The lattice Boltzmann method is applied to study the flow in elastic blood vessels. The volume-flow rate increases considerably when the compliance constant of the blood vessel is below a critical value. There is a region of the compliance constant in which the average volume-flow rate is dramatically enhanced. A harmonic perturbation of the pressure does not change the behaviour of the average volume-flow rate while the harmonic wave attenuates very quickly along the tube when the resonant period is close to that of the input wave. The model, together with the simulation results, is expected to be helpful to understand the mechanism of the blood volume-flow rate related to the compliance constant of the blood vessel, especially on the dependence of the flux of human blood vessel under weather changes, which has medical significance.