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

Area of Science:

• Epidemiology
• Statistical Mechanics
• Network Science

Background:

• Agent-based SIR models capture disease spread.
• Stochastic interactions in networks present complexity challenges.
• Understanding epidemic dynamics requires accounting for network topology.

Purpose of the Study:

• To rigorously formulate agent-based SIR epidemic dynamics as a discrete-state Markov process.
• To derive and analyze a hierarchy of evolution equations for stochastic network propagation.
• To investigate the impact of network structure, stochasticity, and interventions on epidemic dynamics.

Main Methods:

• Formulation of SIR dynamics as a discrete-state Markov process.
• Derivation of evolution equations using indicator functions and marginal probabilities.
• Monte Carlo simulations for validating closures and approximations.
• Analysis of epidemic spread on networks with a focus on COVID-19 in Northern Italy.

Main Results:

• A hierarchy of evolution equations analogous to the BBGKY hierarchy was derived.
• Challenges in achieving closure and the problem of systemic complexity were clarified.
• The interplay of network topology, stochasticity, and infection dynamics was explored.
• The influence of lockdown measures on SIR dynamics within networked agents was illustrated.

Conclusions:

• The derived framework provides a unified perspective on epidemic dynamics in networks.
• The study highlights the critical role of network structure and stochasticity in shaping epidemic outcomes.
• The model offers insights into the effectiveness of interventions like lockdowns in complex network environments.