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The Math Behind Social Distancing

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The math behind social distancing

The Math Behind Social Distancing

As we wait for scientists and healthcare professionals to develop a vaccine for COVID-19, there is another, more readily available tool at our disposal.

Social distancing, defined as measures taken to reduce physical contact, is the first line of defense for containing an infectious disease like COVID-19. That’s because these infections spread when people cough, sneeze, or touch surfaces on which the virus resides.

To help us grasp the impact these measures can actually have, today’s infographic illustrates how a reduction in social exposure can theoretically contain the spread of infection.

Theoretical Potential

The calculations used to create today’s infographic come from Signer Laboratory, a stem cell research lab located in the Moores Cancer Center at the University of California San Diego.

Using a summation formula makes it possible to estimate the number of new infections over a 30 day period, across three scenarios.

Scenario5 Day Period30 Day Period
No social distancing practiced1 person infects 2.5* others406 people infected as a result
50% reduction in social exposure1 person infects 1.25* others15 people infected as a result
75% reduction in social exposure1 person infects 0.625* others2.5 people infected as a result

*For estimations only. It is not possible to infect only a fraction of another person.

To arrive at the figures reported above, Robert A.J. Signer, Ph.D., and his team made a number of key assumptions.

First, they estimated the basic reproduction number (R0) of COVID-19 to be 2.5, a figure supported by recent research. This means that, on average, an infected individual will spread the disease to 2.5 other people.

Next, they assumed that an infected individual will unknowingly spread COVID-19 over the median five day incubation period. After this period, the individual will begin to develop symptoms, immediately self quarantine, and no longer pose a threat.

Finally, they assumed a direct linear correlation between social interactions and R0. This means that when an infected person reduces their physical contact with others by 50%, they also spread the disease by an amount 50% less.

Timing is Everything

While the figures above are the results of mathematical estimations, researchers have actually studied social distancing from a variety of angles.

One study used simulations to determine the magnitude and timing of social distancing measures required to prevent a pandemic. The distancing measures simulated were:

MeasureDetails
School closureTeachers and students spent weekday daytime cycles at home, rather than at school.
Increased case isolationUpon becoming symptomatic, adults (90%) and children (100%) would self quarantine for the duration of the infection.
Workplace non-attendanceEach day, a person had a 50% chance of staying home instead of attending their workplace.
Community contact reductionIndividuals reduced their physical contact with community members by half, each day.
Combination of all fourAll four measures combined. 

The results, for a community of 30,000 people and an epidemic with R=2.5, are charted below. We can define the final illness attack rate as the share of people from an at risk population who ultimately catch the disease.

power of early social distancing

Results showed that when no action was taken, 65% of the population contracted the disease. However, if a combination of all four distancing measures were implemented instead, the attack rates were:

  • 45% (distancing begins after a 4 week delay)
  • 21% (distancing begins after a 3 week delay)
  • 7% (distancing begins after a 2 week delay)

What’s clear is that social distancing was significantly more effective when implemented with minimal delay—the final illness attack rate rose quickest beyond the third week. These findings draw a parallel to the visualizations in today’s infographic, which showed us just how quickly a disease can spread.

Social distancing interventions are important as they represent the only … measure guaranteed to be available against a novel strain of influenza in the early phases of a pandemic.

Kelso, J.K., Milne, G.J. & Kelly, H., BMC Public Health 9, 117 (2009)

We arrive at a similar conclusion when it comes to the types of distancing measures implemented. In the simulations, none of the four measures taken on their own were able to have a similar effect as when they were combined.

We All Have a Part to Play

With the global number of COVID-19 cases still rising, many governments have issued quarantine orders and travel bans.

The math supports these decisions—reducing our physical contact with others, even when we aren’t experiencing any symptoms, is crucial. Studies like the one summarized above also prove that taking action sooner, rather than later, can go a long way in reducing the spread of infection.

The key takeaway from all of this? Social distancing is a powerful disease control tool, but only if we all participate.

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Global COVID-19 Containment: Confirmed Cases, Updated Daily

This continuously updated chart provides a more complete look at the efficacy of COVID-19 containment strategies.

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Global COVID Containment: Confirmed Cases, Updated Daily

Sometimes, it helps to gain a fresh perspective.

Since the pandemic began, there have been innumerable tracking resources made available online, but rarely do they paint a complete picture of a country’s containment progress.

How Much Progress Is Really Being Made

Featured above, this continuously updated chart from Our World in Data provides a more complete look at the efficacy of COVID-19 containment strategies, sorted by country.

It is a variation of the Epidemic Curve (or “epi curve”), showing confirmed COVID-19 cases per country in relation to their testing rates—what’s revealed is the strength of each country’s containment strategy.

Only a fraction of total cases–those confirmed by a test–is known. This is why we spent recent months building the database and the visualization tool to make this variation of the epidemic curve possible.

— Our World in Data

Why Look at it This Way? Adequate Benchmarking

Countries vary widely in how they monitor and report on COVID-19. Cases in this particular chart were confirmed via laboratory testing, and the data covers 66% of the globe’s population.

Depending on a country’s containment efforts, confirmed cases can differ dramatically from total cases. To get a better idea of that difference, Our World in Data looked closer at the extent of testing. As they report, the World Health Organization considers an adequate benchmark to be 10-30 tests per confirmed case. And for those countries that experience larger outbreaks, there must be more tests conducted per confirmed case.

What the COVID Test-to-Case Ratio Tells Us

  • Line Trajectory: In this chart, rising lines show that the average number of laboratory-confirmed cases has increased over time, and vice versa for falling lines. Beyond flattening the curve, the end game is to have all of those lines reach zero.
  • Blue Lines: The darker the blue line, the more likely that the line is an accurate indicator, as thousands of tests have been administered per confirmed case. The more blue lines this chart shows over time, the better for us all.
  • Red Lines: By contrast, the warmer the color of the line, the fewer tests are being administered per confirmed case, and it is less likely to be an accurate measure of COVID-19 cases. Red lines, for example, indicate that only five tests are conducted for every confirmed case, suggesting that the count is not accurate and that many cases are going unreported.

Epi Curve Scenario Examples

Consider these three scenarios in the diagram above, and hover over countries in the main visualization to compare:

  • Country A: Winning the fight against COVID-19.
    These countries, like New Zealand, have steadily increased the number of tests per confirmed case. Country A administers hundreds or thousands of tests per confirmed case. The likelihood of missed cases is far lower, most cases are accounted for, and they can confidently state they are winning the fight against COVID-19.
  • Country B: A severe, prolonged outbreak.
    In comparison, countries like the U.S. have experienced a steady rise in confirmed cases. They also have lower rates of testing—only five tests per confirmed case. Country B cases are likely to be higher than the number reported, a fact that is especially concerning given that the U.S. has already surpassed the rest of the world’s countries in confirmed cases.
  • Country C: A volatile scenario.
    While confirmed cases decrease, there is much room for doubt. In Country C (South Africa for instance), confirmed cases are decreasing, but very few tests are administered. Unfortunately, this indicates there are many unrepresented cases. Country C probably has a larger problem than its downward trajectory would indicate.

Cases Per Million People

From a different angle, we can see daily new COVID-19 cases per capita. This gives us a better sense of how countries compare in terms of confirmed cases.

Countries like Thailand, New Zealand, and South Korea all show relatively low rates of COVID-19 per capita, as well as high levels of testing. Conversely, countries like Spain and Kuwait reveal high levels of confirmed cases per capita and extremely low testing rates.

Another Perspective for Good Measure

For a holistic view of testing, the map below shows us the daily number of tests for each newly confirmed COVID-19 case, based on a rolling 7-day average.

Countries like Norway, Australia, and Canada reveal strong testing-to-confirmed-case ratios. In contrast, countries like Bolivia and the Philippines reveal the probability of out-of-control outbreaks.

Due to lower levels of testing in relation to confirmed cases, countries in red are more likely to leave cases unreported.

Making Sense of the Unknown

Although charts like these allow us to look at relationships between critical variables, there are no guarantees of what will come of this outbreak or any second waves.

The only certainty right now, is uncertainty. But with visualizations like this one—updated daily—we can at least stay up-to-speed with the knowledge curve.

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Markets

How the S&P 500 Performed During Major Market Crashes

How does the COVID-19 market crash compare to previous financial crises? We navigate different contextual factors impacting crashes.

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How the S&P 500 Performed During Major Market Crashes

Like spectacular market peaks, market crashes have been a persistent feature of the S&P 500 throughout time.

Still, the forces underpinning each rise and fall are often less clear. Take the COVID-19 crash, for example. Despite lagging economic growth and historic unemployment levels, the S&P 500 bounced back 47% in just five months, in a stunning reversal.

Drawing data from Macrotrends, the above infographic compares six historic market crashes—examining the length of their recoveries and the contextual factors influencing their durations.

The Big Picture

How does the current COVID-19 crash of 2020 stack up against previous market crashes?

TitleStart — End DateDuration (Trading Days)% Drop
Black Tuesday / Great Crash*Sep 16, 1929 — Sept 22, 1954300 months (7,256 days)-86%
Nixon Shock / OPEC Oil EmbargoJan 11, 1973 — Jul 17, 198090 months (1,899 days)-48%
Black Monday**Oct 13, 1987 — May 15, 198919 months (402 days)-29%
Dot Com BubbleMar 24, 2000 — May 30, 200786 months (1,808 days)-49%
Global Financial CrisisOct 9, 2007 — Mar 28, 201365 months (1,379 days)-57%
COVID-19 Crash***Feb 19, 2020 — Ongoing5 months+ (117+ days)-34%

Price returns, based on nominal prices
*Black Tuesday occurred about a month after the market peak on Oct 29, 1929
**The market hit a peak on Oct 13th, prior to Black Monday on Oct 19,1987
***As of market close Aug 4, 2020

By far, the longest recovery of this list followed the devastation of Black Tuesday, while the shortest was Black Monday of 1987—where it took 19 months for the market to fully recover.

Let’s take a closer look at each market crash to navigate the economic climate at the time.

After the Fall

What were some factors that can help provide context into the crash?

1929: Black Tuesday / Great Crash

Following Black Tuesday in 1929, the U.S. stock market took 7,256 days—equal to about 25 years—to fully recover from peak to peak. In response to the market crisis, a coalition of banks bought blocks of shares, but with negligible effects. In turn, investors fled the market.

Meanwhile, the Federal Reserve Board rose the discount lending rate to 6%. As a result, borrowing costs climbed for consumers, businesses, and the central banks themselves. The tightening of rates led to unintended consequences, with the economy capitulating into the Great Depression. Of course, factors that contributed to its prolonged recovery have been debated, but these are just a few of the actions that had implications at the time.

1973: Nixon Shock / OPEC Oil Embargo

The Nixon Shock corresponded with a series of economic measures in response to high inflation. Soaring inflation devastated stocks, consuming real returns on capital. Around the same time, the oil embargo also occurred, with OPEC member countries halting oil exports to the U.S. and its allies, causing a severe spike in oil prices. It took seven years for the S&P 500 to return to its previous peak.

1987: Black Monday

While the exact cause of the 1987 crash has been debated, key factors include both the advent of computerized trading systems and overvalued markets.

To curtail the impact of the crash, former Federal Reserve chairman Alan Greenspan aggressively slashed interest rates, repeatedly promising to take great lengths to stabilize the market. The S&P took under two years to recover.

2000: Dot Com Bubble

To curb the stratospheric rise of U.S. tech stocks, the Federal Reserve raised interest rates five times in eight months, sending the markets into a tailspin. Virtually $5 trillion in market value evaporated.

However, a number of well-known companies survived, including eBay and Amazon. At the time, Amazon’s stock price cratered from $107 to $11 while eBay lost 75% of its market value. Meanwhile, a number of Dot Com flops included Pets.com, WorldCom, and FreeInternet.com.

2007: Global Financial Crisis

Relaxed credit policies, the proliferation of subprime mortgages, credit default swaps, and commercial mortgage-backed securities were all factors behind the market turmoil of 2007. As banks carved out risky loans packaged in opaque tranches of debt, risk in the market accelerated.

Similar to 1987, the Federal Reserve initiated a number of rescue actions. Interest rates were brought down to historical levels and $498 billion in bailouts were injected into the financial system. Crisis-related bailouts extended to Fannie Mae and Freddie Mac, the Troubled Asset Relief Program (TARP), the Federal Housing Administration, and others.

2020: COVID-19 Crash

In 2020, historic fiscal stimulus measures along with trillions in Fed financing have factored heavily in its swift reversal. The result has been one of the steepest rallies in S&P 500 history.

At the same time, the economy is mirroring Great Depression-level unemployment numbers, reaching 14.7% in April 2020. In short, this starkly exposes the sharp disconnect between the markets and broader economy.

Bearing Witness

History offers many lessons, and in this case, a view into the shape of a post-coronavirus market recovery.

Although the stock market is likely rallying off Fed liquidity, investor optimism, and the promise of potential vaccines, it’s interesting to note that the trajectory of this crash in some ways resembles the initial rebound shown during the Great Depression—which means we may not be out of the woods quite yet.

As the S&P 500 edges 2% shy of its February peak, could the market post a hastened recovery—or is a protracted downturn in the cards?

This graphic has been inspired by this Reddit post.

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