“Flattening the curve” has become a popular expression nowadays, referring to slowing the spread of the new Corona virus (for a reference, https://www.livescience.com/coronavirus-flatten-the-curve.html). In contrast to a “flattened-curve” (South Korea), there are plots of exponential growth (most other countries and South Korea in the early stages of the disease):
First, off, why do many plots start at the day when 100 cases were reached? Before there are a lot of cases, statistically, the spread of the disease can be noisy; that is, say quite a few of those infected early on are socially distant, then the disease might only be transmitted over there few interactions (there’s the separate issue of testing for the disease, but that can be potentially gotten into at a later point). Also, the slow incubation period/ slow time to show symptoms could cause noisiness in the plot because in the early stages, people weren’t tested until there was a reason to. Anyhow, many of the plots in the lower right-hand corner (you are look at each country individually) confirm that the early trend is not as clearly linear as after a many cases have shown up: https://gisanddata.maps.arcgis.com/apps/opsdashboard/index.html#/bda7594740fd40299423467b48e9ecf6 (which I got from here: http://www.cidrap.umn.edu/covid-19/maps-visuals). Also note that the slow incubation period is the reason why social distancing efforts take weeks to be noticed.
To explain the exponential beginnings, we can look at a number of models used to describe the spread of the disease:
(Or for the commonly referred to SEIR model: https://sites.me.ucsb.edu/~moehlis/APC514/tutorials/tutorial_seasonal/node4.html )
Let’s look at the early stages (and also assume that once infected, you cannot be infected again, though in the early stages, we can ignore this relation). Taking to be the number of infected and to be the number of healthy individuals, it is assumed that the rate of infection is proportional to both these values
where the total population is either healthy or infected . Note that even if it is not true that every individual has the same number of contacts, statistically-speaking, the relation often holds. The solution to the differential equation is the sigmoidal function(for reference: https://www.reddit.com/r/dataisbeautiful/comments/fohr58/oc_the_technical_problems_of_fitting_a_logistic/)
For small values ( or ), this curve is exponential:
Another way of seeing this is
when and the solution of that equation is a exponential
the same as the above!
As an exercise, you can plot the approximate and exact solutions and see how they differ (when they are the same and when they differ significantly).