SIR dynamics with Vaccination in a large Configuration Model

TL;DR

This paper analyzes a stochastic SIR epidemic model with vaccination on large sparse networks, deriving scaling limits, optimal control strategies, and the impact of network structure on epidemic outcomes.

Contribution

It introduces a framework for optimal vaccination control in large network-based SIR models, linking network topology to vaccination effectiveness.

Findings

Optimal vaccination strategies reduce infection numbers.

Network degree distribution significantly affects epidemic spread.

The model predicts final epidemic size based on network and disease parameters.

Abstract

We consider a SIR model with vaccination strategy on a sparse configuration model random graph. We show the convergence of the system when the number of nodes grows and characterize the scaling limits. Then, we prove the existence of optimal controls for the limiting equations formulated in the framework of game theory, both in the centralized and decentralized setting. We show how the characteristics of the graph (degree distribution) influence the vaccination efficiency for optimal strategies, and we compute the limiting final size of the epidemic depending on the degree distribution of the graph and the parameters of infection, recovery and vaccination. We also present several simulations for two types of vaccination, showing how the optimal controls allow to decrease the number of infections and underlining the crucial role of the network characteristics in the propagation of the disease and the vaccination program.