A researcher believes that if x thousand individuals among a

A researcher believes that if x thousand individuals among a susceptible population are inoculated, then a function of the form I(x) = a.R^x - b, for constants a, b and R would model the eventual number of infected individuals (in thousands). Let R = 0.6. If the susceptible population is 3000 and no one is inoculated, how many will eventually be sick? Number that will eventually be sick: If the susceptible population is 3000 and everyone one is inoculated, how many will eventually be sick? Number that will eventually be sick: Use the results of parts a) and b) to find values for the constants a and b in the researcher\'s model. Remember that the outputs to the function are in thousands of people. A = b = According to the model, how many individuals will be infected if half of them are inoculated? Give your answer as a number of people. According to the model, at what inoculation level will one thousand individuals become infected? Give your answer as a number of people. Assume that the cost to treat individuals is given by C(x) = 700 middot 0.04 middot (1 + 0.1x^2). I(x) How much does it cost to treat patients if no one gets inoculated? How much does it cost to treat patients if everyone gets inoculated?

Solution

(c) The suspectible population is = 3000 = 3 thousands

==> x = 3

Thus, we have

3 = a (0.6)⁰ - b, and

0 = a (0.6)³ - b ==> a [1 - (0.6)³] = 3,

or, a = 3 / (1 - 0.216) = 3.8265 ==> b = a - 3 = 3.8265 - 3 = 0.8265

(d) Let z thousands be the population, then inoculated population = ^z/₂ thousands = 1500 = 1.5 thousands

Now, infected population I (x) = 3.8265 (0.6)^1.5 - 0.8265 = 0.9519

i.e. 952 persons

(e) Let 1000 thousands be the population, then inoculated population ==> I (z) = 1 thousands

Now, infected population I (z) = 1 = 3.8265 (0.6)^z - 0.8265 = 1

==> 3.8265 (0.6)^z = 1.8265

==> (0.6)^z = 1.8265 / 3.8265 = 0.47733

==> z log (0.6) = log (0.47733)

==> z = log (0.47733) / log (0.6)

= 1.44775 Thousands

= 1448 persons

(f) If no one is inoculated, then the cost

C (3) = 700 (0.04) (1 + 0.9) 3 = 159.6 thousands

(g) If all the persons inoculated, then I (x) = 0

   ==> C (0) = 0