The maths of COVID-19 from Plus magazine

By on 10/02/2021

The maths of COVID-19 from Plus magazine

by Marianne Freiberger

After weeks of tight COVID-19 restrictions we could all do with a breather. Many businesses are being hit hard by a lack of pre-Christmas trade and the rest of us are longing for some time with friends and family. Surely we could afford a stretch of time with more relaxed rules, and then tighten up restrictions again later? What difference can a few days make?

As we recently reported on Plus  magazine, the sad, but mathematically unshakeable, truth is “a lot”. The graph below illustrates why. It shows, for realistic assumptions, how milder rules can allow a steep exponential growth of cases over a relatively short period of time (the rising part of the red curve), which can’t be undone in the same amount of time using stricter rules (the declining part of the red curve). 

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The red curve shows that rise in relative incidence under milder restrictions, and then the slow decline after tougher measures are brought in after seven days (on day D). The coloured intervals denote the various time periods involved in bringing the relative incidence down to a target value. The exact meaning of this graph is explained in this article on Plus magazine.

The red curve shows that rise in relative incidence under milder restrictions, and then the slow decline after tougher measures are brought in after seven days (on day D). The coloured intervals denote the various time periods involved in bringing the relative incidence down to a target value. The exact meaning of this graph is explained in this article on Plus magazine.

A few days of relative freedom come at the price of a longer time spent under tough interventions. This will lead to all the health costs for those who catch COVID-19 as a result. And even just focussing on the economic case, a delay to intervention is only beneficial in the very short term: a longer time will be probably be needed in a higher tier, or worse, another lockdown in future.

You can find out more about this graph and where it comes from on Plus magazine. It illustrates just how essential mathematics is when it comes to understanding the pandemic.

Put briefly, the graph uses an exponential model to predict the course of the disease — such models are suitable for the kind of time period we are looking at, and more than enough to illustrate the principles involved. Of course, the prediction also depends on assumptions on the level of transmission under mild rules and under stricter rules — which brings us another idea we have explained on Plus  and which has now become a household concept: the  reproduction number R.

R0 and herd immunity

One very important parameter in disease modelling is the basic reproduction number of a disease, denoted by R0:  that’s the average number of people an infected person goes on to infect, assuming that everyone in the population is susceptible to catching the disease. As people recover from the disease and (hopefully) become immune, and as interventions such as lockdowns take effect, the number of susceptible people decreases. 

The effective reproduction number R of the disease, that is the average number of people an infected person goes on to effect in practice, becomes smaller than R0 over time as the effect of interventions takes hold. A relatively easy calculation (see this article on Plus magazine) shows that, in order to force the epidemic into decline, one needs to get the effective reproduction number down to less than 1. The model that produces the graph above assumes that R=1.2 for the mild set of rules and R=0.9 for the strict rules. With these values one week of the milder restrictions needs to be paid for with almost two weeks under stricter rules if we want to get the relative incidence of the disease back to where it was when the mild set of rules was introduced.

As we now all too well by now, it’s possible to get R to less than 1 with interventions such as lockdowns, but it’s hard keeping it there. This is where vaccination will (hopefully) come in.  The over-acrching long term aim of vaccination programmes will be to permanently get R down to below 1. When that’s the case then we have so-called herd immunity: not enough people can catch the disease for an epidemic to take hold. Another reasonably straight-forward calculation (see this article on Plus magazine) shows that, in theory, we can get herd immunity by vaccinating a proportion of 1-1/R0 of the population. For an R0 of 2.5, the higher end of the estimates for COVID-19, this means that we need to get at least a proportion of 0.6, so that’s at least 60%,  of the population immune.

Epidemic growth rate

While R is a useful measure of how the epidemic is behaving in many ways, one thing it doesn’t tell us is how quickly things are changing. This is because R is not a rate, there is no timescale involved. For diseases like HIV or TB, where there can be months or years between one person infecting the next person, even R=2 means slow growth over time. However for influenza or measles, where the infection is much faster, R=2 means very rapid growth.

A measure that does capture the time dimension is the growth rate of a disease.  If the growth rate is positive, the number of new cases each day is increasing, if the growth rate is 0, the number of new cases stays constant. What is needed to keep the epidemic under control is for the growth rate to be negative and hence the number of new cases to be decreasing. As an example, if the number of new cases has decreased by 3% since yesterday, then the growth rate r is, approximately, r = -0.03 per day. Our graph above corresponds to a growth rate of roughly 0.03 for the milder set of rules, and -0.02 or the stricter rules.

To find out more about the growth rate and how it is related to R, see this article on Plus magazine.

The models

Finally, we have a quick look at epidemiological models. Our example above uses an exponential model (find out more here), which is perfectly suitable for the time period involved. However, when we are considering longer time periods we need to take into account that the pool of susceptible people will shrink. 

This is why  the leading paradigm in longer term  disease modelling is the so-called SIR model, which has been around for over 100 years.  The general idea here is to divide a population up into classes (for example susceptible (S), infected (I) and recovered (R)) and to describe the way people pass from one class to another by mathematical equations that depend on particular parameters.

By linking up many SIR models representing geographical locations, such as towns, or other sub-populations, such as schools, one can then model the spread of the disease in a whole country. Crucial in determining the parameters are the contact patterns between people — who meets whom and how often — and these are inferred from social mixing studies. By adjusting those contact rates, one can then also simulate the effect of interventions such as school closures or social distancing measures. You can find out more about the maths of epidemiological modelling in a crisis in this Plus article. 

These are just some of the topics related to COVID-19 we have explored on Plus magazine. To read more about these and others — from using artificial intelligence for diagnosis to deciding football leagues when matches have been cancelled— head over to the Plus magazine COVID page.  

About Pierfrancesco Dionigi

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