Recently developed methods for the nonparametric estimation of Hawkes point process models facilitate the use of such models for the description and forecasting of spread of epidemic diseases. Using data from the 2014 Ebola outbreak in West Africa to evaluate how well Hawkes point process can forecast the spread of Ebola cases in Guinea, Sierra Leone, and Liberia. SEIR models fit previously to the same data are evaluated using identical metrics. Hawkes models accurately describe the spread of Ebola in each of the three countries investigated, and result in a 38% reduction in RMSE for weekly case estimation when compared to SEIR models. We also simulate the ability to make near real-time predictions during an actual outbreak by using the first 75% of the data for estimation and the subsequent 25% of the data for evaluation. We demonstrate that the improved fit from Hawkes modeling cannot be attributed to overfitting alone. Methodological extensions to nonparametric Hawkes models are proposed to allow for nonstationary and covariate dependent background rates and triggering for application to crime. An improved model for disease spread over a network is also investigated.