Suppose a new virus, SAS$-221,$ is spreading in Canada. The number of new cases each day, $\backslash ($ y$[$n$]\backslash ),$ has two sources:

$-$ external sources $($e$.$g$.,$ travel$),$ denoted by $\backslash ($ x$[$n$]\backslash ),$ and

$-$ spread within the population.

The latter is roughly determined by the effective reproduction number of the virus, $\backslash ($ R $\backslash ).\backslash ($ R $\backslash )$ is the average number of people an infected person will spread the disease to$.$

$($a$)(1$ point$)$ Write down a difference equation that describes this system, and draw its block diagram.

$($b$)(1$ point$)$ Under what conditions is this system stable? Justify your answer mathematically based on the properties of the system.

$($c$)(1$ point$)$ You are tasked with rolling out a vaccination campaign. Every day you are able to vaccinate a fraction $\backslash ($ k $\backslash )(\backslash (0$