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Volume 14
Issue 6
10.3390/axioms14060470
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Article

Asymptotical Behavior of Impulsive Linearly Implicit Euler Method for the SIR Epidemic Model with Nonlinear Incidence Rates and Proportional Impulsive Vaccination

by
Zhi-Wei Xu
and
Gui-Lai Zhang
*
School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China
*
Author to whom correspondence should be addressed.
Axioms 2025, 14(6), 470; https://doi.org/10.3390/axioms14060470
Submission received: 7 May 2025 / Revised: 4 June 2025 / Accepted: 14 June 2025 / Published: 16 June 2025
(This article belongs to the Special Issue Differential Equations and Inverse Problems, 2nd Edition)
Versions Notes

Abstract

This paper is concerned with the asymptotical behavior of the impulsive linearly implicit Euler method for the SIR epidemic model with nonlinear incidence rates and proportional impulsive vaccination. We point out the solution of the impulsive linearly implicit Euler method for the impulsive SIR system is positive for arbitrary step size when the initial values are positive. By applying discrete Floquet’s theorem and small-amplitude perturbation skills, we proved that the disease-free periodic solution of the impulsive system is locally stable. Additionally, in conjunction with the discrete impulsive comparison theorem, we show that the impulsive linearly implicit Euler method maintains the global asymptotical stability of the exact solution of the impulsive system. Two numerical examples are provided to illustrate the correctness of the results.
Keywords: impulsive linearly implicit Euler method; local stability; global asymptotical stability; disease-free periodic solution impulsive linearly implicit Euler method; local stability; global asymptotical stability; disease-free periodic solution

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MDPI and ACS Style

Xu, Z.-W.; Zhang, G.-L. Asymptotical Behavior of Impulsive Linearly Implicit Euler Method for the SIR Epidemic Model with Nonlinear Incidence Rates and Proportional Impulsive Vaccination. Axioms 2025, 14, 470. https://doi.org/10.3390/axioms14060470

AMA Style

Xu Z-W, Zhang G-L. Asymptotical Behavior of Impulsive Linearly Implicit Euler Method for the SIR Epidemic Model with Nonlinear Incidence Rates and Proportional Impulsive Vaccination. Axioms. 2025; 14(6):470. https://doi.org/10.3390/axioms14060470

Chicago/Turabian Style

Xu, Zhi-Wei, and Gui-Lai Zhang. 2025. "Asymptotical Behavior of Impulsive Linearly Implicit Euler Method for the SIR Epidemic Model with Nonlinear Incidence Rates and Proportional Impulsive Vaccination" Axioms 14, no. 6: 470. https://doi.org/10.3390/axioms14060470

APA Style

Xu, Z.-W., & Zhang, G.-L. (2025). Asymptotical Behavior of Impulsive Linearly Implicit Euler Method for the SIR Epidemic Model with Nonlinear Incidence Rates and Proportional Impulsive Vaccination. Axioms, 14(6), 470. https://doi.org/10.3390/axioms14060470

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