Find the logistic function f with the given properties. HINT [See Example 1.] Exa...

Find the logistic function f with the given properties. HINT [See Example 1.]

Example 1:

A flu epidemic is spreading through the U.S. population. An estimated 150 million people are susceptible to this particular strain, and it is predicted that all susceptible people will eventually become infected. There are 10,000 people already infected, and the number is doubling every 2 weeks. Use a logistic function to model the number of people infected. Hence predict when, to the nearest week, 1 million people will be infected.

Solution: Let t be time in weeks, and let P(t) be the total number of people infected at time t. We want to express P as a logistic function of t, so that

We are told that, in the long run, 150 million people will be infected, so that

N = 150,000,000. Limiting value of P

At the current time (t = 0), 10,000 people are infected, so

Solving for A gives

10,000(1 + A) = 150,000,000

1 + A = 15,000

A = 14,999.

What about b? At the beginning of the epidemic (t near 0), P is growing approximately exponentially, doubling every 2 weeks. Using the technique of Section 2.2, we find that the exponential curve passing through the points (0, 10,000) and (2, 20,000) is

giving us . Now that we have the constants N, A, and b, we can write down the logistic model:

The graph of this function is shown in Figure 18.

Now we tackle the question of prediction: When will 1 million people be infected? In other words: When is P(t) = 1,000,000?

Thus, 1 million people will be infected by about the 13th week.

f (0) = 1, f has limiting value 10, and for small values of x, f is approximately exponential and grows by 50% with every increase of 1 in x.