Question

bilogy a new flu virus is introdused when a stranger visits an isilated vilige of 8000 people. Every infected person infect 2 more each day. write the exponential function to model the number of uninfected people.

Answer 1

Answer: u(n)=8000−3n⋅2

Step-by-step explanation:

Let i(n) be the number of infected people and u(n) the number of uninfected people, both on the n-th day.

Let x(n) be the number of people who got infected on the n-th day. The relation between i(n) and x(n) is given by:

i(n)=i(n−1)+x(n)

Let's assume the stranger leaves before the next day.

Taking the day the stranger arives as the 0-th day and assuming he infects two people on that day, then

i(0)=x(0)=2

u(0)=8000−2=7998

The two people infected now will also infect two other people, meaning that:

x(1)=2i(0)=4

i(1)=i(0)+x(1)=6

u(1)=7998−x(1)=7994=8000−i(1)

Some more relations become clear:

u(n)=u(n−1)−x(n)

u(n)=8000−i(n)

As every person infects two people per day, then

x(n)=2i(n−1)

Therefore

i(n)=i(n−1)+2i(n−1)=3i(n−1)

This a recurrence relation for the number of infected people at a given day:

i(n)={3i(n−1)

if n is not equal to 2

if n=0

Implies

i(1)=3i(0)=6=31⋅2

i(2)=3i(1)=18=32⋅2

i(3)=3i(2)=54=33⋅2

Analogously, we can figure out that

i(n)=3n⋅2

Hence the number of uninfected people on day n is

u(n)=8000−3n⋅2