We derive accurate, closed-form analytical approximations for the phase-plane trajectories of the standard susceptible-infectious-removed (SIR) epidemic model, including host births and deaths. Our approximations for the SIR ordinary differential equations also allow us to provide convenient, accurate analytical approximations for the associated Poinaré map, and the minimum and maximum susceptible and infected host densities in each epidemic wave. Our analysis involves matching asymptotic expansions across branch cuts of the Lambert W function.