In this communication, we first discuss the statistical methods used to estimate the probability distribution function that describes the serial interval distribution of the COVID-19 virus on a given set of data collected on viral shedding in patients with laboratory-confirmed COVID-19. Based on this estimation, we estimate the time-varying reproduction number and transmission rates observed on the island of Mayotte from March 2020 to January 2022.

First, we consider parameter estimation of mixture for a set of exponential family distributions using maximum likelihood estimation. Secondly, we consider parameter estimation of mixture for stable distributions. Our methods are based on the characteristic function using a Gaussian kernel estimator of the density distribution. The choice of the optimal bandwidth parameter was done using a plug-in method. We highlight another estimation procedure for the maximum likelihood framework based on the false position algorithm method to find a numerical root of the log-likelihood through the score functions. In the case of a mixture-stable distribution, the EM algorithm and the Bayesian estimation method have been adapted to propose an efficient tool for parameter estimation within the above estimation procedure.

A simulation study is conducted to evaluate the performance of our algorithms, which are then applied to real data. Our results seem to accurately estimate mixtures of stable distributions. The application concerned the estimation of the reproduction number of COVID-19 in Mayotte. We compare the proposed methods along with a detailed discussion.

In this paper, we discuss the estimation of the serial interval distribution by several methods and the best-fit model with a mixture model. We derive an estimate of the effective reproduction number along time. We fit the transmission rate parameter range values obtained by a mathematical learning model. The results of this presentation can be found in [1] and [2] for more details.