We are living in an epidemic of armchair epidemiology, and far be it for me to contribute by giving my own feverish take as an expert of an unrelated field. Therefore, I solemnly swear that I will make no predictions about the present pandemic. I am not paid enough to make such predictions–and if you did pay me I would consider it my professional duty to find you a better expert.

What I can do for free, is read up on basic epidemiology, and digest the maths for you, dear reader. My sources: Wikipedia’s article on mathematical modeling and compartmental models, and some lecture notes I found. My expertise: during my PhD in physics, I frequently worked models like the one I’m about to discuss, only with electrons instead of people.

The SIR Model

The very first epidemiological that one learns about, is the so-called SIR model. This model divides the population into three groups (“compartments”): susceptible (S), infected (I), and recovered (R). Susceptible people are those who could be infected; infected people are those who are currently infectious; recovered people are those who are no longer infectious, and are immune to infection. “Recovered” can be a bit euphemistic, since one method of “recovery” is dying. Another method of “recovery” is by developing symptoms strong enough that the victim knows to quarantine themself (becoming less infectious).

The SIR model is intended to work for “outbreaks”, such as what we have right now with COVID-19.  There are other kinds of situations, like what we have with the flu.  The flu just kind of sticks around in perpetuity, because people never gain permanent immunity.  The SIR model is built on the assumption that once people recover, they’re not susceptible anymore.  I don’t think anyone knows if this is true for COVID-19.

The SIR model does not model the behavior of individuals. It simply tracks how many people are in each of the three compartments. The SIR model is part of a larger class of “compartmental models”, each of which defines a set of compartments, and tracks the number of people in each compartment via a series of equations. For instance, one compartmental model includes a compartment for asymptomatic carriers, another compartmental model includes the birth of new people.

Compartmental models strike me as quite limited, and fail to capture the effects of, for instance, a class of workers who are at greater risk of exposure to illness. I’m sure epidemiologists must have much more complicated models in their toolset, but we’re just doing baby’s first epidemiological model.

The SIR Equations

The SIR model comes with a set of equations that predict how the number of people in each of the three compartments will change over time. These are written as differential equations:

where

• t is time.
• S is the fraction of the population who are susceptible.
• I is the fraction who are infected.
• R is the fraction who are recovered.
• β is a constant telling us how infectious the disease is.
• γ is a constant telling us how quickly people recover from the disease.

(Note: S, I, and R usually defined as absolute numbers of people, rather than fractions, but my way works too.)

The equations define two processes.  The first process is infection.  The rate of infection is proportional to the rate that infected people come in contact with susceptible people.  It’s assumed that each person is equally likely to come in contact with each other person.

The second process is recovery.  The model assumes each infected person has a constant probability of recovering on any given day.  To my eyes this assumption is not very realistic, but I suspect including a more realistic recovery curve greatly complicates the math while still giving approximately the same results.

Solving the equations

There’s a trick to solving differential equations like these: don’t bother, just use a computer.  I built a basic simulation in Excel, to show how S, I and R develop over time.

I’d like to note several stages, each of which I will characterize mathematically.

1. The exponential stage, where both I and R increase exponentially.
2. The turning point where I starts to decrease.
3. The end state, where almost all people have been infected and recovered.

Exponential growth

During the exponential stage, I and R both increase exponentially.  We expect a doubling rate of ln(2)/(β-γ).  Another neat fact is that R = I * γ / (β – γ).  Or at least, that’s what we expect initially.  As time goes on, the doubling rate will slow down, eventually stopping when we reach the turning point.

You might have noticed that the doubling rate becomes negative if β>γ.  If β>γ, then the number of infected people decreases at all.  So, you don’t have an epidemic at all.  People recover too quickly to infect enough other people.

You may have read in the newspapers that one really important number associated with the epidemic is R₀.  R₀ is the average number of people infected by each infected person before they recover (at least, initially).  The more precise definition is that R₀ = β / γ.  When R₀ < 1 the disease dies out; when R₀ > 1 we have an outbreak on our hands.

I’d also like to clarify the relation between these numbers, and the “cumulative cases” that are usually reported in the news.  The cumulative cases is basically I+R.  The problem with looking at cumulative cases is that as social behavior changes, the value of R₀ changes.  Cumulative cases depends on both the current value of R₀, and the historical value of R₀.  I think a better measure of current growth rates would look at only I–approximately the number of recent new cases.  Although maybe that data is too noisy, I don’t know.

The turning point

What causes the turning point in infections?  It’s herd immunity.  If 50% of the population is in the “recovering” group, then 50% are immune to infection, so it’s effectively like β is 50% smaller than it was at the beginning.

More precisely, the rate of exponential growth is β*S-γ.  The turning point occurs when β*S-γ=0.  For example, suppose R₀ = 2, then the turning point is when S = 50%.  In other words, the turning point occurs when the cumulative cases have overtaken 50% of the susceptible population!  If R₀ = 3, then the turning point occurs when the cumulative cases have overtaken 67% of the susceptible population!  I do not know the R₀ of COVID-19, but I’ve seen estimates in the range of 2-5.

So yes, herd immunity will eventually stop an infectious disease in its tracks, but it takes a very long time, requiring a significant fraction of the population to be infected.  This is why the whole claim about California having herd immunity from earlier infections is absurd on its face.  Herd immunity takes a lot more than that.

The end state

Eventually, almost everyone gets infected, and recovers one way or another.  One thing that surprised me is that it’s only almost everyone–some people escape!  In fact, you can predict how many people will get infected, based on the value of R₀.  I made a chart, based on more Excel simulations (assuming that everyone is initially susceptible).

Aside: I immediately recognize this as a phase transition.  As R₀ crosses the value of 1, it transitions from nothing, to an epidemic.  I am not speaking in metaphor, I am saying it’s literally a phase transition, just like the transitions between solid/liquid/gas.  Phase transition theory is a mathematical theory, and applies to many situations outside of chemistry.  Last year I wrote an article explaining a phase transition in a card game.

If R₀ is at least 2, and if everyone is susceptible, at least 80% of the population will be infected, and there’s not a whole lot we can do about it.  It’s unclear that social distancing measures will be able to change R₀ by enough to escape that eventuality.  This is why most public health messaging emphasizes “flattening the curve”.  Because we’re not sure we can prevent most people from being infected, but we can spread the infections over a longer period of time, reducing hospital overloading.

What makes predictions hard?

Based on what I’ve read, a major difficulty is that we don’t know what numbers to put into the model.  The number of susceptible people, the rate of infection, the rate of recovery, the symptomaticity ratio, and fatality rate… we don’t know any of them.  And they would vary by country, and change with social behavior.

There is also the other difficulty, that we don’t know if this basic model is even sufficient.  Or at least I don’t know.  Ask an expert, folks.  I gotta say, those Medium posts where people just assume that the infection curve is a Gaussian seem pretty goofy.  You couldn’t even use the most basic model like what appears on Wikipedia, eh?

So how did I do?  Did I step too far outside my area of expertise?

My PhD work involved creating excited populations of electrons, and watching them relax over the course of picoseconds.  The rate of relaxation sometimes depends on two-electron scattering, so it would be proportional to the square of the number of electrons.  It’s not quite the same as the SIR model but not too far off either.