Question: Policy makers are interested in modeling the spread of information through a population. For example, agricultural ministries use models to understand the spread of technical innovations or new seed types through their countries. Two models, based on how the information is spread, follow. Assume the population is of a constant size M.
(a) If the information is spread by mass media (TV, radio, newspapers), the rate at which information is spread is believed to be proportional to the number of people not having the information at that time. Write a differential equation for the number of people having the information by time t. Sketch a solution assuming that no one (except the mass media) has the information initially.
(b) If the information is spread by word of mouth, the rate of spread of information is believed to be proportional to the product of the number of people who know and the number who don't. Write a differential equation for the number of people having the information by time t. Sketch the solution for the cases in which
(i) No one
(ii) 5% of the population
(iii) 75% of the population knows initially. In each case, when is the information spreading fastest?