The Equation That Ends Pandemics
Far from being separate entrées at the pandemic mitigation buffet for politicians to choose from according to their appetites, vaccination, masking, hand hygiene, & lockdowns are intimately connected and ordain our fight against Covid.
A legacy article originally published August 17, 2021 prior to the appearance of Omicron.
This article was part 3 in the five part series I wrote on the pandemic last summer and may be the most important one in the series. I encourage you to stick with it to the end, but have constructed it so that if you want you may stop after the second paragraph (not counting this one). Here we go…
There has been a lot of hand waving for the past year and a half with respect to the governing dynamics of a pandemic. This has been a disservice. Although it may require at least 4 semesters of college math to acquire the skills needed to approach these equations oneself, I believe most people without such a background can understand what they tell us. I want to try to remedy this explanatory gap. No prior knowledge of epidemiology is needed to understand this article, nor is having read my first two articles. A passing familiarity with algebra is helpful but I will keep any actual math to an absolute minimum. To motivate the discussion, I want to foreshadow where this journey is taking us. Part 5 of my series was titled “Opinion: Where Things Went Wrong”. In it, I built on the four prior articles including this one to explain the principle of epidemic containment, how containment has been used successfully many times in prior epidemics, and why it failed in the spring of 2020. In this article I will lay out the relationship between vaccination, masking, closures of gathering locations (lockdowns), etc. are all related through a core epidemiological quantity. Foreshadowing the end of our journey, the final paragraph of this post reads as follows:
With delta’s and Omicron’s massive R0's, the only way out for us is to simultaneously decrease S (through vaccination), increase ‘b’ (also through vaccination), decrease ‘a’ (through lockdown), and decrease ‘p’ (through masking, disinfecting, and improved hand hygiene). As shown in Big Covid Post #2, with masking and disinfecting the devil is in the details. These levers are all interrelated and the more people that get vaccinated and the more people that wear masks (properly fitted and handled), the less of a lockdown that is needed.
If this is all you are after, stop here. If you want the whole argument, keep reading and humor me a bit of algebra. I promise to keep it simple. And if you get a little lost in the middle, I will gather us all back together for the punchline. I am working on a revised and expanded version of the opinion piece I referenced above. I hope to publish it in January. If you like this piece and would like to read that one when it is available, follow me and sign up for email updates.
By now most of us have heard repeated talk of a number called R0. We know that it is pronounced ‘R naught’ and reflects the average number of new people that an infected person will subsequently infect over the course of their illness. We’ve heard that for seasonal flu, R0 is close to 1; for the 1918 flu, it was about 2; for measles, it may be as high as 18; for the original Covid strains and early variants it was 2-3; for delta, it is closer to 8 or 9 and Omicron is likely higher.
But doesn’t the concept of R0 as the average number of new infections that an existing infection will cause seem strange? How do we even measure such a thing? And doesn’t the constraint that it only applies at the very beginning of a pandemic when almost no one has had the new disease seem a bit…um…limiting? Well, R0 turns out to be an extremely important number for pandemic dynamics. It even provides insight into how things like masking, disinfecting, distancing, quarantining, and vaccinating affect a pandemic. It tells us whether cases will increase or decrease, and what sort of end-game we can play against any particular pandemic.
The epidemic spread equations are what are called compartmental models. They are similar to equations that come from chemistry which are based on something called the law of mass action. To see why these are called compartmental models, imagine a chemistry experiment where two liquid reagents are housed separate beakers (compartments). The reagents are then mixed together and allowed to react chemically. Maybe we dump the entire contents of the two beakers together all at once or maybe we slowly drip the contents of one into the other. In either case, the reagents encounter one another at some rate. As the chemical reaction progresses, the products of the reaction are collected in yet another compartment. You get the picture: start stuff off in separate compartments, let them interact in a mixing compartment at some “interaction rate”, and collect the products in a new compartment at some “collection rate”.
Let’s apply this idea to a pandemic. At the start of a pandemic consider everyone to be a member of one of three compartments: susceptible, infectious, or resolved. This is the S-I-R model. (Note: resolved is my own term. The original term was “recovered”. But since not everyone recovers [some die] the meaning of R was changed to “removed”. But to me that sounds weird. I like “resolved” because it reflects the fact that the process has arrived at its natural resolution, either death or recovery).
As shown in the figure, this can get much more robust as we add new compartments. We can also add spatial variables or network elements (for example, the model called MSEI(R/S)D with spatial diffusion). But for simplicity, let’s stick with S-I-R.
At the start of a new pandemic of a new infectious disease, almost everyone is in the S compartment. A few people are in the I compartment. R is empty.
At this point people are interacting as they usually do and encounter each other at some “interaction rate” called ‘a’. If you went shopping today and encountered 10 people then ‘a’ for you today was 10 people/day. If you went to church for an hour with 50 people, it was 50 people per hour. To get ‘a’ for a whole population, you “smooth out” all those individual rates.
At the start of the pandemic, most encounters take place between two S people. When two S people encounter each other, nothing happens. But when an S interacts with an I, there is a chance that the S will become infected. We say that S’s who encounter I’s also turn into I’s with some probablility ‘p’.
Then, people who are infected stay infectious for a while before spontaneously turning into R’s by either recovering or dying. The rate at which I’s spontaneously turn into R’s is the resolution rate ‘b’.
So S goes to I at a rate determined by ‘a’ and ‘p’. And I goes to R at a rate determined by ‘b’.
Let’s look at that resolution rate ‘b’ a bit closer. Like any other rate, it is specified as “number of events per unit time”. The reciprocal of any rate is an interval: the time that passes between events. If you’re driving your car, you specify your rate in miles per hour. But if you are a runner you specify your pace in minutes per mile: an inerval. So if the resolution rate ‘b’ is the number of illnesses that resolve in a day (resolutions/day) then the reciprocal of that, 1/b, is the number of days between resolutions. The number of days between resolutions is just the duration of infectiousness.
So S’s are converted to I’s at some rate given by the rate at which people normally encounter one another and the probability that any given encounter will result in transmission; and I’s are converted to R’s at some rate that is the reciprocal of the duration of infection.
The S-I-R equations are a system of very difficult equations called nonlinear differential equations that have to be solved simultaneously. There is no prescribed way to attack nonlinear differential equations. They do not have the kind of “solution” that you think about in a traditional notion of solving a math problem. Rather, mathematicians explore the behavior of these equations by asking specific questions about them. Over time, by asking many questions, the dynamics these equations describe comes to be understood.
When mathematicians set out to explore the behavior of pandemics, one quantity kept popping up in the answers to many questions. This quantity was a ratio containing the number of remaining susceptible people (S) and of all the rate constants, (a, p, and b).
The ratio that kept popping up was:
(S * a * p) / b
Since these numbers always appeared together in the same way, mathematicians gave their aggregate a single name, Rt: the “real-time effective reproduction number”. At the start of the pandemic, when S is a maximum and includes nearly everyone on earth, Rt is called R0, the “basic reproduction number”. After that, as S decreases, Rt also naturally decreases.
Rt and R0 appear when you ask the question, “Will there be an epidemic of this disease at all?” They appear when you ask, “How fast will this epidemic spread early on when it first starts?” They appear when you ask, “What will be the peak number of simultaneously-infected people?” They appear when you ask, “When all this is over, however long that may take, how many people will have been infected?” And most importantly for us now, Rt appears when you ask, “What is the end condition for this pandemic?”
The answer to this last question is that the pandemic will come to an end when:
(S * a * p) / b is sustained less than 1 for a sufficiently long time.
Rearranging this gives:
S < b / (a * p)
In other words, the pandemic ends when the number of remaining susceptible people is less than b / (a * p) and stays that way.
This means that to end this pandemic, we need to either decrease S or increase the aggregate quantity b / (a * p).
Or both.
Decreasing S means decreasing the number of remaining susceptible people. This is herd immunity. It is accomplished either by lots of people getting infected (herd immunity by natural infection); or by lots of people getting vaccinated (herd immunity by vaccination). The fraction of the population who need to be vaccinated to achieve herd immunity by vaccination is given by:
(1–1/R0) / e
where e is the vaccine effectiveness.
For the original Covid strain with an R0 of 3 and a vaccine effectiveness of 95%, the fraction needed to vaccinate was:
fraction to vaccinate = (1–1/3) / 0.95 = 70%
This is where the initial target of vaccinating 70% of the population came from. When you plug in the confidence intervals to reflect uncertainties in the measurements of e and R0, you get the range of 70–85% that Dr. Fauci often quoted one year ago.
However, those strains of Covid are gone now, replaced by delta and omicron.
For delta, R0 is 8–9 and e is 0.9 (90%). (Omicron is still being evaluated).
Plugging in:
fraction to vaccinate = (1–1/9) / 0.9 = 98.8%
The appearance of delta (and now Omicron) means we would have to vaccinate 99% of the population to achieve herd immunity by vaccination! The situation is just as impossible for herd immunity by natural infection. Everyone would have get sick.
So we can’t end the pandemic by reducing S alone. We need to simultaneously increase the aggregate quantity b/(a*p). This can be accomplished by increasing the numerator ‘b’ (the rate of resolution); or decreasing either quantity in the denominator: ‘a’ (the rate at which people encounter one another) or ‘p’ (the probability that an encounter results in transmission).
Lets look at these one at a time:
Increasing ‘b’, the rate of resolution.
The analysis above for vaccination assumed that vaccine recipients moved directly from S to R, bypassing I. While this is true for many vaccines it is not true for all of them, including the Covid vaccines. To be sure, many people who are vaccinated for Covid will go directly from S to R. But some will remain susceptible to a future (mild) breakthrough infection. These vaccinated individuals who suffer a breakthrough infection have a milder and briefer course of illness. How does this fact impact the end-condition.
Remember that 1/b is the duration of infectiousness. Increasing ‘b’ means decreasing 1/b, the duration of infectiousness. This accomplishes the objective. So even when vaccination does not move a person directly from S to R, by shortening the duration of infectiousness from around 17 days to about 5 days it still favorably impact the pandemic end-condition. Some of the available antiviral medications can do this too.
So vaccination reduces S by moving some people people directly to R, but it also increases ‘b’, the rate of resolution of infectiousness in those people who are not not moved directly to R. Either way, vaccination favorably impacts the pandemic end-condition inequality.
Decreasing ‘a’, the rate at which people encounter one another.
This is what physical distancing, closures of gathering locations, and quarantine are all about. Not all gathering locations are created equal, however, and schools are the NUMBER ONE contributor to epidemic spread and the most important one to lock down. Unfortunately, they also have a disproportionately large negative externality.
Decreasing ‘p’, the probability that an encounter results in transmission.
This is where masks, disinfecting, and 6 feet of separation come in. None of these interventions is perfect (though a properly fitted and handled N95 comes pretty close). Fortunately, they don’t have to be. Reducing the probability of transmission by just 50% is just as effective at ending a pandemic as reducing the contact rate between people by 50%.
So what does this mean for us now and how this pandemic ends? Covid will keep going until the condition set forth in the inequality is met. Delta — and now Omicron — is so contagious (the second or third most contagious of any disease ever) that there is no way out by vaccination alone. There is also no way out by distancing, masking, and disinfecting alone short of us all becoming self-sufficient survivalists self-confined to our homes so that no two people ever come into contact with one another again. Even here, because of animal reservoirs and imperfections in the survivalist protocol, it could take years for this to work and when it was all over, we would have to rebuild civilization entirely.
I digress. The point is that an ‘a’ of 0 is not realistic. But the inequality still must be satisfied in order for the pandemic to end. Due to delta’s (and omicron’s) R0, neither herd immunity by natural infection or vaccination alone can achieve that.
With delta’s and Omicron’s massive R0’s, the only way out for us is to simultaneously decrease S (through vaccination), increase ‘b’ (also through vaccination), decrease ‘a’ (through lockdown), and decrease ‘p’ (through masking, disinfecting, and improved hand hygiene). As shown in my second Covid article, when it comes to masking and disinfecting the devil is in the details. These levers are all interrelated and the more people that get vaccinated and the more people that wear masks (properly fitted and handled), the less of a lockdown that is needed.
Response to one criticism:
This article has received the criticism that the pandemic will never end, that Covid will become endemic. It is perhaps a subtle point of epidemiology (but an important one) that endemicity is one mode by which pandemics end. The pandemic phase will be over and the endemic phase will be begin when the “less than” sign in the equation is replaced by an “almost equals sign”. In other words, epidemiologists consider endemicity to be the end of the pandemic.
S < b / (a * p)
becomes
S = b / (a * p)
COMING SOON: Where Things Went Wrong in the Spring of 2020 and How We Avoid Making the Same Mistake In 2022. Follow me for instant notification when it drops.
See Covid: Endgame for more on pandemic endgames.