by Rudolph Kalveks
For several weeks now, I have been tracking the fate of the “Canaries in the Mine” to see what the Coronavirus death statistics (courtesy of Worldometer) can tell us about the progress of the epidemic in the UK, along with a selection of other countries.
The method used is to fit the evolving historic death statistics to a simple Susceptible Infected Recovered/Resolved (“SIR” ) epidemiological model, as explained in the first article in this series. This data fitting exercise identifies four essential parameters that govern an archetypal epidemic in a given country or region. These correspond to the early rate of spread of infections, the rate at which infected individuals recover (or expire), the size of the (fatally) susceptible sub-population, and the date at which the epidemic starts.
Historic death statistics are augmented daily, and so we should not generally expect the parameters obtained from such a data fitting exercise to remain constant over time. The circumstances where the parameters do remain stable are those where the new death statistics match those extrapolated from the historic statistics under the simple model. This requires (i) that an epidemic conform to a simple SIR profile, (ii) that there is no material change in the combined effects of the many surrounding factors that influence the parameters of the SIR model (such as inherent population characteristics, nature of the pathogen, healthcare systems, social structure, etc.), and (iii) that there is sufficient historic data about the epidemic for the simple SIR model parameters to be well determined.
We can assess the extent to which the profile of an epidemic is stable by tracking these parameters. This in turn can provide data driven insights about the effects of historic or prospective government policy decisions.
Suppose we take as a “Null Hypothesis” the suggestion that the profile of a Coronavirus epidemic in a country is determined by the aforementioned surrounding factors and its early dynamics, so that its SIR model parameters are settled, once it has reached a late stage. Does the historic data for the UK epidemic fit with or disprove such a Null Hypothesis? And what does the data tell us about the UK experience compared with other countries?
The parameters, calculated as above, for simple SIR country models based on Worldometer statistics up to July 7 are set out in Table 1. As previously, various 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 1 to 3 below.
Table 1. Key Statistics for Selected Country Models (July 7, 2020).
Doubling days = natural log(2) / alpha. Half-life = natural log(2) / beta. R0=alpha/beta. Gamma = potentially fatally susceptible population.
Importantly, we can contrast these parameters with those calculated (and published here) a month ago, based on Worldometer statistics up to June 7, as set out in Table 2.
Table 2. Key Statistics for Selected Country Models (June 7, 2020).
It can be seen from a comparison of Tables 1 and 2 that in the main European countries (Spain, France, UK, Italy, Germany) and in the USA, the simple SIR models have been stable and their parameters have not changed significantly over the last month, despite the partial relaxation of lockdowns. These countries were amongst the first to be hit by the worldwide Coronavirus pandemic and it appears that they have been the first to reach the late stages of their epidemics. Their parameters for doubling periods (alpha) and recovery half-lives (beta) remain within a few percent of their earlier values. Importantly, if we consider the parameters (gamma) for their fatally susceptible sub-populations, the values have shifted by less than 2%. In Sweden, intriguingly, the outlook appears to be for an 8% lower fatally susceptible sub-population than a month ago – although this is perhaps just data noise following historic data revisions during June.
Thus, in Europe and in the USA, the Null Hypothesis (that each country is in the later stages of an archetypal SIR epidemic governed by fixed parameters) has held surprisingly well over the last month – within a small variation of 1-2% in term of the size of the fatally susceptible sub-populations. It appears that the partial relaxations of lockdown policies and the well-publicised breaches of lockdown restrictions have not had a material adverse effect, and there has not been any material “second wave”.
The situations of countries such as Brazil, India and S.Africa (amongst others worldwide) are different, since the new data has altered the best-fit parameters for their epidemics. The infection has spread much more slowly (as evident in longer early stage doubling periods) and the epidemics have not yet formed clear peaks (as evident in their long modelled infection half-lives). These epidemics are insufficiently well advanced for the simple SIR model parameters to have stabilised, and it would be premature to take the parameters for their fatally susceptible populations as forecasts. We can expect their simple SIR model parameters to continue to evolve in response to new data.
In Australia, a strict lockdown including inbound travel restrictions was implemented early in the epidemic, with the result that it has not penetrated far into the population. Indeed, as can be seen from Figure 2, the simple SIR model identifies that a proportion of the population has so far remained unexposed. Paradoxically, by virtue of the success of the Australian lockdown, it may be premature to take the parameters for the fatally susceptible sub-populations as a forecast for the outcome when travel restrictions are eventually relaxed.
Importantly, across all countries, the SIR model parameters for fatally susceptible populations remain of the order of 0.1% or less of general populations. This calls into question the assumption of the infection fatality rate (“IFR”) in the region of 0.9% that continues to be used by the UK government’s model builders [1]. Other epidemiologists assess the situation differently. Based on a review of seroprevalence studies from various countries worldwide, a recent preprint by Prof. Ioannidis at Stanford [2] estimates country/regional IFRs for Coronavirus in a wide range from only 0.02% up to 0.86%, with a median value of 0.25% (in line with US CDC estimates). The preprint also notes, “an unknown proportion of people may have handled the virus using immune mechanisms (…) that did not generate any serum antibodies”. Thus, the sizes of susceptible populations may be lower than those implied by IFRs estimated from seroprevalence, and could be consistent with the magnitudes of the fatally susceptible populations (gamma) tabulated herein.
Notwithstanding that populations are not homogeneous, so that there may remain local groups of vulnerable individuals who may continue to benefit from continued sheltering, the simple message for UK policymakers is that the historic data from the Coronavirus pandemic does not at present provide evidence to support the continuation of substantial restrictions on the normal functioning of our society and economy.
[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, pp.1-8. [2] 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 7, 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 7, 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 7, 2020). Calculations carried out using Mathematica.
Dr Rudolph Kalveks is a retired executive. His PhD was in theoretical physics.
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