{"id":19587,"date":"2020-05-02T17:56:26","date_gmt":"2020-05-02T16:56:26","guid":{"rendered":"https:\/\/fortran.uk\/?page_id=19587"},"modified":"2020-05-28T12:53:39","modified_gmt":"2020-05-28T11:53:39","slug":"using-published-covid-19-data-to-deduce-real-infection-rates","status":"publish","type":"page","link":"https:\/\/fortran.uk\/using-published-covid-19-data-to-deduce-real-infection-rates\/","title":{"rendered":"Using Published Covid-19 Data to Deduce Real Infection Rates"},"content":{"rendered":"

John Appleyard,\u00a0 New Section 3.1 28th May 2020<\/strong><\/p>\n

1. Introduction<\/h2>\n

\"\"<\/a>In this note, I will show that a straightforward mathematical model\u00a0 linking testing with confirmed cases and deaths is confirmed by analysis of published data.\u00a0 The resulting equation can be used to estimate real infection rates at any point in time,\u00a0 and can even be used to estimate the IFR.\u00a0 For the UK, on 2nd May, the estimated total infection rate was 4.1%, and the IFR was 1%.\u00a0 The model does not predict the future.\u00a0 Instead, it allows us to work out what published data can tell us about the true situation on any given date.\u00a0 The implications of unreported Covid-19 deaths are also discussed.<\/p>\n

Since the middle of May, it’s become apparent that more and more countries are diverging from the simple model described here. The reasons for this divergence, and its significance are discussed in the new section 3.1.<\/p>\n

1.1 Why is that Important?<\/h3>\n

Knowing what is happening is vital to rational decision making.\u00a0 Currently there appears to be a general understanding of the effect of testing on case numbers.\u00a0 This model provides a quantitative framework for that understanding.\u00a0 In cases where the true infection rate is determined accurately by random sampling, it should be possible to use the results to refine this model, allowing more precise estimates of infection rates where sampling evidence is not available.<\/p>\n

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

A detailed mathematical description of the model is available here<\/a>.\u00a0 This section contains a simplified version which describes the model behaviours when the number of\u00a0 confirmed cases is small, and when it is large, but not in the intermediate region.<\/p>\n

If testing were random, the number of cases, C, found by T tests in a population P will be approximately:<\/p>\n

C = I.(T\/P)<\/p>\n

where I is the total number infected.\u00a0 We don\u2019t know I, but it\u2019s related to the number of deaths by the infection fatality rate, f=D\/I, so<\/p>\n

C = (D.T)\/(f.P)<\/p>\n

This is model 1.\u00a0 Ideally, D should be time shifted to account for the delay between infection and death.\u00a0 This will be discussed further in section 6.<\/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 = (D.T)\/(f.n.C)<\/p>\n

C = sqrt((D.T)\/(f.n))<\/p>\n

where n is the constant of proportionality.\u00a0 This is model 2.<\/p>\n

In effect, testing is concentrated on a bubble of contacts surrounding each case, and for small C (model 2), these bubbles are isolated.\u00a0 As C increases, the bubbles increasingly overlap, and eventually model 1 prevails.<\/p>\n

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

The data, downloaded from worldometers.info at the end of the 2nd May 2020, comprises 3 variables for each of 99 countries with more than 100 cases, and at least ten deaths.\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, 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

 <\/p>\n

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

The match spans more than 3 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.\u00a0 \u00a0All data is measured per million of population, but the fit is equally good if raw numbers are used.<\/p>\n

An even better fit can be obtained by using time shifted death data.\u00a0 This note will be updated with details in the next few days.<\/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 (f) is 0.4%, 1% or 2.5%.\u00a0 The curves projecting down from those points show the trajectories predicted by the detailed model which end at each point.<\/p>\n

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