While the novel coronavirus (COVID-19) itself needs no introduction, its emergence into the national spotlight presents an unique opportunity to discuss the mathematics behind the modeling of infectious diseases.

More specifically, in this post, we’ll be exploring one of the simplest ways to model the human-to-human transmission of an infectious disease—an SIR model:

An SIR model is an epidemiological model that computes the theoretical number of people infected with a contagious illness in a closed population over time. The name of this class of models derives from the fact that they involve coupled equations relating the number of susceptible people $S(t)$, number of people infected $I(t)$, and number of people who have recovered $R(t)$.

This post shouldn’t require a background in mathematics to understand, but some familiarity with ordinary differential equations would be helpful.

Understanding the model

The Susceptible-Infected-Recovered (SIR) model takes the total size ($N$) of a particular closed population (the world, a country, a town, etc.) and divides them into three “compartments”:

  • Susceptibles or $S(t)$: All people capable of becoming (but are not yet) infected by the particular disease.

  • Infected or $I(t)$: All people currently infected by the particular disease.

  • Recovered or $R(t)$: All people who have recovered from being infected by the particular disease.

As implied by the $f(t)$ notation, the size of each compartment may fluctuate over time due to various rates of change:

Ignoring natural birth and death rates (for now), the SIR model can be represented by the following system of ordinary differential equations (ODEs):

$$ \begin{align*} \frac{dS}{dt} & = - \beta SI \textrm{, }~ S(0) = S_0 \newline \frac{dI}{dt} & = \beta SI - \gamma I \textrm{, }~ I(0) = I_0 \newline \frac{dR}{dt} & = \gamma I \textrm{, }~ R(0) = R_0 \end{align*} $$

Where $\beta$ (infection rate) and $\gamma$ (recovery rate) are system parameters, and $S(t) + I(t) + R(t) = N$. To get a better idea of what this looks like, let’s do some plotting with Python.

As you can see, the relationship between the ODEs is pretty intuitive: we assume our entire population is susceptible at the beginning but, over time, the number of susceptibles decrease by $\beta SI$. This quantity is then added to the total number of infected, which loses $\gamma I$ to the total number of recovered.

The basic reproduction number ($R_0$)

Now that we have a basic understanding of the model, we can take a closer look at its system parameters. For starters, we’re going to focus on the second of our equations:

$$ \frac{dI}{dt} = \beta SI - \gamma I $$

When we first encounter a novel infectious disease, this equation helps us determine what the potential for an epidemic is—in other words, we want to examine the equation at time $t = 0$:

$$ \frac{dI}{dt} \Bigr\rvert_{t = 0} = \beta S_0 I_0 - \gamma I_0 $$

The driving question then becomes is $\beta S_0 I_0 - \gamma I_0 > 0$? To make our analysis easier, let’s simplify our equation a little bit:

$$ \begin{align*} \beta S_0 I_0 - \gamma I_0 & > 0 \newline \beta S_0 - \gamma & > 0 \newline \beta S_0 & > \gamma \newline \frac{\beta S_0}{\gamma} & > 1 \newline \frac{\beta}{\gamma} S_0 & > 1 \newline \end{align*} $$

The ratio $\frac{\beta}{\gamma}$ is known as $R_0$ (“R-nought”) or the basic reproduction number.

A scene from *Contagion* (2011) depicting various R-nought values.
A scene from Contagion (2011) depicting various R-nought values.

This is the idea: If $R_0 \ge 1$ (the $\Delta$ of infected with respect to $t$) , then the rate of infection is increasing and—without intervention—we’ll face an epidemic. If, on the other hand, $R_0 < 1$ then people are recovering faster than they’re becoming infected and we aren’t dealing with an epidemic.

“Flattening the curve”: A call to action

I’m sure you’ve heard the phrase “flattening the curve” as a call to action to slow the spread of COVID-19. But what does this really mean?

Well, let’s first take a look at what could happen if no interventive measures are enacted in the U.S.

To estimate the values of $\beta$ and $\gamma$, we’ll use the technique outlined by Penn Medicine: Since the CDC is recommending 14 days of self-quarantine, we’ll use $\gamma = \frac{1}{14}$, which means $\beta = (\textrm{g} + \frac{1}{14})$ where $g = 2^{(1/T_d)} - 1$. Using Penn’s cited doubling time ($T_d$) of $8.5$ days, we have $\beta = 0.156392$.

Thus, our estimated $R_0$ for COVID-19 is 2.189.

Now, using the U.S. data as of March 23rd from Worldometer, let’s look at what could happen if we let COVID-19 spread unchecked.

This is the now-infamous “curve” and is very likely similar to models being shown to elected officials around the country. And the conclusion is clear: we must act and we must act now. In a mere 60 days from today (March 23rd, 2020), the U.S. could have over 6 million cases of COVID-19—a number that would completely overwhelm the healthcare infrastructure capacity.

So, now that we know why we must act, what exactly can be done?

To answer this question, we must return to our motivating inequality: $\frac{\beta}{\gamma} S_0 > 1$? In an attempt to make this inequality false, there are two steps that can be taken:

  • We can decrease the infection rate, $\beta$, by taking preventive measures. This is the goal of the many now-implemented “shelter-in-place” ordinances and the measures are outlined in the World Health Organization’s “Do the Five” campaign.

  • We can decrease the susceptible population, $S_0$, through the administration of a vaccine.

In any case, it’s clear that we’re in a largely unprecedented situation.