COVID-19. What will the next few months be like? Will we have to get vaccinated again?
Although the good news of the last few weeks makes us optimistic, we still do not know if the third dose of vaccine will have to extend the third dose of vaccine beyond the most vulnerable groups, nor when we will be able to dispense with the masks. Four factors will determine it: the appearance of new strains, their contagiousness, the effectiveness of the vaccines, and the persistence of the immunity provided by them. We can study all this with the help of a simple equation.
I started studying the epidemiological models by July 2021, with the idea of understanding why the authorities had fixed the target of vaccinating 70% of the population and whether this target was sufficient. The delta variant of COVID-19 was booming, and the urgency to vaccinate as many people as possible was clear. News from the United States about the incidence of COVID-19 in children and young people, and the seriousness of some cases, made it vital to reach these age groups with vaccines.
It was not difficult for me to find the models that had led to setting that 70% target. I discovered that this target was insufficient, given the contagiousness of the coronavirus delta variant. These same models explain what can happen from now on, so let us analyze them.
I have enjoyed writing the following section for those who are passionate about mathematics. The less enthusiastic can ignore the equations but pay special attention to what R₀ is. It is an essential concept for understanding the impact of COVID-19 and its evolution.
Compartmental models: the SIS model
The most classical epidemiological models are called compartmental models. They assume that the population can be distributed in different compartments, each corresponding to a defined state. As the epidemic progresses, each person can suffer a change in her/his state, passing to a different compartment. The simplest possible model divides the population into two sets of people: those susceptible to disease (S) and those infected (I). In a given period, a proportion μ of the susceptible people contract the disease by coming into contact with infected people, and a fraction α of the infected recover.
Given a period, e.g., one week, the increase in the number of infected persons will be equal to the new cases minus the number of sick persons who have recovered in that week.
This idea is represented schematically in the figure below.
What we see is a simple differential equation. Differential calculus is the essential tool to study any time-varying phenomenon (or with respect to another variable), from a disease to the diffusion of a substance in a medium, or the evolution of two related populations. Formulating the equations that describe a phenomenon is easy. Finding a solution… is not so easy, although in this case, it is achievable. For the SIS model, shown here, the condition for the number of infected to decrease, and tend to zero, is something we are familiar with: that R₀, the basic reproductive number, is less than 1.
R₀ is defined as a function of the parameters we have seen ( μ and α) and of the total population, N.
If you find the definition of R₀ confusing, don’t worry. In short, it is the number of new cases that appear for each infected person when no one takes particular precautions. It should not be confused with Rₜ, which is the effective reproductive number, which we have been following during this time. Rₜ is the number of new cases that appear for each infected person, when we have already taken precautions to avoid infection.
The concept of R₀ allows us to assess the contagiousness of each infectious agent. However, its value is not something we can see by studying the causative microorganism (virus/bacteria/protozoa, etc.) with the aid of a microscope or by sequencing its DNA. It is an observed value, and therefore, it can oscillate within certain margins according to the methodology of data collection and evaluation and the peculiarities of each human community.
As a sample, I have compiled the R₀ values of different pathogens, well known.
In general, R₀ is always greater than 1. No infectious disease listed here will stop its spread unless we do something purposely designed to limit contagion.
Being so simple, the SIS model does not consider the possibility that recovered people will become immunized against a second infection. Neither includes the option of vaccinating people to prevent the disease.
If we introduce the scenario of vaccinating the population and recalculate the equations, we can assess how many people we must vaccinate to stop the epidemic. Defining r as the fraction of people that must be immunized to stop the pandemic, the condition is:
If we take into account the effectiveness of vaccines, we have to complicate the expression a little.
Let’s see the curves we can obtain with this equation and what they mean.
Studying “the other curve”. The effect of contagiousness
It is not easy to imagine what the formula with which we closed the previous section means. But if we paint it, everything will be much simpler. The percentage of the population we need to vaccinate depends on the contagiousness of the disease. Below we can see how.
The curve I have represented corresponds to 95% vaccine efficacy, and there you can see our magic number of 70%. For R₀ equal to 3 we need to vaccinate 70% of the population to stop an epidemic. The original strain of the SARS-CoV-2 virus, which caused this pandemic, had a basic reproductive number of between 2.5 and 2.7. Hence the objective of vaccinating 70% of the population.
In the same graph, I have highlighted other R₀ values, and not by chance. Since the virus appeared, mutations have given place to a plethora of variants. Two of them have been particularly successful: the alpha variant, which emerged in late 2020, and the delta variant, from spring 2021. In other words, the UK variant and the Indian variant, although the WHO prefers a more aseptic nomenclature. As I commented in my previous article (why is a more contagious strain so bad news? ) the emergence of a much more contagious variant is a big problem. First, because it quickly displaces the previous variants and becomes dominant. Second, because it means that many more people become ill, which significantly increases the number of severe and fatal cases. And third, because it makes it much more complicated to achieve herd immunity. The graph above shows precisely this.
The graph of the Coronavirus variants makes it easy to understand why I highlighted the values corresponding to R₀ = 5 and R₀ = 8, in addition to R₀ = 3. This is the basic reproductive number of the three coronavirus variants that have successively dominated in Spain and in most of the world. As the contagiousness of the virus increases, we need to immunize a much higher percentage of people to achieve herd immunity. For this reason, the delta variant has radically changed the rules of the game, and now the vaccination target has been raised to 90%.
As the contagiousness of the virus increases, we need to immunize a much higher percentage of people to achieve herd immunity
How the effectiveness of vaccines affects it all
The uncertainty about how the pandemic may evolve is not only related to the virus. We have learned that SARS-CoV-2 mutates frequently. So, new variants of the virus may emerge that pose a more serious difficulty, if only because of their greater contagiousness.
But we cannot forget that at the other end of this “arms race” between the virus and us, there is another element whose response cannot be predicted: our immune system. If the vaccines lose effectiveness, the curve previously shown changes, and we need to vaccinate many more people to achieve herd immunity.
The above graph shows how important it is to know the degree of efficacy of each vaccine to gauge how ambitious the vaccination objectives should be. For R₀ = 3 (the infectivity of the initial Coronavirus strain), if the efficacy goes from 95% to 80%, the vaccination target will change from 70% to 83%.
Perhaps even more worrying is that, with an R₀ = 5 (that of the alpha variant), we would have to reach 100% of the population to control the epidemic if the vaccines were only 80% effective.
According to recently published studies, it seems that vaccines have lost efficacy in the last months. Why has this happened, and how is vaccine efficacy going to evolve?
When we administer a vaccine to a person we provide his or her immune system with “training”. Thanks to it, the immune system will be ready to neutralize the attack of a given virus or bacterium from the minute zero of a new infection. Under normal conditions, the immune system will need to develop its defenses once the virus has already gained access to our body and is attacking our cells. And because it takes time to learn how to block the virus, the virus may be able to crush our internal organs before the immune system can stop it. That is why, with severe or highly contagious diseases, it is essential that the immune system has been trained and can respond immediately. This response consists of making specific antibodies capable of blocking the virus.
The efficacy of a vaccine, shortly after being inoculated, depends on the ability of our immune system to take advantage of that learning. In the coronavirus vaccines, this efficacy was in the order of 95%, meaning that 95% of people were able to generate more than enough antibodies against SARS-CoV-2 to stop the disease.
As time passes from the moment we are vaccinated, our immune system can lose “memory” and its effectiveness in making antibodies against the virus decreases.
Today, months after the first vaccines have been given, this loss of memory of the immune system is beginning to be observed, which translates into a loss of effectiveness of the vaccines.
Therefore, as far as vaccine efficacy is concerned, its diminution over time may be due to two causes:
1) That a new strain of the virus appears, against which the antibodies generated by the immune system thanks to the vaccine are not effective.
2) That the immune system “loses memory” and is no longer able to generate antibodies in sufficient quantity.
If in the following weeks we detect a decrease in vaccine efficacy, the appropriate response will depend on the cause of this decrease. I have summarized it in a table.
At the moment it seems that the situation is the first one. Vaccines are losing effectiveness due to “memory loss” of the immune system, so a third dose will be given to the groups most vulnerable to the virus.
At the same time, the appearance of new variants of the virus is being closely monitored throughout the world, to quickly detect whether the new variants are more contagious than the previous ones. As we have clearly seen, a more contagious strain soon becomes dominant. It is, therefore, necessary to be clear about how to deal with it. In any case, all strains are also studied to check the efficacy of existing vaccines, in case the second problem arises.
As to what will happen in the medium term, there is total uncertainty right now. We do not know whether SARS-CoV-2 will become extinct after a few more outbreaks, as happened with the 1918 influenza virus. Or whether it will become an endemic disease against which we will have to be vaccinated every year with a vaccine adapted to the latest variants. As is the case, in fact, with influenza.
A few thoughts
I wrote this article to explain a simple epidemiological model with two objectives. First, to understand vaccination campaigns, and second, to analyze what we may find in the following months. Analytical models are great tools for understanding a phenomenon, detecting what factors influence it, and to what extent.
Our situation will depend on the appearance of new strains of the virus, their contagiousness, and their sensitivity to existing vaccines
In this way, we have been able to see to what extent our situation will depend on the appearance of new strains of the virus, their contagiousness, and their sensitivity to existing vaccines. We can also see how important it is to continue to spread vaccines in the population, and how we depend on the ability of our immune system to “keep what we have learned.” That is, we are in the hands of a multitude of factors that will determine how the next few months unfold.
However, analytical models suffer from a problem: they assume uniformity in the studied environment, and this is an unrealistic assumption. As long as there are “pockets” of non-vaccinated people, we will have local outbreaks, despite the high percentage of vaccinated people. These outbreaks will affect the unvaccinated, those whose immune system hasn’t been able to develop antibodies, and the most vulnerable people, whose capacity to respond to the virus is minimal.
Since we are dealing with a highly non-linear system, with an exponentially spreading infection (why transmission explosively grows in certain areas?), the effect of pockets of unvaccinated people is much more severe than any intuition can tell us. Explaining this would require even more pages than I have written so far. It will require an article in its own.
The analytical model such as the one I have used has its limitations. But the “elephant in the room” of this article is the data corresponding to an R₀ = 8, which is the contagiousness of the delta variant. With an efficacy of 80% or less, as we are measuring in some vaccines, more than 100% of the population would have to be immunized to stop the pandemic. Since it is impossible to vaccinate more than 100% of the people, how can we control the epidemic?
The answer is easy: the key to limiting the spread of COVID19 is not in the R₀, but in the Rt, the effective contagiousness. The number of new cases per COVID-19 case remains well below 1, thanks to two facts:
1º) Most of the population is vaccinated, which considerably reduces the number of new infections and the transmission of the disease.
2º) We maintain prudent measures to avoid contagion (use of masks indoors, “seclusion” at home in case of suspicion of having the virus, limitation of contacts, screening of contacts when a new infection case is detected,…).
Both of these are very important to overcome such a contagious variant as delta. Let’s get ready for a winter of Vaccines + Masks. And let’s not lose sight of the appearance of new strains and their characteristics.
