Diphtheria $($a contagious bacteria disease$)$ is spreading in a small village with a population size of $1000$ inhabitants. Consider that the population is divided into three categories: susceptible, infected, and recovered $($or removed$).$ Once someone gets the infection this year, they cannot get the infection again. The village has a healthcare system that can facilitate vaccinations to prevent the spread of Diphtheria. The average length of the disease is $\frac{5}{3}$ weeks, over which time the person is deemed infected and can spread the disease. The time period for the model is $t=\{0,1,2,$dots,$n\},$ number of weeks.

$(3\mathrm{.}1)$ Initially at $t=0,$ only $5$ cases are reported. Given that $a=\frac{1}{\frac{?}{?}}$bar $(t)=\frac{1}{(average\text{l}\text{e}\text{n}\text{g}\text{t}\text{h}of\text{d}\text{i}\text{s}\text{e}\text{a}\text{s}\text{e}in\text{w}\text{e}\text{e}\text{k}\text{s})},$ and that after one week, the total number of the infected is $9,$ use the discrete SIR model to compute the value of $r.($Hint: Read the online lesson $7.$ Also, a should be written in decimal form e$.$g$.a=0\mathrm{.}25)$

$34$

APM$1514/101/0/2024$

$(3\mathrm{.}2)$ Solve the discrete SIR model iteratively using a table. On the same system of axis, plot the $S_t,I_t$ and $R_t$ on the y$-$axis and weeks on the x$-$axis. Also, make use of legends. The time period is from $0$ to $18$ weeks. $($Hint: You can draw by hand or use Excel or GNU Octave$)$

$(3\mathrm{.}3)$ Determine the rate of change of I $($infected$)$ at $t=0,t=9$ and $t=18.$

$(3\mathrm{.}4)$ Explain what the dynamics of the three plots and $($:$R_t$