Problem 48 - Precalculus Mathematics for Calculus

Question

The rate \(r\) at which a disease spreads in a population of size \(P\) is jointly proportional to the number \(x\) of infected people and the number \(P-x\) who are not infected. An infection erupts in a small town that has population \(P=5000\). (a) Write an equation that expresses \(r\) as a function of \(x .\) (b) Compare the rate of spread of this infection when 10 people are infected to the rate of spread when 1000 people are infected. Which rate is larger? By what factor? (c) Calculate the rate of spread when the entire population is infected. Why does this answer make intuitive sense?

Step-by-Step Solution

Answer

(a) \( r = k \cdot x \cdot (5000 - x) \); (b) When 1000 are infected, the rate is larger by a factor of about 80.16; (c) When all are infected, the rate is 0.

Step 1: Understanding joint proportionality

The problem states that the rate \( r \) is jointly proportional to \( x \), the number of infected people, and \( P-x \), the number of non-infected people. Joint proportionality means \( r = k \cdot x \cdot (P-x) \), where \( k \) is the constant of proportionality.

Step 2: Writing the equation for r as a function of x

Given that the population is \( P = 5000 \), substitute \( P \) into the joint proportionality equation: \( r = k \cdot x \cdot (5000 - x) \). This equation expresses \( r \) as a function of \( x \).

Step 3: Calculating the rate when 10 people are infected

Substitute \( x = 10 \) into the equation: \( r = k \cdot 10 \cdot (5000 - 10) = 10k \cdot 4990 = 49900k \).

Step 4: Calculating the rate when 1000 people are infected

Substitute \( x = 1000 \) into the equation: \( r = k \cdot 1000 \cdot (5000 - 1000) = 1000k \cdot 4000 = 4000000k \).

Step 5: Comparing the rates

Compare \( 49900k \) with \( 4000000k \). Clearly, when 1000 people are infected, the rate of spread \( 4000000k \) is larger than when 10 people are infected. The factor by which it is larger is \( 4000000 / 49900 \approx 80.16 \).

Step 6: Calculating the rate when the entire population is infected

If the entire population is infected, \( x = 5000 \). Substitute this into the equation: \( r = k \cdot 5000 \cdot (5000 - 5000) = 0 \).

Step 7: Intuitive explanation

When the entire population is infected, there are no more people left to infect, hence the rate of spreading \( r \) is zero. This makes intuitive sense because no new infections can occur.

Key Concepts

Rate of Infection

The rate of infection refers to how quickly a disease spreads within a given population. In this context, it is particularly important to understand how we mathematically define and calculate this rate.
The exercise from the textbook uses the concept of joint proportionality. This means that the rate of infection, denoted as \( r \), depends on two factors:

• The number of infected people, represented by \( x \).
• The number of those who are not infected, represented by \((P-x)\), where \(P\) is the total population size.

Joint proportionality implies we use an equation of the form \( r = k \cdot x \cdot (P-x) \), where \( k \) is a constant that remains the same regardless of \( x \). This reflects real-world scenarios where the more infected people there are, along with those still healthy, the faster the disease is likely to spread. It's a tug-of-war between those spreading the disease and those who are susceptible.

Population Dynamics

Population dynamics is a branch of life sciences that studies the size and age composition of populations as dynamical systems. When we examine how a disease affects a population, knowing the dynamics is crucial.
In this model, we track how the infected and non-infected groups change over time.

• At first, when few people are infected, the rate of spreading is relatively slower since there are fewer infectious contacts.
• As more people get infected, more opportunities for the disease to spread arise, which greatly increases the rate of infection.
• Ultimately, if everyone is infected, the rate of infection falls to zero, as there is no one left to spread the disease to.

The textbook example shows different infection rates with 10 versus 1000 infected individuals.
The factor by which the infection rate is greater when 1000 people are infected compared to 10 people, highlights how rapidly the dynamics can change with an increasing number of infections. Understanding these shifts is essential for effective management and intervention in disease outbreaks.

Mathematical Modeling

Mathematical modeling in the context of infectious diseases provides a way to predict and analyze the behavior of disease spread through a population. The core idea is to use equations, like the one for rate of infection, to represent real-world phenomena accurately.

• It highlights dependencies between parameters such as the number of infected people and the total population.
• The model also involves constants like \( k \), the constant of proportionality, critical for scaling the model to match real-life infection rates.

In our example, the equation \( r = k \cdot x \cdot (5000-x) \) is simple yet powerful – allowing us to explore different what-if scenarios:

• Calculating the rate of spread at various infection levels
• Understanding when the spread of the disease will naturally slow down or stop

By using mathematical modeling, health officials and researchers can improve their understanding and provide valuable insights for strategic planning in public health efforts.