This paper extends a novel piecewise mathematical model of the COVID-19 epidemic using fractional and variable-order differential equations and fractional stochastic derivatives in three intervals of time. The deterministic models are augmented with hybrid fractional order and variable order operators, while the stochastic differential equations incorporate fractional Brownian motion. To probe the behavior of the proposed models, we introduce two numerical techniques: the nonstandard modified Euler Maruyama method for the fractional stochastic model, and the Caputo proportional constant-Grünwald-Letnikov nonstandard finite difference method for the fractional and variable-order deterministic models. Several numerical experiments corroborate the theoretical assertions and demonstrate the efficacy of the proposed approaches.

In the current study, we present a mathematical model for human immunodeficiency virus type-1 (HIV-1) transmission, incorporating Cytotoxic T-Lymphocyte immune impairment within a seasonal environment. The model divides the infected cell compartment into two sub-compartments: latently infected cells and productively infected cells. Additionally, we consider three possible routes of infection, allowing HIV to spread among susceptible cells via direct contact with the virus, latently infected cells, or productively infected cells. The system is analyzed, and the basic reproduction number is derived using an integral operator. We demonstrate that the HIV-free periodic trajectory is globally asymptotically stable if ${\mathcal{R}}_0<1$, while HIV persists when ${\mathcal{R}}_0>1$. Several numerical simulations are provided to validate the theoretical results.