Input-to-state stability and Lyapunov functions with explicit domains for SIR model of infectious diseases

TL;DR

This paper establishes input-to-state stability for the SIR infectious disease model using explicitly constructed Lyapunov functions, providing geometric insights and removing the need for traditional invariance principles.

Contribution

It introduces explicit Lyapunov functions with domains that verify ISS for both disease-free and endemic equilibria in the SIR model, enhancing robustness analysis.

Findings

ISS of the SIR model is proven with explicit Lyapunov functions.

Level sets of Lyapunov functions reflect trajectory flow.

The approach avoids LaSalle's invariance principle.

Abstract

This paper demonstrates input-to-state stability (ISS) of the SIR model of infectious diseases with respect to the disease-free equilibrium and the endemic equilibrium. Lyapunov functions are constructed to verify that both equilibria are individually robust with respect to perturbation of newborn/immigration rate which determines the eventual state of populations in epidemics. The construction and analysis are geometric and global in the space of the populations. In addition to the establishment of ISS, this paper shows how explicitly the constructed level sets reflect the flow of trajectories. Essential obstacles and keys for the construction of Lyapunov functions are elucidated. The proposed Lyapunov functions which have strictly negative derivative allow us to not only establish ISS, but also get rid of the use of LaSalle's invariance principle and popular simplifying assumptions.