1. Introduction<\/h2>\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 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.<\/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 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, 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 The match 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 (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 The conclusion is that all of these factors combine to a factor of up to about 2.<\/p>\n There are 2 interesting exceptions on the chart, namely Qatar and Singapore.\u00a0 In Singapore, the number of cases increased by a factor of 3.8 in a period when the number of tests increased by only 1.3.\u00a0 For Qatar it was 2.4 and 1.4.\u00a0 In both cases, the increase seems to caused by the discovery of infection hot-spots among expatriate workers.\u00a0 For example the Singapore MOH issued a press release on 20th April saying “As of 20 April 2020, 12pm, the Ministry of Health (MOH) has preliminarily confirmed an additional 1,426 cases of COVID-19 infection in Singapore, the vast majority of whom are Work Permit holders residing in foreign worker dormitories. 16 cases are Singaporeans\/ Permanent Residents.”\u00a0 This point is discussed further below.<\/p>\n Venezuela is an obvious outlier.\u00a0 It is reported as having conducted more tests per case than any other country.\u00a0 Second is Hong Kong, a factor of 9.5 behind.\u00a0 Djibouti and Gabon are also outliers with 2 and 1 deaths respectively.<\/p>\n The model predicts that when C is small, it varies with the square root of the number of people who have been tested.\u00a0 That means that we can predict I, the number of cases that would be found if we tested the entire population (P) using<\/p>\n I = C*sqrt(P\/T)<\/p>\n However, this simple formula breaks down as C increases, because it depends on the assumption that there is no overlap between the contacts of individual cases.\u00a0 A more thorough, but somewhat mathematical theory, which allows for that overlap, is described in this note<\/a>.<\/p>\n The more detailed theory requires an estimate of the infection fatality rate (IFR) – the percentage of all infected patients who will eventually die.\u00a0 That number may vary between countries, but is probably a little over 1% in Europe and North America.<\/p>\n Table 1 shows the top 50 countries as of 20th April.\u00a0 There are estimates of the real cases per million for the simple extrapolation, and for the detailed model with different assumptions about the IFR.\u00a0 The multiplier is sqrt(P\/T), and is an indication of the likely error in the figures: a high value means that the model is extrapolating further and may be more error prone.<\/p>\n Table 1 Reported and “Real” <\/strong>Cases per Million – 20th April<\/strong><\/p>\n Although the hypothesis of section 1 appears to fit the data well, it was noted in section 2 that D should be time shifted to account for the delay between infection and death.\u00a0 In this section, I will examine this idea using time series data for the UK and US.Historic data for the UK and US was downloaded from https:\/\/github.com\/tomwhite\/covid-19-uk-data\/blob\/master\/data\/covid-19-indicators-uk.csv and\u00a0https:\/\/covidtracking.com\/data\/us-daily.A preliminary study with varying number of days’ shift applied to the death statistics, established that for the US, a 5 day delay produced a clearly visible improvement in the fit to the data.\u00a0 For the UK, there was no clear trend.<\/p>\n Figure 3 shows the resulting plots of C against T.D for the US.\u00a0 The yellow dots are the unmodified US time series, which shows a noticeable bulge above the straight line.\u00a0 This is to be expected in the exponential growth phase because deaths are systematically understated.\u00a0 The blue line shows the same data, but with a 5 day shift applied to death data, and the bulge is eliminated.\u00a0 Both lines converge to the standard formula as time progresses.The red line shows the prediction of real cases per million on each date.\u00a0 This is based on extrapolation to complete testing, as described in Section 4, and using an IFR of 0.4%.<\/p>\n![\"\"](\"https:\/\/fortran.uk\/wp-content\/uploads\/2020\/04\/Fig1-2.jpg\")
<\/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 deduce real infection rates at any point in time.\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>\n1.1 Why is that Important?<\/h3>\n
2. The Model for Covid-19 Testing<\/h2>\n
3. Comparison with Published Data<\/h2>\n
<\/a><\/p>\n\n
4. Predicting the Real Infection Rate<\/h2>\n
\n\n
\n <\/td>\n <\/td>\n <\/td>\n True Cases %<\/td>\n<\/tr>\n \n <\/td>\n Reported Cases \/ M<\/td>\n Multiplier<\/td>\n Simple Model<\/td>\n IFR=1%<\/td>\n IFR=2.5%<\/td>\n<\/tr>\n \n Andorra<\/td>\n 9,280<\/td>\n 4.43<\/td>\n 6.31%<\/td>\n 9.48%<\/td>\n 7.18%<\/td>\n<\/tr>\n \n San Marino<\/td>\n 13,616<\/td>\n 8.46<\/td>\n 6.04%<\/td>\n 12.30%<\/td>\n 7.62%<\/td>\n<\/tr>\n \n Spain<\/td>\n 4,282<\/td>\n 6.80<\/td>\n 3.04%<\/td>\n 4.48%<\/td>\n 3.44%<\/td>\n<\/tr>\n \n Belgium<\/td>\n 3,450<\/td>\n 7.09<\/td>\n 2.92%<\/td>\n 4.45%<\/td>\n 3.34%<\/td>\n<\/tr>\n \n France<\/td>\n 2,380<\/td>\n 6.58<\/td>\n 2.82%<\/td>\n 3.84%<\/td>\n 3.12%<\/td>\n<\/tr>\n \n Luxembourg<\/td>\n 5,684<\/td>\n 11.87<\/td>\n 2.45%<\/td>\n 2.93%<\/td>\n 2.61%<\/td>\n<\/tr>\n \n UK<\/td>\n 1,838<\/td>\n 13.26<\/td>\n 2.44%<\/td>\n 3.18%<\/td>\n 2.66%<\/td>\n<\/tr>\n \n Ireland<\/td>\n 3,170<\/td>\n 10.00<\/td>\n 2.34%<\/td>\n 2.84%<\/td>\n 2.51%<\/td>\n<\/tr>\n \n USA<\/td>\n 2,395<\/td>\n 6.21<\/td>\n 2.17%<\/td>\n 2.62%<\/td>\n 2.32%<\/td>\n<\/tr>\n \n Isle of Man<\/td>\n 3,528<\/td>\n 11.63<\/td>\n 2.05%<\/td>\n 2.44%<\/td>\n 2.19%<\/td>\n<\/tr>\n \n Channel Islands<\/td>\n 2,807<\/td>\n 7.38<\/td>\n 2.03%<\/td>\n 2.47%<\/td>\n 2.18%<\/td>\n<\/tr>\n \n Switzerland<\/td>\n 3,229<\/td>\n 7.24<\/td>\n 2.01%<\/td>\n 2.48%<\/td>\n 2.16%<\/td>\n<\/tr>\n \n Italy<\/td>\n 2,997<\/td>\n 9.07<\/td>\n 1.97%<\/td>\n 2.83%<\/td>\n 2.21%<\/td>\n<\/tr>\n \n Netherlands<\/td>\n 1,950<\/td>\n 4.30<\/td>\n 1.95%<\/td>\n 2.50%<\/td>\n 2.12%<\/td>\n<\/tr>\n \n Sweden<\/td>\n 1,463<\/td>\n 5.82<\/td>\n 1.70%<\/td>\n 2.09%<\/td>\n 1.83%<\/td>\n<\/tr>\n \n Panama<\/td>\n 1,035<\/td>\n 6.57<\/td>\n 1.56%<\/td>\n 1.68%<\/td>\n 1.60%<\/td>\n<\/tr>\n \n Iran<\/td>\n 994<\/td>\n 7.76<\/td>\n 1.53%<\/td>\n 1.73%<\/td>\n 1.60%<\/td>\n<\/tr>\n \n Iceland<\/td>\n 5,196<\/td>\n 15.42<\/td>\n 1.46%<\/td>\n 1.58%<\/td>\n 1.50%<\/td>\n<\/tr>\n \n Qatar<\/td>\n 2,088<\/td>\n 6.96<\/td>\n 1.39%<\/td>\n 1.43%<\/td>\n 1.41%<\/td>\n<\/tr>\n \n Portugal<\/td>\n 2,046<\/td>\n 7.02<\/td>\n 1.35%<\/td>\n 1.53%<\/td>\n 1.41%<\/td>\n<\/tr>\n \n Ecuador<\/td>\n 574<\/td>\n 8.21<\/td>\n 1.34%<\/td>\n 1.44%<\/td>\n 1.38%<\/td>\n<\/tr>\n \n Germany<\/td>\n 1,755<\/td>\n 7.05<\/td>\n 1.22%<\/td>\n 1.37%<\/td>\n 1.27%<\/td>\n<\/tr>\n \n Turkey<\/td>\n 1,079<\/td>\n 6.15<\/td>\n 1.21%<\/td>\n 1.29%<\/td>\n 1.24%<\/td>\n<\/tr>\n \n Mayotte<\/td>\n 993<\/td>\n 5.69<\/td>\n 1.21%<\/td>\n 1.27%<\/td>\n 1.23%<\/td>\n<\/tr>\n \n Moldova<\/td>\n 632<\/td>\n 2.81<\/td>\n 1.17%<\/td>\n 1.24%<\/td>\n 1.20%<\/td>\n<\/tr>\n \n Austria<\/td>\n 1,643<\/td>\n 15.03<\/td>\n 1.15%<\/td>\n 1.28%<\/td>\n 1.20%<\/td>\n<\/tr>\n \n Serbia<\/td>\n 759<\/td>\n 23.32<\/td>\n 1.10%<\/td>\n 1.16%<\/td>\n 1.12%<\/td>\n<\/tr>\n \n Singapore<\/td>\n 1,370<\/td>\n 13.25<\/td>\n 1.08%<\/td>\n 1.10%<\/td>\n 1.08%<\/td>\n<\/tr>\n \n Denmark<\/td>\n 1,297<\/td>\n 11.19<\/td>\n 1.01%<\/td>\n 1.13%<\/td>\n 1.05%<\/td>\n<\/tr>\n \n Israel<\/td>\n 1,584<\/td>\n 13.98<\/td>\n 0.95%<\/td>\n 1.01%<\/td>\n 0.97%<\/td>\n<\/tr>\n \n Djibouti<\/td>\n 856<\/td>\n 13.70<\/td>\n 0.94%<\/td>\n 0.96%<\/td>\n 0.95%<\/td>\n<\/tr>\n \n Norway<\/td>\n 1,320<\/td>\n 14.18<\/td>\n 0.81%<\/td>\n 0.88%<\/td>\n 0.84%<\/td>\n<\/tr>\n \n Canada<\/td>\n 976<\/td>\n 6.00<\/td>\n 0.80%<\/td>\n 0.88%<\/td>\n 0.83%<\/td>\n<\/tr>\n \n North Macedonia<\/td>\n 588<\/td>\n 7.89<\/td>\n 0.78%<\/td>\n 0.84%<\/td>\n 0.80%<\/td>\n<\/tr>\n \n Peru<\/td>\n 495<\/td>\n 9.67<\/td>\n 0.74%<\/td>\n 0.78%<\/td>\n 0.75%<\/td>\n<\/tr>\n \n Chile<\/td>\n 550<\/td>\n 18.52<\/td>\n 0.70%<\/td>\n 0.73%<\/td>\n 0.71%<\/td>\n<\/tr>\n \n Finland<\/td>\n 698<\/td>\n 13.30<\/td>\n 0.68%<\/td>\n 0.72%<\/td>\n 0.69%<\/td>\n<\/tr>\n \n Armenia<\/td>\n 452<\/td>\n 12.14<\/td>\n 0.67%<\/td>\n 0.70%<\/td>\n 0.68%<\/td>\n<\/tr>\n \n Estonia<\/td>\n 1,157<\/td>\n 14.46<\/td>\n 0.66%<\/td>\n 0.71%<\/td>\n 0.68%<\/td>\n<\/tr>\n \n Romania<\/td>\n 465<\/td>\n 6.41<\/td>\n 0.65%<\/td>\n 0.70%<\/td>\n 0.67%<\/td>\n<\/tr>\n \n Belarus<\/td>\n 663<\/td>\n 14.93<\/td>\n 0.64%<\/td>\n 0.66%<\/td>\n 0.64%<\/td>\n<\/tr>\n \n Montenegro<\/td>\n 497<\/td>\n 26.99<\/td>\n 0.63%<\/td>\n 0.66%<\/td>\n 0.64%<\/td>\n<\/tr>\n \n Dominican Republic<\/td>\n 458<\/td>\n 12.42<\/td>\n 0.63%<\/td>\n 0.67%<\/td>\n 0.64%<\/td>\n<\/tr>\n \n Croatia<\/td>\n 458<\/td>\n 13.85<\/td>\n 0.57%<\/td>\n 0.60%<\/td>\n 0.58%<\/td>\n<\/tr>\n \n Bosnia and Herzegovina<\/td>\n 399<\/td>\n 13.29<\/td>\n 0.53%<\/td>\n 0.56%<\/td>\n 0.54%<\/td>\n<\/tr>\n \n Brazil<\/td>\n 192<\/td>\n 5.95<\/td>\n 0.52%<\/td>\n 0.55%<\/td>\n 0.53%<\/td>\n<\/tr>\n \n Algeria<\/td>\n 62<\/td>\n 22.32<\/td>\n 0.51%<\/td>\n 0.53%<\/td>\n 0.52%<\/td>\n<\/tr>\n \n Czechia<\/td>\n 644<\/td>\n 82.20<\/td>\n 0.51%<\/td>\n 0.54%<\/td>\n 0.52%<\/td>\n<\/tr>\n \n Bahrain<\/td>\n 1,121<\/td>\n 12.73<\/td>\n 0.49%<\/td>\n 0.50%<\/td>\n 0.49%<\/td>\n<\/tr>\n \n Albania<\/td>\n 203<\/td>\n 14.89<\/td>\n 0.45%<\/td>\n 0.47%<\/td>\n 0.46%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n 5. US and UK Time Series with Time-Shifted Death Data<\/h2>\n
29th April – a major update of this section will be posted in the next day or so.<\/strong><\/em><\/h3>\n
<\/a><\/p>\n