Problem 8 - Applied Calculus

Question

In the spring of \(2003,\) SARS (Severe Acute Respiratory Syndrome) spread rapidly in several Asian countries and Canada. Table 4.9 gives the total number, \(P\), of SARS cases reported in Hong Kong \(^{17}\) by day \(t,\) where \(t=0\) is March 17,2003. (a) Find the average rate of change of \(P\) for each interval in Table 4.9 (b) In early April \(2003,\) there was fear that the disease would spread at an ever-increasing rate for a long time. What is the earliest date by which epidemiologists had evidence to indicate that the rate of new cases had begun to slow? (c) Explain why an exponential model for \(P\) is not appropriate. (d) It turns out that a logistic model fits the data well. Estimate the value of \(t\) at the inflection point. What limiting value of \(P\) does this point predict? (e) The best-fitting logistic function for this data turns out to be $$P=\frac{1760}{1+17.53 e^{-0.1408 t}}$$ What limiting value of \(P\) does this function predict? Total number of SARS cases in Hong Kong by day \(t\) (where \(t=0\) is March 17,2003) $$\begin{array}{c|c|c|c|c|c|c|c}t & P & t & P & t & P & t & P \\\\\hline 0 & 95 & 26 & 1108 & 54 & 1674 & 75 & 1739 \\\5 & 222 & 33 & 1358 & 61 & 1710 & 81 & 1750 \\\12 & 470 & 40 & 1527 & 68 & 1724 & 87 & 1755 \\\19 & 800 & 47 & 1621 & & & & \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

(a) Find average rates: e.g., first is 25.4. (b) Rate slowed early April. (c) Exponential isn't suitable due to slowed rate. (d) Inflection around \(t=-2.531\), \(P\approx880\). (e) Limiting \(P=1760\).

Step 1: Calculate Average Rate of Change for Each Interval

To find the average rate of change of a function between two points, use the formula \( R = \frac{P(t_2) - P(t_1)}{t_2 - t_1} \), where \( t_1 \) and \( t_2 \) are the endpoints of the interval. Using the table provided, calculate this for each interval, such as from \( t=0 \) to \( t=5 \): \[ R = \frac{P(5) - P(0)}{5 - 0} = \frac{222 - 95}{5} = 25.4 \]Continue this process for all given intervals.

Step 2: Identify the Slowing Rate of New Cases

Compare the average rate of change of SARS cases over time. The rates calculated in Step 1 should show an increasing trend initially, slowing between certain intervals. Noticeable slowing suggests a potential inflection point. Examine the intervals and identify where the rate first starts to decrease, indicating that the growth of new cases is slowing.

Step 3: Assess Inappropriateness of Exponential Model

An exponential model implies continuous, ever-increasing growth without constraints. Since the calculated rates in Step 1 eventually slow down and cases do not continue to rise indefinitely (i.e., Hong Kong eventually reaches a saturation point), an exponential model is not appropriate for this dataset.

Step 4: Estimate Inflection Point in Logistic Model

The logistic model for \( P \) has the general form \( P = \frac{L}{1 + ae^{-bt}} \). The inflection point in a logistic curve occurs where the rate of growth is fastest and then begins to slow, typically around \( t = -\frac{1}{b} \log(a) \). Using the provided logistic model's parameters, calculate the inflection point.

Step 5: Determine Limiting Value from Logistic Function

In logistic functions, \( L \) represents the carrying capacity or the limiting value of \( P \). Given the function provided, \( P = \frac{1760}{1 + 17.53 e^{-0.1408 t}} \), the limiting value of \( P \) as \( t \to \infty \) is 1760. This is the highest predicted number of cases possible.

Key Concepts

Average Rate of Change

The average rate of change in a function provides insight into how quickly a quantity is increasing or decreasing over a specific interval. It's a fundamental concept in calculus that helps us understand trends within data, such as the spread of a disease.

To calculate the average rate of change, use the formula \( R = \frac{P(t_2) - P(t_1)}{t_2 - t_1} \), where \( P(t_1) \) and \( P(t_2) \) are the values at the beginning and end of the interval, respectively. This formula essentially computes the slope of the line connecting the two points and gives us the average rate at which the number of cases changes between these two points.

• If \( R > 0 \), the quantity is increasing.
• If \( R < 0 \), the quantity is decreasing.
• If \( R = 0 \), the quantity remains constant.

By using this method for SARS data, we determine the speed of change from one day to another and identify whether the spread was accelerating or decelerating during specific timeframes.

Epidemiology Modeling

In the field of epidemiology, modeling helps us understand, track, and predict the spread of diseases. It involves mathematical frameworks that represent the behavior of disease propagation in populations. During the SARS outbreak, models were crucial in analyzing how quickly cases were rising and when they started to decrease.

One popular approach in epidemiology is using the Susceptible-Infected-Recovered (SIR) model, which segments the population into three distinct groups. Modeling helps public health officials allocate resources, implement controls, and ultimately contain diseases.

By examining the average rate of change in SARS cases, we could identify critical points where the spread began to slow down. This allowed epidemiologists to assess the effectiveness of interventions and gauge the trajectory of the outbreak. Including these analyses as part of epidemiology modeling can lead to more effective responses during future outbreaks.

Logistic Growth Model

The logistic growth model is a crucial tool in understanding populations with limits. Unlike exponential models, which suggest unlimited growth, logistic models foresee a saturation point. This is often more realistic as many factors can limit growth, like resource availability or maximum capacity.

The logistic model is expressed as \( P(t) = \frac{L}{1 + ae^{-bt}} \), where \( L \) is the limiting value, \( a \) and \( b \) are constants, and \( t \) is time.

• The inflection point is significant in the logistic model, representing the moment where growth shifts from accelerating to decelerating. Mathematically, it's given by \( t = -\frac{1}{b} \log(a) \).
• The parameter \( L \) denotes carrying capacity, forecasting the maximum number of cases.

Modeling the SARS outbreak with logistic growth provided a realistic prediction about the eventual number of total cases, showcasing how mathematical models can aid in understanding epidemics.

Exponential vs Logistic Models

Exponential and logistic models are both used to describe growth, but they apply to different scenarios. For unrestricted growth, like bacteria in a petri dish with unlimited nutrients, the exponential model fits well. However, it does imply continuous, infinite growth.

The general form of an exponential model is \( P(t) = P_0e^{rt} \), with \( P_0 \) being the initial value and \( r \) the growth rate. This is not suitable for human scenarios like the SARS outbreak since populations face constraints.

The logistic model, \( P(t) = \frac{L}{1 + ae^{-bt}} \), contrasts by involving a carrying capacity, \( L \). This means it predicts reaching a maximum possible number, which matches real-life growth patterns where resources or conditions limit expansion.

• Exponential models are ideal for theoretical growth situations without limits.
• Logistic models are more applicable to real-world scenarios with natural saturation points.

Therefore, recognizing whether to use an exponential or logistic model is essential for accurate forecasting and understanding of any population's growth dynamics.