Problem 42 - Algebra and Trigonometry

Question

Spread of a Disease 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 with 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) Rate is 80 times larger with 1000 infected. (c) The rate is 0 when everyone is infected.

Step 1: Define the Relation

From the problem description, we know that the rate \( r \) is jointly proportional to the number of infected people \( x \) and the number of uninfected people \( P - x \). Joint proportionality implies a mathematical relationship given by \( r = k \cdot x \cdot (P - x) \), where \( k \) is the constant of proportionality.

Step 2: Equation for Rate in Terms of Infected Individuals

Substitute the given population size to express \( r \). For \( P = 5000 \), the equation becomes \( r = k \cdot x \cdot (5000 - x) \). This is the equation of \( r \) in terms of \( x \).

Step 3: Calculate Rates for Different Infections

To compare the rate of spread at 10 and 1000 infections, substitute \( x = 10 \) and \( x = 1000 \) into the equation: For \( x = 10 \), \( r = k \cdot 10 \cdot (5000 - 10) = 49900k \). For \( x = 1000 \), \( r = k \cdot 1000 \cdot (5000 - 1000) = 4000000k \). Comparing the results, \( 4000000k \) is larger than \( 49900k \).

Step 4: Rate Comparison and Factor

To find the factor, divide the rates: \[ \text{Factor} = \frac{4000000k}{49900k} = \frac{4000000}{49900} \approx 80.16. \] The rate with 1000 infected is approximately 80 times larger than with 10 infected.

Step 5: Rate of Spread When Entire Population is Infected

If the entire population is infected, \( x = 5000 \) and therefore \( P - x = 0 \). Substitute in the equation: \[ r = k \cdot 5000 \cdot (5000 - 5000) = k \cdot 5000 \cdot 0 = 0. \] The rate of spread is 0 when everyone is infected, which makes intuitive sense as there are no more uninfected individuals to contract the disease.

Key Concepts

Rate of Spread

When discussing disease spread, the 'rate of spread' refers to how quickly a disease infects people within a population. It's important to understand that this rate is influenced by specific variables:

• The number of currently infected individuals, denoted as \( x \).
• The number of individuals not yet infected, which is \( P - x \) for a population \( P \).

When we say that the rate \( r \) is "jointly proportional," it means that \( r \) depends on the product of both these numbers. In simpler terms, if either increases, the spread rate potentially increases, assuming a constant of proportionality \( k \). Hence, the formula is:
\[ r = k \cdot x \cdot (P - x) \]
This equation shows us the intricate balance in disease dynamics, highlighting the roles of both infected and uninfected people.

Infected Population

The concept of the 'infected population' revolves around understanding how many people in a given community have contracted a disease at a given time. This number is crucial because it directly impacts the rate at which the disease continues to spread.
The infected population, denoted as \( x \), together with the uninfected group \( P - x \), forms a whole population \( P \). For a small town of 5000 people, if 10 are infected, the rate of spread can be calculated using our previous equation:

• Infectious scenario 1: \( x = 10 \) results in \( r = k \cdot 10 \cdot (5000 - 10) = 49900k \)
• Infectious scenario 2: \( x = 1000 \) results in \( r = k \cdot 1000 \cdot (5000 - 1000) = 4000000k \)

Here, a larger infected population (1000 compared to 10) significantly increases the rate of spread, illustrating how the number of infections scales the risk dramatically.

Mathematical Relationship

Mathematical relationships in epidemiology help us make sense of how diseases spread. In the context of joint proportionality, it provides a formulaic approach to understanding such dynamics: \( r = k \cdot x \cdot (P - x) \).
This formula highlights two key insights:

• As the number of infected \( x \) increases, more people are at risk of infection, escalating the spread.
• When everyone is infected (\( x = P \)), the potential for further spread is nullified as there are no more susceptible hosts.

By analyzing these factors, we're able to calculate the specific rates and gain a deeper comprehension of disease dynamics in any given population.