Zero-Inflated Promotion Cure-Rate Models With Frailty and Flexible Activation Schemes

We propose a zero-adjusted promotion cure-rate regression model that jointly accommodates instantaneous failures (times at or near-zero), a non-susceptible cured fraction, competing risks, and latent heterogeneity modeled with a Gamma frailty. The number of latent causes follows a Poisson distribution and their activation times follow a Weibull distribution. We derive closed-form survival and density functions under three activation schemes-minimum, maximum, and random-and allow covariates to act on the zero inflation and cure proportions as well as on the baseline time-to-event distribution. Parameters are estimated by maximum likelihood, and finite-sample 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-rate and zero-adjusted survival models, providing a flexible tool for complex settings with cure fractions, competing risks, and frailty.