Disease dynamics deconstructed

This is a walk through the simplest model of the spread of COVID-19 and what even a simple model can tell us about all those curves we see in the press, exponential growth, herd immunity and so on. It's light on equations, but there are a few for those who like that sort of thing.

Usual caveat: I'm not an epidemiologist. I am a professional mathematician, who teaches mathematical modelling to business decision-makers to help them ground their decisions in data and informed expertise.

Another caveat: This model is pedagogic, it's purpose is to explain some of what we're hearing and seeing in the news. It's too simple to make predictions and the numbers in my examples are pure fiction.

The simplest model

This is the simplest model we can build for the spread of a disease in a fixed population. It's called a compartmental model.

The compartments

If you're susceptible, you can catch the disease: you have contact with infectious people and you haven't had the disease before. That's everyone. No-one had COVID-19 before; that's the whole problem.

In this simple model, we lump together people who have been exposed to the virus, but are still incubating; people who are infectious, but not symptomatic; and people who are actually sick. We call all of these infectious. They are infected and not yet either recovered or dead.

Removed is either recovered or dead. Separating these will improve the book-keeping, but it won't make a huge difference to the model. The point is that once you're recovered or dead, you can't infect anyone else.

The arrows

The model tracks the number of people who move from one compartment to another in a given space of time. The transfer from one compartment to another is what the model is all about.

For illustration, let's say we start with 1 million people, 100 of whom are infectious (so 1 in 10,000)

1) Susceptible to infectious

Let's say people come in to contact each other 25 times a day per person (see footnote 1). So the susceptibles will altogether have 999,900 x 25 contacts in a day.

Of these nearly 25 million encounters, about 1 in 10,000 (roughly 2,500) will be between a susceptible and an infectious. Let's say the probability of being infected if a susceptible comes into contact with an infectious is 20% (see footnote 2). Then we can reasonably expect about 500 additional people to be infected during the course of day 1.

In general, the rate of transmission out of susceptible and into infection is then

So the rate at which susceptibles become infected is proportional to

  • The number of infected (this gives the exponential growth, more on that in a second)

  • The number of contacts per person per day

  • The probability of transmission

  • The proportion of the population that are still susceptible (this is what saturates the growth, more on that in a minute)

The middle two drive the measures we now take. Keeping your distance and washing your hands reduces the probability of transmission if you do meet someone infected; staying home from work, closing the pubs and keeping the kids home all act to reduce the average number of contacts per person per day.

2) Infectious to removed

Assuming our original cohort of infectious weren't all infected at the same time, we may reasonably expect a certain proportion of them, either to have died or to have recovered during the course of day 1

We can estimate the proportion of infectious who die or recover per day. If they're all "infectious" (remember that includes incubation) for about 10 days then on average about a tenth of them will get better per day. (Luckily the mortality rate is so low that the deaths don't very much affect that calculation.)

In general, the transmission from infectious to removed is simply a given proportion of the infectious. To start with, this is a fairly small number, but it will get more important in time as the number of infectious increases.

Day 2 and exponential growth

On day 2, there are 100-10+500=590 people infected and 999,400 x 25 encounters (still very close to 25 million). The proportion of those that are potential transmission contacts has increased almost 6-fold. The probability of transmission is the same, so the number of new infectious cases increases almost 6 times what it was on day 1. Almost 3,000 people are infected on day 2.

This is exponential growth. We are increasing the rate of infection by increasing the number of infectious. It's true, we are also reducing the rate of infection by reducing the number of susceptibles; the problem is that this is measured relative to the total population, so the proportional decrease is very small. On the other hand, the increase in the number of infectious is measured relative to the previous number of infectious, so the proportional increases are enormous.

Growth rate and suppression

The rate at which the number of infectious grows is simply the rate susceptibles become infected less the rate at which the infected die or recover.

When the number of susceptibles is still close to the total population then the growth rate is just the number of contacts per person per day times the probability of transmission (I'll call this product beta) less the proportion who die or recover per day (which I'll call gamma) all multiplied by the number of infectious.

If we can get the number of contacts per person per day multiplied by the probability of transmission (what I've called beta) down under the 10% of infectious who die or recover daily then the growth rate will be negative. Fewer people will be infectious from day to day and eventually the epidemic will die out. This is called suppression and is what they achieved in Wuhan, amongst other places.

The problem with this approach is that as long as there is someone infectious around then removing the measures and letting beta increase again will turn the growth rate back to positive and you're right back where you started.

Reproduction rate

There's been a fair amount of discussion in the press of something called the base reproduction rate, sometimes called R0. This is just beta divided by gamma. If it's greater than one then the rate of infection is greater than the rate of removal. Less than one and it goes the other way.

Peak infectious

Exponential growth is terrifying because it doesn't really matter how many people are infected, it only matters how quickly the contagion multiplies. Exponential growth will make up for any shortfall in numbers in however big a population in a matter of days.

Any exponential growth will eventually infect so many people that the rate of increase in infection starts to notice that there aren't so many susceptible left to infect. In our example, on day 5, there are around 120,000 infected and 2,500 removed and around 880,000 left susceptible. Now there are only about 22 million contacts. Although the proportion of those that are potential transmission contacts is high because of the high proportion of infectious in the population, the susceptible population is now falling rapidly and the rate at which the number of infectious increases will soon start falling too.

Mathematically, as the number of susceptibles decreases relative to the population, the fraction in the equation above gets smaller and smaller and the growth rate falls accordingly. Eventually, it turns over and the increase becomes a decrease. The point at which it does this is the epidemic peak.

Flattening the peak and looking after the sick

Clearly the greater beta is relative to gamma (i.e. the greater R0), the more you have to draw down the number of susceptibles before the growth rate turns over. This means the greater R0, the greater the number of infectious at the peak.

Our main concern in Denmark and the UK is the capacity of our health service to cope with the number of severe cases at the peak of the epidemic. The number of severe cases is roughly a proportion of the number of cases, i.e. the number of infectious. So governments many places are trying to bring down the number of severe cases at the peak by bringing down the peak. They do this by trying to bring down R0.

The methods are the same: reducing the number of contacts and reducing the probability of transmission, but in addition this approach develops a herd immunity (the number of removed), decreasing the number of susceptibles. This has the added effect of increasing the beta that you can get away with without having a positive growth rate.

So what? Governments' tough balancing act

Governments face a dilemma. They want to decrease beta (and so R0) by reducing contact and reducing transmission so that the hospital capacity at peak infectious is not exceeded. There are three big challenges.

First, measures cost money and the stronger the measures the more they cost, so you don't want your measures to be stronger than they need to be. Secondly, when you bring R0 down, you slow the growth rate and you push out the time it takes to get to peak infectious. So you want to bring it down far enough to achieve a manageable peak, but not so far that we stay locked down for an untenable length of time. The third challenge is that the smaller the peak, the smaller the population who are immune at the end of it and the less you are able to unwind your measures and still maintain a negative growth rate.

Governments face a tough balancing act - all the economic pressure is on alleviating the measures as much as possible, to minimize disruption, to minimize the time over which the disruption is incurred and to maximize the latitude to alleviate measures after infections subside. But the human cost - for the victims and the heroic souls who will care for them in hospital is immense. Every individual "removed" from the susceptible pool will suffer various degrees of viral unpleasantness and a significant proportion of them will be removed through death.

Footnote 1: This is a bit oddly formulated, because it isn't the number of contacts each person has in a day. Its the number of contacts divided by the number of people. If it were the former then we would count every contact twice.

Footnote 2: This is low because in this model "infectious" people include incubating people.

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