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• A new stochastic SIRD model with imperfect immunity based on a continuous time Markov process is proposed. • We explore novel stochastic properties like the extinction and alarm time beside the infection and mortality time of a tagged case. • Theorems and recursive algorithms for the computation of the stochastic properties are presented. • An extensive sensitivity analysis reveals monotonous and unimodal tendencies for the considered stochastic indicators. • Health authorities can employ this information to achieve the best balance between health benefits and economic drawbacks. Efficient assessment of epidemic phenomena has an important role in modern epidemiology, while many novel methods propose reliable estimates for epidemic evolution in a population. In this paper, we focus on the stochastic modeling of a novel epidemiological (susceptible-infected-recovered-deceased) SIRD model with imperfect immunity based on a continuous-time Markov process adapted to the characteristics of the epidemic model. We investigate several novel stochastic properties of the SIRD scheme that are both population- and individual-oriented, such as the extinction and alert time of an epidemic, in parallel with the infection and mortality times of a tagged individual. We provide propositions and detailed recursive algorithms for computing probabilities and moments, giving additional information beyond the mean and variance of the stochastic quantities. Important remarks for representing higher-order matrices and system solving are provided, decreasing significantly the operation time of these algorithms. Extensive sensitivity analysis gives light to the influence of the system's parameters, while enhancing the validity of our methodology. Unlike other analyses that focus mainly on fitting the evolution of various diseases, our goal is to complement these studies by highlighting additional noteworthy features that determine an epidemic's future. The proposed methodology is applied on the monkeypox outbreak of 2022 in India and the first COVID infection wave in Barbados. Finally, public health authorities can use the information provided by these indicators to adjust the duration of lockdowns, accordingly, achieving the best possible balance between health benefits and economic disadvantages. Simultaneously, knowledge of mortality probabilities and timing of infection can support early coordination of hospitals and intensive care units, which could notably reduce the high risk of mortality. [ABSTRACT FROM AUTHOR]