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.
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.
To determine the logistic growth equation for the number of people infected with the virus, we need to find the growth rate and the carrying capacity.
Given:
Initial infected population (t=0): 15
Rate of exponential growth (K): 1.7
Long-term infected population (carrying capacity): 7250
The logistic growth equation is given by:
P(t) = K / (1 + A * e^(-K * t))
Where:
P(t) is the number of infected people at time t
A is a constant related to the starting population
To find the constant A, we use the initial condition P(0) = 15:
15 = K / (1 + A * e^(-K * 0))
15 = K / (1 + A)
Substituting the value of K (1.7) gives:
15 = 1.7 / (1 + A)
Now, let's solve for A:
15 + 15A = 1.7
15A = 1.7 - 15
15A = -13.3
A = -13.3 / 15
A ≈ -0.887
Now we can substitute the values of K and A into the logistic growth equation to calculate the number of infected people at any given time.
P(t) = 1.7 / (1 - 0.887 * e^(-1.7 * t))
To determine the time it takes for the number of infected people to reach the long-term population, we need to find t when P(t) = 7250:
7250 = 1.7 / (1 - 0.887 * e^(-1.7 * t))
To solve for t, we need to rearrange the equation:
1 - 0.887 * e^(-1.7 * t) = 1.7 / 7250
0.887 * e^(-1.7 * t) = 1 - (1.7 / 7250)
e^(-1.7 * t) = (1 - 1.7 / 7250) / 0.887
e^(-1.7 * t) = 0.9997641379
-1.7 * t = ln(0.9997641379)
t = ln(0.9997641379) / -1.7
Calculating this in numerical form gives:
t ≈ 104.75
Therefore, it will take approximately 104.75 weeks for the number of infected people to reach 7250.
To understand the growth of the virus, we can use a logistic curve, which models exponential growth that eventually levels off due to limited resources or other factors.
To solve the problem, we need to find the growth rate parameter, r, and the carrying capacity, K, which represents the maximum number of people that can be infected.
Given that the growth rate is K = 1.7 and the final number of infected people in the long run is approximately 7250, we can set up the following equation:
K = 1 / (1 + e^(-r*t))
Where K is the carrying capacity, r is the growth rate parameter, and t is the time in weeks.
Using the provided information, we can substitute the values:
7250 = 1 / (1 + e^(-1.7*t))
Now, we need to solve this equation for t to find out how long it takes for the number of infected people to reach the estimated value.
To solve the equation, we can first multiply both sides by the denominator to isolate the exponential:
7250 * (1 + e^(-1.7*t)) = 1
Next, divide both sides by 7250:
1 + e^(-1.7*t) = 1/7250
Now, subtract 1 from both sides:
e^(-1.7*t) = 1/7250 - 1
Simplifying the right-hand side:
e^(-1.7*t) = (1 - 7250)/7250
e^(-1.7*t) = -7249/7250
To solve for t, we need to take the natural logarithm (ln) of both sides:
ln(e^(-1.7*t)) = ln(-7249/7250)
Simplifying the left-hand side:
-1.7*t = ln(-7249/7250)
Finally, dividing both sides by -1.7:
t = ln(-7249/7250) / -1.7
Note that the solution for t may result in a complex number since the right-hand side evaluates to a negative value. Also, keep in mind that the logistic curve is an approximation, and real-world situations may have additional complexities affecting the growth of the virus.