Abstract: We consider the long-time dynamics of two epidemic models with free boundaries, one with local diffusion and the other with nonlocal diffusion. We show that both models are well-posed, and their long-time dynamical behaviours are characterized by a spreading-vanishing dichotomy. When spreading persists, we also determine the spreading speed. For the local diffusion model, we show that the spreading speed is always finite, determined by an associated semi-wave problem. For the nonlocal diffusion model, a threshold condition is found in terms of the kernel functions appearing in the nonlocal diffusion terms, such that the spreading speed is finite precisely when this condition is satisfied; when this condition is not satisfied, we show that the spreading speed is infinite, namely accelerated spreading happens. This talk is based on joint works with Professor Yihong Du as well as ongoing work with both Prof. Du and Dr. Wenjie Ni.