Modeling the spread of infectious diseases

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Two young mathematicians discuss differential equations.

Check out this dialogue between two calculus students (based on a true story):

Devyn: Riley, check out this book, it has some cool applications of calculus:

The following differential equation can be used to model the spread of an infectious disease: $\mathrm {infect}'(t) = k\cdot \mathrm {infect}(t)\cdot (P-\mathrm {infect}(t))$ where $k$ is a constant, $\mathrm {infect}(t)$ is the number of people infected by the disease on day $t$ , and $P$ is the size of the population vulnerable to the disease.
Riley: Whoa. That’s like a formula for a derivative. Wow. Much calculus.
Devyn: I wonder how you solve equations like this?
Riley: I wonder if we can sometimes just use facts about the derivative to give us an approximation for the function that models the spread of infection?

Suppose your calculus class has had a freak outbreak of the math-philia. Some facts: We have around $200$ students in our class, we are now on the $23$ rd day of the outbreak, and currently $100$ students are infected. Using the differential equation $\mathrm {infect}'(t) = k\cdot \mathrm {infect}(t)\cdot (P-\mathrm {infect}(t))$ we can model the spread of math-philia by setting $k=0.001$ . What is $\mathrm {infect}'(23)$ ? $\mathrm {infect}'(23) = \answer {10}$

Do your best to explain why the equation $\mathrm {infect}'(t) = k\cdot \mathrm {infect}(t)\cdot (P-\mathrm {infect}(t))$ is reasonable.

Don’t worry, just do your best.

Answers will vary.