In this research, a three-dimensional non-linear dynamical behavior of the COVID-19 pandemic model is described using fractional calculus. This model was presented by using the Caputo operators. Actually, a three-compartment fractional-order model of COVID-19 is considered, where A(t) denotes newly reported cases, B(t) represents severe infections, and C(t) represents daily deaths. The model is formulated using the Caputo derivative of fractional order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((\eta \in (0,1])\)</EquationSource> </InlineEquation>, which incorporates memory effects and allows us to capture delays and long-term persistence in epidemic dynamics. Numerical experiments show that reducing <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((\eta )\)</EquationSource> </InlineEquation> strengthens memory and can suppress oscillations or chaotic outbreaks. Parameter values and initial conditions are obtained from publicly available COVID-19 data (Johns Hopkins CSSE database and National Health Commission of China), ensuring biological relevance of the simulations. We investigated the existence and uniqueness of the COVID-19 pandemic model and the conditions under which it provides the uniqueness of the solution. We calculated the equilibrium points and exhibited the stability of the suggested model. Also, for numerical simulation, the Toufik-Atangana method is used. The Toufik–Atangana scheme is adopted because it employs piecewise linear interpolation that ensures stability and accuracy for stiff, nonlinear fractional-order systems, while preserving the memory effect inherent in the Caputo derivative. Its proven effectiveness in fractional epidemic and ecological models makes it particularly suitable for capturing the complex dynamics of COVID-19. The simulated results are analyzed and interpreted in detail, and show that the proposed approaches are innovative and trustworthy. The approach is innovative as it integrates fractional modeling with Lyapunov stability, bifurcation, and chaos analysis within a single framework, and trustworthy because it reliably captures both stable and unstable dynamics of the system without numerical divergence or instability. The approach is computationally efficient and straightforward to implement. In this work, numerical robustness highlights the ability of this scheme to generate stable and accurate solutions of the COVID-19 model even when the system shows nonlinear or chaotic behavior. Bifurcation analysis and graphic representations offer more insight into the suggested model, in addition to identifying the direction, stability, and bifurcation of the COVID-19 model.

Numerical robustness of COVID-19 model with Lyapunov stability analysis

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2026年2月6日

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Ambika Pandey,
Surath Ghosh

摘要

In this research, a three-dimensional non-linear dynamical behavior of the COVID-19 pandemic model is described using fractional calculus. This model was presented by using the Caputo operators. Actually, a three-compartment fractional-order model of COVID-19 is considered, where A(t) denotes newly reported cases, B(t) represents severe infections, and C(t) represents daily deaths. The model is formulated using the Caputo derivative of fractional order \((\eta \in (0,1])\) , which incorporates memory effects and allows us to capture delays and long-term persistence in epidemic dynamics. Numerical experiments show that reducing \((\eta )\) strengthens memory and can suppress oscillations or chaotic outbreaks. Parameter values and initial conditions are obtained from publicly available COVID-19 data (Johns Hopkins CSSE database and National Health Commission of China), ensuring biological relevance of the simulations. We investigated the existence and uniqueness of the COVID-19 pandemic model and the conditions under which it provides the uniqueness of the solution. We calculated the equilibrium points and exhibited the stability of the suggested model. Also, for numerical simulation, the Toufik-Atangana method is used. The Toufik–Atangana scheme is adopted because it employs piecewise linear interpolation that ensures stability and accuracy for stiff, nonlinear fractional-order systems, while preserving the memory effect inherent in the Caputo derivative. Its proven effectiveness in fractional epidemic and ecological models makes it particularly suitable for capturing the complex dynamics of COVID-19. The simulated results are analyzed and interpreted in detail, and show that the proposed approaches are innovative and trustworthy. The approach is innovative as it integrates fractional modeling with Lyapunov stability, bifurcation, and chaos analysis within a single framework, and trustworthy because it reliably captures both stable and unstable dynamics of the system without numerical divergence or instability. The approach is computationally efficient and straightforward to implement. In this work, numerical robustness highlights the ability of this scheme to generate stable and accurate solutions of the COVID-19 model even when the system shows nonlinear or chaotic behavior. Bifurcation analysis and graphic representations offer more insight into the suggested model, in addition to identifying the direction, stability, and bifurcation of the COVID-19 model.