When it started developing an Ebola drug in 2010, Tekmira Pharmaceuticals Corp. wasn’t counting on finding infected humans to test it in. Instead, the company planned to seek approval based on animal studies, which regulators sometimes allow.

Now the current outbreak in West Africa is presenting Tekmira and other drug makers with thousands of potential patients desperate for treatment—and setting off a rush among companies and health officials to find anything that might work.

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