Problem 26:
    Jamie Howard
    Randy Keeling
    Jeremy Phillips

Often the carrying capacity N is determined by the way the data is collected. In particular, the table below lists the number of people infected per 10,000 population during a 1996 flu outbreak in the Alsace region of France.

Estimate the intrinsic growth rate k and the duration of the epidemic.

As one can see, we let y be equal to the number of individuals infected per 10,000 people. Thus, N-y, is the number of those susceptible at a given time, t, where N is equal to the total population sector, in this case 10,000. Thus, the general logistic equation is of the form,

(1)
To find the intrinsic growth rate, k, we allow N=10,000, transform the original data, and apply the least squares algorithm to estimate alpha. We must first separate the variables, giving,

(2)
The left hand integral must be expanded to a partial fraction, leading to,

(3)
and this changes our integral to,

(4)
Evaluating the integral leads to,

(5)
and multiplying both sides of the equation by 10,000 provides,

(6)
We now let  which gives,

(7)
and using logarithmic properties, results in

(8)
We now let  and to develop the linear model

(9)
This transforms the original data into a new data set.

Using the least squares algorithm results in,

(10)
Thus, the integral curve solution is

(11)
And the solution function is,

(12)
Substituting  for P gives,

(13)

Thus, the intrinsic growth rate k = 75.8% per day. Thus, the number of people who develop the flu increases by about 80% per day near the beginning of the epidemic.

To determine the duration of the epidemic, the time required for half the population to become infected is doubled. Therefore, the duration is twice the time t for which y = 1/2(10,000) = 5000. Thus,

(14)
and t = 112 days.

Therefore, the duration of the flu epidemic is, 2t = 2(112) = 224 days.