  The susceptible-infected-recovered (SIR) model has been used extensively to
model disease spread and other processes. Despite the widespread usage of this
ordinary differential equation (ODE) based model which represents the
mean-field approximation of the underlying stochastic SIR process on contact
networks, only few rigorous approaches exist and these use complex semigroup
and martingale techniques to prove that the expected fraction of the
susceptible and infected nodes of the stochastic SIR process on a complete
graph converges as the number of nodes increases to the solution of the
mean-field ODE model. Extending the elementary proof of convergence for the SIS
process introduced by Armbruster and Beck (2016) to the SIR process, we show
convergence using only a system of three ODEs, simple probabilistic
inequalities, and basic ODE theory. Our approach can also be generalized to
many other types of compartmental models (e.g.,
susceptible-infected-recovered-susceptible (SIRS)) which are linear ODEs with
the addition of quadratic terms for the number of new infections similar to the
SI term in the SIR model.
