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A birth-death-sampling model gives rise to phylogenetic trees with samplesfrom the past and the present. Interpreting "birth" as branching speciation,"death" as extinction, and "sampling" as fossil preservation and recovery, thismodel -- also referred to as the fossilized birth-death (FBD) model -- givesrise to phylogenetic trees on extant and fossil samples. The model has beenmathematically analyzed and successfully applied to a range of datasets ondifferent taxonomic levels, such as penguins, plants, and insects. However, thecurrent mathematical treatment of this model does not allow for a group oftemporally distinct fossil specimens to be assigned to the same species. Inthis paper, we provide a general mathematical FBD modeling framework thatexplicitly takes "stratigraphic ranges" into account, with a stratigraphicrange being defined as the lineage interval associated with a single species,ranging through time from the first to the last fossil appearance of thespecies. To assign a sequence of fossil samples in the phylogenetic tree to thesame species, i.e., to specify a stratigraphic range, we need to define themode of speciation. We provide expressions to account for three commonspeciation modes: budding (or asymmetric) speciation, bifurcating (orsymmetric) speciation, and anagenetic speciation. Our equations allow forflexible joint Bayesian analysis of paleontological and neontological data.Furthermore, our framework is directly applicable to epidemiology, where astratigraphic range is the observed duration of infection of a single patient,"birth" via budding is transmission, "death" is recovery, and "sampling" issequencing the pathogen of a patient. Thus, we present a model that allows forincorporation of multiple observations through time from a single patient.

The fossilized birth-death model for the analysis of stratigraphic range data under different speciation concepts