As the Ebola crisis unfolded in West Africa over the spring and summer, public health experts began calling for speedy international intervention to stop the spread of the disease.

The crucial factor in containing Ebola, these experts say, is to keep infected patients isolated to reduce transmission. If that does not happen, the number of cases shoots up rapidly.

These charts are based on an epidemiological model created by the Centers for Disease Control and Prevention, using figures from Liberia and Sierra Leone, two of the countries hit hardest by the epidemic. In the third, Guinea, case counts have been unreliable.

Large-scale intervention did not begin until late August and early September, and the C.D.C. model suggests that was too late to stop the epidemic from killing tens of thousands of people — and perhaps many more — by the end of the year. Had successful interventions been started much earlier, the total number of cases would have remained below 5,000.

“In this case, lack of speed kills,” said Martin Meltzer, the lead author of the C.D.C.’s paper.

At the end of August, the World Health Organization set out an “Ebola Response Roadmap,” with a goal of isolating 70 percent of patients within two months after the start of intervention. The C.D.C. model uses a less ambitious timeline of five months. (Isolation means being treated safely in a hospital, Ebola treatment unit, or home or community setting; it also requires that the dead be given a safe burial.) For intervention that began in August, the predicted current rate of safe treatment in isolation is between 40 and 50 percent.

Because the situation in the three affected countries has been difficult to assess, the model, which was first published on Sept. 26, continues to be updated as new information is gathered. The C.D.C. estimates that the true number of cases may be more than twice the reported counts.