Logistic Problem

Question

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 M, and let P (a function of time, t) 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 k>0) 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. 

dPdt= k(M−P)

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? 

P→ M

At what value of P is the number of people who know the information increasing the fastest? 

When P= 

(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 k>0) 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. 

dPdt= kP(M−P)

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, P→ 0 

If 5 percent initially know, P→  

If 75 percent initially know, P→ 

In the latter two cases, when is the information spreading fastest? 

If 5 percent initially know, when P=  

If 75 percent initially know, when P= 

Answer 1

Both differential equations appear separable. The second equation will require partial decomposition to integrate. Technology can help in graphing each solution since the P can be solved explicitly.

Answer 2

The approach to this problem would fall under "quantitative analysis for solving ODEs". The steps would be elementary (involving little to no direct calculus calculations), but it would be considerable work to type out an answer, or even a guide to getting the answer, here. It would involve graphing the ODEs and drawing conclusions from the sketches to graph the solution. And once you have the solution, you can then answer the other questions.

A helpful resource can be found here: math.sci.ccny.cuny.edu/document/Chap_4.pdf

If you want more direct help, I would encourage you to get a tutor on the platform to guide you.

Good luck!