Reproductive number

Definition

The total number of secondary cases produced by an infected individual.
Though, if we want to be more general, it first premiered in demography for the number of offspring a parent would have. But the idea is similar

Interpretation

If $R > 1$ , epidemic grows!
If R < 1, epidemic dies.

Flavors

$R_{0}$ (R-Naught): This is the average number of secondary cases produced by a primary case in a completely susceptible population. Easier to calculate.
$R_{t}$ (Effective reproductive number): This is the average number of secondary cases produced by a primary in a not completely susceptible population. Harder to calculate.

Calculation of R0

For the simplest SIR Model (General). It is simply: $\frac{β}{γ}$
Next generation matrix (NGM) especially for vectorborne diseases where you have multiple infections (a human infecting a mosquito, a mosquito infecting a human)

Connection to intrinsic growth rate

Related to the Lotka–Euler Equation.

Derivation

The total number of offsprings mothers would produce is:

R = \int_{a = 0}^{i n f} n (a) d a

where $n (a)$ is the total number of female offsprings born to a mother of age $a$

The rate $n (a)$ can be normalized in a probability distribution $g (a)$ :

g (a) = \frac{n (a)}{\int (n (a) d a} = \frac{n (a)}{R}

This $g (a)$ can be considered the generation interval distribution.

Damie: put picture here

The age in this case is the age since infection. Then we can fit this into the Lotka-Euler Equation:

\frac{1}{R} = \int_{a = 0}^{inf} e^{- r a} g (a) d a,

This is the Laplace transform of function g(a)

Also a Moment generating function function

And in the paper. The moment generating function is $M$ and we can state:

R = \frac{1}{M (- r)}

So if we have a standard SIR Model (General) model, we assume that the rate of leaving the infectious period is $γ$ . And the waiting time is exponentially distributed (check Exponential Distribution ) and therefore the mean time that an individual is infectious is $\frac{1}{γ}$

The exponential moment generating function is $M (z) = \frac{γ}{γ - z}$ and if we replace $z$ with $- r$
then $M (- r) = \frac{γ}{γ + r}$ or $R = \frac{γ + r}{γ} = 1 + \frac{r}{γ}$ .

References

Wallinga, Jacco, and Marc Lipsitch. "How generation intervals shape the relationship between growth rates and reproductive numbers." Proceedings of the Royal Society B: Biological Sciences 274.1609 (2006): 599.
- DP: A really nice paper.