It was soon clear that health authorities had other worries, as word emerged that they were isolating not just Dr. Spencer’s fiancée but also two friends who had been with him in the two days before he arrived at the hospital. Dr. Spencer said he had started feeling sluggish on Tuesday.

City officials were making plans to provide case managers for every family or person who might need to be quarantined. Those managers would help with the chores of daily life, such as providing school materials for children or food for people confined to their homes.

New York has some advantage in that it may be able to learn from what happened in Dallas, where two nurses became infected with Ebola after treating Thomas E. Duncan, the patient with the first case of Ebola to be diagnosed in the United States who died on Oct. 8.