deletions | additions       diff --git a/The_notation_N_will_be__.tex b/The_notation_N_will_be__.tex  index 65e4703..1a5709a 100644  --- a/The_notation_N_will_be__.tex  +++ b/The_notation_N_will_be__.tex 
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The notation $N$ will be used for the total population value. It should be noted that as these are the only categories the population may be divided into, $S+I+R=N$ must be case for any value of $t$.  \\In order to determine the \textbf{rate} at which individuals transition between \textit{susceptible and infected} and \texit{infected and removed}, a rate constant must be involved to state the percentage of the population that makes each of these transmissions. If values for $t$ are taken at discrete integer intervals, we can deduce the equation for the rate of transition, i.e. difference in the number of people in the susceptible category for time $t$ and $t+1$, or the number of people that have become infected in day $t$. This can be expressed in the equation below:  \begin{equation}  S_{t+1} - S_t = \frac{\mathrm{d} S}{\mathrm{d} t} = -\gamma -\beta  S_t I_t \end{equation}  where $\gamma$ $\beta$  is the aforementioned rate constant. Gamma $beta$  is given a negative value, as the number of people who are classed as susceptible is decreasing, as they are moving into the infected classification. A similar method may be applied to calculated the \textbf{rate} at which individuals move from the infected to the removed category. The rate of people becoming infected would logically be the same as the rate of decrease in susceptible individuals, as individuals can only from susceptible to infected.  \\However, people who are already infected may have either recovered or died at this time, meaning they are moved into removed classification. This is explained mathematically below:  \begin{align*}  I_{t+1} - I_t &= \frac{\mathrm{d} I}{\mathrm{d} t} = \gamma \beta  S_t I_t && \text{This equation is incorrect as it does not take into account recovered individuals} \\ I_{t+1} - I_t &= \frac{\mathrm{d} I}{\mathrm{d} t} = \gamma \beta  S_t I_t -\beta -\gamma  S_t I_t && \text{ This equation includes individuals changing from infected to removed} \\ \end{align*}