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(b) An influenza epidemic spreads at a rate proportional to the product of the number of people infected and the number not yet infected. Assume that 100 people are infected at the beginning of the epidemic in a community of 20,000 people, and 400 are infected 10 days later. i. Write a differential equation that models the change in the number of infected people, y(t), in t days. Include any and all initial conditions. ii. Moreover, analyze the behavior of the solution without solving the O.D.E. i.e, critical values, stability, and sketch a family of solutions on the direction field, and include it with your solution. iii. Solve the differential equation and write an equation for the number of the people infected in t days. On the same direction field, be sure to include the particular solution. iv. For what value of y will the epidemic be spreading the fastest? v. What time (in days) corresponds to the y value in part (iv)?