  Recently, emerging epidemics like COVID-19 and its variants require
predictive mathematical models to implement suitable responses in order to
limit their negative and profound impact on society. The SIR
(Susceptible-InfectedRemoved) system is a straightforward mathematical
formulation to model the dissemination of many infectious diseases. The present
paper reports novel theoretical and analytical results for a perturbed version
of an SIR model with Gamma-distributed delay. Notably, our epidemic model is
represented by It\^o-L'evy stochastic differential equations in order to
simulate sudden and unexpected external phenomena. By using some new and
ameliorated mathematical approaches, we study the long-run characteristics of
the perturbed delayed model. Within this scope, we give sufficient conditions
for two interesting asymptotic proprieties: extinction and persistence of the
epidemic. One of the most interesting results is that the dynamics of the
stochastic model are closely related to the intensities of white noises and
L'evy jumps, which can give us a good insight into the evolution of the
epidemic in some unexpected situations. Our work complements the results of
some previous investigations and provides a new approach to predict and analyze
the dynamic behavior of epidemics with distributed delay. For illustrative
purposes, numerical examples are presented for checking the theoretical study
