The deterministic Kermack-McKendrick model bounds the general stochastic   epidemic
  We prove that, for Poisson transmission and recovery processes, the classic Susceptible $\to$ Infected $\to$ 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. 
Robert R. Wilkinson, Frank G. Ball, Kieran J. Sharkey
q-bio.PE cond-mat.stat-mech math.PR physics.soc-ph