  This paper studies the infinite-time stability of the numerical scheme for
stochastic McKean-Vlasov equations (SMVEs) via stochastic particle method. The
long-time propagation of chaos in mean-square sense is obtained, with which the
almost sure propagation in infinite horizon is proved by exploiting the
Chebyshev inequality and the Borel-Cantelli lemma. Then the mean-square and
almost sure exponential stabilities of the Euler-Maruyama scheme associated
with the corresponding interacting particle system are shown through an
ingenious manipulation of empirical measure. Combining the assertions enables
the numerical solutions to reproduce the stabilities of the original SMVEs. The
examples are demonstrated to reveal the importance of this study.
