The deterministic Kermack-McKendrick model bounds the general stochastic epidemic

We prove that, for Poisson transmission and recovery processes, the classic Susceptible → Infected → Recovered (SIR) epidemic model of Kermack and McKendrick provides, for any given time t > 0, a strict lower bound on the expected number of suscpetibles and a strict upper bound on the expected number of recoveries in the general stochastic SIR epidemic. The proof is based on the recent message passing representation of SIR epidemics applied to a complete graph.