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

Hiroshi Ito

doi:10.3934/dcdsb.2020338

Article Contents

2021, Volume 26, Issue 9: 5171-5196. Doi: 10.3934/dcdsb.2020338

This issue Previous Article Recurrent solutions of the Schrödinger-KdV system with boundary forces Next Article

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

Hiroshi Ito

Department of Intelligent and Control Systems, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka 820-8502, Japan

Received: April 30, 2020

Revised: August 31, 2020

Early access: November 2020

Published: September 2021

The author is supported by JSPS KAKENHI Grant Number 20K04536

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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.

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Mathematics Subject Classification: Primary: 93D30, 93C10, 93D09; Secondary: 92D25, 34D23.

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Figure 1. Level sets of the ISS Lyapunov function (22) for the disease-free equilibrium with $ \hat{B} = 3 $ (Dash lines); The arrows are segments of trajectories of (8) for $ B(t) = \hat{B} $; The dotted line is $ S = \hat{x}_1 $

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Figure 2. Level sets of the ISS Lyapunov function (37) for the endemic equilibrium with $ \hat{B} = 17 $ (Dash lines); The arrows are segments of trajectories of (8) for $ B(t) = \hat{B} $; The dotted lines are $ S = \hat{x}_1 $, $ I = \hat{x}_2 $ and $ SI = \hat{x}_1\hat{x}_2 $; The lower left area along $ S $-axis cannot be filled with sublevel sets of any Lyapunov functions

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Figure 3. Obstacles in constructing a strict Lyapunov function in terms of level sets: The lines and the arrows are segments of level sets and trajectories, respectively

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