This paper deals with the spreading speed of the disease described by a partially degenerate and cooperative epidemic model with free boundaries. We show that the spreading speed is determined by a semi-wave problem. To find such a semi-wave solution, we prove the existence of a monotone solution to a reduced ODE by an upper and lower solution approach. And then we establish the uniqueness of the semi-wave solution via the sliding method. It is demonstrated that the precise asymptotic spreading speed is less than the minimal speed of traveling waves.