log-normal distributions of variables, eg viral loads, are related to power-law distributions, and the two functions are often used in similar situations. Both power laws and log-normals are distributions for a variable, for example an amount of virus. A log-normal distribution is such that the distribution of the log of the variable is the normal, or Gaussian distribution. The Central Limit Theorem of statistics tells us that generically a normal or Gaussian distribution arises for any property that is the sum of many smaller steps each of which has a randomly varying part. The same theorem implies that a log-normal distribution arises for any property that is the product of many smaller steps, each of which has a random part.
Viral infection proceeds via a virus infecting and taking over a cell that then releases a large but variable number of viruses, each of which may or may not go on to infect another cell and so repeat the cycle. At each step there is a variability and at each step the amount of virus multiples not adds. So log-normal distributions are natural models for the amount of virus.
Here is a plot of the viral load – as measured by qPCR by Takatsuki et al. – plus fits*:
where the exceedance E for a log-normal function of x is
where μ and σ are the fit parameters, they are related to the mean and standard variation**. Note that I only fit the log-normal to the same data in the tail as the power law, but then I extrapolate the log-normal over the whole range of the data. In the large viral-load tail the log-normal function (dashed green curve) does OK, about as well as a power-law fit (blue solid curve) there**. And the extrapolation down 6 orders of magnitude is not great, but it is not terrible either.
We also have data on the transmission of COVID-19, thanks to the users of the NHS’s (pandemic) COVID app. Earlier, both Ferretti and coworkers, and I*** (see earlier post) fitted power laws to the probability of COVID-19 being transmitted PT, as a function of time. And the power-law fit (blue line in figure up top) is pretty good. But a log-normal function (green dashed) is about as good, over the range we have data, as the power-law fit. There is not much to choose between them.
Except that a log-normal function is physical over the complete range. I have fitted the cumulative function of a log-normal which tends to 1. By contrast extrapolating a power law will eventually give you numbers above 1, which are impossible for a probability.
Of course the extrapolating red-dashed curve above is a bit crazy, we only have data up 100 hours or around 4 days, while I extrapolate up to 100,000 hours, or over 11 years! However, it is notable that even after 11 years, the transmission probability is still below one. The extrapolation predicts that that some interactions have such a low risk of transmission that you could spend 11 years with some infected people, and still not get infected. Of course this is a daft timescale, COVID infections last weeks not over a decade, fortunately.
But it may be telling us something about how variable infected people and transmission can be. It may well be that some people (falling in the right-hand tail of the exceedance plot) have such a high viral load they can infect you in 30 minutes, while others (from the left-hand side of the exceedance plot) are unlikely to infect you in years or even decades.
* The fitting and plots are all generated in this Google Colab notebook.
** For a log-normal the relationships between μ and σ, and the mean and standard deviation are a bit messy
*8* The data and power-law fit (but not log-normal) are in a paper of mine from 2024.
Chance & Necessity 