Problem 33 - Applied Calculus

Question

The rate, \(r,\) at which people get sick during an epidemic of the flu can be approximated by $$ r=1000 t e^{-0.5 t} $$ where \(r\) is measured in people/day and \(t\) is measured in days since the start of the epidemic. (a) Write an improper integral representing the total number of people that get sick. (b) Use a graph of \(r\) to represent the improper integral from part (a) as an area.

Step-by-Step Solution

Answer

(a) \( \int_{0}^{\infty} 1000t e^{-0.5t} \, dt \); (b) The area under the curve of \( r(t) \) from 0 to infinity.

Step 1: Identify the Expression for Total Sickness

The rate at which people get sick is given by the function \( r(t) = 1000 t e^{-0.5 t} \). To find the total number of people that get sick over time, we need to integrate this function over all time, from \( t = 0 \) to \( t = \infty \). This is setup as an improper integral.

Step 2: Write the Improper Integral

The integral from 0 to infinity represents the total number of people who get sick over the course of the epidemic. The improper integral is given by:\[ \int_{0}^{\infty} 1000 t e^{-0.5 t} \, dt \]

Step 3: Represent the Integral as Area on Graph

Plot the function \( r(t) = 1000 t e^{-0.5 t} \) against \( t \). The improper integral is represented by the area under this curve from \( t = 0 \) to \( t = \infty \). This area depicts the total number of people that get sick throughout the epidemic.

Key Concepts

Epidemic Modeling

Epidemic modeling plays a crucial role in understanding how diseases spread within a population. By using mathematical models, we gain insights into the dynamics of epidemics, predict future outbreaks, and devise strategies to control them effectively. In this exercise, the function \( r(t) = 1000 t e^{-0.5 t} \) models how the rate of sickness changes over time during a flu epidemic.

• The model assumes the number of new cases follows an exponential decay due to the \( e^{-0.5 t} \) term, reflecting decreased transmission over time.
• Time \( t \) is measured in days, which helps track the epidemic's progression from the start.
• Initially, the rate \( r(t) \) increases as more people get sick, but eventually decreases as fewer people are left to infect.

This information aids public health officials in making informed decisions about measures like vaccination campaigns to mitigate the epidemic's impact.

Rate Function

The rate function is a powerful mathematical tool used to describe how something changes over time. Here, the rate function \( r(t) = 1000 t e^{-0.5 t} \) tells us how many people are getting sick each day during the course of the epidemic.

• At time \( t = 0 \), the rate is zero because the epidemic has just started.
• As \( t \) increases, the rate initially grows because more people are exposed to the flu.
• After a peak, the rate begins to decline, indicating fewer new infections per day as the flu becomes controlled.

The behavior of this function is typical of many epidemics; initially, there is a rapid increase in cases, followed by a slowdown as immunity spreads or interventions take effect. Understanding this pattern is vital in modeling epidemics and predicting when the epidemic will end.

Area Under Curve

In mathematical analysis, the area under a curve can represent the accumulation of a quantity over time, such as the total number of infections during an epidemic. When we have the rate function \( r(t) = 1000 t e^{-0.5 t} \), the area under this curve from \( t = 0 \) to \( t = \infty \) corresponds to the total number of people who will contract the flu during the epidemic.

• This concept involves calculating an improper integral: \( \int_{0}^{\infty} 1000 t e^{-0.5 t} \, dt \).
• The area under the curve suggests that even long after the initial outbreak, some infections might still occur.
• Using graphical representations helps better understand how the epidemic evolves and peaks.

By evaluating this area and understanding the curve's behavior, we learn that the epidemic's total impact can be quantified, offering insights into how resources should be allocated to manage the epidemic effectively.