COVID-19 – Or the end of the beginning? (2)

This piece is the second of a two parts story by mathematician Philippe Lacoude. To read the first part, click on COVID-19 – The beginning of the end? (1)

As we saw previously, for SARS-CoV-2, the base reproduction number, or R₀, is around 3.0.

When all individuals are susceptible, this implies three new cases per patient, on average, and before any specific prophylactic measures are taken.

If nothing were done, we would therefore have to wait until two thirds of the general population are infected before the epidemic stops.

The effective reproduction number

But is this really the case?

No. And a comment must be made on the subject because some politicians are spellbound by this magic number.

R₀ is an extremely intuitive concept: if I am contagious for 15 days, how many people will I infect? Again, this depends on how many people I usually see in a fortnight, so R₀ can only be an average of a distribution of a random variable, as we saw earlier.

Well, just changing the number of contacts per person as well as the risk of each contact (masks, gloves, hand washing, lack of physical contact) will change the number of people infected – on average – by each patient.

In fact, as the epidemic spreads, varying proportions of the population are immune to the disease at any given time. To take this into account, the effective reproduction number, often denoted R_e and sometimes denoted R_t, is defined as the average number of new infections caused by a single individual infected at time t in the partially susceptible population.

A naive value of R_t can be estimated simply by multiplying R₀ by the fraction S(t)/N of the population N which is still susceptible at time t, S(t).

If R₀ is the distribution of a random variable, so is R_t. And if the R₀ has a strong variance, R_t will also have a strong variance. So, everything we said in the first part about the consequences of the high variance of R₀ for SARS-CoV-2 still applies during the course of the epidemic.

This is a simplification because, if there are multiple ways to calculate R₀, there are still even more ways to estimate R_t.

Not only does R_t depend on the susceptible population S(t) – those who are not immune – but also on the behavior of the latter. Thus, although it is by definition impossible to modify R₀ for a given population and culture even through vaccination or other behavioral changes, R_t is modified by these.

Each measure, individual or coercive, has an effect on R_t which is different from the others and different by geographical location. Physical distancing, medical protocols, school closures, isolation,… Researchers are starting to have enough data to be able to measure the effectiveness of each.

Because R_t changes profoundly when behaviors adapt to the reality of the disease, it is perfectly possible to have a high R₀ – and therefore a very contagious disease – without having an epidemic.

The right question

But this is a two-sided remark: if we manage to reduce the epidemic contagion to an R_t below 1.0 by making societal changes (voluntary or coercive), it emerges that this epidemic is “finished” only to the extent that these changes are there. If they disappear, R₀ predominates again!

The correct question is not whether the epidemic is “over”. This is a poorly worded question.

Rather, we would be interested to know if we have reached a point where the disease would no longer spread if some or all of the measures in place were abandoned.

Serology

To answer this question, it is “only” necessary to know if (1-1/R₀)% of the population has been infected and has recovered from the disease. Or, more precisely, if a large enough part of the population is not or no longer susceptible: after all, some people may be immune to SARS-CoV-2.

To do this, it is possible to measure the antibodies still present in the blood of patients who have recovered from the disease.

These antibodies are proteins synthesized by B lymphocytes which appear in several distinct forms, immunoglobulins A (IgA) found in the mucous membranes such as those of the respiratory tract, immunoglobulins M (IgM) and then immunoglobulins G (IgG) which correspond to a secondary but very specific response of the immune system.

It should be noted that these tests now exist in large numbers and that their quality has drastically improved. The first tests, which gave many false positives and false negatives, were gradually replaced by tests that are 99.8% reliable. The tests are now very well documented and understood.

Anyway for the purpose of epidemiological studies, as this Nature article explains, the tests do not need to be as precise as for medical purposes. If their error rate, even high, is known with precision, this is perfectly sufficient for statistical purposes.

Serological results

In Wuhan, a Nature study measured the levels of IgM and IgG in 17,368 patients in the period from March 9 to April 10, 2020. Seropositivity ranged from only 3.2% to 3.8% depending on the study cohorts! Unsurprisingly, seropositivity gradually declined in other cities as one moved away from Wuhan.

Wuhan patients who visited hospitals for dialysis as well as hospital staff had a seroprevalence of 3.3% (with a 95% CI of 2.5 to 4.3%) and 3.8% (with a CI of 2.6 to 5.4%), respectively.

These numbers are so low that we could suspect a test error: yet the researchers had validated their serological test in-house with serum samples taken from 447 patients and had shown an empirical specificity of 99.3% (444 out of 447) and 100% (447 out of 447) for IgG and IgM antibodies, respectively.

In the Western World, the numbers are similarly low, with some geographic exceptions.

To date, approximately 4 to 8% of populations in the United States and Europe would have been infected with SARS-CoV-2:

• According to a study in the Lancet, centered around May 3, 2020, including a very large sample of 61,075 participants, the seroprevalence was 5.0% (with a 95% CI ranging from 4.7 to 5.4%) for point-of-care tests and 4.6% (ranging from 4.3 to 5.0%) by immunoassay, with no difference according to sex but with a lower seroprevalence in children under 10 years old (less than 3.1% by the point-of-care test). There was a substantial geographic variability, with a higher prevalence, from 10.0 to 13.6%, around Madrid.
• In France, according to a study by the Institut Pasteur published in Science, as of May 11, only 2.9 million people had been infected, or 4.4% of the population. This is confirmed by another study which uses microsimulation – and, therefore, a radically different method from the Institut Pasteur and SIR/SEIR models – to conclude that at the same date, 10.9% of the French population had been affected. These values ​​are very insufficient to have herd immunity.
• In the state of Rio Grande do Sul in Brazil, the seroprevalence was 0.048% (with a 95% CI of 0.006 to 0.174%) between April 11 and 13. This percentage rises to 0.135% (with a 95% CI ranging from 0.049 to 0.293%) between April 25 and 27. Finally, it could have reached 0.222% (with a 95% CI ranging from 0.107 to 0.408%) on May 9, 10 and 11. Suffice to say that Brazil, though hard hit, is only at the very beginning of the crisis in some regions.
• In Sweden, which has deliberately decided to play the herd immunity card, public health Swedish authorities issued a preliminary estimate, based on antibodies tests of late April, that shows 7.3% of inhabitants of Stockholm had been infected with SARS-CoV-2, with an overall national rate of around 5.0%.

There are exceptions to these low numbers in heavily affected areas:

• In Santa Clara, researchers at Stanford University have calculated that the serology was 2.8% (with a 95% CI of 1.3 to 4.7%) after weighting the demographics according to the county’s population. These seroprevalence estimates implied that 54,000 people (with a 95% CI of 25,000 to 91,000) had been infected in Santa Clara County by early April, 50 times more than the official 1,000 cases of so. It should be noted that these data have been strongly criticized because of the sampling method of finding volunteers on Facebook, which tends to inflate cases.
• In Miami-Dade, Florida, a Government study suggested that 6.2% of local residents were infected on the date of April 24, 2020. This finding implies around 165,000 infections, more than 16 times the official tally of confirmed cases.
• In a (more or less?) representative sample of 863 adults tested at the beginning of April, researchers at the University of Southern California (USC), working in collaboration with the Public Health Service from the State of California, estimated that “about 4.1% of the adult population [with a 95% CI ranging from 2.8% to 5.6%] in Los Angeles County [had] antibodies against the virus” which was 28 to 55 times more than the tally of confirmed cases at the time of the study.
• Up to 9.9% of the inhabitants of Ile-de-France (with a 95% CI of 6.6% to 15.7%) would have been infected on the date of May 11, 2020. In hard hit eastern France, this figure drops to 9.1% (with a 95% CI of 6.0 to 14.6%) according to the Institut Pasteur study cited above.
• A preprint of an Italian study published on May 11, 2020 attempts to determine the percentage of people in the Milan metropolitan area who were already infected with the virus at the start of the epidemic. The researchers noted a gradual increase, eventually reaching around 7.1% (with a 95% CI ranging from 4.4 to 10.8%) of the population of Lombardy.
• Three thousand customers of supermarkets in New York state were tested randomly on April 20. Almost 13.9% of them were declared positive. In New York City, 21% of those tests came back positive. While this is much higher than the March 23 to April 1 study from the CDC, which found a 6.9% seroprevalence (with a 95 CI of 5.0 to 8.9%), it is almost exactly in line with the April 25 to May 6 sample, which had a seroprevalence of 23.2% for New York City.
• At Mass General, in Boston, a small study of 200 blood samples from Chelsea has shown that nearly 32% were positive around April 10, 2020. These samples were not selected at random but from people who thought they were or had been sick.

To summarize, here are the seroprevalence curves taken from the main model of the Institute for Health Metrics and Evaluation (IHME) of the University of Washington to which I added some of the (sporadic) measurements of the above studies:

We are still not testing!

All these prevalence figures indicate that we are only detecting a small part of infections: according to a study by the University of Göttingen, only 6% of infections have been detected worldwide. This average global number is very variable.

Notably, the least affected countries are those which have best detected patients, such as South Korea (with a detection rate of 49% as of March 17), Japan (25%) or Germany (16%). The dunces of COVID-19 in terms of mortality are also those who have not been able to organize testing like Belgium (2%) or France (2%).

If these figures seem abstract, let’s translate them: out of 50 French or Belgian patients, 49 were walking placidly in public, obviously without a mask…

The result is known: South Korea, Japan and Germany deplore 6 dead, 8 dead, and 109 dead per million inhabitants, respectively. The French and Belgians, 462 and 845 dead per million inhabitants despite confinement measures that bordered on the absurd.

I would be eager to ask the officials of these latter two countries how they explain death rates 81 and 148 times higher, respectively, than those of South Korea.

In the meantime, according to a recent report , in view of the available data, researchers at Harvard University are now convinced that we can find a semblance of normality, respect the foundations of “a free society”, “protect the human life, secure our institutions and prevent the destruction of our economy”, if we give ourselves the means to test massively.

Actual death rate

All these serological tests give a much more precise idea of ​​the number of patients and, consequently, of the lethality of the disease.

According to a Nature paper, the latter would be around 0.66% for China, 0.70% for France, and 1.00% for Spain and Brazil. Despite a considerable decline in this estimate, this would still mean that “COVID-19 is on average 50 to 100 times more deadly than the seasonal flu” as National Geographic reminds us in a recent article on the issue.

These figures should be taken with caution because everything still depends on the “true” serology rate and the real death toll.

The latter is often underestimated by 20 to 50% in many countries. In the United States, excess mortality from March 1 to May 30 was 27065 higher the 95235 reported deaths officially attributed to COVID-19 in the same period. Similarly, excess mortality in France is around 26%, suggesting medical authorities only record a fraction of all COVID-19 deaths.

If we refer to the above studies, the death rate would be 1.3% and 1.2% for, respectively, the city and the state of New York.

Typically, lethality estimates decline with time.

Worrisome implications?

All the large countries in Europe and North America would therefore have around 4 to 8% seroprevalence in their populations, except Germany at 1.0%. Japan’s would be at 0.1%.

If the R₀ was really equal to 3.0 then this result would suggest that we are only less than a tenth of the way through the pandemic (in the absence of a vaccine).

Are there any reasons to doubt it?

I don’t think we can reject the statistical estimates of R₀ or serology:

For R₀, there are far too many methods, data, publications. We are probably not heavily mistaken on a question which is based on universal mathematics and epidemiology fundamentals that work well for all other infectious diseases.

For serology, biological results are perfectly aligned with the predicted data by models like the IHME’s and are consistent with mortality data from one country to another and within the same country.

We must therefore look elsewhere for the reasons which could completely or partially invalidate the implication of an R₀ of 3.0 and a seroprevalence of less than 8%.

Pre-existing natural immunity?

First of all, the question of the group immunity threshold should be re-examined because it is reached when (1-1/R₀)% of the population is no longer susceptible, i.e. 67% if R₀ equals 3,0.

What if a large part of the population was not susceptible anyway?

This is an attractive idea for three reasons:

• First, it has been suggested that some people who have been vaccinated for other diseases are more resistant to SARS-CoV-2 and therefore develop fewer cases of COVID-19.
• Second, SARS-CoV-2 is not the first human coronavirus. In fact, when we say that we have “caught a cold”, we have developed either a rhinovirus (30 to 80% of cases), an adenovirus (5% of cases), or a human coronavirus, mainly 229E and OC43 (in 15% of cases). On average, an adult human has experienced these latter two several times in the life of his immune system. So we might have some natural immunity to SARS-CoV-2 because we have developed one for 229E and OC43.
• Finally, the immune system is used to unknown pathogens.

Therefore, the question is whether we can close the gap between the 67% that we would need (to achieve herd immunity) and the 8% (at best) that we see:

• The vaccination hypothesis is not improbable because certain vaccines, such as Bacillus Calmette-Guérin (BCG) against tuberculosis, confer broad protection against other infectious diseases (see here and there). However, this effect, if it exists, is probably not sufficient at all to reduce the susceptible population sufficiently: we are looking for a significant effect and the first studies do not find any effect at all (see here and there) or a non-significant effect (here). This does not mean that someone will not discover a link between a vaccine and less infection with SARS-CoV-2 but the facts, if they exist, are not yet proven.
• A recent Nature article shows that people with SARS 2003 have immune cells that, 17 years later, can recognize SARS-CoV-2 capsids. Although this does not apply to the 229E and OC43 viruses, it shows a mechanism by which a certain percentage of the population might already be immune. This also documents a possible duration of immunity once one has had SARS-CoV-2 and suggests that it could last a very long time.
• The idea of some natural immunity to SARS-CoV-2 is more promising because several studies have shown that blood plasma donors who had never been exposed to the virus had an immune response (see the numerous references here). There, the proportion is significant because it goes from 1 in 10 to 1 in 2. We have to be cautious because these results are in vitro! The effect on group immunity is uncertain and complex to determine but probably not zero (here and there).

If the latter two assumptions are true, serological studies, all of which rely on IgM and IgG, at this time, underestimate the true seroprevalence.

Serological undervaluation

The immunology of SARS-CoV-2 is extremely complex and still being researched (here and there) but scientists have found that some patients who exhibited mild symptoms of Covid-19 did not appear to have developed IgM and IgG antibodies. However, they showed “strong T cell specific immunity,” according to a report in Science Immunology.

“If, as appears the case, measuring T-cell immunity is a more enduring and reliable marker of adaptive immunity in COVID-19 than antibody [i.e. IgM and IgG], it will be valuable to achieve roll-out for health services of commercial T-cell testing kits,” study authors Rosemary Boyton and Daniel Altmann, professors of immunology at Imperial College London, told The Independent.

In other words, if we measure IgM and IgG antibodies but some patients have recovered from COVID-19 without producing them, our seroprevalence estimates are underestimated.

By how much?

About 20 to 50% depending on the studies (here, here and there).

The age-specific seroprevalence estimate in the CDC’s interactive serology dashboard for commercial laboratory surveys also suggests we are missing some cases as the age-pyramid does not make sense for some localities (like New York City for instance).

This makes it even more difficult to calculate group immunity. However, if we start from 4 to 8%, the possible adjustment is not enough to reach more than 8 to 16% of seroprevalence in the general population, at best.

Limits to error

For the epidemic to be at an end – if an R₀ of 3.0 is realistic – 59 to 63% of people would need to be naturally immune (given that 4 to 8% are already seropositive).

It’s just not compatible with the data we have on cruise ships, warships, prisons and nursing homes:

• In the latter, up to 100% of patients are affected by the disease because of the immuno-senescence.
• On the Diamond Princess, despite the draconian quarantine in the cabins, 712 people out of the 3,711 on board fell ill, or 19%. The actual cases are probably much higher because we now know that the serological tests show many more cases than the PCR tests performed then.
• On the Charles de Gaulle aircraft carrier, at least 1,046 of the 1,760 people on board, or 59%, fell ill despite health measures. Unlike retirement homes and the Diamond Princess, it is a young population, in good physical condition, which does not suffer from immuno-senescence.
• Similarly, COVID-19 is raging in US prisons, and the numbers show significant infection rates. For example, according to the official tally from the Federal Bureau of Prisons, in Seagoville, at least 1,072 of the 1,798 inmates contracted the virus, or 59%, of a young (but male) population.

Finally, medical personnel who are regularly confronted with human coronaviruses 229E and OC43 do not seem to be immune to SARS-CoV-2 if we consider the many fatal infections they have suffered: this is so true that Matt Ridley pointed out that during confinement, COVID-19 had essentially become a nosocomial disease.

A study showed that oncology staff at a large hospital in the east of England were heavily affected: although all participants were negative for nasopharyngeal swab PCR (at the time of the study), 21.4% had positive antibodies for SARS-CoV-2 using the Luminex test.

The law of the excluded middle

In summary, in the current state of research, the hypothesis of a pre-existing natural immunity does not allow to close the gap between the 67% of seroprevalence that we would need and the 8% (at best) that we observe.

Anyone who claims that the crisis is over without providing either a reassessment of R₀ (substantially downward), serology prevalence (substantially upward), or of a strong natural immunity from a large fraction of the population makes a serious error of judgment.

Life in society

Unless we can prove that more people are immune than we think (possible and moderately likely) or that our R₀ estimates are wrong (also possible but much less likely), we are still in the midst of a crisis.

Potentially, this would mean that the new coronavirus could continue to spread for at least 18 months to two more years as we wait for the pharmaceutical industry to ramp up production of hundreds of millions of doses of a still-tentative vaccine.

In that case, “this thing’s not going to stop until it infects 60 to 70 percent of people,” as Mike Osterholm, director of the Center for Infectious Disease Research and Policy (CIDRAP) at the University of Minnesota said.

This does not signify at all that the next two years will be as bad as the last four months because it seems that it is possible to limit the human and economic damage as proved by the Koreans, the Germans and the Japanese.

On the other hand, exceptional prophylactic measures will be with us for a while (such as masks, physical distancing, gloves, rigorous hand washing, etc.).

Large indoor gatherings, international travel, public transport will not immediately return to their 2019 activity levels. In fact, all indoor activities will be affected as it is increasingly evident that the SARS-CoV-2 is mainly transmitted indoors (here and there).

In the short term, we would return to semi-normal. Schools will reopen but not the sports halls, restaurants will serve a limited number of patrons, planes will be a third empty, etc.

This has an impact on economic growth and therefore on the financial markets, all sky-high as they are right now…

Even if it turns out that the epidemic is truly over, we are still heading towards extremely difficult economic times. When airlines, tourism, or even education (in some countries) will be forced to part with a large share of their workforce in the coming months, many people will have difficulty paying their rents and honor their loans.

This post is also available in: FR (FR)