  The basic reproduction number ($R_0$) is a threshold parameter for disease
extinction or survival in isolated populations. However no human population is
fully isolated from other human or animal populations. We use compartmental
models to derive simple rules for the basic reproduction number for populations
with local person-to-person transmission and exposure from some other source:
either a reservoir exposure or imported cases. We introduce the idea of a
reservoir-driven or importation-driven disease: diseases that would become
extinct in the population of interest without reservoir exposure or imported
cases (since $R_0<1$), but nevertheless may be sufficiently transmissible that
many or most infections are acquired from humans in that population. We show
that in the simplest case, $R_0<1$ if and only if the proportion of infections
acquired from the external source exceeds the disease prevalence and explore
how population heterogeneity and the interactions of multiple strains affect
this rule. We apply these rules in two cases studies of Clostridium difficile
infection and colonisation: C. difficile in the hospital setting accounting for
imported cases, and C. difficile in the general human population accounting for
exposure to animal reservoirs. We demonstrate that even the hospital-adapted,
highly-transmissible NAP1/RT027 strain of C. difficile had a reproduction
number <1 in a landmark study of hospitalised patients and therefore was
sustained by colonised and infected admissions to the study hospital. We argue
that C. difficile should be considered reservoir-driven if as little as 13.0%
of transmission can be attributed to animal reservoirs.
