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The primary objective of the current study was to create a mathematical model utilizing fractional-order calculus for the purpose of analyzing the symmetrical characteristics of Wolbachia dissemination among Aedesaegypti mosquitoes. We investigated various strains of Wolbachia to determine the most sustainable one through predicting their dynamics. Wolbachia is an effective tool for controlling mosquito-borne diseases, and several strains have been tested in laboratories and released into outbreak locations. This study aimed to determine the symmetrical features of the most efficient strain from a mathematical perspective. This was accomplished by integrating a density-dependent death rate and the rate of cytoplasmic incompatibility (CI) into the model to examine the spread of Wolbachia and non-Wolbachia mosquitoes. The fractional-order mathematical model developed here is physically meaningful and was assessed for equilibrium points in the presence and absence of disease. Eight equilibrium points were determined, and their local and global stability were determined using the Routh–Hurwitz criterion and linear matrix inequality theory. The basic reproduction number was calculated using the next-generation matrix method. The research also involved conducting numerical simulations to evaluate the behavior of the basic reproduction number for different equilibrium points and identify the optimal CI value for reducing disease spread.

In this paper, a novel age-structured delayed mathematical model to control Aedes aegypti mosquitoes via Wolbachia-infected mosquitoes is introduced. To eliminate the deadly mosquito-borne diseases such as dengue, chikungunya, yellow fever, and Zika virus, the Wolbachia infection is introduced into the wild mosquito population at every stage. This method is one of the promising biological control strategies. To predict the optimal amount of Wolbachia release, the time varying delay is considered. Firstly, the positiveness of the solution and existence of both Wolbachia present and Wolbachia free equilibrium were discussed. Through linearization, construction of suitable Lyapunov–Krasovskii functional, and linear matrix inequality theory (LMI), the exponential stability is also analyzed. Finally, the simulation results are presented for the real-world data collected from the existing literature to show the effectiveness of the proposed model.