Coronavirus

Here is another way of thinking about the course of the Wuhan coronavirus epidemic. It is common to refer to the R0, i.e. the number of persons infected by a person who already has the disease. An alternative, that seems to match the experience of certain countries that have experienced significant COVID-19 cases is as follows.

Rather than try to estimate how many persons are infected by each already infected person, assume that each case will transmit the disease to two other persons, x number of days apart. It makes sense that the harder it is for the virus to spread, the longer the interval between transmissions. Thus, the number of days that it takes an infected person to infect two others gives a measure of how effective interventions to slow the virus are. The calculation is also very straightforward.

If we assume that a person becomes infected, then infects someone in x days, and another person in an additional x days, then no one else, the growth in the number of cases follows a modified Fibonacci sequence, but the ratio of the number of cases the previous day to the present day is the same as the Fibonacci or "golden ratio," i.e. 0.618. (The Fibonacci sequence will give 121393 cases in 25 days assuming daily transmission; the one patient-two-transmissions in 2x days gives 196417 cases in 25 days).

When the virus has broken out in a country before containment measures have been instituted the ratio of new cases to known previous cases is approx 0.45. This implies that each infected person infects two people about 1.37 days apart. (It is also possible to create similar models in which an infected person infects three other people y days apart, four other people z days apart, etc., but it does not affect the underlying analysis. The important point is that as spread of the virus is impeded, the number of days between transmissions increases.)

This approach is justified when the graph of logarithm of cumulative cases approximates a straight line. It implies that the growth in the number of cases is exponential, as is the case with the Fibonacci sequence (N~N0*(1+.0618)^t). In other words, if the logarithmic graph of cases is linear, the disease growth is behaving as expected.

Now we can make the following observation. If we look at the countries that have had a significant number of coronavirus cases, South Korea and China seem to have gotten control of their epidemics, and Taiwan's seems not to have gotten out of control. Here is the interesting thing: The growth in the number of cases seems to have reversed in China on February 4, 2020, and in South Korea on March 1. 2020. In both cases the calculated "x" was about 2.8 days. Again, this is not a real-world number. It is a statistical surrogate for the difficulty involved with transmitting infection between one person and another. It likely reflects a number of factors: how long the virus can remain infective on fomites, the "distancing" between persons in a population, infection control procedures, etc. It should also be noted that the number of 2.8 days occurs when the daily number of new cases is approximately 19% the number of previous day's cases. This does not mean that the number of infections stops or that the epidemic is over; it does suggest that when x=2.8, the number of daily new cases is about to peak.

Of note, in the United States currently, the number is approximately 2.16 days. In Italy it is 3.1, implying that Italy's number of new cases should start to decline.

It is also important to note that this number reflects difficulty in disease transmission depending on the current environment, with current precautions and behaviors. It is not an indication of how the virus will behave if the environment, e.g. containment measures, etc. change.

The number 2.8 (actually 2.77) days is not a scientific constant. It does however have the benefit of quantifying the state of an epidemic against an intuitive metric. It is reasonable that if x is greater than the number of days an infectious person can pass along the disease, the epidemic will die out. But there is some subtlety involved. It is not unreasonable to consider a variable I, which can be thought of as the transmissibility of the disease. It is also not unreasonable to think that this number is constant, starting at some time after infection and abruptly dropping to zero at some later time, indicating the that person is no longer capable of transmitting the disease. Rather it is reasonable to assume that the value I follows a bell-shaped curve over time, reaching a peak at some point, and being highly unlikely to support disease transmission at the tails. The profile of I in a given person is likely to be a factor of the person and the virus, and to be stable over time. If the amount of time required to infect an additional one or two people (from the time the person becomes capable of transmitting the infection, not when he becomes infected) increases, the likelihood of disease transmission goes down, and at a certain point, the epidemic stalls. This is true even in the absence of "herd immunity," or other disease-limiting processes.