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The number of latent causes follows a Poisson distribution and their activation times follow a Weibull distribution. We derive closed\u2010form survival and density functions under three activation schemes\u2014minimum, maximum, and random\u2014and allow covariates to act on the zero inflation and cure proportions as well as on the baseline time\u2010to\u2010event distribution. Parameters are estimated by maximum likelihood, and finite\u2010sample performance is assessed via Monte Carlo experiments. We illustrate the method with data on time to insulin initiation among 390 pregnant women with gestational diabetes, where the random activation scheme provides the best fit by Akaike Information Criterion and Bayesian Information Criterion and yields interpretable covariate effects. The proposed framework unifies and extends existing cure\u2010rate and zero\u2010adjusted survival models, providing a flexible tool for complex settings with cure fractions, competing risks, and\u00a0frailty.<\/jats:p>","DOI":"10.1002\/bimj.70162","type":"journal-article","created":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T13:11:07Z","timestamp":1787404267000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Zero\u2010Inflated Promotion Cure\u2010Rate Models With Frailty and Flexible Activation Schemes"],"prefix":"10.1002","volume":"68","author":[{"given":"Danillo","family":"Assun\u00e7\u00e3o","sequence":"first","affiliation":[{"name":"Institute of Mathematical and Computer Sciences University of S\u00e3o Paulo S\u00e3o Carlos SP Brazil"},{"name":"Department of Statistics Federal University of S\u00e3o Carlos S\u00e3o Carlos SP Brazil"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5387-2457","authenticated-orcid":false,"given":"Pedro L.","family":"Ramos","sequence":"additional","affiliation":[{"name":"Department of Statistics Pontificia Universidad Cat\u00f3lica de Chile Santiago Chile"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6780-2089","authenticated-orcid":false,"given":"Vera","family":"Tomazella","sequence":"additional","affiliation":[{"name":"Department of Statistics Federal University of S\u00e3o Carlos S\u00e3o Carlos SP Brazil"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jeremias","family":"Le\u00e3o","sequence":"additional","affiliation":[{"name":"Department of Statistics Federal University of Amazonas Manaus AM Brazil"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-4579-1588","authenticated-orcid":false,"given":"Lucas","family":"Osses","sequence":"additional","affiliation":[{"name":"Department of Statistics Pontificia Universidad Cat\u00f3lica de Chile Santiago Chile"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Agatha","family":"Rodrigues","sequence":"additional","affiliation":[{"name":"Department of Statistics Federal University of Paraiba Jo\u00e3o Pessoa PR Brazil"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,8,22]]},"reference":[{"key":"e_1_2_11_2_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4348-9_4"},{"key":"e_1_2_11_3_1","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.1952.10501187"},{"key":"e_1_2_11_4_1","doi-asserted-by":"publisher","DOI":"10.1080\/02664763.2019.1597029"},{"key":"e_1_2_11_5_1","doi-asserted-by":"publisher","DOI":"10.1002\/bimj.201800056"},{"issue":"2","key":"e_1_2_11_6_1","first-page":"95","article-title":"The Effect of Frailty Term in the Standard Mixture Model","volume":"4","author":"Calsavara V. 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