Welcome to The Plague Pit

In this issue – number 11 – I’m delighted to present this article by Dr John Cullerne, Undermaster at Winchester College, where he teaches Physics and Maths.

In September 2014, at the height of the West African Ebola outbreak, my General Studies group (we call this Div here at Winchester College) began reading Don Taylor’s The Roses of Eyam, an extraordinary play about villagers in 1660s Derbyshire, who quarantined themselves when the Black Death arrived in their midst. Little did I know that this would begin a journey culminating in this little article on modelling epidemics for non-scientists?

General Studies has a tendency to move off at tangents, and my group was no exception, persuading me to “do the maths” of the Eyam epidemic. However, not all of them were studying maths, so I began a project to explain it in non-technical language. Inspired by William Phillips, the famous Economist, I decided to build a Phillips machine. Phillips taught Economics at LSE in the 1950s, and encountered a problem: how could non-mathematicians learn the concepts needed to describe the movement of money around an economy? He used his engineering background to design a machine that used water to represent money as it flowed around the economic system. Using valves, pulleys, levers and pumps, Phillips created a two-metre tall hydrodynamic computer that modelled an economy.

The head physics technician and I put together a Phillips machine to model the Eyam epidemic (please see the photos at the end of this article). It is a bit “Heath Robinson”, but it illustrates things quite well, I think. This is how it works: there are three reservoirs of water, the top represents those who are susceptible to the disease (S), the middle one represents those who have been infected (I) and the bottom, those who have recovered (R). We will also make a simplifying assumption that those who recover become immune (there is of course a big debate about this for COVID-19 at the moment):

The flow from S into I depends on how much water is in S (representing the susceptible population) and the amount of water in I (representing the infected population). More water in S(a larger susceptible population) provides a greater pressure pushing water into I. Also, the I reservoir is on a lever (see photo), which controls a valve, so more in I causes a yet greater flow from S to I. This is modelling the idea that the infection spread is greater if there are more infected and if there are more who are susceptible to be infected. We will see this important dependence on the susceptible population again a little later on.

You will also notice that the reservoir I supplies water to the “recovered” reservoir, R. The rate at which this happens also depends on the I population as more in I provides a greater pressure to push water into R. This bit models the recovery of those infected and we can see that the whole system, when these elements are combined, gives us a dynamic situation in which the water level in the I reservoir has three possible situations:

1.          The level in I rises if the flow from S is greater than the flow into R – we have an epidemic.

2.          The level in I reaches a maximum when the flow from S exactly matches the flow into R – we have reached the peak.

3.          The level in I will fall when the flow from S is less than the flow into R – we pass beyond the peak. The epidemic ends.

The diagram below may help to visualise this better:

This diagram is a model of the Eyam epidemic in 1666, with the black squares representing the actual numbers of infected at the time, and the red and green dots are our mathematical model using the same principles that have been described by the water analogy above. One point to add, almost no residents infected in Eyam survived, so sadly the R group here represents those who passed away during the epidemic.

We can now answer the question as to why epidemics end at all. It might be a surprise that if we introduce an infection into a susceptible population, not all those susceptible seem to get it. We have made the assumption, for this simple model, that those who recover become immune, so we see that the reason our epidemic ends, is that the number in the susceptible population drops to a threshold value, below which an infected individual becomes more likely to recover than to go on and infect another susceptible. The important number here is the threshold susceptible population, and depends on the likelihood of contact between the susceptible and infected populations. The threshold is smaller; the epidemic starts in a smaller population, if there is a lot of contact. Conversely, the threshold is much larger if we can reduce contacts. In fact, for a given infection and level of contact, the threshold may be estimated. The number of times the initial susceptible population is greater than this threshold is called R_o (pronounced R-naught), and is the number the government scientists keep quoting. It is also interpreted as the average number of others an infected person may go on to infect, and has been as much as 3 for COVID-19. R_ois now less than 1 in the community, so we are below the threshold and so we no longer see the exponential rise in infections.

Indeed, the social distancing measures are designed to change the conditions in the system so that the flow from the S reservoir into I is effectively throttled. Prof. Patrick Valance often stresses how a peak in the UK COVID-19 cases is not a natural peak, but rather a supressed one. This also helps us to better understand why different governments around the World have approached things in such different ways. Imposing social distancing too early would reduce the size of the initial peak, but would prolong the epidemic curve because there would be only a trickle into and out of I, and without a viable vaccination programme to provide some herd immunity safely, economies would not survive the duration. Applying the restrictions too late would lead to an overwhelming of the healthcare system as the flow from S into I runs away with itself in the so-called exponential phase of the epidemic (flow from S into I  >> flow from I into R). The lifting of measures, whilst we have a suppressed system is also, therefore, fraught with dangers – the balancing act will have to be replayed, but with a whole lot of new initial conditions. The conditions in each country are so different that this balancing act has led to very different timings for various measures and the lifting of those measures. 

I was extremely lucky that our work on Eyam caught the eye of an Oxford-based epidemiologist, Dr Robin Thompson, when he came to give a Studium talk, and that link gave me the opportunity to take up an academic visitorship at the Maths Institute at Oxford University. During my time there, I was fortunate to have spent some time with Prof. Dame Angela McLean, who also has a great interest in physical models of epidemics, primarily for teaching purposes. Prof. McLean, as Chief Scientific Advisor to the MoD, has provided some of the scientific presence on the daily Downing Street briefings. She is also one of the foremost scientists at the Institute of Emerging Infections of Humans. She enjoyed my water model and very kindly lent me another example of a physical model of an epidemic, which I used in a lecture I gave in October 2019.

Prof. McLean’s model (called the Reed-Frost Model), uses beads of different colours. There are 80 white beads representing those susceptible to the infection and 80 blue “blocker beads” that will separate out the susceptibiles, and one red infective bead. The beads are put in a jar and mixed, and then poured out into a wooden gutter. For example:

Groups of white susceptibles are separated into “households” randomly by the blue beads, and if we have an event like the one in the picture, all the white beads in the household with the red bead would be turned into red, and the previously red bead is turned into turquoise (recovered). We then return all the beads to the jar, mix up, and then pour out again. By repeating this process over and over we can “run” a number of these epidemics and investigate their nature. Interestingly, on more occasions than you might imagine, the epidemic dies out before it ever gets to start because the red beads, when their number is low, have quite a high chance of being flanked by blue beads. Indeed, social distancing might be modelled in this system by adding more blue beads! This is also a nice illustration of the element of chance in these biological processes. 

I am sure you will appreciate that both physical models described here (Phillips and Reed-Frost), whilst they are useful as ways of explaining some of the inner workings of these things, are not useful when it comes to actually modelling real systems. Here is where coding comes in, and it is the use of computer modelling that has made it possible to provide data to guide governmental decisions. However, the algorithms, or sets of instructions, that are processed by billions of operations per second in supercomputers, are essentially more sophisticated versions of the simple models covered here.

Finally, a point about where we are in this epidemic: Hindsight is a wonderful thing. Could the Containment Phase have been more aggressive? Yes, probably, but the amazing personnel in the NHS and other essential workers, the scientific community, and indeed all of us, have bought us some time. Testing, tracing all contacts, and isolating all potentially infected (Test, Trace, Isolate, or TTI) may well have stopped things getting to this stage earlier on in the epidemic, but were we in a position to be able to administer and implement such a scheme nationally? General Sir Nick Carter said that the Coronavirus has been the greatest logistical challenge in 40 years’ service. We will learn lessons, but we need to get the numbers of infected down to the lowest we can before we lift social distancing measures so that TTI can be effective. Tracing the contacts of large numbers of infected individuals is an impossible task, so we have to return to pre-epidemic levels for this to be viable. TTI will have to be the way we slowly emerge from the lockdown – essentially returning us to the Containment Phase, and human trials of a vaccine, to provide immunity safely, have started in Oxford. The scientific community is working flat out and we must all try to be hopeful and patient.

Phillips’ Machine on the Left and The Eyam Water Model on the right.

Dr John Cullerne

When the Science Museum, London, opened the new Winton Gallery for mathematics in 2016, the exhibits included a working Phillips’ “MONIAC”. I don’t know if it’s still on display – but, if it is, it must be feeling pretty lonely right now. G.S.