The math behind the virus’ dangerous spread

You land a gig for three weeks and are given two options for payment:

Option A: You are paid P100 for the first day, P200 for the second, P300 for the third, and on and on up to 21 days. Each day, you are paid P100 more than the previous day.

Option B: You are paid P1 for the first day, P2 for the second, P4 for the third—also up to 21 days. Each day, you are paid double the amount of the prior day.

At first glance, option A appears more attractive. At the end of the first week, you get P100 + P200 + P300 + P400 + P500 + P600 + P700, which total P2,800.

For the first week, option B gives you only P1 + P2 + P4 + P8 + P16 + P32 + P64, which total P127.

For the second week, option A gives P800 + P900 + P1,000 + P1,100 + P1,200 + P1,300 + P1,400, which total P7,700. Not bad.

But look at option B in the second week: P128 + P256 + P512 + P1,024 + P2,048 + P4,096 + P8,192, for the princely sum of P16,256.

This disparity becomes clea­rer in the third week.

Option A gives P1,500 + P1,600 + P1,700 + P1,800 + P1,900 + P2,000 + P2,100, which total P12,600.

But with option B, you get P16,384 + P32,768 + P65,536 + P131,072 + P262,144 + P524,288 + P1,048,576, for the unbelievable sum of P2,080,768.

The fact that you will already exceed a million pesos on the 21st day (having started with just one peso on the first day and doubling the amount thereafter) goes against intuition.

Option A is easier to visualize: your pay increases by the same amount (P100) every day, an example of linear growth.

But in option B, not just your pay, but the increase in your pay, increases every day. This illustrates exponential growth.

Compound interest is also an instance of exponential growth.

But unfortunately, so are viral epidemics.

Bill Gates

The doubling time of the spread of the new coronavirus, which causes COVID-19, is significant.

In webinars, I use the illustration above to explain why health authorities worldwide are trying to prolong the period it takes for the virus spread to double. The longer the better.

People understand that more humans will be infected if quarantine is not set, but they find it hard to imagine how quickly the virus will spread, precisely because it does so exponentially.

Health groups look at the so-called reproduction rate, which is too complex to explain here. What is important is that we need to keep the virus reproduction rate below 1, as low as possible.

If it goes above 1, then more urgent measures would be needed.

In his Gates Foundation blog last April 23, Bill Gates describes it this way: “If you say that 2 percent of the population is infected and this will double every eight days, most people won’t immediately figure out that in 40 days, the majority of the population will be infected.” We need to work toward this opposite: the virus’ exponential decline.

“If every infection goes from causing 2.0 cases to only causing 0.7 infections, then after 40 days you have one-sixth as many infections instead of 32 times as many. That’s 192 times fewer ca­ses. Here’s another way to think about it: If you started with 100 infections in a community, after 40 days you would end up with 17 infections at the lower [rate] and 3,200 at the higher one. Experts are debating now just how long to keep [the rate] very low to drive down the number of cases before opening up begins.”

We are all tired of the lockdown, with its economic, social, mental and emotional consequences.

But given the math, is the lockdown necessary? “Overwhelmingly, the answer is yes,” Gates says.

“There might be a few areas where the number of cases would never have gotten large numbers of infections and deaths, but there was no way to know in advance which areas those would be. The [lockdown] allowed us to avoid many millions of deaths and extreme overload of the hospitals, which would also have increased deaths from other causes.”