Consider the disease model given by: = , = , = where ,, and represent the proportion of people in a population who are susceptible, infected, and recovered from a disease, and is time in months.

Assume that = 2.5, = 0.6, (0) = 0.99,(0) = 0.01,(0) = 0.

a) Use Eulers method to approximate the solution on the interval 0 10. What is the maximum proportion of people infected at any given time during this time interval, and when does this occur?

b) Over time, epidemiologists gather additional data and then find that the recovery rate is actually closer to = 0.8. What is the maximum proportion of people infected at any given time during the time interval 0 10, and when does this occur?

c) Suppose that a new vaccine is generated, and 50% of the population agrees to take it. Also, 10% of the population has already recovered (and is assumed to be immune). Therefore, we can set (0) = 0.60. Assume that (0) = 0.01, (0) = 0.39. Continue to assume that = 2.5, = 0.8. According to this model, what proportion of people will remain susceptible to this disease? (Hint: plot the solution over a longer time period use your plot to estimate the horizontal asymptote lim ()