How astronomy can help us understand the spread of a pandemic
Article by Michael Burton, Director of the Armagh Observatory and Planetarium
The Hill of Infinity of Armagh, in the Astropark of the Observatory and Planetarium, contains a scale model of the universe, illustrating the place of planet Earth in the immensity of the Cosmos. How can this possibly be related to understanding the pandemic caused by the spread of the COVID 19 coronavirus around our planet? It is because the underlying concepts, between way the scale model of the universe has been constructed and of our understanding of how the growth in the number of cases of the coronavirus occurs, are the same. They are based on the principle of exponential growth.
Let us explain. We have become familiar during the lockdown with the concept of the R-factor for the coronavirus. Of the importance of lowering its value if we wish to slow the spread of infection. The R-factor is a measure of how easily the virus can spread, simplified into a single number. It represents the number of people that a person already infected with the virus will pass it onto, on average. Let’s suppose, for an example, that the R-factor is two, and for simplicity that the transmission occurs within a day (the concepts are the same if the transmission time is longer, but it is easier to illustrate if we stick to a day). Then, after one day, two more people have been infected, making a total of three. After two days, they in turn each affect two further people, making a total of seven cases. And so on. If we keep on multiplying by two and adding, we will find that there are 255 cases by the end of the first week.
In contrast, suppose the R-factor had been kept to one. Now each person only infects one other. Then the number of cases increases by one per day. After a week there are a total of 8 cases. We show these comparisons in the first set of graphs (Fig. 1), it is easy to see that there are about 30 times as many cases when R=2 as when R=1. However R=2 might be too high, you say? But it is easy to calculate the number of cases for whatever the correct value of R is. So we also show the scenario when R=1.5 here as well. In this example, after one week the situation is much better, there are five times fewer cases than when R=2. However, the situation is still six times worse than when R=1.
This is an example of what mathematicians call exponential growth. Numbers are small at first but rapidly rise to dramatic numbers. We illustrate this in the second set of graphs (Fig. 2). The time axis has been extended from one week to one month after the first case occurs. Now the plots are dominated by the scenario when the R-factor is 2. By continuing to double each day we find that there are over 4 billion cases after one month! It’s impossible to even see how many cases there have been for the other two smaller values for the R-factor, so large is the number cases recorded when R=2. This is the power of exponential growth! In practice, the growth cannot keep up at this rate indefinitely. Ultimately, resources will limit the total number of cases. For instance, in the spread of a disease you clearly cannot infect more people than the total population that exists!
However, the key in limiting infection is to slow down the rate of growth so we don’t reach such disastrous levels. The plot in Figure 2 does not help us to understand what is needed. We need to replot the graphs in such a way that we can see how the rate of infection grows for each of the values chosen for the R-factor. So we change the axis showing number of cases from a linear to a logarithmic scale (Fig. 3). With this new scale the horizontal lines are no longer evenly spaced apart, rather along each line there are ten times as many cases as along the line below it. This allows us to use a single plot to show the behaviour for one case as well as for a billion. There is still a dramatic difference seen between the three scenarios. This time, however, by reading off the scales we can see that when the R-value is 2 it is 30,000 times worse than when R=1.5, and an incredible 100 million times worse than when R=1! A truly astronomical difference.
It is also apparent what is meant by the term flattening the curve, as heard repeatably in the daily media briefings from governments. The curve for R=1 is flat, it is not for R=2. If we can achieve a flat curve then we can control the spread of the virus, so limiting the number of people who catch it to relatively small numbers that our public health systems can then cope with.
So how does this all relate to astronomy?! In the Astropark surrounding the Observatory and Planetarium in Armagh there is a scale model of the cosmos that you can walk around, illustrating what the planets, stars and galaxies are. Actually, there are two scale models, one for the Solar System, and one for the Universe itself. The first model uses a linear scale for its representation, the second a logarithmic scale, just as in the two representations we were using to describe the spread of the coronavirus. In the case of the cosmos it is distances between the different types of celestial objects that are growing exponentially, not the number of cases, but the principle is the same.
Along the Solar System walkway is laid out a scale model of the Sun and planets, spaced at their right relative separations (Fig. 4). The Sun is five steps from Earth, representing 150 million kilometres. While these distances seem vast to us, we can send spacecraft to the outer planets, their journey times are measured in years, they are comprehensible. A linear scale allows us to represent them, a gentle stroll in the Astropark then lets us visit all the planets.
However, the walk soon becomes impractical should we decide the leave the Solar System and head to the stars. The nearest star, Proxima Centauri, would be found close to Paris. Perhaps achievable with determination? But in order to reach the centre of our Galaxy we would have to walk a distance that is twenty times further away than the Moon, which is clearly not possible! We have to change scales to describe the cosmos beyond the Solar System. We move from a linear scale to a logarithmic scale, just as we did when describing the spread of the coronavirus.
This is how the cosmos is represented on the journey up the Hill of Infinity, passing from the Solar System, past the stars and nebulae in our Galaxy, through the realm of the galaxies, to the Big Bang at the formation of the universe itself at its top. The journey starts at the Hypercube (Fig 5.), where the display illustrates the concept of logarithms and powers of ten. There are three cubes in the Hypercube – small, medium and large. The large cube is ten times bigger, in every dimension, than the medium cube. Similarly when comparing the medium cube to the small. The principle here is that when walking up the Hill of Infinity a stone band laid in the ground is passed every ten steps. Each band represents being ten times further away than the previous band. So one hundred times further away than two bands away, one thousand times further away than three bands ago, etc. This is the principle of exponential growth being displayed on a logarithmic scale.
We then get a picture of how the cosmos is structured, a sense of the scale of the Solar System and the stars within our Galaxy, and of the scale of the Galaxy within the larger structure of the universe itself (Fig. 6). All the way to the moment of creation in the Big Bang, which also represents the size of the observable universe, the very greatest distance that it is possible to view to. Yet, while the concepts here are utterly different to those being used to understand the spread of the coronavirus, the principles used to display the concepts are the same in both, of exponential increase and logarithms to capture all the scales in the problems at hand.
However, with the Hill of Infinity there is one final treat when one reaches the very top (Fig. 7). Here we find a stone circle, a modern version of the kind our Neolithic ancestors used to mark out the passage of the Sun in our skies, used as the first calendars. Here is a modern version of this calendar, showing where the Sun rises and sets at the equinoxes and solstices, the turning points for the seasons. As well as linking us from the mythology of the past to the science of today, and all that captures our imaginations about the heavens above. Finally, we can turn to take in the view of the City of Armagh, its two great cathedrals, and the beautiful green hills that surround.