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Example 6: Using the Logistic-Growth Model

An influenza epidemic spreads through a population rapidly, at a rate that depends on two factors: the more people who have the flu, the more rapidly it spreads, and the more uninfected people there are, the more rapidly it spreads. These two factors make the logistic model a good one to study the spread of communicable diseases. Clearly, there is a maximum value for the number of people infected: the entire population.

For example, at time [tex]$t=0$[/tex], there is one person in a community of 1,000 people who has the flu. So, in that community, at most 1,000 people can have the flu. Researchers find that for this particular strain of the flu, the logistic growth constant is [tex]$b=0.6030$[/tex].

Tasks:
1. Estimate the number of people in this community who will have had this flu after ten days.
2. Predict how many people in this community will have had this flu after a long period of time has passed.

Solution:

We substitute the given data into the logistic growth model:
[tex]\[ f(x)=\frac{c}{1+a e^{-b x}} \][/tex]



Answer :

GinnyAnswer
To tackle this problem, we will use the logistic growth model formula to predict the spread of the flu in a community. Here's a detailed step-by-step solution:

### Step 1: Given Data
We have the following given data:
- Maximum Population ([tex]\(c\)[/tex]): The maximum number of people that can have the flu, which is 1000.
- Logistic Growth Constant ([tex]\(b\)[/tex]): The constant for this particular strain of flu, which is 0.6030.
- Initial Population ([tex]\(P_0\)[/tex]): At time [tex]\(t=0\)[/tex], there is 1 person with the flu.

### Step 2: Determine the Logistic Growth Model Parameters

The logistic growth model is given by the formula:
[tex]\[ f(x) = \frac{c}{1 + a e^{-bx}} \][/tex]

Here, [tex]\(a\)[/tex] is a parameter that needs to be calculated. It depends on the initial condition where at [tex]\(t=0\)[/tex], the population [tex]\(P_0\)[/tex] with the flu is 1.

### Step 3: Calculate Parameter [tex]\(a\)[/tex]

At [tex]\(t=0\)[/tex]:
[tex]\[ P_0 = \frac{c}{1 + a e^{-b \cdot 0}} \][/tex]
[tex]\[ 1 = \frac{1000}{1 + a} \][/tex]
[tex]\[ 1 + a = 1000 \][/tex]
[tex]\[ a = 999 \][/tex]

### Step 4: Estimate Population After 10 Days

Now we need to find the number of people infected after 10 days ([tex]\(t=10\)[/tex]) using the logistic growth model:
[tex]\[ f(10) = \frac{1000}{1 + 999 e^{-0.6030 \cdot 10}} \][/tex]

Using the exponential calculation inside the formula, we get the estimated number of people after 10 days.

### Step 5: Predict the Population at Equilibrium

After a long period of time ([tex]\(t \rightarrow \infty\)[/tex]), the exponential term [tex]\( e^{-bx} \)[/tex] will approach zero because the exponent is negative and very large.

Therefore, the logistic growth equation simplifies to:
[tex]\[ \lim_{t \to \infty} f(t) = \frac{1000}{1 + 999 \cdot 0} = 1000 \][/tex]

Thus, the entire population of 1000 people will have had the flu after a long period.

### Conclusion

By following these calculations:
1. The logistic growth model parameter [tex]\(a\)[/tex] is 999.
2. The number of people in the community who will have had the flu after 10 days is approximately 293.85.
3. The number of people in the community who will have had the flu after a long period of time is 1000.

So, the results are:
- [tex]\(a = 999\)[/tex]
- Population after 10 days: [tex]\(\approx 293.85\)[/tex]
- Population at equilibrium: [tex]\(1000\)[/tex]