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<p>In SIR epidemic models, fractional-order differential equations have recently been employed to describe long-term memory effects and genetic characteristics that are frequently observed in environments but are not adequately represented by traditional integer-order systems. In this study, we analyze the dynamics of a generalized conformable discrete-time SIR epidemic model. We establish the conditions required for the existence and local stability of both the disease-free equilibrium (<i>DFE</i>) and the endemic equilibrium (<i>EE</i>). The analysis of the SIR epidemic model reveals that the disease-free equilibrium is asymptotically stable when the basic reproduction number <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\mathcal{R}_{0}}\)</EquationSource> </InlineEquation> is less than one, while the endemic equilibrium becomes asymptotically stable when <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\({\mathcal{R}_{0}}\)</EquationSource> </InlineEquation> is greater than one. The dynamical behavior of the proposed SIR epidemic model is explored numerically for both commensurate and incommensurate fractional orders, using phase portraits, bifurcation diagrams, the maximum Lyapunov exponent, chaos control, and the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(0-1\)</EquationSource> </InlineEquation> test. The model exhibits more complex dynamics for incommensurate fractional orders compared to commensurate fractional orders. Compared to classical integer-order structures, our fractional formulation offers a more explanation representation of population oscillations and stability transitions. In addition to providing deeper insights into the dynamics of the considered model, these findings highlight the significance of fractional calculus in expanding our understanding of a generalized conformable incommensurate fractional-order modified SIR epidemic model. Finally, MATLAB simulations are conducted to validate the presented results.</p>

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Dynamical analysis of a generalized conformable incommensurate fractional-order modified SIR epidemic model: stability, chaos and \(0-1\) test

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • Messaoud Berkal,
  • M. B. Almatrafi,
  • Billel Semmar,
  • Juan F. Navarro

摘要

In SIR epidemic models, fractional-order differential equations have recently been employed to describe long-term memory effects and genetic characteristics that are frequently observed in environments but are not adequately represented by traditional integer-order systems. In this study, we analyze the dynamics of a generalized conformable discrete-time SIR epidemic model. We establish the conditions required for the existence and local stability of both the disease-free equilibrium (DFE) and the endemic equilibrium (EE). The analysis of the SIR epidemic model reveals that the disease-free equilibrium is asymptotically stable when the basic reproduction number \({\mathcal{R}_{0}}\) is less than one, while the endemic equilibrium becomes asymptotically stable when \({\mathcal{R}_{0}}\) is greater than one. The dynamical behavior of the proposed SIR epidemic model is explored numerically for both commensurate and incommensurate fractional orders, using phase portraits, bifurcation diagrams, the maximum Lyapunov exponent, chaos control, and the \(0-1\) test. The model exhibits more complex dynamics for incommensurate fractional orders compared to commensurate fractional orders. Compared to classical integer-order structures, our fractional formulation offers a more explanation representation of population oscillations and stability transitions. In addition to providing deeper insights into the dynamics of the considered model, these findings highlight the significance of fractional calculus in expanding our understanding of a generalized conformable incommensurate fractional-order modified SIR epidemic model. Finally, MATLAB simulations are conducted to validate the presented results.