An estimation of the proportion of asymptomatic Omicron infection

Abstract

In this document, I compare data from two COVID vaccine trials carried out in South Africa (the Ubuntu trial and the Sisonke sub-study). Both trials were conducted on unvaccinated subjects, asymptomatic at the time of enrollment. Among them, some were found positive to SARS-CoV-2, but while the variant was identified as Delta in the Sisonke sub-study, the Omicron variant was reported in asymptomatic subjects of the Ubuntu trial. By considering the proportion of SARS-CoV-2 positive subjects in these two populations, the average number of daily confirmed cases, and the percentage of unvaccinated South African citizens at the respective times of enrollment of the two trials, I calculated the proportion of asymptomatic infections for Omicron, as a function of the same parameter for the Delta variant. I found that if we accept a proportion of asymptomatic infections of 17% for Delta, then we conclude that about 60% of all Omicron cases are asymptomatic, in unvaccinated subjects. This points towards significantly lower pathogenicity for the Omicron variant when compared to the Delta variant.

Introduction

The B.1.1.529 (Omicron) SARS-CoV-2 variant was first identified in Botswana and South Africa and was classified by WHO as a variant of concern (VOC) on 26 November 2021 (WHO website). As of 15 December 2021, the Omicron variant had already spread in 77 countries (especially in the United Kingdom, South Africa, and the United States) (Thakur V, Ratho RK, 2021). Omicron is characterized by a large number of mutations and deletions that seem to give it increased antibody escape capacity and enhanced transmissibility (Miller NL et al 2021). Yet, if we look at the number of deaths in South Africa (R), we note that despite a rapid increase in confirmed cases (most Omicron), it doesn’t follow the usual consequent increase (with a phase shift of two weeks). This suggests that Omicron is less dangerous than Delta, also considering that previous infection with Delta does not seem to protect against infection with Omicron (Garret N et al 2021) and that only 30% of the population of South Africa received at least one dose of vaccine (R).

In what follows I present a quantitative method for the comparison between the effect of Omicron and Delta on unvaccinated subjects. In particular, I calculate the ratio between asymptomatic SARS-CoV-2+ subjects and total SARS-CoV-2+ subjects for Omicron ( $r_{o}$ ) as a function of the same ratio for Delta ( $r_{\delta}$ ). We consider only subjects who did not receive any dose of vaccination against SARS-CoV-2. By considering only unvaccinated subjects, we eliminate the confounding effect of vaccination on the pathogenicity of Omicron and Delta. The aforementioned ratio might be considered one of the possible indexes of disease severity, while we wait for further data, according to the following assumption: the higher the number of asymptomatic carriers of a virus divided by the total number of carriers, the lower the pathogenicity of the virus. The derivation of the function $r_{o}\,=\,r_{o}(r_{\delta})$ is described in the Methods section.

Results

The function $r_{o}\,=\,r_{o}(r_{\delta})$ is plotted in Figure 1. It indicates the proportion of Omicron asymptomatic infections as a function of the same proportion for Delta, in unvaccinated subjects. As you can see, the Omicron variant leads to a higher proportion of asymptomatic cases when compared to the Delta one. If we assume for Delta a value $r_{\delta}\,=\,0.17$ (Byambasuren O. et al. 2020), we have that $r_{o}\,\sim\,0.6.$ The diagram has been plotted for two different choices of the ratio $k_{o}/k_{\delta}$ (the meaning of this ratio is explained in the following paragraph), but as you can see, it doesn’t change much whether you consider one value or the other.

**Figure 1.** The ratio of asymptomatic SARS-CoV-2+ subjects and total SARS-CoV-2+ subjects for the Omicron variant expressed as a function of the same ratio for the Delta variant. All subjects are considered unvaccinated.

Methods

Let $N_{pos}$ be the number of unvaccinated individuals positive to SARS-CoV-2 at a given time and $r$ the ratio between the unvaccinated individuals who are positive but are asymptomatic and $N_{pos}$ . Then, the total number of the asymptomatic and unvaccinated carriers of the virus is $N_{pos} r$ . If $N_{tot}$ is the total number of unvaccinated individuals without symptoms in a certain geographical region, then the ratio between asymptomatic and unvaccinated carriers of the virus and asymptomatic and unvaccinated individuals (no matter their positivity to the virus) is $\frac{N_{pos}r}{N_{tot}}$ . We are interested in expressing $r$ in the case of the Omicron variant (let’s say $r_{o}$ ) as a function of the same parameter in the case of the Delta variant (let’s say $r_{\delta}$ ). The following table collects the symbols we have just introduced. In what follows we will add a pedix $o$ for the Omicron variant and a pedix $\delta$ when we refer to the Delta variant.

Between December 2 and December 17, 2021, a total of 330 asymptomatic subjects from South Africa were enrolled in a phase III clinical trial (Ubuntu trial, PACTR202105817814362) to assess the relative efficacy of the COVID-19 vaccine mRNA-1273 (MODERNA). This population is exclusively made up of individuals who had not been previously exposed to any COVID-19 vaccine. This population included persons living with HIV (PLWH) with a median age of 39 years (18-76). Among them, 230 individuals were tested for SARS-CoV-2 by RT-PCR and 31% (71 subjects) turned out to be positive. Among them, 56 samples were successfully subjected to further investigations: all had S gene dropout, suggestive of Omicron infection (Garret N et al 2021). It is worth mentioning that positivity to Omicron does not correlate with previous exposure to SARS-CoV-2, in this population: in other words, previous exposure to other SARS-CoV-2 variants does not protect against Omicron. Positivity to Omicron does not correlate with CD4+ cell count in PLWH either, so it seems that HIV does not interfere with $r$ , the value we are interested in. If we now consider that the average daily number of confirmed new cases in South Africa for the interval December 2-17 is 18745, we can write

$N_{pos-o}\,=\,18745K_{o}\,+\,r_{o}N_{pos-o}\,\Rightarrow\,N_{pos-o}\,=\,\frac{18745K_{o}}{1\,-\,r_{o}}$

where $K_{o}$ is a multiplicative constant that accounts for all the possible errors in the measure of the positive symptomatic cases and also for the fact that we are here considering only unvaccinated subjects, while data available for confirmed cases does not distinguish between vaccinated and unvaccinated. Moreover, the number refers to average daily new cases, while we are in fact considering the number of total positive cases, at that point in time; so $K_{o}$ is also a way to get the latter from the former (assuming linear proportionality). Therefore, we can write

$\frac{N_{pos}r_{o}}{N_{tot-o}}\,=\,0.31\,\Rightarrow\,\frac{r_{o}}{1\,-\,r_{o}}\frac{18745K_{o}}{N_{tot-o}}\,=\,0.31$

Another study carried out in South Africa between June and August 2021 during the Delta outbreak (Sisonke sub-study, NCT04838795) reported a percentage of asymptomatic carriers of 2.4% (39/1604) among unvaccinated subjects (Garret N et al 2021). The average number of daily reported cases, in this case, is 12181; hence we have

$\frac{r_{\delta}}{1\,-\,r_{\delta}}\frac{12181K_{\delta}}{N_{tot-\delta}}\,=\,0.024$

These last two equations give

$\frac{(1\,-\,r_{\delta})r_{o}}{(1\,-\,r_{o})r_{\delta}}\frac{18745}{12181}\frac{K_{o}N_{tot-\delta}}{K_{\delta}N_{tot-o}}\,=\,\frac{0.31}{0.024}\Rightarrow$

$\Rightarrow\frac{(1\,-\,r_{\delta})r_{o}}{(1\,-\,r_{o})r_{\delta}}1.54\frac{K_{o}N_{tot-\delta}}{K_{\delta}N_{tot-o}}\,=\,12.92\Rightarrow$

$\Rightarrow r_{o}\,=\,\frac{12.92}{1.54\frac{1\,-\,r_{\delta}}{r_{\delta}}\frac{K_{o}N_{tot-\delta}}{K_{\delta}N_{tot-o}}\,+\,12.92}$

We are here considering unvaccinated subjects, but while in the period December 2-17 (Omicron outbreak) the percentage of subjects who received at least one dose of vaccine was on average 30.65%, in the time frame June-August (Delta outbreak), the same percentage was only 10.65% (remember that in both cases we consider subjects from South Africa) (R). Then we must assume that $N_{tot-o}\,=\,0.69N_{tot}$ and $N_{tot-\delta}\,=\,0.89N_{tot}$ , therefore we have

$\frac{N_{tot-\delta}}{N_{tot-o}}\,=\,\frac{0.89N_{tot}}{0.69N_{tot}}\,=\,1.29$

This means that $r_{o}$ is given by

$r_{o}\,=\,\frac{12.92}{1.99\frac{1\,-\,r_{\delta}}{r_{\delta}}\frac{K_{o}}{K_{\delta}}\,+\,12.92}$

Note that we are here considering only unvaccinated subjects, while the number of reported cases includes both vaccinated and unvaccinated individuals. This means that $K_{o}$ can’t have the same value of $K_{\delta}$ . One possible choice is to assume

$\frac{K_{o}}{K_{\delta}}\,=\,\frac{k\cdot k_{o}}{k\cdot k_{\delta}}\,=\,\frac{k_{o}}{k_{\delta}}\,=\,\frac{100-30.65}{100-10.65}\,=\,0.78$

where we assumed that k is the error (underestimation) made in measuring positive cases, which is the same in both cases since it depends on the efficiency of the health care system, on the behavior of the population, and other factors that do not change; while $k_{\delta}$ and $k_{o}$ are directly proportional to the percentage of individuals who are unvaccinated and allow for the calculation of the number of positive individuals with no prior vaccination from the number of positive individuals (no matter the vaccination status). Another possible choice is to assume that the confirmed cases are mostly unvaccinated and in this case we would simply have

$\frac{K_{o}}{K_{\delta}}\,=\,1$

All that said, we can now express $r_{o}$ as a function of $r_{\delta}$ and we get the plot in Figure 1. The script in Octave used to plot the figure is the one that follows. The number of confirmed cases has been retrieved from this website, in particular from this CSV file: (download).

% file name = omicron
% date of creation = 2/01/2022
clear all
close all 
% the array of r_delta
steps = 100;
r_delta(1) = 0;                                                            
r_delta(steps) = 1;
inc = ( r_delta(100) - r_delta(1) )/(steps-1)
for i = 2:steps-1
  r_delta(i) = r_delta(i-1) + inc;
endfor  
% the array of r_omicron
for i=1:steps
  ratio_r = (1 - r_delta(i))/r_delta(i);
  ratio_k = 0.78;
  r_omicron (i) = 12.92/((1.99*ratio_r*ratio_k) + 12.92);
  ratio_k = 1.;
  r_omicron2 (i) = 12.92/((1.99*ratio_r*ratio_k) + 12.92);
endfor
% plotting
plot (r_delta, r_omicron, '-k', "linewidth", 1)
hold on
plot (r_delta, r_omicron2, '--k', "linewidth", 1)
ylabel('r_{o} = (asymptomatic AND positive)/positive (Omicron)','fontsize',15); 
xlabel('r_{\delta} = (asymptomatic AND positive)/positive (Delta)','fontsize',15);
axis equal;    
axis([0,r_delta(steps),0,r_omicron2(steps)])                             
grid on
grid minor
legend ('k_{0}/k_{\delta} = 0.78', 'k_{0}/k_{\delta}  = 1', 'location', "northwest", 'fontsize', 15);

Limitations

The main limitation of this method is the fact that the characteristics of the two populations considered (the one from the Ubuntu trial and the one from the Sisonke sub-study) were not taken into account, so we can’t say whether these two populations are comparable with respect to variables such as age, comorbidities, and previous SARS-CoV-2 infection. We don’t know how well these two populations represent the general population either.

Further comments and conclusions

From the present analysis, the Omicron variant shows a higher relative number of asymptomatic cases, when compared to the Delta variant, in unvaccinated subjects. This points towards lower pathogenicity for the new variant.

In Italy, as of 31 December 2021, the prevalence of Omicron was only 20% (R), so its effect on the number of deaths and hospital resource use has yet to be appreciated. At present, with Delta being the most prevalent variant in our country, the large majority of deaths and intensive care occupation is seen among unvaccinated subjects who are accepted in intensive care units 6.5 times more than vaccinated subjects (Figure 2) and die 5.2 times more frequently (Figure 3). If Omicron is really less dangerous than Delta for unvaccinated subjects, we will see a progressive convergence between the yellow curve and the blue one in both Figure 2 and Figure 3. The plots below will be updated as new data become available.

Figure 2. The number of subjects admitted to intensive care unit, in Italy, for 100,000 unvaccinated (yellow), vaccinated (blue), boosted (green) subjects. You can select a specific age range from the bar at the top of the plot.

The equations of this blog post were written using $\LaTeX$ (see this article)

3 thoughts on “An estimation of the proportion of asymptomatic Omicron infection”

Anonymous says:

January 11, 2022 at 6:21 pm

Interesting calculation, Paolo – we’ll take a closer look. Regards, Nigel Garrett

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1. Paolo Maccallini says:
  
  January 11, 2022 at 7:25 pm
  
  Hi, thank you for your comment! I am currently trying to refine this calculation.
  
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Paolo Maccallini says:

January 13, 2022 at 3:16 pm

Featured in this medical lecture by Mobeen Syed: https://www.youtube.com/watch?v=sokweHgXvr0&t=184s

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$N_{pos}$	The number of carriers of the virus who did not receive the vaccine.
$N_{tot}$	The number of asymptomatic individuals who did not receive the vaccine. Their positivity to the virus is not specified.
$r$	The ratio between the asymptomatic carriers of the virus who are unvaccinated and $N_{pos}$ .
$N_{pos}r$	The number of asymptomatic carriers of the virus who did not receive the vaccine.

An estimation of the proportion of asymptomatic Omicron infection

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