📂Dynamics

What is the Basic Reproduction Number in Epidemic Spread Models?

Definition

Basic Reproduction Number $\mathcal{R}_{0}$ represents the speed at which an epidemic spreads, and is essentially the expected number of people that one infected person will infect.

Epidemiological Compartment Models ¹

In a dynamical system expressed by differential equations, the largest real part of the eigenvalue of the Jacobian matrix is referred to as $\mathcal{R}_{0}$.

Explanation

Although the definition alone may seem difficult to understand, thinking about it in terms of specific numbers makes it much easier. For COVID-19, it is estimated to be around $\mathcal{R}_{0} \approx 5.7$², which means, simply put, one can expect an infected person to infect about 5.7 others.

If you can grasp the concept through numbers, then understanding it mathematically shouldn’t be too difficult. If $\mathcal{R}_{0} < 1$, it would be unlikely to expect one person to infect another, hence the epidemic would soon end, while $\mathcal{R}_{0} > 1$ would indicate that the epidemic will likely continue to spread.

Thus, considering the largest eigenvalue of the Jacobian matrix as $\mathcal{R}_{0}$ is very natural. This is because a matrix with an eigenvalue larger than $1$ is considered expansion, whereas one smaller than $1$ is considered contraction in terms of matrix transformation, directly linking to the increase or decrease in the number of infected individuals. For the same reason, $\mathcal{R}_{0}$ is also referred to as the Outbreak Threshold³.⁴

Effective Reproductive Ratio

$\mathcal{R}_{0}$ is meaningful only at the onset of the outbreak when no control policy has intervened, representing the inherent spread of the disease. However, the speed of disease spread changes with various response strategies, such as controlling contact or administering vaccines. In this context, the number estimated by multiplying the ratio of the population $N$ that can get infected, the group $S$, is known as the Effective Reproductive Ratio. It is represented by the following equation. $$ \mathcal{R} := \mathcal{R}_{0} {{ S } \over { N }} $$ In an extreme scenario, if we interpret $\mathcal{R}$, at the onset of an outbreak, $S \approx N$, meaning the Basic Reproduction Number essentially equals the Effective Reproduction Ratio $\mathcal{R} = \mathcal{R}_{0}$. Conversely, in a situation where there are almost no people left to infect $S \approx 0$, no matter how high $\mathcal{R}_{0}$ is, the disease will not be able to spread because there are no susceptible hosts. No matter how excellent a leader is, it’s hard to change the epidemic itself, but implementing effective quarantine policies means keeping this $\mathcal{R}$ below $1$ and maintaining it there.

The calculation method for the Effective Reproductive Ratio can vary greatly depending on the model or assumptions, but essentially reflects similar concepts observed above.