A percolation model with long-range correlations was introduced to investigate the phenomena of epidemic spreading by Monte Carlo simulations. The correlation exponent a and pathogenic ratio s correspond to different spreading methods and pathogenicity of variant epidemics. As the correlation changes from a weak one to a strong one, the patterns change from site percolation to Eden cluster when pathogenic ratio s=1, or Leath percolation cluster when s<1. Corresponding to change of patterns, the fractal dimension increases up to space dimension. The critical behavior in epidemic spreading has been examined based on the model. It is found that correlation has a great influence on the threshold of spreading percolation.