A Law of Large Numbers for SIR on the Stochastic Block Model: A Proof via Herd Immunity

In this paper, we study the dynamics of the susceptible-infected-recovered (SIR) model on a network with community structure, namely the stochastic block model (SBM). As usual, the SIR model is a stochastic model for an epidemic where infected vertices infect susceptible neighbors at some rate $\eta$ and recover at rate $\gamma$, and the SBM is a random graph model where vertices are partitioned into a finite number of communities. The connection probability between two vertices depends on their community affiliation, here scaled so that the average degrees have a finite limit as the network grows. We prove laws of large numbers (LLN) for the epidemic's trajectory to a system of ordinary differential equations over any time horizon (finite or infinite), including in particular a LLN for the final size of the infection. Our proofs rely on two main ingredients: (i) a new coupling of the SIR epidemic and the randomness of the SBM, revealing a vector-valued random variable that drives the epidemic (related to what is usually called the ``force of the infection'' via a linear transformation), and (ii) a novel technique for analyzing the limiting behavior of the infinite time horizon for the infection, using the fact that once the infection passes the herd immunity threshold it dies out quickly and has a negligible impact on the overall size of the infection.