Abstract

This paper introduces a novel probability distribution, the Double Fuzzy Poisson Distribution, designed to address two levels of uncertainty commonly encountered in biological data analysis. The first layer of fuzziness arises from measurement inaccuracies, where observed data are represented as fuzzy numbers. The second layer accounts for uncertainty in the distribution parameter itself, which is also expressed as a fuzzy number, incorporating prior information from previous studies or expert opinions related to the biological phenomenon under investigation. To evaluate the performance of the proposed distribution, extensive Monte Carlo simulations were conducted alongside applications to real-world data involving HIV infection rates. The results demonstrate that the Double Fuzzy Poisson Distribution offers superior accuracy and greater flexibility compared to the traditional Poisson distribution. Specifically, it provides more reliable estimations of infection rates and enhances risk analysis by effectively capturing the inherent uncertainty in medical data. These findings suggest that the proposed model is a robust and adaptable tool for handling biological data characterized by multiple sources of imprecision, making it a valuable addition to the field of biomathematical modeling.