  In the light of several major epidemic events that emerged in the past two
decades, and emphasized by the COVID-19 pandemics, the non-Markovian spreading
models occurring on complex networks gained significant attention from the
scientific community. Following this interest, in this article, we explore the
relations that exist between the non-Markovian SEIS
(Susceptible--Exposed--Infectious--Susceptible) and the classical Markov SIS,
as basic re-occurring virus spreading models in complex networks. We
investigate the similarities and seek for equivalences both for the
discrete-time and the continuous-time forms. First, we formally introduce the
continuous-time non-Markovian SEIS model, and derive the epidemic threshold in
a strict mathematical procedure. Then we present the main result of the paper
that, providing certain relations between process parameters hold, the
stationary-state solutions of the status probabilities in the non-Markovian
SEIS may be found from the stationary state probabilities of the Markov SIS
model. This result has a two-fold significance. First, it simplifies the
computational complexity of the non-Markovian model in practical applications,
where only the stationary distribution of the state probabilities is required.
Next, it defines the epidemic threshold of the non-Markovian SEIS model,
without the necessity of a thrall mathematical analysis. We present this result
both in analytical form, and confirm the result trough numerical simulations.
Furthermore, as of secondary importance, in an analytical procedure we show
that each Markov SIS may be represented as non-Markovian SEIS model.
