Basic reproduction number I'm not sure where to post this, so I thought I'd post it here. If there is another better place to ask this question then please could you let me know. 
Let
$\beta$  = rate of infection, and $\gamma$ = rate of recovery.
I don't understand how the basic reproduction number (BRN) is $\frac{\beta}{\gamma}$?
I know there is something to do with the exponential distribution, and the fact that the mean of the exponential distribution is $ \frac{1}{\lambda}$, so clearly here $ \lambda = \gamma$, but I don't understand why.
Is it: BRN is the number of secondary infections caused by one infection, so I thought it would be $\beta$, I don't get why the $\frac{1}{\gamma}$ comes into it.
Please could someone explain this?
 A: The answer to this doesn't necessarily rely on the distribution, it can be thought of as a simple problem of incoming infections vs. outgoing infections.
If you are trying to fill a bathtub, the water level will only rise if the rate of incoming water outpaces the rate of outgoing water. The same principle is true of an epidemic. An infection must infect people faster than people recover from infection in order for their to be a sustained increase in cases.
Thus: beta > gamma in order for there to be a sustained epidemic, and therefore: beta/gamma must be greater than 1.
This is, for example, while some of the hemorrhagic fevers aren't capable of sustaining epidemics - while their beta is quite high, the rate of recovery (or in this case, death) is so fast that the epidemic runs out of infective individuals to propagate new infections faster than new people are infected.
A: I'm just guessing here but...
The basic reproduction number is the expected number of secondary infections over the lifetime of the initial infection. Let $S$ be the number of secondary infections over the lifetime of the initial infection and $L$ be the lifetime of the initial infection.
$S|L$ can be modeled as a Poisson random variable with parameter $\beta L$ and $L$ can be modeled as an Exponential random variable with parameter $\gamma$.
To find the expected value of $S$, recall the law of total expectation.
$$\textrm{E}\left[S\right]=\textrm{E}\left[\textrm{E}\left[S|L\right]\right]=\textrm{E}\left[\beta L\right]=\frac{\beta}{\gamma}$$
