Anti-clustering in the national SARS-CoV-2 daily infection counts

Poissonian and ${\varphi }_i^{\mathrm{\prime }}$ models: full sequences

We first consider the full count sequence {n_i(j), 1 ≤ j ≤ T_i} for each country i, with T_i valid days of analysis as defined in “SARS-CoV-2 Infection Data”. Our one-parameter model assumes that the counts are predominantly grouped in clusters, each with ${\varphi }_i^{\mathrm{\prime }}$ infections per cluster. Thus, the daily count n_i(j) is assumed to consist of $n_i(j)/{\varphi }_i^{\mathrm{\prime }}$ infection events. We assume that $n_i(j)/{\varphi }_i^{\mathrm{\prime }}$ on a given day is drawn from a Poisson distribution of mean ${\hat{\mu }}_i(j)/{\varphi }_i^{\mathrm{\prime }}$. We set ${\hat{\mu }}_i(j)$ to the median of the 4 neighbouring days, excluding day j and centred on it. For the initial sequence of 2 days, ${\hat{\mu }}_i(j)$ is set to ${\hat{\mu }}_i(3)$, and ${\hat{\mu }}_i(j)$ for the final 2 days is set to $\hat{\mu }$_i(T_i −2). By modelling ${\hat{\mu }}_i$ as a median of a small number of neighbouring days, our model is almost identical to the data itself and statistically robust, with only mild dependence on the choices of parameters. This definition of a model is more likely to bias the resulting analysis towards underestimating the noise on scales of several days rather than overestimating it; this method will not detect oscillations on the time scale of a few days to a fortnight that are related to the SARS-CoV-2 incubation time (Lauer et al., 2020; Yang et al., 2020; Huang et al., 2020b). For any given value ${\varphi }_i^{\mathrm{\prime }}$, we calculate the cumulative probability $P_{ij}^{\mathrm{\prime }}$ that n_i(j)/ ${\varphi }_i^{\mathrm{\prime }}$ is drawn from a Poisson distribution of mean $\hat{\mu }$_i(j)/ ${\varphi }_i^{\mathrm{\prime }}$. For country i, the values $P_{ij}^{\mathrm{\prime }}$ should be drawn from a uniform distribution if the model is a fair approximation. In particular, for ${\varphi }_i^{\mathrm{\prime }}$ set to unity, $P_{ij}^{\mathrm{\prime }}$ should be drawn from a uniform distribution if the intrisic data distribution is Poissonian. Individual values of $P_{ij}^{\mathrm{\prime }}$ (those that are close to zero or one) could, in principle, be used to identify individual days that are unusual, but here we do not consider these further.

We allow a wide logarithmic range in values of ${\varphi }_i^{\mathrm{\prime }}$, allowing the unrealistic subrange of ${\varphi }_i^{\mathrm{\prime }}$ < 1, and find the value ϕ_i that minimises the Kolmogorov–Smirnov (KS) distance (Kolmogorov, 1933; Smirnov, 1948; Justel, Pen & Zamar, 1997; Marsaglia, Tsang & Wang, 2003) from a uniform distribution, i.e. that maximises the KS probability that the data are consistent with a uniform distribution, when varying ${\varphi }_i^{\mathrm{\prime }}$. The one-sample KS test is a non-parametric test that compares a data sample with a chosen theoretical probability distribution, yielding the probability that the sample is drawn randomly from the theoretical distribution. This test uses information from the whole of the reconstructed cumulative distribution function, i.e. the set of $P_{ij}^{\mathrm{\prime }}$ values for a given country i. We label the corresponding KS probability as $P_i^{\mathrm{K}\mathrm{S}}$. We write $P_i^{\mathrm{P}\mathrm{O}\mathrm{I}\mathrm{S}\mathrm{S}}$: = $P_i^{\mathrm{K}\mathrm{S}}$ ( ${\varphi }_i^{\mathrm{\prime }}$ = 1) to check if any country’s daily infection rate sequence is consistent with Poissonian, although this is likely to be rare, as stated above: super-Poissonian behaviour seems reasonable. Of particular interest are countries with low values of ϕ_i. Allowing for a possibly fractal or other power-law nature of the clustering of SARS-CoV-2 infection counts, we consider the possibility that the optimal values ϕ_i may be dependent on the total infection count N_i. We investigate the (ϕ_i, N_i) distribution and see whether a scaling type relation exists, allowing for a corrected statistic ψ_i to be defined in order to highlight the noise structure of the counts independent of the overall scale N_i of the counts.

Standard errors in ϕ_i for a given country i are estimated once ϕ_i has been obtained by assuming that ${\hat{\mu }}_i(j)$ and ϕ_i are correct and generating 30 Poisson random simulations of the full sequence for that country. Since the scales of interest vary logarithmically, the standard deviation of the best estimates of log₁₀ ϕ_i for these numerical simulations is used as an estimate of σ(log₁₀ϕ_i), the logarithmic standard error in ϕ_i.

Subsequences

Since artificial interference in daily SARS-CoV-2 infection counts for a given country might be restricted to shorter periods than the full data sequence, we also analyse 28-, 14- and 7-day subsequences. These analyses are performed using the same methods as above (“Poissonian and ${\varphi }_i^{\mathrm{\prime }}$ Models: Full Sequences”), except that the 28-, 14- or 7-day subsequence that minimises ϕ_i is found. The search over all possible subsequences would require calculation of a Šidàk–Bonferonni correction factor (Abdi, 2007) to judge how anomalous they are. The KS probabilities that we calculate need to be interpreted keeping this in mind. Since the subsequences for a given country overlap, they are clearly not independent from one another. Instead, the a posteriori interpretation of the results of the subsequence searches found here should at best be considered indicative of periods that should be considered interesting for further verification.

Logarithmic median model

Each country’s time series is by default modelled with the mean of the expected Poisson distribution for n_i(j)/ ${\varphi }_i^{\mathrm{\prime }}$ on a given day being $\hat{\mu }$_i(j)/ ${\varphi }_i^{\mathrm{\prime }}$, where $\hat{\mu }$_i(j) is the median of n_i in the 4 neighbouring days, excluding day j and centred on it. As an alternative, we replace $\hat{\mu }$_i(j) on day j by v_i(j): = exp(median(ln(n_i))) calculated over the same set of neighbouring days. That is, we replace the usual linear median by a logarithmic median. This might better model the growing and decaying exponential phases of the infection count sequence.

Negative binomial model

The negative binomial distribution forbids underdispersion, but is worth considering, given its epidemiological motivation for the step from primary to secondary infections (Lloyd-Smith et al., 2005; Endo et al., 2020; He et al., 2020). For the counts of a given country i, we define an overdispersion parameter ${\omega }_i^{\mathrm{\prime }}$, where the binomial probability mass function for a given infection count k, considered as k failures, compared to r successes, with a probability p of success, is

(1) $$P(k;n,p)=\left(\begin{array}{c}k+r-1\\ k\end{array}\right)(1-p{)}^k{p}^r$$

(2) $$p:={\displaystyle \frac{{\omega }_i^{\mathrm{\prime }}}{1+{\omega }_i^{\mathrm{\prime }}}.}$$

On day j, with a modelled count of _i(j), we set

(3) $$r:={\omega }_i^{\mathrm{\prime }}{\hat{\mu }}_i(j),$$giving $\hat{\mu }$_i(j) as the mean of the distribution and $\hat{\mu }$_i(j) (1 + ${\omega }_i^{\mathrm{\prime }}$) as the variance. The preferred value of ${\omega }_i^{\mathrm{\prime }}$ (that yielding the lowest Kolmogorov–Smirnov test statistic when comparing the set of cumulative probabilities with a uniform distribution, as in “Poissonian and ${\varphi }_i^{\mathrm{\prime }}$ Models: Full Sequences”) is then ω_i. Thus, ω_i should behave similarly to ϕ_i to represent typical cluster size when both are greater than unity, while at low values (below unity), ω_i will be unable to represent distributions that are underdispersed with respect to the Poisson distribution, and will instead rapidly approach zero (the Poisson limit).

Does anti-clustering exist in grouped data?

The temptation to make ‘unnoticeable’ modifications that hide an increase in data from day j to day j + 1 might be less likely to occur on greater timescales. Moreover, some of the phenomena contributing to the intrinsic and administrative components of ${\varphi }_i^{\mathrm{\prime }}$ should be independent of time scale, while others should depend on the time scale. To provide clues for this type of analysis, the n_i(j) data have been summed in pairs and triplets of days, ignoring any one- or two-day remainder at the end of a sequence. These were analysed using the same algorithm as above for the full sequences (“Poissonian and ${\varphi }_i^{\mathrm{\prime }}$ Models: Full Sequences”).

Akaike and Bayesian information criteria

In each case we calculate the Akaike (1974) and Bayesian (Schwarz, 1978) information criteria, defined

(4) $$AIC:=2k-2\sum _i\ln L_i$$

(5) $$BIC:=\ln (N^{\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}})k-2\sum _i\ln L_i,$$respectively. The number of free parameters k is defined as the number of countries satisfying the criteria for a sequence to be analysable (“SARS-CoV-2 Infection Data”), since there is one free parameter allowed to vary individually for each country. The number of data points for BIC is set to the total number of days N^days in the sequences over all k countries. The ${\varphi }_i^{\mathrm{\prime }}$ model, and the logarithmic median and negative binomial alternatives, each have the same values of k and N^days. The 2-day and 3-day alternatives can be expected to have slightly smaller numbers of countries k whose sequences satisfy the analysis criteria, and much smaller numbers N^days of days, since in reality these no longer represent single days. The maximum likelihoood is defined L_i: = $P_i^{\mathrm{K}\mathrm{S}}$, i.e. the Kolmogorov–Smirnov probability that the observed values for the country are drawn from a rescaled Poisson (or negative binomial) distribution, as defined above.

Full infection count sequences

Figure 2 shows, unsurprisingly, that only a small handful of the countries’ daily SARS-CoV-2 counts sequences have noise whose statistical distribution is consistent with the Poisson distribution, in the sense modelled here: $P_i^{\mathrm{P}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{s}}$ (red circles) is close to zero in most cases. Specifically, 63 countries (80.8%) are inconsistent with the Poisson distribution at a significance of $P_i^{\mathrm{P}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{s}}$ < 0.01 and 66 countries (84.6%) are non-Poissonian at $P_i^{\mathrm{P}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{s}}$ < 0.05. On the contrary, the introduction of the ${\varphi }_i^{\mathrm{\prime }}$ parameter, optimised to ϕ_i for each country i, provides a sufficiently good fit in most cases, especially for the countries with low clustering ϕ_i. While some of the probabilities ( $P_i^{\mathrm{K}\mathrm{S}}$ (ϕ_i), green X symbols) in Fig. 2 are low in countries with the highest numbers of infections, these countries also have high ϕ_i, so are not of interest as indicators of the absence of noise. Among countries with ϕ_i < 10.0, the lowest probability $P_i^{\mathrm{K}\mathrm{S}}$ is that of Algeria with $P_i^{\mathrm{K}\mathrm{S}}$ = 0.17, i.e., the ϕ_i model is consistent with the data. In contrast, the negative binomial model ${\varphi }_i^{\mathrm{N}\mathrm{B}}$ (see “Alternative Analyses” below), which is super-Poissonian by definition, and cannot model sub-Poissonian behaviour, yields $P_i^{\mathrm{K}\mathrm{S}}$ = 0.01 for Algeria. Consistently with this, the Poissonian model for Algeria gives $P_i^{\mathrm{P}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{s}}$ = 0.005. The full sequence for Algeria is only fit by the ${\varphi }_i^{\mathrm{\prime }}$ model, which allows sub-Poissonian behaviour.

The consistency of the ϕ_i model with most of the data justifies continuing to Fig. 3, which clearly shows a scaling relation: countries with greater overall numbers N_i of infections also tend to have greater noise in the daily counts n_i(j). A Theil–Sen linear fit (Theil, 1950; Sen, 1968) to the relation between log₁₀ϕ_i and log₁₀N_i has a zeropoint of −1.10 ± 0.44 and a slope of 0.48 ± 0.07, where the standard errors (68% confidence intervals if the distribution is Gaussian) are conservatively generated for both slope and zeropoint by 100 bootstraps. By using a robust estimator, the low ϕ_i cases, which appear to be outliers, have little influence on the fit. The fit is shown as a thick green line in Fig. 3.

This ϕ_i–N_i relation is consistent with ${\varphi }_i\propto \sqrt{N_i}$. To adjust the ϕ_i clustering value to take into account the dependence on N_i, and given that the slope is consistent with this simple relation, we propose an empirical definition of a normalised clustering parameter

(6) $${\psi }_i:={\varphi }_i/\sqrt{N_i},$$

so that ψ_i should, by construction, be approximately constant. While the estimated slope of the relation could be used rather than this half-integer power relation, the fixed relation in Eq. (6) offers the benefit of simplicity.

This relation should not be confused with the usual Poisson error. By the divisibility of the Poisson distribution, the relation ${\varphi }_i\propto \sqrt{N_i}$ that was found here can be used to show that

$$\sigma [{\hat{\mu }}_i(j)/{\varphi }_i]\sim \sqrt{{\hat{\mu }}_i(j)/{\varphi }_i}$$

(7) $$\Rightarrow \sigma [{\hat{\mu }}_i(j)]\sim {\varphi }_i\sqrt{{\hat{\mu }}_i(j)/{\varphi }_i}\propto N_i^{1/4}{\hat{\mu }}_i(j{)}^{1/2},$$where σ[x] is the standard deviation of random variable x. If we accept ${\hat{\mu }}_i(j)$ as a fair model for n_i(j) and that n_i(j) is proportional to N_i, then we obtain

(8) $$\sigma [n_i(j)]\propto n_i^{3/4}.$$

Figure 4 shows visually that ψ_i appears to be scale-independent, in the sense that the dependence on N_i has been cancelled, by construction. The countries with the 10 lowest values of ψ_i are Algeria, Belarus, Nicaragua, Turkey, Russia, Tajikistan, Croatia, Syria, Saudi Arabia, and Iran. Detailed SARS-CoV-2 daily count noise characteristics for the countries with lowest ϕ_i and ψ_i are listed in Table 1, including the Kolmogorov–Smirnov probability that the data are drawn from a Poisson distribution, $P_i^{\mathrm{P}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{s}}$, the probability of the optimal ϕ_i model, $P_i^{\mathrm{K}\mathrm{S}}$, and ϕ_i and ψ_i.

The approximate proportionality of ϕ_i to $\sqrt{N_i}$ for the full sequences is strong and helps separate low-noise SARS-CoV-2 count countries from those following the main trend. However, the results for subsequences shown below in “Subsequences of Infection Counts” suggest that this N_i dependence may be an effect of the typically longer durations of the pandemic in countries where the overall count is higher.

Subsequences of infection counts

Figures 5–7 show the equivalent of Fig. 2 for sequences of lengths 28, 14 and 7 days, respectively. The Theil–Sen robust fits to the logarithmic $({\varphi }_i^{28},N_i)$; $({\varphi }_i^{14},N_i)$; and $({\varphi }_i^7,N_i)$ relations are zeropoints and slopes of 0.57 ± 0.43 and 0.06 ± 0.08; 0.52 ± 0.47 and 0.01 ± 0.09 ; and −0.10 ± 0.83 and 0.02 ± 0.13, respectively. There is clearly no significant dependence of ${\varphi }_i^d$ on N_i for any of these fixed length subsequences, in contrast to the case of the ϕ_i dependence on N_i for the full count sequences. Thus, the empirical motivation for using ψ_i (Eq. (6)) to discriminate between the countries’ full sequences of SARS-CoV-2 data is not justified from the information gained from the subsequences alone. Tables 2–4 show the countries with the least noisy sequences as determined by ${\varphi }_i^{28}$, ${\varphi }_i^{14}$ and ${\varphi }_i^7$, respectively.

Tables 2 and 3 show that the lists of countries with the strongest anti-clustering are similar to one another. Thus, Fig. 8 shows the SARS-CoV-2 counts curves for countries with the lowest ${\varphi }_i^{28}$, and Fig. 9 the curves for those with the lowest ${\varphi }_i^7$. Both figures exclude countries with total counts N_i ≤ 10000, in which low total counts tend to give low clustering. It is clear in these figures that several countries have subsequences that are strongly sub-Poissonian—with some form of anti-clustering, whether natural or artificial.

Countries in the median of the ${\varphi }_i^{28}$ and ${\varphi }_i^7$ distributions have their curves shown in Fig. 6 for comparison. It is visually clear in the figure that the counts are dispersed widely beyond the Poissonian band, and that the ${\varphi }_i^{28}$ and ${\varphi }_i^7$ models are reasonable as a model for representing about 68% of the counts within one standard deviation of the model values.

Alternative analyses

Figure 11 (left) shows that the logarithmic median model (“Logarithmic Median Model”) of the counts gives almost identical best estimates to those of the primary model, i.e. ${\psi }_i^{\mathrm{L}\mathrm{M}}$ ≈ ψ_i, but Table 5 shows very strong evidence favouring the original, arithmetic median model.

Figure 11 (right) shows that the negative binomial model (“Negative Binomial Model”) roughly gives ${\psi }_i^{\mathrm{N}\mathrm{B}}$ ∼ ψ_i (i.e. ω_i ∼ ϕ_i), tending to ${\psi }_i^{\mathrm{N}\mathrm{B}}$ < ψ_i, especially for the least clustered cases. The error bars are very big for ${\psi }_i^{\mathrm{N}\mathrm{B}}$ for several countries. Table 5 again shows very strong evidence favouring the original model over the negative binomial model.

Figure 12 shows that the counts grouped (summed) in pairs and triplets (“Does Anti-Clustering Exist in Grouped Data?”) yield ${\psi }_i^{2\mathrm{d}}$ and ${\psi }_i^{3d}$ with more scatter and generally larger error bars than that of ψ_i, and ${\psi }_i^{2\mathrm{d}}$ and ${\psi }_i^{3d}$ are mostly greater than ψ_i. Whether the AIC and BIC evidence (Table 5) for 2- and 3-day grouped data can be directly compared to that of the main analysis depends on whether the grouped data can be considered to be the same observational data as the original data, modelled with fewer free parameters. Since the characteristic of study is the noise, not the signal, the validity of this direct comparison is doubtful. Nevertheless, if the values of the AIC and BIC evidence are considered literally, then the 2-day grouping would yield a worse model than the model of the daily data, while the 3-day grouping would yield a better model than that for the daily data. The comparison of these different analyses could potentially be used to obtain a deeper understanding of the complex dynamics of this pandemic. The epidemiologically relevant sociological parameters of countries around the world are highly diverse (varying in population density, patterns of social contact, tendency to obey or disobey official health guidelines such as lockdown measures, demographic profiles, quality and availability of health services, communication patterns, frequency of COVID-19 comorbidity conditions, climate (Afshordi et al., 2020)), so comparison of the clustering behaviour on these different time scales might help to separate out some of these contributions.

Low clustering, high N_i

Turkey and Russia have total infection counts of about 3 million, similar to those of several other countries, but have managed to keep their daily infection rates much less noisy—by about a factor of 10 to 100—than would be expected from the general pattern displayed in the figures. These two countries appear as an isolated pair in the bottom-right of both Figs. 4 and 5, and appear in all four tables of low ψ_i (Table 1) and low subsequence ϕ_i (Tables 2–4). Russia has the very modest value of ϕ_i = 7.24 × 10^±0.067 and Turkey has ϕ_i = 6.46 × 10^±0.057, despite their large total infection counts. This would require that both intrinsic clustering of infection events and administrative procedures work much more smoothly in Russia and Turkey than in other countries with comparable total infection counts. Tables 2 and 3 and Fig. 8 show that the Russian and Turkish official SARS-CoV-2 counts indeed show very little noise compared to more typical cases (Fig. 10). There appear to be weekend dips in the Russian case (see “Weekend Dips in the Counts” below). Since these are included in the analysis, an exclusion of the weekend dips would lead to an even lower clustering estimate. At the intrinsic epidemiological level, if the Russian and Turkish counts are to be considered accurate, then very few clusters—in nursing homes, religious gatherings, bars, restaurants, schools, shops—can have occurred. Moreover, laboratory testing and transmission of data through the administrative chain from local levels to the national health agency must have occurred without the clustering effects that are present in the data for the United States, Brazil, India, and other countries with high total infection counts N_i > 2 million, for which ϕ_i is typically above 100. International media interest in Russian COVID-19 data has mostly focussed on controversy related to COVID-19 death counts (Cole, 2020), with apparently no attention given so far to the modestly super-Poissonian nature of the daily counts, in contrast to the strongly super-Poissonian counts of other countries with high total infection counts. How did Russia and Turkey achieve low ϕ_i (super-Poissonian), i.e. low clustering?

Low clustering, medium N_i

Some cases of interest appear among the countries with officially lower total infection counts. The Belarus (BY) case is present in all four tables (Tables 1–4). The least noisy Belarusian counts curve appears in Figs. 8 and 9. As with the other panels in the daily counts figures, the vertical axis is set by the data instead of starting at zero, in order to best display the information on the noise in the counts. With the vertical axis starting at zero, the Belarus daily counts would look nearly flat in this figure. They appear to be bounded above by the round number of 1,000 SARS-CoV-2 infections per day, which, again, as in the case of India, could appear to be a psychologically preferred barrier. Media have expressed scepticism of Belarusian COVID-19 related data (Kramer, 2020; AFN, 2020). The Albanian case (Figs. 8 and 9) also could be interpreted as hitting a psychological barrier of a decimal round number, an artificial cap of 300 infections per day, in mid-October 2020.

One remaining case of a coincidence is that the lowest noise 7-day sequence listed for Poland (Table 4) is for the 7-day period starting 20 June 2020, with ${\varphi }_i^7=0.16\times {10}^{\pm 0.13}$. This is a factor of about 300 below Poland’s clustering value for the full sequence of its SARS-CoV-2 daily infection counts, ϕ_i = 45.71 × 10^±0.057, which Fig. 3 shows is typical for a country with an intermediate total infection count. On 28 June 2020, there was a de facto (of disputed constitutional validity, Wyrzykowski, 2020; Letowska & Pacewicz, 2020) first-round presidential election in Poland. Figure 9 shows that the counts for Poland during the final pre-first-round-election week did not scatter widely throughout the Poissonian band. A decimal-system round number also appears in this figure: the daily infection rate is slightly above about 300 infections per day and drops to slightly below that. This appears to be the same psychological daily infection count attractor as for Albania. The intrinsic clustering of SARS-CoV-2 infections in Poland together with testing and administrative clustering of the confirmed cases appear to have temporarily disappeared just prior to the election date, yielding what is best modelled as an incident of sub-Poissonian counts.

JHU CSSE data

The JHU CSSE data give mostly similar results to the C19CCTF data. These are presented and briefly discussed in Appendix A.

Weekend dips in the counts

One sociological contribution to noise not mentioned above is that in several countries, the official daily counts are lower on or immediately after weekends. Credible factors include fewer medical and laboratory workers available to carry out tests and fewer administrators registering, collecting and transmitting data. A dip in the counts on weekends would tend to add noise to the daily count time series, making the above results conservative. These dips can be quantified using the one-dimensional discrete fast Fourier transform (FFT). With the usual FFT convention, we transform n_i(j) into f_i(j) at j days, where f_i(0) is the mean and a weekly dip should appear as a negative value at f_i(7). We define a weekend dip w_i for country i by subtracting the mean of the neighbours and normalising:

(9) $$w_i:=1+\pi {\displaystyle \frac{f_i(7)-[f_i(6)+f_i(8)]/2}{f_i(0)}.}$$

This should correspond to a multiplicative factor, i.e., w_i = 0.85 means a 15% dip.

Figure 17 shows the distribution of w_i (mean ± std. error: 1.001 ± 0.015; std. dev.: 0.137; median: 0.999; interquartile range: 0.104). Unexpectedly, not only are there several countries with dips, but there are also several countries with a strong excess signal on the 7-day time scale. There is no reason to expect the overall distribution to be Gaussian. The Shapiro–Wilk statistic (Shapiro & Wilk, 1965) is W = 0.806, rejecting the possibility of the distribution being Gaussian to extremely high significance: p = 9.82 10⁻⁹. Future work in studying the noise characteristics of a pandemic could take into account this weekly component of daily infection statistics.