by Sam L. Savage
All infectious disease epidemics start life with exponential growth. For example, suppose that each person infected with a disease in the first month infects one other person by the second month. Then those two people will infect two more, and so on, and the number infected will double with each time period. This exponential growth obviously can’t go on forever because eventually you run out of people, or at least susceptible people. So, in the end, the total number infected over time resembles an S curve as shown in Figure 1 below.
When someone brought this problem to my attention during the Ebola scare, I built a Monte Carlo simulation, available on our Models page, which reflects the uncertainty in R0 as shown in Figure 2.
For all infectious diseases, the flawed path associated with the average growth rate systematically underestimates the severity early in the epidemic (before month 12 in the above example) and overestimates the severity later in the epidemic (after month 12). For this example, in month 9, you expected 5% of the population to be infected, but on average you will observe 10%. Then at month 18, you expected 35% but only observed 30%. Although a single case does not win a statistical argument, the Ebola epidemic of 2014-2015 fits the bill perfectly. It started out with fears of “We’re all going to die! We’re all going to die!” and ended up with the development of effective medications, and the realization that “they did not have many cases left to test it on.”
As to the model, like all nonlinear difference equations, this one can go chaotic for some input values. This model was inspired by by James Gleick’s book “Chaos: Making a New Science.”
For a chaotic time, download the model and set the parameters to the values suggested.