Question

Logistic equation for spread of rumors 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 \(y^{\prime}(t)=k y(1-y)\) for \(t \geq 0,\) where \(k\) is a positive real number and \(t\) is measured in weeks. 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

Now, we can describe the long-term behavior of the rumor function depending on the initial value \(y_{0}\): 1. For \(y_{0}=0.1\): The rumor will spread relatively slowly at the beginning, as only a small fraction of the population has heard the rumor. However, as time goes on, the rate of spread will increase until it reaches a saturation point and slows down again, eventually reaching the entire population. 2. For \(y_{0}=0.7\): The rumor will spread relatively quickly at the beginning, as a significant portion of the population has already heard the rumor. As time goes on, the rate of spread will still increase but at a slower pace since most of the population has already heard it. The spread will eventually reach the entire population, just like in the case of \(y_{0}=0.1\). In both cases, the long-term behavior is the same: the rumor will spread until the entire population knows about it. However, the initial value \(y_{0}\) determines the speed at which the rumor spreads in the beginning.

Step-by-step solution

Solve the initial value problem

We need to solve the logistic differential equation: \(y^{\prime}(t)=k y(1-y)\) with the initial condition \(y(0)=y_{0}\) To do this, we can use the separation of variables method. First, rewrite the equation as: \(\frac{y^{\prime}(t)}{y(1-y)}=k\) Now, integrate both sides: \(\int{\frac{y^{\prime}(t)}{y(1-y)}} dt = \int k dt\) This can be rewritten using the substitution method: \(u=y(1-y)\) \(-2du=(1-2y)dy\) And we get: \(-\frac{2}{y(1-y)}dy = kdt\) Now, integrate both sides: \(-\int\frac{2}{y(1-y)} dy = \int k dt\) \(-\int\left(\frac{1}{y}+\frac{1}{1-y}\right) dy=kdt\) Now, integrate and we get, \(-(\ln |y| - \ln |1-y|) + C_1 = kt + C_2\) Now simplify: \(\ln{\left|\frac{1-y}{y}\right|}=kt+C\) Raise to the power of \(e\): \(\frac{1-y}{y}=e^{kt+C}\) Now, put the initial condition: \(\frac{1-y_{0}}{y_{0}}=e^{C}\) So, \(\frac{1-y}{y}=\frac{1-y_{0}}{y_{0}}e^{kt}\) Now, solve for \(y\): \(y(t)=\frac{y_{0}e^{kt}}{1-y_{0}+y_{0}e^{kt}}\)

Graph the solutions for the given values of \(k\) and \(y_{0}\)

For \(k=0.3\) weeks\({}^{-1}\), we have: 1. For \(y_{0}=0.1:\) \(y(t)=\frac{0.1e^{0.3t}}{1-0.1+0.1e^{0.3t}}\) 2. For \(y_{0}=0.7:\) \(y(t)=\frac{0.7e^{0.3t}}{1-0.7+0.7e^{0.3t}}\) To graph these solutions, you can use any graphing tool or software, by plugging the functions above and visualizing the results.

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

The long-term behavior of the rumor function can be found by analyzing the limit as \(t\) approaches infinity. Since the logistic equation reaches an equilibrium, we need to find when the rate of growth becomes nearly zero. This happens when \(y\) approaches the limiting value \(L\): \(\lim_{t\to\infty}{y(t)}=L\) Plugging the expression for \(y(t)\): \(L=\lim_{t\to\infty}{\frac{y_{0}e^{kt}}{1-y_{0}+y_{0}e^{kt}}}\) As \(t\) tends to infinity, the exponential term \(e^{kt}\) will dominate. Therefore, \(L=\lim_{t\to\infty}{\frac{y_{0}e^{kt}}{y_{0}e^{kt}}}=1\) So, regardless of the initial value \(y_{0}\), the long-term behavior of the rumor function is that it will approach \(1\) as \(t\) increases. This means that eventually, the entire population will know the rumor, as expected for a logistic growth model.

Key concepts

Initial Value Problem

Initial value problems (IVPs) are a fundamental concept in differential equations, particularly in the application of real-world scenarios like modeling the spread of rumors. In an IVP, you're given a differential equation together with the value of the unknown function at a specific point, known as the initial condition. This information is vital as it allows you to find a unique solution to the differential equation that passes through the given initial point.

For the exercise on rumor spreading, the IVP is defined by the logistic differential equation \(y^{\backprime}(t)=k y(1-y)\) with the initial condition \(y(0)=y_{0}\). The solution of this IVP gives the proportion of individuals who know the rumor at a given time, \(t\), considering the initial proportion, \(y_{0}\), that knows the rumor. By solving this initial value problem, sociologists can predict how quickly a rumor spreads within a population.

Separation of Variables

Separation of variables is a technique used to solve differential equations, including the logistic differential equation occurring in our exercise. The essence of this method is to manipulate the equation so that each variable appears on a different side of the equation, which then allows for integration of both sides separately.

In our context, this means taking the logistic growth rate \(y^{\backprime}(t)=k y(1-y)\) and rearranging it into a form where \(y\) and its derivatives are on one side and \(t\) is on the other, allowing us to integrate and find a general solution. It is a powerful method because it transforms the problem into integral forms that are often much simpler to solve.

Modeling the Spread of Rumors

Modeling the spread of rumors is a fascinating application of logistic differential equations. In such models, the population is typically divided into two groups: those who are aware of the rumor and those who aren't. As more people learn about the rumor, the rate at which the rumor spreads changes; it typically grows quickly at first and then slows down as it runs out of new people to reach.

The logistic model captures this dynamic by incorporating the term \(y(1-y)\), which increases when \(y\) (the fraction of the population that knows the rumor) is small and decreases as \(y\) grows, reflecting the reduced potential for the rumor to spread among those who are still uninformed. This exercise utilizes a simplified logistic model to portray how rumors permeate through a community over time.

Long-term Behavior of Differential Equations

Understanding the long-term behavior of solutions to differential equations is crucial for forecasting and planning based on models like those used for spreading rumors. For logistic equations, the long-term behavior can often lead to an equilibrium, a state where the rate of growth is almost null. In the context of rumor spreading, this steady state is when the entire population has eventually heard the rumor.

The math involved includes taking the limit of the solution as \(t\) approaches infinity, which for the logistic model results in \(L=1\). It demonstrates that, given enough time, rumors will spread throughout the entire population, assuming the model's assumptions hold true. This limiting value underscores the equilibrium characteristic of logistic equations and offers insights into various phenomena where saturation effects are observed.