Chapter 4.2, Problem 44E

As an epidemic spreads through a population, the number of infected people, I, is expressed as a function of the number of susceptible people, S, by

$I=k\ln \left(\frac{S}{S_0}\right)-S+S_0+I_0,\mathrm{for}k,S_0,I_0>0.$

1. (a) Find the maximum number of infected people.
2. (b) The constant k is a characteristic of the particular disease; the constants S₀ and I₀ are the values of S and I when the disease starts. Which of the following affects the maximum possible value of I? Explain.
   • The particular disease, but not how it starts.
   • How the disease starts, but not the particular disease.
   • Both the particular disease and how it starts.