A new flu virus is introduced when a stranger visits an isolated village of 8000 people. Every infected person infects two more each day. How do you write an exponential function to model the number of UNINFECTED people?

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Answer 1 · 2018-07-26T23:05:23

#u(n) = 8000-3^n*2#

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)#

#:. 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!= 0),(2 " if " n =0) :}#

#=> i(1) = 3i(0)=6=3^1*2#

# i(2) = 3i(1)=18=3^2*2#

#i(3) = 3i(2)=54 = 3^3*2#

Analogously, we can figure out that

#i(n) = 3^n*2#

Hence the number of uninfected people on day n is

#u(n) = 8000-3^n*2#