Abstract
In this paper, a mathematical dynamical system involving both deterministic (with or without delay) and stochastic "SIR" epidemic model with nonlinear incidence rate in a continuous reactor is considered. A profound qualitative analysis is given. It is proved that, for both deterministic models, if R-d > 1, then the endemic equilibrium is globally asymptotically stable. However, if R-d <= 1, then the disease-free equilibrium is globally asymptotically stable. Concerning the stochastic model, the Feller's test combined with the canonical probability method were used in order to conclude on the long-time dynamics of the stochastic model. The results improve and extend the results obtained for the deterministic model in its both forms. It is proved that if R-s > 1, the disease is stochastically permanent with full probability. However, if R-s <= 1, then the disease dies out with full probability. Finally, some numerical tests are done in order to validate the obtained results.