| Abstract: |
| This study refines the Caputo fractional derivative for non-singular nonlinear functions, unifying Riemann-Liouville and Caputo derivatives. We introduce an improved fractional derivative operator to establish stability outcomes for a model addressing cytoplasmic incompatibility in \textit{Aedes Aegypti} mosquitoes. This work builds on existing research where the Caputo-Fabrizio operator was used in logistic growth equations, confirming solution existence, uniqueness, and $\alpha$-exponential stability. However, previous studies overlooked the non-singular aspects of nonlinear growth factors, which are crucial for our model. We extend existing results to a new fractional-order operator using singular and non-singular kernel functions and their well-posedness properties. Our mathematical model aims to increase cytoplasmic incompatibility in \textit{Aedes Aegypti} by releasing \textit{Wolbachia}-infected mosquitoes, reducing mosquito populations and the incidence of diseases like Dengue, Zika, Chikungunya, and Yellow Fever. We establish conditions for delay-dependent exponential stability using a Lyapunov-Kraskovskii functional and the linear matrix inequality framework. Finally, a numerical example with comparative analysis validates the theoretical outcomes of the improved fractional operator within the population model using real-world data. |
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