A discrete-time variable-order fractional compartmental model with time delay for varicella-zoster virus transmission dynamics

Abstract

This paper develops a discrete-time variable-order fractional (VOF) compartmental model for the transmission dynamics of varicella-zoster virus (VZV), incorporating a constant discrete time delay to account for the latent period between exposure and infectiousness. The population is divided into eight epidemiological classes, including multiple infectious stages. Using the Caputo nabla fractional difference (CNFD) of variable order, the model captures memory effects and long-range dependencies inherent in biological systems. We establish the existence and uniqueness of solutions via contraction mapping on a weighted complete metric space. The basic reproduction number \(\mathcal{R}_0\) is derived using the next-generation matrix method, and explicit expressions for the disease-free equilibrium (DFE) and the unique endemic equilibrium (EE) are provided. Stability analysis proves that the DFE is locally and globally Mittag-Leffler stable (LMLS and GMLS) when \(\mathcal{R}_0 < 1\), while the EE is LMLS and GMLS whenever \(\mathcal{R}_0 > 1\). Numerical simulations validate the theoretical results and illustrate the influence of the VOF and the time delay on system dynamics. The proposed framework extends classical stability criteria for integer-order delayed epidemic models to variable-order fractional discrete systems, offering a flexible and biologically faithful tool for understanding VZV transmission and informing public health strategies.