Yesterday we went through some mathematics of exponential and geometric growth applied to COVID-19. Today epidemic models. This is more complex because we need to understand something about differential equations, but they're actually not so hard.
Let's call cases at time t, N(t). In a short time, dt, the number of cases change from N(t) to N(t+dt). The rate of change (cases per time) is then [N(t+dt) - N(t)]/dt. Now, as dt approaches 0 we call this the "derivative of N with respect to t" and write it as dN/dt.
So a derivative is just the rate of change. If dN/dt=0, then there is no change. dN/dt > 0 means growth. dN/dt < 0 means decline. This will be important. Now in epidemic models we don't just have "N" to worry about. We may have susceptible (S), infected (I), and recovered (R).
There are many variants of epidemic models. They are commonly called SIR models, but for COVID19 you might use a SEIR model to have an Exposed but not infectious class. If immunity is lost, you could have a SIRS model since recovered can become susceptible again. Today just SIR
These models begin with Kermack and McKendrick in early 1930s. One of my PhD mentors @MarcMangel1 says "one might argue that since these days it is so easy to blindly solve a set of equations on the computer, it is even more important now to be able to think about them carefully"
Our first assumption is that S + I + R = N. Meaning no demographics except death due to pathogen (included in R). Fair for COVID19. Stronger assumption is that S, I, and R randomly encounter each other in what is called "mass action transmission". OK assumption for learning.
We assume a recovery rate ν (greek letter "nu"), which is the easier parameter to estimate since it it the inverse of the infectious period. The transmission rate β is more difficult to estimate and think about. The dynamics for S, I, and R can be written as 
The rate at which susceptibles become infecteds flows out of S and into I. The recovery rate flows out of I and into R. Now take a moment to think. What will happen here? S can only decline. R can only increase. I is the most interesting. Let's explore the dynamics of infecteds.
Let's ask when the number of infected is stable (dI/dt = 0) because on either side of that condition I will increase or decrease. Of particular interest is what happens when we add the first infected to the population. The condition leading to dI/dt = 0 is called a "nullcline"
This is the first profound lesson. If S>ν/β, then dI/dt>0 and the number of infections grows. This is true both when the first infected is introduced and when an epidemic rampages through and removes many S. What if we had a strategy to push S<ν/β? We do. It's called vaccination!
Note that it doesn't matter how big I is. The epidemic grows depending on a threshold level for S. Before COVID was introduced everyone was S. So S=N, I=0, and R=0. Enough vaccinations would reduce S below ν/β and provide "herd immunity". Herd immunity is turning enough S into R
There was confusion about the UK strategy mentioning herd immunity. Without vaccinations, to achieve S<ν/β requires that all those susceptibles become infected before recovery, but that requires hundreds of thousands of deaths as a large portion of the population is infected.
Before we look at the actual dynamics of S, I, and R. Let's talk about R0 (basic reproductive number). R0 is the number of new Is produced by the first I in the population. R0 is about the initial population when the first infected comes around.
If R0>1, then an introduced pathogen spreads. If R0<1 it doesn't. So R0>1 is exactly the same condition for spread as S<ν/β we just derived! The size of R0 tells us how fast the epidemic grows (or not) at first. Surely R0 and our threshold S=ν/β are related somehow??
Before we move on, let's scale our variables by the total population. Since S + I + R = N, we divide by N to get S/N + I/N + R/N = 1. It turns out the equations we just wrote can be reinterpreted as proportion of a population in S, I, or R just with β having different units now.
We scaled variables to explore the dynamics so that S, I, and R all all between 0 and 1 to avoid plotting very large numbers. With scaled variables, our threshold condition for epidemic growth S>ν/β becomes 1>ν/β because 100% are S, which is the same as β/ν>1. Turns out R0=β/ν!
If we intervened right away based on lessons from other countries, we might have reduced transmission, β, below recovery, ν, early. R0 (like our threshold) also tells us to vaccinate over 1 - 1/R0 proportion of the population for "herd immunity" to prevent an epidemic.
Let's explore dynamics of the proportion of susceptible, infected, and recovered individuals for different values of R0 by varying transmission, β. We'll use ν=1/14=0.07, assuming 14-day infectious period. Let's start with R0=2.5, so β=2.5ν. We'll assume that I=0.000001 of N.
This is the epidemic wave. At it's peak about 23% of the population has an active infection. That's over 75 million people. That peak is where dI/dt=0 because for a moment I stopped growing and will now start declining. All the while S declines and R grows. What if R0=2?
We delayed the peak substantially and we "flattened the curve" so that 15% of the country or 49 million people have the infection. *Do NOT quote these as real numbers*. This is just illustrative of how dynamics change with declining R0. If R0=1.5 we get a very different picture.
Flattening the curve by reducing transmission, and thus R0, substantially delays and reduces the peak, but also reduces the total number that are ever infected. R0=1.5 has a peak with 6% of the population having an active infection. Ideally we reduce R0 enough to stop this thing.
Obviously this related directly to analyses about hospital beds. The cumulative proportion that get infected is given by what R converges to.
projects.propublica.org/graphics/covid…
projects.propublica.org/graphics/covid…
