The primary aim of this study is to develop and evaluate a new epidemiological model that effectively captures the transmission dynamics of typhoid fever in Mayotte, a French administrative department situated in the Indian Ocean. This model uses medical information on the disease and a dataset of hospital admissions provided by the Regional Health Agency. To achieve our goal, we propose a parametric approach that employs a two-dimensional pure jump continuous-time Markov process to count the persons exposed and infected by the disease, and estimate critical model parameters, such as person-to-person and environmental contamination rates, incubation, and recovery rates. Our estimation methodology is highly innovative, addressing two key challenges associated with the available data. Firstly, observations are available only at fixed dates (daily hospitalizations), and secondly, on these dates, the total number of infected persons is not observed, only newly reported cases are counted, making parameter estimation more complex. To overcome the dataspecific challenges, we derive explicit expressions for the parameter estimators using the moments of the exposed-infected process. Subsequently, we adapt the Baum-Welch algorithm to estimate the transition matrix for the hidden multi-chain Markov model, and employ Monte Carlo simulation to estimate these parameters. The proposed model and estimation methodology have significant implications for understanding and controlling the transmission of typhoid fever in Mayotte, and potentially in other regions with similar epidemiological characteristics. The findings from this study could be instrumental in guiding public health policies aimed at mitigating the spread of typhoid fever in Mayotte.