The total number of people infected with a virus often grows like a logistic curve. Suppose that 15 people originally have the virus, and that in the early stages of the virus (with time,t , measured in weeks), the number of people infected is increasing exponentially with K=1.7 . It is estimated that, in the long run, approximately 7250 people become infected.
To determine the equation that describes the growth of the virus, we can use the logistic growth model. The logistic growth model is used when a population starts with exponential growth and eventually levels off due to limiting factors. The equation is given by:
P(t) = c / (1 + a * e^(-b * t))
Where:
P(t) is the number of people infected at time t
c is the maximum number of people infected (asymptote)
a is the initial growth rate
b is the rate at which the growth rate decreases
We are given that the initial number of people infected is 15 and the asymptote (long-term infected population) is 7250. We need to find the values of a and b to complete the equation.
Let's use the given information to solve for a and b.
1. Initial condition: P(0) = 15
Plugging this into the equation:
15 = c / (1 + a * e^(-b * 0))
15 = c / (1 + a * e^0)
15 = c / (1 + a)
2. Asymptote: P(t) approaches 7250 as t approaches infinity
So, when t approaches infinity, P(t) = 7250
7250 = c / (1 + a * e^(-b * ∞))
7250 = c / (1 + a * e^(-∞))
7250 = c / (1 + a * 0)
7250 = c / 1
7250 = c
Now we have two equations:
15 = c / (1 + a)
7250 = c
Substituting the value of c from the second equation into the first equation:
15 = 7250 / (1 + a)
Simplifying:
15 + 15a = 7250
15a = 7250 - 15
15a = 7235
a = 7235 / 15
a ≈ 482.33
Now that we have the value of a, we can calculate b using the natural logarithm:
b = ln((1 + a) / a)
b = ln((1 + 482.33) / 482.33)
b ≈ ln(1.00207)
b ≈ 0.00107
Finally, we have determined the logistic growth equation for the virus:
P(t) = 7250 / (1 + 482.33 * e^(-0.00107 * t))
This equation describes how the number of people infected with the virus changes over time.