by Rudolf Kalveks
Since the onset of autumn it has become increasingly clear that SARS-CoV-2 (‘coronavirus’) epidemics cannot generally be analysed as waves with single peaks. The virus, which had largely disappeared from Europe by the middle of summer, made a comeback during the autumn. So, what conclusions should we draw about coronavirus dynamics from the historic time series data? And do these justify ongoing Government responses?
Figure 1. Cumulative Death Statistics for Selected Countries. Cumulative coronavirus deaths, expressed as a percentage of country populations, are plotted on a logarithmic scale, as time series up to December 21st, 2020. Source: Worldometer.
As previously, we focus on the ‘Canaries in the Mine’, or the coronavirus death statistics for a selection of European and other countries, published by Worldometer. It has been shown that these statistics exaggerate the impact of the coronavirus. Several countries classify deaths simply according to the presence of coronavirus, rather than apportioning causation to other contributing pathologies. This problem is compounded by the PCR test, which is notorious for false positives (as discussed elsewhere). Nonetheless, the death statistics are considerably more relevant than the ‘cases’ identified by PCR tests alone (which continue to provide the basis for much of the prevailing government and media narrative).
Before proceeding, it is helpful to recap the coronavirus fatalities for our selection of countries. The curves shown in Figure 1 remind us that cumulative fatalities tend to reach a plateau at around 0.1% of a population (or less). During the autumn, a few countries have started to test this plateau and these dynamics merit investigation.
In previous posts, local epidemics were analysed using the three population ‘SIR’ (Susceptible – Infected – Recovered/Resolved) model of epidemiology. Prior to the autumn, such simple models fitted the observed data in many European countries surprisingly well; the profiles of epidemics could be described by three simple epidemiological parameters (corresponding to the size of a susceptible population, the initial doubling period and the typical ‘half-life’ of the recovery period). Notably, these SIR models did not need to invoke explicit effects from variations in lockdown policies to match observed data.
Our approach to understanding the dynamics of the autumn peaks in Europe has been to superimpose second SIR country models onto the first ones. In essence, each country is treated as having distinct population subsets. One subset was susceptible to the spring peak; a second subset becomes susceptible in the autumn. The subsets are disjoint since those recovered from the spring peak have immunity and are no longer susceptible. Given the low level of infections during the summer, the observed data for each country can be separated naturally into distinct spring/summer and autumn/winter time series by identifying inflection points. These series can then be analysed using separate SIR models, and the results aggregated to capture the overall dynamics.
The results of such models are summarised graphically in Figures 2 to 5, along with their key parameters in Table 1. The parameters have been estimated using the curve fitting techniques described in previous posts. Two stage models are provided for the European countries and for Australia. For the others, single stage SIR models are shown, as previously. The parameters in the most recent peak can be expected to change with new data and the outliers may normalise.
These models should be viewed as ‘effective’ or simplified models, whose parameters combine the underlying virology with the effects of modifying factors, such as non-pharmaceutical interventions (‘lockdowns’), seasonal influences and other population or geographic inhomogeneities.
Figures 2 and 3 show that in the European countries, the two stage models fit the observed death statistics very well. In the autumn peak the parameters contain initial doubling periods that are typically longer and R0 values that are typically lower than in the spring peak; this would be consistent with partial herd immunity after the spring. The estimated sizes of the susceptible populations indicate that the autumn peaks may be of a similar order of magnitude to the spring. Importantly, however, the charts in Figure 4 indicate that the autumn peaks of infections (orange sub-populations) have passed or (in the case of Germany) should soon pass. Correspondingly, the charts in Figure 5 indicate current R values at or below 1.
Table 1. Key Parameters for Selected Country Models (December 21, 2020). For two stage SIR models, parameters for each stage are shown as “spring | autumn”. R0= Half-life / Initial doubling period. The parameters for the latest peak are based on incomplete histories and remain sensitive to new data.
It is notable that Sweden, which introduced few lockdown restrictions, enjoyed a summer lull followed by a small autumn peak, while other European countries, which implemented many and varying lockdown restrictions during the year, have found themselves faced by larger autumn peaks. The models indicate that the overall susceptibility in Sweden, taking spring and autumn together, may end up relatively low within Europe.
The key question is what has driven the summer lull followed by the autumn peaks in Europe. It is implausible that lockdowns were responsible for the summer lull, since the subsequent tightening of lockdown policies has not prevented the autumn peaks. Is there any plausible mechanism other than the commonplace observation that there is a high level of natural seasonal variation in (the susceptibility to and transmission of) respiratory infections? As Prof. Ioannidis notes,2 “Seasonality may also play a role in the dissipation of the epidemic wave.”
We should recall the findings in the Lancet,1 that even the most draconian (and impractical) combinations of lockdown policies only reduce R to around 65% of its initial R0 value after seven days, and to around 48% if continued for a month. Once lockdown restrictions are limited, the R values revert.
Coronaviruses can start with an R0 in the range of five to ten times (as seen from Table 1), so that a much greater reduction of 80%-90% would be necessary to reduce R to below one. Simple arithmetic shows that for readily transmissible viruses, lockdown restrictions are insufficient, and that the spread of infections can only be halted by some combination of herd immunity (whether by vaccination or by recovery following infection) and seasonality.
Turning briefly to the other countries: the coronavirus dynamics in India, Brazil and South Africa can all still be captured by simple SIR models. Australia has thus far been largely successful in avoiding the coronavirus through lockdowns, presumably aided by geographic and climatic factors.
The USA is complicated by both geographic and temporal inhomogeneities, so that the one stage SIR model fit with data is not so good and its parameters do not have simple interpretations. In the USA as a whole, there is no decoupling between the spring and autumn peaks. However, it can be observed on Worldometer that (i) individual states are typically affected by one or other seasonal peaks, but not always both, (ii) the states hit hardest by the spring peak have come off lightly in the autumn peak, and (iii) the cumulative fatalities in individual states fall in a range up to 0.1%-0.2% of state populations. While it should in principle be possible to obtain more insight about the dynamics in the USA by constructing multi-stage SIR models at a state level, we continue for now with the one stage country model parameters.
Combining seasonal peaks, the modelled fatally susceptible populations typically fall in a range of 0.1%-0.2% of country populations, or less. Herd immunity takes effect once fatalities in a population approach the Infection Fatality Rate, so these parameters appear consistent with the studies of seroprevalence data by Prof. Ioannidis,2 who has estimated a median population IFR of 0.25%, driven primarily by vulnerable older and/or medically compromised segments.
By contrast, these SIR models do not support the assumed IFRs averaging 0.85% that were used in the Ferguson model,3 where it is notable that the role of seasonality is not even considered. It is perplexing that the Ferguson work has not been publicly updated to reflect actual data and observed seasonality – and yet continues to guide UK government policy.
We have seen that it is not necessary to invoke explicit variations in lockdown effects to fit SIR model parameters to the empirical data. The observed death statistics can be interpreted in terms of the virology of herd immunity, augmented by seasonality, playing out against a background of population characteristics. Whilst lockdowns may play a secondary role in flattening seasonal peaks and/or shifting susceptible individuals between peaks, the empirical evidence for lockdowns leading to a meaningful and sustainable reduction in population susceptibility remains questionable.
1 Li, Y., Campbell, H., Kulkarni, D., Harpur, A., Nundy, M., Wang, X., Nair, H. and for COVID, U.N., 2020. The temporal association of introducing and lifting non-pharmaceutical interventions with the time-varying reproduction number (R) of SARS-CoV-2: a modelling study across 131 countries. The Lancet Infectious Diseases.
2 Ioannidis, J., 2020. The infection fatality rate of COVID-19 inferred from seroprevalence data. medRxiv.
3 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 December 21st, 2020. The solid blue curves represent one or two stage SIR models. 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 one or two stage SIR models. 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 December 21st, 2020. Calculations carried out using Mathematica.
Figure 4. Model Sub-Populations The one or two stage 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 (December 21st, 2020). Calculations carried out using Mathematica.
Figure 5. Model Epidemiological R. The horizontal scale counts days from the first recorded death, with the vertical red line indicating the most recent data (December 21st, 2020). Calculations carried out using Mathematica.
Dr Rudolph Kalveks is a retired executive. His PhD was in theoretical physics.
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