In this paper, we introduce a novel family of statistical models called a new alpha power type-1 family of distributions. Three sub-cases of the family are discussed. Based on the novel family, a special model, explicitly, a new alpha power type-1-Weibull distribution is studied in depth. The new model has very interesting patterns of failure rates like increasing, decreasing, bathtub, and parabola-down. Hence, it is so flexible. Based on the comparison analysis, among five well-known models, it has an impact on health data analysis. Furthermore, the count data models capable of handling overdispersion and zero-inflation are discussed and applied the real health data. The zero-inflated negative binomial model in the frequentist approach has shown its popularity in handling both overdispersion and zero-inflation simultaneously, while the discrete Weibull model with the logit(q) link in the Bayesian approach outperformed its counterparts.