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The math of coronavirus! How fast does an infection spread? Let's crunch some numbers

Coronavirus outbreak: Just like the tail of an exponential curve, in each country, the number of cases with time shows a very slow rise leading to complacence only to grow at a very rapid rate soon after

Dhanada K Mishra | April 2, 2020 | Updated 22:13 IST
The math of coronavirus! How fast does an infection spread? Let's crunch some numbers
Coronavirus pandemic: The USA, the UK and India are a few days if not weeks behind Italy and only time will tell how successful each of these countries has been with their varied strategies in arresting the pandemic

At the time of writing, the world is in the middle of a global pandemic that is causing untold miseries on hapless millions all over. Cutting across national boundaries, racial differences, wealth inequalities, social barriers, and numerous other dividers, the SARS-CoV-2 virus that causes the COVID-19 disease has brought the mightiest of nations on to their knees.

With hospitals overflowing, healthcare workers overworked, entire populations under lockdown, the exponential growth of infections shows no impact of all the counter-measures. There have been viruses such as Ebola and Zika which have been much more fatal almost certain to kill the host without fail.

However, the novel coronavirus in sparing the lives of almost 85% of its victims has posed a much bigger danger to humanity than its predecessors. The fact that the majority of those infected show no symptoms while still able to spread the virus has meant that the pandemic has been the most difficult to control.

Also Read: Lockdown may help flatten coronavirus curve in India, says study

Without a vaccine or any known cure, man is at his wits' end trying to buy time while scientists are working hard to come up with a solution that most estimate will take up to a year and a half to materialise.

One of the most tragic aspects of the unfolding crisis has been the inability of the nation after nation to appreciate the danger inherent in the nature of the disease best characterised by its exponential function.

Globally, it took just over sixty days for the number of COVID-19 infections to reach 1,00,000, around 12 to 14 days for them to reach 2,00,000, yet only 4 days for the next 1,00,000, only 3 days for the subsequent 1,00,000 and recently the infections have grown by 1,00,000. As of now, there are over 6,00,000 people infected with around 24,800 deaths worldwide making the fatality rate about 4.6% as per the data maintained by the Corona Virus Resource Center of Johns Hopkins University.

The late Professor Al Bartlett, a nuclear physicist and professor emeritus at the University of Colorado at Boulder once said "The greatest shortcoming of the human race is our inability to understand the exponential function." Had he been alive today, the ongoing coronavirus crisis would have provided the fittest case to prove his point. He was fond of proving how the exponential function is deceptive, using a wide variety of cases where it is applicable ranging from population growth to use of non-renewable fossil fuels.

The current policy paralysis in tackling the COVID-19 pandemic, so sadly illustrated in countries like Italy, Iran and Spain with many such as UK and USA heading along the same path, proves Professor Bartlett's apocalyptic comment.

Also Read: Coronavirus: India records biggest jump in number of deaths as 12 die in 24 hours

The exponential function

The simplest way to understand the exponential function is to know the difference between simple interest and compound interest. If you start with 100 rupees in the bank and the bank offers 10% simple annual interest, then your money will grow to Rs 200 in 10 years (doubling period).

The same Rs 100 when compounded at 10% interest annually would double in 7 years. If you keep doubling every 7 years, the original quantity becomes very large in a few doubling cycles. For example, over a period of 70 years (one's lifetime), the same Rs 100 at a compound interest of 10% will yield 100 times more than what you will earn at simple interest.

Another famous example closer to the corona pandemic is from Professor Bartlett's lecture where he gives the example of bacteria growth where the pathogen divides itself into two every minute. In the example, the bacteria are contained in a small bottle and the bottle fills up with it in exactly 60 minutes. At what time would the bottle be half full? The answer is not at the 30 min mark but a counter-intuitive 59 minutes after the start.

That is exactly one minute before the bottle completely fills up. It is half full or half empty (depending on how you would like to see it) only a minute before the hour. After 55 minutes into the bacteria growth process, the bottle would have been only 3.125% full or 96.875% empty. It is the same phenomenon that makes global warming and climate change, due to the greenhouse gas build-up process, so difficult to understand and appreciate.

What is R0 and why is it important?

In the now-famous 2011 movie 'Contagion', there is a term 'R0' (pronounced R naught) referred to frequently by the epidemiologists trying desperately to figure out how contagious the virus was.

This term which is also known as the 'Reproduction Number' was not invented by Hollywood, rather borrowed from science. It is used to describe the intensity of an infectious disease outbreak.

By definition, R0 stands for the number of cases, on average, an infected person will cause during his/her infectious period. In each of the previous disease outbreaks, during the 2003 SARS outbreak, the 2009 H1N1 influenza pandemic and the 2014 Ebola epidemic in West Africa, the R0 estimates played an important part in characterising the epidemic and modelling its likely spread.

It is something epidemiologists are racing to calculate for SARS-CoV-2, the virus that causes COVID-19. For example, the R0 of highly contagious disease like measles is 12-18 depending on population density and life expectancy, among other factors, whereas influenza or common flu has a relatively low R0 of between 2-3.

Measles outbreaks are therefore often explosive, whereas flu spreads in a less dramatic fashion. When R0 is less or equal to 1, the epidemic is controlled.

With COVID-19, we have seen reactions ranging from complete complacence followed by diffident actions to outright panic. None other than President Donald Trump tweeted early into the epidemic that a far larger number of people died in the USA from influenza than the likely toll of COVID-19.

The same Donald Trump has now declared an emergency and proclaimed wartime powers as the outbreak seems to have overrun the healthcare system and the economy in the most powerful country on earth.

As one looks at the graphs of the number of cases and the death toll, it becomes clear (Italy being the best example) that unless drastic measures of social distancing including a complete and effective lockdown of the entire populations are taken as was done in Wuhan albeit, after an initial delay, the disease is likely to overwhelm a country's healthcare system and ruin the economy.

Just like the tail of an exponential curve, in each country, the number of cases with time shows a very slow rise leading to complacence only to grow at a very rapid rate soon after.

The USA, the UK and India are a few days if not weeks behind Italy and only time will tell how successful each of these countries has been with their varied strategies in arresting the pandemic.

(The author is an academician currently visiting Hong Kong University of Science and Technology as a Research Scholar.)

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