A researcher believes that if x thousand individuals among a susceptible population are inoculated

Question

A researcher believes that if x thousand individuals among a susceptible population are inoculated (made no longer susceptible to a disease), then a function of form

I(x)=a⋅R^x−b

for constants a, b, and R would model the eventual number of infected individuals (also in thousands). Let R=0.9

(a) If the susceptible population is 5000 and no one is inoculated, how many will eventually be sick?  . What point does that imply should be on the graph of the function I(x)? ( , )

(b) If the susceptible population is 5000 and everyone is inoculated, how many will eventually be sick?  . What point does that imply should be on the graph of the function I(x)? ( , )

(c) Use the results of parts (a) and (b) to find approximate values for the constants a=  and b=  in the researchers model.

(d) According to the model, how many susceptible individuals will be infected if half of them are inoculated?  .

Answer 1

I(x) = a(0.9)^x - b, where x = number (in thousands) that are inoculated and I(x) = number (in thousands) who get sick.

(a). If x = 0, then everyone will eventually be sick. So, a - b = 5 (0,5) is on the graph

(b). If x = 5, then no one gets sick. So, (0.9)⁵a - b = 0 (5,0) is on the graph

(c). 0.59049a - b = 0

a - b = 5

Subtract the equations to get -0.40951a = -5. So a = 12.2097 and b = 7.2097.

(d). I(x) = 12.2097(0.9)^x - 7.2097

Evaluate I(2.5).