  This paper studies the spread dynamics of a stochastic SIRS epidemic model
with nonlinear incidence and varying population size, which is formulated as a
piecewise deterministic Markov process. A threshold dynamic determined by the
basic reproduction number $\mathcal{R}_{0}$ is established: the disease can be
eradicated almost surely if $\mathcal{R}_{0}<1$, while the disease persists
almost surely if $\mathcal{R}_{0}>1$. The existing method for analyzing ergodic
behavior of population systems has been generalized. The modified method
weakens the required conditions and has no limitations for both the number of
environmental regimes and the dimension of the considered system. When
$\mathcal{R}_{0}>1$, the existence of a stationary probability measure is
obtained. Furthermore, with the modified method, the global attractivity of the
$\Omega$-limit set of the system and the convergence in total variation to the
stationary measure are both demonstrated under a mild extra condition.
