  In this paper, we further investigate the global dynamics of a stochastic
differential equation SIS (Susceptible-Infected-Susceptible) epidemic model
recently proposed in [A. Gray et al., SIAM. J. Appl. Math., 71 (2011),
876-902]. We present a stochastic threshold theorem in term of a
\textit{stochastic basic reproduction number} $R_0^S:$ the disease dies out
with probability one if $R_0^S<1,$ and the disease is recurrent if
$R_0^S\geqslant1.$ We prove the existence and global asymptotic stability of a
unique invariant density for the Fokker-Planck equation associated with the SDE
SIS model when $R_0^S>1.$ In term of the profile of the invariant density, we
define a \textit{persistence basic reproduction number} $R_0^P$ and give a
persistence threshold theorem: the disease dies out with large probability if
$R_0^P\leqslant1,$ while persists with large probability if $R_0^P>1.$
Comparing the \textit{stochastic disease prevalence} with the
\textit{deterministic disease prevalence}, we discover that the stochastic
prevalence is bigger than the deterministic prevalence if the deterministic
basic reproduction number $R_0^D>2.$ This shows that noise may increase
severity of disease. Finally, we study the asymptotic dynamics of the
stochastic SIS model as the noise vanishes and establish a sharp connection
with the threshold dynamics of the deterministic SIS model in term of a
\textit{Limit Stochastic Threshold Theorem}.
