Several major epidemic events over the past two decades have highlighted the importance of developing and studying non-Markovian compartmental models. Sellke [1983] introduced an ingenious construction for the SIR epidemic process to study the final size of epidemics. In this paper, we extend this construction to the $SEI_1I_2RS$ model. This model is chosen for its compactness, while including parallel infectious stages ($I_1$ and $I_2$) and cycles (aka loops) due to reinfection. Our methodology easily generalizes to a general class of stochastic compartmental models in closed populations, including SIR-like models (a series of compartments in one row), SEIAR-like models (parallel compartments), but also models with cycles. Our construction inherits from Sellke construction its ability to handle both Markovian and non-Markovian frameworks. Also, it naturally leads to a representation of the epidemic process under the form of a deterministic function of uncertain parameters (such as epidemic parameters) and variables modeling internal noise. Based on this representation, we propose a global sensitivity analysis of the $SEI_1I_2RS$ model. With our methodology we are able to quantify epistemic uncertainty due to the lack of knowledge on epidemic parameters and statistical uncertainty induced by stochasticity of the model. Finally we provide numerical experiments in both Markovian and non-Markovian frameworks.