examine various epidemic situations in which there are more than one vector and/or host. We show that in those more complex systems it is not possible to deduce a single R0 but rather a threshold for infection persistence which is a composite of several quantities closely related to the classical expression of R0. Another definition of R0 given by Diekmann, Heesterbeek and Metz, and denoted in this paper R0NGO is discussed and applied as an alternative to calculate the thresholds for infection establishment.