Question

Suppose a student carrying a flu virus returns to an isolated college campus of 1000 students. Determine a differential equation for the number of people x(t) who have contracted the flu if the rate at which the disease spreads is proportional to the number of interactions between the number of students who have the flu and the number of students who have not yet been exposed to it.

Short answer

The differential equation is $\frac{dx}{dt}=kx\left(1000-x\right)$.

Step-by-step solution

Consider the given condition

It is given that a student carrying a flue virus return to the isolated campus, where total number of students in the campus are 1000.

Find the differential equation

The required differential equation is of the form $\frac{dx}{dt}=kx\left(n+1-x\right)$, where $n+1=x+y$, which is the total number of people who are infected and who are not infected. Then from the given information, total number of students are 1000. Then

x + y = 1000 = n + 1

Use the obtained value in the differential equation $\frac{dx}{dt}=kx\left(n+1-x\right)$.

$\frac{dx}{dt}=kx\left(1000-x\right)$

So, the required differential equation is $\frac{dx}{dt}=kx\left(1000-x\right)$.