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The reconstruction of phylogenetic trees based on viral genetic sequence datasequentially sampled from an epidemic provides estimates of the pasttransmission dynamics, by fitting epidemiological models to these trees. To ourknowledge, none of the epidemiological models currently used in phylogeneticscan account for recovery rates and sampling rates dependent on the time elapsedsince transmission.  Here we introduce an epidemiological model where infectives leave theepidemic, either by recovery or sampling, after some random time which mayfollow an arbitrary distribution.  We derive an expression for the likelihood of the phylogenetic tree ofsampled infectives under our general epidemiological model. The analyticconcept developed in this paper will facilitate inference of pastepidemiological dynamics and provide an analytical framework for performingvery efficient simulations of phylogenetic trees under our model. The main ideaof our analytic study is that the non-Markovian epidemiological model givingrise to phylogenetic trees growing vertically as time goes by, can berepresented by a Markovian "coalescent point process" growing horizontally bythe sequential addition of pairs of coalescence and sampling times.  As examples, we discuss two special cases of our general model, namely anapplication to influenza and an application to HIV. Though phrased inepidemiological terms, our framework can also be used for instance to fitmacroevolutionary models to phylogenies of extant and extinct species,accounting for general species lifetime distributions.

Phylogenetic analysis accounting for age-dependent death and sampling with applications to epidemics