The Imperial College Study: Part 2

[Links to the full series]

Part 1
Part 2
Part 3A
Part 3B

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Now I will start looking at the parameters that the Imperial College study used to characterize the coronavirus in its simulations. I'll start by looking at the basic reproduction number: the now-famous r(0) ("r naught").

1. What is r(0)?

R(0) is defined as the number of secondary infections a single instance of an infection will cause if exposed to a completely vulnerable population. It arises from a combination of multiple elements:

•   how naturally infective the particular virus is
•   how long an infected person is contagious *while* being still present in a population
•   how many people an infected person can be expected to contact during the infective period

Importantly, while the first two elements are virus dependent, the third is societally dependent. The more mobile and mixing a population is, the higher the r(0) will be. "Social distancing" is therefore a way of changing the r(0) of a particular disease by altering that third element.

From the perspective of a microsimulation, however, epidemiologists can just take the r(0) as a constant which determines the chance (on average) that any given non-infected individual coming into contact with an infected individual will become infected. This is because the microsimulation intrinsically accounts for degrees of social mixing by its rules that govern individual behaviors.

2. How is r(0) calculated in general?

This is an oversimplification, but I am aware of two methods of calculating an r(0).

First, an r(0) is said to be proportional to the early doubling rate of infections during an epidemic. During this early time period, there are so many more non-infected people compared to infected people that each instance of an infection will have essentially a "clean slate" of people to infect. Later on, an increasing percentage of infected individuals will come into contact with already-infected individuals, thus slowing the growth rate. Furthermore, in the earliest phases of an epidemic, the disease will be spreading without a lot of symptoms and without causing major alarm, so that social mixing will be normal as a whole. Later on, fear of the disease spreading through society will itself change mixing behavior and therefore the r(0) itself.

Second, you can examine a timeline of infection and death rates after the fact from a given epidemic event, and then run microsimulations for that population using different r(0) values until you reproduce the observed curve in the simulation.

3. How was the r(0) used in the Imperial College study calculated in particular?

The Imperial College simulation used the results from two different studies, each taking one of these approaches.

The first study was done by the China CDC (available here: https://www.nejm.org/doi/full/10.1056/NEJMoa2001316) looked at the first 425 cases reported in Wuhan and reconstructed an infection timeline by interviews which established when symptoms first occurred in each case. From this they saw an initial doubling time of the disease of about 7.5 days, which calculated out to an r(0) of 2.2, with a 95% confidence range of 1.4 to 3.9.

The second study was done by Julien Riou and Christian Althaus and funded by the Swiss National Science Foundation (available here: https://www.eurosurveillance.org/…/1560-7917.ES.2020.25.4.2…). They ran 2 million separate epidemic simulations with varying parameters and looked for those parameters that could reproduce the timeline of the infections as of January 29th, which at the time included 5,997 confirmed cases in China and 68 confirmed cases exported to other countries. Importantly, they allowed the number of cases *actually* in China in their simulations to vary substantially, in order to account for potential massive misreporting of the Chinese data. The results of this study was an r(0) of 2.2, with a 95% confidence range of 1.4 to 3.8.

4. How confident can we be in the results of these two studies?

Both studies that establish an r(0) of 2.2 for Covid-19 acknowledge the fact that they are operating on limited data, collected in a crisis situation. They both, therefore, should be treated as preliminary best estimates given the data that we had in mid February. Nevertheless, the fact that they agree very closely after taking very different approaches to the estimation does count for something.

Furthermore, I think there is good reason to believe that the true r(0) isn't lower than 2.2 given the newer European data that we have. If you look at infection growth rates for all of the European countries and for the United States since community spread started occurring in late February / early March, you can see a pretty consistent slope of the lines, yielding a doubling time of 3-4 days. This is significantly *faster* than the doubling time that was estimated for the Chinese outbreak. I believe that this makes an r(0) of significantly less than 2.2 very difficult to justify. I think the Imperial College agreed, because they used a base doubling time of 6.5 days (somewhere between current European numbers and the Chinese numbers) and a consequent r(0) value of 2.4. They did also explore values with a range from 2.0 to 2.6 to account for uncertainty.

So I think the calculations that the Imperial College did are a very reasonable interpretation of the best data that we have. I think newer studies taking the rapidly evolving new reports of infections given much greater worldwide testing would be good to do, but I very much doubt at the moment these studies would come to different conclusions right now. If anything, they would probably raise the estimated r(0), not lower it.