Disease Control

7. Disease control

7.1 Introducing the disease reproduction number R₀

A standard approach to analyzing this kind of disease model is to calculate the so-called disease reproduction number, R₀ (“R-zero” or “R-naught”). R₀ is defined as the number of new infections that would result from the introduction of a single infectious individual (bird or mosquito) into an entirely susceptible population of birds and mosquitoes.

What does the reproduction number mean in practice? If the value of R₀ is less than one, it means that one infectious individual would generate, on average, less than one new infections. (In other words, only a few such individuals would cause a new infection but most would not, making the average number of new infections less than one.) In this case, the disease would die out. On the other hand, the value of R₀ could be greater than one. In this case, an infectious individual would generate, on average, more than one new infection, and a disease outbreak would occur. Thus, R₀ = 1 is known as the threshold value of R₀.

Given a realistic disease model with R₀ greater than one, a public health agency might well want to do something to try to reduce the chance of a disease outbreak. The reproduction number can provide some guidance as to how to go about this.

R₀ is calculated from the equations for the disease model, and is an expression composed of the model’s parameters and the initial values of the variables. In many cases, these parameters and initial values can be manipulated by humans to increase or decrease R₀. Thus, examining the expression for R₀ can help us figure out the best way to try to prevent a disease outbreak.

For our model, the R₀ calculation gives the expression:

. (Equation 7)

Here, the subscript 0 in N_M₀ and N_B₀ indicates the initial values of these variables, on day 0. (In contrast, the subscript 0 in R₀ does not indicate day 0 – it’s just the conventional abbreviation for the disease reproduction number.