# Could the absence of Covid-19 reinfections be just chance?

As of today there is no confirmed case in which a recovered Covid-19 patient suffered a new infection. Nevertheless, reinfections involving other coronaviruses are quite frequent, so there is no reason to believe immunity to Sars-CoV-2 is life-lasting. Therefore, figuring out how long immunity lasts is of great interest.

Perhaps even more important is knowing what effect reinfection would have: whether the reinfectee could transmit the disease, whether he would suffer serious symptoms, etc. But there’s nothing I can contribute on that front, so in this article I’ll focus on immunity’s length.

The first confirmed Covid-19 cases were detected in late 2019 in China. ¿Could we therefore say with certainty that immunity lasts at least 7 months? The answer is no, for statistical reasons. In the article I show how based on data from Sweden, a country that seen continued infections for over four months, we can barely state that immunity lasts 90 days. perhaps in the end human beings will be immune to Sars-CoV-2 for decades, but that’s something we cannot know for now.

**Covid-19 definitions: case and active case**

When we hear about a Covid-19 case, or learn from the news that someone “tested positive”, the source is normally referring to the PCR test. The point of this test is to find out whether the infectee has an active infection, which is to say that he can spread the disease to others.

Of course a person’s infection happens before he takes the test, normally by a few days. But, since we can’t be sure of the date someone was infected, in practice we tend to use the quantity of positive PCR results on a given day as the number of “cases” or “infections” for that day.

In general, a PCR-positive person who is then quarantined has to test negative before being released. And even if someone tests negative in the first PCR, many governments impose a quarantine regime all the same for people arriving from abroad and similar cases; normally this quarantine lasts 14 days. the reasoning behind this policy is that, after 14 days, authorities can be almost certain that the infectee is no longer contagious (he’s ceased to be an active case).

One must point out that the PCR test can give a positive result simply because it picks up remains of viral RNA, even though the individual in question is no longer infectious. A study by Singapore’s National Center for Infectious Disease concluded that, 11 days after detecting an infection through PCR testing, it was no longer possible to grow viral samples taken from the patients. In other words: the infectees could no longer infect others.

You may have read that PCR tests almost never produce false positives, but it would be more accurate to distinguish two situations: people who were never infected and those who have recovered from infection. It’s true that in the first scenario it’s very rare to see a false positive from PCR, but it happens more commonly with former infectees, as discussed in the paragraph above. That’s exactly what happened to South Korean patients who tested positive after a negative PCR: these turned out to be not reinfections, but false positives from detecting viral RNA.

**Chances of reinfection**

Suppose a person became infected on January 1st. By the 14th of that month he had overcome the disease and developed immunity. Nevertheless, said immunity lasted only three months: by April 14th that person was susceptible reinfection. Immunity is not a black-or-white issue, but in this analysis for simplicity I will assume just that: either you are immune, or you are susceptible. Within such a framework, what are chances that an ex-infectee catches the disease again?

In the article I assume that the ex-infectee has the same probability of getting infected as the rest of the population. In practice, what I use is not the probability of infection (which is unknown) but the quantity of positive PCR tests as a percentage of population. Following the above example, let’s assume that in the month April the country involved is seeing one positive PCR tests every day per 10,000 people. That would mean each day the probability of reinfection for the ex-infectee is 1 divided by 10,000, that is to say 0.01%. On this basis, one can calculate how the chances of seeing reinfections grow over time.

Here the reader will see why the absence of reinfections so far is not surprising: the countries that had the most infections in January and February, mainly China and Korea, now have very few cases (at least officially). Hence, even if all the infectees from the beginning of the year had lost immunity already, their probability of getting reinfected is minimal because the average Chinese or Korean has few chances of encountering the virus today.

The fact that we have not seen reinfections after six months of contagion around the world does not say much about the length of immunity, because over that period the areas with infections have been changing. One cannot know how many ex-infectees who have travelled from China or Korea to Israel and Mexico, but it’s safe to say the number must be low indeed. In order to check how long immunity lasts it’s necessary to analyze a country that has seen infections sustained over several months. There are few candidates, and at least one has problems of data credibility (Iran). Among the rest, the one I know best is Sweden.

Before crunching the numbers, a clarification. Obviously the chances of infection are different for each person, and ex-infectees may not be average members of society. Rather, if they got infected in the first place they probably still have a greater propensity for infection (for instance, they may be more social). But it’s hard to quantify this intuition, so I simply assume a uniform probability of infection.

**Sweden’s figures: so far we can say that immunity lasts 80–90 days**

Sweden has had PCR-confirmed Covid-19 cases every say since February 26th. It’s a large quantity: as of July 3rd the number of infections was equivalent to 0.7% of the country’s population.

I assume cases remain active for fourteen days (the one in which they are detected through PCR testing and thirteen more). This seems a longer period than in reality, but it means that the estimate of the length of immunity is conservative.

By way of example, a person whose infection was detected on February 26th would have remained an active case until March 10th. If this were followed by 90 days of immunity, the ex-infectee would have become vulnerable to reinfection on June 9th. To get a sense of how unlikely it would be to see a reinfection, that day there were 964 cases confirmed by means of PCR, among 10,340,000 inhabitants of Sweden. Put other way: the ex-infectee had only a 0.009% chance of getting reinfected.

To illustrate the calculation a bit more, let’s see the following day. There was another confirmed case on February 27th, and for this reason by June 10th there were not one but two people susceptible to reinfection in Sweden. That day there were 1,486 cases, thus expected reinfections are 2 * 1,486 / 10,340,000 = 0.029. Adding up June 9th and 10th, expected reinfections were under 0.04.

These chances rise in later days, as the pool of individuals susceptible to reinfection grows. But as you may imagine it’s better to do the math with the help of a computer. The result:

· If one assumes immunity lasts 90 days, we ought to have seen 1.21 reinfections by now.

· On the other hand, assuming 85 days of immunity expected reinfections are 2.27.

**Then we should have seen one or two reinfections already, shouldn’t we?**

The question is, given that expect reinfections with a given frequency, what are the chances of seeing a specific quantity (in this case zero). To answer you have to use the Poisson distribution. More generally, this distribution tells us whether the difference between expected and observed frequencies is significant or not. Imagine an intersection at which, on average, there are three traffic accidents a year. How unusual would it be if a year there was only one accident? Or seven accidents? Or ten?

This distribution has only one parameter: lambda, the expected frequency. In the case of 1.21 expected reinfections, yes, the most likely outcome we should see is one reinfection (36% probability). But there’s nearly a 30% chance of not seeing any reinfection.

Going back to the 85-days scenario, the expected number of reinfections was 2.27. Even in that case there is a 10% probability of seeing zero reinfections.

Only by going down to 80 days can we be more or less certain. In that situation, the expected quantity of reinfections is just over 4, and the chance of seeing zero reinfections is below 2%.

In short: the data from Sweden only allow one to say with certainty that immunity lasts 80–90 days.

**Data**

The Swedish public health agency has an official website with coronavirus data.

In the article I used data as of Friday, July 3rd. The Excel file is stored in this Google Drive folder.

The R code used in the article is in the same folder.