this is for holding javascript data
James Moore-Stanley added This_is_however_only_an__.tex
almost 8 years ago
Commit id: 2247868381d9681fa3f99a18605e364c496537e9
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\\This is however, only an approximation of $R_0$. One of the main problems with this approximation is that it does not take the period of time when one is moved into the removed category. Before the next equation is explained, it should be noted that the use of S, I and R as a fraction of the population, or as the real values are interchangeable. In the equation below, we are dealing with \textbf{fractions} of the population, not the whole population, hence instead of $N$ being used, $n$ is used, and since we are using fractions of the population, $n$ can be said to equal 1 (100\% of the population).
\\A new notation $\rho$ will be introduced. $\Rho$ is related to $R_0$, as $\rho$ is the reciprocal of $R_0$. This manipulation is shown below:
\begin{align*}
R_0 &= \frac{\beta}{\gamma} && \text{This is the equation for $R_0$} \\
\rho &= \frac{n \gamma}{\beta} && \text{Note that in this equation, $\gamma$ is on top, and $\beta$ is on the bottom. Remember that $n$ is equal to 1, so we can discard it.} \\
\end{align*}
We can therefore state the equation:
\begin{equation}
\rho = \frac{1}{R_0}
\end{equation}
