From Agent-Based Markov Dynamics to Hierarchical Closures on Networks: Emergent Complexity and Epidemic Applications

TL;DR

This paper formulates agent-based SIR epidemic models as Markov processes on networks, deriving a hierarchy of equations that elucidate complexity and inform epidemic control strategies, validated through simulations and applied to COVID-19 data.

Contribution

It introduces a rigorous Markovian framework for agent-based epidemic dynamics, deriving hierarchical equations and analyzing systemic complexity and control measures.

Findings

Hierarchy of evolution equations clarifies closure challenges

Monte Carlo simulations validate simplified closures

Network structure influences epidemic propagation and control

Abstract

We explore a rigorous formulation of agent-based SIR epidemic dynamics as a discrete-state Markov process, capturing the stochastic propagation of infection or an invading agent on networks. Using indicator functions and corresponding marginal probabilities, we derive a hierarchy of evolution equations that resembles the classical BBGKY hierarchy in statistical mechanics. The structure of these equations clarifies the challenges of closure and highlights the principal problem of systemic complexity arising from stochastic but generally not fully chaotic interactions. Monte Carlo simulations are used to validate simplified closures and approximations, offering a unified perspective on the interplay between network topology, stochasticity, and infection dynamics. We also explore the impact of lockdown measures within a networked agent framework, illustrating how SIR dynamics and structural complexity of the network shape epidemic with propagation of the COVID-19 pandemic in Northern Italy taken as an example.