Logistic Differential Equation - Calculus

Question

A virus spreads through a population according to the logistic differential equation  where y is the proportion of the population (0 ≤ y ≤ 1) that has the virus at time t. What proportion of the population has the virus when it is spreading the fastest?

Answer 1

dy/dt is a concave down parabola (in y), with zeros at y = 0 and y = 1. So its vertex, at y = .5, is a global max.

The virus is spreading the fastest when y = .5, ie when exactly 1/2 the population is infected.