Modeling<i>Wolbachia</i>Spread in Mosquitoes Through Delay Differential Equations

Abstract

Dengue fever is the most common mosquito-borne viral disease. A promising control strategy targets the mosquito vector Aedes aegypti by releasing the mosquitoes infected by the endosymbiotic bacterium Wolbachia to invade and replace the wild population. With infection, Wolbachia reduces the mosquito's dengue transmission potential and brings infected females a reproductive advantage through cytoplasmic incompatibility. As Wolbachia often induces fitness costs, it is important to analyze how the reproductive advantage offsets the fitness costs for the success of population replacement. In this work, we develop a model of delay differential equations to study Wolbachia infection dynamics. We prove that, when the infection does not alter the mean life span, Wolbachia can spread into the whole population as long as the infection frequency stays strictly above a threshold value for a period no less than the prereproductive time $\tau$. For the other cases, we find that such a threshold value cannot be well defined. Our numerical simulation shows that our model can generate predictions well fitting with the experimental data. It also reveals the striking phenomena that the minimal releasing of infected mosquitoes sufficient for fixation is insensitive to $\tau$, but the waiting time increases almost linearly with $\tau$. However, when $\tau$ is fixed but the ratio of infected males over females varies, the waiting time decreases rapidly when the ratio increases moderately, but responds rather weakly when the ratio increases further.