Standstill. Lock-down. Quarantine. Another zoonosis. The world is grappling with a new crisis. In the last few weeks, our world has changed. We are reading how the COVID-19 disease is spreading, the different measures taken around the world. We are learning concepts of
epidemiology, thanks to the print, electronic and the online media. We are getting a glimpse of the simulation models used by epidemiologists to give an estimate of the pandemic; the SEAR model, the hammer and dance model etc. The simulation results help to plan the actions to control the pandemic. Variations and results of the real world actions could then be used to refine the model and plan further actions.
We, the lesser mortals, those who are not epidemiologists, could try to interpret the pandemic data, without going into the sophisticated models. What we need is a tryst with some basic mathematics and statistics. During the earlier stages of the pandemic, many of my friends were asking about the meteoric rise of the spread in Italy and later in Europe in quick time. If we analyse this ‘meteoric’ rise, we will be concerned but not surprised.
There is a concept called exponential growth which is manifested in different events in nature – in biology, physics, economics etc. The growth continues until the resources favoring the growth are exhausted. On the contrary, if a quantity decreases exponentially over time, we call it an exponential decay.
Many of us are already familiar with the term ‘exponentially’ and use it liberally. But we don’t always realize its implications in real life. When a quantity follows an exponential growth, starting with small numbers, we get shocked when we stumble upon big numbers in a short period of time, even if it is mathematically consistent. This is exactly what is happening with the spread of COVID-19. Let’s see a bit more in detail to view this pandemic as a mathematical reality. There are excellent resources available in the media regarding this.
When a quantity demonstrates exponential growth, the rate of increase of the quantity is proportional to the quantity itself. Hence a viral epidemic is an excellent example as new cases are caused by the existing cases.
If we consider the quantity, in this case the number of infected cases, as a function, then we can say that the quantity is an exponential function over time i.e. time is the exponent. Simply put, this means that the number of cases on a given day is the product of the number of cases of the previous day with a constant greater than 1.
We can write, from the definition of exponential growth, the number of cases
Nt = N0 (1 + r)t , where t is the number of days and r is the growth rate, N0: number of cases at t=0
After 1 day (t=1),
N1 = N0 (1 + r) = N0 + N0 * r
So, the increase of cases is the term N0 * r . The fact that the number of cases N itself is a factor in its increase, makes the process go fast.
Now, let’s see what is the rate r, in this case. If an infected individual is exposed to M number of people in a given day and if each one of those exposures have a probability p of becoming infected, then the increase of cases in a given day is M * p * N.
So, our equation becomes,
N1 = N0 + M * p * N0 = N0 (1 + M * p)
Nt = N0(1 + M*p)t
Lets try to interpret the data (source: Worldometer). In the bottom left plot (source: Worldometer.info), we can see the number of cases (vertical y axis) in the world outside China, as a function of days (horizontal x axis). This is what an exponential growth curve looks like. This becomes easier to interpret if we see the same plot in the logarithmic scale (bottom right), where the exponential growth would be a straight line. In the log scale, each step on the y axis corresponds to a multiplication by 10. On January 29, the cases outside China were 102. It took about 20 days to go from 100 to 1k (Feb 18), 13 days to go from 1k to 10k (March 2) and 14 days to go from 10k to 100k (March 16). We can find out analytically that the average number of days required to grow by a factor of 10 is just more than 2 weeks.
If we extrapolate these results, we could infer that it would reach 1 million cases in the first half of April, and 10 million by the 3rd week of April. In the real world, measures are being taken to avoid this.
We can do this analysis for the countries individually. Initially, in Italy, the growth rate was more than 30%, which came down to around 20% and subsequently even less. Countries with a lesser number of cases shouldn’t be complacent. Considering their growth rate, they might be only a few days away from reaching a much higher number. In India it seems that the exponential growth has started (third plot left).
Now, the question is when and how does this slows down mathematically?
Coming back to our equation, Nt = N0(1 + M*p)t , this slows down when M or p diminishes i.e. the term (1+M*p) diminishes towards 1. Then the exponential curve will move towards a logistic curve (bottom right).
There will be a point called the inflection point when the number of new cases per day stops increasing and remains constant before decreasing. So, when the growth factor (ratio between the new cases in a day to the new cases the previous day) is 1, we can say that the epidemic has hit the inflection point. Roughly, this would mean that the maximum number of cases will be about twice the number of cases where the inflection point is hit. Any subtle change of the growth factor is significant, since it would affect the number of cases in a very short period of time. In the current pandemic, countries even with excellent health infrastructure have found it overwhelming, given the elevated number of cases in a short period of time, especially those requiring intensive care. So, measures are required to reduce the growth rate, when the final number of infections would be big but the increase would be gradual over an extended period of
time. This would reduce the impact on the health system. This is what the experts term as ‘flattening the curve’.
So, we need to take measures to take down the growth factor towards 1, i.e. to reduce M or p towards zero in the equation, where M is the number of people exposed to an infected person and p is the probability that those people will get that infection
If the pandemic is uncontrolled, the exponential curve will anyway deviate from a true exponential when we approach the total population. In that case, the decay will also be rapid, since there wouldn’t be many unaffected people remaining.
Lets see how the growth rate reduces with controlled measures.
If we had a vaccine for COVID-19, and the M people had it, then the probability p would approach 0, reducing the growth rate.
Since we don’t have a vaccine, the most efficient way is NOT to be near an infected person, which would mean M=0.
In the initial stage of an infection, this would mean to isolate the people infected. But when the epidemic is spreading fast, this would mean a lock-down. Staying at home, isolated from the outside world, taking the precautions. That’s exactly what the world is doing right now. There is no room for complacency in this regard. A lapse could create havoc as witnessed in several countries in the initial stages of the epidemic.
In the real world, there are several other factors which affect the probability of getting the infection, p. This includes the type of virus, how it spreads etc. An infected person might meet a person who is already infected and so will not contribute to a ‘new’ case. Also, we need to take into account people who have already recovered or deceased. All these factors are studied by the epidemiologists to refine their models and to plan actions, which might be different for different countries. The number of cases reported and tested may not show the actual situation. There might be cases, where there will be mild symptoms and would recover without being reported. So there might be a difference between the actual and the predicted results.
To conclude, even if mathematically the ‘cases’ are just numbers, in the real world those are human lives and cannot be quantified. It is our duty to continue doing what is expected of us, to come out of this crisis.