It is of considerable interest to policy makers to model the
spread of information through a population. For example, various
agricultural ministries use models to help them understand the spread of
technical innovations or new seed types through their countries.
Two models, based on how the information is spread, are given
below. Assume the population is of a constant size , and
let (a function of time, ) be the number of people in
that population who have the information of interest.
(a)
If the information is spread by mass media (TV, radio,
newspapers), the rate at which information is spread is believed to be
proportional (with constant of proportionality ) 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 .
Sketch a solution assuming that no one (except the mass media)
has the information initially. What is the limiting value of the
population that knows the information?
At what value of is the number of people who know the
information increasing the fastest?
When
(b)
If the information is spread by word of mouth, the rate of
spread of information is believed to be proportional (again,
with constant of proportionality ) 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 .
Sketch the solution for the cases in which
(i) no one; (ii) 5 percent of the population; and (iii) 75 percent
of the population initially knows the information.
What is the limiting value for the population that knows the
information in each case?
If no one initially knows,
If 5 percent initially know,
If 75 percent initially know,
In the latter two cases, when is the information spreading fastest?
If 5 percent initially know, when
If 75 percent initially know, when
You can earn partial credit on this problem.