Theoretical and Numerical Simulation of a Discrete Fractional Order HIV Model with Latency

Abstract

This paper presents a novel discrete fractional-order mathematical model to describe the dynamics of Human Immunodeficiency Virus (HIV) infection, incorporating the critical role of latent reservoirs in CD4⁺ T-cells. The model captures interactions among healthy T-cells, actively infected cells, latently infected cells, and free virus particles. By employing the Caputo fractional difference operator, the model accounts for memory effects and hereditary properties inherent in biological systems, which are often overlooked in traditional integer-order models. We establish the existence, uniqueness, boundedness, and non-negativity of solutions using Banach's fixed-point theorem and other analytical techniques. The basic reproduction number (R₀) is derived, and local stability conditions for both disease-free and endemic equilibria are determined via linearization. A sensitivity analysis identifies viral production and infection rates as the most influential parameters on R₀. Numerical simulations demonstrate how the fractional order (σ ) modulates the infection dynamics, showing that stronger memory effects (lower σ ) lead to a dampened acute viral burst but a higher chronic viral set point. The results underscore the utility of discrete fractional-order models in capturing key features of HIV progression, including latency and history-dependent behavior.