$($b$)$ Ebola $($a hemorrhagic fever$)$ is spreading in a small village with a population size of $2000$ 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 bas a healthcare system that can facilitate vaccinations to prevent the spread of Ebola. The average length of the disease is $\frac{11}{6}$ 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.

$($i$)$ Initially at $t=0,$ only $8$ cases are reported. Given that $a=\frac{1}{\frac{?}{?}}$bar $(t)=$ that after one week, the total number of the infected is $10,$ use the discret sir model to come the value of $r.$

$($ii$)$ At $t=0,$ determine the rate of change of S $,$ the rate of change of I $,$ and the rate of change of