Dr Rudolph Kalveks
FT 29/7/2020: “Europe battles to contain surge in Covid-19 cases. Experts surprised at how fast the lifting of restrictions led to a rise in infections.”
Telegraph 1/8/2020: “The virus warning light is flashing.”
Let us see what the “Canaries in the Mine” (i.e., the coronavirus death statistics, courtesy of Worldometer) tell us about the actual development of the epidemic in Europe and other parts of the world.
First of all, bearing in mind the usual caveats about reliability, we should recap the death statistics in our selection of countries. These are summarised as time series in Figure 1 below, where a logarithmic scale has been used. (An upward sloping straight line on such a graph would indicate an exponential growth rate).
Figure 1. Cumulative Death Statistics for Selected Countries (July 31, 2020).
Cumulative coronavirus deaths, expressed as a % of country populations, are plotted on a logarithmic scale, as time series up to July 31, 2020. Source: Worldometer.
The curves show that when the penetration of coronavirus in a country reaches a ceiling, typically represented by a fatality rate below 0.1% of its population, its spread slows to a standstill, with few further fatalities arising. This certainly appears to have been the case in mainland Western Europe, where the average daily death rates from the coronavirus have now declined to single figures (in terms of individuals) in every country, notwithstanding the relaxation of many of the lockdown restrictions imposed early during the pandemic. The “Canaries” are telling us empirically that Western Europe has reached or is close to herd immunity. Prof. Gupta at Oxford and other epidemiologists explain this in terms of the immune mechanisms at work in addition to antibodies, such as T-cells. 
By using cases rather than fatalities to judge prevalence, the “experts” at the FT and Telegraph are replicating the widespread mistake of using a systematically biased measurement. As the extent of testing has increased over time, and has become better targeted through ‘test and trace’ systems, there has been an inevitable increase in the ratio of cases identified (which include many mild or asymptomatic presentations) to eventual fatalities. This problem was foreshadowed early on by Prof. Ioannidis of Stanford who observed that the Infection Fatality Rate depends on both its numerator (deaths) and its denominator (infections) – and that medical authorities have little idea of the latter, which could be rather large. 
Nonetheless, many commentators and perplexingly the UK Government continue to spread the “second wave” narrative. As to when the “second wave” will actually appear in the UK death statistics, the answer invariably seems to be “mañana”!
A “second wave” in an epidemic would be signaled by a significant upward inflection in the gradient of the cumulative deaths curve, so that the fatality rate increased instead of declining. One putative example in Figure 1 is Australia. Here the initial lockdown policy was so effective that nearly all the population remained unexposed, and thereby vulnerable to a re-introduction of the virus, as has happened in the last few weeks. Perhaps, however, this should be taken as the definition of an actual “mañana wave” – being a consequence of policies that defer herd immunity until tomorrow.
The “Canaries” in Figure 1 are also telling us that other countries around the world – for example, Brazil, India and South Africa – are not yet at similarly advanced stages to Western Europe; their cumulative fatalities are at proportionately lower levels and their epidemics do not yet appear close to their eventual ceilings.
As elaborated in earlier notes, we can assess the extent to which the dynamics of an epidemic are stable by tracking the empirical parameters for an epidemiological SIR model extracted from the death statistics. Thus, we can test our previous Null Hypothesis that the profile of the coronavirus epidemic in the UK has been largely determined by initial conditions and its early dynamics (implying that ongoing late stage Government interventions have little impact).
The parameters for simple SIR country models, calculated as previously, updated using Worldometer statistics up to July 31st are set out in Table 1. Additional graphics illustrating (i) the fit of the model with the data, (ii) the three model sub-populations and (iii) the R values implied by the model are shown in Figures 2 to 4 below.
Table 1. Key Statistics for Selected Country Models (July 31, 2020).
Doubling days = natural log(2) / alpha. Half-life = natural log(2) / beta. R0 = alpha/beta. Gamma = potentially fatally susceptible population.
We find that the simple SIR models in the main European countries (Spain, France, UK, Italy, Germany), have been remarkably stable over the last two months, despite the continued relaxation of lockdowns. The important parameters (gamma in Table 1) for their fatally susceptible sub-populations have shifted by at most 2% from the calculations previously carried out using the Worldometer statistics up to June 7th. And in Sweden, the outlook appears to be for a fatally susceptible sub-population 9% smaller than two months ago! All this supports the perspective that the epidemics in Western Europe have essentially reached their natural limits.
The situation in the USA is, however, less straightforward. The model parameter for the fatally susceptible has increased by 10% over the last two months, and a recent upward inflection in the death statistics can be seen in Figure 2. Detailed investigation shows that while many North Eastern states (New York, New Jersey, Connecticut, Massachusetts…) have reached stable late stages in their epidemics, similar to Western Europe, while many Southern and Western states are still at earlier stages – their cumulative fatalities are proportionately lower and not yet reaching their ceilings. We can ask whether early lockdowns may have had a similar effect to that witnessed in Australia, such that for many US states outside the North East the eventual achievement of herd immunity has simply been deferred – another example of actual “mañana” waves?
As noted above, other countries such as Brazil, India and South Africa, also remain at an earlier stage. In part this can be attributed to the later arrival of the pandemic, but mostly this follows from the slow growth rate of the epidemic in these countries (i.e. long doubling periods in Table 1). Presumably, this in turn reflects differences in the characteristics of local populations and geographies. However, the decay and susceptibility parameters for these countries have not yet stabilized and the simple SIR model still provides unclear visibility as to how far the coronavirus will ultimately penetrate their populations.
Finally, while it cannot be assumed that the many different country populations worldwide can be naturally aggregated into a single SIR model, so that the “global” parameters should be treated with caution, it must be said that the increase in the global susceptibility parameter to over two million presages a substantial further spread of the epidemic before worldwide herd immunity is eventually established.
 Lourenco, J., Pinotti, F., Thompson, C. and Gupta, S., 2020. The impact of host resistance on cumulative mortality and the threshold of herd immunity for SARS-CoV-2. medRxiv.
 Le Bert, N., Tan, A.T., Kunasegaran, K., Tham, C.Y., Hafezi, M., Chia, A., Chng, M.H.Y., Lin, M., Tan, N., Linster, M. and Chia, W.N., 2020. SARS-CoV-2-specific T cell immunity in cases of COVID-19 and SARS, and uninfected controls. Nature, pp.1-10.
 Ioannidis, J., 2020. The infection fatality rate of COVID-19 inferred from seroprevalence data. medRxiv.
Figure 1. Model Fit with Data. The orange data points are cumulative deaths, as reported daily by Worldometer, starting from the first recorded death until July 31, 2020. The solid curves represent the minimal SIR model. Calculations carried out using Mathematica.
Figure 2. Model Sub-Populations. The three SIR model sub-populations are Susceptible (blue), Infected (orange) and Resolved (green). The vertical scale counts cumulative deaths. The horizontal scale counts days from the first recorded death, with the vertical red line indicating the most recent data (July 31st, 2020). Calculations carried out using Mathematica.
Figure 3. Model Epidemiological R. The horizontal scale counts days from the first recorded death, with the vertical red line indicating the most recent data (July 31st, 2020). Calculations carried out using Mathematica.
Dr Rudolph Kalveks is a retired executive. His PhD was in theoretical physics.