Based on Hoppensteadt and Peskin (1992). The following model (the Reed–Frost model) is often used to model the spread of an infectious disease. Suppose that at the beginning of period 1, the population consists of five diseased people (called infectives) and 95 healthy people (called susceptibles). During any period there is a 0.05 probability that a given infective person will encounter a particular susceptible.

If an infective encounters a susceptible, there is a 0.5 probability that the susceptible will contract the disease. An infective lives for an average of 10 periods with the disease. To model this, assume that there is a 0.10 probability that an infective dies during any given period. Use @RISK to model the evolution of the population over 100 periods. Use your results to answer the following questions. [Hint: During any period there is probability 0.05(0.50) 5 0.025 that an infective will infect a particular susceptible. Therefore, the probability that a particular susceptible is not infected during a period is (1 – 0.025)n, where n is the number of infectives present at the end of the previous period.]

a. What is the probability that the population will die out?

b. What is the probability that the disease will die out?

c. On the average, what percentage of the population is infected by the end of period 100?

d. Suppose that people use infection “protection” during encounters. The use of protection reduces the probability that a susceptible will contract the disease during a single encounter with an infective from 0.50 to 0.10. Now answer parts a through c under the assumption that everyone uses protection.