One stochastic epidemic, three views: everything that happened, the transmission history behind the samples, and the tree you could actually reconstruct.
Effective reproductive number
Re = λ / (μ + ψr)
—
offspring per infection
Total removal rate
D = μ + ψr
—
becoming uninfectious
Sampling proportion
S = ψr / (μ + ψr)
—
removals that are observed
r is the treatment probability. A host sampled at rate ψ is removed from the
infectious population with probability r, and stays infectious otherwise. At r = 1 sampling
always removes and the three quantities reduce to the familiar λ/(μ+ψ), μ+ψ and
ψ/(μ+ψ). Below 1 the removal rate is μ + ψr, matching BEAST's
birthRate / (samplingRate * treatmentProbability + deathRate), and hosts sampled without
being removed can reappear as sampled ancestors in the reconstructed tree.
Every lineage the process ever produced — including those that died unsampled. One row per lineage, ordered by birth.
Sampled lineage
Unsampled lineage
Sampled and removed
Sampled, still infectious
The transmission history behind the samples: every sampled lineage plus the unsampled ancestors that carried the infection between them. Side branches that led nowhere are gone, but unsampled ancestors remain — they are real hosts who were never observed.
Sampled lineage
Unsampled ancestor
Sampled and removed
Sampled, still infectious
The reconstructed tree — what you could infer from the samples alone. Subtrees with no sampled descendant are pruned, and any unsampled node left with a single surviving child is suppressed, its two branches merging into one. Sampled nodes are never suppressed: a host sampled while remaining infectious, whose descendants are also sampled, survives as a sampled ancestor sitting on a branch rather than at its tip.
At r = 1 there are no sampled ancestors, and the tree is strictly bifurcating with n tips and exactly n−1 internal nodes. Below 1 that identity becomes bifurcations = tips − 1, with the sampled ancestors counted separately.