Analysis & Comment

Opinion | The Math That Explains the End of the Pandemic

By Zoë M. McLaren

Dr. McLaren is an associate professor at the University of Maryland, Baltimore County, who studies policies to combat infectious disease epidemics, including Covid-19.

The United States has vaccinated more than half of its adults against Covid-19, but it could be months until the country has vaccinated enough people to put herd immunity within reach (and much of the world is still desperately waiting for access to vaccines).

Places with rising vaccination rates, like the United States, can look forward to case numbers coming down a lot in the meantime. And sooner than you might think. That’s because cases decline via the principle of exponential decay.

Many people learned about exponential growth in the early days of the pandemic to understand how a small number of cases can quickly grow into a major outbreak as transmission chains multiply. India, for example, which is in the grips of a major Covid-19 crisis, is in a phase of exponential growth.

Exponential growth means case numbers can double in just a few days. Exponential decay is its opposite. Exponential decay means case numbers can halve in the same amount of time.

U.S. cases, 14-day average

300k

200k

Exponential

decay

Exponential

growth

100k

April

April

July

Oct.

Jan. 2021

U.S. cases, 14-day average

300k

200k

Exponential

growth

Exponential

decay

100k

April

April

July

Oct.

Jan. 2021

By The New York Times

Understanding exponential dynamics makes it easier to know what to expect in the coming phase of the pandemic: Why things will improve quickly as vaccination rates rise and why it’s important to maintain some precautions even after case numbers come down.

Exponential decay will cause infections to plummet

Total active cases

Cases fall faster when

numbers are high

But fall more slowly

as cases come down

Time

Total active cases

Cases fall faster when

numbers are high

But fall more slowly

as cases come down

Time

By The New York Times

Every case of Covid-19 that is prevented cuts off transmission chains, which prevents many more cases down the line. That means the same precautions that reduce transmission enough to cause a big drop in case numbers when cases are high translate into a smaller decline when cases are low. And those changes add up over time. For example, reducing 1,000 cases by half each day would mean a reduction of 500 cases on Day 1 and 125 cases on Day 3 but only 31 cases on Day 5.

The end of the pandemic will therefore probably look like this: A steep drop in cases followed by a longer period of low numbers of cases, though cases will rise again if people ease up on precautions too soon.

This pattern has already emerged in the United States: It took only 22 days for daily cases to fall 100,000 from the Jan. 8 peak of around 250,000, but more than three times as long for daily cases to fall another 100,000. This pattern has also been borne out among the elderly, who had early access to vaccination, and in other countries, such as Israel, that have gotten their Covid-19 epidemics under control.

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