Question 3.5.3: Modeling the Spread of the Flu The function f(t) = 30,000/1 ...
Modeling the Spread of the Flu
The function
f(t)=\frac{30,000}{1+20 e^{-1.5 t}}
describes the number of people, f(t), who have become ill with influenza t weeks after its initial outbreak in a town with 30,000 inhabitants.
a. How many people became ill with the flu when the epidemic began?
b. How many people were ill by the end of the fourth week?
c. What is the limiting size of f(t), the population that becomes ill?
a. The time at the beginning of the flu epidemic is t = 0. Thus, we can find the number of people who were ill at the beginning of the epidemic by substituting 0 for t.
f(t)=\frac{30,000}{1+20 e^{-1.5 t}} This is the given logistic growth function.
f(0)=\frac{30,000}{1+20 e^{-1.5(0)}} When the epidemic began, t = 0.
=\frac{30,000}{1+20} e^{-1.5(0)}=e^0=1
≈ 1429
Approximately 1429 people were ill when the epidemic began.
b. We find the number of people who were ill at the end of the fourth week by substituting 4 for t in the logistic growth function.
f(t)=\frac{30,000}{1+20 e^{-1.5 t}} Use the given logistic growth function.
f(4)=\frac{30,000}{1+20 e^{-1.5(4)}} To find the number of people ill by the end of week four, let t = 4.
≈ 28,583 Use a calculator.
Approximately 28,583 people were ill by the end of the fourth week. Compared with the number of people who were ill initially, 1429, this illustrates the virulence of the epidemic.
c. Recall that in the logistic growth model, f(t)=\frac{c}{1+a e^{-b t}}, the constant c represents the limiting size that f(t) can attain. Thus, the number in the numerator, 30,000 , is the limiting size of the population that becomes ill.