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

Christian Borgs, Karissa Huang, Christian Ikeokwu

Abstract

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 $η$ and recover at rate $γ$, 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.

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

Abstract

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 and recover at rate , 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.

Paper Structure

This paper contains 32 sections, 32 theorems, 246 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and Statements of Results
  3. Notation
  4. Model Definition
  5. Statements of Results
  6. Related Work
  7. Preliminaries
  8. Poisson Stochastic Block Model Coupling
  9. Coupling the SIR dynamic and the Poisson Stochastic Block Model
  10. Law of Large Numbers on Bounded Time Intervals
  11. Formal statement for arbitrary initial conditions
  12. Proof of Theorem \ref{['thm:lln']}
  13. Proof of Theorem \ref{['thm:LLN']}
  14. Law of Large Numbers for the Final Size
  15. Solution to Differential Equations
  16. ...and 17 more sections

Key Result

Theorem 2.6

Consider the SIR epidemic on $G\sim \mathrm{PSBM}(\bm V,W)$ or $G\sim \mathrm{SBM}(\bm V,W)$, with the initial state of the infection obeying the conditions from Assumption ass:initial-SandI. Let $s_k(t)$, $i_k(t)$, and $x_k(t)$ be the unique solutions to with initial conditions $s_k(0)$ and $x_k(0)=i_k(0)$, and let $0<t_0<\infty$, and let $\varepsilon>0$ be arbitrary. Then as $n \to \infty,$ whe

Figures (2)

  • Figure 1: A simulation of our system of ordinary differential equations on a two-community stochastic block model. We vary parameter values of $\varepsilon$, the initial fraction of infected individuals going from $0.1, 0.01, 0.001$ from the left column to the right column; $W$, the contact matrix; and $s_1(0)$ and $s_2(0)$, the initial proportion of susceptible individuals in each community. For all simulations, $\eta = \gamma = 1/2$.
  • Figure 2: Simulations for one hundred stochastic SIR trajectories of the fraction of infected individuals on a stochastic block model with two communities of the same size. From left to right, we increase the number of nodes in the network keeping the ratio $\frac{n_1}{n_2}$ the same. The solid blue and orange curves are the solution to the differential equations with the same parameters.

Theorems & Definitions (88)

  • Definition 2.1: Stochastic Block Model (SBM)
  • Remark 2.2
  • Definition 2.3: Poisson Stochastic Block Model (PSBM)
  • Remark 2.4
  • Theorem 2.6: Law of Large Numbers for Finite Time
  • Remark 2.7
  • Theorem 2.8
  • Definition 2.9: Poisson Multi-type Branching Process
  • Definition 2.10: Infection Tree for arbitrary graphs
  • Remark 2.11
  • ...and 78 more