Stability Analysis and Numerical Simulation of Fractional-Order Models for Wolbachia Transmission in Aedes Aegypti Mosquitoes with Seasonal Effects

Abstract

Mosquito-borne diseases have historically impacted people and remain a global problem. Wolbachia represents an innovative vector management strategy capable of diminishing mosquito populations and reducing the threat of mosquito-borne diseases. In order to provide optimal and efficient control, Wolbachia should be released at each step of the mosquito life cycle. Employing fractional calculus enables the capture of memory effects and hereditary characteristics of this process. In this study, we develop and analyze a fractional-order mathematical model to investigate Wolbachia transmission dynamics, which accounts for imperfect maternal transmission and infection loss. The model considers Wolbachia infected and uninfected subpopulations of Aedes aegypti mosquitoes, assuming equal numbers of adult males and females. We establish the positivity and boundedness of solutions for nonnegative initial conditions. The invasive reproduction number R0 w w has been found to determine whether the Wolbachia infection spreads. We consider two fractional-order models: one that ignores the impacts of seasons on the mosquito populations and another that incorporates these effects. Furthermore, the stability of the model is analyzed using Lyapunov functions and Ulam-Hyers stability theories. The models are numerically solved using the Adams-Bashforth-Moulton technique,
demonstrating the effects of model parameters and fractional-order values. Based on the results, we find that the fractional order α = 0.5 is optimal, and the corresponding conditions could be applied in real-world experiments to increase the population of Wolbachia-infected mosquitoes. These results highlight the importance of fractional-order modeling.