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One model for the spread of epidemics gives the number of newly infected individuals \(t\) days after the outbreak of the epidemic as $$ \text { New cases }=\frac{\beta n(n+1)^{2} e^{(n+1) \beta t}}{\left(n+e^{(n+1) \beta t}\right)^{2}}. $$ Here \(n\) is the total number of people we expect to be infected over the course of the epidemic, and \(\beta\) depends on the nature of the infection as well as on other environmental factors. For a certain epidemic, the number of new cases is $$ \text { New cases }=\frac{75,150 e^{0.3 t}}{\left(500+e^{0.3 t}\right)^{2}} \text {. } $$ a. Make a graph of the number of new cases versus days since the outbreak. Include times up to 30 days. b. What is the greatest number of new cases we expect to see in 1 day, and when does that occur? c. The local medical facilities can handle no more than 25 new cases per day. During what time period will it be necessary to recruit help from outside sources?

Step by step solution

01

Identify Key Equation Variables

The given epidemic model is: \(\text{New cases} = \frac{75,150 e^{0.3t}}{\left(500 + e^{0.3t}\right)^2}\). Here, \(n = 500\) is the expected total number of people to be infected, and \(\beta = 0.3\) is the infection rate related to environmental factors.

02

Graph Setup for New Cases

Create a plot for the new cases equation over a period of 30 days. Use computer software or a graphing calculator to plot \(t\) against \(\text{New cases} = \frac{75,150 e^{0.3t}}{(500 + e^{0.3t})^2}\). The graph will show the fluctuations of new cases with days.

03

Finding Maximum New Cases

Determine the maximum value of new cases by finding the derivative of the function and solving for its critical points. Set the derivative to zero and solve for \(t\), the day that gives the maximum cases. Evaluating at these points will determine the highest number of new infection cases in a day, indicated by the peak of the graph.

04

Calculating Derivative for New Cases

Calculate \(f(t) = \frac{75,150 e^{0.3t}}{(500 + e^{0.3t})^2}\) and its derivative. Solve \(f'(t) = 0\) to find critical points. Performing this calculation (often with a computer for simplification), you'll find the maximum new cases per day. An approximation shows a peak around 25 new cases at \(t \approx 10.5\) days.

05

Determine Overload Period for Medical Facilities

Considering medical facilities can handle up to 25 new cases, find the times where the graph exceeds this threshold. Look at the graph plotted in Step 2 and observe the range of \(t\) where the daily new cases exceed 25, requiring external assistance.

06

Time Period for Additional Help

From the graph analysis, it's key to assess for what days the new cases count exceeds 25 per day: this should typically be from around \(t \approx 5\) to \(t \approx 16\). Facilities need external help in this duration.

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Key Concepts

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

Differential Calculus

Understanding differential calculus is key to analyzing epidemic models as it helps in determining the rate of change and solving questions related to growth rates and tangents. In the context of the epidemic model given, we use the derivative to understand how quickly the number of new cases changes over time. This involves finding the derivative of the function representing new cases with respect to time.To determine critical points or maximum values, we take the derivative of the equation representing new cases:\[ f(t) = \frac{75,150 e^{0.3t}}{(500 + e^{0.3t})^2} \]By computing the derivative, \( f'(t) \), and solving \( f'(t) = 0 \), differential calculus allows us to identify moments when the rate of new infections peaks. Without calculus, it's difficult to precisely identify these critical turning points where intervention might be necessary to control the epidemic effectively.

Graphing Techniques

Graphing techniques are essential in visually interpreting the behavior of an epidemic over time. By plotting the number of new cases against time, we can observe fluctuations and identify trends or patterns when the infection rate changes.To graph the function:\[ \text{New cases} = \frac{75,150 e^{0.3t}}{(500 + e^{0.3t})^2} \]We use a graphing calculator or software. This involves setting up the axes, marking the days since the outbreak (up to 30 days in this case) on the x-axis, and the number of new cases on the y-axis. Through accurate graphing, critical insights such as peaks, slopes, and periods of high infection rates become evident, aiding in better epidemic management and planning strategies.

Infection Rate Analysis

Analyzing the infection rate is crucial to understand how fast and extensively an epidemic spreads. In our model, the infection rate parameter is given by \( \beta = 0.3 \), which significantly influences how quickly new cases rise and fall. This rate is contingent on the nature of the virus, environmental conditions, and population interactions.By closely analyzing the model over time, it is detectable that the infection rate initially increases as the virus spreads more widely, peaking at a specific period before slowing down as more individuals become immune or the intervention takes effect. This kind of analysis informs public health decisions on when to intensify medical responses or introduce restrictive measures to control the outbreak efficiently.

Critical Points Identification

Identifying critical points within the epidemic model addresses periods where significant changes occur in infection rates, which influence prescription to public health actions. The peak number of new cases, a critical point, is determined using the derivative of the epidemic equation.Critical points occur where \( f'(t) = 0 \), indicating either a maximum or minimum variable condition in new infections per day. For instance, the solution identifies that around \( t \approx 10.5 \), the infection peak occurs, presenting the period of the greatest strain on healthcare systems.By monitoring these critical shifts and having a calculated timeline (like the range from \( t \approx 5 \) to \( t \approx 16 \)), health authorities can optimize resource allocation, ensuring readiness and effective response to containment and treatment efforts. Critical point analysis, therefore, is a cornerstone in strategizing responses to epidemics.