Suppose an infectious disease spreads through an elementary school at a rate proportional to the product of the percentage of pupils who have the disease and the percentage of pupils who have not yet contracted the disease. Suppose that at the beginning of the epidemic 5% of the pupils have the disease. Let f(t) be the percentage of pupils who have the disease at time t; give the differential equation satisfied by f(t). Does the following accurately describe this situation: $y ^ { \prime }$ = ky(100 - y); y(0) = 0.05, where k is a positive constant?

Question

Suppose an infectious disease spreads through an elementary school at a rate proportional to the product of the percentage of pupils who have the disease and the percentage of pupils who have not yet contracted the disease. Suppose that at the beginning of the epidemic 5% of the pupils have the disease. Let f(t) be the percentage of pupils who have the disease at time t; give the differential equation satisfied by f(t). Does the following accurately describe this situation:

y ^ { \prime }

= ky(100 - y); y(0) = 0.05, where k is a positive constant?

Quizplus · Accepted Answer

The given differential equation is $y ^ { \prime }$ = ky(100 - y), where y is the percentage of pupils who have the disease at time t. This equation is satisfied by the function f(t) = 100/(1 + Ce^(-kt)), where C is a constant. The initial condition f(0) = 0.05 gives C = 19.

Suppose an Infectious Disease Spreads Through an Elementary School at a Rate