Question

Eliminating a Disease Suppose that in any given year, the number of cases of a disease is reduced by 20\(\% .\) If there are \(10,000\) cases today, how many years will it take (a) to reduce the number of cases to \(1000\)? (b) to eliminate the disease; that is, to reduce the number of cases to less than \(1\)?

Short answer

It will take approximately \(11.5\) years to reduce the number of cases to \(1000\) and more than \(23\) years to eliminate the disease.

Step-by-step solution

Identify the initial number of cases and the rate of reduction

The initial number of cases is given as \(10,000\) and the rate of reduction is given as \(20\%\), or \(0.20\) in decimal form.

Formulate exponential decay formula

An exponential decay formula can be expressed as \(N = N_0(1 - r)^t\) where \(N\) is the final amount, \(N_0\) is the initial amount, \(r\) is the rate of reduction, and \(t\) is the time period. Here, \(N_0 = 10,000\) and \(r = 0.20\).

Find the time to reduce the number of cases to 1000

Substitute \(N = 1000\) into the formula and solve for \(t\). This results in the equation \(1000 = 10,000 (1 - 0.20)^t\). Solving this equation for \(t\) gives \(t = \frac{log(0.1)}{log(0.8)} \approx 11.5\) years.

Find the time to eliminate the disease

Substitute \(N < 1\) into the formula and solve for \(t\). This results in the inequality \(1 > 10,000 (1 - 0.20)^t\). Solving this inequality for \(t\) gives \(t > \frac{log(0.0001)}{log(0.8)} \approx 23.3\) years. Therefore, it will take more than 23 years to eliminate the disease.

Key concepts

Exponential Decay Formula

When analyzing diseases and their cases, understanding the exponential decay formula is crucial. Exponential decay describes the process of reducing an amount by a consistent percentage rate over time. In epidemiology, it can be used to model how the number of disease cases decreases.

The general formula for exponential decay is given by:
\[ N = N_0(1 - r)^t \]
where:

• \( N \) is the final amount or, in the context of disease, the remaining number of cases.
• \( N_0 \) is the initial amount or the initial number of cases.
• \( r \) is the rate of reduction, expressed as a decimal (for a 20% reduction, \( r = 0.20 \)).
• \( t \) is the time period over which the reduction occurs.

To calculate the number of years to reduce disease cases from 10,000 to 1,000 using a 20% annual reduction rate, you apply the formula with \( N_0 = 10,000 \), \( N = 1,000 \), and \( r = 0.20 \). The variable you solve for is \( t \), the time it will take to reach the desired reduced number of cases.

Logarithmic Equations

Solving the exponential decay problem requires understanding logarithmic equations, as they are the inverse operations of exponentials. When you are given an equation like \( 1000 = 10,000(1 - 0.20)^t \) and asked to solve for \( t \), logarithms become the tool of choice.

The base of the logarithm you use to solve for \( t \) depends on the base of the exponential equation. In the case of our exercise, it’s most convenient to use the natural logarithm or base 10 logarithm:

\[ t = \frac{log(\frac{N}{N_0})}{log(1 - r)} \]
Applying logarithms to both sides of the equation factors out \( t \), thus allowing you to solve for the unknown time. Be mindful that when your solution for \( t \) involves an inequality (like when finding the time to reduce cases to less than one), the inequality direction must be maintained throughout your calculations.

Rate of Reduction

The rate of reduction is a key component in modeling how quickly something decreases. In our disease case exercise, the reduction is consistent—20% every year. It's important to note that 'rate of reduction' and 'reduction rate' are two phrases describing this concept.

This rate of reduction determines how the number of cases dwindles over time and is represented by \( r \) in the exponential decay formula. In infections or disease contexts, higher rates of reduction are favorable as they signify more rapid decreases in cases.

When interpreting results, remember that while the mathematical model may predict a certain timeline for disease eradication, actual outcomes can differ due to various factors such as changes in infection rates, population behaviors, and intervention strategies. This concept of the rate of reduction extends beyond diseases and is applicable to topics like radioactive decay, depreciation of assets, and population decline.