Problem 47 - Calculus and its applications

Question

Spread by skin-to-skin contact or via shared towels or clothing, methicillin- resistant Staphylococcus aureus (MRSA) can easily infect growing numbers of students at a university. Left unchecked, the number of cases of MRSA on a university campus weeks after the first 9 cases occur can be modeled by $$ N(t)=\frac{568.803}{1+62.200 e^{-0.092 t}} $$ a) Find the number of infected students beyond the first 9 cases after 3 weeks, 40 weeks, and 80 weeks. b) Find the rate at which the disease is spreading after 20 weeks. c) Explain why an unrestricted growth model is inappropriate but a logistic equation is appropriate for this situation. Then use a calculator to graph the equation.

Step-by-Step Solution

Answer

After 3 weeks: 17 cases; 40 weeks: 568 cases; 80 weeks: 569 cases. The rate at 20 weeks is approximately 2.66 cases per week. Logistic model reflects limited growth to max capacity.

Step 1: Calculate Infections After 3 Weeks

To find the number of infected students after 3 weeks, substitute \( t = 3 \) into the function \( N(t) = \frac{568.803}{1+62.200 e^{-0.092 t}} \). Calculate \( e^{-0.092 \times 3} \) first, then solve the fraction. Substituting \( t=3 \):\[ N(3) = \frac{568.803}{1+62.200 e^{-0.276}} \]Calculate \( e^{-0.276} \) and then complete the fraction operations to find \( N(3) \).

Step 2: Calculate Infections After 40 Weeks

Using the same method as Step 1, substitute \( t = 40 \) into the function. Substitute and calculate:\[ N(40) = \frac{568.803}{1+62.200 e^{-3.68}} \]Calculate \( e^{-3.68} \), then perform the arithmetic operations to solve for \( N(40) \).

Step 3: Calculate Infections After 80 Weeks

Substitute \( t = 80 \) into the equation.\[ N(80) = \frac{568.803}{1+62.200 e^{-7.36}} \]First, calculate \( e^{-7.36} \) and then solve the fraction to find \( N(80) \).

Step 4: Calculate the Rate of Spread After 20 Weeks

To find the rate of spread, take the derivative of \( N(t) \) to get \( N'(t) \). The derivative of the logistic function \( N(t) = \frac{L}{1 + Ae^{-kt}} \) is given by \( N'(t) = \frac{kLAe^{-kt}}{(1 + Ae^{-kt})^2} \).Here, \( L = 568.803 \), \( A = 62.200 \), \( k = 0.092 \). Substitute \( t = 20 \) into \( N'(t) \). Calculate \( e^{-0.092 \times 20} \) first, and then apply it to find \( N'(20) \).

Step 5: Explain Why Logistic Model is Appropriate

An unrestricted exponential model assumes unlimited growth, which is not realistic for a university population as there are finite students. A logistic model accounts for a carrying capacity, in this case, 568.803, which represents the maximum potential number of infections given constraints such as population size and natural recovery processes. The logistic model considers the slowing growth as more students get infected until saturation.

Key Concepts

MRSA Spread Among University Students

Methicillin-resistant Staphylococcus aureus (MRSA) is a type of bacteria resistant to many antibiotics. It's a growing concern, especially in crowded settings like universities where students are in close contact through shared items and living spaces.
This makes MRSA a prime candidate for study using models that predict disease spread.

When an initial infection appears, like the 9 cases mentioned, it can escalate if unchecked.
The rate of infection among large groups, such as a university campus, can be modeled mathematically to forecast future infections.

• Initially, MRSA spreads rapidly due to a lack of immunity in the population.
• Over time, interventions or natural limits, known as carrying capacities, hinder the continual growth of cases.

Such models help public health officials implement strategies to mitigate MRSA outbreaks by predicting critical points to intervene, thus protecting vulnerable student populations.

Differential Equations in Modeling MRSA Spread

Differential equations are equations involving rates of change, and they're crucial to understanding phenomena where quantities change over time.
The spread of MRSA is modeled using a particular type of differential equation known as a logistic growth equation.

This equation is \( N(t)=\frac{L}{1+Ae^{-kt}} \), where:

• \( L \) is the carrying capacity or maximum number of cases,
• \( A \) is a constant related to initial conditions,
• \( k \) is the growth rate of the infection.

The logistic growth model reflects real-world conditions, where growth slows as it reaches a maximum capacity. In the case of MRSA, this reflects limited resources like susceptible hosts or intervention measures like antibiotics.
By solving these equations, we can calculate specific values such as the number of cases at any given time, as well as the rate at which the disease is spreading.
This is a powerful tool for planning public health responses.

Concept of Carrying Capacity in Disease Models

Carrying capacity is a fundamental concept in ecology and mathematical modeling. It represents the maximum population size of a species that an environment can sustain indefinitely.
For an infection like MRSA on a university campus, carrying capacity is effectively the limit of how many can become infected before the spread naturally begins to slow and stabilize.

This concept is crucial because:

• It accounts for factors like limited number of susceptible individuals.
• It considers resources such as medical care and space.
• It predicts a slowing of spread as population reaches this maximum.

In the logistic growth model used for MRSA, the carrying capacity is 568.803, indicating the theoretical maximum number of individuals that can be infected under current conditions.
This understanding helps health authorities determine when and how aggressively to deploy interventions to manage and reduce the infection rate among students.