recursion equation modelling flu epidemic

[fredg1414]

I am in advanced alegebra trig and I have a homework problem that I cannot seem to figure out.

Suppose that in a closed community with population P, a flu epidemic begins and that the number of people newly exposed to the flu on a given day is proportional to the number not yet exposed on the previous day.

1) If "f_n" represents the number of people exposed to the flu n days after it begins, explain how the description above leads to the recursion equation "f_n - (f_{n - 1}) = k(P - (f_{n - 1})).

2) Suppose that, in a college community of 2500 students, the flu begins with 100 students exposed to the flu; in other words, that f₀ = 100. On the next day, f₁ = 220. Find the value of k, and then show that f_n = 0.95f_{n - 1} + 125.

Thank you!

 

[pka]

Let’s be clear on terms.
\(\displaystyle \L
f_n\) is the total exposed after n days.
\(\displaystyle \L
f_n - f_{n - 1}\) is the number exposed on the nth day.
Then on the nth day \(\displaystyle \L
\left( {P - f_{n - 1} } \right)\) is the number not exposed.

Therefore using proportionally \(\displaystyle \L
\left( {f_n - f_{n - 1} } \right) = k\left( {P - f_{n - 1} } \right)\).

Now solve for k in
\(\displaystyle \L
\left( {f_1 - f_0 } \right) = k\left( {P - f_0 } \right) \Rightarrow \quad (220 - 100) = k(2500 - 100)\).