  The present paper is devoted to the investigation of the long time dynamics
for a double free boundary system with nonlocal diffusions, which models the
infectious diseases transmitted via digestive system such as fecal-oral
diseases, cholera, hand-foot and mouth, etc...We start by proving the existence
and uniqueness of the Cauchy problem, which is not a trivial step due to
presence of couple dispersals and new types of nonlinear reaction terms. Next,
we provide simple conditions on comparing the basic reproduction numbers
$\CR_0$ and $\CR^*$ with some certain numbers to characterize the global
dynamics, as $t\to\infty$. We further obtain the sharp criteria for the
spreading and vanishing in term of the initial data. This is also called the
vanishing-spreading phenomena. The couple dispersals yield significant obstacle
that we cannot employ the approach of Zhao, Zhang, Li, Du \cite{WWZ1} and Du-Ni
\cite{DN}. To overcome this, we must prove the existence and the variational
characterization for the principal eigenvalue of a linear system with nonlocal
dispersals, then use it to obtain the right limits as the dispersal rates and
domain tend to zero or infinity. The maximum principle and sliding method for
the nonlocal operator are ingeniously employed to achieve the desired results.
