Analysis of a fractional-order for Zika virus transmission with waning immunity and extrinsic incubation: Equilibria, stability, and numerical simulations

Abstract

This paper presents a six-compartment fractional-order mathematical model for the transmission dynamics of Zika virus (ZIKV), formulated as an SIR--SEI system governed by the Caputo fractional derivative of order $\rho\in(0,1]$ with an auxiliary dimensional parameter $\psi$. The human population is partitioned into susceptible, infectious, and recovered classes, while the mosquito population is partitioned into susceptible, exposed, and infectious classes, thereby incorporating both a temporary human immunity period and the extrinsic incubation period of ZIKV in \textit{Aedes aegypti}. Transmission occurs through direct human-to-human sexual contact and through infectious mosquito bites, with the mosquito population governed by a biologically consistent natural death rate corresponding to a 14-day mean lifespan. The basic reproduction number $\mathcal{R}_0$ is derived analytically via the next-generation matrix method, yielding a closed-form threshold that separates the contributions of direct sexual transmission and the human--mosquito--human vector-borne cycle; the extrinsic incubation period enters through the survival probability factor $\sigma_v/(\sigma_v+k_2)$. Positivity and uniform boundedness of all solutions are established using the fractional comparison principle and Mittag-Leffler function estimates, and the feasible region is shown to be positively invariant. Existence and uniqueness of the solution are proved via Lipschitz continuity of the kernel functions and a Volterra integral reformulation of the system. The disease-free equilibrium is shown to be locally asymptotically stable if and only if $\mathcal{R}_0<1$, and unstable when $\mathcal{R}_0>1$. A unique endemic equilibrium exists for $\mathcal{R}_0>1$ and is shown to be locally asymptotically stable, with a spiral-sink structure arising from complex conjugate eigenvalues of the Jacobian. Numerical simulations are carried out using the Adams--Bashforth--Moulton predictor-corrector scheme for five fractional orders $\rho\in\{1.00,\,0.95,\,0.90,\,0.85,\,0.80\}$ under both endemic ($\mathcal{R}_0=2.0$) and disease-free ($\mathcal{R}_0=0.042$) parameter regimes. The results demonstrate the fractional memory effect: decreasing $\rho$ slows convergence to equilibrium algebraically rather than exponentially, without altering the equilibrium values. In the endemic scenario, damped oscillations of period approximately 260 days are observed, driven by the waning-immunity timescale and the complex eigenstructure of the endemic equilibrium.