Problem 4 - Applied Calculus

Question

(a) In a school of 150 students, one of the students has the flu initially. What is \(I_{0}\) ? What is \(S_{0}\) ? (b) Use these values of \(I_{0}\) and \(S_{0}\) and the equation $$\frac{d I}{d t}=0.0026 S I-0.5 I$$ to determine whether the number of infected people initially increases or decreases. What does this tell you about the spread of the disease?

Step-by-Step Solution

Answer

Initially, the number of infected decreases, indicating the spread of the disease is not increasing.

Step 1: Define Initial Values

Identify the initial conditions based on the information given. \( I_0 \) represents the number of initially infected students. Since one student has the flu initially, \( I_0 = 1 \). The total number of students is 150, so the number of susceptible students initially, \( S_0 = 150 - I_0 = 149 \).

Step 2: Substitute Initial Values into the Differential Equation

Use the given differential equation \( \frac{dI}{dt} = 0.0026 S I - 0.5 I \). Substitute \( I_0 = 1 \) and \( S_0 = 149 \) into the equation to analyze the initial change in the number of infected students. Substitute to get \( \frac{dI}{dt} = 0.0026 \times 149 \times 1 - 0.5 \times 1 \).

Step 3: Calculate the Initial Change in I

Compute the right side of the substituted differential equation. This gives \( \frac{dI}{dt} = 0.0026 \times 149 - 0.5 \). Calculate this to see if the result is positive or negative. \(0.0026 \times 149 = 0.3874\), so \( \frac{dI}{dt} = 0.3874 - 0.5 = -0.1126 \).

Step 4: Analyze the Result

Since \( \frac{dI}{dt} = -0.1126 \), the change in the number of infected students \( I \) is negative initially. Thus, the number of infected students decreases at the beginning. This implies that initially, the spread of the disease is decreasing, meaning the infection is not spreading initially.

Key Concepts

Initial Conditions

Initial conditions in a mathematical model are like the starting line in a race. They provide the specific values needed to begin solving a differential equation problem. For this model, the initial conditions are crucial because they determine how the disease dynamics begin.

• We denote \(I_0\) as the initial number of infected individuals. In the exercise, there is 1 student who has the flu at the start, so \(I_0 = 1\).
• The initial number of susceptible individuals, \(S_0\), includes all those who could potentially be infected. Here, \(S_0 = 149\) because there are 150 students in total, minus the one already infected.

Initial conditions set the stage for how we anticipate the disease to spread through the student body initially. They are like the baseline numbers that all predictive calculations are built upon. By understanding these, we can effectively use the differential equations to predict future outcomes.

Infectious Disease Modeling

Infectious disease modeling is an approach used to understand how diseases like the flu spread through populations. It uses mathematical frameworks to simulate the interactions between individuals within a group, helping predict future outbreaks and inform public health strategies.
One common method of modeling is through the use of differential equations, which represent rates of change. For instance, the differential equation given in the exercise:\[\frac{dI}{dt} = 0.0026 SI - 0.5 I\]This equation represents how the number of infected people changes over time based on:

• The interaction between susceptible and infected individuals (\(0.0026 SI\)), which can increase infections.
• The rate of recovery or removal from the infectious state (\(-0.5 I\)).

By incorporating initial conditions into this equation, we can evaluate the first rate of change, giving insights into whether the disease will initially spread or decline.

Susceptible-Infected Model

The Susceptible-Infected (SI) model is one of the simplest forms of infectious disease modeling. It divides the population into two groups:

• **Susceptible (S):** Individuals who can catch the infection.
• **Infected (I):** Individuals who are currently infected and can spread the disease.

In our exercise, the SI model uses the differential equation that relates to the initial number of susceptible and infected students. The aim is to predict how the infection progresses.
The SI model assumes that once infected, individuals remain infected. However, in practice, other models can include additional compartments such as "Recovered" or "Exposed." For educational purposes, the SI model provides a straightforward way to grasp basic epidemic dynamics and understand the initial phase of an outbreak.
By focusing on these two groups, the SI model helps us refine how quickly, or not, the disease may spread, driving strategic interventions and responses to control the outbreak effectively.