SIR on locally converging dynamic random graphs (2501.09623v1)

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.

• The paper introduces dynamic local convergence, enabling local graph properties to approximate epidemic trajectories.
• It leverages dynamic random intersection and marked union graphs to realistically simulate evolving social contacts.
• Simulation studies validate the framework, showing improved prediction of disease spread in time-varying networks.

An Exploration of Dynamic Random Intersection Graphs and Their Applications in Epidemic Modelling

The paper entitled "Dynamic random intersection graph: Dynamic local convergence and giant structure," authored by Milewska, van der Hofstad, and Zwart, makes a substantial contribution to the understanding of epidemic spread on dynamic networks. It explores Susceptible-Infected-Recovered (SIR) epidemic models on dynamic random graphs, where edges evolve continuously over time, reflecting real-world social networks more accurately than static models.

Overview

The authors examine the dynamic local convergence of random graphs, highlighting its relevance in approximating epidemic trajectories in networks with changing structures. A particular focus is given to dynamic random intersection graphs, where connections between vertices are influenced by evolving group memberships, simulating a more realistic social contact pattern.

The paper builds upon the established concept of static local convergence and extends it to a dynamic context, offering a new perspective on how local graph properties influence global epidemic dynamics. The notion of dynamic local convergence is a relatively recent development, here refined by considering the full trajectory of vertex interactions over time, captured in union graphs with detailed temporal marks.

Main Contributions

1. Dynamic Local Convergence: The study introduces and characterizes a form of convergence specific to dynamic graphs—dynamic local convergence—ensuring that local neighborhood structures in these graphs converge to a limit. This aspect is pivotal in approximating epidemic processes on dynamic networks through their limits.
2. Application to Epidemic Modelling: By leveraging the dynamic local convergence, the paper establishes that the epidemic spread on dynamic random graphs approximates the spread on their local limits. This insight offers a powerful technique for simplifying complex epidemic simulations on dynamic networks by substituting them with their tractable local limits.
3. Marked Union Graphs: It introduces marked union graphs as a tool to capture the cumulative connectivity over a specified time frame, including ON/OFF states and their temporal sequences. This innovation addresses the challenges posed by the dynamic nature of social contacts, which static models fail to consider.
4. Simulation Studies: The paper supplements theoretical findings with comprehensive simulation outcomes. These simulations validate the theoretical assumptions and demonstrate the practicality of modeling epidemics using the dynamic local limit approximation.

Implications and Future Directions

The implications of this research are significant for both theoretical and practical domains of network science and epidemiology. The dynamic graph approach offers a more nuanced understanding of disease dynamics, improving predictions and informing public health strategies, especially as societies become increasingly interconnected.

For future directions, the framework can be extended to consider other network types and additional complexities such as demographic factors, prevention strategies, and adaptive behaviors during epidemics. Additionally, incorporating other disease models like SEIR (Susceptible-Exposed-Infected-Recovered) could further enhance the applicability of the model.

In summary, this work bridges a critical gap in epidemic modeling using dynamic graphs, providing a robust methodological foundation for understanding how local interactions in time-varying networks affect the overall course of epidemics. Researchers are encouraged to build upon these findings to explore more intricate network structures and apply these insights to real-world epidemic data.