Using asymptotics for efficient stability determination in epidemiological models

Area of Science:

• Dynamical Systems Theory
• Mathematical Epidemiology
• Computational Mathematics

Background:

• Local stability analysis is crucial for understanding dynamical systems.
• Traditional methods (Routh-Hurwitz) become computationally intensive for systems with >3 components.
• Parameter-dependent stability analysis requires methods that avoid explicit value substitution.

Purpose of the Study:

• To develop and present efficient methods for local stability analysis of dynamical systems.
• To address the computational challenges of analyzing larger systems (4-6 components).
• To provide tools and guidelines for applying asymptotic approximation in stability analysis.

Main Methods:

• Utilized asymptotic approximation, leveraging small parameters common in epidemiological models (ratio of timescales).
• Developed general tools and guidelines for applying this simplification method.
• Demonstrated the approach through two case studies in epidemiological modeling.

Main Results:

• The proposed asymptotic approximation significantly simplifies stability analysis for larger systems.
• The method is efficient and introduces minimal cost in terms of accuracy.
• Provided practical examples showcasing the effectiveness of the described tools and guidelines.

Conclusions:

• Asymptotic approximation offers a computationally feasible alternative for stability analysis in complex dynamical systems.
• The presented methodology is particularly beneficial for epidemiological models with disparate timescales.
• This work provides a valuable framework for researchers needing to perform parameter-dependent stability analyses efficiently.