Dengue is a viral illness transmitted by Aedes mosquitoes and is a significant global threat. In this study, we developed a model of the dengue epidemic that incorporates larvicide and adulticide, as well as the harmonic mean incidence rate under fractal-fractional derivatives. We examined various theoretical aspects of the model, including nonnegativity, boundedness, existence, uniqueness, and stability. We computed the basic reproduction number $ \Re _{0} $ using the next-generation matrix. The model has two disease-free equilibriums, a trivial equilibrium, and a biologically realistic, along with one endemic equilibrium point. These findings enhanced our understanding of dengue transmission, providing valuable insights for awareness campaigns, control strategies, intervention approaches, decision support, guiding public health planning, and resource allocation to manage dengue effectively.

In this study, we introduce the dynamics of a Hepatitis B virus (HBV) model with the class of asymptomatic carriers and conduct a comprehensive analysis to explore its theoretical aspects and examine the crossover effect within the HBV model. To investigate the crossover behavior of the operators, we divide the study interval into two subintervals. In the first interval, the classical derivative is employed to study the qualitative properties of the proposed system, while in the second interval, we utilize the ABC fractional differential operator. Consequently, the study is initiated using the piecewise Atangana–Baleanu derivative framework for the systems. The HBV model is then analyzed to determine the existence, Hyers–Ulam (HU) stability, and disease-free equilibrium point of the model. Moreover, we showcase the application of an Adams-type predictor-corrector (PC) technique for Atangana–Baleanu derivatives and an extended Adams–Bashforth–Moulton (ABM) method for Caputo derivatives through numerical results. Subsequently, we employ computational methods to numerically solve the models and visually present the obtained outcomes using different fractional-order values. This network is designed to provide more precise information for disease modeling, considering that communities often interact with one another, and the rate of disease spread is influenced by this factor.

In this research paper, we dedicate our interest to an investigation of the sufficient conditions for the existence of solutions of two new types of a coupled systems of hybrid fractional differential equations involving $\varphi$-Hilfer fractional derivatives. The existence results are established in the weighted space of functions using Dhage’s hybrid fixed point theorem for three operators in a Banach algebra and Dhage’s helpful generalization of Krasnoselskii fixed- point theorem. Finally, simulated examples are provided to demonstrate the obtained results.

Ebola virus disease (EVD) is a severe and often fatal illness posing significant public health challenges. This study investigates EVD transmission dynamics using a novel fractional mathematical model with five distinct compartments: individuals with low susceptibility (${\mathbb{S}}_1$), individuals with high susceptibility (${\mathbb{S}}_2$), infected individuals ($\mathbb{I}$), exposed individuals ($\mathbb{E}$), and recovered individuals ($\mathbb{R}$). To capture the complex dynamics of EVD, we employ a $\Phi$-piecewise hybrid fractional derivative approach. We investigate the crossover effect and its impact on disease dynamics by dividing the study interval into two subintervals and utilize the $\Phi$-Caputo derivative in the first interval and the $\Phi$-ABC derivative in the second interval. The study determines the basic reproduction number ${\mathit{R}}_0$, analyzes the stability of the disease-free equilibrium and investigates the sensitivity of the parameters to understand how variations affect the system’s behavior and outcomes. Numerical simulations support the model and demonstrate consistent results with the theoretical analysis, highlighting the importance of fractional calculus in modeling infectious diseases. This research provides valuable information for developing effective control strategies to combat EVD.

This paper focuses on using piecewise derivatives to simulate the dynamic behavior and investigate the crossover effect within the coupled fractional system with delays by dividing the study interval into two subintervals. We establish and prove significant lemmas concerning piecewise derivatives. Furthermore, we extend and develop the necessary conditions for the existence and uniqueness of solutions, while also investigating the Hyers–Ulam stability results of the proposed system. The results are derived using the Banach contraction principle and the Leary–Schauder alternative fixed-point theorem. Additionally, we employ a numerical method based on Newton’s interpolation polynomials to compute approximate solutions for the considered system. Finally, we provide an illustrative example demonstrating our theoretical conclusions’ practical application.