A sex-structured model with birth pulse and release strategy for the spread of Wolbachia in mosquito population.

Dengue fever is one of the most important diseases causing illness and death all over the world, which brings tremendous threat to peoples' life and property security, especially in the undeveloped areas. The main vector, Aedes aegypti, must be controlled to prevent the transmission of dengue. There are a variety of methods to control it. Wolbachia is an innovative bacterium which breaks the dengue transmission cycle for its characteristics of cytoplasmic incompatibility and maternal transmission. In this paper, a sex-structured model with birth pulse is established to study the spread of Wolbachia in mosquito population. The results show that if the maternal transmission is perfect, Wolbachia will spread successfully. Moreover, all the mosquitoes will be infected with Wolbachia. If the maternal transmission is imperfect, there are two locally asymptotically stable periodic solutions. One is Wolbachia-extinction periodic solution, and the other is part replacement periodic solution. Numerical simulations show that the initial occupancy of Wolbachia-infected mosquitoes has an important effect on the success of part replacement strategy. If the initial occupancy is relatively large, the part replacement strategy can be successful. Furthermore, in consideration of the fact that the initial occupancy cannot be always large enough in the wild nature, to release Wolbachia-infected mosquitoes artificially into the wild nature becomes necessary. Therefore, we add a release strategy into the sex-structured model with birth pulse for further analysis. The condition to ensure the stability of the Wolbachia total replacement periodic solution is obtained. Finally, the effect of the release quantity is simulated numerically.