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In this paper, we formulated a system of delay differential equations, including larval and adult stages, interfered by &lt;italic&gt;Wolbachia&lt;\/italic&gt;-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.&lt;\/p&gt;<\/jats:p>","DOI":"10.3934\/mbe.2024328","type":"journal-article","created":{"date-parts":[[2024,11,1]],"date-time":"2024-11-01T11:43:09Z","timestamp":1730461389000},"page":"7454-7479","source":"Crossref","is-referenced-by-count":0,"title":["A stage structured model for mosquito suppression with immigration"],"prefix":"10.3934","volume":"21","author":[{"given":"Mugen","family":"Huang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zifeng","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zixin","family":"Nie","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"2321","reference":[{"key":"key-10.3934\/mbe.2024328-1","unstructured":"<i>World Health Organization<\/i>, Global strategy for dengue prevention and control 2012\u20132020, 2012. 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