The probability of epidemic burnout in the stochastic SIR model with vital dynamics

We present a new approach to computing the probability of epidemic ``burnout'', i.e., the probability that a newly emergent pathogen will go extinct after a major epidemic. Our analysis is based on the standard stochastic formulation of the Susceptible-Infected-Removed (SIR) epidemic model including host demography (births and deaths), and corresponds to the standard SIR ordinary differential equations (ODEs) in the infinite population limit. Exploiting a boundary layer approximation to the ODEs and a birth-death process approximation to the stochastic dynamics within the boundary layer, we derive convenient, fully analytical approximations for the burnout probability. We demonstrate---by comparing with computationally demanding individual-based stochastic simulations and with semi-analytical approximations derived previously---that our fully analytical approximations are highly accurate for biologically plausible parameters. We show that the probability of burnout always decreases with increased mean infectious period. However, for typical biological parameters, there is a relevant local minimum in the probability of persistence as a function of the basic reproduction number $\R_0$. For the shortest infectious periods, persistence is least likely if $\R_0\approx2.57$; for longer infectious periods, the minimum point decreases to $\R_0\approx2$. For typical acute immunizing infections in human populations of realistic size, our analysis of the SIR model shows that burnout is almost certain in a well-mixed population, implying that susceptible recruitment through births is insufficient on its own to explain disease persistence.