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

  • 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

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

Suppression

Mitigation

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.

Mathematical modelling for business and the business of mathematical modelling. See stochastic.dk/articles for a categorized list of all my articles on medium.

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