10.7: SIR and the Spread of Disease

Main Points
Mathematics has an application in everything in everyday life, including to ability to predict an epidemic and therefore the amount of vaccine that would be needed with respect to a population which is of course growing exponentially; (however for the SIR model to work population is help constant).
The S-I-R model:
S - susceptibles (these are not sick yet but could be, future)
I - Infecteds (these are sick already, present)
R - Recovered (these have already been sick, past)
The 'S' reduce over time as they get sick when they come in contact with the 'I's. The two are thus proportional. (infections and contact) thus giving the formula: dS/dt = - (rate at which S gets sick) = -aSI. (the newly sick are then aSI). Those recovering are proportional to those infected: bI; therefore: dI/dt = rate ('S's get sick - 'I's cured) = aSI - bI.
The recovered are no longer susceptible and therefore also proportional to those infected: dR/dt = bI. ('a' is how infectious the disease is and 'b' is rate at which 'I's become 'R's.)
The threshold population value is computed by b/a.
A vaccination should be administered according to the threshold population value (TPV). The entire population save TPV should be vaccinated to avoid an epidemic.

Challenges
I dont understand how the 192 of the dI/dS equation was obtained. And why is there a peak only went the initial value is greater than 192? Shouldn't all initial values no matter what it is produce a peak with its trajectory? I do not exactly understand the significance of a threshold population value.