  Heterogeneous Markovian Susceptible-Infected-Susceptible (SIS) epidemics with
a general infection rate matrix $\widetilde{A}$ are considered. Using a
non-negative matrix factorization to approximate $\widetilde{A}$, we are able
to identify when a metastable state can be expected, and that the metastable
distribution, under certain conditions, will feature a normal distribution with
known expectation and covariance. Furthermore, we model a heterogeneous
Markovian SIS epidemic, that starts with a fraction of initially infected nodes
different from that in the metastable state, by approximating its behaviour by
a standard linear stochastic differential equation (SDE) in sufficiently high
dimensions. By exploiting the knowledge of the covariance matrix from the SDE,
we demonstrate significant accuracy improvements over the first-order
mean-field approximation NIMFA.
