Exponential Effective Population Growth

Let denote the initial effective population size at the time of infection , and assume that the effective population size grows exponentially with growth rate such that the effective population size at current time (i.e., time after infection) is .

Unconstrained Coalescence

Cumulative Density Function (CDF) — P(t)

Remember that the current time is , and is backward time from . Thus, the effective population size at time is . The CDF of the next arrival time is as follows:

Inverse CDF — t(P)

Constrained Coalescence

NEED TO FIX

Cumulative Density Function (CDF) — P(t|T)

Inverse CDF — t(P|T)