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Epidemics models in Networks

Tiago Pereira

TL;DR

Epidemics on networks are analyzed through a hierarchy of models that connect disease dynamics to the spectral properties of contact graphs. The authors develop SIS/SIR frameworks, extend them to networks, age-structured populations via contact matrices, and spatially explicit mobility models, showing that the dominant eigenvalue of key matrices governs the epidemic threshold and early growth rate. They demonstrate that heterogeneity, such as hubs or age-based mixing, can dramatically lower thresholds and localize spread, with zero-threshold behavior in large power-law networks. The combined approach provides a principled way to interpret historical outbreaks, fit models to data, and inform interventions by tying network structure to dynamical outcomes. The work highlights how linear stability and Perron–Frobenius theory offer a unifying lens to understand and anticipate epidemic behavior in complex populations.

Abstract

These lectures are based on material which was presented in the 2025 Summer school at Fundação Getulio Vargas. The aim of this series is to introduce graduate students with a little background in the field of dynamical systems and network theory to epidemic models. Our goal is to give a succinct and self-contained description of the models

Epidemics models in Networks

TL;DR

Epidemics on networks are analyzed through a hierarchy of models that connect disease dynamics to the spectral properties of contact graphs. The authors develop SIS/SIR frameworks, extend them to networks, age-structured populations via contact matrices, and spatially explicit mobility models, showing that the dominant eigenvalue of key matrices governs the epidemic threshold and early growth rate. They demonstrate that heterogeneity, such as hubs or age-based mixing, can dramatically lower thresholds and localize spread, with zero-threshold behavior in large power-law networks. The combined approach provides a principled way to interpret historical outbreaks, fit models to data, and inform interventions by tying network structure to dynamical outcomes. The work highlights how linear stability and Perron–Frobenius theory offer a unifying lens to understand and anticipate epidemic behavior in complex populations.

Abstract

These lectures are based on material which was presented in the 2025 Summer school at Fundação Getulio Vargas. The aim of this series is to introduce graduate students with a little background in the field of dynamical systems and network theory to epidemic models. Our goal is to give a succinct and self-contained description of the models
Paper Structure (43 sections, 25 theorems, 240 equations, 1 figure, 1 table)

This paper contains 43 sections, 25 theorems, 240 equations, 1 figure, 1 table.

Table of Contents

  1. Classical compartimental models
  2. Infectious Diseases and Isolation
  3. The SIS Model
  4. Biological vs. effective reproduction numbers
  5. Main Lessons
  6. The SIR Model
  7. Lecture 3: Fitting SIR models to Data
  8. The Black Death
  9. Spread of Influenza A
  10. Coronavirus in Brazil
  11. Networks Lecture by Zheng Bian
  12. Networks
  13. Spectral radius of a network
  14. Random graphs from a given expected degree sequence
  15. Spectral radius upper bound for $G(\boldsymbol{\mathrm{w}})$
  16. ...and 28 more sections

Key Result

Proposition 1

The number of walks of length $k$ in $G$ starting from $i$ and ending at $j$ is given by $(A^k)_{ij}$. The total number of walks of length $k$ is If $G$ is connected, then the exponential growth rate of $N_k$ is given by

Figures (1)

  • Figure 1: $I$ (dots) e prediction (full lines) as a function of time.

Theorems & Definitions (41)

  • Proposition 1: Number of walks
  • proof
  • Remark 1
  • Proposition 2
  • proof
  • lemma thmcounterlemma: Subgraph lemma
  • lemma thmcounterlemma: Perron-Frobenius; Horn2012 Theorem 8.3.1
  • lemma thmcounterlemma: Neighborless node
  • proof
  • theorem 1: $\sqrt{m}$ lower bound of graph spectral radius
  • ...and 31 more