  In this paper, we study a reaction-diffusion vector-host epidemic model. We
define the basic reproduction number $R_0$ and show that $R_0$ is a threshold
parameter: if $R_0\le 1$ the disease free steady state is globally stable; if
$R_0>1$ the model has a unique globally stable positive steady state. We then
write $R_0$ as the spectral radius of the product of one multiplicative
operator $R(x)$ and two compact operators with spectral radius equalling one.
Here $R(x)$ corresponds to the basic reproduction number of the model without
diffusion and is thus called local basic reproduction number. We study the
relationship between $R_0$ and $R(x)$ as the diffusion rates vary.
