Question

Five mice in a stable population of 500 are intentionally infected with a contagious disease to test a theory of epidemic spread that postulates the rate of change in the infected population is proportional to the product of the number of mice who have the disease with the number that are disease free. Assuming the theory is correct, how long will it take half the population to contract the disease?

Answer 1

To determine how long it will take for half the population to contract the disease, we can set up a differential equation. The differential equation can be solved using integration and partial fraction decomposition to find an expression for the number of infected mice as a function of time. From this, we can determine the value of the constant of proportionality and calculate the time it will take for half the population to contract the disease.

Step-by-step explanation:

To determine how long it will take for half the population to contract the disease, we can set up a differential equation. Let's use 'I' to represent the number of mice infected with the disease, and 'N' to represent the total population of mice. According to the theory, the rate of change of the infected population (dI/dt) is proportional to the product of the infected mice (I) and the disease-free mice (N - I). So, we have:

dI/dt = k * I * (N - I),

where 'k' is a constant of proportionality. To solve this differential equation, we need to find an expression for I as a function of time (t). Given that initially 5 mice are infected (I = 5) and the total population is 500 (N = 500), we can integrate the equation:

∫(1 / (I * (N - I))) dI = ∫k dt,

Using partial fraction decomposition and integrating, we get:

ln(I / (N - I)) = kt + C,

where C is another constant. Rearranging the equation, we have:

I / (N - I) = e^(kt+C).

Since initially 5 mice are infected (I = 5), we can solve for the constant C:

5 / (500 - 5) = e^(k * 0 + C),

Simplifying, we get:

5 / 495 = e^C.

Taking the natural logarithm of both sides, we find:

ln(5 / 495) = C,

Substituting this value back into the equation, we have:

I / (N - I) = e^(kt + ln(5 / 495)).

Now, we need to determine the value of k. The time it takes for half the population to contract the disease is the time when I = N / 2. Substituting this into the equation, we get:

(N / 2) / (N - (N / 2)) = e^(kt + ln(5 / 495)),

which simplifies to:

1 / 2 = e^(kt + ln(5 / 495)).

Taking the natural logarithm of both sides, we find:

ln(1 / 2) = kt + ln(5 / 495),

Simplifying, we have:

ln(2) = kt + ln(5 / 495),

Rearranging the equation, we get:

kt = ln(2) - ln(5 / 495),

which further simplifies to:

kt = ln(2 * 495 / 5),

Finally, we can solve for t:

t = (1 / k) * (ln(990) - ln(5)).