$2\mathrm{.}10.$ In this exercise we consider variations of the situation in Exercise $2\mathrm{.}8.$

$($a$)$ Suppose that the product only keeps customers active for an average of $4$

days instead of $20.$ Will a product bust occur?

$($b$)$ Suppose, in addition to the change in $b,$ that $a=0\mathrm{.}00005.$ Now will a

product bust occur?

$($c$)$ In an SIR model where $S(0)=40,000$ and $b=\frac{1}{14},$ how big does a need

to be to avoid a product bust?

$($d$)$ In a product bust, are there any new customers? Explain.

$2\mathrm{.}11.($a$)$ Construct a SIR model for a product that keeps people's interest for an

average of $20$ days, where an average potential customer meets about $0\mathrm{.}01\%$

of the active population each day, and where it takes eight contacts with

an Active before a Potential becomes Active.

$($b$)$ How many Potentials are needed to avoid a product bust?

$2\mathrm{.}12.$ One way to create more interest in a product is to provide rewards for Actives

who refer Potentials. By creating an incentive for Actives to reach out to Poten$-$

tials, a reward program can increase the transmission coefficient.

$($a$)$ Suppose, in the setting of Exercise $2\mathrm{.}9,$ that a reward program is put into

effect that doubles the chance that a Potential will become Active. What

is the new transmission coefficient?

$($b$)$ Compute the new threshold if the reward program in part $($a$)$ is put into

effect, and compare it to the original threshold in Exercise $2\mathrm{.}9.$

$($c$)$ Suppose that, for a different product, $S(0)=20,000,b=\frac{1}{14},$ and $a=$

$0\mathrm{.}000003.$ In order to be profitable, this product must have a threshold

under $15,000.$ Will this product be profitable?

$($d$)$ Suppose that, for the product in part $($c$),$ the company initiates a reward

program, hoping to bring the threshold under $10,000.$ By what factor must

the reward program alter the transmission coefficient to achieve this goal?

$2\mathrm{.}13.$ Consider two products with the same transmission coefficient a and the same

initial conditions. They differ only in the average length of time that someone

stays Active. Which product has the higher threshold level: the one with the

higher average usage time or the lower average usage time? Explain.