by Rudolph Kalveks

We should always keep in mind the rationale for focusing on reported deaths, a.k.a. “the Canaries in the Mine”, in order to assess the state of the Coronavirus epidemic. The sensible reason for constructing models around death statistics rather than identified cases was given early on by the UK government’s advisers – “Reported deaths are likely to be far more reliable than case data…”^{1} The many factors that contribute to problems with case data, such as increased testing rates, false positives, the conflation of asymptomatic with serious presentations, and double counting, have all been echoed by numerous healthcare professionals in recent weeks and need not be elaborated here.

Nonetheless, it is clear from the Worldometer death statistics that there has been some resurgence of the virus in European countries from late summer onwards. Should we be concerned that this is a “second wave”? Or are we merely seeing natural fluctuations in response to the many factors that influence the evolution of an epidemic, such as (a) inhomogeneous populations, (b) changes in virulence and (c) changes in social behaviour (whether in response to government restrictions, or otherwise)? Any such fluctuations must inevitably be compounded by the vagaries of national reporting systems.

So, how can we objectively distinguish a “wave”, which may represent a cause for concern, from a natural “ripple”, which can be expected to subside without impacting the population at large?

The approach taken here is to look at the deviations between the actual statistics and those generated by a simple epidemiological SIR model, as described in previous posts. How significant are these deviations and do they follow a pattern?

But first let us recap the Coronavirus fatalities for a selection of countries. The curves shown in Figure 1 remind us that cumulative fatalities typically reach a plateau at or below 0.1% of a population. This pattern has remained unchanged for several months and justifies the comparison of the observed data with the curves generated by simple SIR models, which characteristically exhibit such strong initial growth followed by a plateau.

**Figure 1. Cumulative Death Statistics for Selected Countries.** Cumulative Coronavirus deaths, expressed as a % of country populations, are plotted on a logarithmic scale, as time series up to September 30, 2020. Source: Worldometer.

Table 1 contains a summary of the parameters for basic SIR epidemiological models that have been fitted to the Worldometer death statistics for a selection of countries, using the curve fitting techniques described in previous posts. The key parameters are the initial doubling period (evident in the early phase of an epidemic), the subsequent recovery period from infections (which can be expressed as a half-life), and the size of the susceptible population. When the prevailing doubling period exceeds the half-life, an epidemic fades away and “herd immunity” is achieved.

**Table 1. Key Parameters for Selected Country Models (September 30th, 2020).** Doubling days – natural log(2)/alpha. Half-life = natural log(2)/beta. R_{0}=alpha/beta. Gamma = potentially fatally susceptible population.

These models continue to be fairly stable for those countries that are advanced in their epidemics. Thus, the parameters for the fatally susceptible in the main European countries, at around 0.06% of general populations, have changed little since this table was first posted on June 7th. In Spain, France, Italy and Germany, the susceptible populations remain within 3% of those calculated four months ago. In Sweden and the UK (in the latter case helped by definitional changes) the susceptible populations are indeed about 10% smaller!

In other countries around the world, where the pandemic hit later, or has spread more slowly, the parameters for the SIR model parameters have evolved (and can be expected to continue to evolve) in response to new data. Nonetheless, visual inspection of Figure 2 illustrates a generally good fit between the historic data and the SIR models. Deviations are however clearly visible in the cases of Australia and the US and, more recently, Spain and France.

In Figure 3 these deviations are scaled relative to country populations and expressed as parts per million. Thus, in the case of the UK, at the time of writing, the graph for the UK shows that the deviation between actual and modeled cumulative deaths stands at around 6.5 per million, or 440 people.

These charts illustrate the significance of the “ripples” relative to the basic epidemic wave that is affecting each country. Although their irregular sawtooth behavior makes it problematic to anticipate turning points in the ripples, there are a number of features that could merit a detailed epidemiological examination. For example, besides irregular fluctuations and data noise, several of the charts exhibit an element of periodicity over a cycle of three months or so. Does this periodicity arise from the superposition of multiple epidemics that are separated by transmission delays, or alternatively, does it arise from delayed social reactions to changes in the severity of an epidemic? It could also be speculated that the summer holiday season may have served to synchronise ripples across regions within Europe, most notably in Spain.

Recalling the empirically observed “herd immunity” plateau, which appears at a cumulative fatality rate of around 0.1% of a population, or 1000 per million, we can see that the magnitudes of the ripples observed thus far in our selection of countries have been bounded within ranges of 10 to 50 per million – a factor of 20 to 100 times smaller than the basic epidemic wave.

Suppose a lockdown zealot were to insist that we view the increase in cases in Europe since the start of August as a separate “second wave”. While this two month period does not contain a sufficiently long time series to calculate all the SIR parameters with any degree of confidence, we could extract information about the implied doubling period for this “second wave”. Considering Italy, UK, Spain, France and Germany, we can estimate doubling periods during August and September that are around *three* times longer than those observed in the relevant SIR models in Table 1 (which range between 1.8 and 2.8 days). This entails that any sub-populations involved in the mooted Coronavirus resurgence start significantly closer to “herd immunity” than in March and that its impact could be expected to be correspondingly smaller.

Thus, we continue to have the perplexing situation in the UK where an observational analysis across multiple countries puts the ripples (and their plausible development) at (one or two) orders of magnitude *smaller* than the first epidemic wave, while government advisers persist in presenting scenarios where the “second wave” could be an order of magnitude *greater* than the first!

1 Flaxman, S., Mishra, S., Gandy, A., Unwin, H.J.T., Mellan, T.A., Coupland, H., Whittaker, C., Zhu, H., Berah, T., Eaton, J.W. and Monod, M., 2020. Estimating the effects of non-pharmaceutical interventions on COVID-19 in Europe. Nature, 584(7820), pp.257-261.

**Figure 2. Model Fit with Data. **The orange data points are cumulative deaths, as reported daily by Worldometer, starting from the first recorded death until September 30th, 2020. The solid blue curves represent the minimal SIR model. Calculations carried out using *Mathematica*.

**Figure 3. Variation Between Actual and Model Data.** The vertical scale shows the excess (or deficit) of actual cumulative deaths over the SIR model. The differences are scaled relative to country populations and expressed as deaths per million. The horizontal scale counts days from the first recorded death until September 30th, 2020. Calculations carried out using *Mathematica*.

**Figure 4. 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 (September 30th, 2020). Calculations carried out using *Mathematica*.

*Dr Rudolph Kalveks is a retired executive. His PhD was in theoretical physics.*