A campus has 5000 students. After the Christmas break one student returns carrying a flu virus. Assume that the rate at which the virus spreads is proportional to the number of infected students P(t), but also to the number 5000?P(t) of students not yet infected. Here time is measured in days.
(a) Our assumptions mean that the function P(t) satisfies the initial value problem, dP dt = kP(5000? P), P(0) = 1 where the parameter k is initially unknown. Find the solution of this IVP by solving the DE as a separable equation.
(b) After seven days 50 students are infected. Use this to determine the parameter k .
(c) How long does it take before 10 percent of the students are infected?