  We show that the basic reproduction number of an SIS patch model with
standard incidence is either strictly decreasing and strictly convex with
respect to the diffusion coefficient of infected subpopulation if the patch
reproduction numbers of at least two patches in isolation are distinct or
constant otherwise. Biologically, it means that fast diffusion of the infected
people reduces the risk of infection. This completely solves and generalizes a
conjecture by Allen et al. ({\it SIAM J Appl Math}, 67: 1283-1309, 2007).
Furthermore, a substantially improved lower bound on the multipatch
reproduction number, a generalized monotone result on the spectral bound the
Jacobian matrix of the model system at the disease-free equilibrium, and the
limiting endemic equilibrium are obtained. The approach and results can be
applied to a class of epidemic patch models where only one class of infected
compartments migrate between patches and one transmission route is involved.
