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SIR on locally converging dynamic random graphs

Marta Milewska, Remco van der Hofstad, Bert Zwart

TL;DR

The paper develops a rigorous framework for studying SIR epidemics on dynamic random graphs by introducing dynamic local convergence through time-marked union graphs. A central result shows that, under a strengthened dynamic convergence condition, the epidemic on the evolving graph converges to the epidemic on its dynamic local limit, enabling analysis on tractable limiting objects such as time-marked trees. The backward process on the time-marked union graph yields a practical mechanism to compute infection paths and marks, and the authors illustrate the theory with dynamic ER and random-intersection graph models, supported by simulations that demonstrate faster epidemic progression under graph dynamics and strong accuracy of the local-limit approximation. This work extends static local-convergence results to dynamic networks, providing a versatile toolkit for epidemic modelling on evolving contact structures and suggesting future directions including more complex compartments and adaptive rewiring. The results have potential applications in public health policy by enabling efficient, local-structure-based predictions on realistic, time-varying networks.

Abstract

In this paper, we study the trajectory of a classic SIR epidemic on a family of dynamic random graphs of fixed size, whose set of edges continuously evolves over time. We set general infection and recovery times, and start the epidemic from a positive, yet small, proportion of vertices. We show that in such a case, the spread of an infectious disease around a typical individual can be approximated by the spread of the disease in a local neighbourhood of a uniformly chosen vertex. We formalize this by studying general dynamic random graphs that converge dynamically locally in probability and demonstrate that the epidemic on these graphs converges to the epidemic on their dynamic local limit graphs. We provide a detailed treatment of the theory of dynamic local convergence, which remains a relatively new topic in the study of random graphs. One main conclusion of our paper is that a specific form of dynamic local convergence is required for our results to hold.

SIR on locally converging dynamic random graphs

TL;DR

The paper develops a rigorous framework for studying SIR epidemics on dynamic random graphs by introducing dynamic local convergence through time-marked union graphs. A central result shows that, under a strengthened dynamic convergence condition, the epidemic on the evolving graph converges to the epidemic on its dynamic local limit, enabling analysis on tractable limiting objects such as time-marked trees. The backward process on the time-marked union graph yields a practical mechanism to compute infection paths and marks, and the authors illustrate the theory with dynamic ER and random-intersection graph models, supported by simulations that demonstrate faster epidemic progression under graph dynamics and strong accuracy of the local-limit approximation. This work extends static local-convergence results to dynamic networks, providing a versatile toolkit for epidemic modelling on evolving contact structures and suggesting future directions including more complex compartments and adaptive rewiring. The results have potential applications in public health policy by enabling efficient, local-structure-based predictions on realistic, time-varying networks.

Abstract

In this paper, we study the trajectory of a classic SIR epidemic on a family of dynamic random graphs of fixed size, whose set of edges continuously evolves over time. We set general infection and recovery times, and start the epidemic from a positive, yet small, proportion of vertices. We show that in such a case, the spread of an infectious disease around a typical individual can be approximated by the spread of the disease in a local neighbourhood of a uniformly chosen vertex. We formalize this by studying general dynamic random graphs that converge dynamically locally in probability and demonstrate that the epidemic on these graphs converges to the epidemic on their dynamic local limit graphs. We provide a detailed treatment of the theory of dynamic local convergence, which remains a relatively new topic in the study of random graphs. One main conclusion of our paper is that a specific form of dynamic local convergence is required for our results to hold.
Paper Structure (38 sections, 11 theorems, 131 equations, 3 figures, 1 algorithm)

This paper contains 38 sections, 11 theorems, 131 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Dynamic random graphs
  3. SIR epidemic model
  4. Informal description of local time-marked union convergence
  5. Main result
  6. Some dynamic random graph models
  7. Examples
  8. Simulation study
  9. Discussion and conclusion
  10. Dynamic local convergence
  11. The metric space of rooted graphs
  12. Local time-marked union convergence
  13. Marked union graph
  14. Local time-marked union convergence
  15. Relationship to dynamic local convergence
  16. ...and 23 more sections

Key Result

Theorem 1.2

Let $(G_n^s)_{s\in[0,T]}$ be a dynamic graph sequence converging in probability in a local time-marked union sense to the limiting time-marked union graph on $\mathcal{G}_{\star}$. Consider an SIR epidemic process on $(G_n^s)_{s\in[0,T]}$, in which every vertex is initially infected independently wi with where $\mu$ on the right-hand side refers to the law of the limiting time-marked union graph,

Figures (3)

  • Figure 1: Comparison of the SIR epidemic time progression on the dynamic Erdős-Rényi random graph with $\gamma=3$ and $n=25 000$ and its dynamic local limit. The parameters $I$ and $R$ represent the rates of $D_I$ and $D_R$, respectively, which are assumed to follow exponential distributions. We have performed $500$ runs of simulations in each case, for various values of $\rho$, $I$ and $R$, explained under the plots.
  • Figure 2: Comparison of the SIR epidemic time progression on static and dynamic Erdős-Rényi random graphs with $\gamma=3$ and $\rho=0.01$. The parameters $I$ and $R$ represent the rates of $D_I$ and $D_R$, respectively, which are assumed to follow exponential distributions. We have performed $500$ runs of simulations for both static and dynamic graphs for various values of $I$ and $R$, explained under the plots.
  • Figure 3: Comparison of the SIR epidemic curve on local limits of static and dynamic random intersection graphs with $\mathbf{E}[D_n]=2.052$ and $\rho=0.1$. The parameters $I$ and $R$ represent the rates of $D_I$ and $D_R$, respectively, which are assumed to follow exponential distributions. We have performed $500$ runs of simulations for both static and dynamic graphs for various values of $I$ and $R$, explained under the plots.

Theorems & Definitions (44)

  • Definition 1.1: Dynamic random graph
  • Theorem 1.2: Dynamic convergence of the epidemic processes
  • Definition 1.3: Dynamic Erdős-Rényi random graph
  • Definition 1.4: Alternative dynamic Erdős-Rényi random graph
  • Definition 1.5: Dynamic random intersection graph
  • Definition 1.6: Configuration model with rewiring of the edges
  • Definition 2.1: Rooted graph, isomorphism and $r$-neighbourhood
  • Definition 2.2: Metric on rooted graphs
  • Definition 2.3: Union graph
  • Definition 2.4: Marked graphs
  • ...and 34 more