A stage structured model for mosquito suppression with immigration

The incompatible insect technique based on Wolbachia is a promising alternative to control mosquito-borne diseases, such as dengue fever, malaria, and Zika, which drives wild female mosquitoes sterility through a mechanism cytoplasmic incompatibility. A successful control program should be able to withstand the perturbation induced by the immigration of fertilized females from surrounding uncontrolled areas. In this paper, we formulated a system of delay differential equations, including larval and adult stages, interfered by Wolbachia-infected males. We classified the release number of infected males and immigration number of fertile females, to ensure that the system displays globally asymptotically stable or bistable dynamics. The immigration of fertile females hinders the maximum possible suppression efficiency so that the wild adults cannot be reduced to a level below $ A^*_\infty $. We identified the permitted most migration number to reduce the wild adults to a target level. To reduce up to $ 90\% $ of wild adults in the peak season within two months, an economically viable strategy is to reduce the immigration number of wild females less than $ 0.21\% $ of the carrying capacity of adults in the control area.