Abstract
This work is devoted to analyzing an impulsive control synthesis to maintain the self-sustainability of Wolbachia among Aedes Aegypti mosquitoes. The present paper provides a fractional order Wolbachia invasive model. Through fixed point theory, this work derives the existence and uniqueness results for the proposed model. Also, we performed a global Mittag-Leffler stability analysis via Linear Matrix Inequality theory and Lyapunov theory. As a result of this controller synthesis, the sustainability of Wolbachia is preserved and non-Wolbachia mosquitoes are eradicated. Finally, a numerical simulation is established for the published data to analyze the nature of the proposed Wolbachia invasive model.
Highlights
In the 19th century, fractional calculus (FC) theory has been built by some famous mathematicians like Grunwald, Letnikov, Riemann, Liouville, Euler and Caputo [1,2,3]
While constructing the model we have considered the total of 10 stages such as non-Wolbachia eggs(We ), non- Wolbachia larvae (Wl ), non-Wolbachia pupae (Wp ), non-Wolbachia adult female (W f ), non- Wolbachia adult male (Wa ), Wolbachia infected eggs ( Ie ), Wolbachia infected larvae ( Il ), Wolbachia infected pupae ( I p ), Wolbachia infected adult female ( I f ), Wolbachia infected adult male ( Ia )
Our results shows that this method will increase the self-sustainability of Wolbachia bacterium among
Summary
In the 19th century, fractional calculus (FC) theory has been built by some famous mathematicians like Grunwald, Letnikov, Riemann, Liouville, Euler and Caputo [1,2,3]. In [38], the authors discussed the linear feedback control strategy of a mathematical model containing only three stages such as aquatic, female Wolbachia infected and uninfected mosquitoes. The author utilized two various types of controls like vaccination for humans and Wolbachia infected mosquitoes’ release for mosquitoes. In [41], the authors proposed an age structured fractional order mathematical model to control the Aedes Aegypti mosquitoes via Wolbachia bacterium using the Linear Matrix Inequality (LMI) approach. A novel mathematical model, which considers the total of ten stages in Aedes Aegypti mosquitoes (combining both Wolbachia infected and Wolbachia uninfected) is proposed and the possible optimal stages to release the Wolbachia are discussed, and the most important concept of Wolbachia invasion and Wolbachia gain are adopted.
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