India can’t wait 12 months for a vaccine — here’s why.

Photo by Sujeeth Potla on Unsplash

In my last article, I discussed the act of flattening the curve and its purpose. If you haven’t read it, I highly advise you do so (The link for the first paper is here. The link for the second one is here).

It’s difficult for a country like India to flatten the curve, but in theory, when it is done, the healthcare system is able to give a satisfactory amount of care to those in need. But what’s important to note is that even if the healthcare system could handle the epidemic, there exists a mortality rate that is far too high to go by unignored. It is true that flattening the curve is more vital relatively speaking, but should a preventative measure or cure not exist, the toll will be massive.

Many people seem to misunderstand the role of flattening the curve. The simplified takeaway is that it is not a substitute for a vaccine or cure, and will not impact the current mortality rate.

The curve above represents the number infected at a given time over the duration of the epidemic. However, the total number of cases does not change when the curve is flattened. The same amount of total cases will occur over a longer period of time when the curve is flattened, contrary to the fallacy that we are reducing the number of cases overall.

So if flattening the curve won’t affect the current mortality rate, how severe will the crisis be?

International sources claim it can take anywhere from 12 to 18 months to develop a vaccine, and a cure will not be produced in the foreseeable future.

You may be familiar with the concept of herd immunity. It is defined as “a situation in which a sufficient proportion of a population is immune to an infectious disease (through vaccination and/or prior illness) to make its spread from person to person unlikely” by the Center for Disease Control. The premise is that once the majority of a population becomes immune, the whole population will develop an indirect resistance.

Once enough of the population develops an immunity, about 60–80% of a closed system (this can range from an apartment complex to even a country), that community becomes immune. As the recovered population increases, the infectious population decreases. Moreover, the recovered population is incapable of spreading the disease further, hence, the vulnerable population is less likely to come into contact with the disease.

It’s difficult to give precise numbers because there are numerous variables in the spread of disease: population density, how easily a disease spreads, the measures to prevent spread, etc. However, the trend is quite simple.

To understand how a society will develop herd immunity, imagine this: In an office, a young man develops a case of Z-Virus. Oblivious to his condition, he accidentally spreads it to three of his colleagues. The next day, he decides to stay home because of the sickness, but his colleagues go to work and spread the virus to another three people each. The next day, the spread continues exponentially. A few days in, however, the same young man who initially was infected recovers and returns to work. Even after coming in contact with an infected colleague, he remains healthy because he has developed immunity. As more people recover and return to work with immunity, it becomes less likely that an infected person happens to come into contact with a susceptible person. After enough time, the “R0” factor — the number of people that an infected individual spreads the virus to — will decrease from three down to two (because one out of the three interactions is with immune or already infected colleagues). From there, R0 drops to one and eventually zero. So many workers have immunity, that it is unlikely for the very few infected workers to come into contact with the very few vulnerable workers.

Since India is so densely populated, the percentage recovered likely needs to high close to 75%, to compensate for the more common human interaction. Now, coming back to the original question: how severe will the crisis be? If the country is unable to distribute vaccines or alternate preventative measures, herd immunity will naturally prevent all of the population from being infected. The mortalities will not just be the rate multiplied by the population, but a constant of 0.75 will be introduced into the expression to represent the percent of the population that will actually be infected.

If India’s only defense mechanism was herd immunity, the death count would still be catastrophic. The reality is that the discussion of herd immunity as a solution may be viable in other countries with remarkable healthcare (leading to a low mortality rate), but for a country as populated as India with a high mortality rate, it seems that dependence on this as a strategy is all but impossible.

This leads to the thought that in order to decrease the mortality rate, India must be able to produce and distribute vaccines fast enough so that less than 75% of the population is infected; this is the bare minimum, and we need significantly less than 75% to prevent these many deaths. Then, at the current growth rate, how many cases will we witness by the time we develop and have ready a vaccine? Will it be too late?

To understand the growth, we’ll look at the latest data on Coronavirus growth. I’ll once again be using the compound daily growth rate formula (CDGR). Instead of using the full duration of the lockdown, I will be using data on the growth in the last 14 days. This is because the rate of growth has actually slowed over the course of the quarantine.

The growth rate is vastly lower with the introduction and effect of the lockdown, but we have to see whether it is sufficient enough to allow officials the time to develop a vaccine. What will the number be in 12 months? Instead of using a standard exponential curve, I’ll be using a logistic equation. This takes into account a constant called the “carrying capacity” — a limiting value on the number infected — while still demonstrating the exponential growth of a virus.

This curve is much more confounding than the exponential curve, but the use of this curve is essential since it takes into account that the growth of the virus will slow down and eventually plateau in the long run. The standard exponential curve will only be a reasonable approximation in the short run.

To clarify, the constant k in the equation above represents a rate without practical limitations. We already found this when calculating the growth rate over the past 14 days. The assumption, however, is made that this rate is constant when in reality it may change slightly just in a matter of days, weeks, or even months. Evolving circumstances make the rate somewhat volatile.

By manipulating the equation, we will be able to find the growth with respect to time.

This is a lot of complex mathematics, and it’s okay if this is incomprehensible. All you need to know is that we took an equation that allows one to derive the rate of change of the population infected. However, we want to see the population infected with respect to time. After several manipulations, we were able to find the relationship between P and t. Since we’ve already found m and k, we can substitute their values.

The constant A can be found by substituting data from a specific day. I will consider May 27th to be Day 0.

This leaves us with the following as the final equation establishing the relationship between time in days and the population infected with Day 0 being May 27th.

Notice that the plateau occurs shortly after 200 days. We know that the number infected approaches the threshold for herd immunity by 365 days, meaning that even the best-case scenario of 12 months is too long. This means that at the current rate, the vaccine will be ready too late. The population will theoretically already develop immunity, and the 29M+ mortalities will already have been realized.

The way to solve this problem is by changing the unrestrained growth rate, k. If this value is decreased further, the curve will stretch. It will take longer for the infected population to approach the carrying capacity. In order to half the number of mortalities, k must be found for the point in which the population infected is equal to half of the carrying capacity by 365 days.

From now on, if the average unrestrained daily growth rate can be brought down to 2.4% for a year, the number of mortalities can be brought down by half. This equation can be used to find the unrestrained daily growth rate given a specific time and population. What this does tell us is that the current 5% CDGR is still far too high and must be reduced further to make the wait for vaccines worthwhile.

To summarize, flattening the curve alone will just stretch the duration of prevalence of the Coronavirus. In addition to supporting the healthcare system, the purpose of flattening the curve is to also provide experts enough time to develop a preventative measure, almost certainly to be a vaccine. However, the curve has not been flattened enough still, and tens of millions will die at the current rate. The vaccine will be ready to distribute far too late and the 75% of the population that will be infected will already have recovered (the other 25% will develop an indirect resistance). Using the differential logistic equation, we can derive an approximation of the growth of Coronavirus in the long-term. We can then use this approximation to find out how much the unrestrained growth rate must be to reduce the number of deaths because the immunization will be ready in enough time.

The truth is that these calculations are approximations based on public data and theoretical equations. There are so many active variables that the approximations may prove to be incorrect. When looking at a longer period of time, precise predictions are unlikely to be correct but are usually close. The predicted general trend/relationship, however, does match the situation. I can say confidently that lockdown is helping, but for a country like India that has so many people that are so close together, the growth rate will have to be even lower than it is right now.

The human mind is trained to think proportionally. If you could have a $2 discount off a notebook, you’d be much happier than a $2 discount off a car. Even though you save the same amount of money, $2 is just too insignificant in comparison to the price of the car. This thought process would be a blunder for the current crisis. 29 million lives are too significant to be just a small percentage. Remind yourself constantly of the number and not the proportion because only then will you remember the true severity and need for change.


Student in Delhi NCR. Using math to better understand the events of the large blue marble we stand on.

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