[Paper Review] SIR epidemics on evolving graphs

Yufeng Jiang, Remy Kassem+3 more|arXiv (Cornell University)|Jan 19, 2019

Complex Network Analysis Techniques27 references4 citations

TL;DR

This paper studies the evoSIR model on Erdős-Rényi random graphs, where susceptible individuals rewire away from infected neighbors at rate ρ. It proves that the critical infection rate λc and the probability of a large epidemic are identical for evoSIR and the delSIR model (where edges are deleted instead of rewired), and shows the final size of a large delSIR epidemic undergoes a continuous transition, while simulations suggest evoSIR exhibits a discontinuous transition at λc.

ABSTRACT

We consider evoSIR, a variant of the SIR model, on Erd\H os-Renyi random graphs in which susceptibles with an infected neighbor break that connection at rate $ρ$ and rewire to a randomly chosen individual. We compute the critical infection rate $λ_c$ and the probability of a large epidemic by showing that they are the same for the delSIR model in which $S-I$ connections are deleted instead of rewired. The final size of a large delSIR epidemic has a continuous transition. Simulations suggest that the final size of a large evoSIR epidemic is discontinuous at $λ_c$.

Motivation & Objective

• To analyze the evoSIR model, a variant of the SIR epidemic process on evolving random graphs where susceptible individuals rewire from infected neighbors.
• To determine the critical infection rate λc for the onset of large-scale epidemics in evoSIR.
• To compare the final size behavior of evoSIR with the delSIR model, in which edges are deleted instead of rewired.
• To investigate whether the final size of a large epidemic in evoSIR exhibits a continuous or discontinuous phase transition at λc.

Proposed method

• Models the epidemic process on Erdős-Rényi random graphs G(n, μ/n) with mean degree μ.
• Introduces two variants: evoSIR, where S-I edges are rewired to random individuals at rate ρ, and delSIR, where S-I edges are deleted at rate ρ.
• Uses martingale techniques and branching process approximations to derive the critical threshold λc.
• Applies a continuous-time Markov chain approach with infinitesimal mean and variance calculations to show convergence to a deterministic limit.
• Derives a differential equation system for the evolution of susceptible and average degree in unexplored vertices under fixed-time rewiring.
• Uses generating functions and the law of large numbers to prove convergence of the normalized number of infected individuals to a deterministic function.

Experimental results

Research questions

• RQ1 What is the critical infection rate λc for the evoSIR model on Erdős-Rényi random graphs?
• RQ2 Are the critical thresholds and large epidemic probabilities the same for evoSIR and delSIR models?
• RQ3 Does the final size of a large epidemic in evoSIR exhibit a continuous or discontinuous phase transition at λc?
• RQ4 How does rewiring behavior affect the final epidemic size compared to the standard SIR model?
• RQ5 What is the limiting behavior of the epidemic process as the graph size n → ∞?

Key findings

• The critical infection rate λc for evoSIR is identical to that of the delSIR model, implying equivalent thresholds for large-scale epidemics.
• The final size of a large delSIR epidemic undergoes a continuous phase transition at λc.
• Simulations suggest that the final size of a large evoSIR epidemic exhibits a discontinuous transition at λc.
• The normalized number of infected individuals U[ns]/n converges almost surely to e^{-sμEτ} as n → ∞, establishing a deterministic limit.
• The system of differential equations for u_s and v_s under fixed-time rewiring leads to a solution of the form u = A / (B + (A - B)e^{At}), showing the dynamics of susceptible fraction over time.
• The quantity αu_s + (1−α)v_s is conserved over time, where α is the rewiring probability, leading to a closed-form expression for v_s in terms of u_s.