Effective population size
Given a clock-like tree, one of the quantities we can estimate is the effective population size over time assuming a coalescent model.
This captures the rate at which the tree branches:
Small effective population size = high branching rate and short branches
Large effective population size = low branching rate and long branches
What is the effective population size?
Not simply the number of infected individuals
For simple models
$N_e=I \tau /2$
$I$ = number of infected individuals
$\tau$ = average time between infections in the population ('generation time')
As both change over the course of an epidemic, it is often hard to interpret effective population size epidemiologically
However, during the epidemic growth phase, the rate of change of $N_e$ reflects the exponential growth rate of infected individuals
Coalescent models vs. birth-death-sampling models
Coalescent models assume the population size is changing deterministically
Birth-death models allow for stochastic fluctuations in the population, but have not yet been extended to complex, nonlinear, structured populations