A researcher believes that if x thousand individuals among a susceptible populat

Question

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?

Explanation / Answer

(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

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