It remains unclear when the White House began contact tracing. If the effort did not begin right away, or go far back enough, infections may have been missed, experts said.

If the White House had started immediately, “then early control could have held back numbers of infections and further need for ongoing tracking,” said Jeanne Marrazzo, an infectious-diseases expert at the University of Alabama at Birmingham.

Several White House staffers and administration officials expressed anger and bewilderment that the White House had not undertaken a more robust contact-tracing effort sooner. They said many people — including White House residence staff who do not have the stature of a lawmaker or a top political aide — had not been contacted despite possible exposures, putting them and others at risk in a still-growing outbreak. The officials spoke on the condition of anonymity to discuss sensitive matters.

The CDC began offering help last Friday, after President Trump announced he had tested positive, only to be repeatedly spurned, according to a CDC official who spoke on the condition of anonymity to discuss internal matters. On Wednesday, an arrangement was made for “some limited CDC involvement,” the official said.