In the spring of 2003, SARS (Severe Acute Respiratory...

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) $$\begin{array}{c|c|c|c|c|c|c|c}t & P & t & P & t & P & t & P \\\hline 0 & 95 & 26 & 1108 & 54 & 1674 & 75 & 1739 \\5 & 222 & 33 & 1358 & 61 & 1710 & 81 & 1750 \\12 & 470 & 40 & 1527 & 68 & 1724 & 87 & 1755 \\19 & 800 & 47 & 1621 & & & & \\\hline\end{array}$$

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)
$$\begin{array}{c|c|c|c|c|c|c|c}t & P & t & P & t & P & t & P \\\hline 0 & 95 & 26 & 1108 & 54 & 1674 & 75 & 1739 \\5 & 222 & 33 & 1358 & 61 & 1710 & 81 & 1750 \\12 & 470 & 40 & 1527 & 68 & 1724 & 87 & 1755 \\19 & 800 & 47 & 1621 & & & & \\\hline\end{array}$$

Key Concepts

Average Rate of Change

The average rate of change is a measure of how a function’s output changes between two points relative to the change in the input variable. In many applied contexts, such as analyzing trends in epidemiological data, it helps to quantify the speed at which a variable (like the number of disease cases) increases or decreases over a specific period, effectively approximating the overall behavior of the change over an interval.

Exponential Growth

Exponential growth describes a process in which the rate of change of a quantity is proportional to the current amount, leading to rapid increases, particularly in early stages. However, although this model is useful for early, unconstrained growth phases, it is often inappropriate for long-term predictions in real-world scenarios where limitations constrain growth.

Logistic Growth Model

The logistic growth model is used to describe processes that eventually level off due to limiting factors. This model starts with near-exponential growth that slows as it approaches a maximum value, called the carrying capacity, making it a more realistic model for phenomena such as population growth or disease spread in situations where resources or susceptible individuals are limited.

Inflection Point

The inflection point of a curve is the point at which the function changes from concave upward to concave downward, or vice versa. In the context of logistic growth, this point typically represents the moment of maximum growth rate; before the inflection point, the growth is accelerating, and after the point, it is decelerating as the limiting effects become more pronounced.

Limiting Value (Carrying Capacity)

The limiting value, or carrying capacity, is the maximum value that a function, particularly in a logistic model, approaches as time progresses. In systems such as epidemiological models, it represents the upper bound of the quantity being measured, reflecting the fact that while growth may initially be rapid, it cannot continue indefinitely and will plateau due to natural constraints.

Transcript

00:01 All right, hello everyone.

00:02 So i've written down the table you're given and i've added two lines.

00:09 So the first thing to do is to take the difference between the number of cases in one period and that of the previous period.

00:18 So for instance here, 222 minus 95 gives you 127 and that basically gives you the number of cases, total number of cases between the two intervals.

00:31 So again here, 417 minus 22 gives you 248 and so on.

00:37 But now the question asks you the average number of cases in each interval and then to do that you have to divide by the number of days in each interval.

00:47 And the first interval has five days and all the other ones have seven days.

00:53 So you then divide here 127 by 5 to obtain 25 .4.

00:59 And that basically tells you that on average, you have 25 .4 new cases in the first five days.

01:08 And then the following seven days, so between the fifth day and the 12th day, you have 35 .4 cases on average as you divide 248 by 7.

01:25 And then you continue for all the different values.

01:30 So exactly it's the same thing.

01:32 You divide this number.

01:33 So, for instance, this number here, 308 by 7 to obtain 44.

01:41 So i've just written down values.

01:44 I stopped a bit before because it basically is the same thing for later values.

01:51 And that gives you the answer for question a.

01:55 Now regarding this the next question.

02:01 So when did the epidemiologists have evidence to indicate the rate of new cases become to slow? well, you then simply have to take the average rate and see when that starts to decline.

02:15 And it's more or less here, okay? because so at t equals 19.

02:23 So unfortunately, we don't have a day by day data.

02:29 We only have every weekly data set.

02:33 But we can say that approximately after 20 days, the rate of inflection started to slow down as the average number of cases per day starts to decline.

02:49 Okay.

02:51 Right.

02:52 So why would an expansion exponential model be inappropriate? so there are two ways of seeing this.

03:00 Firstly, well, as you can see, it slows down, whereas an exponential model simply grows forever and grows at increasingly fast rate.

03:12 So in a way, the exponential model works quite well in the early phases of this...

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