[Paper Review] A geometric analysis of the SIRS epidemiological model on a homogeneous network

Hildeberto Jardón-Kojakhmetov, Christian Kuehn+2 more|arXiv (Cornell University)|Nov 4, 2020

Mathematical and Theoretical Epidemiology and Ecology ModelsMedicine53 references36 citations

TL;DR

This paper applies Geometric Singular Perturbation Theory (GSPT) to a fast-slow SIRS model on homogeneous networks, derived via pair-approximation moment closure. By analyzing the slow manifold and critical manifold with the entry-exit function, it demonstrates that network structure enables stable limit cycles—unlike in homogeneous mixing—under specific conditions (n ∈ [3,5]), revealing complex periodic dynamics absent in standard models.

ABSTRACT

We study a fast-slow version of an SIRS epidemiological model on homogeneous graphs, obtained through the application of the moment closure method. We use GSPT to study the model, taking into account that the infection period is much shorter than the average duration of immunity. We show that the dynamics occurs through a sequence of fast and slow flows, that can be described through 2-dimensional maps that, under some assumptions, can be approximated as 1-dimensional maps. Using this method, together with numerical bifurcation tools, we show that the model can give rise to periodic solutions, differently from the corresponding model based on homogeneous mixing.

Motivation & Objective

• To analyze the SIRS model on networks using geometric singular perturbation theory (GSPT), overcoming limitations of homogeneous mixing.
• To investigate how network structure, particularly homogeneous graphs, alters epidemic dynamics compared to well-mixed populations.
• To identify parameter regimes where periodic solutions emerge due to network-induced time-scale separation.
• To validate analytical findings with numerical bifurcation tools and geometric arguments based on layer and slow flows.
• To extend the entry-exit function method to higher-dimensional non-standard fast-slow systems arising from network moment closure.

Proposed method

• Applies moment closure at the pair level to reduce the infinite-dimensional network ODE system to a tractable ODE model on homogeneous graphs.
• Uses a fast-slow decomposition where infection dynamics are fast (short infectious period) and immunity waning is slow.
• Employs GSPT to analyze the system’s critical manifold and slow flow, identifying regions of phase space where standard two-time-scale structure emerges.
• Utilizes the entry-exicit function to study transitions across the critical manifold, especially where stability changes across a hyperplane.
• Constructs a 2D map via layer and slow flow iterations to approximate periodic orbits, with transverse intersection of entry and exit intervals indicating limit cycles.
• Combines numerical bifurcation analysis with geometric intuition to confirm existence of stable limit cycles in parameter regimes.

Experimental results

Research questions

• RQ1 Can the SIRS model on a homogeneous network exhibit stable periodic solutions, unlike the homogeneous mixing case?
• RQ2 What role does network structure—specifically node degree n—play in enabling complex dynamics such as limit cycles?
• RQ3 How does the entry-exit function behavior on a non-standard fast-slow system with a stability-switching critical manifold affect epidemic persistence?
• RQ4 Under what conditions does the slow flow on the critical manifold lead to periodic orbits via geometric return maps?
• RQ5 Can the geometric argument from 3D systems in prior work be extended to higher-dimensional network-based SIRS models?

Key findings

• The SIRS model on a homogeneous network exhibits stable limit cycles for node degrees n ∈ [3, 5], a regime not observed under homogeneous mixing.
• Numerical bifurcation analysis confirms the existence of a stable limit cycle for β ≈ 2, with periodic orbits emerging when the entry and exit intervals in the geometric map intersect transversally.
• The critical manifold exhibits a stability switch across a hyperplane, enabling the entry-exit mechanism essential for periodic orbit formation.
• For n ≥ 6 or n ≤ 2, the system converges globally to the endemic equilibrium, indicating a sharp threshold in network structure for oscillatory behavior.
• The geometric method successfully predicts periodic dynamics without integrating the full stiff system, relying only on layer and slow flow solutions.
• The analysis confirms that network structure fundamentally alters epidemic dynamics, enabling persistent oscillations even when the well-mixed model only converges to equilibrium.