Disease dynamics distilled

The absolute basics of what you need to know about the mathematics of how the novel Coronavirus will spread and how we might stop it.

The novel Coronavirus is, well, novel, so to start with at least, no one is immune to it. Everyone is susceptible to start.

Some of us, perhaps many of us, will get sick, then we are infected. Some of us will recover and — for a while at least, develop immunity, some of us will die, either way we won’t get sick again; we have been removed from the pool of the susceptible. As we get sick and recover or die, there will be fewer susceptible remaining to get sick.

Early stages

To start with, the more infected cases there are, the more there are to infect new cases. The ratio of new cases to existing cases is the called growth rate. The growth rate depends on

• The number of people with whom, on average, someone infected comes into contact
• The probability that someone infected infects someone new if they come into contact

which multiplied together give the rate at which new cases become infected, and

• The rate at which infected people recover or die

which you subtract off.

If the growth rate is positive, i.e. if the rate of infection is greater than the rate of recovery and fatality, then the number of infected cases will grow, exponentially. If it is negative then it will fall, exponentially. See this outstanding video for an illustration and explanation of exponential growth.

The magic R0 that we hear so much about is the rate of infection divided by the rate of removal (recovery and death). If it’s greater than 1 then the number of infected cases grows and if it’s less than 1, it falls.

Later stages

Now, if the infection spreads and grows as more and more people get sick, recover or die, there are fewer and fewer people for that larger and larger number of infected cases to infect.

The growth rate is now

• The infection rate there was at the start multiplied by the proportion of the population still susceptible
• Minus the removal rate

So as we develop herd immunity, i.e. as the number of people who can’t be infected any more grows, the growth rate falls. Eventually, it falls so far it becomes negative and the number of infected cases begins to fall. This is logistic growth (see again the outstanding video above).

The point at which it turns is when the growth rate is zero. If your infection rate is high then you need a big draw-down on the proportion of the population that is still susceptible to get the growth rate down to zero. That’s a lot of people who’ve been sick and a fair few who’ve died.

More people will get infected, but the infection rate is now lower than the recovery rate. Eventually the contagion dies out. When it does, a lucky proportion of the population will never have been sick.

Implications

Isolation reduces contact between known infected and the remaining susceptible population. Social distancing reduces the rate of contact between unwitting infected cases and the remaining population. Washing hands and all the other hygiene exercises reduce the probability of infection. All of these reduce the rate of infection.

Suppression

If you can early on reduce the rate of infection below the rate of removal, you turn the exponential growth into an exponential decline and the infection goes away. The problem here is that you have to keep those measures in place otherwise the growth rate will reverse again.

Mitigation

By reducing the rate of infection you can reduced the draw-down you need to get to the turning point in the figure above. You both reduce the proportion of the population who have been infected when the curve turns, but also the proportion of the population who are sick at the peak, thus also the proportion of the population who need hospitals and intensive care at the peak of the crisis.

Now you can actually start easing back on measures because all those infected will recover / die allowing an increase in infection rate without necessarily turning the growth rate positive. This is the herd immunity mitigation strategy we hear so much about.

A key component of this strategy is the relationship between the proportion of the population who are sick at the peak and the proportion of the population who avoid sickness altogether. In this, very simple model, this is just a function of R0 and the results are shown below. Even quite low peaks result in a relatively large proportion of the population getting sick. This is good news with respect to the easing of measures, but the human cost is horrendous.

Originally published at https://www.stochastic.dk on March 23, 2020.