Opinion | The Math That Explains the End of the Pandemic

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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.

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.

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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.

Reaching herd immunity is a key goal. It drives cases toward zero by slowing the spread of the virus through a combination of vaccination and infection-acquired immunity to maintain exponential decay — even as society resumes normal activities.

But contrary to popular belief, reaching herd immunity doesn’t prevent all outbreaks, at least not initially. It simply means so few people are susceptible to infections that any outbreaks that do happen tend to be snuffed out and case counts decline. Over time, outbreaks themselves become less and less common.

It is possible to bring Covid-19 case numbers down quickly via exponential decay even before vaccination rates reach herd immunity. We just need to keep transmission rates below the tipping point between exponential growth and exponential decay: where every person with Covid-19 infects fewer than one other person. Every single thing people can do to slow transmission helps — including wearing masks, getting tested and avoiding crowded indoor spaces — especially given concerns about current and future variants, since it could be what gets us past the threshold into exponential decay.

As more and more people get vaccinated, people can gradually ease precautions while cases continue to decline. Keeping cases down gets easier over time until — and this is the beauty of vaccine-driven herd immunity — it’s almost effortless, once enough people are vaccinated, to keep cases sustainably low. That’s the power of exponential decay.

You shouldn’t expect the road to herd immunity to be smooth, though. It’s natural for people to want to ease precautions when cases fall and to feel reluctant to step up precautions when cases rise again. The tricky part is that it can be hard to know how much to ease up while keeping cases trending downward so exponential growth doesn’t get out of control, as is happening in India.

Fortunately, the exponential dynamics that lead to wild swings in case numbers when cases are high lead to far less dramatic swings when cases are low. And as more and more people are vaccinated, the swings will also shrink, since fewer people are susceptible to infection.

Every vaccination helps keep us in the realm of exponential decay. So does everything else people do to slow the spread of the virus, like masking and distancing. Synchronizing these efforts magnifies their impact by making it nearly impossible for the virus to spread and breaking many transmission chains at once.

The United States is still a long way from reaching herd immunity, but things could improve a lot before then. The worst of the pandemic may be over sooner than you think.

Zoë McLaren (@ZoeMcLaren) is an associate professor in the School of Public Policy at the University of Maryland, Baltimore County. She studies health and economic policy to combat infectious disease epidemics, including H.I.V., tuberculosis and Covid-19.

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