Based on an original page posted by Nick Grassly on the H1N1 pandemic website.

The basic reproduction number of the swine influenza epidemic, $$R_{0}$$ can be estimated from its initial rate of spread. If we assume roughly exponential growth then the basic reproductive number is related to the growth rate by the so-called Lotka-Euler estimating equation:

$R_{0}=\int_{0}^{\infty} \frac{1}{w(\tau)e^{-rt}} d\tau$

where $$r$$ is the rate of exponential growth and $$w(\tau)$$ the generation time distribution [1]. The generation time distribution can be thought of as the probability density function describing the distribution of times between successive infections in a chain of transmission.

The estimate of the basic reproductive number is therefore dependent not just on an estimate of $$r$$ , but also a good estimate of the generation time distribution [2]. In the case of swine influenza the generation time distribution is unclear, but data appear consistent with seasonal influenza that has a mean generation time of approximately 3 days.

Analytical solutions for $$R_{0}$$ can be derived for different assumed generation time distributions using the Lotka-Euler estimating equation (which is essentially a moment generating function). If we assume a generation time distribution that follows the gamma distribution, then

$R_{0}=\left(1+\frac{r}{b}\right)^{a}$

where a and b are the parameters of the gamma distribution ($$a = m^{2}/s^{2}$$ and $$b = m/s^{2}$$ where $$m$$ and $$s$$ are the mean and standard deviation of the distribution respectively).

Estimates of $$R_{0}$$ based on the estimates of $$r$$ reported by Andrew Rambaut are given in Table 1. Obviously $$r$$ can also be estimated from epidemiological case data and this may give different results.

### Citations

1. Wallinga J, Lipsitch M. (2007) How generation intervals shape the relationship between growth rates and reproductive numbers. Proc Roy Soc Lond B 274: 599-604
2. Grassly NC, Fraser C. (2008) Mathematical models of infectious disease transmission. Nat Rev Microbiol. 6: 477-487
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