The traditional notion of herd immunity has never applied to COVID-19 and is mathematically impossible with the current vaccines available.

COVID-19 has all of the necessary ingredients to evade herd immunity and ensure long-term staying power. We are facing an unparalleled global challenge that could reduce the life expectancy of our entire species. This highly contagious, airborne virus is fueled by asymptomatic carriers and is around to stay.

A little over a year ago, the nationwide vaccine rollout resulted in Americans being fixated on the percentage vaccinated. This ever-changing percentage, often referred to as the COVID-19 herd immunity threshold has been heavily marketed as the official endgame of the pandemic.

The problem is that the percentage vaccinated in a specific region at a given time and the herd immunity threshold of that region are not the same things.

What is herd immunity?

If a population reaches herd immunity for a given infectious disease, then the pathways of virus spread are limited so that individuals without immunity are not likely to interact with infectious individuals and become infected.

For example, when we go grocery shopping we aren’t concerned with catching diseases such as poliomyelitis, smallpox, rubella, measles, or diphtheria. This is because we have reached herd immunity for these diseases.

All of these victories have required vaccination. Herd immunity has not been achieved for most major infectious diseases on this population scale through a natural infection process.

Considering the risk of long COVID, the frail immunity after infection and the frequent emergence of new variants which are masters of immune escape, letting the virus run unchecked in an effort to reach herd immunity is a losing strategy. This plan of action will result in a lasting tsunami of chronic illness that will leave millions disabled. 

Impenetrable walls in reaching herd immunity

Image by torstensimon from Pixabay 

The COVID-19 vaccines have proven to be effective against symptomatic infections and disease severity, however, they were never going to give us sterilizing immunity and they do not block transmission. Moreover, natural immunity fails to elicit a robust and broad immune response.

However, the main issue is that the herd immunity threshold is mostly a theoretical concept and has limited practical applications. In reality, it’s more of a gradient and we knew when we got there with the diseases that we achieved this population level of protection.

Letting the virus run unchecked in an effort to reach herd immunity is a losing strategy.

It’s not some time-invariant percentage that we should be chasing. It’s a moving target and values estimated from past outbreaks usually do not apply to current outbreaks. In the case of COVID-19, the notion of a herd immunity threshold takes on more of a transient meaning. Any uniform percentage or cut-off is just a mirage. An overall improvement in the numbers, e.g. cases, hospitalizations, and disease-induced deaths will tell us when we are temporarily in a low-risk period.

This mysterious percentage is calculated using a formula that depends on a widely misunderstood and misinterpreted dimensionless number called the basic reproduction number (R0), pronounced R-naught. This fundamental epidemiological metric provides an estimation of the expected number of secondary cases directly generated from a typical case in a fully susceptible population and is the key to understanding the concept of herd immunity. 

For example, in influenza seasonal strains, “the R0” is approximately 1.3, so if someone is infected with the flu then they will infect 1.3 others on average. The standard epidemiological interpretation is that if R0 is greater than one then the disease spreads and if it is less than one then the disease dies out. However, this metric has many shortcomings and the often overlooked mathematical fact is that R0<1 is only a necessary, but not sufficient condition for disease elimination, as a backward bifurcation may occur. This phenomenon is called subthreshold persistent, and it basically means that in some cases, even if R0 is below one, then the disease can still circulate. 


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R0 is widely used by scientists and public health officials to quantify the intensity of an outbreak and to estimate herd immunity. The problem is that it’s rarely observed in the field and is calculated using mathematical models. Once presented with an estimated R0, many scientists and public health officials claim that the interpretation is straightforward. However, due to many underlying mathematical subtleties, the R0 resulting from an overlapping majority of mathematical modeling efforts fails to measure the average number of secondary cases in a fully susceptible population. It is merely a theoretical threshold which best serves as a measure of disease strength and cannot be used in estimating herd immunity. 

All of the issues related to estimating and interpreting R0 are inherited in the herd immunity threshold. Due to the functional dependence on R0, the herd immunity threshold can vary based on several biological, socio-behavioral, and environmental factors. This often misinterpreted quantity which is derived utilizing a relatively deep mathematical theoretical framework is then used to estimate the herd immunity threshold. This common problem has two main consequences.

  • The estimated herd immunity threshold is incorrectly estimated due to a misinterpretation of the estimated R0. As a result, the percentage has no epidemiological meaning and is essentially useless.
  • The estimates are very susceptible to manipulation and misinformation. 

Since the very beginning of the pandemic, there has been a large amount of variation in the percentages brought to public attention.

Since it can be calculated from a wide array of different mathematical modeling approaches, every estimation needs to be interpreted in the context of the model.

The moment you confront a model with data, you lose generalization. This is why a uniform percentage applying to the entire county is improbable and not realistic. 

COVID-19 has all of the necessary ingredients to evade herd immunity and ensure long-term staying power. We are facing an unparalleled global challenge.

Mathematical models rely on simplifying assumptions, which are not accurate reflections of reality. When it comes to mathematical modeling, it’s just the name of the game.

Reality is inherently too complex to be fully captured by equations and when it comes to modeling a new virus, this fact is exaggerated. Many models assume great degrees of homogeneity when it comes to age, gender, and interaction. For example, when we go to work every day or to socialize with friends, the people we are around are not decided by putting a bunch of random marbles in a barrel, mixing them around, and then randomly choosing. Any oversimplified, toy calculations claiming to estimate the COVID-19 herd immunity threshold should be taken with a grain of salt. 

To complicate matters even more, in reality, the entire population is not susceptible, due to natural immunity, vaccines, and preventative measures. The average amount of secondary cases changes with time. This reduces the real-world application potential of R0 to an even greater degree. The actual amount of secondary infections at a given time is given by the effective R0, often denoted Re. 

Image by Alexandra_Koch from Pixabay 

The classical definition of herd immunity originates from mathematical models accounting for the impact of vaccination. Here is an example of the typical mathematical reasoning commonly used to estimate the critical vaccination coverage and herd immunity threshold.

If we want the disease to die out then we need to bring the effective reproduction number below one, that is Re<1. Re is usually obtained through a simplified formula involving the basic reproduction number R0, the proportion of vaccinated individuals and the vaccine efficacy. A little basic algebra shows that if we want Re<1, then we need the proportion of vaccinated individuals to be at least equal to the critical vaccination coverage. 

Now if the vaccine has 100% efficacy and vaccines are selected uniformly in the community then we arrive at the classical formula for the herd immunity threshold. However, no vaccine has 100% efficacy, so in practice, this variable will be less than one.  This is the most common method used to calculate this epidemiological threshold. The equations say that we need at least the proportion given by the critical vaccination coverage to reach this invisible safety net. For example, let’s assume that “the reproduction number” for the highly contagious Omicron variant is 9.5 and that vaccine efficacy is 95%, regardless of the vaccine. Then the herd immunity threshold is approximately 94%. Assuming the same vaccine efficacy and an estimated R0=5.1 of the Delta variant then the herd immunity threshold is approximately 85%. The pattern emerges that the higher the R0, the larger the herd immunity threshold. This basically means that the more contagious a virus is, the more difficult it is to reach herd immunity. Moreover, a higher vaccine efficacy results in a smaller herd immunity threshold, as expected. 

Additionally, for COVID-19 to be endemic this means that Re=1, or we are at critical vaccination coverage for a sufficient amount of time. This means that the average number of secondary cases per primary case is one. Mathematically speaking, if Re is stably at one for a sufficient period of time, then the disease is said to be in endemic equilibrium. Clearly, this is not what is occurring on the ground. COVID-19 is not even remotely close to an endemic status yet. What is being marketed as endemic is simply the level of disease spread we are willing to accept.

3 major problems with calculating COVID-19 herd immunity

  • Where did the estimated R0 originate from? Depending on the mathematical modeling approach, R0 is often not “the reproduction number” of the disease, but just a theoretical threshold that lacks epidemiological meaning.
  •  Vaccine efficacy fluctuates, especially with Omicron. It decreases with respect to time and increases with additional doses, so in reality, it’s a time-dependent function, not a number.  
  • The proportion vaccinated can still transmit the virus and get infected, so the transmission chains are not completely broken. There is growing evidence that vaccinated individuals have similar viral loads in the nasopharynx as unvaccinated, even in asymptomatic cases.   

The above issues render the estimations of COVID-19 herd immunity to be of little practical value. Although mathematically, thresholds such as R0 are of great importance. Real-world applications are limited and are mainly reduced to classifying diseases on a categorical level.

We need to let go of the fantasy of COVID-19 herd immunity, it’s never going to happen, at least not in the canonical sense.

To effectively combat this virus, we need mountains of more data and transparency so that data-driven decisions can be made. We cannot vaccinate our way out of this pandemic.

Vaccines alone have proved to be insufficient and this is precisely why we are in desperate need of the nationwide implementation of mitigation strategies focused on the dominant transmission routes and mass testing, i.e. airborne mitigation and detecting asymptomatic carriers.

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Dr. Jacob B. Aguilar

Dr. Aguilar currently holds the academic rank of Assistant Professor in the Department of Mathematics in the School of Computing, Artificial Intelligence, Robotics, and Data Science and is the Associate...

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