Math 132 Lesson R: Population models and the spread of rumors Application

Transcription

1 Math 132 Lesson R: Population models and the spread of rumors Application Population models are not limited to births and deaths. All kinds of quantifiable things pass through populations, from income to diseases. In this lesson, we will look at the transfer of information through a population. Mathematical models As always, we will need to be explicit about our assumptions and definitions. How do rumors spread? From the people who know to the people who don't know. So it makes sense to think of our population in terms of two subpopulations: those who have heard the rumor and those who haven't. For the sake of brevity, let's call them the ignorant and the informed. Since these populations will be changing in time, we'll have one independent variable (time) and two dependent variables (for the two populations). Symbols How do the populations change? An ignorant person hears the rumor, and moves from the category "ignorant" to the category "informed". That event simultaneously decreases P and increases Q. How does that transfer happen? Learning the rumor can only happen by an interaction between an ignorant person and an informed person. If two informed people talk to each other, nobody new learns the rumor. Similarly, if two ignorant people talk to each other, they will still be ignorant. Assumptions quantity definition role t time (hrs) independent variable P(t) number of people who have not heard the rumor (ignorant) dependent variable Q(t) number of people who have heard the rumor (informed) dependent variable 1. Every person has either heard the rumor or not. There is no forgetting. Everyone is either ignorant or informed. 2. The total population is constant. 3. Interactions between ignorant and ignorant do not spread the rumor. Interactions between informed and informed do not spread the rumor. Interactions between ignorant and informed do spread the rumor, changing an ignorant person to an informed person. 4. People interact with each other at random. 5. People interact with the same number of people per time unit. At this point we could construct a model where time is either discrete or continuous. There are some situations where using discrete time is very natural; for example, in speed dating, people talk at fixed time intervals. Let us work with continuous time. 6. Time is continuous.

2 Word Equations We are now ready to formulate our word equation. The population dynamics are governed by the each population. otice that since each person is either ignorant or informed, the rate of increase of the informed population is exactly balanced by the rate of decrease of the ignorant population. informed population = ignorant population So we only need to track one of them. Let's go with the the informed population. informed population = rate of communication * one is ignorant The probability of two things occurring is the probability of the one thing * the probability of the other. The probabilities get multiplied. one is ignorant = one is ignorant * the other is informed The someone you interact with is in one category or the other is the same as the proportion of the population that is in that category. an indiv is ignorant = number of ignorant people total population and similarly for the informed. Assumption 2 said that the total population is constant. ( total pop) = ( ignorant pop) + ( informed pop) = const Since Assumption 5 says that people interact with each other at a constant rate, the rate of communication will be a constant. So will the total population. These are parameters. Symbols, revised Our symbols are now: quantity definition role t time (hours) independent variable P(t) number of people who have not heard the rumor (ignorant) dependent variable Q(t) number of people who have heard the rumor (informed) dependent variable total population parameter r rate of communication (conversations/hour) parameter

3 Translations ( total pop) = ( ignorant pop) + ( informed pop) = const = P(t) + Q(t) an indiv is ignorant an indiv is informed = number of ignorant people total population = P(t) number of informed people = total population = Q(t) one is ignorant = one is ignorant * the other is informed one is ignorant = P(t) * Q(t) informed population = rate of communication * one is ignorant dq(t) = r * P(t) * Q(t) We can eliminate P since P = -Q. dq(t) = r * Q(t) * Q(t) This is our differential equation model for the spread of rumors. If we also have an initial condition, a specific value for Q(0), we will have a complete model. Solutions Analytical To solve a differential equation, we need to find a function which satisfies the differential equation and its initial condition. This can in general be quite challenging, but Maple has a large knowledge base built in. It can solve a vast number of differential equations whose solutions have already been found over the last couple of centuries. We first type our differential equation and give it a name:

4 otice that Maple slightly reorganizes our expressions, but they are mathematically equivalent. Then we use Maple command dsolve to solve the differential equation: Because the inverse of a derivative is an integral, there is a constant of integration, which Maple arbitrarily calls _C1. If we have an initial condition, we can use that to determine _C1: This will be our solution, so let's give it a name: What does it look like? To plot a function, Maple has to know all the parameter values. We use subs to plug in specific values. Suppose that in a population of 100, one person knows the rumor at time 0. Suppose there's one communication per hour.

5 In 100 hours, only 2 more people hear the rumor. But in the next 100 hours, we get up to 7 people knowing the rumor. It looks like we have exponential growth, just like in our birth models. But if we go a bit further, we see that the spread of a rumor is not exactly like exponential growth:

6 umerical What if we had a differential equation that didn't have an analytical solution? We could still get a solution numerically, for example by Euler's Method (which we used in the heating/cooling lesson). It could be implemented in a spreadsheet. It is also built into Maple and other programs. To numerically solve our differential equation, first we have to load Maple's library of tools for differential equations, DETools. ote the capitalization. The library needs to be loaded once per Maple session. Then its commands are available to use. It only needs to be loaded once per session. As with the plot command, the DEplot command has to have specific values for the parameters, or it won't know what to plot. So we have to plug in parameter values first: Then we have to give DEplot information about the differential equation, the range of t, and the initial condition: The yellow curve is the same curve we got from the analytical solution. The red arrows indicate the slope dq for each value of t and Q. otice that the slopes are aways tangent to the solution curve.

7 Equilibrium otice from the graphs that Q levels off over time. That is because eventually everyone has heard the rumor, and there is no one left to hear it. The graph of Q(t) asymptotes to the steady-state value Q =. When that happens, the the population is 0. That is, dq = 0 when Q = in the rumor model. Any value of the dependent variable Q, for which dq = 0 is called an equilibrium. We can see from the differential equation dq(t) = r * Q(t) * Q(t) that dq will = 0 when Q = or when Q = 0. Thus we say that Q = 0 and Q = are equilibria. The term "steady state" means the same thing as "equilibrium". Other population models Here are some other well-known population models. The Exponential model is the model we analyzed previously. Its solutions are exponentials. The Logistic model describes populations which have a maximum sustainable population, like fish in a lake. The Harvesting model describes a population which is harvested, like fish in a lake. otice that the Logistic model is almost the same as our rumor model. Exponential model dy(t) Logistic model quantity definition role t time independent variable y(t) population dependent variable r per capita population growth rate parameter M maximum population size parameter H harvest rate parameter dy(t) Harvesting model dy(t) = r * y(t) = r * M y(t) M = r * M y(t) M * y(t) * y(t) H