Existence and stability of two periodic solutions for an interactive wild and sterile mosquitoes model

A Fractional-Order Density-Dependent Mathematical Model to Find the Better Strain of Wolbachia

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.

Periodic Orbits of a Mosquito Suppression Model Based on Sterile Mosquitoes

In this work, we investigate the existence and stability of periodic orbits of a mosquito population suppression model based on sterile mosquitoes. The model switches between two sub-equations as the actual number of sterile mosquitoes in the wild is assumed to take two constant values alternately. Employing the Poincaré map method, we show that the model has at most two T-periodic solutions when the release amount is not sufficient to eradicate the wild mosquitoes, and then obtain some sufficient conditions for the model to admit a unique or exactly two T-periodic solutions. In particular, we observe that the model displays bistability when it admits exactly two T-periodic solutions: the origin and the larger periodic solution are asymptotically stable, and the smaller periodic solution is unstable. Finally, we give two numerical examples to support our lemmas and theorems.