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 . 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 . 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.

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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