  In this paper, we consider a compartmental SIRS epidemic model with
asymptomatic infection and seasonal succession, which is a periodic
discontinuous differential system. The basic reproduction number
$\mathcal{R}_0$ is defined and valuated directly for this model, and the
uniformly persistent of the disease and threshold dynamics are obtained.
Specially, global dynamics of the model without seasonal force are studied. It
is shown that the model has only a disease-free equilibrium which is globally
stable if $\mathcal{R}_0\le 1$, and as $\mathcal{R}_0>1$ the disease-free
equilibrium is unstable and the model has an endemic equilibrium, which is
globally stable if the recovering rates of asymptomatic infective and
symptomatic infective are close. These theoretical results provide an intuitive
basis for understanding that the asymptomatic infective individuals and the
disease seasonal transmission promote the evolution of epidemic, which allow us
to predict the outcomes of control strategies during the course of the
epidemic.
