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Keywords = Schur–Cohn stability

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16 pages, 1382 KiB  
Article
Modeling the Transmission Dynamics of Coronavirus Using Nonstandard Finite Difference Scheme
by Ihsan Ullah Khan, Amjid Hussain, Shuo Li and Ali Shokri
Fractal Fract. 2023, 7(6), 451; https://doi.org/10.3390/fractalfract7060451 - 31 May 2023
Cited by 2 | Viewed by 1587
Abstract
A nonlinear mathematical model of COVID-19 containing asymptomatic as well as symptomatic classes of infected individuals is considered and examined in the current paper. The largest eigenvalue of the next-generation matrix known as the reproductive number is obtained for the model, and serves [...] Read more.
A nonlinear mathematical model of COVID-19 containing asymptomatic as well as symptomatic classes of infected individuals is considered and examined in the current paper. The largest eigenvalue of the next-generation matrix known as the reproductive number is obtained for the model, and serves as an epidemic indicator. To better understand the dynamic behavior of the continuous model, the unconditionally stable nonstandard finite difference (NSFD) scheme is constructed. The aim of developing the NSFD scheme for differential equations is its dynamic reliability, which means discretizing the continuous model that retains important dynamic properties such as positivity of solutions and its convergence to equilibria of the continuous model for all finite step sizes. The Schur–Cohn criterion is used to address the local stability of disease-free and endemic equilibria for the NSFD scheme; however, global stability is determined by using Lyapunov function theory. We perform numerical simulations using various values of some key parameters to see more characteristics of the state variables and to support our theoretical findings. The numerical simulations confirm that the discrete NSFD scheme maintains all the dynamic features of the continuous model. Full article
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10 pages, 244 KiB  
Article
Continued Fraction Expansions of Stable Discrete-Time Systems of Difference Equations
by Ziad Zahreddine
Symmetry 2022, 14(6), 1226; https://doi.org/10.3390/sym14061226 - 13 Jun 2022
Cited by 1 | Viewed by 1902
Abstract
We provide a systematic procedure for generating the coefficients of the continued fraction expansion of the test function associated with the characteristic polynomial of a stable system of difference equations. We illustrate the feasibility of the procedure, and we provide an application on [...] Read more.
We provide a systematic procedure for generating the coefficients of the continued fraction expansion of the test function associated with the characteristic polynomial of a stable system of difference equations. We illustrate the feasibility of the procedure, and we provide an application on the stability of two-dimensional digital filters. Full article
(This article belongs to the Topic Dynamical Systems: Theory and Applications)
9 pages, 232 KiB  
Article
Symmetric Properties of Routh–Hurwitz and Schur–Cohn Stability Criteria
by Ziad Zahreddine
Symmetry 2022, 14(3), 603; https://doi.org/10.3390/sym14030603 - 18 Mar 2022
Cited by 5 | Viewed by 3754
Abstract
It is often noticed in the literature that some key results on the stability of discrete-time systems of difference equations are obtained from their corresponding results on the stability of continuous-time systems of differential equations using suitable conformal mappings or bilinear transformations. Such [...] Read more.
It is often noticed in the literature that some key results on the stability of discrete-time systems of difference equations are obtained from their corresponding results on the stability of continuous-time systems of differential equations using suitable conformal mappings or bilinear transformations. Such observations lead to the search for a unified approach to the study of root distribution for real and complex polynomials, with respect to the left-half plane for continuous-time systems (Routh–Hurwitz stability) and with respect to the unit disc for discrete-time systems (Schur–Cohn stability). This paper is a further contribution toward this objective. We present, in a systematic way, the similarities, and yet, the differences between these two types of stability, and we highlight the symmetry that exists between them. We also illustrate how results on the stability of continuous-time systems are conveyed to the stability of discrete-time systems through the proposed techniques. It should be mentioned that the results on Schur–Cohn stability are known to be harder to obtain than Routh–Hurwitz stability ones, giving more credibility to the proposed approach. Full article
(This article belongs to the Topic Dynamical Systems: Theory and Applications)
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