A one-locus model describing the evolutionary dynamics of resistance against insecticide in Anopheles mosquitoes

Vector control is regarded an efficient approach for preventing and reducing the spread of malaria. For several decades, insecticides have been used to suppress the population of Anopheles mosquitoes throughout the globe. It turns out that continual usage of a single compounded insecticide has massively contributed to the rise of resistant mosquitoes. Information on the evolution of insecticide resistance is considered an essential aspect in the control of Anopheles mosquitoes. In this study, a mathematical model to describe the dynamics of Anopheles mosquitoes is constructed, highlighting genetic classification based on their resistance status with respect to insecticides. Focusing on a one-locus case related to a specific insecticide class, the model is constructed according to a scenario in which the intermediate form of resistance called heterozygous resistance can be generated from random matings between susceptible and resistant mosquitoes, therefore carrying both genes. The model is governed by a three-dimensional system of differential equations that describes the interaction between homozygous wild, heterozygous, and resistant subpopulations, with three corresponding fitness levels related to their intrinsic growth rates. Criteria for the existence and stability of all existing equilibria are obtained. It is generally concluded from the model that the fitness levels heavily determine to which steady state the model solution converges. Numerical simulations indicate that long-term implementation of high doses of insecticide can increase the proportion of resistant mosquitoes significantly. Accordingly, understanding the fitness levels is very important for selecting proper intervention strategies as well as for predicting the long-term effects of the implementation of certain insecticides.