  We introduce non-Markovian SIR epidemic spreading model inspired by the
characteristics of the COVID-19, by considering discrete- and continuous-time
versions. The incubation period, delayed infectiousness and the distribution of
the recovery period are modeled with general functions. By taking corresponding
choice of these functions, it is shown that the model reduces to the classical
Markovian case. The epidemic threshold is analytically determined for arbitrary
functions of infectivity and recovery and verified numerically. The relevance
of the model is shown by modeling the first wave of the epidemic in Italy, in
the spring, 2020.
