Epidemics Last year’s epidemic of Martian flu began with a single case in...

Epidemics 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 40% 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. HINT [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.