Question

Suppose a student carrying a flu virus returns to an isolated college campus (no student leaves the campus) of 1000 students. If it is assumed that the rate at which the virus spreads is proportional not only to the number to infected students but also to the number of students not infected, determine the number of infected students after 6 days if it is further observed that after 4 days x(4)=50. Use the logistics equation, one can write a mathematical model of this situation as follows. dx/dt = kx(1000 - x), x(0) = 1

Answer 1

276 students will be infected after 6days

Explanation:

The mathematical model is:

dx/dt = kx(1000-x), x(0) = 1

Since

dx/dt = 1000kx-kx^2, a = 1000k, b = k, P = 1.

Hence

x(t) = 1000k/ {k+ (1000k-k)*e^1000kt}

= 1000/ (1+999e^1000kt)

From the question, it is given that x(4) = 50

Then solve for the value of k from the above equation:

50 = 1000/ {1+ 999e^-1000k(4)}

k = -1/4000 ln (19/999)

x(t) = 1000/ {1+999e^0.25ln(19/999)t} , x(6) = 1000/ {1+999e^0.25ln(19/999)(6)

x(t) = 276

276 students will be infected after 6days

Answer 2

the number of infected students after 6 days = 276 students

Explanation:

The solution is presented in the attached images

Solving the final quadratic equation,

x² - 100x + 199736.6 = 0

x = 276 or 724

724 represents the number of well students.