What is $R_0$?

In an epidemic, we measure the rate of spread of the infectious agent by its reproduction number. For instance, if one infected person spreads their infection to two people, each of those two people will spread their infections to two additional individuals each, respectively. This is a case where $R_0$ is 2. For any value of $R_0$ greater than 1, there will be exponential growth.

What does exponential growth mean?

Here's an example. Take a regular piece of paper. It's about 0.1mm thick I would say. Fold that piece of paper in half. What happens to the thickness now? It doubles! It's now 0.2mm thick. Fold that paper again, it's 0.4mm thick. A handy way of expressing this is $2^N$ where N represents the number of times you fold the paper. You'd multiply this value by the thickness of the paper to find the overall thickness after folding N times.

To demonstrate the power of exponential growth, let's fold this piece of paper 42 times.

  • Paper Thickness: 0.1mm = 0.0001m = 1E-4m
  • Number of Folds (N) = 42
  • Thickness after folding 42 times: $2^{42}$ 1E-4m = 4.4E12 1E-4m = 4.4E8m

In other words, folding a piece of paper 42 times results in a thickness of 440 million meters! To put this in perspective, the distance from the Earth to the Moon is only 384.4 million meters!

Back to $R_0$

This reproduction number is time-based, so that zero subscript denotes the reproduction number at t = 0. This is the basic reproduction number, otherwise known as "R nought". This is a good starting point, however, this value does not take into consideration factors that influence the value of this number. So from here on, we will refer to it as $R_t$.

What factors influence $R_t$?

Remember the SIR model??

This is a compartmental model which means the entire population is accounted for. S represents the portion of the population that is susceptible to the infection via a transmission rate denoted by $\beta$. Once infected, these individuals enter the portion of the population that is infected, denoted by I. $/gamma$ denotes the recovery rate of infected individuals which would send them to the portion of the population that has recovered, denoted by R. Detailed explanation of SIR here

$\beta$, by the way, is defined as the product of probabilities of the number of contacts with an infected individual ($\kappa$), and disease transmissibility ($\tau$). Since this is a probability, we divide this product by the total population (N). Giving us:

$\beta = \frac{\kappa \tau}{N} $

$\gamma$ is defined by the inverse of the duration (D) of the infection. So if the duration of infection is 3 weeks, the recovery rate is 1/3 of the population (for that week).

The sum of each of these population portions equals the total population, which we can represent as N. So: N = S + I + R

How to find the exact number of S, I, and R for a specific time?

We set up a system of ordinary differential equations as follows:

  • $S'_t = - \frac{\beta I_t}{N}S_t$
  • $I'_t = \frac{\beta I_t}{N}S_t - \gamma I_t$
  • $R'_t = \gamma I_t$

What this shows us is that the number of susceptible individuals can only decrease. The number of infected individuals can either increase or decrease depending on the sign of $\gamma$. The number of recovered individuals can only increase.

What do I mean by the number of infected individuals can either increase or decrease? Let's take a closer look at the second equation. At the beginning of the epidemic (before anyone is infected), the total population is just susceptible. This means S = N. So let's plug in N for $S_t$:

$I'_t = \frac{\beta I_t}{N}N - \gamma I_t = I_t(\beta - \gamma)$

So we need:

$(\beta - \gamma) > 0 $ which means $ \beta > \gamma $ which makes $\frac{\beta}{\gamma} > 1$

What are ordinary differential equations?

Each of the aforementioned terms (S, I, R) are coupled non-linear quadratic ordinary differential functions with time being the common independent variable. "ordinary" is used merely to provide a contrast with "partial" differential equations which have more than one independent variable.

In other words: S, I, and R are not constants. They're derivatives.

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