Time scales theory has been in use since the 1980s with many applications. Only very recently, it has been used to describe within-host and between-hosts dynamics of infectious diseases. In this study, we present explicit and implicit discrete epidemic models motivated by the time scales modeling approach. We use these models to formulate the basic reproduction number, which determines whether an outbreak occurs, or the disease dies out. We discuss the stability of the disease-free and endemic equilibrium points using the linearization method and Lyapunov function. Furthermore, we apply our models to swine flu outbreak data to demonstrate that the discrete models can accurately describe the epidemic dynamics. Our comparison analysis shows that the implicit discrete model can best describe the data regardless of the data frequency. In addition, we perform the sensitivity analysis on the key parameters of the models to study how these parameters impact the basic reproduction number.


Mathematics and Statistics


National Science Foundation, Grant DEB-2030479

Keywords and Phrases

Difference equations; Global stability; Influenza outbreak in rural campus; Local stability; Lyapunov; Positive equilibrium points; SIR epidemic models

International Standard Serial Number (ISSN)

1432-1416; 0303-6812

Document Type

Article - Journal

Document Version


File Type





© 2023 Springer, All rights reserved.

Publication Date

01 Jan 2024