A zombie virus has broken out and has infected a large portion of the world. The

Question

A zombie virus has broken out and has infected a large portion of the world. The spread of the virus can be modeled by the following function where Z is measured in billions of zombies and t is measured in years:

Z(t)= 10/ 1+2.8e^ -.0224t

a. What is the initial number of zombies?

b. What is the growth/decay rate?

c. How long will it take until 5 billion people are infected?

d. If the current population of the world is 7.6 billion people, and the current population growth rate is 1.2%; come up with a logistic function that models the growth rate, H(t) for humans.

e. How long will it take for all humans to become zombies (assuming that the growth rate for both humans and zombies continues)? The carrying capacity for humans is the same as it is for zombies.

Explanation / Answer

a) initial numbe of zombies t =0

=> Z(0) = 10/( 1 + 2.8 e0)=10/(1+2.8)= 10/3.8=2.6315 billion

b) Growth/decay rate - Take derivative

d(Z(x)/dx= d(10/(1 + 2.8e-0.0224t)/dx = -700 e0.0224/(14x + 5e0.0224)2

c) Z(t) = 5 . 109 =10/(1 + 2.8e-0.0224t)

=> 1 + 2.8e-0.0224t= 1/5.108

=> 2.8e-0.0224t= 0.000000002 - 1

=> e-0.0224t = -0.35714285

taking log

=> t = 46 years

d) Current Population H(t) = 7.6 * ( 1 + 1.2% )t= 7.6(101.2%)t=7.6(1.012)t

where t isin years and population is in billions

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