{"id":19525,"date":"2020-04-22T15:47:45","date_gmt":"2020-04-22T14:47:45","guid":{"rendered":"https:\/\/fortran.uk\/?page_id=19525"},"modified":"2023-11-21T13:55:22","modified_gmt":"2023-11-21T13:55:22","slug":"a-simple-model-for-interpreting-published-covid-19-data-23april","status":"publish","type":"page","link":"https:\/\/fortran.uk\/a-simple-model-for-interpreting-published-covid-19-data-23april\/","title":{"rendered":"ZZ A Simple Model for Interpreting Published Covid-19 Data (23rd April)"},"content":{"rendered":"

This version is no longer current – please see<\/strong><\/span><\/h2>\n
https:\/\/fortran.uk\/using-published-covid-19-data-to-deduce-real-infection-rates\/<\/a><\/h5>\n

John Appleyard,\u00a0 updated 23rd April 2020<\/p>\n

Disclaimer: I trained as a physicist to PhD level many years ago, and my career required good numeracy. However, I have no special expertise in epidemiology or statistics.<\/em><\/p>\n

1. Introduction<\/h2>\n

In this note, I will show that a simple model of linking testing with confirmed cases and deaths is confirmed by analysis of published data.\u00a0 The resulting equation can be extrapolated to predict true infection rates.\u00a0 It is argued that, although the model must break down for high infection rates, the way in which it breaks down can be anticipated.<\/p>\n

2. The Model for Covid-19 Testing<\/h2>\n

If testing were random, we would expect the number of cases found (c) to be proportional to the number of tests (t), and the proportion of the population that is infected.\u00a0 The second factor is unknown, but the number of deaths (d) is probably a reasonable proxy, so we have:<\/p>\n

c \u221d d.t<\/p>\n

A better proxy for the infection rate now might be the death rate in a few days time, but initially, we will use the simpler model.<\/p>\n

In the early stages of the pandemic, testing is not random, but is designed to find as many cases as possible.\u00a0 It focuses on a small part of the population comprising the contacts of existing cases, medical staff, and those with symptoms.\u00a0 If the size of that reduced population is proportional to c, then the number of cases found will be<\/p>\n

c \u221d d.t\/c<\/p>\n

c \u221d sqrt(d.t)<\/p>\n

In effect, testing is concentrated on a bubble of contacts surrounding each case, and for small c, these bubbles are isolated.\u00a0 As c increases, the bubbles increasing overlap, and eventually encompass the entire population.\u00a0 At that point testing effectively becomes random, with c \u221d t.<\/p>\n

To summarise, the hypothesis is that c should vary with sqrt(d.t) for small c, and with t for large c.<\/p>\n

3. Comparison with Published Data<\/h2>\n

The data, downloaded from worldometers.info at the end of the 20th April 2020, comprises 3 variables for each of 115 countries with more than 100 cases, and at least one death.\u00a0 China is excluded because testing data is not available.\u00a0 The variables are the number of confirmed cases (c),\u00a0 deaths (d), and tests (t) per million of population.\u00a0 \u00a0Some historic data, from the UK, US, Qatar and Singapore is also plotted as coloured traces.<\/p>\n

Figure 1 shows that the data fits very well to the equation<\/p>\n

c = 1.667 *sqrt(d*t)<\/p>\n

\"\"<\/a><\/p>\n

The prediction spans almost 4 orders of magnitude, but even so, almost all countries are within the tramlines, which indicate a factor of 2 either side of the predicted value.<\/p>\n

The red dots at the top of the chart represent the hypothetical situation in which the number cases approaches saturation (c = 1 million), and the eventual infection fatality rate is 0.4%, 1% or 2.5%.\u00a0 The hypothesis is that, in that situation, a time series trace would deviate upwards from the best fit line, so that it could reach the point corresponding to the actual infection fatality rate.<\/p>\n

There are many reasons to expect data from different countries to differ, including:<\/p>\n