The issues which beleaguered the first weeks and months of the rollout haven’t just disappeared. Some state and local health systems still report confusion about when new supplies of the vaccine will reach them, contributing to disparities in who is able to get a shot.

But that will likely change in the next few months, according to experts such as former Food and Drug Administration (FDA) commissioner Scott Gottlieb. In fact, the dynamic could reverse itself: In the foreseeable future, there may not be enough vaccine demand to measure up with an ever-increasing supply.

Gottlieb, during an appearance on CNBC’s Squawk Box Monday said he believes that a significantly higher number of Americans will be eligible for COVID vaccines by April.

“I think we’re going to run out of demand sooner than we think. I suspect that in some point in March and certainly by the end of March, we’re going to have to make this [vaccine] generally available,” Gottlieb, a physician and pharmaceutical industry guru, told the show’s hosts.

“That doesn’t mean everyone can go and get a vaccine on April 1, but I think everyone is going to be able to go online and get an appointment sooner than we think.”

In Gottlieb’s estimation, that means that eligibility guidance for who can get a COVID vaccine should be opened up sooner rather than later. It also means that the federal government should ramp up efforts to secure vaccines other than the currently available, two-dose Pfizer and Moderna shots (such as Johnson & Johnson’s, which is based on a different kind of vaccine technology and may only require one dose) to tackle the current supply problem.

As Eric Topol, director of the Scripps Research Translational Institute, pointed out, addressing the supply problem is currently the most urgent action item. But in comments over the weekend, he agreed with Gottlieb that a supply glut was foreseeable.

Why would the supply-and-demand dynamic flip in just months? For one thing, America is still grappling with vaccine skepticism, and enthusiasm about getting a COVID vaccine varies strongly by age. Older Americans who are at higher risk for a deadly COVID case are far more eager to get their shots as soon as possible, according to recent survey data from the AARP. Those over 60 are the most willing to jump at an immunization opportunity.

High-risk groups are among the earliest tranches of people who qualify for a vaccine. But as more and more Americans are considered eligible for one, the overall public demand may fall, which could force health systems and local health departments to figure out a strategy for maximizing the number of people who get a vaccine as manufacturing capacity grows.

What Gottlieb is saying, essentially, is that expediting that process by opening up eligibility sooner rather than later is the best way to address what’s likely to be a supply-and-demand relationship in rapid flux.