The susceptible-infected-susceptible (SIS) epidemic model is presented which can be used to study the epidemic spreading behavior in the local-world evolving network model. Local-world evolving model displays a transition from the exponential network to the scale-free network with respect to the connectivity distribution. From theoretical analyses and numerical simulations, we find that the epidemic spreading behavior on local-world networks also takes on some kind of transitional behaviors. The transitional behavior is further verified by comparing the spreading behavior of local-world network with that of random and scale-free networks. Some feasible control strategies are also proposed to keep from the epidemic spreading on networks.