The total number of people infected with a virus often grows like a logistic curve. Suppose that...

Question:

The total number of people infected with a virus often grows like a logistic curve. Suppose that 15 people originally have the virus, and that in the early stages of the virus (with time,t , measured in weeks), the number of people infected is increasing exponentially with K=1.7. It is estimated that, in the long run, approximately 7250 people become infected.

Use this information to find a logistic function to model this situation.

Logistic Population Curves:

The logistic curve is a sigmoidal curve that can be used to function. The resulting curve by plotting a graph against the time and the population is termed as the logistic curve.

The general form of the logistic curve is {eq}p=\frac{A}{1+Be^{-kt}}\\ \text{Where}\\ p=\text{Population}\\ A=\text{Population Maximum}\\ k=\text{the logistic growth rate}\\ B=\text{Ratio between Growth and Initial Population}, \ B=\frac{A-p_0}{p_0} {/eq}

Answer and Explanation:

Given,

{eq}p_0 = 15\\ A = 7250 {/eq}

Then,

{eq}\begin{align*} B&=\frac{A-p_0}{p_0}\\ B &= \frac{{7250 - 15}}{{15}}\\ B&= \frac{{7235}}{{15}}\\ B&= 482.33 \end{align*} {/eq}

If the general logistic curve is

{eq}p=\dfrac{A}{1+Be^{-kt}} {/eq}

Then our function is:

{eq}p = \dfrac{{7250}}{{1 + 482.3{e^{ - 1.7t}}}} {/eq}


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Logistics of Goods and Services

from Business 100: Intro to Business

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