Chapter 7, Problem 19RE

To determine

To calculate : how long does it take for $80\%$ of the population to become infected

Answer to Problem 19RE

It takes $15$ days

Explanation of Solution

Given information : the question is

In an isolated town of $5000$ inhabitants, $160$ people have a disease at the beginning of the week and $1200$ have it at the end of the week

Calculation : let $P$ be the number of sick people

  $\begin{array}{l}\frac{dP}{dt}=rP\left(5000-P\right)\\ \frac{dP}{dt}=5000rP\left(1-\frac{P}{5000}\right)\end{array}$

This is a logistic equation carrying capacity $K=5000$

  $\begin{array}{l}P\left(t\right)=\frac{K{P}_0}{P_0+\left(K-P_0\right)e^{-5000rt}}\\ P\left(t\right)=\frac{5000\times 160}{160+\left(5000-160\right)e^{-5000rt}}\\ P\left(t\right)=\frac{20000}{121e^{-5000rt}+4}\\ P\left(t\right)=1200\&t=7\\ 1200=\frac{20000}{121e^{-5000r\times 7}+4}\\ 5000r=0.322\\ P\left(t\right)=\frac{20000}{121e^{-0.322t}+4}\\ P\left(t\right)=0.8\times 5000=4000\\ 4000=\frac{20000}{121e^{-0.322t}+4}\\ t\approx 15\end{array}$