91Ó°ÊÓ

As an epidemic spreads through a population, the number of infected people, \(I\), is expressed as a function of the number of susceptible people, \(S,\) by \(I=k \ln \left(\frac{S}{S_{0}}\right)-S+S_{0}+I_{0}, \quad\) for \(k, S_{0}, I_{0}>0\) (a) Find the maximum number of infected people. (b) The constant \(k\) is a characteristic of the particular disease; the constants \(S_{0}\) and \(I_{0}\) are the values of \(S\) and \(I\) when the disease starts. Which of the following affects the maximum possible value of \(I ?\) Explain. The particular disease, but not how it starts. How the disease starts, but not the particular disease. Both the particular disease and how it starts.

Step by step solution

01

Set up functions and constants

We start with the given function: \[ I = k \ln\left(\frac{S}{S_0}\right) - S + S_0 + I_0 \] Here, \(k\), \(S_0\), and \(I_0\) are positive constants.

02

Find the derivative of I with respect to S

To find extrema (maximum or minimum points), we need to calculate the derivative of \(I\) with respect to \(S\).\[ \frac{dI}{dS} = \frac{k}{S} - 1 \]

03

Set the derivative to zero to find critical points

To find critical points, set \(\frac{dI}{dS} = 0\):\[ \frac{k}{S} - 1 = 0 \]Solve for \(S\):\[ \frac{k}{S} = 1 \] \[ S = k \] This gives us a critical point at \(S = k\).

04

Determine if the critical point is a maximum

To determine if this critical point is a maximum, check the second derivative:\[ \frac{d^2I}{dS^2} = -\frac{k}{S^2} \]Evaluate at \(S = k\):\[ \frac{d^2I}{dS^2} = -\frac{k}{k^2} = -\frac{1}{k} \]< 0Since the second derivative is negative, \(S=k\) is a maximum for \(I\).

05

Calculate the maximum number of infected people

Substitute \(S = k\) back into the expression for \(I\):\[ I_{max} = k \ln\left(\frac{k}{S_0}\right) - k + S_0 + I_0 \] This calculates the maximum.

06

Address which constants affect the maximum

The maximum value of \(I\) is affected by \(k\) and \(S_0\) in the equation for \(I\). Hence, both the way the disease starts (\(S_0\)) and the characteristic of the disease (\(k\)) influence the peak.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Epidemic Model

An epidemic model is a mathematical way to represent the spread of a disease within a population. It helps in understanding how quickly a disease can spread and reach a peak. By observing factors such as the number of susceptible and infected individuals, we can predict the future course of the epidemic. In our exercise, the model is represented with the function \[ I = k \ln\left(\frac{S}{S_0}\right) - S + S_0 + I_0 \] where:

• \(S\) is the number of susceptible individuals.
• \(I\) is the number of infected individuals.
• \(k\) is a positive constant specific to the disease.
• \(S_0\) and \(I_0\) represent the initial values of \(S\) and \(I\) respectively.

This model captures how the susceptibles contribute to the growth in the number of infected individuals. It also includes parameters that define how an epidemic starts, useful for planning interventions.

Differentiation

Differentiation is a fundamental concept in calculus, used to determine the rate at which one quantity changes with respect to another. In the context of our epidemic problem, differentiation helps in finding the point where the number of infected individuals reaches a maximum. When we differentiate the function of \(I\) with respect to \(S\), we aim to find how the infection level changes as the number of susceptible individuals shifts. The derivative is determined as follows:\[\frac{dI}{dS} = \frac{k}{S} - 1\]This derivative represents the slope of the function. Setting it to zero helps identify the critical points, which are candidates for maximum or minimum values.

Critical Points

Critical points in a function occur where the derivative is zero or undefined. They are significant as they may represent maximum or minimum values of the function. In the epidemic model, setting \(\frac{dI}{dS} = 0\) gives:

• \[\frac{k}{S} - 1 = 0\]
• Solving this yields \(S = k\)

The value \(S = k\) is therefore a critical point in this model. Understanding these points is crucial, as they help determine the peak infection level which is essential for planning health responses during an epidemic.

Second Derivative Test

The second derivative test is a method used to determine if a critical point is a maximum or a minimum. After finding a critical point, the second derivative is computed to discover the curvature of the graph at that point. For our epidemic model:\[ \frac{d^2I}{dS^2} = -\frac{k}{S^2} \]Evaluating it at the critical point \(S = k\), we find:\[ \frac{d^2I}{dS^2} = -\frac{1}{k} \]The resulting negative value indicates the graph is concave down at \(S = k\), confirming it is a maximum point. Thus, this analysis helps determine the peak in the number of infections, which is crucial for effective epidemic surveillance and control.