Homework 12

Yuri

The equation given:

P(t)={\frac {1}{1+e^{-t}}}

We can set up this equation to be more specific:

P(t)={\frac {K}{1+re^{-at}}}

To change the height of the horizontal asymptote on the right:

Change the value of K

To change the y-intercept to any number between 0 and K

Change the coefficient of $e^{-}t$
Change the value of r

BONUS

To change slope of the curved part, slope can go from very close to zero to almost vertical:

Change the coefficient in front of t
Change the value of a

We have an excel graph showing these changes:

MODEL- Tanya

Another model which uses the same logistic formula is the growth of number of H1N1 cases in 2009. H1N1 virus is still having an exponential growth but the overall population of the world is still about the same, hence the ratio is getting smaller and smaller. As the population will remain the same, eventually this disease will kill too many people and the growth will taper off. The maximum will be reached at this is called the Stationary phase.

P(t)={\frac {K}{1+re^{-at}}}

K= the height of the horizontal asymptote which also the carrying capacity
a= the rate by which the growth is occurring
1= Initial number of people
T= Time (days/ months/years)

Eg if K= 1000
A= 3.02%
R=1
At what value of t will the population affected by H1N1 will reach 95% of the carrying capacity

P(t)={\frac {1000}{1+1e^{-0.03t}}}

{950+950e^{-0.03t}}=1000

{950e^{-0.03t}}=50

e^{-0.03t}={\frac {1}{19}}

(lne^(-0.03t))=ln(1/19)
(-0.03t)=ln(1/19)
t=98.15