  In order to explore the impact of periodically evolving domain on the
transmission of disease, we study a SIS reaction-diffusion model with logistic
term on a periodically evolving domain. The basic reproduction number
${\mathcal{R}}_0$ is given by the next generation infection operator, and
relies on the evolving rate of the periodically evolving domain, diffusion
coefficient of infected individuals $d_I$ and size of the space. The
monotonicity of ${\mathcal{R}}_0$ with respect to $d_I$, evolving rate
$\rho(t)$ and interval length $L$ are derived, and asymptotic property of
${\mathcal{R}}_0$ if $d_I$ or $L$ is small enough or large enough in
one-dimensional space are discussed. ${\mathcal{R}}_0$ as threshold can be used
to characterize whether the disease-free equilibrium is stable or not. Our
theoretical results and numerical simulations indicate that small evolving
rate, small diffusion of infected individuals and small interval length have
positive impact on prevention and control of disease.
