Variance of the SIS Epidemic on Networks: A Diffusion Approximation

2026/07/03 by Lucija Nora Farkaš, Sebastian Morel Balbi, Hrvoje Štefančić +2 · 1 voice
Physics and Astronomy · Mathematics · Biochemistry, Genetics and Molecular Biology · #cond-mat.stat-mech #math.PR #physics.soc-ph #q-bio.QM

paper · pdf

Abstract

Functional laws of large numbers (FLLNs) describe the mean-field trajectory of epidemics on networks, but say nothing about the fluctuations around it. These fluctuations are governed by moments of the degree distribution not relevant at the level of the mean. A rigorous functional central limit theorem (FCLT) exists for the susceptible--infected (SI) process on configuration-model graphs, but no analogue exists for SIS, where recovery reintroduces vertices into the susceptible pool with partially known neighborhoods, breaking the clean neighborhood distribution the SI derivation relies on. We develop a tractable variance approximation for Markovian SIS on configuration-model graphs, combining Gleeson's approximate master equation (AME) framework with a van Kampen system-size expansion in the spirit of the SI FCLT. We derive a closed drift and diffusion matrix for a reduced susceptible/SI-edge/SS-edge count vector and obtain the time-dependent covariance via the associated Langevin/Lyapunov equation. Validation against Gillespie simulation across Poisson, regular, and power-law networks shows close agreement, with deviations near the epidemic threshold and in strongly heterogeneous networks.

Citations

• Threshold and quasi-stationary distribution for the susceptible-infectious-susceptible model on networks
• Central limit theorems for SIR epidemics and percolation on configuration model random graphs
• A stochastic SIR network epidemic model with preventive dropping of edges
• The large graph limit of a stochastic epidemic model on a dynamic multilayer network
• Heterogeneous network epidemics: real-time growth, variance and extinction of infection
• Law of large numbers for the SIR epidemic on a random graph with given degrees
• Law of large numbers for the SIR epidemic on a random graph with given degrees
• Dynamics of stochastic epidemics on heterogeneous networks
• Approximating the epidemic curve
• Binary-State Dynamics on Complex Networks: Pair Approximation and Beyond
• Large graph limit for an SIR process in random network with heterogeneous connectivity

Discussions

• Another one fresh from the oven. For our project, we need to know how errors behave. There are not many papers on the problem. After the experience, it became clear that the reason for that is that th [bsky, 0 points, 0 comments]

Related

• Exact Markovian SIR and SIS epidemics on networks and an upper bound for the epidemic threshold
• Epidemic threshold and localization of the SIS model on directed complex networks
• Estimating the covariance structure of heterogeneous SIS epidemics on networks
• Analysis of Exact and Approximated Epidemic Models over Complex Networks
• Exact mean field dynamics for epidemic-like processes on heterogeneous networks