The baby who died last week was treated at Rady Children's Hospital in San Diego.
The highly contagious illness has symptoms similar to the common cold, but a persistent cough that lasts weeks may indicate the illness, which is also known as pertussis.
More than 5,270 people have been infected in California, but whooping cough is a concern nationwide, points out CBS News Medical Correspondent Dr. Jennifer Ashton.
"The babiess died unnecessarily," she told co-anchor Chris Wragge on "The Early Show" Thursday. "This is a bacterial infection caused by a bacteria called pertussis. It's spread via airborne respiratory droplets. People cough, sneeze, it goes in the air. And the very vulnerable are infants, because they have not yet been vaccinated. … If you've ever heard that barky type of cough, you know what it sounds like. You will not forget it. But a lot of times, you can have pertussis, or whooping cough and not have that cough."
Whooping cough symptoms, Ashton notes, "start in adults or children just like the common cold. You'll get some cough, some runny nose. In about one-to-two weeks, it kind of moves into a very violent cough. When adults get this, they have these fits of coughing where they almost can't breathe. Again, children will get that loud, barky noise. And it's very important to remember that adults and infants under age six months may not have that barky cough. They can have this. They can be contagious. And you might not know it."
Vaccinations, Ashton adds, are "the big problem. Under the age of two months, infants and newborns cannot be vaccinated. They depend on adults to have their vaccination to protect them. It's called the cocoon effect.
"It's normally given at about two months. Anyone with contact with a newborn, whether that's a sibling, a grandparent, a health care worker, day care worker should be revaccinated, because the vaccines you and I got really only cover us for five years. Again, if you or I get sick, not a big deal. If we pass it to a newborn, it could be deadly."