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

## TL;DR

The paper studies how epidemics spread through networks using mathematical models and simulations, focusing on complexity and lockdown effects.

## Contribution

A novel hierarchical closure framework for agent-based epidemic dynamics on networks is introduced, bridging statistical mechanics and stochastic processes.

## Key findings

- Hierarchical evolution equations for SIR dynamics on networks were derived, similar to the BBGKY hierarchy.
- Monte Carlo simulations validated simplified closures and highlighted the role of network topology in epidemic spread.
- Lockdown measures were shown to significantly alter epidemic dynamics in networked populations, as demonstrated with the Northern Italy example.

## 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.