Question

Sociologists model the spread of rumors using logistic equations. The key assumption is that at any given time, a fraction \(y\) of the population, where \(0 \leq y \leq 1,\) knows the rumor, while the remaining fraction \(1-y\) does not. Furthermore, the rumor spreads by interactions between those who know the rumor and those who do not. The number of such interactions is proportional to \(y(1-y) .\) Therefore, the equation that describes the spread of the rumor is \(y^{\prime}(t)=k y(1-y)\) where \(k\) is a positive real number. The number of people who initially know the rumor is \(y(0)=y_{0},\) where \(0 \leq y_{0} \leq 1\) a. Solve this initial value problem and give the solution in terms of \(k\) and \(y_{0}\) b. Assume \(k=0.3\) weeks \(^{-1}\) and graph the solution for \(y_{0}=0.1\) and \(y_{0}=0.7\) c. Describe and interpret the long-term behavior of the rumor function, for any \(0 \leq y_{0} \leq 1\)

Short answer

To summarize, we have solved the logistic differential equation modeling the spread of rumors among a population, given by \(y^{\prime}(t)=ky(1-y)\) with the initial condition \(y(0)=y_{0}\). The solution to this initial value problem is: \(y(t)=\frac{y_{0}e^{kt}}{1-y_{0}+y_{0}e^{kt}}\) By graphing this function for specific values of \(k\) and \(y_{0}\), we observed that the function approaches 1 as time goes to infinity. This indicates that, in the long run, the rumor will reach almost the entire population, regardless of the initial value \(0\leq y_{0}\leq1\).

Step-by-step solution

Solve the initial value problem

First, we need to separate the variables and integrate both sides. The equation is given by: \(y^{\prime}(t)=ky(1-y)\) We can separate the variables as follows: \(\frac{1}{y(1-y)}y^{\prime}(t)=k\) Now we integrate both sides with respect to \(t\): \(\int\frac{1}{y(1-y)}y^{\prime}(t)dt=\int kdt\) Let \(u = y\) and \(v = 1-y\). Then, \(du = dv = dy\). Applying partial fractions, we get: \(\frac{1}{u(1-v)} = \frac{A}{u}+\frac{B}{1-v}\) By setting u = 0, we get A = 1. And setting v = 1, we get B = -1. Thus \(\frac{1}{y(1-y)} = \frac{1}{y} - \frac{1}{1-y}\) Therefore, our integral becomes \(\int\left(\frac{1}{y}-\frac{1}{1-y}\right)dy=kt+C\) Now we will solve for the constant \(C\) using the initial condition \(y(0)=y_{0}\): \(\int\left(\frac{1}{y_{0}}-\frac{1}{1-y_{0}}\right)dy_0=0+C\) So, \(C=\ln\frac{y_{0}}{1-y_{0}}\) Now we have: \(\int\left(\frac{1}{y}-\frac{1}{1-y}\right)dy=kt+\ln\frac{y_{0}}{1-y_{0}}\) This can be simplified to: \(\ln\frac{y}{1-y}=kt+\ln\frac{y_{0}}{1-y_{0}}\) Now we can solve for \(y(t)\) as a function of \(k\) and \(y_{0}\): \(\frac{y}{1-y}=\frac{y_{0}}{1-y_{0}}e^{kt}\) and finally, \(y(t)=\frac{y_{0}e^{kt}}{1-y_{0}+(y_{0})e^{kt}}\)

Graph the solution for specific values of k and y_0

Now let's graph the solution when \(k=0.3\) weeks\(^{-1}\), \(y_{0}=0.1\) and \(y_{0}=0.7\). For \(y_{0}=0.1\), we have: \(y(t)=\frac{0.1e^{0.3t}}{1-0.1+0.1e^{0.3t}}\) For \(y_{0}=0.7\): \(y(t)=\frac{0.7e^{0.3t}}{1-0.7+0.7e^{0.3t}}\) Plot these functions using suitable plotting software (such as Desmos or Wolfram Alpha). We can note that both functions start at their respective initial values and approach 1, with different rates of growth depending on the initial value.

Analyze and interpret the long-term behavior of the rumor function

When analyzing the long-term behavior of the solution \(y(t)=\frac{y_{0}e^{kt}}{1-y_{0}+y_{0}e^{kt}}\), we can see that as \(t\) goes to infinity, the exponent term \(e^{kt}\) approaches infinity as well. However, the denominator \(1-y_{0}+y_{0}e^{kt}\) is also going to infinity. Since both the numerator and the denominator approach infinity, the value of the fraction approaches 1. In the context of the problem, this implies that over a long period, almost the entire population would know the rumor regardless of the initial value \(0\leq y_{0}\leq1\).

Key concepts

Rumor Spread Model

The study of how rumors spread is intriguing and mathematicians often use the logistic equation to model this phenomenon. The logistic equation is given by \(y'(t) = k y(1-y)\), where \(y\) represents the fraction of the population that knows the rumor at any given time, \(t\). This equation relies on the assumption that the spread of the rumor is influenced by interactions between people who know the rumor and those who do not. Therefore, the spread is proportional to \(y(1-y)\), where \(1-y\) is the fraction of the population who hasn't heard the rumor yet.
The coefficient \(k\) is a positive constant that denotes the rate at which the rumor spreads. This mathematical model helps us understand that, as more people learn the rumor, the speed at which it spreads increases until it reaches a point of saturation in the population.
In simpler terms, at first, only a few people know the rumor, so it spreads slowly. As more people become aware, the spread accelerates, reaching a peak, and then slows down as fewer people are left who don't know the rumor yet.

Initial Value Problem

An initial value problem is a differential equation that comes with a specific condition known as the initial condition. In the rumor spread model, the given initial condition is that a fraction \(y_0\) of the population knows the rumor at time \(t = 0\). This is represented by \(y(0) = y_0\).
Solving the initial value problem involves finding a function for \(y(t)\) that satisfies both the logistic differential equation and the initial condition. Using techniques such as variable separation and integration, we derive the solution:
\[y(t) = \frac{y_0 e^{kt}}{1 - y_0 + y_0 e^{kt}}\]
This function precisely describes how the rumor spreads over time, taking into account the initial fraction of people who knew it. The beauty of this solution lies in its ability to predict the dynamics of rumor spreading based on initial understanding. Let's explore how this evolves into long-term behavior.

Long-Term Behavior Analysis

The long-term behavior of the rumor function, \(y(t) = \frac{y_0 e^{kt}}{1 - y_0 + y_0 e^{kt}}\), can be understood by examining what happens as time \(t\) becomes very large. As \(t\) goes to infinity, the exponential term \(e^{kt}\) grows immensely large, dominating the expression.
The numerator \(y_0 e^{kt}\) and the denominator \(1-y_0+y_0 e^{kt}\) both increase, but since they have similar exponential terms, their ratio approaches 1. This implies that in the long run, the fraction \(y(t)\) of the population that knows the rumor approaches 1.
What does this mean in simpler terms? It means that eventually, almost everybody in the population will know the rumor, regardless of how many people knew it initially. The logistic model elegantly shows that initial conditions impact the speed, but not the eventual outcome of rumor spread. This insight is crucial for understanding how information dissemination saturates a population over time.