Structure-Preserving Schemes for a Fractional SVIR Epidemic Model with a Hybrid Mittag-Leffler-Caputo-Fabrizio Operator

Abstract: This paper proposes and analyzes a fractional-order SVIR epidemic model based on a hybrid Mittag-Leffler-Caputo-Fabrizio (MLCF) fractional operator with a nonsingular kernel. This model captures short- and long-term memory effects in epidemic transmission dynamics. The positivity and boundedness of the solutions are proven through an integrated formulation of the MLCF operator and a fractional Gronwall inequality. The basic reproduction number $\mathcal{R}_0$, equilibrium points, and their local and global stability properties are rigorously investigated through Jacobian analysis, logarithmic Lyapunov functionals, and a fractional LaSalle invariance principle. To approximate the model, a $θ$-weighted nonstandard finite difference (NSFD) method is developed. This method preserves the continuous system's key qualitative properties, including positivity and boundedness, and is unconditionally stable in the fully implicit case. Consistency and first-order convergence are also proven. Numerical experiments, together with sensitivity and bifurcation analyses, illustrate the impact of fractional memory parameters on epidemic evolution and demonstrate the effectiveness of the proposed approach.

• The paper develops a hybrid Mittag-Leffler-Caputo-Fabrizio SVIR model and proves positivity, boundedness, and global stability, with the classical epidemic threshold governing disease persistence.
• The proposed fully implicit θ-weighted nonstandard finite-difference scheme preserves non-negativity and population bounds, achieves first-order convergence, and remains unconditionally stable for real non-positive spectra.
• Sensitivity and bifurcation analyses show that fractional orders mainly change transient dynamics while transmission dominates infection levels, although the kernel-order effect at high transmission rates remains unresolved.

Overview

This paper develops and analyzes a fractional-order SVIR (susceptible–vaccinated–infected–recovered) epidemic model driven by a hybrid Mittag-Leffler-Caputo-Fabrizio (MLCF) operator (2606.19045). The MLCF derivative combines the non-singular exponential kernel of Caputo-Fabrizio with Mittag-Leffler-type memory, parameterized by two orders: $\nu \in (0,1)$ controlling the memory kernel strength and $\alpha \in (0,1]$ governing the Mittag-Leffler kernel. For $\alpha = 1$ the operator reduces to the classical Caputo-Fabrizio derivative, so the framework strictly generalizes existing non-singular fractional epidemic models while capturing both short- and long-term memory effects within a single formulation.

The paper makes three contributions: a complete qualitative analysis of the continuous model (positivity, boundedness, equilibria, local and global stability), a structure-preserving $\theta$-weighted nonstandard finite difference (NSFD) scheme for the MLCF operator with proven positivity, boundedness, unconditional stability ($\theta=1$), and first-order convergence, and numerical evidence—sensitivity and bifurcation analyses—demonstrating that the fractional orders affect transient dynamics but not the epidemic threshold.

The hybrid MLCF operator and model formulation

The MLCF derivative of $u \in H^1(a,b)$ is defined via an integral against the kernel $$E_\alpha\!\bigl(\tfrac{\nu}{1-\nu}\kappa^\alpha\bigr)$$, where $$E_\alpha(z) = \sum_{k\ge0} z^k/\Gamma(\alpha k + 1)$$ is the one-parameter Mittag-Leffler function, with normalization $M(\nu)$ satisfying $M(0)=M(1)=1$. The key analytical tool is an integrated form of the operator: writing $\alpha \in (0,1]$0 and $\alpha \in (0,1]$1 with $\alpha \in (0,1]$2, any solution satisfies

$\alpha \in (0,1]$3

with $\alpha \in (0,1]$4, $\alpha \in (0,1]$5, and $\alpha \in (0,1]$6 following from complete monotonicity of $\alpha \in (0,1]$7. This representation is the workhorse for both the positivity proof (via a generalized Gronwall inequality) and the discrete analysis.

The SVIR system extends an integer-order COVID-19-inspired model with vaccination rate $\alpha \in (0,1]$8, vaccine effectiveness $\alpha \in (0,1]$9, disease mortality $\alpha = 1$0, and immunity loss $\alpha = 1$1. The total population satisfies $\alpha = 1$2.

Qualitative analysis

Positivity and boundedness are established using the integrated form together with a comparison principle and a fractional Gronwall inequality; each compartment remains non-negative for non-negative initial data, and $\alpha = 1$3.

Threshold dynamics. Via the next-generation matrix method,

$\alpha = 1$4

Notably, $\alpha = 1$5 is independent of both fractional orders $\alpha = 1$6 and $\alpha = 1$7—a structural consequence of the fact that equilibria of the MLCF system coincide with those of the integer-order system.

Local stability. Using the Matignon criterion $\alpha = 1$8 applied to the Jacobian eigenvalues, the disease-free equilibrium is locally asymptotically stable for $\alpha = 1$9 and unstable for $\theta$0; all eigenvalues at the DFE are real, so the criterion holds trivially for any $\theta$1. At the endemic equilibrium, the Routh-Hurwitz conditions on the characteristic polynomial are verified explicitly, with $\theta$2 proportional to $\theta$3, yielding local stability for $\theta$4.

Global stability. Logarithmic Lyapunov functionals combined with a chain-rule inequality for the MLCF derivative and a fractional LaSalle invariance principle establish global asymptotic stability of the DFE for $\theta$5 on the invariant region $\theta$6 and of the endemic equilibrium for $\theta$7 on $\theta$8. The central implication is that classical threshold behavior survives intact under the hybrid memory operator: memory alters convergence rates, not stability type or bifurcation location.

One caveat: the paper's chain-rule inequality proof is deferred by reference rather than given in full, and the LaSalle principle is invoked from prior work on related operators; the transfer of these tools to the MLCF setting rests on the completely-positive-kernel property being argued rather than axiomatized.

The $\theta$9-weighted NSFD scheme

The MLCF derivative is discretized with a two-point forward difference and trapezoidal quadrature of the Mittag-Leffler kernel, producing weights $\theta=1$0 and $\theta=1$1, plus a recursive memory accumulator $\theta=1$2 that keeps per-step cost linear in time. Mickens's denominator function $\theta=1$3 replaces $\theta=1$4, and the right-hand side is $\theta=1$5-weighted between implicit and explicit evaluations; the resulting nonlinear system is solved by Newton iteration (typically 3–5 iterations).

Three properties are proven rigorously:

• Positivity: an induction argument built on the telescoping identity $\theta=1$6 shows every compartment stays non-negative.
• Boundedness: the same identity applied to $\theta=1$7 yields $\theta=1$8, mirroring the continuous bound exactly.
• Unconditional stability and first-order convergence: on the Dahlquist test equation with $\theta=1$9, the amplification factor $u \in H^1(a,b)$0 for $u \in H^1(a,b)$1, giving unconditional stability; truncation error analysis gives $u \in H^1(a,b)$2 consistency, and a Gronwall-type induction under Lipschitz continuity of $u \in H^1(a,b)$3 establishes uniform convergence $u \in H^1(a,b)$4.

The convergence proof contains a step the authors acknowledge as incomplete: after deriving the error recursion, they note it "can be used to prove a Gronwall type inequality" but instead invoke a standard result for one-step methods with memory, so the final convergence theorem leans on that stability-plus-consistency shortcut rather than closing the Gronwall estimate directly. Additionally, the unconditional stability claim is restricted to real non-positive eigenvalues of the linearization, which is argued to hold "for typical epidemic models" but is not proven for the full nonlinear system.

Sensitivity and bifurcation results

Local sensitivity (1% perturbations) and global sensitivity via PRCC with Latin Hypercube Sampling ($u \in H^1(a,b)$5 over $u \in H^1(a,b)$6, $u \in H^1(a,b)$7, $u \in H^1(a,b)$8) yield a consistent picture: $u \in H^1(a,b)$9 dominates the long-term infected level, with strong positive PRCC, while $$E_\alpha\!\bigl(\tfrac{\nu}{1-\nu}\kappa^\alpha\bigr)$$0 and $$E_\alpha\!\bigl(\tfrac{\nu}{1-\nu}\kappa^\alpha\bigr)$$1 have PRCC values near zero—quantitatively confirming that memory parameters do not affect the endemic state.

Bifurcation diagrams in $$E_\alpha\!\bigl(\tfrac{\nu}{1-\nu}\kappa^\alpha\bigr)$$2 exhibit a transcritical bifurcation at $$E_\alpha\!\bigl(\tfrac{\nu}{1-\nu}\kappa^\alpha\bigr)$$3 whose location is unchanged by the fractional orders; only curve steepness varies. Two observations qualify the clean "memory affects transients only" narrative:

• For larger $$E_\alpha\!\bigl(\tfrac{\nu}{1-\nu}\kappa^\alpha\bigr)$$4 (approaching 1), simulations show higher infection levels and stronger oscillations around the endemic state, indicating that near the integer-order limit the dynamics become more sensitive to $$E_\alpha\!\bigl(\tfrac{\nu}{1-\nu}\kappa^\alpha\bigr)$$5.
• In contrast to $$E_\alpha\!\bigl(\tfrac{\nu}{1-\nu}\kappa^\alpha\bigr)$$6, the kernel order $$E_\alpha\!\bigl(\tfrac{\nu}{1-\nu}\kappa^\alpha\bigr)$$7 exerts a mild but noticeable influence on the steady state itself for large $$E_\alpha\!\bigl(\tfrac{\nu}{1-\nu}\kappa^\alpha\bigr)$$8: smaller $$E_\alpha\!\bigl(\tfrac{\nu}{1-\nu}\kappa^\alpha\bigr)$$9 slightly reduces $$E_\alpha(z) = \sum_{k\ge0} z^k/\Gamma(\alpha k + 1)$$0. This partially contradicts the theoretical claim that equilibria are independent of the fractional parameters and suggests the observed effect may be a finite-time artifact of slow algebraic convergence rather than a true steady-state shift—the paper does not resolve this discrepancy.

Stability-domain computations confirm that $$E_\alpha(z) = \sum_{k\ge0} z^k/\Gamma(\alpha k + 1)$$1 yields unconditional stability while the region shrinks rapidly as $$E_\alpha(z) = \sum_{k\ge0} z^k/\Gamma(\alpha k + 1)$$2, justifying the fully implicit choice. A practical limitation is noted: the Mittag-Leffler series evaluation overflows for small $$E_\alpha(z) = \sum_{k\ge0} z^k/\Gamma(\alpha k + 1)$$3 at long horizons, restricting the sensitivity study to $$E_\alpha(z) = \sum_{k\ge0} z^k/\Gamma(\alpha k + 1)$$4 and issuing warnings for $$E_\alpha(z) = \sum_{k\ge0} z^k/\Gamma(\alpha k + 1)$$5.

Limitations and open questions

Several points remain open. First, the mild dependence of the computed endemic level on $$E_\alpha(z) = \sum_{k\ge0} z^k/\Gamma(\alpha k + 1)$$6 at large $$E_\alpha(z) = \sum_{k\ge0} z^k/\Gamma(\alpha k + 1)$$7 conflicts with the exact independence of equilibria from fractional orders and warrants investigation into whether very slow Mittag-Leffler convergence contaminates long-time asymptotics. Second, the convergence proof relies on a generic one-step-with-memory result rather than completing the derived Gronwall estimate, and unconditional stability is established only for real non-positive spectra. Third, the global sensitivity ranges exclude $$E_\alpha(z) = \sum_{k\ge0} z^k/\Gamma(\alpha k + 1)$$8 due to numerical overflow of the Mittag-Leffler series, leaving the strongly memory-dominated regime unexplored computationally. Finally, whether the transcritical bifurcation can degenerate (e.g., backward bifurcation) under modified model structures within the MLCF framework is not addressed.

Conclusion

The paper provides a self-contained treatment of a hybrid-memory fractional SVIR model: rigorous well-posedness, full threshold and stability theory showing that $$E_\alpha(z) = \sum_{k\ge0} z^k/\Gamma(\alpha k + 1)$$9 retains its classical role, and a provably structure-preserving NSFD scheme with linear-cost memory updates and unconditional stability in the fully implicit case. The numerical studies substantiate the main qualitative claim—that fractional memory reshapes transient epidemic behavior without altering the persistence threshold—while leaving the quantitative role of the kernel order $M(\nu)$0 at large transmission rates as the most concrete open question.