1. The logistic function is given by:
N(t) = K * (N0 / (N0 + (K - N0) * e^(-rt)))
Where:
N(t) is the number of people infected at time t,
K is the carrying capacity (the maximum number of people that can be infected),
N0 is the initial number of people infected,
r is the growth rate, and
e is the base of the natural logarithm (approximately 2.71828).
2. We are given the following information:
K = 7250 (the long-run number of people infected),
N0 = 15 (the initial number of people infected),
r = 1.7 (the exponential growth rate), and
t is measured in weeks.
3. We want to find the number of people infected at different time points, so we will use the logistic function:
N(t) = 7250 * (15 / (15 + (7250 - 15) * e^(-1.7t)))
4. To find the number of people infected at different time points, simply substitute t with the desired time (in weeks) and solve for N(t). For example, to find the number of people infected after 1 week (t = 1), plug in t = 1 in the equation:
N(1) = 7250 * (15 / (15 + (7250 - 15) * e^(-1.7(1))))
N(1) ≈ 119.62
After 1 week, approximately 119.62 people will be infected. Similarly, you can calculate the number of people infected after 2 weeks, 3 weeks, and so on.
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.
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