Last year's epidemic of Martian flu began with a single case in a total susceptible population of 10,000. The number of cases was increasing initially by 38% per day. Find a logistic model for the number of cases of Martian flu and use your model to predict the number of flu cases 3 weeks into the epidemic. Round your answer to the nearest integer.
A) cases
B) cases
C) cases
D) cases
E) cases

Question

Last year's epidemic of Martian flu began with a single case in a total susceptible population of 10,000. The number of cases was increasing initially by 38% per day. Find a logistic model for the number of cases of Martian flu and use your model to predict the number of flu cases 3 weeks into the epidemic. Round your answer to the nearest integer.
A) cases 
B) cases 
C) cases 
D) cases 
E) cases

Quizplus · Accepted Answer

We can use the logistic equation, where t is time in days and N(t) is the number of cases:

dN/dt = rN(1 - N/K)

Here, r represents the growth rate and K is the carrying capacity (maximum number of cases possible). Since the epidemic has just started, we can assume that K is much larger than the current number of cases, so we can approximate K as infinity.

Given that the number of cases is increasing by 38% per day, we can find the growth rate as follows:

r = 0.38 = 0.38/1.38 (since 1 + r = 1.38, or 138%)

Plugging in this value for r and K = infinity, we get:

dN/dt = 0.38N

This is a separable differential equation, so we can solve it using integration. Separating variables and integrating from t = 0 (when there is 1 case) to t = 21 (3 weeks):

(1/N) dN = 0.38 dt
ln(N) = 0.38t + C
N(t) = e^(0.38t+C) = Ce^(0.38t)

Using the initial condition that N(0) = 1, we can solve for C:

N(0) = C = 1

Therefore, the model for the number of cases is:

N(t) = e^(0.38t)

To find the number of cases 3 weeks into the epidemic (21 days), we plug in t = 21:

N(21) = e^(0.38*21) ≈ 18,412.5

Rounding to the nearest integer, we get:

N(21) ≈ 18,413

Therefore, the answer is B, 18,413 cases.

Last Year's Epidemic of Martian Flu Began with a Single