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Chapter 4: Problem 69

Suppose that the spread of a flu virus on a college campus is modeled by the function $$y(t)=\frac{1000}{1+999 e^{-0.9 t}}$$ where \(y(t)\) is the number of infected students at time \(t\) (in days, starting with \(t=0\) ). Use a graphing utility to estimate the day on which the virus is spreading most rapidly.

Short Answer

Expert verified
The virus spreads most rapidly around day 7.7.

Step by step solution

01

Analyze the Function

The given function is \( y(t) = \frac{1000}{1 + 999 e^{-0.9t}} \), which is a form of the logistic growth model. Logistic growth models are commonly used to describe the spread of diseases like the flu. To find when the virus is spreading most rapidly, we need to determine where the growth rate of infected students is maximum in terms of \( t \).
02

Identify the Point of Maximum Growth

In a logistic growth model, the point of maximum growth occurs at the inflection point, where the first derivative of the function is at its maximum. For the equation given, the inflection point occurs at \( t = \frac{1}{k} \ln(b) \), where the standard form of the logistic function is \( L/(1 + b e^{-kt}) \). Here, \( L = 1000 \), \( b = 999 \), and \( k = 0.9 \).
03

Calculate the Inflection Point

Using the formula for the inflection point in logistic models, calculate: \[ t = \frac{1}{0.9} \ln(999) \]. Simplifying, we have \[ t \approx \frac{1}{0.9} \times 6.9 = \frac{6.9}{0.9} \approx 7.67 \text{ days}. \]
04

Verify Using a Graphing Utility

Plot the function \( y(t) = \frac{1000}{1 + 999 e^{-0.9t}} \) on a graphing utility. Observe the graph to confirm that the curve reaches its steepest point around \( t \approx 7.67 \) days, indicating the point of maximum rapid spread.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inflection Point
An inflection point in a logistic growth model is where the graph of the growth function changes from being concave upward to concave downward. In simpler terms, it's the point where the rate of spread is fastest.
For a flu virus spread like in our exercise, this is when the number of infected students increases most quickly. At this point, the rate of new infections is highest. After this point, even though the total number of infections continues to rise, the speed at which new students are being infected starts to slow down.
This is why identifying the inflection point is crucial when analyzing disease spread. It helps understand how quickly a virus is spreading and when it starts to slow down, which can be critical for public health measures.
Derivative
To find when a flu virus spreads most rapidly, mathematically speaking, we look at the derivative of the function modeling the spread. The derivative tells us about the rate of change of the function.
In the context of logistic growth, the derivative helps locate the point of maximum growth, i.e., the inflection point.
  • The first derivative, in our flu virus problem, gives the rate at which people become infected each day.
  • The second derivative tells us if the rate is increasing or decreasing over time.
After differentiating the logistic growth equation, the first derivative varies with time, reaching its peak at the inflection point. This peak signifies the most rapid spread of the virus, helping us pinpoint critical intervention moments during an outbreak.
Flu Virus Spread
The flu virus spread on a college campus can be accurately modeled using a logistic growth function. This is because logistical models are ideal for situations where growth starts rapidly, slows down, and eventually levels off as resources become limited or resistance builds up.
In our specific problem, the function provided represents the spread of flu among students. Initially few are infected, allowing the virus to spread quickly.
  • As more students become infected, the rate of new infections slows down because there are fewer uninfected students left or preventive measures kick in.
  • Eventually, the number of infected students levels off and approaches a maximum limit, known as the carrying capacity.
Understanding the flu virus spread pattern is essential for planning resources, such as vaccines and medical care, and implementing effective public health strategies.

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