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Walking through the pandemic modelling, exponential growth and herd immunity maze

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Dr Michael Gering is a strategy consultant and has consulted in a variety of industries both in South Africa and in Europe. He has a PhD in theoretical physics and an MBA from the Open Business School. Professor Mojalefa Ralekhetho is a graduate of Columbia University (Politics) and New School for Social Research (Economics). He is a management consultant and executive leadership coach. He has held executive and senior positions in universities (TUT, CUT) and government (DMRE, FSPG), and currently teaches a graduate course in economics at the Institute for Economic Research in Innovation at Tshwane University of Technology (TUT).

Is the concept of herd immunity against Covid-19 helpful? Are the models being used to predict the spread of the virus accurate? Scientists tend to have blind spots and one of them is being so rooted in a paradigm that they struggle sometimes to understand data outside their framework.

The Covid-19 pandemic has resulted in a prolonged and repeated exposure to the simple model of epidemiology, with its concepts of exponential growth and exponential decay. The simple model has built in a concept of herd immunity, the point where an outbreak is no longer able to sustain itself and rapidly decays.

It is not difficult to see, and is now common cause, that the simple model of epidemiology overestimates both growth of an outbreak and its peak. However, the concept of herd immunity is intimately tied to the model and new predictions of “herd immunity” are not all derived within its paradigm and must therefore be treated with caution.

The basic epidemiological model is known as the SIR model, where S stands for the portion of a population that is susceptible, I the portion of the population that is infected and R the portion that has recovered. The rate of infection depends on the number currently infected (those who can infect) and the number of people who are still susceptible (those who can get infected). The number of people recovering depends on who is infected and their recovery rate.  This sets up three simple equations, which draw on concepts from high school algebra and, for the more ambitious, a spot of first-year university calculus.

The basic SIR model predicts a rapid rise in the number of infected people, reaching a maximum and then dying off rapidly to zero. The model predicts that once enough people have been infected, the circulation of the disease fades and disappears. The growth and the subsequent decay are both exponential, with the exponent being the reproduction number R(t): when R is greater than one, the infections explode; when R is less than one, they disappear. The SIR model predicts that epidemics either explode or fade away. Much is made of the reproduction number at the core of this model; many websites plotting the unfolding of the epidemic calculate and plot on the basis of this number.

The basic SIR model also introduces the concept of herd immunity, the tipping point where the exponential growth becomes exponential decay. The basic model also makes a clear prediction: that point is reached when roughly 70% of the population is infected.

The SIR model was developed in the 1920s, and a lot of science has happened since then. Nevertheless, it would be fair to say that there has been an explosion of people who have dusted off their high school algebra, delighted in the knowledge that the word ‘exponential’ carries a literal as well as a figurative meaning, and who have taken it upon themselves to explain herd immunity and the projections of the SIR model as though they were fact rather than what they are: a simple model requiring scientific and empirical corroboration.

There have also been more cautious voices, such as our very own Nobel laureate Michael Levitt, who would have learnt his high school algebra in Pretoria. In the first months of the pandemic, Levitt, who won the Nobel prize in chemistry for his modelling work on complex chemical systems, plotted the outbreaks. He commented that, in practice, the growth never was exponential. Later evidence shows that the tipping point of an outbreak, while still at unconscionably high levels, was always much less than the one concrete prediction of the simple SIR model.

Populations are not uniform (homogeneous). In fact, they are lumpy (heterogeneous) in at least three ways: transmissibility, susceptibility and connectivity. Each of these potentially has a big impact on the disease trajectory.

In fact, some really neat work has been done by a team led by Gabriela Gomes, professor of mathematics and statistics at the University of Strathclyde, over a period of more than a decade.

Professor Gomes and her team modelled outbreaks by assuming that people are differently susceptible; some catch the disease more easily than others. As the outbreak progresses, people with lesser and greater susceptibility will both stand a chance of being infected. However, a greater proportion of susceptible people will be infected and a smaller fraction of less susceptible people. Over time, the population becomes overall less susceptible. We come across such models in junior high school when we study dynamic equilibria such as evaporation.

Of course, no one knows how susceptibility is actually distributed in a population. What scientists typically do is to parametrise the distribution and fit the data. One day, in the future, we may find out how to measure this directly and then we will be able to check the parameterisation; that is the way science progresses. Professor Gomes has modelled many outbreaks, used this to estimate variabilities in susceptibility and shown that this hypothesis accounts for the data far more accurately than the simple SIR model.

This is one of the three mechanisms. People have varying susceptibility; they live in clusters which have varying exposure to other clusters and they take different measures to reduce their exposure. There is some evidence that in epidemics with a propensity for superspreading events, the cluster effect is important.

The effect of homogeneity will require an understanding of the data. The Gomes model predicts a tipping point that, while still unacceptably high, is much lower than the prediction of the SIR model. What does the data say about the effect of heterogeneity on the Covid pandemic?

The concept of “herd immunity” is an exponential concept based on the SIR model. Reach herd immunity and the virus disappears like the morning mist. Until then it explodes. In practice, the curves were never exponential. Far from rapidly exploding or rapidly decaying, new infections tick along.

The US, for example, ticks along with a relatively constant infection rate. From April to June it was about 25,000 per day. Between August and September more like 40,000. It would be a brave scientist to put an exponential curve to these numbers.

In fact, the reproduction number R(t) is one of the widely quoted numbers in the pandemic reporting, with the assumption that R greater than one will result in an explosion and R less than one in a disappearing pandemic. There are websites calculating R with great sophistication using Bayesian models to get more and more accurate calculations of R(t).

One such website is https://rt.live/ giving R(t), the reproduction number for all 50 US states. At the time of writing, all 50 states had reproduction rates close to unity. In all but two, unity was within the calculational error. The empirical data looks much more like a flat line and much less like an exponential. Exponential growth is mathematically an explosion that grows off itself until it burns itself out and fades away. The data is not exploding, and it is not dying out either.

The spread of a virus appears to be more like a widespread fire which flares up in places and which does not seem about to burn itself out. Populations are connected in a lumpy way. People interact in local, regional and geographical clusters of different sizes, which are differently permeable. And, within those clusters, people distance themselves differently. With the virus circulating, the spread is more like a fire flaring and dying out in many different clusters simultaneously but leaving flammable material that might well flare up again.

So, what does this mean? We do have to be careful of policy decisions based on a simple modelling where we might not have fully understood the inherent assumptions. Scientists tend to have blind spots and one of them is being so rooted in a paradigm that they struggle sometimes to understand data outside their framework. Models might suggest that outbreaks will no longer explode uncontrollably, but what affects the steady state, the burn rate, and what if that is unacceptably high?

In summary, we suspect that the SIR model is not as helpful once the virus is circulating freely and its metaphors might be misleading. Herd immunity calculations may be the point where the virus no longer explodes but may not be the point where the virus dies out.

Underlying parameters such as heterogeneous susceptibility can’t be measured directly, so we need new models and methods for calculating their policy impact.

The underlying level of infection will depend critically on the extent of the social measures in place. DM

Dr Michael Gering is a strategy consultant and has consulted in a variety of industries both in South Africa and in Europe. He has a PhD in theoretical physics and an MBA from the Open Business School.

Professor Mojalefa Ralekhetho is a graduate of Columbia University (Politics) and New School for Social Research (Economics). He is a management consultant and executive leadership coach. He has held executive and senior positions in universities (TUT, CUT) and government (DMRE, FSPG), and currently teaches a graduate course in economics at the Institute for Economic Research in Innovation at Tshwane University of Technology (TUT).

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  • Bernhard Kirschner says:

    Great explanation, but to assume that the social measures, or to put it bluntly restrictions will determine the level of infection is an unproven assumption. Many are looking at Ivor Cummins graphs which show that the mortality levels and trends often have no relationship to restrictions. After all, it was mortality that was foremost when restrictions our governments imposed restrictions to “save lives”. Additionally looking at the graphs of many countries the rate of increase was most faster than the speed of decrease, which illustrates as you say the limitations of the basic SIR model.
    What may be missing from the above models is an extra variable, the less-discussed adaptive immunity. Adaptive immunity increases with greater exposure to the virus, or what doesn’t kill you makes you stronger. As the amount of virus in the community increases, initially, there is more infection, with increasing acceleration as numbers reach a critical point. With greater contact with the virus, most will develop robust resistance to the virus, so infection numbers then start falling, but slower than the rapid rise. Immunity in most minds means reasonably long term protection, but in this case, the adaptive immunity may be relatively short, so perhaps a better word would be resistance. This resistance might only last until the next wave or next year.
    Additionally, there will be many more who will be less sick due to growing resistance both in the first wave and subsequent waves.
    We might consider why there was no discussion on asymptomatic patients earlier in the year, particularly in Wuhan. Earlier patients may have had suffered more severe disease because of the novelty of the virus with little adaptive immunity. Many later patients having been incrementally exposed to the virus only became asymptomatic.

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