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<p>This article develops and analyzes a fractional-order SEIJR epidemic model incorporating the Caputo derivative to account for memory effects in disease transmission. The population is divided into susceptible, exposed, two infectious classes and recovered individuals, with infection dynamics governed by a generalized nonlinear incidence function reflecting heterogeneous infectiousness. We first establish the mathematical well-posedness of the model by proving the existence, uniqueness, positivity, and boundedness of solutions under biologically feasible conditions, ensuring that the system is mathematically consistent and biologically meaningful. Equilibrium analysis establishes both disease-free and endemic states. The basic reproduction number <i>R</i><sub>0</sub> is derived using the next-generation matrix approach and serves as the key threshold parameter. When <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(R_0 &lt; 1\)</EquationSource> </InlineEquation>, the disease-free equilibrium is locally and globally asymptotically stable, ensuring the eradication of the disease. In contrast, when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(R_0 &gt; 1\)</EquationSource> </InlineEquation> and under suitable conditions, the endemic equilibrium is locally and globally asymptotically stable, capturing the persistence of infection. These results are rigorously proven using the Routh-Hurwitz criteria, Lyapunov functions, LaSalles invariance principle, and fractional calculus lemmas. Numerical simulations complement the theoretical findings: the Taylor approach is employed to illustrate local asymptotic stability, while the Grünwald-Letnikov method adapted to Caputo derivatives validates global stability and highlights the role of memory effects. Sensitivity analysis further reveals that transmission and inflow rates strongly increase <i>R</i><sub>0</sub>, while recovery and natural death rates reduce it. Control strategies, derived from Pontryagins Maximum Principle and implemented using the fractional Euler scheme combined with the forward–backward sweep method, are evaluated by cost-effectiveness analysis, showing that the joint application of vaccination, treatment, and isolation measures is both epidemiologically dominant and economically optimal.</p>

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Optimal control and stability investigation of a fractional SEIJR approach with a generalized nonlinear transmission rate

  • 期刊论文

  • 发表日期：

    2025年11月28日

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  • Ahmed Jidou El Bechir,
  • Mohamed Saad Elemine Vall,
  • Yahya Mohamed

摘要

This article develops and analyzes a fractional-order SEIJR epidemic model incorporating the Caputo derivative to account for memory effects in disease transmission. The population is divided into susceptible, exposed, two infectious classes and recovered individuals, with infection dynamics governed by a generalized nonlinear incidence function reflecting heterogeneous infectiousness. We first establish the mathematical well-posedness of the model by proving the existence, uniqueness, positivity, and boundedness of solutions under biologically feasible conditions, ensuring that the system is mathematically consistent and biologically meaningful. Equilibrium analysis establishes both disease-free and endemic states. The basic reproduction number R0 is derived using the next-generation matrix approach and serves as the key threshold parameter. When \(R_0 < 1\) , the disease-free equilibrium is locally and globally asymptotically stable, ensuring the eradication of the disease. In contrast, when \(R_0 > 1\) and under suitable conditions, the endemic equilibrium is locally and globally asymptotically stable, capturing the persistence of infection. These results are rigorously proven using the Routh-Hurwitz criteria, Lyapunov functions, LaSalles invariance principle, and fractional calculus lemmas. Numerical simulations complement the theoretical findings: the Taylor approach is employed to illustrate local asymptotic stability, while the Grünwald-Letnikov method adapted to Caputo derivatives validates global stability and highlights the role of memory effects. Sensitivity analysis further reveals that transmission and inflow rates strongly increase R0, while recovery and natural death rates reduce it. Control strategies, derived from Pontryagins Maximum Principle and implemented using the fractional Euler scheme combined with the forward–backward sweep method, are evaluated by cost-effectiveness analysis, showing that the joint application of vaccination, treatment, and isolation measures is both epidemiologically dominant and economically optimal.