Disease dynamics distilled

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
  • The rate at which infected people recover or die

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


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.


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.


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.



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Graeme Keith

Graeme Keith


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